Torque Control.indd

Torque Control.indd
TORQUE CONTROL
Edited by Moulay Tahar Lamchich
Torque Control
Edited by Moulay Tahar Lamchich
Published by InTech
Janeza Trdine 9, 51000 Rijeka, Croatia
Copyright © 2011 InTech
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Image Copyright demoded, 2010. Used under license from Shutterstock.com
First published Februry, 2011
Printed in India
A free online edition of this book is available at www.intechopen.com
Additional hard copies can be obtained from [email protected]
Torque Control, Edited by Moulay Tahar Lamchich
p. cm.
ISBN 978-953-307-428-3
free online editions of InTech
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Contents
Preface
Part 1
IX
Different Techniques for the Control
of Asynchronous Motors and Double Feed
or Double Star Induction Machines 1
Chapter 1
Torque Control of CSI Fed Induction Motor Drives
Aleksandar Nikolic
Chapter 2
Direct Torque Control Based Multi-level Inverter and
Artificial Intelligence Techniques of Induction Motor 29
Lamchich Moulay Tahar and Lachguer Nora
Chapter 3
Direct Torque Control using Space Vector
Modulation and Dynamic Performance of the Drive,
via a Fuzzy Logic Controller for Speed Regulation 51
Adamidis Georgios, and Zisis Koutsogiannis
Chapter 4
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase 83
Kasmieh Tarek
Chapter 5
Control of a Double Feed and Double Star
Induction Machine Using Direct Torque Control
Leila Benalia
Part 2
Oriented Approach of Recent Developments
Relating to the Control of the Permanent
Magnet Synchronous Motors 127
Chapter 6
Direct Torque Control of
Permanent Magnet Synchronous Motors 129
Selin Ozcira and Nur Bekiroglu
Chapter 7
Torque Control of PMSM
and Associated Harmonic Ripples
Ali Ahmed Adam, and Kayhan Gulez
155
3
113
VI
Contents
Part 3 Special Controller Design and Torque Control
of Switched Reluctance Machine 199
Chapter 8 Switched Reluctance Motor
Jin-Woo Ahn
201
Chapter 9 Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
Tian-Hua Liu
253
Preface
Modern electrical drive systems are composed principally of; motors, power electronics components, transformers, analog/digital controllers and sensors or observers. The
improvements and the enormous advances in the field of power electronic (semiconductor components), converters technology and software/implementation technology
have enabled advanced and complex control techniques. With these advances and robust control algorithms improved, considerable research effort is devoted for developing optimal techniques of speed and torque control for induction machines.
Also, torque control has been, for a long time, a remarkable field for industrial and
academic research. Different speed and torque control techniques were developed for
many types of induction machines and for various applications. As it can be confirmed
from the increasing number of conferences and journals on speed and torque control
of induction motors, it is certain that the optimal and robust torque control are a significant guidance for technology development especially for applications based speed
variators.
This book is the result of inspirations and contributions from many researchers (a collection of 9 works) which are, in majority, focalised around the Direct Torque Control and may be comprised of three sections: different techniques for the control of
asynchronous motors and double feed or double star induction machines, oriented
approach of recent developments relating to the control of the Permanent Magnet Synchronous Motors, and special controller design and torque control of switched reluctance machine.
In the first section, composed on chapters 1 to 5, the recent developments of the induction machines control are widely developed: after presentation of the machine model
(asynchronous, doubly fed or double star), and control method adopted (specially Field
Oriented Control, Direct Torque Control or their association) some ideas for improved
performance are provided. In this object, the impact study, of different structures of
static power converters (Current Source Inverter, Multi-level inverter), on the dynamic
and system performance is exploited. In addition, the contributions made by the use
of artificial intelligence techniques either for the design of the controllers or in the
development of switching tables, in the case of the Direct Torque Control, are amply
explained. Furthermore the study of special phenomena (flux weakening phase) or
X
Preface
other types of machines for specific applications (Double feed or Double star induction
machines) is presented.
The second section, composed of two chapters, is on behaviour of Permanent Magnet
machine controlled by direct torque control method. Different techniques are used to
improve torque ripple reduction and harmonic noises in PMSM. Also, as an improvement approach, different filters types are discussed and a RLC low pass is tested in
order to eliminate the harmonics.
The last section includes two research articles on development of reluctance motors.
The investigations are focused on the controller design and the implementation of
sensorless reluctance drive with direct torque control and also on the torque ripple
which can be minimized through magnetic circuit design or switched reluctance motors control.
Finally, in my capacity, as the Editor of this book, I would like to thank and appreciate
the chapter authors, who ensured the quality of their best works submitted. Most of
the results presented in the book have already been presented on many international
conferences and will make this book useful for students and researchers who will contribute to further development of the existing technology.
I hope all will enjoy the book.
February 10, 2011
Pr. M.T. Lamchich
Department of Physic
Laboratory of Electronic and Instrumentation
Faculty of Sciences Semlalia - University Cadi Ayyad
Marrakech - Morocco
Part 1
Different Techniques for the Control of
Asynchronous Motors and Double Feed or
Double Star Induction Machines
1
Torque Control of CSI Fed
Induction Motor Drives
Aleksandar Nikolic
Electrical Engineering Institute “Nikola Tesla”, Belgrade
Serbia
1. Introduction
An electric drive is an industrial system which performs the conversion of electrical energy
to mechanical energy (in motoring) or vice versa (in generator braking) for running various
processes such as: production plants, transportation of people or goods, home appliances,
pumps, air compressors, computer disc drives, robots, music or image players etc. About
50% of electrical energy produced is used in electric drives today.
Electric drives may run at constant speed or at variable speed. Nowadays, most important
are variable speed drives, especially in Europe where according to the Ecodesign for
Energy-Using Products Directive (2005/32/EC) (the "EuP Directive") and its regulation
regarding electric motors (Regulation 640/2009/EC) on 1 January 2015 - motors with a rated
output of 7.5-375kW must meet higher energy efficiency standards, or meet the 2011 levels
and be equipped with a variable speed drive.
The first motor used in variable speed applications was DC motor drive, since it is easily
controllable due to the fact that commutator and stator windings are mechanically
decoupled.
The cage rotor induction motor became of particular interest as it is robust, reliable and
maintenance free. It has lower cost, weight and inertia compared to commutator DC motor
of the same power rating. Furthermore, induction motors can work in dirty and explosive
environments. However, the relative simplicity of the induction motor mechanical design is
contrasted by a complex dynamic structure (multivariable, nonlinear, important quantities
not observable).
In the last two decades of the 20th century, the technological improvements in power
semiconductor and microprocessor technology have made possible rapid application of
advanced control techniques for induction motor drive systems. Nowadays, torque control
of induction motor is possible and has many advantages over DC motor control, including
the same system response and even faster response in case of the latest control algorithms.
Two most spread industrial control schemes employs vector or field-oriented control (FOC)
and direct torque control (DTC).
Current source inverters (CSI) are still viable converter topology in high voltage high power
electrical drives. Further advances in power electronics and usage of new components like
SGCT (Symmetric Gate Commutated Thyristor), gives the new possibilities for this type of
converter in medium voltage applications. Power regeneration during braking, what is a
one of main built-in feature of CSI drives, is also merit for high power drives. Despite the
4
Torque Control
above advantages, the configuration based on a thyristor front-end rectifier presents a poor
and variable overall input power factor (PF) since the current is not sinusoidal, but
trapezoidal waveform. Also, the implementation of CSI drive systems with on-line control
capabilities is more complex than for voltage source inverters (VSI), due to the CSI gating
requirements. Regarding mentioned disadvantages, CSI drives are of interest for research in
the field of torque control algorithms, such as vector control (or FOC) and direct torque
control (DTC). This chapter will present basic FOC and DTC algorithms for CSI drives and
show all features and disadvantages of those control schemes (sluggish response, phase
error, large torque ripples, need for adaptive control, etc.). Using recent analysis tools like
powerful computer simulation software and experiments on developed laboratory
prototype two new FOC and DTC solutions will be presented in the chapter. The proposed
FOC enables CSI drive to overcome mentioned inconveniences with better dynamic
performances. This enhancement relies on fast changes of the motor current, without phase
error, similar to the control of current regulated voltage source PWM inverter. The realized
CSI drive has more precise control, accomplished by the implemented correction of the
reference current. This correction reduces the problem of the incorrect motor current
components produced by the non-sinusoidal CSI current waveform. On the other side,
proposed DTC algorithm is completely new in the literature and the only such a control
scheme intended for CSI induction motor drives. Presented DTC is based on the constant
switching frequency, absence of coordinate transformation and speed sensor on the motor
shaft. Furthermore, since flux estimator is based only on DC link measurements, there is not
necessity for any sensor on the motor side which is one of main drive advantages. In this
case, by combination of vector control and basic DTC, a robust algorithm is developed that
has a faster torque response and it is simpler for implementation.
2. Characteristics of current source inverters
The most prevailing industrial drive configuration in low voltage range is based on IGBT
transistors as power switches and voltage-source inverter (VSI) topology. On the other side,
the induction motor drives with thyristor type current-source inverter (CSI, also known as
auto sequentially commutated inverter, Fig. 1) possess some advantages over voltage-source
inverter drive, but it has a larger torque ripples since the current wave-form is not
sinusoidal. Furthermore, due to the nature of the CSI operation, the dynamic performance
that exists in VSI PWM drives could not be achieved. But, CSI permits easy power
regeneration to the supply network under the breaking conditions, what is favorable in
large-power induction motor drives. At low voltage range (up to 1kV) this type of inverter is
very rare and abandoned, but this configuration is still usable at high power high voltage
range up to 10kV and several MW. In traction applications bipolar thyristor structure is
replaced with gate turn-off thyristor (GTO). Nowadays, current source inverters are very
popular in medium-voltage applications, where symmetric gate-commutated thyristor
(SGCT) is utilized as a new switching device with advantages in PWM-CSI drives (Wu,
2006). New developments in the field of microprocessor control and application in electrical
drives gives possibility for employment of very complex and powerful control algorithms.
Torque control of CSI fed induction motor drives becomes also viable and promising
solution, since some of CSI control disadvantages could be overcome using improved
mathematical models and calculations.
5
Torque Control of CSI Fed Induction Motor Drives
Ld
T1'
T1
T3
T3'
T5'
T5
C1
C3
C5
380V, 50Hz
D1
D3
D5
D4
D2
L1
L2
L3
T2
C4
T6
T4'
M
3~
C2
T2'
W
T4
V
U
D6
C6
AC MOTOR
T6'
Fig. 1. Basic CSI topology using thyristors as power switches
Two most important torque control schemes are presented, namely FOC and DTC. Both
control schemes will be shown with all variations known from literature, including those
proposed by previous research work of author. All presented torque control algorithms,
basic and proposed by author, are analyzed and verified by simulations and experiments.
3. Vector control
In the past, DC motors were used extensively in areas where variable-speed operation was
required, since their flux and torque could be controlled easily by the field and armature
current. However, DC motors have certain disadvantages, which are due to the existence of
the commutator and brushes. On the other side, induction motors have less size for the same
power level, has no brushes and commutator so they are almost maintenance free, but still
has disadvantages. The control structure of an induction motor is complicated since the
stator field is revolving. Further complications arise due to the fact that the rotor currents or
rotor flux of a squirrel-cage induction motor cannot be directly monitored.
The mechanism of torque production in an AC and DC machine is similar. Unfortunately,
that similarity was not emphasized before 1971, when the first paper on field-oriented
control (FOC) for induction motors was presented (Blaschke, 1971). Since that time, the
technique was completely developed and today is mature from the industrial point of view.
Today field oriented controlled drives are an industrial reality and are available on the
market by several producers and with different solutions and performance.
3.1 Basic vector control of CSI drives
Many strategies have been proposed for controlling the motion of CSI fed induction motor
drives (Bose, 1986; Novotny & Lipo, 1988; Wu et al., 1988; Deng & Lipo, 1990; Vas, 1990).
The vector control has emerged as one of the most effective techniques in designing highperformance CSI fed induction motor drives. Compared to the PWM VSI drives CSI has
advantage in the reversible drives, but it has a larger torque ripples since the current waveform is not sinusoidal. Furthermore, due to the nature of the CSI operation, the dynamic
performance that exists in PWM drives are not achieved with the existing vector control
algorithms.
The well-known ("basic") structure of a CSI fed induction motor drive with indirect vector
control is shown in Fig. 2 (Bose, 1986; Vas, 1990).
6
Torque Control
supply
CSI
α
-
i *sd
PI controller
i *sq
resolver
i dc
M
ring counter
+
θe
i s*
Φ
3∼
ΔΦ +
+ θ rf
lead circuit
1/s
ωe
ω *s
slip
calculator
ωr
+ +
Fig. 2. Indirect vector control of a CSI fed induction machine
This method makes use of the fact that satisfying the slip relation is a necessary and sufficient
condition to produce field orientation, i.e. if the slip relation is satisfied (denoted as slip
calculator in Fig. 2), current d component will be aligned with the rotor flux. Current commands
are converted to amplitude and phase commands using resolver (rectangular to polar
coordinate transformation). The current amplitude command is directly employed as the
reference for the current PI controller intended for controlling the input converter (three phase
full wave bridge rectifier). The phase command is passed through a lead circuit such that phase
changes are added into the inverter frequency channel, since these instantaneous phase changes
are not contained in the slip frequency command signal coming from the slip calculator.
3.2 Proposed vector control of CSI drives
In the vector controlled CSI drives found in (Bose, 1986; Wu et al., 1988; Deng & Lipo, 1990;
Vas, 1990; Novotny & Lipo, 1996) and shown in previous chapter, the problems of the speed
response are reported. This is influenced by the instantaneous phase error and, as a result,
these configurations have slower torque response compared to the current regulated PWM
drives. In addition to the phase error, the commutation delay and the non-sinusoidal supply
that is inhered in CSI operation must be generally compensated for, to achieve acceptable
vector control. To overcome these disadvantages the phase error elimination and the
reference current correction should be performed.
In this chapter, the vector control algorithm that eliminates the two drawbacks is shown
(Nikolic & Jeftenic, 2006). The suggested algorithm produces the performance of the CSI
drive that exists in the PWM vector controlled drives. That enables this simple and robust
configuration to be used in applications where reversible operation is a merit.
The necessity for the phase error elimination can be explained with the help of the following
phasor diagram:
i*
i*
sd
Ψr
sq1
ΔΦ
i*
s1
Fig. 3. Phasor diagram with shown phase error
i*
sq2
i*
s2
i*
s
7
Torque Control of CSI Fed Induction Motor Drives
When the torque command is stepped from isq1* to isq2* (with a constant isd*), the current
vector should instantaneously change from is1* to is2*. The slip frequency should also change
immediately. The resolver does give the correct amplitude and the new slip frequency will
be obtained by the slip calculator. However, although the phase change ΔΦ is added by a
lead circuit as shown in Fig. 2, since the instant phase changes are not contained in the slip
frequency command signal coming from the slip calculator (Bose, 1986; Deng & Lipo, 1990;
Novotny & Lipo, 1996), the stator current command will correspond to the vector is* in
Fig. 3, and there will be a phase error in the vector control system. This would result in an
instantaneous loss of the field-orientation that produces a very sluggish response of both
flux and torque. This problem could be overcome by the proposed algorithm, which unifies
features of both PWM and CSI converter. The resolver is still used to calculate the rectifier
reference current, but for the inverter thyristors control, a method used in the current
controlled PWM inverter is implemented. Instead of a lead circuit (shown in Fig. 2), the new
algorithm includes a synchronous to stator transformation (T-1) to transfer the d-q
commands to the three-phase system. This is essential for achieving a fast torque response,
since the torque value is determined by the fundamental harmonic of the stator current.
For correct firing of the thyristors in the inverter, the switching times should be properly
determined to ensure that the phase angle of the motor current matches the phase angle of
the reference currents in a-b-c system. The reference sinusoidal currents obtained as a result
of transformation T-1 are divided by the value obtained on the resolver output to produce
currents of unity amplitudes. Introduction of these currents into the comparator with trigger
level equal to 0.5 gives the proper thyristor conduction time of 120 degrees. This is
illustrated in Fig. 4, where ia* is unity sinusoidal current, ia is scaled CSI output current and
ia1 is fundamental stator current.
1.5
ia1
Currents [p.u.]
1.0
0.5
ia
ia*
0.0
-0.5
-1.0
-1.5
0
60
120
180
240
phase angle [degrees]
300
360
Fig. 4. Waveforms of the reference current and fundamental stator current
The algorithm possesses the additional advantage regarding the practical realization. In
digital control system the lead circuit divides the difference of the two succeeding samples
with the sampling time. Since the sampling time is small, this operation produces the
computational error. The phasor diagram from Fig. 3 with removed phase error is presented
in Fig. 5.
Without the phase error (ΔΦ = 0), the step of the torque command produces the new stator
current command (is* = is2*). Due to non-sinusoidal currents of the CSI, the average values of
the motor d-q currents isd_av and isq_av and the resulting stator current vector is greater than the
corresponding references shown in Fig. 5. To improve proposed algorithm and avoid
improper resultant d-q motor currents, the rectifier reference current correction is performed.
8
Torque Control
i *sq1
i*sd
isd_av
ΔΦ =0
*
i*s
i
=
* sq2
i s2
i*s1
isq_av
is
Ψr
Fig. 5. Phasor diagram without phase error
In the vector controlled induction motor drive fed by a CSI a problem of incorrect copying
of the d-q references to the motor exist. As stated earlier, the reason is non-sinusoidal
current waveform produced by a CSI. The ideal CSI current is a quasi-square waveform
(shown in Fig. 4). The Fourier analysis of this waveform gives the expression:
ia =
1
1
⎛
⎞
⋅ I d ⋅ ⎜ sin(ωt ) − sin(5ωt ) − sin(7ωt ) + "⎟
π
5
7
⎝
⎠
2 3
(1)
The previous relation shows that the fundamental component of AC output current has the
amplitude 10 percent greater than the value of DC link current. For correct reproduction of
the d-q references and satisfactory vector control it is not sufficient to adjust only the phase
of the fundamental motor current and the phase angle of the generated commands. The finetuning of the motor currents in d-q frame is required.
To avoid supplementary hardware and software, a procedure that relies only on the values
calculated off-line is proposed. The corresponding relation between the mean values of the
motor currents in d-q frame and the commanded d-q currents is calculated. For proposed
correction it is not sufficient to use the difference between currents of 10% from (1), because
the correction depends on the phase angle of the d-q components and the inverter
commutation process. At lower speed, the commutation process could be neglected since it
is much shorter than the motor current cycle. Taking all this in consideration, the rectifier
reference current is corrected concerning the reference amplitude, the phase angle and the
commutation duration. The rectifier reference current formed in that manner is now
introduced to the current controller to obtain suitable motor d-q currents and achieve
desired vector control.
The calculation starts from the fundamental reference current from the resolver:
* 2
* 2
is* = (isd
) + (isq
)
(2)
and the phase angle (also obtained from the resolver):
(
*
*
Φ = arctan isd
/ isq
)
(3)
Since the inverter commutation process is not neglected, the waveform of the inverter
output current is represented by a trapezoidal approximation analyzed in (Cavalini et al.,
1994) with adequate precision. Trapezoidal waveform is very near to the real current cosine
waveform due to the short commutation period, as explained in (Bose, 1986). This
approximation assumes that during the commutation period the inverter current rises with
9
Torque Control of CSI Fed Induction Motor Drives
a constant rate of change. For rated rectifier current Id the current rate of change during the
commutation is equal to Id/tc, where tc is corresponding commutation time calculated from
the values of the commutation circuit components. The adequate commutation angle μ
could be obtained as a product of the inverter frequency ωe and particular commutation
time. This time interval is determined from the current rate of change Id/tc and the reference
value of the DC link current is*, therefore the commutation angle is:
μ = ωe ⋅
tc *
⋅ is
Id
(4)
Since the inverter current is periodical, the trapezoidal waveform in all three phases could
be represented on a shorter angle interval with the following equations:
π
⎧
θ < Φ−
⎪0
3
⎪
π⎞
π
⎪ Id ⎛
θ < Φ− + μ
⎪ μ ⋅ ⎜θ − Φ + 3 ⎟
3
⎠
⎪ ⎝
⎪
π
θ < Φ+
ia (θ , μ, Φ) = ⎨I d
3
⎪
⎪
Id ⎛
π⎞
π
⎪I d − ⋅ ⎜θ − Φ − ⎟ θ < Φ + + μ
μ
3
3
⎝
⎠
⎪
⎪0
θ <π
⎪
⎩
(5)
ib (θ , μ , Φ ) = ia (θ −
2π
, μ , Φ)
3
(6)
ic (θ , μ , Φ) = ia (θ −
4π
, μ , Φ)
3
(7)
where θ ranges from 0 to π, μ is the commutation angle and Φ is the phase angle obtained
from (3). The instantaneous values of d-q currents are solved by a three phase to d-q frame
transformation T:
⎡ia (θ , μ , Φ )⎤
⎡isd (θ , μ , Φ ) ⎤ 2
⎥
⎢
T
=
⋅
⋅
⎥
⎢
⎢ib (θ , μ , Φ ) ⎥ =
(
,
,
)
i
θ
μ
Φ
3
sq
⎦⎥
⎣⎢
⎢⎣ic (θ , μ , Φ ) ⎥⎦
(8)
⎡ia (θ , μ , Φ )⎤
2 ⎡sin(θ ) sin(θ − 2π / 3) sin(θ − 4π / 3) ⎤ ⎢
⎥
= ⋅⎢
⎥ ⋅ ⎢ib (θ , μ , Φ ) ⎥
3 ⎣cos(θ ) cos(θ − 2π / 3) cos(θ − 4π / 3) ⎦
⎢⎣ic (θ , μ , Φ ) ⎥⎦
The average values of the currents in d and q axis obtained from (8) on the range from 0
to π be:
isd _ av ( μ , Φ ) =
1 π
⋅ ∫ isd (θ , μ , Φ ) d θ
π
0
(9)
10
Torque Control
isq _ av ( μ , Φ ) =
1 π
⋅ ∫ isq (θ , μ , Φ ) d θ
π
(10)
0
The amplitude of the motor current vector in polar coordinates could be determined using
the average values obtained from (9) and (10):
2
2
is ( μ , Φ ) = isd
_ av ( μ , Φ ) + isq _ av ( μ , Φ )
(11)
The difference between reference amplitude calculated from (2) and the resulting stator
amplitude obtained from (11) is shown in Fig. 4. To avoid this difference, the corresponding
correction factor fcor is introduced as a ratio of the reference (2) and the actual motor current
(11):
f cor ( μ , Φ ) =
is*
is ( μ , Φ )
(12)
For simulation and practical realization purposes, the correction factor fcor is computed from
(2) – (12), and placed in a look-up table with the following restrictions:
•
isd* is constant,
•
isq* is changed only to its rated value with is* limited to 1 p.u.
•
for given references, all possible values of Φ and μ are calculated using (3) and (4),
respectively.
The rectifier reference current that provides the correct values of motor current d-q
components is now:
iref = is* ⋅ f cor ( μ , Φ )
(13)
The interdependence between correction factor fcor, commutation angle μ and phase angle Φ
is presented in Fig. 6 as a 3-D graph.
0.9
Correction fac
tor, fcor [p.u.]
1.0
0.8
0.7
0.4
0.3
Co
mm
uta
0.2
tio
na
ng
le,
μ
3.0
2.5
2.0
1.5
0.1
[ra
d]
ad]
, Φ [r
angle
e
s
a
h
P
1.0
0.5
0.0
0.0
Fig. 6. Correction factor, commutation angle and phase angle interdependence
11
Torque Control of CSI Fed Induction Motor Drives
1.0
0.8
Motor currents inq-axis [A]
Motor currents ind-axis [p.u.]
The calculated results of the current correction in d-axis and q-axis are presented in Fig. 7a
and Fig. 7b respectively. The corrected currents are given along with references and motor
average d-q currents (values without correction). The flux command is held constant
(0.7 p.u.), while torque command is changed from –0.7 p.u. to 0.7 p.u.
0.8
0.6
isd
0.4
isd
0.2
0.0
-0.8
-0.6
corrected
= isd* [p.u.]
not corrected
-0.4
-0.2 0.0
0.2
0.4
*
Torque command,
i sq [p.u.]
[p.u.]
0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
isq
-0.6
isq
-0.8
-0.8
0.8
a)
-0.6
corrected
*
= isq [p.u.]
not corrected
-0.4
-0.2 0.0
0.2
0.4
*
Torque command,
i sq [A]
b)
[p.u.]
0.6
0.8
Fig. 7. Calculated motor current corrected in d-axis and q-axis (a,b respectively)
From the previous analysis the new resolver with current correction is formed as shown in
Fig. 8. This structure is used both in the simulations and the experiments. The new resolver
is consisted of the block “Cartesian to polar” (the coordinate transformation) and the block
“Correction” that designates the interdependence given in Fig. 6. As stated before, this
interdependence is placed in a 3-D look-up table using (2)-(12).
isd* isq*
Correction is*
Φ
is*
x
Cartesian
to polar
slip
calculator
tc/Id
Φ
fcor
μ
x
Resolver with correction
ω ωe
*
s
+ +
ωr
iref
Fig. 8. New resolver with current correction
To analyze dynamic performances of the proposed CSI drive, the torque response of the
"basic" structure shown in Fig. 2 is compared to the response of the new vector control
algorithm. This is done by simulations of these two configurations' mathematical models in
Matlab/Simulink. The first model represents the drive with basic arrangement and the
second is the drive with new control algorithm. The simulation of both models is done with
several initial conditions. Magnetizing (d-axis) current for rated flux has been determined
from the motor parameters and its value (0.7p.u.) is constant during simulations. The rated
q-axis current has been determined from the magnetizing current and the rated full-load
current using (2). At first, simulations of both models are started with d-axis command set to
0.7p.u, no-load and all initial conditions equal to zero. When the rotor flux in d-axis
approaches to the steady state, the machine is excited. This value of d-axis flux is now initial
12
Torque Control
for the subsequent simulations. For the second simulation the pulse is given as a torque
command, with the amplitude of 0.2p.u. and duration of 0.5s. With no-load, the motor will
be accelerated from zero speed to the new steady-state speed (0.2p.u.), which is the initial
condition for the next simulation. Finally, the square wave torque command is applied to
both models with equal positive and negative amplitudes (±0.2p.u) and the observed
dynamic torque response is extracted from the slope of the speed (Lorenz, 1986). The square
wave duty cycle (0.9s) is considerably greater than the rotor time constant (Tr = 0.1s), hence
the rotor flux could be considered constant when the torque command is changed. Fig. 9
shows torque, speed and rotor flux responses of both models. It could be noticed that the
torque response of the basic structure is slightly slower (Fig. 9a), while the proposed
algorithm gives almost instantaneous torque response (Fig. 9b). This statement could be
verified clearly from the speed response analysis. In both cases the torque command is the
same. In the new model this square wave torque command produces speed variations from
0.2p.u. to 0.6p.u. with identical slope of the speed. But, in the basic model at the end of the
first cycle the speed could not reach 0.6p.u. for the same torque command due to the fact
1.2
0.6
Ψrd
1.0
0.4
Rotor flux [p.u.]
Motor torque and speed [p.u.]
0.8
0.2
0.0
-0.2
-0.4
0.8
0.6
0.4
0.2
Ψrq
-0.6
0.0
-0.8
0.0
0.5
1.0
1.5
time [s]
a)
2.0
2.5
0.0
1.0
1.5
time [s]
c)
2.0
2.5
1.0
1.5
time [s]
d)
2.0
2.5
1.2
0.6
Ψrd
1.0
0.4
Rotor flux [p.u.]
Motor torque and speed [p.u.]
0.8
0.5
0.2
0.0
-0.2
-0.4
0.8
0.6
0.4
0.2
-0.6
Ψrq
0.0
-0.8
0.0
0.5
1.0
1.5
time [s]
b)
2.0
2.5
0.0
0.5
Fig. 9. Torque, speed and rotor flux of the basic structure (a), (c) and of the proposed
algorithm (b), (d)
13
Torque Control of CSI Fed Induction Motor Drives
that torque response is slower. Also, in the next cycle (negative torque command) the speed
does not return to 0.2p.u. for the same reason. From different slopes of the speed in these
two models it could be concluded that proposed algorithm produces quicker torque
response. The rotor q-axis flux disturbance in transient regime that exists in the basic model
(Fig. 9c) is greatly reduced by the proposed algorithm in the new model (Fig. 9d). It could be
seen that some disturbances also exist in the case of d-axis flux, but they are almost
disappeared in the new model.
To illustrate the significance and facilitate the understanding of theoretical results obtained
in the previous section, a prototype of the drive is constructed. The prototype has a standard
thyristor type frequency converter digitally controled via Intel’s 16-bit 80C196KC20
microcontroller. Induction motor used in laboratory is 4kW, 380V, 50Hz machine. The speed
control of the drive and a prototype photo are shown in Fig. 10. Simplicity of this block
diagram confirms that the realized control algorithm is easier for a practical actualization.
The proposed circuit for the phase error elimination is at first tested on the simulation
model. The simulation is performed in such a manner that C code for a microcontroller
could be directly written from the model. The values that are read from look-up tables in a
real system (cosine function, square root) are also presented in the model as tables to
properly emulate calculation in the microcontroller.
Fig. 11a shows waveforms of the unity sinusoidal references (ia* and ib*) while Fig. 11b
indicates inverter thyristors switching times with changed switching sequence when the phase
is changed (0.18s, marked with an arrow). On these diagrams it could be observed that
thyristors T1 and T2 are switched to ON state when unity references ia* and ib* reach 0.5 p.u.,
respectively. Fig. 11c,d represents the instant phase variation of the currents in a and b phases
after the reference current is altered. The corresponding currents without command changes
are displayed with a thin line for a clear observation of the instant when the phase is changed.
ωref +
ωr
Speed
controller
_
Microcontroller
isq* isd*
Slip
calculator
Resolver
(Fig. 7)
iref
+
_
is*
+ ωr
ωe
Id
Current
controller
is*
3~
3
Uc
Id
θe
2=
ia,b,c
arccos
α
*
+ ωs
isq* isd*
1/s
*
Firing circuit
without
phase error
ωr
6
M
3~
Rectifier
LDC
E
CSI
Fig. 10. CSI fed induction motor drive with improved vector control algorithm: control block
diagram (left), laboratory prototype (right)
14
Torque Control
*
ib
*
6
* *
ib i
a
4
0.5
Current,ia [A]
Current [p.u.]
1.0
ia
0.0
-0.5
0
-2
-1.0
-1.5
0.00
2
-4
-6
0.05
0.10
0.15
0.20 0.25
time [s]
a)
0.30
0.35
0.40
0.0
0.1
0.2
time [s]
c)
0.3
0.4
0.0
0.1
0.2
time [s]
d)
0.3
0.4
6
4
T2
2
Current,ib [A]
T1
T3
T4
0
-2
T5
-4
T6
-6
0.00
0.05
0.10
0.15
0.20 0.25
time [s]
b)
0.30
0.35
0.40
Fig. 11. Results of the phase error elimination (a,b - simulation, c,d - experimental)
The effects of the reference current correction are given by the specific experiment. To
estimate d and q components, the motor currents in a and b phases and the angle θe between
a-axis and d-axis are measured. This angle is obtained in the control algorithm (Fig. 10) as a
result of a digital integration:
θ e ( n ) = θ e ( n − 1) + ωe ⋅ Ts
(14)
where n is a sample, Ts is the sample time and ωe is excitation frequency. The integrator is
reset every time when θe reaches 0 or 360 degrees. The easiest way for acquiring the value of
this angle is to change the state of the one microcontroller's digital output at the instants
when the integrator is reset. On the time range between two succeeding pulses the angle is
changed linearly from 0 to 360 degrees (for one rotating direction). Since only this time
range is needed for determine the currents in d and q axis, the reset signal from the digital
output is processed to the external synchronization input of the oscilloscope. In that way the
motor phase currents are measured only on the particular time (angle) range. The
corresponding currents in d-q axes are calculated from (8) using for θe, ia and ib
experimentally determined values.
The experimental results are given in Fig. 12 with disabled speed controller.
15
4.5
4.5
4.0
4.0
Motor currents inq-axis [A]
Motor currents ind-axis [A]
Torque Control of CSI Fed Induction Motor Drives
3.5
3.0
corrected
isd
[A]
not corrected
isd
[A]
*
isd [A]
2.5
2.0
1.5
1.0
0.5
0.0
3.5
3.0
2.5
2.0
isq
1.5
isq
1.0
1.6
1.8
2.0 2.2 2.4 2.6 2.8 3.0
*
Torque command,
i sq [A]
a)
3.2
3.4
[A]
not corrected
[A]
*
0.5
isq [A]
0.0
1.4
corrected
1.4
1.6
1.8
2.0 2.2 2.4 2.6 2.8 3.0
*
Torque command,
i sq [A]
3.2
3.4
b)
Fig. 12. Experimental results of the motor current correction in d-axis and q-axis (a,b
respectively)
The flux reference was maintained constant at 2.96A (0.7p.u.) and torque command was
changed from 1.5A (0.35p.u.) to 3.3A (0.78 p.u.). The inverter output frequency is retained
the same during experiment (≈20Hz) by varying the DC motor armature current. From
Fig. 12 it could be seen that for the proposed algorithm average values of d-q components in
the p.u. system are almost equal to corresponding references. On the other side, in the
system without correction there is a difference up to 15%, which confirmed the results
obtained from calculations shown in Fig. 7. This difference produces steady state error, what
makes such a system unacceptable for vector control in high performance applications.
On the Fig. 13 the motor speed and rotating direction changes are shown with enabled
speed controller. The reference speed is swapped from -200min-1 to +200min-1.
300
100
-1
Motor speed [min ]
200
0
-100
-200
-300
0
5
10
15
20
time [s]
Fig. 13. The motor speed reversal
In Fig. 14 the influence of the load changes to the speed controller is presented. As a load,
DC machine (6kW, 230VDC, controlled by a direct change of the armature current via 3phase rectifier) is used. At first, the induction motor works unloaded in a motor region (M)
with the reference speed of –200min-1 that produces the torque command current isq1* = 1.57A. After that, the DC machine is started with its torque in the same direction with
rotating direction of the induction motor. That starts the breaking of the induction motor
and it goes to the generator region (G). In this operating region the power from DC link
16
Torque Control
returns to the supply network. The reference torque command current changes its value and
sign (isq2* = 1.72A). When DC machine is switched off, the induction motor goes to the motor
region (M) and the reference torque command current is now isq3* = -1.48A.
-1
Motor speed [min ]
-150
G
M
-175
M
-200
isq1
-225
*
isq2
*
isq3
*
-250
0
5
10
15
20
time [s]
Fig. 14. The load changes at motor speed of –200min-1
4. Direct torque control
The direct torque control (DTC) is one of the actively researched control schemes of
induction machines, which is based on the decoupled control of flux and torque. DTC
provides a very quick and precise torque response without the complex field-orientation
block and the inner current regulation loop (Takahashi & Noguchi, 1986; Depenbrok, 1988).
DTC is the latest AC motor control method (Tiitinen et al., 1995), developed with the goal of
combining the implementation of the V/f-based induction motor drives with the
performance of those based on vector control. It is not intended to vary amplitude and
frequency of voltage supply or to emulate a DC motor, but to exploit the flux and torque
producing capabilities of an induction motor when fed by an inverter (Buja et al., 1998).
4.1 Direct torque control concepts
In its early stage of development, direct torque control is developed mainly for voltage
source inverters (Takahashi & Noguchi, 1986; Tiitinen et al., 1995; Buja, 1998). Voltage space
vector that should be applied to the motor is chosen according to the output of hysteresis
controllers that uses difference between flux and torque references and their estimates.
Depending on the way of selecting voltage vector, the flux trajectory could be a circle
(Takahashi & Noguchi, 1986) or a hexagon (Depenbrok, 1988) and that strategy, known as
Direct Self Control (DSC), is mostly used in high-power drives where switching frequency is
need to be reduced.
Controllers based on direct torque control do not require a complex coordinate transform.
The decoupling of the nonlinear AC motor structure is obtained by the use of on/off control,
which can be related to the on/off operation of the inverter power switches. Similarly to
direct vector control, the flux and the torque are either measured or mostly estimated and
used as feedback signals for the controller. However, as opposed to vector control, the states
of the power switches are determined directly by the estimated and the reference torque and
flux signals. This is achieved by means of a switching table, the inputs of which are the
17
Torque Control of CSI Fed Induction Motor Drives
torque error, the stator flux error and the stator flux angle quantized into six sections of 60°.
The outputs of the switching table are the settings for the switching devices of the inverter.
The error signal of the stator flux is quantized into two levels by means of a hysteresis
comparator. The error signal of the torque is quantized into three levels by means of a three
stage hysteresis comparator (Fig. 15).
SA
+
ψ*
n*
+
-
speed
controller
optimal
S
switching B
selection
table
SC
T e*+
δψ
ψest
T est
motor
model
isa
isb
isc
θme
ωme
ψsa
ψsb
Te
polar
coordinate
transform.
Fig. 15. Basic concept of direct torque control
The equation for the developed torque may be expressed in terms of rotor and stator flux:
Te =
M
Ls ⋅ Lr − M 2
G
G
⋅ ψ s ⋅ ψ r ⋅ sin(δψ )
(15)
where δΨ is the angle between the stator and the rotor flux linkage space phasors. For
constant stator and rotor flux, the angle δΨ may be used to control the torque of the motor.
For a stator fixed reference frame (ωe = 0) and Rs = 0 it may be obtained that:
ψs =
1 t
∫ u s ⋅ dt
Tn 0
(16)
The stator voltage space phasor may assume only six different non zero states and two zero
states, as shown in Fig. 16. The change of the stator flux vector per switching instant is
therefore determined by equation (16) and Fig. 16. The zero vectors V0 and V7 halt the
rotation of the stator flux vector and slightly decrease its magnitude. The rotor flux vector,
however, continues to rotate with almost synchronous frequency, and thus the angle δΨ
changes and the torque changes accordingly as per (15). The complex stator flux plane may
be divided into six sections and a suitable set of switching vectors identified as shown in
Table 1, where dΨ and dTe are stator flux and torque errors, respectively, while S1,…,6 are
sectors of 60° where stator flux resides.
Further researches in the field of DTC are mostly based on reducing torque ripples and
improvement of estimation process. This yields to development of sophisticated control
algorithms, constant switching schemes based on space-vector modulation (Casadei et al.,
2003), hysteresis controllers with adaptive bandwidth, PI or fuzzy controllers instead of
hysteresis comparators, just to name a few.
18
Torque Control
q
V3 (010)
V4 (011)
V2 (110)
V0 (000)
V1 (100)
V7 (111)
V5 (001)
d
V6 (101)
Fig. 16. Voltage vectors of three phase VSI inverter
dΨ
1
0
dTe
1
0
-1
1
0
-1
S1
-π/6,
π/6
V2
V0
V6
V3
V7
V5
S2
π/6,
π/2
V3
V7
V1
V4
V0
V6
S3
π/2,
2π/3
V4
V0
V2
V5
V7
V1
S4
2π/3,
-2π/3
V5
V7
V3
V6
V0
V2
S5
-2π/3,
-π/2
V6
V0
V4
V1
V7
V3
S6
-π/2,
-π/6
V1
V7
V5
V2
V0
V4
Table 1. Optimal switching vectors in VSI DTC drive
4.2 Standard DTC of CSI drives
Although the traditional DTC is developed for VSI, for synchronous motor drives the CSI is
proposed (Vas, 1998; Boldea, 2000). This type of converter can be also applied to DTC
induction motor drive (Vas, 1998), and in the chapter such an arrangement is presented. The
induction motor drives with thyristor type CSI (also known as auto sequentially
commutated inverter) possess some advantages over voltage-source inverter drive. CSI
permits easy power regeneration to the supply network under the breaking conditions,
what is favorable in large-power induction motor drives. In traction applications bipolar
thyristor structure is replaced with gate turn-off thyristor (GTO). Nowadays, current source
inverters are popular in medium-voltage applications (Wu, 2006), where symmetric gatecommutated thyristor (SGCT) is utilized as a new switching device (Zargari et al., 2001) with
advantages in PWM-CSI drives.
DTC of a CSI-fed induction motor involves the direct control of the rotor flux linkage and
the electromagnetic torque by applying the optimum current switching vectors.
Furthermore, it is possible to control directly the modulus of the rotor flux linkage space
vector through the rectifier voltage and the electromagnetic torque by the supply frequency
of CSI. Basic CSI DTC strategy (Vas, 1998) is shown in Fig. 17.
19
Torque Control of CSI Fed Induction Motor Drives
~
Ψr*
+
Rotor flux
controller
Controlled
rectifier
LF
Torque
comparator
Optimal
switching
vectors
Te* +
-
CSI
Ψr position
Ψrest
Ψ r & Te
Estimator
Teest
IM
Fig. 17. DTC of CSI drive based on hysteresis control
The stator flux value, needed for DTC control loop, is not convenient to measure directly.
Instead of that, the motor flux estimation is performed. In the voltage-based estimation
method, the motor flux can be obtained by integrating its back electromotive force (EMF).
The EMF is calculated from the motor voltage and current (17) and the only motor
parameter required is the stator winding resistance. In practice, this simple integration is
replaced by more sophisticated closed-loop estimators using filtering techniques, adaptive
integration or even observers and Extended Kalman filters (Holtz, 2003).
t
(
)
ψ ss = ∫ u ss − Rs ⋅ iss dt + ψ ss0
0
(17)
For DTC of CSI fed induction motor drive, the appropriate optimal inverter currentswitching vectors (Fig. 18) are produced by using an optimal current-switching table
similarly to the table given for VSI drive (Table 2). The main difference is that in CSI exist
only one hysteresis comparator for torque and only one zero switching current vector.
b
i2
i3
5π/6
π/2
π/6
-5π/6
i4
i5
Fig. 18. Current vectors in CSI
a
-π/6
-π/2
c
i1
i6
20
Torque Control
S2
π/3,
2π/3
i3
i0
i1
S1
0,
π/3
i2
i0
i6
dTe
1
0
-1
S3
2π/3,
π
i4
i0
i2
S4
-π,
-2π/3
i5
i0
i3
S5
-2π/3,
-π/3
i6
i0
i4
S6
-π/3,
0
i1
i0
i5
Table 2. Optimal switching vectors in CSI DTC drive
4.3 Proposed DTC of CSI drives
In DTC schemes, the presence of hysteresis controllers for flux and torque determines
variable-switching-frequency operation for the inverter. Furthermore, using DTC schemes a
fast torque response over a wide speed range can be achieved only using different switching
tables at low and high speed. The problem of variable switching frequency can be overcome
by different methods (Vas, 1998; Casadei et al., 2003). In (Casadei et al., 2003), a solution
based on a stator flux vector control (SFVC) scheme has been proposed. This scheme may be
considered as a development of the basic DTC scheme with the aim of improving the drive
performance. The input commands are the torque and the rotor flux, whereas the control
variables are the stator flux components. The principle of operation is based on driving the
stator flux vector toward the corresponding reference vector defined by the input
commands. This action is carried out by the space-vector modulation (SVM) technique,
which applies a suitable voltage vector to the machine in order to compensate the stator flux
vector error. In this way it is possible to operate the induction motor drive with a constant
switching frequency.
In proposed DTC CSI drive shown in Fig. 19 the inputs are rotor flux and torque as in VSI
presented in (Casadei et al., 2003), but now as a control variable the stator flux angle αs is
used (Nikolic & Jeftenic, 2008).
LDC
Supply
UDC
IDC
α
PI current
controller
Modified optimal
switching table
- IDC
+
is_ref
isq*
Resolver
isd*
PI rotor
flux
controller
αs
position
+ θe
isq* calculator
Ψr*
M
3~
αs
Rotor flux
Φs +
Rotor flux
and torque
estimator
Te*
PI torque
controller
- Ψr
+
+
Ψr*
CSI
Te
Te_ref
Fig. 19. Proposed constant-switching DTC strategy in CSI fed induction motor drive
21
Torque Control of CSI Fed Induction Motor Drives
Although this configuration could remind on field-oriented control, the main difference is
absence of coordinate transformation since it is not necessary to use coordinate
transformation to achieve correct firing angle as in vector control of the same drive (Nikolic
& Jeftenic, 2006). Identical result would be obtained when phase angle Φs between d-q
current references and rotor flux vector angle θe = arctan(Ψrβ/Ψrα) are summed and
resulting angle αs is than used to determine sector of 60 degrees where resides rotor flux
vector. In that way, phase angle Φs acts as a torque control command. When reference
torque is changed, isq* is momentary changed. Phase angle Φs “moves” stator current vector
is in direction determined by the sign of torque reference and its value accelerate or
decelerate flux vector movement according to the value of the reference torque (Fig. 20).
β
is_ref1
i2 (αs1)
2
i1 (αs)
i6 (αs2)
1
Ψr
3
αs1
αs2
4
6
α
is_ref2
5
Fig. 20. Selecting proper current vector in proposed DTC algorithm
This modification implies somewhere different switching table for activating inverter
switches from that shown in Table 2. Now αs (angle between referent α-axis and reference
current vector is) determines which current vector should be chosen: i2 for torque increase, i6
for torque decrease or i1 for keeping torque at the current value.
Current vector
i1
i2
i3
i4
i5
i6
Angle range (degrees)
αs > 0° and αs ≤ 60°
αs > 60° and αs ≤ 120°
αs > 120° and αs ≤ 180°
αs > 180° or αs ≤ -120°
αs > -120° and αs ≤ -60°
αs > -60° and αs ≤ 0°
Table 3. Optimal switching table in proposed DTC
It is necessary to emphasize the importance of zero space vectors. In VSI there are two zero
voltage vectors: V0 denotes case when all three switches from the one half of inverter are
switched ON while V7 represent state when switches are OFF. Contrary, in CSI (using analogy
to the VSI) zero current vector i0 represent case when all thyristors are OFF. That could lead to
both torque and motor speed decrease. Due to the nature of commutation in CSI, it is
convenient to keep the selected current vector at instants when zero current vector is chosen.
22
Torque Control
The voltage and the current of CSI fed induction motor, necessary for stator flux calculation,
can be reconstructed from the DC link quantities knowing the states of the conducting
inverter switches. In one duty cycle of the output current CSI has six commutations. In that
case six intervals of 60 degrees can be defined in which the current and the voltage changes
its values. In every interval the current from DC link flows through two inverter legs and
two motor phase windings. The motor line voltage is equal to the DC voltage on the inverter
input reduced for the voltage drop on the active semiconductors, i.e. serial connection of the
thyristor and diode in each inverter leg (Fig. 1). This voltage drop is forward voltage and for
diodes it is about 0.7V-0.8V and for thyristors it is about 1V-1.5V. In this algorithm the
average value of the overall forward voltage is used (2V), but for the practical realization it
is chosen from the semiconductors datasheets or determined experimentally. It can be
generally concluded that the voltage drop on the corresponding thyristor-diode par could
have the following values in dependence of the conducting thyristor Tx, where x = 1,…,6:
VTDx = VF, when Tx is conducting,
VTDx = 0.5⋅UDC, when conducts thyristor from the same half-bridge where Tx is,
VTDx = UDC – VF, when conducts thyristor from the same inverter leg where Tx is.
These results are used for the voltage calculation in all conducting intervals, and they are
summarized in Table 4. Prior to the flux estimation, the currents and voltages given in the
Table 4 should be converted to α-β stationary frame.
The resistance of the stator windings, needed for stator flux calculation, can be easily
determined from the simple experiment when the motor is in the standstill. When only
thyristors T1 and T6 conducts, the DC current will flow through motor phases a and b. Since
the motor is in the standstill, the only voltage drop is on the stator resistance Rs:
(18)
U ab = 2 ⋅ Rs ⋅ ia
when the windings are Y-connected. Generally, for the motor voltage value calculated from
Table 4 and any type of the motor winding connection, the stator resistance is:
Rs =
U ab
U
− 2 ⋅ VF
= DC
k s ⋅ ia
k s ⋅ I DC
(19)
where ks = 1 for Delta connection and ks =2 for Y connection. Relation (19) can be easily
implemented in the control software if the thyristors T1 and T6 are switched ON prior the
motor start and the stator resistance is determined from the measured DC link current and
voltage and the knowing voltage drop on the thyristor-diode par using Table 4.
1
2
3
4
5
6
Active
ia
Thyristors
IDC
T1,T6
IDC
T1,T2
T3,T2
0
T3,T4
–IDC
T5,T4
–IDC
T5,T6
0
ib
Uab
Ubc
0
–IDC
–IDC
0
IDC
IDC
UDC – 2⋅VF
0.5⋅UDC – VF
–0.5⋅UDC + VF
–UDC + 2⋅VF
–0.5⋅UDC + VF
0.5⋅UDC – VF
–0.5⋅UDC + VF
0.5⋅UDC – VF
UDC – 2⋅VF
0.5⋅UDC – VF
–0.5⋅UDC + VF
–UDC + 2⋅VF
Table 4. Motor current and voltage determined only by DC link measurements
23
Torque Control of CSI Fed Induction Motor Drives
The main feedback signals in DTC algorithm are the estimated flux and torque. They are
obtained as outputs of the estimator operating in stator reference frame. This estimator at
first performs electro-motive force (EMF) integration (17) to determine the stator flux vector
and than calculates the flux amplitude and find the sector of 60 degrees in α-β plane where
flux vector resides, according to the partition shown in Fig. 20. After the stator current and
voltage are determined by previously explained reconstruction of stator voltages and
currents, pure integrator in (17) yields flux vector, which components are subsequently
limited in amplitude to the magnitude values of the stator flux references. The trajectory of
flux vector is not circular in the presence of DC offset. Since its undisturbed radius equals
Ψs*, the offset components tend to drive the entire trajectory toward one of the ±Ψs*
boundaries. A contribution to the EMF offset vector can be estimated from the displacement
of the flux trajectory (Holtz, 2003), as:
off
EMFαβ
=
(
1
⋅ Ψαβ max + Ψαβ min
Δt
)
(20)
where the maximum and minimum values in (20) are those of the respective components
Ψsα and Ψsβ, and Δt is the time difference that defines a fundamental period. The signal
EMFoff is fed back to the input of the integrator so as to cancel the offset component in EMF.
The input of the integrator then tends toward zero in a quasi-steady state, which makes the
estimated offset voltage vector equal the existing offset Ψs0 in (17). The trajectory of Ψs is
now exactly circular, which ensures a precise tracking of the EMF offset vector. Since offset
drift is mainly a thermal effect that changes the DC offset very slowly, the response time of
the offset estimator is not at all critical. It is important to note that the dynamics of stator
flux estimation do not depend on the response of the offset estimator (Holtz, 2003).
The estimated rotor flux is calculated from the stator flux estimate using motor parameters
and reconstructed stator current:
L ⋅ L − L2m
Lr ˆ
ˆ
Ψ
⋅ Ψsαβ − s r
⋅ isαβ
r αβ =
Lm
Lm
(21)
and its position in α-β reference frame is determined by:
⎛ Ψrβ ⎞
(22)
⎟
θ e = arctan⎜⎜
⎟
⎝ Ψrα ⎠
Finally, from the estimated stator flux and reconstructed current vector the motor torque is:
Te =
(
3
p ⋅ isβ Ψsα − isα Ψsβ
2
)
(23)
where the stator flux and current vectors are given in stationary α-β frame, and p denotes
the number of poles.
The simulation model is developed in Matlab/SIMULINK, using SimPowerSystems block
library that allows a very real representation of the power section (rectifier, DC link, inverter
and induction motor). All electrical parameters (inductance of DC link, motor parameters)
are the same as in real laboratory prototype, also used for testing previosly explained FOC
algorithm. Rated flux is 0.8Wb and rated torque is 14Nm. Rectifier reference current is
limited to 12A and reference torque is limited to 150% of rated torque (20Nm).
24
Torque Control
3
3
2
2
1
1
Torque [Nm]
Torque [Nm]
Comparison between the basic and proposed DTC of CSI induction motor drive are shown
in Fig. 21, using the same mathematical model of CSI drive as used for FOC algorithm.
Proposed DTC shows much better torque response from motor standstill.
0
-1
0
-1
-2
-2
-3
-3
0
5
10
15
time [s]
20
25
30
0
5
10
15
20
25
30
time [s]
Fig. 21. Torque response for basic (left) and proposed (right) DTC algorithm
Dynamical performances of DTC algorithm are analyzed at first with rated flux and zero
torque reference, than drive is accelerated up to 1000rpm. The speed is controlled in closedloop via digital PI controller (proportional gain: KP = 5, integral gain: KI = 0.5, torque limit =
20Nm, controller sampling time: Ts = 4ms). The speed reference is set using succeeding
scheme: +1000rpm at t = 0.25s, than –1000rpm at t = 0.6s. As could be seen from Fig. 22, a
fast torque response is achieved with correct torque reference tracking and slow rotor flux
ripple around the reference value (<1.5%).
For experimental purposes, the same laboratory prototype of CSI drive is used as explained
before. The CSI feeds a 4kW induction motor and, as a mechanical load, the 6kW DC
machine with controlled armature current is used (Nikolic & Jeftenic, 2008). The presented
algorithm is not dependent on the motor power or the type of switching devices and it
could be applied to any current source converter topology. The low-power induction motor
and standard type thyristors are used just for the simplicity of the laboratory tests.
The torque response is analyzed both with direct torque demand and under the closed-loop
speed control. Speed controller is implemented with soft start on its input and sample time
of 20ms. Torque limit on the controller output is ±5Nm and is determined in such a way that
under the maximum torque value slip is equal to maximal slip for current control:
1
= 0,0405
smax =
(24)
ωe ⋅ Tr
where ωe is synchronous frequency (314rad/s) and Tr is rotor time constant (78.7ms).
Since rotor flux is not measured but determined by estimation, its value is checked with that
obtained from simulation. The comparison between simulated and estimated rotor flux
whith zero speed (torque) reference and rated flux reference are given in Fig. 23 (a). Good
performance of the flux estimator, necessary for proper direct torque control, could be
observed from Fig. 23 (b), where flux trajectory is shown starting from zero to its rated
value. Almost circular flux trajectory with equal amplitudes in both α and β axes assures
correct offset compensation.
25
Torque Control of CSI Fed Induction Motor Drives
Rotor flux [Wb]
1,0
0,8
0,6
0,4
0,2
0,0
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
Motor speed [rpm]
1500
1000
500
0
-500
-1000
-1500
40
Torque [Nm]
30
20
10
0
-10
-20
-30
-40
time [s]
Fig. 22. Simulation results for the proposed DTC method
Ψβ [Wb]
1,0
0,8
1,0
0,6
0,4
0,8
Ψs [Wb]
0,2
0,6
0,0
-0,2
0,4
-0,4
Flux from simulated model
Estimated stator flux
-0,6
0,2
-0,8
-1,0
0,0
0,0
0,2
0,4
0,6
0,8
1,0
-1,0 -0,8 -0,6 -0,4 -0,2
0,0
0,2
time [s]
Ψα [Wb]
(a)
(b)
Fig. 23. Rotor flux response (a) and its trajectory (b) during motor start-up
0,4
0,6
0,8
1,0
26
Torque Control
Motor speed and torque response when the speed control loop is closed is shown in Fig. 24.
Response tests are performed during motor accelerating from 0rpm to 300rpm, than from
300rpm to 500rpm and back to 300rpm and 0rpm.
3,0
600
2,5
500
2,0
400
1,5
300
1,0
200
0,5
100
0,0
0
-0,5
-100
-1,0
0
5
10
15
20
25
time [s]
(a)
30
35
40
45
0
5
10
15
20
25
30
35
40
45
time [s]
(b)
Fig. 24. Motor speed (a) and torque (b) response under different speed references
5. Conclusions
In this chapter two main torque control algorithms used in CSI fed induction motor drives
are presented, namely FOC and DTC. The first one is precise vector control (FOC)
algorithm. The explained inconveniences of the vector controlled induction motor drives fed
by a CSI could be overcome with the new vector control algorithm. The main advantage of
the suggested algorithm compared to that known from the literature is better dynamic
performances of the proposed CSI drive. This enhancement relies on the fast changes of the
motor current, without phase error, similar to the control of current regulated voltage source
PWM inverter. The realized CSI drive has more precise control, accomplished by the
implemented correction of the reference current. This correction reduces the problem of the
incorrect motor d-q currents values produced by the non-sinusoidal CSI current waveform.
Next, the two different methods of direct torque control in CSI fed induction motor drive are
presented. Contrary to the well-known hysteresis control derived from VSI drive, new DTC
algorithm based on the constant switching frequency is proposed. Merit of such a solution
in comparison to the vector control of the same drive is absence of coordinate
transformation and speed sensor on the motor shaft. Furthermore, since flux estimator is
based only on DC link measurements, there is not necessity for any sensor on the motor side
which is one of main drive advantages. In this case, by combination of vector control and
basic DTC, a robust algorithm is developed that has a faster torque response and it is
simpler for implementation. Moreover, algorithm is less sensitive to the parameter variation
than standard FOC on the same drive. Contrary to the slip calculation using rotor time
constant, proposed algorithm uses stator resistance for flux calculation and its value could
be checked every time when motor is stopped using explained method for reconstruction
Torque Control of CSI Fed Induction Motor Drives
27
based only on DC link measurements. Other motor parameters (windings and mutual
inductances) are used only when flux reference is changed and their values have no
influence on the performance of the flux estimator due to the offset compensation. The
validity of all presented torque control algorithms was proven by simulations and
experimental results on developed laboratory prototype of CSI drive.
6. References
Blaschke, F. (1971). A new method for the structural decoupling of A.C. induction machines,
Proceedings of IFAC Symposium on Multivariable Technical Control Systems, pp. 1–15,
ISBN 0720420555, Duesseldorf, Germany, October 1971, American Elsevier, New
York
Bose, BK. (1986). Power Electronics and AC Drives, Prentice-Hall, ISBN 0-13-686882-7, New
Jersey, USA
Novotny, DW & Lipo, TA. (1996). Vector Control and Dynamics of AC Drives, Oxford
University Press, ISBN 978-0-19-856439-3, New York, USA
B, Wu; SB, Dewan & Sen PC. (1988). A Modified Current-Source Inverter (MCSI) for a
Multiple Induction Motor Drive System. IEEE Transactions on Power Electronics,
Vol. 3, No. 1, January 1988, pp. 10-16, ISSN 0885-8993
Lorenz, RD. (1986). Tuning of Field-Oriented Induction Motor Controllers for HighPerformance Applications, IEEE Transactions on Industry Applications, Vol. IA-22,
No. 2, March 1986, pp. 293-297, ISSN 0093-9994
Deng, D & Lipo, TA. (1990). A Modified Control Method for Fast Response Current Source
Inverter Drives, IEEE Transactions on Industry Applications, Vol. IA-22, No. 4, July
1986, pp. 653-665, ISSN 0093-9994
Vas, P. (1990). Vector Control of AC Machines, Clarendon Press, ISBN-10: 0198593708, ISBN13: 978-0198593706, Oxford, New York, USA
Cavallini, A.; Loggini, M. & Montanari, GC. (1994). Comparison of Approximate Methods
for Estimate Harmonic Currents Injected by AC/DC Converters, IEEE Transactions
on Industial Electronics, Vol. 41, No. 2, April 1994, pp. 256-262, ISSN 0278-0046
Nikolic, A. & Jeftenic, B. (2006). Precise Vector Control of CSI Fed Induction Motor Drive,
European Transactions on Electrical Power, Vol.16, March 2006, pp. 175-188, ISSN
0170-1703
Zargari, N. R.; Rizzo, S. C.; Xiao, Y.; Iwamoto, H.; Satoh, K. & Donlon, J. F. (2001). A new
current-source converter using a symmetric gate-commutated thyristor (SGCT),
IEEE Transactions on Industry Applications, Vol.37, 2001, pp. 896-903, ISSN 0093-9994
Takahashi, I. & Noguchi, T. (1986). A New Quick-Response and High-Efficiency Control
Strategy of an Induction Motor, IEEE Transactions on Industry Applications, Vol. 22,
No. 5, September/October 1986, pp. 820-827, ISSN 0093-9994
Depenbrok, M. (1988)., Direct Self-Control (DSC) of Inverter-Fed Induction Machine, IEEE
Transactions on Power Electronics, Vol. PE-3, No. 4, October 1988, pp. 420-429, ISSN
0885-8993
Tiitinen, P.; Pohjalainen, P.& Lalu, J. (1995). The next generation motor control method:
Direct Torque Control (DTC), European Power Electronics Journal, Vol.5, March 1995,
pp. 14-18, ISSN 09398368
28
Torque Control
Buja, G.; Casadei, D. & Serra, G. (1998). Direct Stator Flux and Torque Control of an
Induction Motor: Theoretical Analysis and Experimental Results, in Proceedings of
the IEEE International Conference on Industrial Electronics IECON '98, pp. T50-T64,
Vol. 1, ISBN 0-7803-4503-7, Aachen, Germany, August/September 1998, IEEE, New
Jersey
Vas, P. (1998). Sensorless Vector and Direct Torque Control, Oxford University Press, ISBN 019-856465-1, New York, USA
Boldea, I. (2000). Direct Torque and Flux (DTFC) of A.C. Drives: A Review. Proceedings of the
9th Conference EPE-PEMC 2000, pp. 88-97, ISBN 80-88922-18-6, Kosice, Slovakia,
September 2000, EPE-PEMC, Budapest
Casadei, D.; Serra, G.; Tani, A.; Zarri, L. & Profumo, F. (2003). Performance Analysis of a
Speed-Sensorless Induction Motor Drive Based on a Constant-Switching-Frequency
DTC Scheme, IEEE Transactions on Industry Applications, Vol.39, 2003, pp. 476-484,
ISSN 0093-9994
Wu, B. (2006). High-power converters and AC drives, John Wiley & Sons, Inc., Hoboken, ISBN13 978-0-471-73171-9, ISBN-10 0-471-73171-4, New Jersey, USA
Nikolic, A. & Jeftenic, B. (2008). Different Methods for Direct Torque Control of Induction
Motor Fed From Current Source Inverter, WSEAS Transactions on Circuits and
Systems, Volume 7, Issue 7, January 2008, pp. 738-748, ISSN 1109-2734
Holtz, J. (2003). Drift- and Parameter-Compensated Flux Estimator for Persistent ZeroStator-Frequency Operation of Sensorless-Controlled Induction Motors, IEEE
Transactions on Industrial Applications, Vol.39, 2003, pp. 1052-1060, ISSN 0093-9994
2
Direct Torque Control Based Multi-level
Inverter and Artificial Intelligence
Techniques of Induction Motor
Lamchich Moulay Tahar and Lachguer Nora
University Cadi Ayyad/ Faculty of Sciences Semlalia/Department of Physic/Laboratory of
Electronic and Instrumentation
Morocco
1. Introduction
With the enormous advances in converters technology and the development of complex and
robust control algorithms, considerable research effort is devoted for developing optimal
techniques of speed control for induction machines (IM). Also, induction motor control has
traditionally been achieved using field oriented control (FOC). This method involves the
transformation of the stator currents into a synchronously rotating dq reference frame that is
aligned with one of the stator fluxes, typically the rotor flux. In this reference frame, the
torque and flux producing components of the stator currents are decoupled, such that the daxis component of the stator current controls the rotor flux magnitude and the q-axis
component controls the output torque.
The implementation of this system however is complicated and furthermore FOC, in
particularly indirect method, which is widely used, is well known to be highly sensitive to
parameters variations due to the feed-forward structure of its control system.
Another induction motor control technique known as a Direct Torque Control (DTC) was
introduced in the mid 1980s, by Takahachi and Noguchi, for low and medium power
applications; also Direct Self Control (DSC) was proposed by Depenbrock for high power
applications.
DTC has a relatively simple control structure yet performs at least as good as the FOC
technique. It is also known that DTC drive is less sensitive to parameters de-tuning (only
stator resistor is used to estimate the stator flux) and provides a high dynamic performances
than the classical vector control (fastest response of torque and flux).
This method allows a decoupled control of flux and torque without using speed or position
sensors, coordinate transformation, Pulse Width Modulation (PWM) technique and current
regulators. This type of command involves nonlinear controller type hysteresis, for both
stator flux magnitude and electromagnetic torque, which introduces limitations such as a
high and uncontrollable switching frequency. This controller produces a variable switching
frequency and consequently large torque and flux ripples and high currents distortion.
The DTC is mostly used in the objective to improve the reduction of the undulations or the
flux’s distortion, and to have good dynamic performances. It’s essentially based on a
localization table which allows selecting the vector tension to apply to the inverter according
30
Torque Control
to the position of the stator flux vector and of the direct control of the stator flux and the
electromagnetic torque.
The general structure of the asynchronous motor with DTC and speed regulation and using
multilevel inverter is represented by the following figure.
Fig. 1. General structure of the asynchronous motor with DTC and speed regulation
Also, the use of multi-level inverters and artificial techniques contribute to the performances
amelioration of the induction machine control. In fact, the use of three level inverter (or
multi-level inverter) associated with DTC control can contribute to more reducing
harmonics and the ripple torque and to have a high level of output voltage.
Also, in last years, much interest has focused on the use of artificial intelligence techniques
(neural networks, fuzzy logic, genetic algorithms,…) in identification and non linear control
systems. This is mainly due to their ability learning and generalisation.
It become a number of papers appeared in literature interest to improving the performance
of DTC applied to induction motor drive.
Among the different control strategies that were applied to achieve improved performance
include:
•
The switching frequency is maintained constant by associating the DTC to the space
vector modulation;
•
The space voltage is divided into twelve sectors instead of six with the classic DTC, and
used some changes of the switching table.
Many researches have been performed using the multi-level inverter and, for example, some
articles described a novel DTC algorithm suited for a three level inverter, and proposed a
very simple voltage balancing algorithm for the DTC scheme.
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
31
Also, different other strategies using the artificial intelligence techniques were introduced,
in order to achieve the objective that improving the performance of DTC:
•
The direct torque control using a fuzzy logic controller to replace the torque and stator
flux linkage hysteresis loop controller, space vector modulation, and fuzzy stator
resistance estimator is more developed;
•
The artificial neural network replacing the convectional switching table in the DTC of
induction motor is also widely detailed.
In this chapter, all these points will be deeply developed and some simulation results, using
Matlab/Simulink environment and showing the advantages of these approaches, will be
presented. In the 1st section, we present the description of DTC method applied to the
induction motor, as well as the simulation results will be illustrate the effectiveness of this
method. In 2nd section, in the objective to improve the performance of DTC, the technique of
multi-level inverter fed induction motor has been analyzed and simulation results show the
performance of this approach. In 3rd section, we present the fuzzy logic direct torque control
with two approaches: pulse width modulation and space vector modulation, also a model of
artificial neural network is applied in DTC.
In the latest sections, the association of three-level inverter with fuzzy/Neural speed
corrector for direct torque control of induction motor is developed.
2. Direct flux-torque control fundamentals
The direct torque control is principally a non-linear control in which the inverter switching
states are imposed through a separate control of stator flux and electromagnetic torque of
the motor. The inverter command is instantaneous and it replaces then the decoupling
through the vectorial transformation. One of the most important characteristics of the DTC
is the non-linear regulation of stator flux and electromagnetic torque with variables
structures or by hysteresis.
The flux regulation is imperative for an efficient control of the induction machine torque and
in the DTC, the stator flux regulation is chosen because it’s easier to estimate, and partly it has
a faster dynamics than the rotor flux. By adjusting the stator flux, we also adjust the rotor flux.
As in the other control methods, which use a direct regulation of the flux, the flux nominal
value is imposed as a constant reference, for speeds lower than the nominal value. For
higher speeds, a flux reference value, decreasing proportionally with speed; is imposed. On
the other hand, the quality of rotation speed, and/or position, control of the modern
actuators depends directly on the toque control.
2.1 Stator flux control
The IM equations, in a stator reference frame, are defined by:
⎧
⎪ Vs
⎪
⎪⎪
⎨ Vr
⎪
⎪φs
⎪
⎪⎩φr
= R s Is +
dφs
dt
= 0 = R r Ir +
= L s Is + M sr
dφr
- jω φr
dt
Ir
= L r Ir + M sr Is
(1)
32
Torque Control
where Rs and Rr are the stator and rotor resistances.
L s and L r are the mutual stator and rotor inductances.
The stator flux is estimated from the measure of stator current and voltage and their
transformation in the αβ subspace. So:
t
Φ sα = ∫ (Vsα − Rs I sα )dt
0
t
Φ sβ = ∫ (Vsβ − Rs I sβ )dt
(2)
0
The stator flux module and the linkage phase are given by:
Φs = Φ s2α + Φ s2β
α s = arctg(
φsβ
)
φsα
(3)
On a sampling period Te , and by neglecting the term ( Rs I s ) in equation of stator flux, valid
hypothesis for high speeds, the evolution of this last one is given by the vector Vs during
Te:
ΔΦ s = Φ s − Φ s 0 = V sTe
(4)
Φ s 0 is the initial stator flux at the instant t0 .
So, the variation of the stator flux is directly proportional to the stator voltage, thus the
control is carried out by varying the stator flux vector by selecting a suitable voltage vector
with the inverter.
A two level hysteresis comparator could be used for the control of the stator flux. So, we can
easily control and maintain the flux vector Φ s in hysteresis bound as shown in Figure.2.
The output of this corrector is represented by a Boolean variable cflx which indicates
directly if the amplitude of flux must be increased ( cflx = 1) or decreased ( cflx = 0 ) so as to
maintain: (Φ s )réf − Φ s ≤ ΔΦ s , with (Φ s )réf the flux reference value and ΔΦ s the width of the
hysteresis corrector.
Fig. 2. Flux hysteresis corrector
2.2 Torque control
The electromagnetic torque expression is defined as follws, where γ represents the angle
between the rotor and stator flux vectors:
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
Γelm = p
Lm
σ L s Lr
Φ s Φ r sin( γ )
33
(5)
where p is the number of pole pair
Lm: mutual inductance
σ: leakage coefficient (Blondel coefficient)
We deduct that the torque depends on the amplitude and the position of stator and rotor
flux vectors.
On the other hand, the differential equation linking the stator flux and the rotor flux of
motor is given by:
dΦ r
Lm
1
+(
− jω )Φ r =
Φs
dt
στ r
στ r L s
(6)
From this equation, the flux Φ r tracks the variations of the flux Φ s with a time constant
στ r .
In controlling perfectly the stator flux vector, from the vector Vs , in module and in position,
we can control the amplitude and the relative position of the rotor flux vector and
consequently the electromagnetic torque. This is possible only if the command period Te of
the voltage Vs is very lower to time constant στ r .
The expression of the electromagnetic torque is only obtained from the stator flux
components Φ sα , Φ sβ and currents I sα , I sβ :
Γelm = p(φsα i sβ - φsβ i sα )
(7)
For the control of the electromagnetic torque, we can use a three level hysteresis comparator
which permits to have the two senses of motor rotation. The output of this corrector is
represented by a Boolean variable Ccpl indicating directly if the amplitude of the torque
must be increased, decreased or maintained constant ( ccpl = 1, - 1, 0 ) .
Fig. 3. Three level hysteresis comparator
2.3 Control strategy of DTC based two-level voltage inverter
Direct Torque Control of IM is directly established through the selection of the appropriate
stator vector to be applied by the inverter. To do that, in first state, the estimated values of
stator flux and torque are compared to the respective references, and the errors are used
through hysteresis controller.
The phase plane is divided, when the IM is fed by two-level voltage inverter with eight
sequences of the output voltage vector, into six sectors.
34
Torque Control
V3(010)
V2(110)
S3
V4(011)
S2
V0
S4
S5
V1(100)
V7
S1
S6
V5(001)
V6(101)
Fig. 4. Stator vectors of tensions delivered by a two level voltage inverter
When the flux is in a sector (i), the control of flux and torque can be ensured by the
appropriate vector tension, which depends on the flux position in the reference frame, the
variation desired for the module of flux and torque and the direction of flux rotation:
Φs increase, Γelm
increase
Φs increase, Γelm
decrease
Φs decrease,
Γelm increase
Φs decrease, Γelm
decrease
Vi+1
Vi-1
Vi+2
Vi-2
Vector tension
selected
Table 1. Selection of vector tension
α
Φs decrease
Γelm increase
Vi+2
e
Φs increase
Γelm increase
Vi+1
d
f
π/3
h
g
β
c
Vi-2
Φs decrease
Γelm decrease
V0 ,V7
Φs cste
Γelm decrease
Vi-1
Φs increase
Γelm decrease
Fig. 5. Selection of vector tension
The null vectors (V0, V7) could be selected to maintain unchanged the stator flux.
According to the table 2, the appropriate control voltage vector (imposed by the choice of
the switching state) is generated:
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
Cflx
1
0
35
ccpl
S1
S2
S3
S4
S5
S6
1
V2
V3
V4
V5
V6
V1
0
V7
V0
V7
V0
V7
V0
-1
V6
V1
V2
V3
V4
V5
1
V3
V4
V5
V6
V1
V2
0
V0
V7
V0
V7
V0
V7
-1
V5
V6
V1
V2
V3
V4
Table 2. Voltage vector selected (for each sector Si)
The following figure shows the selected voltage vector for each sector to maintain the stator
flux in the hysteresis bound.
Fig. 6. Selection of vector tension
2.4 Simulation results
Simulations were performed to show the behavior of the asynchronous motor fed by twolevel inverter and controlled by Direct Torque Control.
The torque reference value is deduced from the regulation of the IM speed using a PI
corrector. We have chosen to present the results corresponding to the rotation speed
evolution, the electromagnetic torque, the flux evolution in the αβ subspace and the stator
currents.
The obtained simulation results show that:
•
trajectory of the stator flux, represented by its two components in the αβ phase plane, is
in a circular reference (Figure 7)
phase current obtained by this strategy is quasi-sinusoidal (Figure 7)
•
speed track its reference with good performance (Figure 8)
•
overshoot on torque is limited by saturation on the reference value (Figure 8)
•
36
Torque Control
1.5
50
40
1
30
20
Stator currents (A)
Stator flux
0.5
0
-0.5
10
0
-10
-20
-30
-1
-40
-1.5
-1.5
-1
-0.5
0
0.5
1
-50
1.5
0
30
30
20
20
10
10
0
0
-10
-10
-20
-20
-30
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
-30
0.7
3
0.5
1
1.5
2
2.5
Time(s)
3
3.5
4
4.5
5
3.02
3.04
3.06
3.08
3.1
3.12
3.14
3.16
3.18
3.2
Fig. 7. Stator flux in the αβ phase plane and stator current time evolution
1000
35
900
30
800
25
600
Torque (N.m)
speed (rpm)
700
500
400
300
20
15
10
200
5
100
0
0
0.5
1
1.5
2
2.5
time (s)
3
3.5
4
4.5
5
0
0
0.5
1
1.5
2
2.5
Time(s)
3
3.5
4
4.5
5
Fig. 8. Time evolution of speed and electromagnetic torque
3. DTC of Induction motor fed by multilevel inverter
Multilevel inverter present a big interest in the field of the high voltages and the high
powers of the fact that they introduce less distortion and weak losses with relatively low
switching frequency.
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
37
Three level inverter (or multilevel) can be used in the command DTC, what allows to reduce
advantage the harmonics, to have a high level of output voltage and can contribute to more
reducing harmonics and the ripple torque. In that case, the space of voltages is subdivided
into twelve sectors (instead of six with the classic DTC) and by considering the method of
the virtual vectors, three sections with small, medium and large vectors can be exploited.
We can also subdivide the space of voltages into only six sectors by adopting a technique
which employs only twelve active voltage space vectors, corresponding to the small and
large vectors and consequently without using the null or the medium space vectors.
3.1 Vectors tensions and phase level sequences of a three level inverter
The structure of the so called diode clamped three level inverter associated with the
asynchronous motor is shown by figure 9.
Fig. 9. Three level inverter structure
To analyze the potential generated by this three states inverter, every arm is schematized by
three switches which permit to independently connect the stator inputs to the source
potentials (represented by E/2, 0 and –E/2).
The interrupters (IGBTs) are switched in pairs consisting of (C11, C12), (C12, C 11 ) and
( C 11 , C 12 ). When, as example, the upper pair (C11, C12) is turned, the output is connected to
the positive rail of the DC bus.
By making a transformation into αβ (or dq) subspace, a resulting voltage vector is defined
and associated to the spatial position of the stator flux. Then, the different states number of
this vector is 19, since some of the 27 possible combinations produce the same voltage
vector. There are three different inverter states that will produce the zero voltage vector and
two states for each of the six inner voltage vectors (called small vector).
The figure 10 shows the various discreet positions, in the αβ subspace, of the tension vector
generated by a three level inverter.
Fig. 10. Tension vectors generated by a three level inverter
38
Torque Control
3.2 Selection of voltages vectors for the control of the stator flux amplitude
As noted previously, the space evolution of the stator flux vector could be divided into
twelve sectors i (Figure 11), instead of six with the classical DTC, with i= [1, 12] of 30° each,
or into six sectors without using the medium vectors.
When the stator flux vector is in a sector i, the control of the flux and the torque can be
assured by selecting one of 27 possible voltages vectors.
The difference between each of the inverter states that generate the same voltage vectors is
in the way the load is connected to the DC bus. The analysis of the inverter states show that:
the large vectors, such as V24 (+--), correspond to only the positive and negative rails of
•
the DC bus are used and consequently have no effect on the neutral point potential;
in the case of the medium vectors, the load is connected to the positive rail, neutral
•
point and negative rail. The affect on the neutral point depends on the load current;
•
there are two possible states of each of the small voltage vectors which can be used to
control the neutral point voltage. As an example, small vector V22 (+00) causes capacitor
C1 to discharge and C2 to charge and as a result the voltage of the neutral point starts to
rise.
Depending on the stator flux position (sector) and the values of the outputs of torque and
flux controllers, ε Φs and ε Γelm respectively, the optimal vector is selected, from all available
vectors. The first sector could be chosen between -15° and 15° or 0° and 30°. Figure 11
present the space plane for the second case.
Fig. 11. Selection of vectors tensions Vs corresponding to the control of the magnitude Φ s
for a three level inverter.
3.3 Elaboration of the control switching table
The elaboration of the command structure is based on the hysteresis controller output
relating to the variable flux (Cflx) and the variable torque (Ccpl) and the sector N
corresponding to the stator flux vector position.
The exploitation of the first degree of freedom of the inverter, is made by the choice of
vectors apply to the machine among 19 possibilities, during a sampling period. For the
rebalancing of the capacitive middle point, the phase level sequence is chosen among all the
possibilities associated with every voltage vector adopted. This establishes the second
degree of freedom which must be necessarily used.
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
39
The switching table is elaborated depending on the technique adopted for the switching
states choice.
3.3.1 Switching table based on a natural extension of classical DTC
This control scheme, which uses only twelve active voltage space vectors corresponding to
the sections with small and large vectors and without using the null and medium space
vectors, is a natural extension of classical DTC for a two level inverter.
We can consider the case where stator flux is achieved by using two-level hysteresis
comparator and electromagnetic torque by using 4-level hysteresis. The inverter state is
considered as high if the output of torque comparator is high or equal to two and otherwise,
the state is low.
We can note that the choice of one of the two same states corresponding to the level low is
relating to the capacitor voltage balancing.
Table 3 represents, in this case, the switching table.
Φs
Γelm
S1
S2
S3
S4
S5
S6
V2H
V3H
V4H
V5H
V6H
V1H
V2L
V3L
V4L
V5L
V6L
V1L
V6L
V1L
V2L
V3L
V4L
V5L
V6H
V1H
V2H
V3H
V4H
V5H
V3H
V4H
V5H
V6H
V1H
V2H
V3L
V4L
V5L
V6L
V1L
V2L
V5L
V6L
V1L
V2L
V3L
V4L
V5H
V6H
V1H
V2H
V3H
V4H
Table 3. Switching table with twelve active voltage space vectors
As shown by figure 10, the high vectors V1H , V2H , V3H , V4H , V5H and V6H are represented
respectively by the configuration states of the inverter (+--),(++-),(-+-),(--+),(-++) and (+-+).
3.3.2 Switching table with twelve sectors
The space voltage vector diagram, for the three-level inverter, is divided into twelve sectors
by using the diagonal between the adjacent medium and long vector.
According to the errors of torque and the stator flux linkage, the optimal vector is selected,
from all 19 different available vectors (figure 12). The first sector is then chosen between -15°
and 15°.
40
Torque Control
Fig. 12. Space voltage vector diagram (case of twelve sectors).
In analysing the effect of each available voltage vector, it can be seen that the vector affects
the torque and flux linkage with the variation of the module and direction of the selected
vector. For example, to increase the torque and flux V3, V4 and V5 can be selected, but the
action on the increasing torque and flux respectively of V5 and of V3 is the biggest.
Table 4 represents one of the solutions adapted to choice the optimal selected voltage vector
for each sector. In this case, stator flux and torque are achieved by using respectively three
levels and four levels hysteresis comparator. This technique doesn’t use the null voltage
vector for dynamics raisons.
Φs
1
0
-1
Γelm
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
2
V5
V6
V8
V9
V11
V12
V14
V15
V17
V18
V2
V3
1
V3
V5
V6
V8
V9
V11
V12
V14
V15
V17
V18
V2
-1
V18
V2
V3
V5
V6
V8
V9
V11
V12
V14
V15
V17
-2
V17
V18
V2
V3
V5
V6
V8
V9
V11
V12
V14
V15
2
V7
V7
V10
V10
V13
V13
V16
V16
V1
V1
V4
V4
1
V4
V4
V7
V7
V10
V10
V13
V13
V16
V16
V1
V1
-1
V16
V1
V1
V4
V4
V7
V7
V10
V10
V13
V13
V16
-2
V13
V16
V16
V1
V1
V4
V4
V7
V7
V10
V10
V13
2
V8
V9
V11
V12
V14
V15
V17
V18
V2
V3
V5
V6
1
V9
V11
V12
V14
V15
V17
V18
V2
V3
V5
V6
V8
-1
V12
V14
V15
V17
V18
V2
V3
V5
V6
V8
V9
V11
-2
V14
V15
V17
V18
V2
V3
V5
V6
V8
V9
V11
V12
Table 4. Switching table with twelve sectors
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
41
This approach and others, for establishing of the optimal switching table and taking into
account all the factors such as the capacitors balance, system dynamic and system reliability,
must be deeply analysed and tested. Also, it’s difficult, in this case, to select the optimal
voltage vectors; however, the use of artificial intelligence techniques will add their
superiority to some extent.
4. Direct torque control based fuzzy / neural network
By analyzing the structure of the switching table, we can notice that it can be translated in
the form of fuzzy rules. Consequently, we can replace the switching table and the hysteresis
comparator by a fuzzy system. The fuzzy character of this system allows flexibility in the
choice of the fuzzy sets of the input and the capacity to introduce knowledge of the human
expert there.
Also, as the DTC uses algorithms to select a large number of statements inverter switches,
neural networks can accomplish this task after a learning phase. The neural network selector
inputs will be proposed as the position of the flux stator vector, the error between its estimated
value and the reference one, and the difference between the estimated and reference values of
electromagnetic torque. The next figure shows an example of this structure.
Fig. 13. Switching table based Fuzzy / ANN technique
4.1 Direct torque control based fuzzy logic
The principle of fuzzy direct torque control (FDTC) consists to replace, in conventional DTC,
the torque and stator flux hysteresis controllers and the switching table by a fuzzy system.
In this case, two approaches can be presented to illustrate the strategy of FDTC of the
induction motor fed by two-level inverter.
We can consider three variables input fuzzy logic controllers; the stator flux error,
electromagnetic torque error and angle of stator flux, however, the choice of the output
deferred according to the approach utilized. The output could be the voltage space vector,
for FDTC based PWM, or the magnitude and argument of voltage vector for space vector
modulation.
4.1.1 FDTC based Pulse Width Modulation (PWM)
The fuzzy logic controller blocks using PWM inverter is shown in the following figure.
These blocks are composed of two main parts: fuzzification and fuzzy rules base, since no
42
Torque Control
defuzzification is needed because, in this case, the output of fuzzy controller is the actual
PWM voltage vector sequence and these states are directly the results of fuzzy rules.
ΔΓelm
ΔΦs
θ
Fuzzy logic controller
Vs
Fig. 14. Fuzzy logic controller based PWM
4.1.1.1 Fuzification
Based on the switching table of the conventional DTC, the universe of discourse for each
three inputs of the fuzzy logic controller has been divided into: two fuzzy sets (NP), for
stator flux error, three fuzzy sets (NZP), for electromagnetic torque error, and seven fuzzy
sets ( θ 0 , θ1 ,..., θ 7 ) for angle of flux stator.
These fuzzy sets are defined by the delta and trapezoidal membership functions and are
presented by the following figure.
Fig. 15. Membership functions
4.1.1.2 Fuzzy rules base
The table can be expressed by fuzzy rules given by:
The ith rule Ri : if ΔΦ is Ai and ΔΓ is Bi and θ is Ci then n is Ni.
Ai, Bi and Ci are the fuzzy sets of the variables ΔΦs, ΔΓelm and θ
n is the inverter switching state.
The inference method used is Mamdani’s procedure based on min-max decision.
These rules are resumed by the following table.
θ1
θ2
θ3
θ4
θ5
θ6
θ7
ΔΦs
ΔΓelm
P
N
P
N
P
N
P
N
P
N
P
N
P
N
P
V5
V6
V6
V1
V1
V2
V2
V3
V3
V4
V4
V5
V5
V6
Z
V0
V7
V7
V0
V0
V7
V7
V0
V0
V7
V7
V0
V0
V7
N
V3
V2
V4
V3
V5
V4
V6
V5
V1
V6
V2
V1
V3
V2
Table 5. Fuzzy rules
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
43
4.1.2 FDTC based Space Vector Modulation (SVM)
Using space vector modulation permit, in addition to the advantages obtained by the fuzzy
logic controller (reduction of the torque, stator flux and current ripples and to get a fast
torque response), to maintain constant the switching frequency. With this strategy two
fuzzy controller of Mamdani could be used to control the magnitude and argument of
voltage vector reference. For this technique, two controllers (next figure) are used
concerning the variables magnitude and argument of vector tension.
ΔΓelm
⎜Vs⎢
Magnitude
controller
d/dt
Δθ
Argument
controller
ΔΦs
++
Arg(Vs)
θ
Fig. 16. Fuzzy logic controller based SVM
In the following figure, the membership functions of the variables ΔΦs and ΔΓelm are
presented.
Fig. 17. Membership functions for ΔΦs and ΔΓelm
We consider, in this case, two fuzzy sets functions (D: Decrease, I: Increase) for the stator
flux and electromagnetic torque errors and three membership functions (N: Negative, Z:
Zero, P: Positive) for the variation of the electromagnetic torque error.
The fuzzy rules of the argument fuzzy controller are presented in the following table.
ΔΦs
Δθ
ΔΓelm
Dec
Inc
Dec
μ(-2π/3)
μ(-π/3)
Inc
μ(2π/3)
μ(π/3)
Table 6. Fuzzy rules of argument controller
44
Torque Control
μ(θ) is the membership function for the output variable of argument fuzzy controller
defined as represented by the following figure.
Fig. 18. Membership function for output argument controller
The voltage vectors in conventional DTC have constant amplitude in opposite with FDTC
based space vector modulation where this amplitude is modified versus the torque and its
derivative. Then, the fuzzy rules of the amplitude fuzzy controlled take form:
If ΔΓelm decrease and d(ΔΓelm)/dt is negative then magnitude vector is small.
Consequently, these different rules are resumed in the following table, where the fuzzy sets
used are, N: Negative, M: Medium, Z: Zero, P: Positive, S: Small and L: Large.
d( ΔΓ elm )
dt
V
ΔΓelm
N
Z
P
Dec
S
S
M
Inc
M
L
L
Table 7. Fuzzy rules of amplitude fuzzy controller
Finally, the fuzzy sets of output magnitude fuzzy controller are defined by delta and
trapezoidal membership functions as shown by this figure.
Fig. 19. Membership function for output magnitude controller
4.2 Direct torque control based artificial neural networks
Among the other intelligence techniques can improving the performance of system control
and are recently showing good promise for applications in power electronics and motion
control system, the use of Artificial Neural Network (ANN).
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
45
Different techniques based ANN are exploited for the control of IM; particularly, in the field
of the IM Direct Torque Control, many types of these techniques are adopted. The most
popular of ANN, used in DTC, is the multilayer feed forward network, trained by the back
propagation algorithm, which is composed on the input layer, output layer, and several
hidden layers.
Also, as the switching table has an important role in the DTC, for increasing the execution
speed of the system, ANN is applied to emulate the classical switching table of the DTC
obtaining the optimal switching patterns.
The switching table has as inputs the electromagnetic torque error, the stator flux error and
the angle of the flux, and as output the voltage space vector to be generated by the inverter.
Since this switching lookup table only depends on these inputs and not on the parameters of
the IM, it can be trained off-line. Therefore, the inputs of switching table will be converted to
digital signals, for reducing the training patterns and increasing the execution speed of the
training process. Thus, one bit (1 or 0) represents the flux error, two bits (11 for state 1 , 00
for state 0 or 01 for state -1) the torque error and three bits the region of stator flux.
The structure of the ANN as a part of DTC is presented by figure 20, which has six inputs
nodes corresponding to the digital variables (three for angle flux, one for flux error and two
for torque error), six neurons in the first hidden layer, five neurons in the second hidden
layer and three neurons in the output layer.
S1
Position
S2
Flux
Torque
Sigmoid
neuron
layer
Sigmoid
neuron
layer 1
Sigmoid
neuron
layer 2
S3
Fig. 20. Structure of the ANN
After completion of the training procedure, the network performance off-line with an
arbitrary input pattern will be tested to ensure successful training. After that, the weights
and biases are down-loaded to the prototype network substituting the traditional switching
lookup table as a part of DTC.
An example of the ANN combined with Fuzzy inference system for the control of the IM
speed will be presented in the next section.
5. Control of asynchronous motor speed based on a fuzzy / neural corrector
These last years, a most interest concerned the use of the artificial intelligence techniques
(neural networks, fuzzy logic, genetic algorithms) which have the potential to provide an
improved method of deriving non-linear models, have self adapting capabilities which
make them well suitable to handle non-linearities, uncertainties and parameter variations.
46
Torque Control
The simplest of these methods are based on the learning of an already existing conventional
controller; others methods operate a learning off-line of the process inverse model or of a
reference model either completely on-line.
5.1 Description of the technique adopted for IM Speed control
As example, we have chosen to develop the case where a conventional neural controller
(CNC) associated with a reference model (MRAS) for the learning phase is used to control
the IM speed.
The parameters of the CNC are adjusted by minimising the error (e=u’-u) between the
outputs of the MRAS and CNC as shown in the following figure.
Fig. 21. Neural corrector for the IM speed control
Once the learning phase is carried out, the weights obtained are used for the neural
controller, in the second phase, without the reference model.
Neural network, coupled with the fuzzy logic, speed control will be so efficient and robust.
In this case, the reference model is represented by a fuzzy logic corrector (FLC) with two
inputs: the error and the derivative of the error (next figure).
Ω*
Ωm
FLC
+-
Γ*
d
dt
ANN
Γm
+
-
Delay
ΔW(k)
Back
propagation
algorithm
Fig. 22. IM Speed control based Fuzzy / neural corrector
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
47
The neural corrector architecture, shown by figure 23, presents 4 inputs, 3 neurons for the
hidden coat with activation function type sigmoid and one output with linear activation
function.
Fig. 23. Architecture of neural corrector
As it has been noted, a corrector type PI (Proportional Integral), for the reference model,
which parameters are adapted by a fuzzy inference system, is used (Figure 24).
Fig. 24. Controller with PI structure adapted by fuzzy inference system
The PI parameters (Kp, Ki) are calculated by using the intermediate values (K’p and K’i)
given by the fuzzy controller as follows:
K p = (K pmax − K pmin )K 'p + K pmin
K i = (K imax − K imin )K i' + K imin
(8)
where the gains values are defined by using the Ziegler-Nichols method.
Both parameters (K’p, K’i), corresponding to the output of the system based on fuzzy logic,
are meanwhile normalised in the range [0 1].
5.2 Simulation results
Simulations were performed to show the performances of the technique used in this section
and based on fuzzy / neural corrector for the control of the IM speed. The following figure
presents the model structure tested in the Matlab / Simulink environment.
We have chosen to present the results corresponding to the rotation speed evolution, the
electromagnetic torque, the flux evolution in β phase plane and the stator current temporal
evolution.
48
Torque Control
Source
Discrete,
Ts = 5e-006 s.
powergui
-K-
Conn 2
t
Clock
g
+
Conn 1
Conn 3
A
A
B
B
C
C
I_abc
N
-
Thr ee -Level Bri dge
is _ abc
Tm
V _abc
V _Com
A1
A
A2
B
A3
C
vs _ qd
m
m
wm
Te
couple
IS
couple
Vs _ be t a
Vs -a lp ha
flux _alpha
tensions_statorique
IS _ be t a
Trans _ concor di at
flux _beta
IS _ al p ha
Vs
Scope
courants_statorique
Vs_dq
FLUX
TETA
Sector
sector
Flux _angle
flux _est
Gates
H Phi
phi _salpha
Isalpha
phi _sbeta
Vsalpha
teta
Isbeta
flux
Vsbeta
H _flux 1
H _Te
H Te
vitesse
couple_ref1
couple _ref
w_mes
flux _ref
C_ANN
ph_ref
flux & couple Estimator
w_ref
ANN Speed Controler
1.5
60
1
40
0.5
20
Stator current (A)
Stator flux
Fig. 25. Stator flux in the β phase plane and stator current time evolution
0
-0.5
-1
0
-20
-40
-1.5
-1.5
-1
-0.5
0
0.5
1
-60
1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
0.9
1
Fig. 26. Stator flux in the (( phase plane and stator current time evolution
1000
30
800
20
600
10
200
Torque (N.m)
Speed (rpm)
400
0
-200
-400
0
-10
-20
-600
-30
-800
-1000
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
0.9
1
-40
0
0.1
0.2
Fig. 27. Time evolution of speed and electromagnetic torque
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
0.9
1
Direct Torque Control Based Multi-level Inverter
and Artificial Intelligence Techniques of Induction Motor
49
We can note that mechanical speed track the reference values imposed and corresponding to
two senses of the motor rotation (Figure 27). Also the currents are sinusoidal with acceptable
value of total harmonic distortion and the flux components trajectory is circular and limited
to the reference value in the hysteresis bound.
6. Conclusion
The Direct Torque Control (DTC) is an important alternative method for the induction
motor drive, with its high performance and simplicity. The DTC applied to induction
machine fed by a 3-level diode clamped inverter presents good performance and
undulations reduction. In this case, some techniques were developed in order to replace the
conventional DTC switching table adapted for a NPC inverter.
Another issue is concerned with the application of artificial intelligence techniques (fuzzy
system, and neural network) to the induction motor control with DTC. Particularly, the
application of these techniques for the selection of optimal voltage vectors is presented.
Also, the control of the speed induction motor is realized by the technique of the artificial
neural networks (ANN) with reference model. A controller based PI adapted by a fuzzy
inference is used as reference model. Attention is focused on the dynamic performance of
ANN speed control. The effectiveness of the proposed scheme control is demonstrated by
simulation using the blocks PSB of Matlab / Simulink.
Finally, in this chapter, we can conclude that the DTC method applied to an induction motor
fed by a three level NPC inverter and based artificial intelligence techniques present most
interest and contribute to improvement of system response performances.
The first investigations, presented here, of the induction machine control prove its
effectiveness and its high dynamics. It will be completed in a future work by considering
others control techniques.
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Torque Control
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International Symposium, Vol.2, pp 625-630, Jul.1997.
Yang, J., Ryan, M. & Power, J. (1994). Using Fuzzy Logic,” Prentice Hall, 1994.
Kumar, R., Gupta, R.A., Bhangale, S.V. & Gothwal, H. (2008). Artificial Neural Network
based Direct Torque Control of Induction Motor Drives. IETECH Journal of Electrical
Analysis, Vol.2, N°3, pp 159-165, 2008.
Toufouti, R., Mezian, S. & Benalla, H. (2007). Direct Torque Control for Induction Motor
using Intelligent Technique. Journal of Theoretical and Applied Information Technology,
Vol.3, N°3, pp 35-44, 2007.
Dreyfus, G., Martinez, J., Samuelides, M., Gordon, M.B., Badran, F., Thiria, S. & Hérault, L.
(2002). Réseaux de neurons : Méthodologie et applications. Editions Eyrolles, 2002.
Grabowski, P.Z., Kazmierkowski, M.P., Bose, B.K. & Blaabjerg, F. (2000). A simple Direct
torque Neuro Fuzzy control of PWM Inverter fed Induction motor drive. IEEE
Trans. Electron. 47 N° 4, pp 863-870, Aug 2000.
Viljamaa, P. (2000). Fuzzy gain scheduling and tuning of multivariable fuzzy control
methods of fuzzy computing in control systems. Thesis for the degree of doctor of
technology, Temper University of technology, Finland, 2000.
Barbara H. K. (2001). Stator and Rotor Flux Based Deadbeat Direct Torque Control of
Induction Machines. IEEE Industry Applications Society, Annual Meeting, Chicago,
September 30-October 4, 2001.
Casadei, D., Profumo, Serra, G. & Tani, A. (2002). FOC And DTC:Tox Viable Schemes For
Induction Motors Torque Control. IEEE trans.Power Electronics. On PE, Vol.17, N°.5,
Sept 2002.
Schibili, N., Nguyen, T. & Rufer, A. (1998). Three-Phase Multilevel Converter for HighPower Induction Motors. IEEE trans. On Power Elect. Vol. 13 N°.5, 1998.
Roboan, X. (1991). Variateur de vitesse pour machine asynchrone, Contrôle de la vitesse sans
capteur mécanique. Thèse Doctorat de L’INPT, Toulouse, 1991.
Ould Abdeslam, D., Wira, P., Mercklé, J., Chapuis, Y.A. & Flieller, D. (2006). Stratégie
neuromimétique d'identification et de commande d'un filtre actif parallèle. Revue
des Systèmes, Série Revue Internationale de Génie Electrique (RS-RIGE), vol. 9, no. 1, pp
35-64, 2006.
Ould abdeslam, D. (2005). Techniques neuromimétiques pour la commande dans les
systèmes électriques: application au filtrage actif parallèle. Thèse de doctorat d’état en
Electronique, Electrotechnique et Automatique, Université de Batna, 2005.
3
Direct Torque Control using Space
Vector Modulation and Dynamic Performance of
the Drive, via a Fuzzy Logic Controller for
Speed Regulation
Adamidis Georgios, and Zisis Koutsogiannis
Democritus University of Thrace
Greece
1. Introduction
During the last decade, a lot of modifications in classic Direct Torque Control scheme
(Takahashi & Noguchi, 1986) have been made (Casadei et al., 2000), (Reddy et al., 2006),
(Chen et al., 2005), (Grabowski et al., 2005), (Romeral et al., 2003), (Ortega et al., 2005). The
objective of these modifications was to improve the start up of the motor, the operation in
overload conditions and low speed region. The modifications also aimed to reduce the
torque and current ripple, the noise level and to avoid the variable switching frequency by
using switching methods with constant switching frequency.
The basic disadvantages of DTC scheme using hysteresis controllers are the variable
switching frequency, the current and torque ripple. The movement of stator flux vector
during the changes of cyclic sectors is responsible for creating notable edge oscillations of
electromagnetic torque. Another great issue is the implementation of hysteresis controllers
which requires a high sampling frequency. When an hysteresis controller is implemented
using a digital signal processor (DSP) its operation is quite different to the analogue one.
In the analogue operation the value of the electromagnetic torque and the magnitude of the
stator flux are limited in the exact desirable hysteresis band. That means, the inverter can
change state each time the torque or the flux magnitude are throwing the specified limits.
On the other way, the digital implementation uses specific sample time on which the
magnitudes of torque and flux are checked to be in the desirable limits. That means, very
often, torque and flux can be out of the desirable limits until the next sampling period. For
this reason, an undesirable torque and flux ripple is occurred.
Many researchers are oriented to combine the principles of DTC with a constant switching
frequency method for driving the inverter by using space vector modulation. This requires
the calculation in the control schemes of the reference voltage vector which must be
modulated in the inverter output. Therefore, the Direct Torque Control with Space Vector
Modulation method (DTC-SVM) is applied (Koutsogiannis & Adamidis 2007). Since we
know the reference voltage vector it is easy to perform the modulation by applying specific
switching pattern to the inverter (Koutsogiannis & Adamidis 2006). In the DTC scheme a
speed estimation and a torque control are applied using fuzzy logic (Koutsogiannis &
Adamidis 2006). An improvement of DTC with a parallel control FOC is observed (Casadei
52
Torque Control
et al., 2002). The use of the rotor flux magnitude instead of the stator flux magnitude,
improves the overload ability of the motor. This control is sensitive to the machine’s
parameters during transient operations.
Also, the DTC-SVM can be applied using closed loop torque control, for minimization of
torque ripple (Wei et. al., 2004). In this case estimation of stator and rotor flux is required.
Therefore, all the parameters of the induction motor must be known (Reddy et al., 2006). A
new method was developed that allows sensorless field-oriented control of machines with
multiple non-separable or single saliencies without the introduction of an additional sensor
(Zatocil, 2008). In this paper, the closed loop torque control method is applied which
improves the torque response during dynamic and steady state performance. A lot of papers
for the speed control of electrical drives, which uses different strategies based on artificial
intelligence like neural network and fuzzy logic controller, have presented. For the fuzzy PI
speed controller its robustness and disturbance rejection ability Gadou et. Al., 2009) is
demonstrated. In this paper fuzzy logic for the speed estimation of the motor and the
method DTC-SVM with closed loop torque control will be applied. This paper is further
extended through a further improvement of the system control by controlling the
magnitudes of torque and flux using closed loop control. The simulation results were
validated by experimental results.
2. Overview of the classic DTC scheme
The classic DTC scheme is shown in figure 1.
Fig. 1. Classic DTC scheme.
DTC based drives require only the knowledge of the stator resistance Rs. Measuring the
stator voltage and current, stator flux vector can be estimated by the following equation:
G
G
G
ψ s = ∫ (Vs − Rs I s ) dt
(1)
the stator flux magnitude is given by,
G
Ψ s = Ψ 2as + Ψ 2β s
(2)
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
53
where the indicators α,β indicates the α-β stationary reference frame. The stator flux angle is
given by,
Ψ
θ e = sin −1 Gβ s
(3)
Ψs
and the electromagnetic torque Te is calculated by,
Te =
3⎛ P⎞
⎜ ⎟ Ψα s iβ s − Ψ β siα s
2⎝ 2 ⎠
(
)
(4)
where P is the number of machine poles.
In the DTC scheme the electromagnetic torque and stator flux error signals are delivered to
two hysteresis controllers as shown in figure 1. The stator flux controller imposes the time
duration of the active voltage vectors, which move the stator flux along the reference
trajectory, and the torque controller determinates the time duration of the zero voltage
vectors, which keep the motor torque in the defined-by-hysteresis tolerance band. The
corresponding output variables HTe, HΨ and the stator flux position sector θΨs are used to
select the appropriate voltage vector from a switching table scheme (Takahashi & Noguchi,
1986), which generates pulses to control the power switches in the inverter. At every
sampling time the voltage vector selection block chooses the inverter switching state, which
reduces the instantaneous flux and torque errors.
In practice the hysteresis controllers are digitally implemented. This means that they
function within discrete time Ts. Consequently, the control of whether the torque or the flux
is within the tolerance limits, often delays depending on the duration of the sampling
period. This results in large ripples in the torque and the current of the motor. In conclusion,
the abrupt and undesirable ripples in the electromagnetic quantities appear when the
control of the values of the torque and the flux takes place at times when their values are
near the allowed limits. This means that a voltage vector will be chosen which will continue
to modify these quantities in a time Ts, even though these limits have been practically
achieved. Accordingly, in the next control which will be carried out after time Ts, these
quantities will be quite different from the desirable values. Another reason why the
G
electromagnetic torque of the motor presents undesirable ripples is the position of the ψ s in
each of the six sectors of its transition. In general, an undesired ripple of the torque is
G
observed when the ψ s moves towards the limits of the cyclic sectors and generally during
the sectors’ change. Furthermore, the torque ripple does not depend solely on the systems
G
conditions but on the position of ψ s in the sector as well. Therefore, we can establish that
there are more control parameters which could affect the result of the motor’s behavior.
3. DTC-SVM with closed-loop torque control
In this section, the DTC-SVM scheme will be presented which uses a closed loop torque
control. The block diagram of this scheme is shown in figure 2.
The objective of the DTC-SVM scheme, and the main difference between the classic DTC, is
to estimate a reference stator voltage vector V*S in order to drive the power gates of the
inerter with a constant switching frequency. Although, the basic principle of the DTC is that
the electromagnetic torque of the motor can be adjusted by controlling the angle δΨ between
54
Torque Control
Fig. 2. DTC-SVM with closed-loop torque control
the stator and rotor magnetic flux vectors. Generally, the torque of an asynchronous motor
can be calculated by the following equation.
Te =
3 ⎛ P ⎞ Lm
Ψ r Ψ s sin δψ
⎜ ⎟
2 ⎝ 2 ⎠ Lr L's
(5)
Where L's = Ls Lr − L2m . The change in torque can be given by the following formula,
ΔTe =
G
G
3 ⎛ P ⎞ Lm
Ψ r Ψ s + ΔΨ s sin Δδψ
⎜ ⎟
2 ⎝ 2 ⎠ Lr L's
(6)
where the change in the stator flux vector, if we neglect the voltage drop in the stator
resistance, can be given by the following equation,
G
G
ΔΨ s = Vs Δt
(7)
where Δt=Ts, is the sampling period.
Generally, the classic DTC employs a specific switching pattern by using a standard
switching table. That means the changes in the stator flux vector and consequently the
changes in torque would be quite standard because of the discrete states of the inverter. That
happens because the inverter produces standard voltage vectors.
The objective of the DTC-SVM scheme, and the main difference between the classic DTC, is
to estimate a reference stator voltage vector V*S and modulate it by SVM technique, in order
to drive the power gates of the inerter with a constant switching frequency. Now, in every
sampling time, inverter can produce a voltage vector of any direction and magnitude. That
means the changes in stator flux would be of any direction and magnitude and consequently
the changes in torque would be smoother.
According to above observations, and bearing in mind figure 2, we can see that torque
controller produces a desirable change in angle ΔδΨ between stator and rotor flux vectors.
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
55
(a)
(b)
Fig. 3. Principle of Space Vector Modulation (SVPWM)
(a)reference stator vector
(b) modulation of space vector during one switching period which is equal to sampling time
of the DTC-SVM method.
The change in angle ΔδΨ is added in the actual angle of stator flux vector, so we can estimate
the reference stator flux vector by using the following formula, in stationary reference
frame.
G
G j ω t +Δδ
ψ s* = ψ s* e ( e ψ )
(8)
Applying a phasor abstraction between the reference and the actual stator flux vector we
can estimate the desirable change in stator flux ΔΨS. Having the desirable change in stator
flux, it is easy to estimate the reference stator voltage vector:
G
G
G
ΔΨ s
Vs* =
+ Rs I s
TS
(9)
56
Torque Control
If the reference stator voltage vector is available, it is easy to drive the inverter by using the
SV-PWM technique. So, it is possible to produce any stator voltage space vector (figure 3).
As it mentioned before, in the classic DTC scheme, the direction of stator flux vector changes
G
Δψ S are discrete and are almost in the same direction with the discrete state vectors of the
G
inverter. Consequently, in DTC-SVM, stator flux vector changes Δψ S can be of any
G
direction, which means the oscillations of ψ S would be more smoother.
4. Simulation results of DTC and DTC-SVM
The DTC schemes, that are presented so far, are designed and simulated using
Matlab/Simulink (figure 4). The proposed scheme is simulated and compared to the classic
one. The dynamic and also the steady state behavior is examined in a wide range of motor
speed and operating points.
(a)
(b)
Fig. 4. Simulink models of (a) classic DTC and (b) DTC-SVM.
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
57
For simulation purposes, an asynchronous motor is used and its datasheets are shown in the
following table I. The nominal values of the asynchronous motor in the simulation system
are the same with the nominal values of the asynchronous motor in the experimental
electrical system.
P = 4 (2 pair of poles), f = 50 Hz
Rs = 2,81 Ω
Ls = 8,4 mH
230V/ 400V
R’r = 2,78 Ω
L’ r = 8,4 mH
P = 2,2 kW, Nr = 1420 rpm
Lm = 222,6 mH
J = 0.0131
kgm2
Table I. Nominal values of motor.
For the simulations a particular sampling period TS _ DTC for torque and flux was chosen as
well as the proper limits ΗΒψ and ΗΒΤe for the hysteresis controllers, in order to achieve an
average switching frequency which shall be the same with the constant switching frequency
produced by the DTC-SVM control. During the simulation, the dynamic behavior of the
system has been studied using both the DTC and the DTC-SVM method.
4.1 Steady state operation of the system
The results of the simulations are presented in the figure 5, where the electromechanical
magnitudes of the drive system are shown, for both control schemes in various operation
points. In more detail, in figure 5 the operation of the system for low speed and low load is
shown and figure 6 shows the motor operation in normal mode. All the electromechanical
quantities are referred to one electrical period based on the output frequency of the inverter.
The average number of switching for the semiconducting components of the inverter during
the classic DTC is almost the same with the number of switching of the DTC-SVM method
where the switching frequency is constant. In fact, for the classic DTC flux variation of the
hysteresis band equal to ΗΒΨ=0.015 was chosen, which is almost 2% of the nominal flux and
for the torque the hysteresis band controller was chosen to be ΗΒTe=0,65, which means 3% of
the nominal torque. These adjustments led to an average switching number of inverter states
equal to 17540 per second, for the classic DTC, while for the DTC-SVM a switching
frequency equal to 2.5kHz was chosen, namely 15000 switching states per second.
The classic DTC has some disadvantages, mainly in the low speed region with low
mechanical load in the shaft, where the current ripple is very high, compared to DTC-SVM
(figure 5). Also, the classic DTC has variable switching frequency, where it is observed that
the switching frequency is high in low speed area and low in high speeds. In practice, it is
not easy to change the sampling period of the hysteresis controllers with respect to the
operation point of the drive system. For this reason, a value of the sampling period is chosen
from the beginning, which shall satisfy the system operation in the complete speed range.
The high ripple observed in the classic DTC electrical magnitudes during the operation in
low speed area, is due to the fact that many times, instead of choosing the zero voltage
vector for the inverter state, in order to reduce the torque, the backwards voltage vector is
chosen, which changes the torque value more rapidly.
Figure 6 shows the motor operation in normal mode. The switching frequency is also at the
same value in order to have a right comparison. Current ripple has also a notable reduction in
DTC-SVM compared to classic DTC. Also, at this operating point it can be seen that in classic
58
Torque Control
DTC the torque ripple of the electromagnetic torque which is resulted by the cyclic sector
changes of stator flux vector and produces sharp edges, is now eliminated by using DTC-SVM.
Classic DTC
DTC-SVM
(a)
(b)
Fig. 5. Steady state of the motor in an operation point where the motor has the 10% of the
nominal speed and 10% of nominal load, with HBψ = ±0.015 , HBTe = ±0.65
(a) Classic DTC with hysteresis band controllers and TS _ DTC = 12 μ sec the sampling time for
discrete implementation. Inverter produces 16780 states/sec.
(b) DTC with space vector modulation. Switching frequency is equal to 2.5kH and inverter
produces 15000 states/sec.
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
Classic DTC
DTC-SVM
Ψs vector α - β axis (pu)
Vs vector α - β axis (pu)
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
0
1
Is vector α - β axis (pu)
1
1
0.5
0.5
0
0
-0.5
-0.5
1
0.5
0
0
-0.5
-0.5
1
0
0.9
-1
0
1
Is vector α - β axis (pu)
1
1
0.5
0
0
-0.5
-0.5
-1
-1
10
Time (sec)
15
0
1
-1
0
1
1.1
1
0.9
0.8
5
1
0.5
1
1
0
0
-1
-1
20
0
5
-3
x 10
10
Time (sec)
15
10
Time (sec)
15
10
Time (sec)
15
20
-3
x 10
1
Current ia (pu)
1
0.5
0
-0.5
0.5
0
-0.5
-1
-1
0
5
10
Time (sec)
15
0
20
5
-3
x 10
1
1
0.5
0.5
Vab (pu)
Vab (pu)
-1
-1
Ψr vector α - β axis (pu)
Torque (pu)
Torque (pu)
0
1.1
0.8
Current ia (pu)
1
0.5
-1
-1
V*ref vector α - β axis (pu)
Ψs vector α - β axis (pu)
-1
-1
0
1
Ψr vector α - β axis (pu)
-1
59
0
-0.5
20
-3
x 10
0
-0.5
-1
-1
0
5
10
Time (sec)
15
20
0
-3
x 10
(a)
5
20
-3
x 10
(b)
Fig. 6. Steady state of the motor in an operation point where the motor has the 100% of the
nominal speed and 100% of nominal load, with HBψ = ±0.015 , HBTe = ±0.65 for,
(a) Classic DTC. (b) DTC with space vector modulation.
4.2 Dynamic performance of the system
In figure 7 the simulation results are presented for the dynamic case where the mechanical
load is changing while the reference speed must remain constant. The case of this simulation
is very rare and extreme where the motor suddenly loses the 80% of its load (from 100% to
20%) while the speed must remain constant.
60
Torque Control
DTC-SVM
1.1
Speed (pu)
Speed (pu)
Classic DTC
1
0.9
0
0.1
0.2
0.3
0.4
Torque (pu)
Torque (pu)
0.5
0
0
0.1
0.2
0.3
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
1.2
Flux (pu)
Flux (pu)
0.2
Te
Te*
TL
0
1
0.8
ψs
ψs*
1
0.8
0.1
0.2
0.3
0.4
Current IS (pu)
0
Current IS (pu)
0.1
1
0.4
1.2
1.5
1
0.5
0
0.1
0.2
0.3
0
-1
0
0.1
0.2
0.3
0.1
0.2
Time (sec)
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
Time (sec)
0.3
0.4
0
1
0
-1
0
-1
0
0.3
1
Voltage (pu)
0
0.2
1
0.4
1
0.1
0.5
0.4
1
0
1.5
Current ia (pu)
0
Current ia (pu)
0.9
1.5
1
-1
ωr*
ωr
1
0
1.5
Voltage (pu)
1.1
0.4
(a)
(b)
Fig. 7. Load change: (a) Classic DTC, (b) DTC-SVM.
Figure 8 shows the dynamic performance of the drive system due to the reference speed
step commands operation. During the transient operation of the drive system, in both cases,
the ripple in electromechanical magnitudes is shown. It must be noted at this point that the
speed controller, which is used for the simulations, is a fuzzy PI controller. As it is shown in
figure 8 the ripple of the electromechanical magnitudes is higher in the classic DTC method
in comparison to the DTC-SVM method.
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
DTC-SVM
1
Speed (pu)
Speed (pu)
Classic DTC
0.8
0.6
0.4
0
0.1
0.2
0.3
0.8
0.6
0.4
0.4
0
Torque (pu)
Torque (pu)
1
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
1.2
Flux (pu)
Flux (pu)
0.3
Te
Te*
TL
0
1
0.8
ψs
ψs*
1
0.8
0.1
0.2
0.3
0.4
Current IS (pu)
0
Current IS (pu)
0.2
1
0.4
1.2
1.5
1
0.5
0
0.1
0.2
0.3
0
-1
0
0.1
0.2
0.3
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
Time (sec)
0.3
0.4
1
0.5
0
0.4
1
0
1.5
Current i a (pu)
0
Current ia (pu)
0.1
2
0
1
0
-1
0.4
1
Voltage (pu)
1
Voltage (pu)
ωr*
ωr
1
2
0
-1
61
0
0.1
0.2
Time (sec)
0.3
0.4
0
-1
a)
Fig. 8. Speed control response: (a) Classic DTC (b) DTC – SVM.
b)
62
Torque Control
5. Experimental results
The implementation of the system is carried out with the development system dSPACE and
the control panel R&D DS 1104 and the software package Matlab/Simulink. Also the
experimental model, consists of one AC motor feed controlled converter and one DC motor
feed converter which operates as a load for the system. In table II the datasheet of the
experimental system kit is shown.
In figures 9a and 9b the oscillograms of the speed, torque, flux and the current components
id and iq of the machine are shown for reference speed variation, for both cases of control
method classic DTC and DTC-SVM. In figures 10a and 10b we see the oscillograms of the
electromechanical quantities of the system during load loss with classic DTC method and
DTC-SVM method respectively. Where wmegaM is the actual value, wref is the reference
value and the wmega estim is the estimated value of the angular frequency of the motor.
Also the Torque is the actual value of the torque and the Torque calc is the calculated value
of the electromagnetic torque. In figure 10 the Ia ref is the reference and Ia is the actual
value of the DC motor’s current, which is performed as a load in the experimental model.
From the oscillograms it is shown that the control has more advantages in case of DTC-SVM
method compared to the classic DTC.
Asynchronous Motor
DC-Motor
Converter
Nominal Power
PN = 2,2 kW
PN = 4,2 kW
PN = 4 kW
Nominal Voltage
UN = 400 V
UAN = 420 V
IN = 7 A
Nominal Current
IN = 4,85 A
IAN = 12,5 A
Nominal Speed
nN = 1420 min-1
nN = 2370 min-1
Nominal power factor
cosφN = 0,82
Number of poles
p=4
Stator ohmic resistance
Rotor ohmic resistance
R1 = 2,82 Ω
R’ = 2,78 Ω
Stator inductance
Rotor
inductance
stator
of
Ls = 8.4 mH
L’r= 8.4 mH
Excitation voltage
UEN = 310 V
Excitation Current
IEN = 0,93 A
Nominal Frequency
fN = 4 kHz
Table II. Datasheets of the asynchronous motor, DC-motor and converter during the
implementation
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
(a)
(b)
Fig. 9. Electromechanical quantities in transient operation of the system using, a) classic
DTC, b) DTC-SVPWM.
63
64
Torque Control
(a)
(b)
Fig. 10. Electromechanical quantities in transient operation of the system using,
a) classic DTC, b) DTC-SVPWM.
6. Speed regulation using a fuzzy logic controller
So far, two methods were described for controlling the electromagnetic torque of an
asynchronous motor drive. When we need to regulate the speed of such a drive a speed
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
65
controller is needed. The speed controller takes the error signal between the reference and
the actual speed and produces the appropriate reference torque value. That means, the drive
changes mode from torque control to speed control. So, now the mechanical load on motor
shaft defines the electromagnetic torque of the motor. In torque control mode the
mechanical load on motor shaft defines the rotor speed. In figure 11 we can see the block
diagram of the proposed drive, in speed control mode. A reference speed signal ωr* or, in
other words, the speed command is given. The actual speed ωr is estimated or measured
with a speed encoder. This depends on the precision requirements of each application. The
speed is estimated directly from state equations. The dynamic a-b frame state equations of a
machine can be operated to compute speed signal directly [2], [4]. Consequently, the speed
computation is given by:
ωr =
1 ⎡⎛
d
d
⎞
L
⎤
m
(ψ ar iβ s −ψ β r ias )⎥
⎢⎜ψ ar ψ β r − ψ β r ψ ar ⎟ −
dt
dt
ψ r2 ⎣⎝
⎠ Tr
⎦
(10)
Where: Tr = Lr Rr
This method of speed computation requires the knowledge of the machine parameters Lr ,
Lm , and Rr which are the rotor inductance, the magnetizing inductance and rotor
resistance respectively.
The speed controller can be a classic PI controller or a fuzzy PI controller. In
[Koutsogiannis], a detailed presentation and comparison of the two controllers is presented
and operates with a classic DTC drive. In this paper the fuzzy PI controller is also used for
the comparison between the classic DTC and DTC-SVM.
Fig. 11. Speed regulation using a speed controller.
As it will be described in the next section, the error between the estimated speed ωr and the
reference command speed ωr* is delivered to the speed controller, which calculates the
reference electromagnetic torque Te* .
66
Torque Control
The corresponding output variables HTe , Hψ and the stator flux position sector θψ s are
used to select the appropriate voltage vector from a switching table, which generates pulses
to control the power switches in the inverter. At every sampling time the voltage vector
selection block chooses the inverter switching state, which reduces the instantaneous flux
and torque errors.
6.1 Classic PI controller
A classic Proportional plus Integral (PI) controller is suitable enough to adjust the reference
torque value Te* . Nevertheless, its response depends on the gains K p and K i , which are
responsible for the sensitivity of speed error and for the speed error in steady state. During
computer analysis we use a controller in a discrete system in order to simulate a digital
signal processor (DSP) drive system. Its block diagram is shown in Fig.12, where T is the
sampling time of the controller.
Fig. 12. Block diagram of a discrete classic PI speed controller.
The response of the PI speed controller, in a wide range area of motor speed, is very
sensitive to gains Kp and Ki and it needs good tuning for optimal performance. High values
of the PI gains are needed for speeding-up the motor and for rapid load disturbance
rejection. This results to an undesired overshoot of motor speed. A solution is to use a
variable gain PI speed controller [Giuseppe]. However, in the case of using a variable gain
PI speed controller, it is also necessary to know the behaviour of the motor during start up
and during load disturbance rejection in several operation points, in order to determine the
appropriate time functions for PI gains. This method is also time-consuming and depends
on the control system philosophy every time. To overcome this problem, we propose the use
of a fuzzy logic PI controller.
6.2 Fuzzy PI controller
Fuzzy control is basically an adaptive and nonlinear control, which gives robust
performance for a linear or nonlinear plant with parameter variation. The fuzzy PI speed
controller has almost the same operation principles with the classic PI controller. The basic
difference of the two controllers is that the output of the fuzzy PI controller gives the change
in the reference torque value dTe* , which has to be summed or intergraded, to give the Te*
value (Fig.13). The FL controller has two inputs, the speed error E = ωr* − ωr and the change
in the speed error CE , which is related to the derivative dE dt of error. In a discrete system,
assuming dt = T , where T is the constant sampling time of the controller, CE = ΔΕ . Fuzzy
controller computes, for a specific input condition of the variables, the output signal.
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
67
Fig. 13. Basic block diagram of a fuzzy PI speed controller.
All in all, the fuzzy controller is an input/output static nonlinear system which maps the
pair values of E and CE according to fuzzy rules (Table III) and gives the following form:
K1E + K 2CE = dTe*
(11)
Where, K 1 and K 2 are nonlinear gain factors.
The analytical block diagram of the fuzzy PI controller is shown in Fig. 14. The input
variables E and CE are expressed in per unit values. This is achieved by dividing the
variables by specific scale factors. The output will also be expressed in per unit values. The
advantage of fuzzy control in terms of per unit variables is that the same control algorithm
can be applied to all the plants of the same family. Generally, the scale factors can be
constant or programmable. Programmable scale factors control the sensitivity of operation
in different regions of control.
The fuzzy rules of the controller are based in linguistic expressions from the physical
operation of the system. As mentioned before, the output of the controller is the change in
the reference torque value dTe* . The fuzzy sets of linguistic expressions of the variables and
the membership functions (MFs) of these variables are shown in Fig.15 (a),(b). As mentioned
before, the output of the controller is the change in the reference torque value dTe*. The MFs
of the output variable in per unit values are shown in Fig.15(c). The definition of the MFs
depends on the system behavior. All the MFs are asymmetrical because near the origin
(steady state), the signals require more precision.
The next step in the analysis of fuzzy speed controller is the definition of fuzzy rules. The
fuzzy rules for the speed controller are shown in Table III. We can see that the top row and
the left column of the matrix indicate the fuzzy sets of the variables e and ce , respectively,
and the MFs of the output variable dTe* ( pu) are shown in the body of the matrix. There may
be 7 x 7 = 49 possible rules in the matrix, where a typical rule reads as:
IF e( pu) = PS , AND ce( pu) = NM , THEN ΔTe* ( pu) = NVS
Fig. 14. An analytical discrete block diagram of the fuzzy PI controller.
(12)
68
Torque Control
(a)
(b)
(c)
Fig. 15. Membership functions of the input variables (a) speed error e(pu) (b) change in
speed error ce(pu) and of the output variable (c) change in reference torque value ΔΤe* of
the fuzzy PI speed controller.
e(pu)
ce (pu)
NB
NM
NS
Z
PS
PM
PB
NB
NM
NS
Z
PS
PM
PB
NB
NB
NB
NM
NS
NVS
Z
NB
NB
NM
NS
NVS
Z
PVS
NB
NM
NS
NVS
Z
PVS
PS
NM
NS
NVS
Z
PVS
PS
NS
NVS
Z
PVS
PS
PM
PB
NVS
Z
PVS
PS
PM
PB
PB
Z
PVS
PS
PM
PB
PB
PB
where, PB = Positive Big, PM = Positive Medium, PS = Positive Small, PVS = Positive Very Small, Z =
Zero, NVS = Negative Very Small, NS = Negative Small, NM = Negative Medium, NB = Negative Big.
Table III. Fuzzy Rules
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
69
The implication method that we used in simulations is the Mamdani type. There are many
types of fuzzy logic controllers, but now the classical structure of Mamdani type is used.
Fig. 16. DTC block diagram with fuzzy logic controller.
The rule matrix and MF description of the variables are based on the knowledge of the
system. Summing up, the setting of the fuzzy controller depends on the system
requirements for optimal performance. When the fuzzy speed controller is well tuned its
performance is excellent in a wide range of motor speed. Fig 16 shows the block diagram of
Direct Torque Control Induction Motor Drive using a Fuzzy Speed Controller.
6.3 Simulation results
In this paragraph, the simulation results of a system using the classic PI speed controller
(Fig.17.I) and the fuzzy PI speed controller (Fig.17.II) are presented. For the needs of
simulation, we used an 160kW, 400V, 50Hz, 1487rpm, Rs = 0.01379Ω induction motor
which is fed by a VS inverter using the DTC method. In more detail the parameters of motor
are shown in Table IV,
P=4(2 pair of poles), f=50 Hz
230V/400V
P=160kW, N r = 1487 rpm
J=0.02 Kgm
2
Rs=0,0137 Ω
Rr = 0.00728Ω
Ls=0,007705 H
Ls=0,007705 H
Lm=0,00769 H
F=0,05658 Nms
Where J is the machine's inertia, and F is the friction factor.
Table IV. Induction Machine Parameters
To simulate the drive we used Matllab/Simulink software. The DTC sampling time was
30μs and the speed controller sampling time was 3ms. The reference stator flux magnitude
was constant at 1.02 Wb. Fig.17(I) shows the dynamic performance of the DTC-SVM drive
using a classic PI speed controller. The results of Fig.17 show that the startup of the motor,
until it reaches the command speed value 600 rpm, is made with 400 Nm initial load torque.
When the motor runs at 600 rpm/400Nm steady state operation, a step speed command of
70
Torque Control
1200 rpm is given to the drive and the motor reaches again another operation point
(1200rpm/400Nm). Finally, the controllers are tested to step load torque disturbance. It is
easy, therefore, to come to the conclusion that fuzzy speed controller has a remarkably
better response than the classic PI speed controller.
The system was also investigated during the starting period and its control under different
commutative periods. In this fig. 17 it is shown that the torque of the motor has lower ripple
when the speed estimation is being carried out using a fuzzy PI controller.
I. Conventional PI controller
II. Fuzzy Logic controller
1500
1000
(a)
Torque (Nm)
Torque (Nm)
1500
1000
500
(a)
Te
Te*
TL
0
0
0.5
1
1.5
2
2.5
3
500
Te
Te*
TL
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
1500
1000
(b)
Motor Speed (rpm)
Motor Speed (rpm)
1500
1000
(b)
500
ωr
ωr*
0
0
0.5
1
1.5
2
2.5
3
500
ωr
ωr*
0
600
400
Current i a (A)
Current i a (A)
600
200
(c)
0
200
(c)
-200
-400
-600
400
0
0.5
1
1.5
2
2.5
3
0
-200
-400
-600
1.2
1.2
1
(d)
0.8
(d)
0.6
0.4
0
0.5
1
1.5
Time (s)
(a)
2
2.5
0.6
0.4
ψs
ψs*
0.2
0
Flux (Wb)
Flux (Wb)
1
0.8
ψs
ψs*
0.2
3
0
3
Time (s)
(b)
Fig. 17. Motor control response with steps of speed command and load torque. (a)
Electromagnetic torque Te , speed controller output Te* , load torque TL, (b) actual motor
speed ωr , reference speed ωr* ,(c) stator current isa in phase a (d) stator flux magnitude Ψ s ,
and reference value Ψ s* .
Fig. 18 shows, in more detail, the comparison of the motor speed response using the two
different speed controllers, during steps of speed command ωr* and load torque. To
investigate the system for the classic PI controller more than one pairs of values Kp and Ki
have been used. The two controllers were tested in a wide range of engine speed. Extending,
namely, from a very low speed to a very high speed of the motor. It was observed, that the
fuzzy PI controller has better performance than the classic PI controller.
In fig. 19 we observe that the acceleration of the motor using the classic PI speed controller is
almost the same, independently of command step, and generally a linearity is observed,
which depends only on the load on the axis of motor. In other words we have the maximum
acceleration of the motor under these conditions. This means that when we have a small
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
71
1400
1200
motor speed (rpm)
1000
800
Classic PI
Fuzzy PI
600
ωr*
400
200
0
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 18. Motor speed control response with steps of speed command and load torque.
1.1
Classic PI
1.05
1
Rotor Speed (pu)
0.95
Fuzzy PI
0.9
0.85
0.8
0.75
0.7
0.65
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Time (sec)
Fig. 19. Dynamic behaviour of classic PI and Fuzzy PI controller during motor startup. Load
in the shaft of the motor equal with 50% nominal and various step changes of speed.
72
Torque Control
Fuzzy PI
2
2
1.5
1.5
Torque (pu)
Torque (pu)
Classic PI
1
0.5
0
0.3
0.35
0.4
0.45
0.5
0.55
0.5
0.35
0.4
1
0.9
0.8
0.7
0.35
0.4
0.45 0.5
Time (sec)
0.55
0.6
0.8
0.7
0.35
0.4
2
2
1.5
1
0.5
0.35
0.4
0.45
0.45 0.5
Time (sec)
0.55
0.6
(a2)
Torque (pu)
Torque (pu)
0.6
0.9
0.3
0.5
0.55
Te
Te*
TL
1
0.5
0
0.3
0.6
0.35
0.4
0.45
0.5
0.55
0.6
1.1
Rotor Speed (pu)
1.1
Rotor Speed (pu)
0.55
1
1.5
1
0.9
0.8
0.7
0.3
0.5
ωr*
ωr
(a1)
0
0.3
0.45
1.1
Rotor Speed (pu)
Rotor Speed (pu)
1.1
0.3
1
0
0.3
0.6
Te
Te*
TL
0.35
0.4
0.45 0.5
Time (sec)
(b1)
0.55
0.6
ωr*
ωr
1
0.9
0.8
0.7
0.3
0.35
0.4
0.45 0.5
Time (sec)
0.55
0.6
(b2)
Fig. 20. Simulation results of the speed controller response in various speed step commands.
(1) Classic PI controller, (2) Fuzzy PI controller. (a) 30%, (b) 20%
load in the shaft of the motor and the step is small, then an overshoot in the speed of the
motor is observed. On the contrary, with the fuzzy PI of controller, we observe an
acceleration that depends on the step of command and the load on the shaft. In fig. 20 an
analytical comparison of the dynamic performance of the control system is presented. The
system behavior can be studied when the motor speed increases, while the load torque in
the motor shaft remains constant at 50% of the nominal load. In more detail, the dynamic
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
73
performance of the two speed controllers, classic PI and fuzzy PI, is presented during
increase of the motor speed by 30%, 20% and 10% step commands of the nominal speed
respectively. In this figure, the improvement in motor acceleration and the change in motor
torque using the fuzzy PI controller can be seen. Classic PI controller shows an undesirable
overshoot of the actual speed. On the other hand, fuzzy PI controller has a smoother
response. The output of each controller is the value of the reference electromagnetic torque
Te* . The change in motor speed is the result of applying the produced reference torque to
the DTC scheme.
7. Direct torque control for three level inverters
7.1 Control strategy of DTC three-level inverter
The applications of inverter three or multiple level inverters have the advantage of reducing
the voltage at the ends of semiconductor that mean the inverters can supply engines with
higher voltage at the terminals of the stator. Also, the three level inverters show a bigger
number of switching states. A three level inverter shows 33=27 switching states. This means
an improvement in the higher harmonics in the output voltage of the inverter and hence
fewer casualties on the side of the load and less variation of electromagnetic torque. In
direct torque control for a two-level inverter there is no difference between large and small
errors of torque and flux. The switching states selected by the dynamics of drive system
with the corresponding change of desired torque and flux reference is the same as those
chosen during the operation in steady state. For the three-level voltage inverter is a
quantification of the input variables. In this case, increasing the torque on the control points
of the hysteresis comparators in five (Figure 21) and the three magnetic flux (Figure 22).
Also divided the cycle recorded by electromagnetic flow of the stator in a rotating, in 12
areas of 30º as shown in Figure 23. This combined with the increased number of operational
situations, for three-level inverter is 27 and is expressed in 19 different voltage vectors can
be achieved better results. Figure 24a shows the 19 voltage vectors for the three level voltage
source inverter of figure 25, and the vector of magnetic flux of the stator Ψs. It should be
noted that in Figure 24a vectors V1, V2, V3, V4, V5, V6 shown each for two different
operating conditions and the zero vector V0 for three different situations. The angle the
vector i in relation to the axis a is less than 30º. The possible changes in magnetic flux stator
which can be achieved using the voltage vectors of operating conditions shown in 24b.
From Figures 24.a and 24b seems to change the value of stator flux Ψs in a new value
should be selected the following voltage vectors. If an increase in the flow can be achieved
by applying one of the voltage vectors V9, V2, V8, V1, V7, because in this case, the new
vector of stator flux will be correspondingly Ψs+ΔΨ9, Ψs+ΔΨ2, Ψs+ΔΨ8, Ψs+ΔΨ1,
Ψs+ΔΨ7. By the same token if we can achieve a reduction of magnetic flux should
implement one of the voltage vectors V14, V5, V15, since in this case the new vector of
stator flux is είναι Ψs+ΔΨ14, Ψs+Ψ5, Ψs+ΔΨ15, which is less than the original Ψs. Also
for the electromagnetic torque, taking into account the equation 6, if is necessary very sharp
increase in torque, then we can apply one from the voltage vectors V11, V3, V12 because it
will grow along with the flow and the angle between the vectors δ of stator flux and the
rotor. If a reduction of the torque is needed we can apply one from the voltage vectors V6,
V17, V18. By the same token if is required large increase in flow and a slight increase in
torque can do a combination of the above and apply the vector V8 or if stator magnetic flux
is constant and requires a small reduction of the torque is needed can be chosen one from
74
Fig. 21. Hysteresis comparator 5 level for the electromagnetic flux
Fig. 22. Hysteresis comparator 3 level for the magnetic flux
Fig. 23. Sectors of Statorsmagnetic flux.
Torque Control
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
(a)
75
(b)
Fig. 24. a) voltage vectors of 3 level voltage b) changes of the stator’s flux with the vector of
each switching state.
Fig. 25. Three- level voltage source inverter
zero voltage vectors V0. Of course the number of vectors that can bring the desired change
in magnetic flux in stator and electromagnetic torque varies to the angle the vector of
magnetic flux on the axis A. As is natural in such cases there are other suitable candidate
voltage vectors. The correct choice of the vectors, depending on the desired change in the
flow and torque that we want to do, depending on the sector in which the vector of the flow,
76
Torque Control
it is the biggest challenge to build such a table in direct torque control for drive systems
powered by three-level voltage inverters. So the inverter three-level table is not widely
accepted for pulsing as in the case of two-level inverters.
Based on the above logic while taking into account the intersection of Figure 3 in which may
be in the vector of the stator magnetic flux, it became the table I.
Flux(ψS) Torque(Te)
-1
0
1
S1
S2
S3
S4
S5
S6
S7
S8
S9
-2
V0
V0
V0
V0
V0
V0
V0
V0
V0
S10 S11 S12
V0
V0
V0
-1
V3
V4
V4
V5
V5
V6
V6
V1
V1
V2
V2
V3
0
V3
V4
V4
V5
V5
V6
V6
V1
V1
V2
V2
V3
1
V11 V12 V13 V14 V15 V16 V17 V18
V7
V8
V9
V10
2
V11 V12 V13 V14 V15 V16 V17 V18
V7
V8
V9
V10
-2
V0
V0
V0
V0
V0
V0
V0
V0
V0
V0
V0
V0
-1
V2
V3
V3
V4
V4
V5
V5
V6
V6
V1
V1
V2
0
V2
V3
V3
V4
V4
V5
V5
V6
V6
V1
V1
V2
1
V10 V11 V12 V13 V14 V15 V16 V17 V18
V7
V8
V9
2
V10 V11 V12 V13 V14 V15 V16 V17 V18
V7
V8
V9
-2
V0
V0
V0
V0
V0
V0
V0
V0
V0
V0
V0
V0
-1
V2
V3
V3
V4
V4
V5
V5
V6
V6
V1
V1
V2
0
V2
V3
V3
V4
V4
V5
V5
V6
V6
V1
V1
V2
1
V9
V11 V11 V13 V13 V15 V15 V17 V17
V7
V7
V9
2
V9
V11 V11 V13 V13 V15 V15 V17 V17
V7
V7
V9
Table Ι
7.2 Simulation of the system in the computer
The drive system considered consists of three-phase asynchronous motor, three phase three
level voltage inverter and control circuit with hysteresis comparators electromagnetic torque
and flux of Figures 21 and 22 respectively. The system design was done by computer
simulation with Matlab / Simuling. Figure 26 shows the general block diagram of the
simulation.
By simulating the drive system on the computer can pick up traces of electromechanical
sizes in both permanent and transition state in the system. From the curves can be drawn for
the behavior of both the load response and the response speed. Details of the induction
motor and inverter with three levels that will make computer simulations are shown in
Tables II and III respectively.
7.3 Simulation resuls
In this text we will present the waveforms of electromechanical changes in the size of the
load. To investigate the behavior of the electric drive system in response to load change
incrementally load of 25 Nm to 30Nm, then by 30Nm to 25 Nm, maintaining the engine
speed steady at 1000 rpm. Figure 27 shown the electromagnetic torque and Figure 8, the
engine speed according to the time when the transition state in which they affect the load.
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
Fig. 26. Block diagram DTC Three-level Inverter in the Simulink with speed estimator.
Nominal power
P = 4000W
Stator phase voltage
V = 460 V
Ohmic resistance of stator
Rs = 1.405 Ω
Ohmic resistance of rotor
Rr = 1.395 Ω
Main magnetic induction
Lm = 172.2x10-3 H
Stator leakage inductance
Lls = 5.84x10-3 H
Motor leakage inductance
Llr = 5.84x10-3 H
Leakage torque
J = 0.0131 Kg.m2
Coefficient of friction
F = 0.002985 Nms
Number of poles
P =4
(two pairs of poles)
Table ΙΙ. Nominal details of induction motor
Semiconductor
IGBT with antiparallel diodes
Ohmic resistance Snubber
Rs = 1000 Ω
Capacitance Snubber
Cs = infinite
Internal resistance semiconductor
Ron = 0.001 Ω
IGBT voltage crossing
Vf = 0.8 V
Diode voltage crossing
Vf = 0.8 V
Table ΙΙΙ. Nominal details of inverter
77
78
Fig. 27. Electromagnetic flux, reference flux and load flux versus time
Fig. 28. Speed reference and actual speed versus time
Torque Control
Direct Torque Control using Space Vector Modulation
and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
79
Fig. 29. Electromagnetic stator flow versus time.
By changing the load observed a slight, temporary change of speed. Figure 9 shows the
change of the stator flux versus time and Figure 30 is the change of magnetic flux in the
stator three-axis system that is α,β system versus time. Figure 31 shows the change of the
vector current in the stator system. In this figure shows the change of the modulus of vector
power to change the load. When the torque load is reduced and the measure of the vector
current and increase the vector of power when the load increases.
Fig. 30. Electromagnetic flow in the stator ιν α,β system is a function of time
80
Torque Control
Fig. 31. Current in the stator in α,β reference system
8. Conclusion
This paper has presented a modified Direct Torque Control method for PWM-Inverter fed
asynchronous motor drive using constant switching frequency. Constant-switchingfrequency is achieved by using space vector modulation and finally, an SVM based DTC
system, compared to the classic DTC scheme for torque control. DTC-SVM schemes improve
considerably the drive performance in terms of reducing torque and flux pulsations, reliable
startup and low-speed operation, well-defined harmonic spectrum, and radiated noise.
Therefore, DTC-SVM is an excellent solution for general-purpose asynchronous motor
drives. On the contrary, the short sampling time required by the classic DTC schemes makes
them suited to very fast torque- and flux-controlled drives because of the simplicity of the
control algorithm. When a speed control mode instead of torque control is needed, a speed
controller is necessary for producing the reference electromagnetic torque value. For this
purpose a fuzzy logic based speed controller is used. Fuzzy PI speed controller has a more
robust response, compared to the classic PI controller, in a wide range area of motor speed.
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and Dynamic Performance of the Drive, via a Fuzzy Logic Controller for Speed Regulation
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Induction Motor Drive, in 35th Annual IEEE Power Electronics Specialists
Conference (PESC’04), June 20-25, , Aachen, Germany, pp. 3481-3485.
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Induction Motor Drive, IEEE Trans. Ind. Electron., Vol. 47, No. 4, pp. 863-870, Aug..
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Torque-Ripple Reduction, IEEE Trans. Ind. Electron., vol.50, pp.487–492,Jun..
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current limitation, IEEE Industrial Electronics Society, IECON 2005. 32nd Annual
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Electronics, Vol. 48, No. 6, December 2001, pp. 1148 – 1157.
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December, pp. 1660-1668.
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Modulation and dynamic performance of the drive via a Fuzzy logic controller for
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Hybrid Space Vector Pulsewidth Modulation to Reduce Ripples and Switching
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Source Inverter, Asian Power Electronics Journal, Vol. 1, No. 1, Aug 2007.
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Symposium on Power Electronics, Electrical Drives, Automation and Motion
4
Induction Motor Vector and Direct Torque
Control Improvement during the
Flux Weakening Phase
Kasmieh Tarek
Higher Institute for Applied Sciences and Technology
Syria
1. Introduction
Some industrial applications, such as spindle, traction, and electric vehicles, need a high
speed for the fixed rating power, Fig. 1. To achieve this goal a suitable control method based
on the flux weakening is usually applied. This gives an economic solution for the power
converter and the motor (Grotstollen, H. & Wiesing, J, 1995).
Fig. 1. Motor torque according to speed range
For the induction motor, its magnetic state changes during the flux weakening phase. It goes
from the saturation to the linear region, since the rating magnetic point is at the knee of the
magnetizing curve of the iron. Therefore, the change of the magnetic state of the motor
should be taken into account in the control law.
Field Oriented Control (FOC) and Direct Torque Control (DTC) are based on a linear twophase model of the induction motor (Vas, P. & Alakula, M. 1990). This model considers that
the magnetic state of the motor is fixed, and all control parameters are calculated according
to this state. When decreasing the flux level, the motor inductances increase. The change of
real inductances values of the motor influences the desired current and speed dynamics.
In this chapter, the linear two-phase model of the induction motor is re-examined, and a
new non linear two-phase model of the induction motor is developed. This model takes into
account the variation of the saturation level (Kasmieh, T. & Lefevre, Y, 1998). The calculation
84
Torque Control
of this model needs the motor inductances values at each calculation step. The inductances
curves, as functions of the magnetic state of the motor, can be obtained using a finite
elements calculation program.
Unlike many models developed for the induction motor that take into account the variation
of the saturation level (Vas, P, 1981), the model presented in this chapter does not introduce
the inductances time derivatives. This leads to an easy computation algorithm, using
iteration method at each calculation step. The derived model is validated by comparing its
dynamic behavior to the dynamic behavior of a finite elements model.
Based on the new model, a complete sensitivity study of the classical FOC and DTC techniques
is presented. The FOC is highly dependent on the motor parameters. During the flux
weakening phase, the inductances values increase. This influences the dynamic behavior fixed
by the controllers, which is calculated for the rating inductances values. To overcome this
problem an adaptive FOC is introduced. At each sampling period, the magnetic state of the
motor is calculated by iteration, and then the controllers are tuned to this new magnetic state.
Concerning the DTC, this control law is less sensitive to the variation of the saturation level
(Kasmieh, T, 2008). The DTC is based on applying the good voltage stator vector in order to
achieve the desired stator fluxes and torque variations. The main problem of the DTC lies in is
the accuracy of the stator fluxes calculation at each sampling period. Usually, this calculation is
easily done by using the stator electric equation. The performance of this estimator is highly
dependent on the value of the stator winding resistor, which varies with the motor
temperature. A more complicated flux estimator can be derived from the rotor electric
equation (Kasmieh, T, 2008). This estimator is less sensitive to the variation of the rotor
resistor, but more sensitive to the variation of the saturation level. To overcome this problem,
an adaptive flux estimator is presented in this chapter. The estimator parameters are tuned
according to the saturation level of the motor. This new estimation method increases the
computation time of the DTC, but it remains smaller than the computation time of the FOC.
2. Magnetic state study of the induction motor using finite elements
calculation program
The goal is to determine the main variable that influences the magnetic state of the induction
motor, and to establish new flux-current relationships in the two-phase reference that take
into account the influence of the magnetic saturation level variation. The study is done for a
two pole pairs (p=2) 45(KW) induction motor using a finite elements calculation program.
Fig. 2 shows the cross section of the studied 45(KW) induction motor. The motor has two
cages of 40 bars each.
Fig. 2. Cross section of the studied 45(KW) induction motor
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
85
The induction motor is modeled as a magnetic circuit of 3 stator phases and m rotor phases,
(m=10 in the case of the 45(KW) induction motor), Fig. 3.
Fig. 3 Axes of multi-phase model of an induction motor
The flux-current relationships can be written as follows:
⎡ [ L ss ]3,3
[M sr (θ)]3,m ⎤
⎥ .[ I ]
[φ]m + 3 = ⎢
[L rr ]m ,m ⎥⎦ m + 3
⎢⎣[ M rs (θ)]m ,3
(1)
The elements of the vector [ φ]m + 3 are the stator and rotor fluxes, the elements of the vector
[I ]m + 3 are the stator and rotor currents, [L ss ]3,3 is the stator inductance matrix, [L rr ]m ,m is the
rotor inductance matrix, [ M sr (θ)]3,m is the stator to rotor mutual inductance matrix,
[M rs (θ)]m ,3 is the rotor to stator mutual inductance matrix and [L rr ]m ,m is the rotor
inductance matrix.
For the non-linear case where the magnetic saturation effect is taken into account, the stator
and rotor fluxes are functions of the motor angle θ, the stator and the rotor currents.
[φ]m + 3 = [φ]m + 3 (θ, [I ]m + 3 )
(2)
In this case, it is difficult to find the fluxes analytically, but they can be calculated using a
finite elements calculation program, in which the magnetic characteristics of the motor
material can be introduced, Fig. 4. The calculation of the fluxes as functions of the motor
Fig. 4. Iron magnetic characteristics of the 45(KW) motor
86
Torque Control
angle θ and the currents is possible using the concept of equivalent saturated inductances
deduced from the saturation curve of the motor material.
A finite elements calculation program is used to determine the main variable that influences
the magnetic state of the induction motor in a two-phase reference related to the rotor, Fig. 5.
Fig. 5. Two-phase reference related to the rotor
The currents are applied in the two-phase reference, and then the currents [ I ]m + 3 are
calculated using the inversed Park transformation. These currents are injected in the motor
finite elements model, and the program calculates the fluxes [ φ]m + 3 . The fluxes in the twophase reference are finally calculated by applying the Park transformation. The calculation
procedure is shown in Fig. 6.
Finite
⎡i sd ⎤
⎡φsd ⎤
T2 ,3 .P(pθ)
⎢ i ⎥ P( −pθ).T3,2
⎢φ ⎥
⎡ [ φs (θ)]3 ⎤
Elements
⎢ sq ⎥ − − − − − − − → ⎡ [ I s (θ)]3 ⎤
⎢ sq ⎥
−
−
−
−
−
−
−
−
−
−
−
→
⎢
⎥
⎢
⎥
⎢ i rd ⎥
⎢ φrd ⎥
Pr ogram
⎢⎣[ I r (θ)]m ⎥⎦
⎢⎣[ φr (θ)]m ⎥⎦
Tm ,2
T2 ,m
⎢ ⎥
⎢ ⎥
i
−
−
−
−
−
−
−
−
−
−
−
→
⎢⎣ rq ⎥⎦
⎣⎢ φrq ⎦⎥
Fig. 6. Calculation procedure of the two-phase fluxes
Where
T3,2 =
⎡
⎢ 1
⎢
2⎢ 1
−
3⎢ 2
⎢
⎢ 1
⎢⎣ − 2
⎤
0 ⎥
⎥
3 ⎥
,
2 ⎥
⎥
3⎥
−
2 ⎥⎦
Tm ,2
1
0
⎡
⎤
⎢
⎥
π
π
2
2
⎢
⎥
cos( )
sin( )
⎢
⎥
m
m
2 ⎢
⎥
.
.
=
⎥
m⎢
.
.
⎢
⎥
⎢
⎥
π
π
2
2
⎢ cos((m − 1) ) sin((m − 1) )⎥
m
m ⎦⎥
⎣⎢
are
the
⎡ cos(ψ ) sin(ψ ) ⎤
Concordia matrices, and P( ψ ) = ⎢
⎥ is the rotation matrix.
⎣ − sin( ψ ) cos( ψ )⎦
The simulations show that the magnetic state of the induction motor depends on the
modulus of the magnetizing current vector I m = (i sd + i rd )2 + (i sq + i rq )2 . The magnetizing
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
87
current vector can be written using complex representation as follows:
I m = (i sd + i rd ) + j.(i sq + i rq ) . Fig. 7 shows the magnetic state of the motor for different twophase currents that give the same value of I m .
Fig. 7. Magnetic state of the motor for different two-phase currents of the same I m
88
Torque Control
The flux-current relationship can then be written, in the two-phase reference, as follows:
⎡L s ( I m )
0
M( I m )
0
⎡φsd ⎤ ⎢
⎢φ ⎥ ⎢ 0
Ls ( Im )
0
M( I m
⎢ sq ⎥ = ⎢
⎢ φrd ⎥ ⎢ M( I )
0
Lr ( Im )
0
m
⎢ ⎥ ⎢
⎢⎣ φrq ⎥⎦ ⎢
M( I m )
0
Lr ( Im
⎢⎣ 0
⎤
⎥ ⎡i sd ⎤
)⎥ ⎢ i sq ⎥
⎥ . ⎢ ⎥ = φ = ⎡M( I )⎤ . I
m
⎥ ⎢i ⎥ [ ] ⎣
⎦[ ]
⎥ ⎢ rd ⎥
⎥ ⎢ i rq ⎥
)⎥ ⎣ ⎦
⎦
(3)
It is possible to obtain the cyclic inductances curves as functions of I m , by injecting one
two-phase current. Fig. 8 shows the cyclic inductances curves as function of this injected
current.
Fig. 8. Cyclic inductances curves
From these curves, inductances lookup tables are established. The values of the injected
current can be associated to I m (Kasmieh, T. & Lefevre, Y. 1998).
It is important to mention that the saturation harmonics disappear from the two-phase
fluxes. This issue can be demonstrated taking into accounts the saturation third harmonics
of the fluxes:
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
φsa
φsb
φsc
89
=
a.cos(pθ) + b.cos(3pθ)
2π
6π
= a.cos(pθ − ) + b.cos(3pθ − )
3
3
4π
12 π
= a.cos(pθ − ) + b.cos(3pθ −
)
3
3
Two-phase stator fluxes can be obtained by applying Park transformation of an angle pθ:
2π
4π ⎤
⎡
cos(pθ) cos(pθ − ) cos(pθ − ) ⎥ ⎡ φsa ⎤
⎡φsd ⎤
2⎢
3
3 . ⎢φ ⎥
⎢
⎥ ⎢ sb ⎥
⎢φ ⎥ =
2
4
π
3
⎢ − sin(pθ) − sin(pθ − ) − sin(pθ − π )⎥ ⎢ ⎥
⎣ sq ⎦
φ
3
3 ⎦⎥ ⎣ sc ⎦
⎣⎢
4π
8π
⎡ 3a a + b
⎤
⎢ 2 + 2 (cos(2pθ) + cos(2pθ − 3 ) + cos(2pθ − 3 )) + ⎥
⎢
⎥
⎢ b (cos(4pθ) + cos(4pθ − 8π ) + cos(4pθ − 16 π ))
⎥ ⎡ 3⎤
⎡φsd ⎤
⎥ ⎢
2 ⎢2
3
3
⎥
⎢φ ⎥ =
⎢
⎥=⎢ 2⎥
b
−
a
4
π
8
π
3
⎣ sq ⎦
⎢
⎥
(sin(2pθ) + sin(2pθ − ) + sin(2pθ − )) +
⎢ 0 ⎦⎥
⎢
⎥ ⎣
2
3
3
⎢
⎥
b
8π
16 π
⎢ − (sin(4pθ) + sin(4pθ − ) + sin(4pθ −
⎥
))
2
3
3
⎣
⎦
The demonstration can be extended to the general expressions of the saturated fluxes.
The next paragraph presents a new dynamic model of the induction motor that takes into
account the variation of the saturation level. The resolution of the non-linear equations of
the model is done by iteration.
3. Establishment of a saturated two-phase model of the induction motor
Since the magnetic state of the induction motor depends on the modulus of the magnetizing
current vector, thus, the new equations that describe the dynamic behavior of a variable
saturation level motor in the two-phase reference are:
The electric equations:
dΦ sd dψ
.Φ sq
−
dt
dt
dΦ sq dψ
dΦ s
dψ
vsq = R s .i sq +
.Φ sd
Vs = R s .I s +
.Φ s
+ j.
+
dt
dt
dt
dt
or
dΦ rd d(ψ − pθ)
dΦ r
d(ψ − pθ)
0 = R r .i rd +
.Φ rq
−
+ j.
.Φ r
0 = R r .I r +
dt
dt
dt
dt
dΦ rq d(ψ − pθ)
0 = R r .i rq +
.Φ rd
+
dt
dt
vsd = R s .i sd +
(4)
The flux-current relationships:
[φ] = ⎡⎣M( Im )⎤⎦ .[I ]
or
Φ s = L s .I s + M.I r
Φ r = L r .I r + M.I s
(5)
90
Torque Control
The mechanical equation: j
dΩ
= Tem − Tr
dt
where
Tem = p
M
M
(Φ r ∧ I s ) = p ( Φ rd .i sd − Φ rq .i sd )
Lr
Lr
(6)
is the electromagnetic torque and Tr is the resistive torque. Rs and Rr are the stator and the
rotor windings resistances. Vsd and Vsq are the stator two-phase voltages and J is the rotor
inertia.
The resistive torque is the sum of the viscosity resistive torque, and a resistive torque
Ts : Tr = f.Ω + Ts , where f is the viscosity factor. Usually, the variations of Ts are considered
smaller than the variation of the velocity when controlling the motor. Note that the complex
quantity X = xd + j.xq is used to represents the vectors in the D, Q reference.
The numeric resolution of the new saturated two-phase model equations is done avoiding
the complicated development of the equations as currents deferential equations. The
following differential equations can simply be written.
d [Φ ]
dt
⎡ Rs
⎢ σ.L
s
⎢
⎢
0
⎢
A( I m ) = ⎢
⎢ M.R r
⎢−
⎢ σ.L s .L r
⎢
0
⎢
⎣
+ A( I m ).[ Φ ] = [ v ]
0
Rs
σ.L s
0
−
M.R r
σ.L s .L r
−
M.R s
σ.L s .L r
0
Rr
σ.L r
d(p.θ)
−
dt
(7)
⎤
⎥
⎥
M.R s ⎥
−
⎥
σ.L s .L r ⎥
(I )
d(p.θ) ⎥ m
⎥
dt
⎥
⎥
Rr
⎥
σ.L r ⎦
0
(8)
The matrix A is written for a two-phase reference related to the stator Ψ=0.
σ = 1 − M 2 /(L s .L r ) is the dispersion factor which is never equal to zero because the leakage
inductances.
The new non-linear model of the induction motor is described by equations (3), (7) and the
expression of the electromagnetic torque. This model is called the saturated two-phase
model.
The numeric resolution procedure of these equations starts from an initial state. At each
calculation step equation (7) is solved using for example Runge-Kutta 4 (RK4) method. This
will give a new flux vector that describes a new magnetic state of the motor. Then, the
corresponding current vector must be determined by resolving equation (3). In fact,
equation (3) is a non-linear equation. The matrix M depends on the modulus magnetizing
current vector. The resolution of this equation can be done by a non-linear iterative
resolution method, like substitution method.
Equation (7) can be written as follows:
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
d [Φ ]
dt
91
= [F([ Φ ]t , [ I ]t )] = [ F ]t
(9)
where [ F ]t is a function of the two-phase fluxes and currents.
The RK4 method gives an approximated numerical solution of equation (9). The fluxes at the
instant t+Δt are calculated using equation (10).
4
[Φ ]t + Δt = [Φ ]t + ∑ bi .Δt [F]i
(10)
i =1
where
Δt
Δt
[F]1 = [Φ ]t + [F ]t
2
2
Δt
Δt
[Φ ]2 = [Φ ]1 + [F ]2 = [Φ ]1 + ⎡⎣F([Φ ]1 , [I]1 )⎤⎦
2
2
Δt
Δt
[Φ ]3 = [Φ ]2 + [F]3 = [Φ ]2 + ⎡⎣F([Φ ]2 , [I ]2 )⎤⎦
2
2
Δt
Δt
[Φ ]4 = [Φ ]3 + [F]4 = [Φ ]3 + ⎡⎣F([Φ ]3 ,[I]3 )⎤⎦
2
2
[Φ ]1 = [Φ ]t +
1
1
1
1
, b2 = , b3 = , b 4 = .
6
3
6
6
To be able to calculate [ Φ ]i + 1 , the currents [ I ]i must be calculated by solving the non-linear
and b1 =
equation
[φ]i = ⎡⎣M( Im i )⎤⎦ .[I ]i .
Finally, Fig. 9 shows the calculation procedure of the
saturated two-phase model of the induction motor.
The resolution of the non-linear equations of the flux-current relationships can be done
using a non-linear iterative resolution method. The substitution method searches the
intersection point between
[I ]1 = [I]t .
( ⎡⎣M( I
m
)
)⎤ .[ I ] (t) and [ φ] starting from the first iteration
t +Δt
⎦
The next iteration is calculated from the previous iteration: [ I ]i + 1 = [ I ]i + Δ [ I ] ,
−1
(
)
where Δ [ I ] = ⎡M( I m )⎤ . [ Φ ]t + Δt − M( I m ).[ I ]i . In fact the Inductance matrix can be
i ⎦
i
⎣
inversed, since the leakage inductances cannot be zero:
1
M
⎡
⎤
−
0
0
⎢ σ.L
⎥
σ
.L
.L
s
s
r
⎢
⎥
⎢
1
M ⎥
0
0
−
⎢
⎥
−1
σ.L s
σ.L s .L r ⎥
⎡M( I m )⎤ = ⎢
(I )
⎣
⎦
⎢
⎥ m
Mr
1
0
0
−
⎢
⎥
σ.L r
⎢ σ.L s .L r
⎥
⎢
⎥
M
1
0
0
−
⎢
⎥
σ.L s .L r
σ.L r ⎦
⎣
Fig. 10. shows the substitution calculation procedure for vectors dimension equal to one.
92
Torque Control
Fig. 9. Calculation procedure of the saturated two-phase model of the induction motor
Fig. 10. Substitution calculation procedure
The iteration procedure is stopped when achieving a suitable error of the modulus of the
flux vector.
The execution of the calculation procedure of the Fig. 9 gives the results shown in Fig. 11.
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
93
Fig. 11. Dynamic behavior of the saturated two-phase model of the induction motor
The comparison between the saturated two-phase model and the finite elements model is
shown in Fig. 12. It is clear that it gives closer results to the finite elements model results
than the results of the linear model.
Fig. 12. Saturated two-phase model, linear model and finite element model results
comparison
4. Field oriented control law improvement during the flux weakening phase
The vector control law or field-oriented control (FOC) law of an induction motor has
become a powerful and frequently adopted technique world-wide. It is based on the twophase model, Park model. The aim of this control is to give the induction motor a dynamic
behavior like the dynamic behavior of a direct current motor. This can be done by
controlling separately the modulus and the phase angle of the flux (Blaschke, F. 1972).
Using this control technique, the electrical and mechanical dynamic responses of the
induction motor are determined by fixing the coefficients of the current loops controllers,
flux loop controller and the velocity loop controller. Usually, these coefficients are calculated
for the rating values of the cyclic inductances, which correspond to the rating saturation
level. In fact, this level is achieved by applying the rating flux value as a reference value to
the flux loop.
Some industrial applications require the induction motor to operate at a high speed over the
rating speed. The method used to reach this speed is to decrease the reference value of the
flux in order to work at the rating power. This decrease can cause a coupling between the
two-phase axes D and Q, so FOC does not work properly (Kasmieh, T. & Lefevre, Y. 1998).
94
Torque Control
Many published papers have studied the effects of the variation of the saturation level on
FOC law (Vas, P. & Alakula, M. 1990) (Vas, P. 1981), but few attempts have been made to
develop a FOC law that takes into account this variation.
In this paragraph the sensitivity of the classical FOC law to the variation of saturation level
of an induction motor is studied. Then, a new indirect vector control law in accordance to
the rotor flux vector that takes into account this variation is developed. This law is based on
the saturated two-phase model found in the previous sections.
The simulations are done using an electromechanical simulation program called "A_MOS",
Asynchronous Motor Open Simulator, (Kasmieh, T. 2002), Fig. 13.
Fig. 13. The main window of “A-MOS” Software
The resolution algorithm of the non-linear model is implemented in this programmed. The
user can write his own control algorithm.
4.1 Classical FOC law
The strategy of the FOC in accordance with the rotor flux vector is adopted. This strategy
leads to simpler equations than those obtained with the axis D aligned on the stator flux
vector or with the magnetizing flux vector (Vas, P. & Alakula, M. 1990).
The development of the FOC equations in accordance to the rotor flux vector can be done by
t
t
supposing φr = ⎡⎣φrd , φrq ⎤⎦ = [ φr ,0 ] , Fig. 14. The expression of the motor torque is reduced to:
Tem = p
M
.Φ r .i sd
Lr
(11)
Since the rotor flux vector turns at the synchronized speed ωs , the electric equations become:
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
95
dΦ sd
− ωs .Φ sq
dt
dΦ sq
v sq = R s .i sq +
+ ωs .Φ sd
dt
dΦ r
0 = R r .i rd +
dt
dθ
0 = R r .i rq + (ωs − p ).Φ r
dt
v sd = R s .i sd +
(12)
Fig. 14. Two-phase reference in accordance with the rotor flux vector
4.1.1 Stator voltages and stator fluxes equations
The stator voltages of equation (12), and the stator fluxes expressions can be written using
complex representation ( X = xd + j.xq ):
dΦ s
+ j.ωs .Φ s
dt
Φ s = L s .I s + M.I r
Vs = R s .I s +
By adding and subtracting the term
M2
.I s in the stator flux vector expression, the
Lr
magnetizing rotor current vector is introduced I mr :
Φ s = σL s .I s +
M2
L
M2
.(I s + r .I r ) = σL s .I s +
.(I mr ) .
Lr
M
Lr
Since the rotor flux vector is aligned on the magnetizing rotor current vector:
Φ r = Φ r = L r .I r + M.I s = M.I m r , the stator flux vector can be written as a function of the stator
current vector and the rotor flux.
Φ s = σL s .I s +
M
.Φ r
Lr
Substituting (13) in the expression of the stator voltage vector:
(13)
96
Torque Control
Vs = R s .I s + σL s .
dI s M dΦ r
+ .
+ j.ωs .Φ s
dt L r dt
(14)
4.1.2 Rotor voltages and rotor fluxes equations
dθ
The pulsation (ωs − p ) is the rotor pulsation ωr , thus the rotor electric equations become:
dt
dΦ r
dt
0 = R r .i rq + ωr .Φ r
0 = R r .i rd +
(15)
From the rotor fluxes expressions, the rotor currents are expressed as functions of the rotor
flux and the stator currents:
φrd = L r .i rd + M.i sd
φrq = L r .i rq + M.i sq
i rd =
⇒
φr = L r .i rd + M.i sd
0 = L r .i rq + M.i sq
⇒
φr M
− .i sd
Lr Lr
i rq = −
M
.i sq
Lr
(16)
(17)
4.1.3 Transfer functions of the induction motor
In order to establish the FOC strategy, the transfer functions of the motor are developed.
The inputs of the transfer functions are v sd and vsq , and the outputs the variables that
determine the motor torque Φ r and i sd .
Transfer functions on D axis:
It is possible to control the rotor flux via the stator current on the D axis. This can be
demonstrated from the rotor electric equation on the D axis and from equation (16):
dΦ r
R
M
= − r .Φ r + R r . .i sd
dt
Lr
Lr
(18)
Developing equation (14) on the axis D yields to:
vsd = R s .i sd + σL s .
di sd M dΦ r
+ .
− ωs .Φ sq
dt L r dt
By substituting equation (17) in the expression of Φ sq , the following equation is obtained:
φsq = L s .i sq + M.i rq = L s .i sq −
M2
M2
M2
.i sq = (L s −
).i sq = L s .(1 −
).i sq = L s .σ.i sq .
Lr
Lr
L r .L s
The D stator voltage expression becomes:
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
vsd = R s .i sd + σL s .
97
di sd M dΦ r
+ .
− ωs .L s .σ.i sq
dt L r dt
(19)
By replacing (18) in (19), the stator voltage of the D axis can be written as follows:
v sd = R sr .i sd + σL s .
di sd
+ Ed
dt
(20)
2
⎛M⎞
M
where R sr = R s + R r . ⎜ ⎟ , and the electrical force Ed = −R r . 2 .Φ r − ωs .L s .σ.i sq represents
L
L
r
⎝ r⎠
the coupling between the two axes D and Q.
Transfer functions on Q axis:
By developing equation (14) on the axis Q, the stator voltage of the same axis is obtained:
vsq = R s .i sq + σL s .
di sq
+ ωs .Φ sd
dt
From equation (13) the D stator flux is: Φ sd = σL s .i sd +
M
.Φ r . By replacing Φ sd in the
Lr
previous expression, vsq becomes:
vsq = R s .i sq + σL s .
di sq
dt
+ ωs .σL s .i sd + ωs .
M
.Φ r
Lr
(21)
Φ r can be written as a function of the stator current on the Q axis by substituting the
expression of irq, equation (17), in the rotor electric equation on the Q axis:
Φr = Rr .
M
i sq
ωr .L r
(22)
By replacing (22) in (21) :
vsq = R s .i sq + σL s .
vsq = R s .i sq + σL s .
di sq
dt
di sq
dt
2
+ ωs .σL s .i sd +
ωs ⎛ M ⎞
. ⎜ ⎟ .R r .i sq
ωr ⎝ L r ⎠
+ ωs .σL s .i sd +
ω + ωr ⎛ M ⎞
. ⎜ ⎟ .R r .i sq
ωr ⎝ L r ⎠
2
Finally vsq can be written as follows:
vsq = R sr .i sq + σL s .
2
di sq
⎛M⎞
+ ωs .σL s .i sd + ω. ⎜ ⎟ .R r .i sq = R sr .i sq + σL s .
+ Eq
dt
dt
⎝ Lr ⎠
di sq
(23)
The electrical force Eq represents the coupling between the two axes D and Q.
The equations (18), (20) and ( 23) describe the transfer functions of the induction motor if the
D axis is aligned on the rotor flux vector, Fig. 15.
98
Torque Control
Fig. 15. Transfer functions of the induction motor (D axis is aligned on the rotor flux vector)
4.1.4 Establishment of the classical FOC law
It is important to mention that the transfer functions shown on Fig. 15 are valid if the axis D
is rotating with the rotor flux vector. Taking into account this hypothesis the control scheme
of Fig. 16 can be built.
The two axes D and Q are decoupled by estimating the electric forces Ed and Eq:
2
Ede = −R r .
M e
e
e
m
m ⎛M⎞
m
.Φ r − ωes .L s .σ.i m
⎟ .R r .i sq . The index e is for the
sq and E q = ωs .σL s .i sd + ω . ⎜
L
Lr 2
⎝ r⎠
estimated variables, and the index m is for the measured variables.
Φ er is calculated by solving numerically the equation ( 18). The value of Φ er is also used as a
feedback for the rotor flux control closed loop.
M m m
i sq . ω = p.Ω m = p.dθ /dt is the
Φ er .L r
electric speed of the motor that can be measured using a speed sensor, and p is the pole
pairs number.
For the induction motor, L r / R r is ten times bigger than σ.L s / R sr , so it is possible to do
ωes is calculated from equation (18): ωes = ωm + R r .
poles separation by doing an inner closed loop for the current and an outer closed loop for
the rotor flux.
From Fig. 16, it is clear that the D axis closed loops are for controlling the amplitude of the
rotor flux, and the closed loop of the Q axis is for controlling the stator current, thus for
controlling the motor torque, equation (11).
In practice, the three phase currents are measured, and then the two phase currents are
calculated using Park transformation of an angle Ψ. The angle Ψ is estimated by integrating
M
ωes = ωm + R r . e i m
sq . After calculating the control variables v sd and v sq , the three phase
Φ r .L r
control variables vsa , vsb and vsc are found using the inversed Park transformation.
4.2 Sensitivity of the classical FOC law to the variation of the saturation level
the FOC algorithm is implemented in “A_MOS“ program. The controller parameters are
fixed according to rating values of the induction motor cyclic inductances. The simulation
results of fig. 17 show that during the flux weakening phase, the rotor flux does not follow
its reference and the dynamic response of the speed is disturbed. This due to the fact that the
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
99
Fig. 16. FOC law scheme
Fig. 17. Simulation results of the dynamic behavior of the induction motor modeled by the
saturated two-phase model, and controlled by the classical FOC law
cyclic inductances values of the motor become different from the cyclic inductances values
introduced in the controllers.
In the next paragraph, the classical FOC law is developed in order to take into account the
variation of the saturation level. The new control law is called the saturated FOC law.
4.3 New saturated FOC law
To simplify the study, stator and rotor leakage inductances ( L sf and L rf ), are supposed to be
constant. Only the mutual cyclic inductance M is considered to be variable with the
modulus of magnetizing current vector, where L s = M + L sf and L r = M + L rf .
From expression (13), The derivative of the stator flux vector is:
M
d( )
dΦ s
dI s M dΦ r
d(σL s )
dI
M dΦ r
dM L2rf
dM L rf
Lr
= σ.L s .
+ .
+ Is .
+ Φr .
= σ.L s . s + .
+ Is .
.( ) + Φ r .
.
dt
dt L r dt
dt
dt
dt L r dt
dt L2r
dt L2r
100
Torque Control
Finally the expression of the stator flux vector derivative is:
dΦ s
dI
M dΦ r
L dM
= σ.L s . s + .
+ (I s .L rf + Φ r ) 2rf .
dt
dt L r dt
L r dt
(24)
The stator voltage vector is then modified to:
Vs = R s .I s + σ.L s .
dI s M dΦ r
L dM
+ .
+ (I s .L rf + Φ r ) 2rf .
+ j.ωs .Φ s
dt L r dt
L r dt
(25)
As previous, the resistance Rsr can be introduced. The stator voltages on the D and Q axes
are:
vsd = R sr .i sd + σ.L s .
= R sr .i sd + σ.L s .
vsq = R sr .i sq + σ.L s .
di sd
M
L dM
− R r . 2 . Φ r − σ.L s .ωs .i sq + (i sd .L rf + Φ r ) 2rf .
dt
Lr
L r dt
di sd
+ Ed
dt
di sq
dt
+ ω.
(26)
di sq
M
L2 dM
+ σ.L s .ωs .i sd = R sr .i sq + σ.L s .
+ Eq (27)
. Φ r + i sq . 2rf .
Lr
L r dt
dt
where Ed and Eq are electrical forces and equal to:
Ed = R r .
M
L dM
,
. Φ r + σ.L s .ωs .i sq − (i sd .L rf + Φ r ). 2rf .
L2r
L r dt
Eq = −ω.
M
L2 dM
.
. Φ r − σ.L s .ωs .i sd − i sq . 2rf .
Lr
L r dt
The obtained transfer functions are approximately the same as in the linear case. The main
difference is that the parameters of these transfer functions are time variant. Terms containing
dM
L
appear in the expressions of Ed and Eq. Anyhow, this term can be neglected since r is
dt
Rr
bigger than 10
σL s
for induction machines, so the expressions of Ed and Eq become:
Rs
Ed ≈ R r .
M
M
. Φ r + σ.L s .ωs .i sq , Eq ≈ −ω. . Φ r − σ.L s .ωs .i sd .
L2r
Lr
The idea of the saturated FOC is to tune the coefficients of the controllers according to the
value of I m . At each sampling period I m is calculated, and the corresponding cyclic
inductances are found from look up tables to update the controller’s coefficients.
The expression of I m is I m = (i sd + i rd )2 + (i sq + i rq )2 . isd and isq can be measured at each
sampling period. ird can be calculated from the first rotor equation ( 15), and irq from the
equation (17) using a non-linear resolution method as the substitution method.
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
101
Fig. 18 shows the strategy of the new FOC law. The blocks with dashed lines are the blocks
necessary for calculating the modulus of magnetizing current vector. At each sampling
period the controller’s coefficients are updated according to the new values of the cyclic
inductances.
Calculation
of ird & irq
Tables of
cyclic inductances
Calculation
of |Im|
isq
isd
Ψr
(Ls, Lr, M)
Ed
ref
|Φ r|
RΦ
|Φr|
ref
sd
i
+
− isd
ref
ω
+
Rω
−
C em
+
−
(Ls, Lr, M)
d |Φ r| R r
(Misd- |Φr|)
dt = Lr
ref
RI
Eq
ref
Lr isq
pM|Φ r| +
− isq
RI
+
−
Saturated
two-phase
model of
the induction
machine
Fig. 18. Saturated FOC law
Fig. 19 presents simulation results of the dynamic response of the 45KW induction motor
controlled by the new saturated FOC control. This simulation is done for the same inputs of
figure 5. It is clear that the performance of the machine is clearly improved.
Fig. 20. Simulation results with saturated FOC
5. Stator flux estimation improvement during the flux weakening phase for
the Direct Torque Control Law
Thirteen years after developing the FOC law by F. Blaschke in 1971 (Blaschke, F. 1972), I.
Takahashi and M. Depenbrock presented a new technique for the induction motor torque
102
Torque Control
control called Direct Torque Control (DTC), (Noguchi, T. & Takahashi, I. 1984), Depenbrock,
M. & Steimel A. 1990). DTC is based on applying the appropriate voltage space vector in
order to achieve the desired flux and torque variations.
DTC permits to have very fast dynamics without any intermediate current control loops.
The DTC is based on the fact that the variations of the stator flux vector are directly
controlled by the stator voltage vector for high speed:
Vs = R s .I s +
dΦ s dΦ s
≈
dt
dt
(28)
5.1 Direct Torque Control Law for an induction machine with a voltage source inverter
drive
A small variation of the stator flux vector is in fact the product of the stator voltage vector
and the sampling period ΔT :
ΔΦ s = Vs .ΔT
(29)
Usually, the motor is driven by a voltage source inverter. The stator voltage vector for such
an inverted has only 8 positions, Fig. 21. From Fig. 21 If the stator flux vector is in sector i,
then its magnitude is increased when applying Vi , Vi + 1 or Vi − 1 . To decrease Φ s , the vector
Vi + 2 , Vi − 2 or Vi + 3 can be applied.
Fig. 21. Stator Voltage space vector for a voltage source inverter
In order to search what does the stator voltage space vector act on the motor torque, its
expression can be rewritten starting from equation ( 6) and taking into account the fluxcurrent relationships as follows:
Tem = p.Φ s ∧ I s
Tem = p.
M
M
.Φ s ∧ Φ r = p.
. Φ s . Φ r .sin θsr
Ls.Lr − M2
Ls.Lr − M2
where θsr is the angle difference between Φ s and Φ r .
(30)
(31)
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
103
It is important to mention that the rotor flux vector time constant is bigger than the time
constant of the stator flux vector. This can be demonstrated by writing the transfer function
from the stator flux vector to the rotor flux vector. For a two-phase reference related to the
rotor: ψ = pθ , the rotor electric equation becomes: 0 = R r .I r +
relationships: I r =
dΦ r
. From the flux-current
dt
M
Φr
−
.Φ s . By substituting the expression of I r in the rotor electric
σ.L r σ.L s .L r
equation, the following transfer function is obtained:
M Lr
Φr
=
Φ s 1 + σ.τr .p
(32)
where τr = R r L r is the rotor time constant. From equation (32), it is clear that the stator flux
vector changes slowly compared to the stator flux vector.
Going back to the expression of the motor torque, equation (31), if the stator flux vector
modulus is maintained constant, then the motor torque can be rapidly changed and
controlled by changing the angle θsr . Thus the tangential component of ΔΦ s = Vs .ΔT is for
controlling the torque, and its radial component is for controlling Φ s .
For a stator flux vector existing in sector i, the following stator voltage vector can is applied
in order to have the desired variations of the stator flux vector modulus and the motor
torque.
Vs
Increase
Decrease
Φs
Vi , Vi + 1 or Vi − 1
Vi + 2 , Vi − 2 or Vi + 3
Tem
Vi + 1 or Vi + 2
Vi − 1 or Vi − 2
Table 1. Stator voltage vector for the desired variations of Φ s and Tem
The vectors Vi and Vi + 3 are not considered for controlling the torque because they increase
the torque for the positive 30 degree half sector, and decrease it for the negative 30 degree
half sector. They can be used if 12 sectors are considered for dividing the total locus.
By analyzing Table 1, it is possible to do a decoupled control of Φ s and Tem. For all the six
sectors, Table 2 shows the good stator voltage vector that gives the desired variations of
Φ s and Tem.
Fig. 22 shows the scheme of the DTC.
There are two different loops for controlling the stator flux vector modulus and the motor
torque. The reference values of Φ s and Tem are compared with the estimated values. The
resulting errors are fed into the two-level and three-level hysteresis comparators
respectively. The outputs of the hysteresis comparators and the position of the stator flux
vector are used as inputs for the look up table (selection table of Table 2).
104
Torque Control
Φs
FI
FD
Tem
S1
S2
S3
S4
S5
S6
TI
V2
V3
V4
V5
V6
V1
=
V0
V7
V0
V7
V0
V7
TD
V6
V1
V2
V3
V4
V5
TI
V3
V4
V5
V6
V1
V2
=
V7
V0
V7
V0
V7
V0
TD
V5
V6
V1
V2
V3
V4
Table 2. Stator voltage vector for the desired variations of Φ s and Tem in all sectors
Fig. 22. Scheme of the DTC law
Usually, the estimation of the stator flux vector is done using the stator electric equation:
Φs
e
t +Δt
(
= Vs − R s .I s
)
t
.Δt + Φ s
e
t
(33)
The accuracy of this flux estimator is highly dependent on the value of the stator winding
resistor, which varies with the motor temperature.
This chapter proposes a new estimation technique that uses the rotor electric equation. It
shows that it is less sensitive to the variation of the rotor resistor, but more sensitive to the
variation of the saturation level. To overcome this problem, an adaptive estimator is
proposed, based on a previous saturation phenomenon study.
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
105
5.2 Direct Torque Control Law for an induction machine for a fixed chopping
frequency voltage source inverter
It is possible to develop the expression of a continuous optimal stator voltage vector that
gives the desired variations of Φ s and Tem (C.A, Martins.; T.A, Meynard.; X, Roboam. &
opt
opt
Des
and vsq
that give the desired ΔTem
Δt
A.S, Carvalho2, 1999). The control voltages vsd
and Δ Φ s
Des
Δt are searched.
The expression of the motor torque derivative is:
disq dΦsq
dTem
dΦ
di
.isd − Φsq . sd )
= p( sd .isq + Φsd .
−
dt
dt
dt
dt
dt
The expressions of
(34)
dΦ sq
dΦsd
and
can be found from the stator electric equations in the
dt
dt
fixed reference:
dΦ sd
= vsd − R s .isd
dt
dΦ sq
= vsq − R s .isq
dt
(35)
By writing the expressions of isd and isq from the flux-current relationships, the derivatives
of these currents versus time are:
disd
1 ⎛ dΦsd M dΦ rd ⎞
.⎜
=
− .
⎟
dt
Lr dt ⎠
σ.Ls ⎝ dt
disq
1 ⎛ dΦsq M dΦ rq ⎞
.⎜
=
− .
⎟
dt
Lr dt ⎟⎠
σ.Ls ⎜⎝ dt
(36)
The rotor electric equations give the expressions of the rotor fluxes derivatives versus time:
dΦ rd
d(pθ)
d(pθ)
R
.Φ rq − R r .i rd = −
.Φ rq − r . ( Φ sd − L s .i sd )
=−
dt
dt
dt
M
dΦ rq d(pθ)
d(pθ)
Rr
=
.Φ rd − R r .i rq =
.Φ rd − . Φ sq − L s .i sq
dt
dt
dt
M
(
(37)
)
The final expressions of the stator fluxes derivatives can be obtained by substituting Φ rd and
Φ rq by their expressions using stator variables:
dΦ rd
d(pθ)
d(pθ) L r
R
=−
.Φ rq − R r .i rd = −
. (Φ sq − σ.L s .i sq ) − r . ( Φ sd − L s .i sd )
dt
dt
dt M
M
dΦ rq d(pθ)
d(pθ) L r
Rr
.Φ rd − R r .i rq =
. (Φ sd − σ.L s .i sd ) − . Φ sq − L s .i sq
=
dt
dt
dt M
M
(
By replacing (38) in (36), the stator currents derivatives become:
)
(38)
106
Torque Control
disd
1 ⎛ dΦ sd M ⎛ d(pθ) L r
R
⎞⎞
=
.⎜
− .⎜ −
. (Φsq − σ.Ls .isq ) − r .( Φsd − Ls .isd ) ⎟ ⎟
σ.Ls ⎝ dt
dt
Lr ⎝
dt M
M
⎠⎠
(39)
1 ⎛ dΦsq M ⎛ d(pθ) L r
R
⎞⎞
=
.⎜⎜
− .⎜
. (Φsd − σ.Ls .isd ) − r . Φ sq − Ls .isq ⎟ ⎟⎟
dt
σ.Ls ⎝ dt
Lr ⎝ dt M
M
⎠⎠
disq
(
)
The motor torque derivative is finally obtained as a function of stator voltage, stator current
and stator flux components.
dTem
= p(vsd .K sq − vsq .K sd + K1 )
dt
(40)
with
K sd = isd −
Φ sq
L
Φ sd
, K sq = isq −
, R s' = R s + s .R r , Φ s =
σ.Ls
Lr
σ.Ls
2
3
( Φsd )2 + ( Φsq )
2
and
dθ
3.p.
R s'
dt . Φ 2 + p. dθ (Φ .i − Φ .i ) =
K1 = −
Tem −
s
sd sd
sq sq
σ.Ls .p
2.σ.Ls
dt
dθ
3.p.
R s'
dt . Φ 2 + p. dθ (Φ .(Φ − σ.L .i ) − Φ .(Φ − σ.L .i ))
−
Tem −
s
sd
sd
s sd
sq
sq
s sq
σ.Ls .p
2.σ.Ls
dt
Using the stator electric equations, the derivative of Φ s =
2
3
( Φsd )2 + ( Φsq )
2
can be
found:
d Φs
dt
=
2
3. Φ s
( Φsd .vsd + Φsq .vsq − R s .(Φsd .isd + Φsq .isq ))
(41)
opt
opt
Finally, the optimal control vsd
and vsq
are obtained by replacing the desired variations
Des
during the sampling period ΔTem
Δt and Δ Φ s
the derivatives
opt
vsd
=
Des
Δt in equations (40) and (41) instead of
d Φs
dTem
and
.
dt
dt
3
⎛
.Φs .⎜ Δ Φs
2
⎝
Des
((
)
⎞
Des
Δt ⎟ .K sd + R s .K sd .(Φsd .isd + Φ sq .isq ) + Φsq . ΔTem
Δt / p − K1
⎠
Φsd .K sd + Φsq .K sq
)
(42)
At each sampling period the stator currents are measured and the stator fluxes are estimated
from the stator electric equations. Actual values of Φ s and Tem are then calculated. Using
the reference values for the motor torque and for the modulus of the stator flux vector, the
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
107
desired variations during a period of Δt are calculated and used in equation (42) to find the
opt
opt
and vsq
.
optimal values of the control vsd
This control strategy can be implemented using a fixed chopping frequency source voltage
inverter.
5.3 Sensitivity study of the DTC stator flux estimator to the variation of the stator
resistor
The classical stator flux estimator used generally for the DTC is based on the stator electric
d Φs
. It is clear that this
equation written in a fixed two-phase reference: Ψ=0, Vs = R s Is +
dt
estimator is highly affected by the stator resistor variations, due to the motor temperature
variations, especially for low speed applications.
The DTC for fixed chopping frequency of the voltage source inverter is implemented in
A_MOS program. Fig. 23 shows simulation results of a 45(KW) induction machine
controlled by the DTC law with the previous estimator. A difference of 15% between the
motor stator resistor and its value implemented in the control estimator is considered.
Fig. 23. Stator electric equation estimator results with 15% increase for the stator resistor
The difference may cause oscillations to the motor speed, and this problem is more
important for low speed.
6. New stator flux estimator for the DTC
If the motor speed is available, the stator fluxes can be calculated from the flux currents
relationships:
108
Torque Control
M
.Φ
Lr rd
M
Φsq = σLs .isq +
.Φ
Lr rq
Φsd = σLs .isd +
(43)
At each sampling period the stator currents are measured and the rotor fluxes are calculated
using the rotor electric equations:
dΦ rd
L
d(pθ)
⎛Φ
⎞ d(pθ)
= − R r .i rd −
Φ rq = −R r .⎜ sd − s .isd ⎟ −
Φ rq
dt
dt
M
dt
⎝ M
⎠
dΦ rq
dt
= − R r .i rq +
⎛ Φsq Ls
⎞ d(pθ)
d(pθ)
Φ rd = −R r .⎜⎜
Φ rd
− .isq ⎟⎟ +
dt
M
dt
⎝ M
⎠
(44)
The calculation of the stator fluxes using equations ( 43) and ( 44) does not require the stator
resistor, thus any change in its value has no influence. In fact, the estimator uses the value of
the rotor resistor which determines the time constant of the rotor flux. It is obvious that the
accuracy in measuring the rotor resistor has no big effect on estimating the stator flux vector
using the two previous equations. This is due to the fact that the stator fluxes time constant
is smaller than the time constant of the rotor fluxes, as it was shown previously. Fig. 24
shows that for an increase of 15% in the rotor resistor value, the DTC with the new estimator
gives better results.
Fig. 24. New estimator results with 15% increase for the rotor resistor
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
109
The new method of estimating the stator fluxes requires the knowledge of a greater number
of motor parameters. It is clear that the new estimator will be more sensitive to the variation
of the induction motor saturation level, since it uses the cyclic inductances.
Fig. 25 shows simulation results of the induction motor controlled by the DTC with the new
stator fluxes estimator. During the flux weakening phase a big difference between the
desired stator flux and the real one was obtained. This influences the dynamic behavior of
the speed when applying a load torque of 110(Nm) at 3(s).
Fig. 25. New estimator sensitivity to the variation of saturation level
It is possible to overcome this problem by tuning the new estimator parameters at each
sampling period according to the magnetizing current modulus that can be estimated as
described in the section of the saturated FOC.
The scheme of the DTC using this estimator is presented in Fig. 26.
Fig. 27 shows the improvement of dynamic behavior of the induction motor after
implementing the adaptive estimator of Fig. 26.
110
Fig. 26. Adaptive stator fluxes estimator
Fig. 27. 45(KW) motor behavior with the adaptive stator flux estimator
Torque Control
Induction Motor Vector and Direct Torque Control
Improvement during the Flux Weakening Phase
111
It is possible to combine the two previous estimators. The first one can be used for high
speed, while the second estimator can be used for low speed range. In this case, there is no
need to program the adaptive estimator of the stator fluxes, since it will not work during the
flux weakening phase.
7. Conclusion
This chapter has presented a full study of the magnetic state variation of the induction
motor. Using a finite elements calculation program, it was possible to establish a two-phase
model that takes into account the variation of the saturation level. A very simple resolution
method of this new model was presented. The dynamic response of the new model was
validated by comparing it to the dynamic response of the induction motor given by the
finite element calculation program. After establishing the new model it was possible to
review the advanced control laws like the FOC and the DTC laws. A new saturated FOC law
was developed in order to enhance the dynamic behavior of the motor during the flux
weakening phase, because of the difference between the motor cyclic inductances values
and the values of the cyclic inductances introduced in the controllers. Concerning the DTC
law, it was shown that a small error in the stator resistor value will highly influence the
stator flux estimation, which is done using the stator electric equation. A new stator fluxes
estimator was developed using rotor electric equations. This estimator is less sensitive to the
motor temperature variation, but it is more sensitive to the variation of the saturation level.
An adaptive solution was proposed to tune the estimator parameters according to the
saturation level of the motor. Nevertheless the adaptive part added to the DTC algorithm,
its computation time remains very small comparing to the FOC algorithm that takes into
account the variation of the saturation level. It is important to mention that it is possible to
combine the classical estimator and the new estimator according to the speed range. The
classical estimator can be used at high speed, but at low speed, it is better to use the new
stator flux estimator.
8. References
Grotstollen, H. & Wiesing, J. (1995). Torque capability and control of saturated induction
motor over a wide range of flux weakening, Transaction on Industrial Electronics,
Vol. 42, No. 4, (August 1990) page numbers (374-381).
Vas, P. & Alakula, M. (1990). Field oriented control of saturated induction motors, IEEE
Transaction on Energy Conversion, Vol. 5, No. 1, (March 1990) page numbers (218224), ISSN 0885-8969.
Kasmieh, T. & Lefevre, Y. (1998). Establishment of two-phase non-linear simulation model
of the dynamic operation of the induction motor, EPJ European Physical Journal, Vol.
1, No. 1, (January 1998) page numbers (57-66).
Vas, P. (1981). Generalized transient analysis of saturated a.c motors, Archiv fur
Elektrotechnik, Vol. 64, No. 1-2, (June 1981) page numbers (57-62).
Kasmieh, T. (2008). Adaptive stator flux estimator for the induction motor Direct Torque
Control, Proceedings of SPEEDAM 2008, pp. 1239-1241, Ischia, June 2008,
Italy.
112
Torque Control
Blaschke, F. (1972). The principal of field orientation as applied to the new trans-vector
closed-loop control system for rotating field machines, Siemens Review, (May
(1972).
Kasmieh, T.( 2002), Presentation of a powerful opened simulator for the saturated induction
motor traction system, Proceedings of SPEEDAM 2002, (June 2002), pp. A1 24-A1 37,
Ravello, June 2002, Italy.
Noguchi, T. & Takahashi, I. (1984). Quick torque response control of an induction motor
based on a new concept, IEEE Tech, Vol. RM84-76, (September 1984) page numbers
(61-70).
Depenbrock, M. & Steimel A.(1990). High power traction drives and convertors. Proc. of
Elect. Drives Symp.’90, pp. 1–9, Capri,1990, Italy.
C.A, Martins.; T.A, Meynard.; X, Roboam. & A.S, Carvalho2. (1999). A predictive sampling
scale model for direct torque control of the induction machine fed by multilevel
voltage-source inverters. European Physical Journal-Applied Physics, AP. 5, (1999)
page numbers (51-61).
5
Control of a Double Feed and
Double Star Induction Machine
Using Direct Torque Control
Leila Benalia
Department of electrical Engineering
Batna University, Rue Chahid Med El Hadi boukhlouf
Algeria
1. Introduction
DTC is an excellent solution for general-purpose induction drives in very wide range The
short sampling time required by the TC schemes makes them suited to a very fast torque
and flux controlled drives as well the simplicity of the control algorithm. DTC is inherently
a motion sensor less control method.
2. Objective of the work
This chapter describes the control of doubly fed induction machine (DFIM) and the control
of doubly star asynchronous machine (DSAM), using direct torque control (DTC).
3. Principe du control direct du couple
Direct torque control is based on the flux orientation, using the instantaneous values of
voltage vector.
An inverter provides eight voltage vectors, among which two are zeros (Roys & Courtine,
1995), (Carlos et al., 2005). This vector are chosen from a switching table according to the
flux and torque errors as well as the stator flux vector position. In this technique, we don’t
need the rotor position in order to choose the voltage vector. This particularity defines the
DTC as an adapted control technique of ac machines and is inherently a motion sensor less
control method (Casdei et al., 2001), (Kouang-kyun et al., 2000).
4. Double feed induction machine (DFIM)
In the training of high power as the rolling mill, there is a new and original solution using a
double feed induction motor (DFIM). The stator is feed by a fixed network while the rotor
by a variable supply which can be either a voltage or current source.
The three phase induction motor with wound rotor is doubly fed when, as well as the stator
windings being supplied with three phase power at an angular frequency ωs , the rotor
windings are also fed with three phase power at a frequency ωrr .
114
Torque Control
Under synchronous operating conditions, as shown in (Prescott & Alii., 1958), (Petersson.,
2003) , the shaft turns at an angular velocity ωr , such that:
ωr = ωs + ωrr
The sign on the right hand side is (+) when the phase sequences of the three phase supplies
to the stator and rotor are in opposition and (-) when these supplies have the same phase
sequence. The rotational velocity of the shaft, ωr , is expressed in electric radians per second,
to normalize the number of poles.
4.1 Double feed induction machine modelling
Using the frequently adopted assumptions, like sinusoid ally distributed air-gap flux
density distribution and linear magnetic conditions and considering the stator voltages
( vsα , vsβ ) and rotor voltages ( vrα , vr β ) as control inputs, the stator flux
( Φ sα , Φ sβ ), and the rotor current ( irα , irβ ) as state variables. In the referential axis fixed in
relation to the stator, the following electrical equations are deduced:
⎡ I sα ⎤ d ⎡ Φ sα ⎤
⎢ I ⎥ + ⎢Φ ⎥
⎣ sβ ⎦ dt ⎣ sβ ⎦
(1)
⎡ I rα ⎤ d ⎡ Φ rα ⎤ ⎡ 0 ω ⎤ ⎡ Φ rα ⎤
⎢ I ⎥ + ⎢Φ ⎥ + ⎢
⎥
⎥⎢
⎣ r β ⎦ dt ⎣ r β ⎦ ⎣ −ω 0 ⎦ ⎣Φ r β ⎦
(2)
⎡Vsα ⎤ ⎡ Rs
⎢V ⎥ = ⎢
⎣ sβ ⎦ ⎣ 0
⎡Vrα ⎤ ⎡ Rr
⎢V ⎥ = ⎢
⎣ rβ ⎦ ⎣ 0
0⎤
Rr ⎥⎦
0⎤
Rs ⎥⎦
Expressions of fluxes are given by:
⎧ Φ sα
⎪Φ
⎪ sβ
⎨
⎪ Φ rα
⎪Φ r β
⎩
= ls I sα
= ls I sβ
= lr I rα
= lr I r β
+ MI rα
+ MI r β
+ MI sα
+ MI sβ
(3)
The mathematical model is written as a set of equations of state, both for the electrical and
mechanical parts:
•
dX
= X = AX + BU
dt
(4)
Where:
⎡ I rα ⎤
⎢I ⎥
rβ ⎥
X =⎢
⎢ Φ sα ⎥
⎢
⎥
⎣⎢Φ sβ ⎦⎥
The matrices A and B are given by:
and
⎡Vsα ⎤
⎢V ⎥
sβ
U =⎢ ⎥
⎢Vrα ⎥
⎢ ⎥
⎣⎢Vrβ ⎦⎥
(5)
Control of a Double Feed and Double Star Induction Machine Using Direct Torque Control
⎡ −1
⎢ T 'δ
⎢ s
⎢− ω
r
⎢
A = ⎢
M
⎢
⎢ Ts
⎢
⎢ 0
⎣
⎡
⎢−
⎢
B= ⎢
⎢
⎢
⎢
⎢⎣
1−δ
δ MT s
1−δ
ωr
−
δM
ωr
−1
T s'δ
1
−
Ts
0
M
Ts
0
1−δ
δM
0
−
0
1
0
1−δ
δM
0
1
J
1−δ
⎤
ωr⎥
δM
⎥
1−δ ⎥
δ MT s ⎥
⎥
0
⎥
⎥
1 ⎥
−
T s ⎥⎦
1
Lrδ
0
0
0
dΩ
=Cem-Cr-KfΩ.
dt
⎤
0 ⎥
⎥
1 ⎥
Lrδ ⎥
0 ⎥
⎥
0 ⎥⎦
115
(6)
(7)
(8)
Where J is the moment of inertia of the revolving parts, Kf is the coefficient of viscous
friction, arising from the bearings and the air flowing over the motor, and Cr is the load
couple.
The equation of the electromagnetic torque is:
Ce =
3 pM
(Φ sα I r β − Φ sβ I rα )
2 Ls
(9)
The block diagram for the direct torque and flux control applied to the double feed
induction motor is shown in figure 1.The stator flux Ψref and the torque Cemref magnitudes
are compared with respective estimated values and errors are processed through hysteresisband controllers.
Stator flux controller imposes the time duration of the active voltage vectors, which move
the stator flux along the reference trajectory, and torque controller determinates the time
duration of the zero voltage vectors, which keep the motor torque in the defined-by
hysteresis tolerance band (Kouang-kyun et al.,2000), ( Xu & Cheng.,1995). Finally, in every
sampling time the voltage vector selection block chooses the inverter switching state, which
reduces the instantaneous flux and torque errors (Presada et al., 1998).
5. Simulation results machine
Figure 2 refer in order, to the variation in magnitude of the following quantities, speed, flux
and electromagnetic torque obtained while starting up the induction motor initially under
no load then connecting the nominal load. During the starting up with no load the speed
reaches rapidly its reference value without overtaking, however when the nominal load is
applied a little overtaking is noticed and the command reject the disturbance. The excellent
dynamic performance of torque and flux control is evident.
116
Torque Control
DFIM
Uc
isa
Sa Sb
isb
Sc
PARK
Transformation
Switching
Table
ccpl
Load
Vsα Vsβ i sα i sβ
cflx
N
φ ref
1
+- φ sest
0
1
0
Network
3 2
4
1
56
Estimated
Stator flux
φ sα φ sβ
i sα
i sβ
-1
C emest
_
Estimated
electromagnetic
Torque
Cemref
Fig. 1. DTC applied to double feed induction machine
6. Robust control of the IP regulator
a) Speed variation
Figure 3 shows the simulation results obtained for a speed variation for the values: (Ωref =
157, 100 and 157 rad/s), with the load of 3 N.m applied at t =0.8s. This results show that the
variation lead to the variation in flux and the torque. The response of the system is positive,
the speed follow its reference value while the torque return to its reference value with a little
error.
b) Speed reversal of rated value
The excellent dynamic performance of torque control is evident in figure 4, which shows
torque reversal for speed reversal of (157, -157 rad/s), with a load of 5N.m applied at t=1 s.
The speed and torque response follow perfectly their reference values with the same
response time. The reversal speed leads to a delay in the speed response, to a peak
oscillation the current as well as a fall in the flux magnitude which stabilise at its reference
value.
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
Control of a Double Feed and Double Star Induction Machine Using Direct Torque Control
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
Fig. 2. Simulation results obtained with an IP regulator
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 3. Robust control for a speed variation
117
118
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
Torque Control
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 4. Robust control under reversal speed
c) Robust control for load variation
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
The simulation results obtained for a load variation (Cr = 3 N.m, 6 N.m) in figure 5, show
that the speed, the torque and the flux are inflated with this variation. Indeed the torque and
the speed follow their reference values.
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 5. Robust control under load variation
Control of a Double Feed and Double Star Induction Machine Using Direct Torque Control
119
d) Robust control of the regulator under stator resistance variation
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
In order to verified the robustess of the regulator under motor parameters variations we
carried out a test for a variation of 50% in the value of stator resistance at tile t= 1.5s. The
speed is fixed at 157 rad/s and a resistant torque of 5 N.m is applied at t= 1s. Figure 6 shows
the in order the torque response, the current, the stator flux and the speed. The results
indicate that the regulator is very sensitive to the resistance change which results in the
influence on the torque and the stator flux.
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 6. Robust control under stator resistance variation
7. Double star induction machine (DSIM)
For the last 20 years the induction machines with a double star have been used in many
applications for their performances in the power fields because of their reduced pulsation
when the torque is minimum (Kalantari et al.,2002).The double stator induction machine
needs a double three phase supply which has many advantages. It minimise the torque
pulsations and uses a power electronics components which allow a higher commutation
frequency compared to the simple machines. However the double stator induction machines
supplied by a source inverter generate harmonic which results in supplementary losses
(Hadiouche et al., 2000). The double star induction machine is not a simple system, because
a number of complicated phenomena’s appears in its function, as saturation and skin effects
(Hadiouche et al., 2000).
The double star induction machine is based on the principle of a double stators displaced by
α=300 and rotor at the same time. The stators are similar to the stator of a simple induction
machine and fed with a 3 phase alternating current and provide a rotating flux.
Each star is composed by three identical windings with their axes spaced by 2π/3 in the
space. Therefore, the orthogonality created between the two oriented fluxes, which must be
120
Torque Control
strictly observed, leads to generate decoupled control with an optimal torque (Petersson.,
2003).
This is a maintenance free machine.
The machine studied is represented with two stars windings: As1Bs1Cs1 et As2Bs2Cs2 which
are displaced by α = 30° and thee rotorical phases: Ar Br Cr.
B s2
B s1
A rRotor
θ A s2
Br
Star N°2
α
Star N°1
A s1
C s1
C
Cr
Fig. 7. Double star winding representation
8. Double star induction machine modeling
The mathematical model is written as a set of state equations, both for the electrical and
mechanical parts:
d
⎡⎣Vabc ,s 1 ⎤⎦ = [ Rs 1 ] ⎡⎣ I abc ,s 1 ⎤⎦ + ⎡⎣Φ abc , s 1 ⎤⎦ .
dt
d
⎡⎣Vabc ,s 2 ⎤⎦ = [ Rs 2 ] ⎡⎣ I abc ,s 2 ⎤⎦ + ⎡⎣Φ abc ,s 2 ⎤⎦ .
dt
d
⎡⎣Vabc ,r ⎤⎦ = [ Rr ] ⎡⎣ I abc ,r ⎤⎦ + ⎡⎣Φ abc ,r ⎤⎦ .
dt
J
dΩ
=Cem-Cr-Kf Ω.
dt
(1)
(2)
Where:
J is the moment of inertia of the revolving parts.
Kf is the coefficient of viscous friction, arising from the bearings and the air flowing over the
motor.
Cem is the electromagnetic torque.
The electrical state variables are the flux, transformed into vector [ Φ ] by the “dq”
transform, while the input are the “dq” transforms of the voltages, in vector [V].
d
[ Φ ] = [ A ] .[ Φ ] + [ B ] .[ v ]
dt
(3)
Control of a Double Feed and Double Star Induction Machine Using Direct Torque Control
121
⎡ Φ ds 1 ⎤
⎢Φ ⎥
⎢ ds 2 ⎥
⎢ Φ qs 1 ⎥
[Φ ] = ⎢ Φ ⎥
⎢ qs 2 ⎥
⎢
⎥
⎢ Φ dr ⎥
⎢ Φ qr ⎥
⎣
⎦
(4)
⎡ vds 1 ⎤
⎢v ⎥
ds 2
[V ] = ⎢⎢ v ⎥⎥
qs 1
⎢
⎥
⎢⎣ vqs 2 ⎥⎦
The equation of the electromagnetic torque is given by
Cem= p
Lm
(Φdr (iqs1+iqs2)-Φqr (ids1+ids2))
Lm + Lr
(5)
The flux equation is:
⎛ Φ ds1 Φ ds2 Φ dr ⎞
⎟
+
+
⎜
L r ⎠⎟
L s2
⎝ L s1
(6)
⎛ Φ qs1 Φ qs2 Φ qr ⎞
⎟
+
+
Φ mq = La ⎜
Ls2
Lr ⎟⎠
⎝ L s1
(7)
Φ md = La
Given that the “dq”axes are fixed in the synchronous rotating coordinate system, we have:
⎡ a11
⎢a
⎢ 21
⎢a
[ A] = ⎢ a31
⎢ 41
⎢a
⎢ 51
⎣⎢ a61
a12
a22
a13
a23
a14
a24
a15
a25
a32
a42
a33
a43
a34
a44
a35
a45
a52
a62
a53
a63
a54
a64
a55
a65
⎡1
⎢0
⎢
⎢0
[ B] = ⎢0
⎢
⎢0
⎢
⎣⎢0
Where:
a11= a33 =
a12=a34=
Rs 1La Rs 1
−
Ls 1
L2s 1
Rs 1La
Ls 1Ls 2
0 0 0⎤
1 0 0 ⎥⎥
0 1 0⎥
⎥
0 0 1⎥
0 0 0⎥
⎥
0 0 0 ⎦⎥
a16 ⎤
a26 ⎥⎥
a36 ⎥
⎥
a46 ⎥
a56 ⎥
⎥
a66 ⎦⎥
(8)
(9)
122
Torque Control
ωs a13= a24= - a31= - a42=
a14= a16 =a23 = a26= a32=a35=a41=a45 =a53= a54=a61=a62 = 0
R L
R L
a15= a36= s 1 a , a21= a43= s 2 a
Lr Ls 1
Ls 1Ls 2
a22= a44=
Rs 2 La Rs 1
−
Ls 1
L2s 2
a51= a63=
Rr La
RL
, a52= a64= r a
Lr Ls 1
Lr Ls 2
a55= a66=
Rr La Rr
−
Lr
L2r 2
,
,
a25= a46=
Rs 2 La
Lr Ls 2
a56= - a65= ω gl
Figure 8 shows the block diagram for the direct torque and flux control applied to the
double star induction motor shown in.
DSIM
Uc
isb
isa
Sa Sb
Sc
PARK
Transformation
Switching
Table
VsαVsβ
i sβ
φ ref
N cflx
+
_
Estimated
Star flux
φ sest
4
Network
32
56
1
φ sα φ sβ
C
_ emest
+
i sβ i s α
Estimated
electromagnetic
Torque
Cemref
Fig. 8. DTC applied to double star induction machine
Load
Control of a Double Feed and Double Star Induction Machine Using Direct Torque Control
123
9. Simulation results
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
Figure 9 refer in order, to the variation in magnitude of the following quantities, speed,
electromagnetic torque, current and flux obtained while starting up the induction motor
initially under no load then connecting the nominal load. During the starting up with no
load the speed reaches rapidly its reference value without overtaking, however when the
nominal load is applied a little overtaking is noticed and the command reject the
disturbance. The excellent dynamic performance of torque and flux control is evident.
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 9. Simulation results obtained with an PI regulator
10. Control of the regulator
a) Speed variation
Figure.10 shows the simulation results obtained for a speed variation for the values: (Ωref =
314 and 260 rad/s), with the load of 5 N.m applied at t=1.5s.
These results shows that the variation lead to the variation in flux and the torque. The
response of the system is positive, the speed follow its reference value while the torque
return to its reference value with a little error.
b) Robust control for load variation
Figure.11 shows the simulation results obtained for a load variation (Cr = 5 N.m, 2.5 N.m).
As can be seen the speed, the torque, the flux and current are influenced by this variation.
The torque and the speed follow their reference values.
We can see that the control is robust from the point of view load variation.
124
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
Torque Control
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
Fig. 10. Robust control for a speed variation
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 11. Robust control under load variation
Control of a Double Feed and Double Star Induction Machine Using Direct Torque Control
125
c) Robust control of the regulator under star resistance variation
Angular
speed (rad/s)
Electromagnetic
torque (N.m)
In order to verified the robustess of the regulator under motor parameters variations we
carried out a test for a variation of 50% in the value of star resistance at time t= 1.5s. The
speed is fixed at 314 rad/s and a resistant torque of 5N.m is applied at t= 1s. Figure 6 shows
in order the torque response, the current, the stator flux and the speed. The results indicate
that the regulator is very sensitive to the resistance change which results in the influence on
the torque and the stator flux
Time (s)
Stator flux( Wb)
Time (s)
Time (s)
Fig. 12. Robust control under stator resistance variation
11. Conclusion
This chapter presents a control strategy for a double feed induction machine and double star
induction machine based on the direct control torque (DTC) using a PI regulator. The
simulation results show that the DTC is an excellent solution for general-purpose induction
drives in a wide range of power.
The main features of DTC compared to the classical flux oriented control FOC can be
summarized as follows:
•
DTC has a simple and a robust control structure.
•
DTC operates with closed torque and flux loops but without current controllers.
•
DTC needs stator flux and torque estimation and, therefore, is not sensitive to rotor
parameters.
•
DTC is inherently a motion sensor less control method.
The simulation results show that the DTC is an excellent solution for general-purpose
induction drives in a very wide power range. The short sampling time required by the DTC
scheme makes it suited to very fast torque and flux controlled drives beside the simplicity of
the control algorithm.
126
Torque Control
However the DTC presents two major problems:
•
The absence of the harmonic of the couple restraint (electromagnetic compatibility,
audible noise, variation of the acoustic quality).
•
The excitation of some mechanical resonant modes which lead to a serious ageing of the
system.
The DTC applies the same control effort to regulate flux as it does for the torque. Finally, we
believe that the DTC principle will continue to play a strategic role in the development of
high performance motion sensor less AC drives.
12. References
Carlos Ortega, Antoni Arias, Xavier del Toro. (2005). Novel direct torque control for
induction motors using schort voltage vectors of matrix converters. IEEE
Trans.ind.Appl, pp 1353- 1358, 2005.
Casdei.D, Serra.G, Tani.A,.( 2001). The use of matrix converters in direct torque control of
induction machines. IEEE Trans.on Industrial Electronics, Vol 48,N 6,.
Hadiouche. D, H.Razik, A.Rezzoug.(2000). Stady and simulation of space vector PWM
control of Double-Star Induction Motors . IEEE-CIEP, Acapulco, Mexico, pp 42-47.
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an arbitrary shift angle between its three phase windings. EPE-PEMC, Kosice.
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a dual three-phase induction machine. Electrimacs, pp 18-21.
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induction motor with reduction of torque ripple. IEEE Trans.ind.Appl, pp 1087 -1092
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vol 31,no. 3, pp 636-642.
Part 2
Oriented Approach of Recent Developments
Relating to the Control of the Permanent
Magnet Synchronous Motors
6
Direct Torque Control of
Permanent Magnet Synchronous Motors
Selin Ozcira and Nur Bekiroglu
Yildiz Technical University
Turkey
1. Introduction
Modern electrical drive systems consist of; power electronics components, transformers,
analog/digital controllers and sensors or observers. The improvements in the
semiconductor power electronic components have enabled advanced control techniques
with high switching frequency and the high efficiency. Complex control algorithms have
been widely used and got simplified in drivers due to the developments in software
technology. DC, asynchronous and synchronous motors are frequently used motor types
with these driver systems. New kinds of motors are developed like linear motors, step
motors, switching reluctance motors, and permanent magnet synchronous motors.
Permanent magnet synchronous motors are used where in general high demands are made
with regard to speed stability and the synchronous operation of several interconnected
motors. They are suitable for applications where load-independent speeds or synchronous
operation are required under strict observance of defined speed relations within a large
frequency range.
As the technology gets improved, studies on PMSM such as direct torque control method
have been improved as well. DTC has many advantages such as faster torque control, high
torque at low speeds, and high speed sensitivity. The main idea in DTC is to use the motor
flux and torque as basic control variables, identical to the DC drives. In order to emulate the
magnetic operating conditions of a DC motor, the information regarding the rotor status is
required to perform the field orientation process of the flux-vector drive. This information
should be obtained by feeding the rotor speed and angular position back by using a pulse
encoder. Encoders are costly and they add more complexity to the overall system.
Several methods have been developed to obtain the rotor position and angular speed from
the electrical measurements and computations in order to eliminate the need for sensors.
The idea behind such methods is to manipulate the motor equations in order to express
motor position and speed as functions of the terminal quantities. In (Adreescu & Rabinovici,
2004) self tuning speed controller and Luenberger observer was proposed, however it is still
necessary to use an encoder for rotor position detection. Most of the methods; however,
work only when the rotor is anisotropic and if the dependence of the inductance on the rotor
position is accurately known; moreover, the speed is not estimated, thus the drive is not
completely without mechanical sensors (Yan et al, 2005). On the other hand, it can be
reported that the method in (Bolognani et al, 1999) can a viable solution for the online
determination of the rotor position and speed of a PMSM. In (Ichikawa et al, 2003) extended
130
Torque Control
electromotive force model is used to estimate the rotor position. However, this method is
based on expert knowledge. Some researchers have proposed a combination of current and
flux linkage estimation (Rahman & Toliyat, 1996), (Toliyat et al, 2002) or methods including
Kalman filters, fuzzy logic, and neural network observers to obtain the rotor position angle
(Dan et al, 2004), (Grzesiak & Kazmierkowski, 2007). Recently, researchers have tried to
reduce the torque pulses and harmonics in PMSM. In addition, (Sozer et al, 2000) presented
an inverter output filter system for PWM derives to reduce the harmonics of surface
mounted permanent magnet synchronous motor (SPMSM), it shows some effectiveness in
reducing switching harmonics, but very large circulating current between inverter output
and filter elements is required to reshape the motor terminal voltage which may violate the
inverter's current limit.
Efficiency is important due to the energy scarcity of the world and higher performance is
needed for modern motion control applications. PMSM offer efficiency advantages over
induction machines when employed in variable speed drives. Since much of the excitation
in the PMSM is provided by the magnets, the PMSM will have smaller losses associated
with the magnetizing component of the stator current. The stator current may be almost
purely torque producing in a PMSM drive while in an induction machine drive there is
always a large magnetization current present. Due to the synchronous operation of the
PMSM, rotor losses are greatly reduced. The application of the PMSMs for low speed
operation in direct drives is an economic alternative for the induction motors with
gearboxes. Since the speed of the direct drive PMSMs is lower than the speed of the
induction motors with the gearboxes, the risk of torque harmonics appearing at the
mechanical resonances is increased in the speed range of normal operation. Permanent
magnet motors have been used for decades in low power applications such as servo drives
and domestic appliances. Recently, the PMSM drives have been developed further and are
used in industrial applications requiring high torque at low speed. PMSM drives are replacing
standard induction motors with gearboxes in, for example, paper and textile industry, special
applications for marine (Laurila, 2004). However, PM synchronous motors can not be fed
directly from the mains supply and need to be driven by the AC motor drives. Similar to the
induction machine, vector control method is employed for the PM synchronous motors to
obtain high bandwidth torque control performance. For vector control, the rotor flux angle
needs to be known by the AC motor drive. Therefore, sensors (e.g. an incremental encoder) on
the motor shaft are utilized to sense the rotor flux angle and AC motor drives use this angle
information for vector coordinate transformations. With vector coordinate transformations, the
AC motor in the control coordinates is converted to a DC motor where torque control is a
simple issue of current control to be achieved by the current regulator. Thus, the vector control
method enables high bandwidth torque control of an AC machine, which brings high
bandwidth of speed and position control (Geyer et al, 2010).
Although machine drives with modern control techniques have brought high performance
and robustness to the motion control area, research has continued for developments in the
AC motors and drives technology. The motivation is to improve the technology for high
efficiency and for high performance. The literature study regarding to the direct torque
control of the permanent magnet synchronous machines is presented in this chapter.
The theory of the direct torque control was developed by (Depenbrock, 1985) for the first
time for asynchronous motors.
(Takahashi and Noguchi, 1986) developed the direct torque control only using the torque
routines for asynchronous motors.
Direct Torque Control of Permanent Magnet Synchronous Motors
131
(Depenbrock, 1988) presented the direct self control theory. According to this theory, the
variations in operation frequency are obtained by the algebraic calculations from the
Heyland-Ossanna circular diagram based on the torque demand feed-back.
(Pillay and Krishnan 1989) modeled the permanent magnet synchronous motor by statespace variables and accomplished the torque analysis. This study was a very significant step
for the permanent magnet synchronous machine studies.
(Adnanes, 1990) developed the torque analysis of the permanent magnet synchronous
machine in per unit mode and obtained the detailed the mathematical relationship between
the flux and torque.
(Raymond and Lang, 1991) presented the real-time adaptive control of the PMSM by using
the Motorola 68020 microprocessor. The motor model was linearized for the proposed
controller and the nonlinear effects of the inverter and system dynamics that can not be
modeled were omitted. The mechanical parameters were estimated so that they could be
taken into account for the controller.
(Pelczewski et al., 1991) performed the optimal model tracking control of the PMSM. In
order to have the controller doing the computations, motor model and its linearization were
required.
(Matsui and Ohashi, 1992) proposed a DSP based adaptive controller for PMSM. Therefore,
they proved that the DSPs can be implemented in motor controls as well.
(Chern and Wu, 1993) presented the position control of the PMSM by using the variable
oriented controller. The controller instantaneously does the calculations depending on the
unknown load and motor parameters. The system model is required and the computations
require very long periods of time.
(Ogasawara and Akagi, 1996) realized the position estimation of the PM motors for zero and
low speed conditions according to the saliency.
(Zhong et al., 1997) completed one of the first academic studies in the field of direct torque
control of the permanent magnet synchronous machines.
(Zhong et al., 1999) proposed the direct torque control of the permanent magnet
synchronous motor using two level torque hysteresis controller.
(Rahman et al., 1999) achieved the direct torque control by using a method based on
obtaining the d and q-axis voltages using certain coefficients.
(Luukko, 2000) developed the switching table for the direct torque control by adding the
zero vectors to the vector selection algorithm.
(Vaez Zadeh, 2001) experimentally achieved the constant torque control on a vector
controller by using the TMS320C31 DSP. In this study, the torque did not respond well in
terms of the desired torque value and response time, since the DSP technology was not
sufficient to implement the dynamic behavior of the motor.
(Tan et al., 2001) and Martins et al. (2002) proposed to reduce the torque ripples and fix the
switching frequency in AC drive systems by using a multi-level inverter. These methods
result in better waveforms, reduce the distortions, and are capable of operating in lower
switching frequency. However, on the other hand, they require more number of switching
devices. Moreover, the control strategies of these methods are very complicated.
(Dariusz et al., 2002) implemented the space vector modulation by using a DSP and
achieved the direct torque control.
(Balazovic, 2003) published a technical guide that describes the torque control and vector
control of the permanent magnet synchronous motor. This publication has been referred in
many similar studies of the industry and academia.
132
Torque Control
(Tang et al., 2004) used the space vector modulation in order to reduce the torque ripples
and they got good results. Developed control algorithm required two PI controllers and the
estimation of the reference voltage and the switching sequence of the selected vectors.
(Zhao et al., 2004) achieved the control of the very high speed (200000 rpm) permanent
magnet synchronous machine by using a DSP.
(Popescu et al., 2006) investigated the torque behavior of a single phase permanent magnet
synchronous motor.
(Jolly et al., 2006) performed the control of the permanent magnet synchronous machine in
constant power region.
(Luukko et al., 2007) presented the different rotor and load angle estimation methods for the
direct torque control. They directly calculated the load angle from the PMSM equations. In
these calculations, they used the tangent function. When the results of the DSP controlled
inverter and the motor test setup are investigated, it is seen that the rotor angle estimation
has oscillations. The error between the actual and estimated rotor angle variations get larger
in the periods when the oscillations are larger. It is revealed that PID coefficients should be
kept very high in order to compensate this error.
(Chen et al., 2007) designed an output filter for a direct torque controlled inverter. The filter
is composed of an RLC filter and an isolation transformer. This study is interesting since it
includes both the transformer design and the soft switching techniques in power electronics.
Furthermore, it is true that the transformer and RLC based filter will add significant cost to
the system instead as compared to the developments in the generation of inverter switching
signals in terms of controls.
(Noriega et al., 2007) designed a fuzzy logic controller for the DTC. They used the torque
error and the stator current for the fuzzy logic membership functions. In addition to the
simulation studies, they used an AC motor drive setup called Platform III by implementing
fuzzy logic functions to the software of this setup. Both simulation and the experimental
results show that the stator current is not in a wave form and it has some uncertain and
random shapes. As compared to the PI controlled DTC method, stator current is much more
distorted.
(Wang et al., 2007) developed the reference flux vector calculation in space vector
modulation for DTC. They extracted the voltage as a trigonometric function of the period
and using the frame transformations, they calculated the usage periods of the zero vectors
depending on the angular frequency of the current. However, this complicated control
structure has been implemented in simulations but not experimentally completed. In torque
graphs, there are long delay periods between the actual and calculated values.
(Zhao et al., 2007) developed a fourth order sliding mode observer for the surface mounted
permanent magnet synchronous motor. According to the motor parameters, it is seen that
the motor was a high power low speed motor. In simulations, the model available in
Matlab&Simulink library was implemented. The authors stated that fuzzy logic can be
employed to address the chattering issues commonly occur in sliding mode controls.
(Swierczynski et al., 2008) applied the DTC method onto a high power PMSM by using an
inverter that is driven by the space vector modulation method. In this study, they used
DSPACE 1103 control unit in which the Matlab Simulink simulation models can be directly
applied. In this study, it was not required to design a speed controller. This is mainly due to
the fact that the vehicle operator can control the speed according to the different driving
conditions.
Direct Torque Control of Permanent Magnet Synchronous Motors
133
(Yutao et al., 2008) used the radial basis neural network functions to exploit the reference
torque in a rectangular step functions. They attempted to reduce the torque vibrations and
make it in a rectangular shape which is not in fact.
(Cui et al., 2008) researched on a high performance DTC system based on a DSP. They
performed simulation and experimental studies. However, in their studies, the controls
execution time completed in 130 μs.
(Li et al., 2008) used zero vectors in space vector modulation for DTC. Zero vectors are
theoretically used in asynchronous motor's direct torque control. They tried to increase the
application duration of the vectors that are used to enlarge the torque angles in low speed
operation of the PMSM applications. However, in low speeds, using zero vectors for long
period of time causes the fast changes in stator flux and the limit values are enforced.
Moreover, switching losses of this implementation will be higher since 8 vectors are used
instead of 6.
(Jilong et al., 2008) proposed an improved Kalman filter in order to sensor-less estimation of
the rotor's initial position in DTC. For this reason, they employed the high frequency signal
injection method. Since the high frequency signal is weak, it would not help the rotor's
motion. Therefore, rotor speed is assumed to be zero. The computation intensity is very high
due to the fact that the voltage and current quantities are obtained through differential
inequalities depending on the speed.
(Guo et al., 2009) applied the space-vector modulation in a matrix converter for DTC
application in naval vehicles. In this study, the signals for the matrix converter is generated
by a DSP where the dual space modulation method was used. However, it is seen that the
current drawn by the matrix converter has very high total harmonic distortion.
(Sanchez et al., 2009) achieved the direct torque control without using a speed sensor but
using only current and voltage senor in order to determine the stator voltage vectors. In
their results that they used a closed loop controller, they indicated that the calculated speed
data oscillates too much.
(Siahbalaee et al., 2009) studied the copper losses by flux optimization for their direct torque
controlled PMSP in order to reduce the torque and flux oscillations.
(Liu et al., 2009) tried to use the predictive control method in direct torque control. There are
limited number of studies in the literature using predictive control in direct torque control.
Experimental results were obtained by using a DSP. However, in the experimental results,
they implemented complicated trigonometric functions. In the experimental results, much
more flux drop can be observed as compared to their simulation results. They could apply
the proposed application by reducing the flux reference.
(Inoue et al., 2010) linearized the torque control system of a direct torque controlled buried
magnet synchronous motor and they acquired the torque response depending on a constant
gain coefficient. They calculated the PI coefficients that are updated according to the
estimated torque values by a new gain scheduling method.
(Geyer et al., 2010) achieved the direct torque control of the PMSM by implementing a
model predictive control algorithm that reduces the switching frequency and hence the
switching losses. The proposed algorithm could reduce the switching losses by 50% and the
THD by 25%.
2. Permanent magnet synchronous motor technology
Permanent magnet synchronous motors are different from the wound field synchronous
motors. However the stator structure of a permanent magnet synchronous motor is similar
134
Torque Control
to the wound field synchronous motors, the difference is only between the rotor structures.
In the wound field synchronous motors, field is created on the rotor by separate excitation
through the brushes (slip rings), where the field of the permanent magnet synchronous
motors is created by the permanent magnets placed on the rotor. Therefore, permanent
magnet synchronous motors are brushless motors. Since they are brushless, they are more
robust than the DC motors; and since the field is created by permanent magnets and there
are not any rotor currents, they are more efficient than the induction motors, where the rotor
field currents cause rotor copper losses. However costs of PMSM are higher than DC and
induction motors because of the high permanent magnet and production costs. Also, the
reliability of these motors is questionable under certain circumstances such as magnetic
property loss due to high working temperatures etc. Nevertheless, such properties as high
efficiency, high torque, high power, small volume, and accurate speed control make
permanent magnet synchronous motors preferred for chemical fiber industry (spinning
pumps, godets, drive rollers), texturing plants (draw godets), rolling mills (roller table
motors), transport systems (conveyor belts), glass industry (transport belts), paper
machines, robotic automation, electrical household appliances, ship engines and escalators.
Permanent magnet synchronous motors are classified mainly into two groups with respect
to their rotor structures as; Surface Mount Permanent Magnet (SMPM) Synchronous Motors
and Interior Permanent Magnet (IPM) Synchronous Motors. SMPM motors have the
permanent magnets mounted on the outer surface the rotor, and IPM motors have the
permanent magnets buried in the rotor core. SMPM motors are also classified into two types
with respect to the stator winding as; concentrated winding and distributed winding.
Concentrated winding SMPM motor’s back-emf waveform is trapezoidal; distributed
winding SMPM motor’s back-emf waveform is sinusoidal. Concentrated winding SMPM
motors are called as Brushless DC (BLDC) motors and driven with trapezoidal signals. The
distributed winding SMPM motors are called as Permanent Magnet AC (PMAC) motors and
driven with sinusoidal signals. PMAC motors are also designated as servo motors or
brushless AC motors. PMAC motors are generally built with strong magnetic material
Samarium Cobalt (SmCo5, Sm2Co17) and Neodymium Iron Boron (NdFeB). They have high
dynamic performance, high efficiency, robustness, high torque density and significantly
better short-time overload capability than induction motors (400% to 150%). PMAC motors
are mostly employed in high performance servo (robotics, machining, etc.) applications.
Some PMAC motors are built with low cost permanent magnet materials (Ferrites) to be
used in low cost (fan) applications. IPM motors are newly developed motors with high
torque density, high efficiency characteristics as the SMPM motors and additionally provide
field weakening operation, which is impossible with the SMPM motors. IPM motors are
preferred in the industrial applications such as adjustable speed drives as a replacement for
the squirrel cage induction motors, to improve the efficiency and the performance. In
contrast to the induction motors, IPM motors also have the advantage of providing position
control loop with accuracy, without a shaft encoder (Omer & Hava, 2010). Standard
induction motors, designed to run at 750-3000 rpm, are not particularly well suited for low
speed operation. Normally gearboxes are used to reduce the speed from, for example, 1500
rpm to 600 rpm. A gearless PMSM drive of 600 rpm replaces as a solution, in contrast to
conventional using of 1500 rpm induction motor. Because a gearbox takes up space and
needs maintenance as well as considerable quantities of oil. Eliminating the gearbox saves
space, installation costs and improved efficiency of the drive.
Direct Torque Control of Permanent Magnet Synchronous Motors
135
2.1 Permanent magnet synchronous motor design types
The location of the magnets on the rotor and their specifications determine the performance
of the motor therefore various designs are possible. The simple representations of the
frequently used designs are given below. Other designs are derived from these two.
1. Placing the magnets on the rotor surface (SMPM)
2. Placing the magnets inside the rotor (IPM)
2.1.1 Placing the magnets on the rotor surface (Surface mounted magnet rotor)
Magnets are mounted on the rotor in forms of strips or arcs. The rotor configuration shown
in Fig.1 adopts surface-mounted magnets which are often glued onto rotor surface. Since the
relative permeability of the magnets is almost the same as for air, such design possesses very
small rotor saliency and these are sometimes referred to as “nonsalient” designs. By filling
the gaps between the magnets partially with iron, a significant rotor saliency can be
achieved which offers the possibility to utilize the reluctance torque. The greatest drawback
of this common design is the low endurance of the magnets to the centrifugal forces.
Therefore these motors are preferred in low-speed applications to avoid detachment of the
magnets. These motors are commonly known as surface permanent magnet motors
(SMPM). A simple representation is shown in Fig.1.
Fig. 1. Magnets placed on the rotor surface
2.1.2 Placing the magnets inside the rotor
Air gap induction of the previous design is limited and the magnets being exposed to high
centrifugal forced under high speeds have lead to different designs. Here, the magnets are
placed in the cavities bored in the rotor. The magnets are surrounded by magnetic materials
instead of air.
The magnets now have a better resistance to centrifugal forces therefore they are more
suitable for high speed applications. The efficiency values of these motors are also higher
than other magnet motors. The main disadvantage is their high costs. Placement of the
magnets in the rotor is a high tech process that requires fine labour. These motors are
commonly known as interior permanent magnet synchronous motor (IPMSM). These
magnet motor designs are mainly in two types.
2.1.2.1 Radially placed interior magnet structure
As seen in Fig.2 magnets are placed around the rotor axis buried and radially magnetized.
These motors have small air gap, and low armature reaction. Flux density in the air gap can
136
Torque Control
be higher than inside the magnet, thus the low-cost Ferrit magnets can be utilized for high
torque density. The surface where the magnets come in contact with the rotor is coated with
a non magnetic material to avoid magnetic short-circuit. However these materials have high
costs.
Fig. 2. Radially placed interior magnet structure
2.1.2.2 Symmetrically buried magnet structure
Permanent magnets are again buried in the rotor but are placed pointing the main axis. The
most important feature of this design is to constitute induction at the poles independent
from the working point of magnets. Through to this feature, air gap induction can be
increased to high levels. Since the magnets are buried in the rotor, they have a great
resistance to the centrifugal forces. Symmetrically buried magnet structure is shown in
Fig.3.
Fig. 3. Symmetrically buried magnet structure
When buried magnets are used, L sd ≠ L sq and the electromagnetic torque also contains a
reluctance torque. The fact that three-phase symmetrical sinusoidal quantities are
transformed into two DC components through the well known Park transformation has
made modelling of PMSM in the rotor-fixed dq reference frame used almost exclusively for
control purposes.
137
Direct Torque Control of Permanent Magnet Synchronous Motors
2.2 Model of PMSM on the rotor reference frame and motor equation
For high dynamic performance, the current control is applied on rotor flux (dq) reference
system that is rotated at the synchronous speed. Stator magnetic flux vector ψ s and rotor
magnetic flux vector ψ M can be represented on rotor flux (dq), stator flux (xy) reference
system as shown in Fig.4. The angle between the stator and rotor magnetic flux (δ), is the load
angle that is constant for a certain load torque. In that case, both stator and rotor fluxes rotate
at synchronous speed. However under different loads, δ angle varies. Here, by controlling the
stator current variation or the δ angle variation, the increase of the torque can be controlled.
β
y
q
α
isq
is
isy
θr
x
isx
δ
ψs
ψM
isd
d
Fig. 4. Stator and rotor magnetic fluxes in different reference systems
The mathematical equations are given below (Vas, 1998). Stator current vector can be
represented on rotor flux (dq) reference system as (isd) (isq) and the electromagnetic torque is
related with these vectors.
ψ sd = L sdi sd + ψ M
(1)
ψ sq = L sq i sq
(2)
u sd = R si sd +
d
ψ sd − ωr ψ sq
dt
(3)
u sq = R si sq +
d
ψ sq + ωr ψ sd
dt
(4)
L sq
R
d
1
i sd =
u sd − s i sd +
ωr i sq
dt
L sd
L sd
L sd
(5)
R
L
ψ ω
d
1
i sq =
u sq − s i sq + sd ωr i sd − M r
dt
L sq
L sq
L sq
L sq
(6)
(
Te = 3 p ψ sd i sq − ψ sq i sd
2
(
)
)
Te = 3 p ⎡ψ M i sq − L sq − L sd i sd i sq ⎤
⎦
2 ⎣
(7)
(8)
138
Torque Control
fb
fβ
fq
θr
fa
fα
fc
fd
Fig. 5. Rotating reference frames
To simplify the modelling of the DTC drive, it is common practice to transform all variables
from the three-phase system (abc) to an orthogonal (dq) reference frame with a direct (d)
and quadrature (q) axis, where f represents the voltage, current or magnetic flux and θr is
the rotor angle. abc → αβ , Clarke Transformation (9) and αβ → dq , Park Transformation
(10) can be applied regarding reference frame theory. Here, equation (11) presents these
both transformations in one matrix, abc → dq .
⎡a⎤
⎡α ⎤ 2 ⎡ 1 cos ( 2 π / 3 ) cos ( 4 π / 3 ) ⎤ ⎢ ⎥
⎢ β ⎥ = ⎢0 sin 2 π / 3 sin 4 π / 3 ⎥ ⎢ b ⎥
(
)
(
)⎦ ⎢ ⎥
⎣ ⎦ 3⎣
⎣c⎦
(9)
⎡α ⎤ ⎡ cos ( θr ) − sin ( θr ) ⎤ ⎡d ⎤
⎥⎢ ⎥
⎢ β ⎥ = ⎢ sin θ
⎣ ⎦ ⎣ ( r ) cos ( θ r ) ⎦ ⎣ q ⎦
(10)
⎡a ⎤
cos ( θ r -4π/3 ) ⎤ ⎢ ⎥
⎡d ⎤ 2 ⎡ cos ( θ r ) cos ( θ r -2π/3 )
b
⎢ q ⎥ = ⎢ -sin θ
( r ) − sin ( θr -2π/3 ) − sin ( θr -4π/3 ) ⎥⎦ ⎢⎢ ⎥⎥
⎣ ⎦ 3⎣
⎣c⎦
(11)
In Eq. (12), by using the vector representation shown in Fig.5, Eq. (13) is obtained and using
the Park transformation, Eq. (14) is obtained.
⎡fd ⎤ ⎡cos δ − sin δ⎤ ⎡fx ⎤
⎢⎣fq ⎥⎦ = ⎣⎢sin δ cos δ ⎦⎥ ⎢⎣fy ⎥⎦
sin δ =
cos δ =
(12)
ψ sq
ψs
ψ sd
ψs
(13)
139
Direct Torque Control of Permanent Magnet Synchronous Motors
(
)
(
)
3
Te = p ⎡ ψ sd i sxsinδ+i sy cosδ -ψ sq i sx cosδ-i sy sinδ ⎤
⎦
2 ⎣
Te =
2 ⎤
ψ sdψ sq
ψ sq
3 ⎡ ψ sd ψ sq
ψ2
p ⎢i sx
+ i sy sd − i sx
+ i sy
⎥
2 ⎣⎢
ψs
ψs
ψs
ψ s ⎦⎥
Te =
3
p ψ s i sy
2
(14)
It is clear that the electromagnetic torque is directly proportional to the y-axis component of
the stator current (Zhong et al, 1997). Dependency on less number of parameters is the main
advantage of the stator current control. It is possible to say that in a practical application, the
estimation technique shown in equation (8) requires knowledge of inductances. The
estimated instantaneous electric torque is easily compared with a reference value to achieve
a fast torque control. At the same time, the stator flux linkage is compared with the reference
value to ensure sufficient magnetization of the motor. The torque of the PMSM is controlled
by monitoring and controlling the armature current since electromagnetic torque is
proportional to the current.
3. Direct torque control of permanent magnet synchronous motors
In general, there are two control methods used for the PMSM; field oriented control and
direct torque control. The AC drives in which flux oriented control (FOC) is used field
control leads to flux control. Here, rotor flux space vector is calculated and controlled by
using the angular velocity which is derived from the speed feedback and the stator current
vector. The greatest drawback of the flux vector control is the need for a tachogenerator or
an encoder for high accuracy. This need definitely increases the costs of the system.
The basic principle of DTC is to directly select the stator voltage vectors according to the
errors between the reference and actual values of the torque and stator flux. Torque and flux
are resolved and directly controlled using nonlinear transformations on hysteresis
controllers, without performing coordinate transformations. A double layer hysteresis band
controller is utilized for stator flux control and a three-layer hysteresis band controller is
used for torque control. DTC is an alternative to field oriented control method in high
performance applications due to the advantages of reduced computations (Swierczynski et
al, 2008) Since the torque and flux estimators in DTC requires and relies on the parameters
identification and accuracy of the estimations, the estimation of the electromagnetic torque
is essential for the entire system performance.
In classical PWM and flux vector controlled drives, voltage and frequency are used as basic
control variables and that are modulated and then applied to the motor. This modulator
layer needs an additional signal processing time and restricts the torque and speed
response. The key notion behind DTC is to directly steer the stator flux vector by applying
the appropriate voltage vector to the stator windings. This is done by using a pre-designed
switching table to directly update the inverter’s discrete switch positions whenever the
variables to be controlled, the electromagnetic torque and the stator flux, exceed the
hysteresis bounds around their references. The switching table is derived on the basis of the
desired performance specifications on the controlled variables also include the balancing of
the inverter’s neutral point potential around zero.
140
Torque Control
3.1 Two - level inverter nonlinear continuous-time model
An equivalent representation of a three-phase two-level inverter driving a PMSM is shown
in Fig.6. At each phase, the inverter can produce two different voltages − udc 2 , udc 2
where udc denotes the voltage of the dc-link. The switch positions of the inverter can
therefore be fully described using the three integer variables v a , v b , v c ∈ {-1, 1} where each
variable corresponds to one phase of the inverter, and the values -1, 1 correspond to the
phase potentials − udc 2 , udc 2 respectively.
There are 2 3 = 8 different vectors of the form vabc = [ va , v b , v c ]
T
Using Eq. (11) these vectors can be transformed into the dq frame resulting in vectors of the
T
form vdq = ⎡⎣ vd , vq ⎤⎦ as shown in Fig.7 where they are mapped into the two-dimensional dq
plane. Even though they are commonly referred to as voltage vectors, this term describes the
switch positions rather than the actual voltages applied to the machine terminals. The
voltage vectors can be divided in two groups: six long vectors forming the outer hexagon
and two zero vectors. The zero vectors correspond to the switch combinations (+1,+1,+1)
and (-1,-1,-1), and short-circuit the machine terminals.
+
udc
2
1
sa
va
1
sb
0
−
1
sc
vb
0
vc
0
udc
2
A
C
PMSM
B
Fig. 6. Three-phase two-level inverter
3.2 Determination of the voltage space vector
The main principle of DTC is determination of correct voltage vectors using the appropriate
switching table. The determination process is based on the torque and stator magnetic flux
hysteresis control. Stator magnetic flux can be calculated using equation (15).
ψs =
t+Δt
∫ ( us − R s is ) dt
(15)
t
Eq. (15) shows that the stator magnetic flux and the voltage space vector are in the same
direction. Therefore, amplitude and direction control of the stator magnetic flux is possible
by selecting the suitable voltage space vectors. Voltage vector plane is divided into six parts
as shown in Fig.7. Two adjacent vectors that yield the lowest switching frequency are
selected in order to increase or decrease the amplitude respectively.
141
Direct Torque Control of Permanent Magnet Synchronous Motors
v4
v4
v5
v4
θ3
v3
v5
v6 θ v
4 4
v5
v6
v3
θ2
v3
v2
v2
v1 θ1
v0 v7
v5
θ5
v6
θ6
v6
v1
v2
v2
v1
v3
v1
Fig. 7. Vectors of space vector modulation
Here, when the stator magnetic flux is moved clockwise in section 1, voltage space vector v2
is selected in order to increase the stator magnetic flux amplitude and voltage space vector
v3 is selected in order to decrease the amplitude. When the stator magnetic flux moves
clockwise, if still in section 1, v6 is used to increase the amplitude and v5 is used to decrease
the amplitude. The torque of the permanent magnet synchronous motor can be controlled
using DTC by means of controlling the stator magnetic flux rotation speed in cases where
the stator magnetic flux amplitude is kept constant (Ozcira et al, 2008). Since the magnets on
the rotor are continuously rotating, stator magnetic flux does not change when v0 and v7
zero vectors are used (Rahman et al, 1998). Zero vectors can be used to estimate the initial
position of the rotor by a fixed active voltage vector while limiting current by applying zero
vectors. In this position data is not used; therefore, zero vectors are not used within DTC for
PMSM. Table 1. shows the suggested switching sequences. For these different states, the
hysteresis controllers can be used as flux and torque hysteresis controllers.
ψ
T
1
1
0
-1
1
0
0
-1
θ
θ(1)
V2
(110)
V7
(111)
V6
(101)
V3
(010)
V0
(000)
V5
(001)
θ(2)
V3
(100)
V0
(000)
V1
(001)
V4
(110)
V7
(111)
V6
(011)
θ(3)
V4
(101)
V7
(111)
V2
(011)
V5
(100)
V0
(000)
V1
(010)
θ(4)
V5
(001)
V0
(000)
V3
(010)
V6
(101)
V7
(111)
V2
(110)
θ(5)
V6
(011)
V7
(111)
V4
(110)
V1
(001)
V0
(000)
V3
(100)
θ(6)
V1
(010)
V0
(000)
V5
(100)
V2
(011)
V7
(111)
V4
(101)
Table 1. Switching vectors
In Table 1. ψ denotes stator magnetic flux hysteresis controller output, Τ denotes torque
hysteresis controller output, and θ represents magnetic flux sector. These vectors are
142
Torque Control
selected in order to provide the stator flux error within 2 Δψ s bandwidth and the actual
torque error in the 2 Δt e bandwidth at each switching period. The flux hysteresis controller
output is dψ s . If an increase is needed for flux, it is assumed that dψ s = 1 and when a
decrease is needed, it is assumed that dψ s = 0 . Two level hysteresis controller is determined
by Eq. (16).
⎧⎪1,
dψ s = ⎨
⎪⎩0,
ψ s ≤ ψ sref − Δψ s
ψ s ≥ ψ sref + Δψ s
(16)
The torque hysteresis controller output is dt e . If an increase is needed at torque, it is
assumed that dt e = 1 and when a decrease is needed, it is assumed that dt e = −1 . If there is
no change at actual torque, dt e = 0 . Three level hysteresis controller is determined by (17)
for clockwise rotation and counter clockwise rotation by (18).
⎪⎧1,
dt e = ⎨
⎪⎩0,
t e ≤ t eref − Δt e
t eref ≥ t eref
⎪⎧−1,
dt e = ⎨
⎪⎩0,
t e ≤ t eref + Δt e
t eref ≥ t eref
(17)
(18)
In direct torque method, three level torque comparator is used to select whether the inverter
output voltage vector should be a torque-increasing vector or a torque-reducing vector. The
appropriate vector is then applied for the duration of the sampling period. At low speed the
torque increasing vectors are very effective at increasing the torque, whereas the torque
reducing vectors are less effective. In contrast, at high rotor speeds, the torque-increasing
vectors are less effective, whereas the torque reducing vectors are more effective. The result
of this is that, at low speed, the torque tends to make a considerable excursion above the
maximum torque hysteresis limit.
3.3 Determination of the stator flux space vector’s sector
Stator flux space vector’s sector (θ) should be known in order to select the appropriate
switching vector. Space vector’s angle determines the sector. Equation (19) helps to find the
sector. This equation can be used when the switching signals will be generated.
θ = arctan
ψ sd
ψ sq
(19)
In practice this mathematical operation is too complex for a DSP. Therefore, another
alternative way can be used to determine θ. Equation (20) is calculated and due to result of
this operation a Table 2 is constituted.
3 ψ sq − ψ sd
(20)
Direct Torque Control of Permanent Magnet Synchronous Motors
ψsd
ψsq
3 ψ sq − ψ sd
Sector(θ)
+
+
+
+/+
+
+/-
+
+
+
+
1.
2.
3.
4.
5.
6.
143
Table 2. Stator flux space vector’s sector
3.4 Stator flux control by using LP filter
In order to eliminate the errors of a pure integration in voltage model, a low pass (LP) filter
should be used. Equation (15) defines stator magnetic flux in general form.
(
)
ψ s = Vs − R s Is /jωe
(21)
In equation (21) this general form is given in sinusoidal form where ωe denotes stator flux
angular frequency.
(
)
ψ s′ = Vs − R s Is /( jωe + ωc )
(22)
ωc denotes the low pass filter’s cut off frequency. In equation (22) ψ′s represents estimated
stator flux but estimated flux isn’t equal to stator flux ψ s which used in equation (21). By
using equation (21) and equation (22) stator flux can be obtained.
ψ s =ψ s′ − j ( ωc / ωe ) ψ s′
(23)
For this equation in the case of ωe ωc LP filter converges to the pure integrator. However
maintain the clarity equation (24) can be derived.
ψ s′
ωe
∠ϕ=
∠ϕ
ψs
ωe 2 + ωc 2
(24)
π
− arctan ( ωe / ωc )
2
(25)
ϕ=
By choosing a low cut off frequency, amplitude and phase errors decrease; however, as LP
filter efficiency decreases, DC shift in current and voltage cannot be eliminated.
3.5 Observerless scheme for sensorless speed control based on DTC
The study includes low pass filter - flux estimator which utilizes voltage model on a direct
torque controlled permanent magnet synchronous motor. Flux estimation techniques used
for the high performance motor drivers are based on using the flux model, voltage model or
both of these models together. Flux estimation with current model is used at low frequencies
144
Torque Control
and requires the stator current and the rotor speed data. The drawbacks of this method are
its susceptibility to the changes of the rotor parameters at high speeds and its need of a
speed sensor. For sensorless flux estimation, the voltage model is preferred. This model has
a very high accuracy at high speeds. However, at low speeds due to the very low stator
voltage, decrease of the ohmic voltage and component voltage diminish the error of the
integration process increases. Accuracy of the flux estimation depends on the accuracy of
the measured current, voltage and real parameters. In practice, the structure of the flux
sensors and the error due to noise or a small DC shift of the transitional region, results in
error accumulation at the integrator. In order to avoid that effect, an integrator with a LP
filter is used instead of a pure integrator. DTC performance is highly sensible to determined
voltage vector and is highly dependant to stator flux space vector being accurately
estimated.
Current model isn’t affected by the variations of the stator resistance however is sensible to
changes of the rotor parameters at high speeds. Estimation process of the voltage model
does not require rotor speed data. Therefore voltage model is preferred in sensorless
applications and operation at moderate or high speeds. In voltage model, errors of current
and voltage measurements affect the accuracy of the integral function. The errors occur
because of the phase delay of the sensors, the errors of the transformer gain, shift of the
measurement system and fault of the quantal errors of the digital system. Variation of the
stator resistance with temperature also causes errors.
In order to avoid the shift of a pure integration in voltage model, a low pass (LP) filter
integrator is used. LP filter eliminates the shifting but causes phase and amplitude errors.
Therefore, the driver performance decreases especially at frequencies close to the cut-off
frequency of the filter. Further study is available to improve the flux estimation using LP
filter. The method utilizes an adaptive control system that depends on the force and the
stator flux being orthogonal. Using an adaptive control complicates the simple DTC system.
In this study a low pass filter which utilizes voltage model, is presented. The effects of the
LP filter to the flux estimation performance and the practical aspects are analyzed in detail.
Simulation results show that a low pass filter which utilizes voltage model on a direct
torque controlled PMSM driver, can achieve a robust control.
The torque of the permanent magnet synchronous motor is controlled by monitoring and
controlling the armature current since electromagnetic torque is proportional to the current.
In DTC, torque and flux can be controlled independently since the stator flux is controlled
directly with the stator voltage using Eqs. (26) and (27). Stator flux can be quickly controlled
and the motor performance can be kept high. Moreover, torque is estimated by using
current data provided by Eq. (28).
t +Δt
∫ (uβ − R siβ ) dt
(26)
∫ ⎡⎣(uα − R si α ) + ψ M ⎤⎦ dt
(27)
ψβ =
t
ψα =
t +Δt
t
(
Test = p ψ α iβ − ψβi α
)
(28)
Direct Torque Control of Permanent Magnet Synchronous Motors
145
Here, current and voltage data on the α-β axis of the PMSM are the input variables. Position
and speed data resulted from the estimation process are conveyed to the proper evaluations
within the DTC control system. Estimation process of the voltage model is independent
from the rotor speed data.
ωest =
ψα
dψβ
dψ α
− ψβ
dt
dt
ψ α2 + ψβ2
(29)
In this speed sensorless algorithm, only stator phase currents and inverter output voltages
are measured. Torque and speed are estimated in an open-loop without using an observer,
as given in the Eqs. (28), (29). A low-pass filter is accompanied to the results obtained from
(29) in order to monitor the speed estimation. On the other hand, this method gives only the
synchronous speed, not the mechanical speed for induction motor due to the slip speed.
This implementation is applicable for general purpose PMSM drives.
Voltage vector for the DTC is determined by comparison of the estimated stator flux and
torque values with their reference values. For high performance driver systems of
permanent magnet synchronous motors, it is basically desired that the drive system should
not be affected by parameter variations. The other criterion is that the speed must reach the
reference value as quickly as possible when speed reference changes. Traditional PI and PID
controllers are widely used as speed controllers in driver systems. Here, the speed controller
consists of both a PI controller and an acceleration compensator.
Observerless scheme for sensorless DTC method is highly sensible to determined voltage
vector and is highly dependent to the stator flux space vector. Therefore, stator flux space
vector should be accurately estimated. The measured input values to the DTC control are
only motor current and voltage. The voltage is defined from the DC-bus voltage and
inverter switch positions. Since the inverter operates at high switching frequencies,
undesired harmonic components can be produced.
4. Low-Pass Filter for Harmonics
It is desirable that the voltages and currents provided to the motor terminals do not include
harmonic components. These harmonics may cause many unwanted effects such as
electromagnetic interference (EMI) noise and high dv/dt which affect the motor control
system as well (Chen et al, 2007). In such hysteresis controlled inverters, when the error per
sampling time is large, the voltage vectors can no longer remain within the small hysteresis
band. The LP filter can be used to affect the inverter switching frequency in order to
decrease the stress on the inverter, so that PMSM is not affected by the variations and noises
in entire system. Fig.8 shows the RLC filter, which plays the main role in reducing the high
dv/dt of line to line voltages at motor terminals.
The transfer function of the filter is given by
Vo
R 1C 1s + 1
=
Vi C 1L 1s2 + (r1 + R 1 )C 1s + 1
To obtain over damping response, the filter resistances is selected to be,
(30)
146
Torque Control
Vi
r1
L1
Vo
C1
R1
Fig. 8. RLC Low pass filter for harmonics
R 1 + r1 >
4L 1
C1
(31)
L 1C 1
(32)
Cut off frequency ωc is given by Eq. (32).
ωc = 1
Here, the cut off frequency is approximately calculated to be 300 Hz and LP filter
parameters are given in Table 3.
L1
C1
R1
r1
22 mH
500 μF
200 Ω
0.2 Ω
Table 3. Filter parameters
By using (30) and the filter parameters, (33) is obtained. Here, the LP block is constructed
with (33). In order to reduce ohmic losses, the series resistance r1 is normally of small value,
as shunt resistance R1 is selected high enough to limit the currents drawn by the filter
(Ozcira et al, 2008).
Vo
1
=
Vi 11.10 −5 s + 1
(33)
The filter system is characterized by affecting inverter switching frequency in such a way to
decrease switching stresses. The filter system uses dissipative elements to reshape motor
voltage waveform to provide voltage to the motor windings.
5. Model verification of direct torque control algorithm
Direct torque control is used as the control method in simulations thus the operating
principle of the DTC is described briefly. In principle the DTC is a hysteresis control - of the
stator flux linkage and the torque that directly selects one of the six non-zero and two zero
discrete voltage vectors of the inverter. The principal operation of the DTC is shown as a
block diagram in Fig.9. The system includes a flux estimator which utilizes a voltage model
on a direct torque controlled PMSM, and speed can also be estimated by using calculations
with aid of the current and voltage data. The voltage and current signals are inputs to an
Direct Torque Control of Permanent Magnet Synchronous Motors
147
accurate motor model which produces the exact value of the stator flux and torque. Motor
torque and flux comparators compare the actual values to the reference values that are
produced by torque and flux reference controllers. The outputs from these controllers are
updated every 10μs and they indicate whether the torque or flux has to be varied.
Depending on outputs, the switching logic directly determines the optimum inverter
switching signals (Bekiroglu & Ozcira, 2010). DTC method was applied and both flux and
torque were controlled by hysteresis controllers; thus, the delays related to the PWM
modulator were eliminated. Besides, a low-pass filter was implemented to reduce the
harmonics and noise in the PMSM.
Fig. 9. Block diagram of the direct torque control (Luukko, 2000)
The control logic is implemented by a MATLAB® Function block that includes the software
developed for this study. This function block determines the optimal switching vectors
according to their sectors. The inverter is operated based on the different IGBT states given
in Table 1 and Table 2. Since the inverter operates at high switching frequencies, undesired
harmonic components can be produced. Simulink® model of low voltage – high power
surface mounted PMSM is developed according to the dq model. The inverter DC bus
voltage is 510V. During t = [ 0, 0.4 ] interval, ω = 13 rad/s and at t = 0.4s ω = −13 rad/s is
applied as the reference speed. According to observe adaptation of the actual speed at
t = 0.2s 60 Nm is applied to load torque value.
Motor parameters are; Pn = 18kW , I n = 50A , R s = 0.43Ω , L d = 25mH , L q = 25mH ,
ψ M = 1.58Wb , J = 2.16kgm 2 , Bm = 0.005538 , p = 10
148
Torque Control
Fig.10 shows the speed control Simulink® diagram of the direct torque control for
permanent magnet synchronous motor.
Fig. 10. Closed loop speed control system Simulink diagram of the PMSM with DTC.
6. Simulation results
The system dynamic responses are shown below with a sampling time 10μs.
Fig. 11. Torque dynamic response
Direct Torque Control of Permanent Magnet Synchronous Motors
Fig. 12. Torque dynamic response
Fig. 13. Current harmonics without LP filter
Fig. 14. Current harmonics without LP filter
149
150
Torque Control
Fig. 15. Stator magnetic flux vector trajectory
7. Conclusion
DTC is intended for an efficient control of the torque and flux without changing the motor
parameters and load. Also the flux and torque can be directly controlled with the inverter
voltage vector in DTC. Two independent hysteresis controllers are used in order to satisfy
the limits of the flux and torque. These are the stator flux and torque controllers. DTC
process of the permanent magnet synchronous motor is explained and a simulation is
constituted. It is concluded that DTC can be applied for the permanent magnet synchronous
motor and is reliable in a wide speed range. Especially in applications where high dynamic
performance is demanded DTC has a great advantage over other control methods due to its
property of fast torque response. In order to increase the performance, control period should
be selected as short as possible. When the sampling interval is selected smaller, it is possible
to keep the bandwidth smaller and to control the stator magnetic flux more accurately. Also
it is important for the sensitivity to keep the DC voltage in certain limits.
As an improvement approach, a LP filter can be added to the simulation in order to
eliminate the harmonics. In simulation, certain stator flux and torque references are
compared to the values calculated in the driver and errors are sent to the hysteresis
comparators. The outputs of the flux and torque comparators are used in order to determine
the appropriate voltage vector and stator flux space vector.
When results with and without filters are compared, improvement with the filters is
remarkable, which will effect the voltage in a positive manner. Choosing cut off frequency
close to operational frequency decreases DC shift in the stator voltage. However, this leads
to phase and amplitude errors. Phase error in voltage leads to loss of control. Amplitude
error, on the other hand, causes voltage and torque to have higher values than the reference
values and field weakening can not be obtained due to voltage saturation. Hence, cutoff
frequency of LP filter must be chosen in accordance to operational frequency.
Direct Torque Control of Permanent Magnet Synchronous Motors
151
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7
Torque Control of PMSM and
Associated Harmonic Ripples
1Fatih
Ali Ahmed Adam1, and Kayhan Gulez2
University, Engineering Faculty, Electrical-Electronics Eng. Dept., 34500
Buyukcekmece-Istanbul,
2Yildiz Technical University, Electrical-Electronics Eng. Faculty, Control and Automation
Engineering Dept., 34349 Besiktas- Istanbul,
Turkey
1. Introduction
Vector control techniques have made possible the application of PMSM motors for high
performance applications where traditionally only dc drives were applied. The vector
control scheme enables the control of the PMSM in the same way as a separately excited DC
motor operated with a current-regulated armature supply where then the torque is
proportional to the product of armature current and the excitation flux. Similarly, torque
control of the PMSM is achieved by controlling the torque current component and flux
current component independently.
Torque Control uses PMSM model to predict the voltage required to achieve a desired output
torque or speed. So by using only current and voltage measurements (and rotor position in
sensor controled machine), it is possible to estimate the instantaneous rotor or stator flux and
output torque demanded values within a fixed sampling time. The calculated voltage is then
evaluated to produce switching set to drive the inverter supplying the motor. PMSM torque
control has traditionally been achieved using Field Oriented Control (FOC). This involves the
transformation of the stator currents into a synchronously rotating d-q reference frame that is
typically aligned to the rotor flux. In the d-q reference frame, the torque and flux producing
components of the stator current can separately be controlled. Typically a PI controller is
normally used to regulate the output voltage to achieve the required torque.
Direct Torque Control (DTC), which was initially proposed for induction machines in the
middle of 1980’s (Depenbrock, 1984 and 1988; Takahashi, 1986), was applied to PMSM in the
late 1990's (French, 1996; Zhong, 1997). In the Direct Torque Control of the PMSM, the
control of torque is exercised through control of the amplitude and angular position of the
stator flux vector relative to the rotor flux vector. Many methods have been proposed for
direct torque control of PMSM among which Hysteresis based direct torque control (HDTC)
and Space Vector Modulation direct torque control (SVMDTC).
In 2009 Adam and Gulez, introduced new DTC algortim for IPMSM to improve the
performance of hysteresis direct torque control. The algorithm uses the output of two
hysteresis controllers used in the traditional HDTC to determine two adjacent active vectors.
The algorithm also uses the magnitude of the torque error and the stator flux linkage
position to select the switching time required for the two selected vectors. The selection of
156
Torque Control
the switching time utilizes suggested table structure which, reduce the complexity of
calculation. The simulation and experimental results of the proposed algorithm show
adequate dynamic torque performance and considerable torque ripple reduction as well as
lower flux ripple, lower harmonic current and lower EMI noise reduction as compared to
HDTC. Only two hysteresis controllers, current sensors and built-in counters
microcontroller are required to achieve torque control.
Torque ripple and harmonic noise in PMSM are due to many factors such as structural
imperfectness associated with motor design, harmonics in control system associated with
measurement noises, switching harmonics and harmonic voltages supplied by the power
inverter which constitute the major source of unavoidable harmonics in PMSM. These
harmonics cause many undesired phenomena such as electromagnetic interference “EMI”
and torque ripples with consequences of speed oscillations, mechanical vibration and
acoustic noise which, deteriorate the performance of the drive in demanding applications
(Holtz and Springob 1996). These drawbacks are especially high when the sampling period
is greater than 40μs (Zhong, et al. 1997).
Recently many research efforts have been carried out to reduce the torque ripples and
harmonics in PMSM due to inverter switching with different degree of success. Yilmaz
(Yilmaz, et al. 2000) presented an inverter output passive filter topology for PWM motor
drives to reduce harmonics of PMSM, the scheme shows some effectiveness in reducing
switching harmonics, but however, very large circulating current between inverter output
and filter elements is required to reshape the motor terminal voltage which violate current
limitation of the inverter. Many researchers (Hideaki et al, 2000; Darwin et al., 2003; Dirk et
al , 2001) have addressed active filter design to reduce or compensate harmonics in supply
side by injecting harmonics into the line current which have no effect on the current
supplying the load. Satomi (Satomi, et al. 2001) and Jeong-seong (Jeong-seong, et al. 2002)
have proposed a suppression control method to suppress the harmonic contents in the d-q
control signals by repetitive control and Fourier transform but, however, their work have
nothing to do with switching harmonics and voltage harmonics provided by the PWM
inverter supplying the motor. Se- Kyo, et al. (1998), Dariusz et al. (2002), and Tang et, al.
(2004) have used space vector modulation to reduce torque ripples with good results;
however, their control algorithm depends on sophisticated mathematical calculations and
two PI controllers to estimate the required reference voltage and to estimate the switching
times of the selected vectors. Holtz and Springob (1996, 1998) presented a concept for the
compensation of torque ripple by a self- commissioning and adaptive control system.
In this chapter, two different methods to improve torque ripple reduction and harmonic
noises in PMSM will be presented. The first method is based on passive filter topology
(Gulez et al., 2007). It comprises the effects of reducing high frequency harmonic noises as
well as attenuating low and average frequencies. The second method is based on active
series filter topology cascaded with two LC filters (Gulez et al., 2008).
Modern PMSM control algorthims
2. Algorithm 1: Rotor Field Oriented Control “FOC”
The control method of the rotor field-oriented PMSM is achieved by fixing the excitation
flux to the direct axis of the rotor and thus, it is position can be obtained from the rotor shaft
by measuring the rotor angle θr and/or the rotor speed ωr.
Consider the PMSM equations in rotor reference frame are given as:
157
Torque Control of PMSM and Associated Harmonic Ripples
⎡ vsd ⎤ ⎡ R + pLsd −Pωr Lsq ⎤ ⎡isd ⎤ ⎡ 0 ⎤
⎥⎢ ⎥ + ⎢
⎢v ⎥ = ⎢
⎥
⎢⎣ sq ⎥⎦ ⎣⎢ Pωr Lsd R + pLsq ⎦⎥ ⎣isq ⎦ ⎣ Pωrψ F ⎦
Te =
3
P(ψ F isq + (Lsd − Lsq )isd isq ))
2
(1)
Where,
vsd, vsq: d-axis and q-axis stator voltages;
isd, isq: d-axis and q-axis stator currents;
R: stator winding resistance;
Lsd, Lsq: d-axis and q-axis stator inductances;
p=d/dt: differential operator;
P: number of pole pairs of the motor;
ωr: rotor speed;
ΨF: rotor permanent magnetic flux;
Te: generated electromagnetic torque;
To produce the largest torque for a given stator current, the stator space current is controlled
to contain only isq.
And since for PMSM Ld ≤ Lq, the second torque component in Eq.(1) is negative with
positive values of isd and zero for SPMSM. Thus, to ensure maximum torque, the control
algorithm should be such that isd is always zero, which result in simple torque expression as:
Te=3/2 PψF isq =3/2 ψF | is| sin(α-θr)
(2)
The stator windings currents are supplied from PWM inverter, using hysteresis current
controller. The actual stator currents contain harmonics, which, produce pulsating torques,
but these may be filtered out by external passive and active filters, or using small hysteresis
bands for the controllers.
2.1 Implementation of rotor field oriented control
The block diagram of rotor-field oriented control of PMSM in polar co-ordinate is shown in
Fig.1 (Vas, 1996). The stator currents are fed from current controlled inverter. The measured
stator currents are transformed to stationary D-Q axis. The D and Q current components are
then transformed to polar co-ordinate to obtain the modulus |is| and the phase angle αs of the
stator-current space phasor expressed in the stationary reference frame.
Fig. 1. Rotor Field Oriented Control of PMSM
158
Torque Control
The rotor speed ωr and rotor angle θr are measured; and the position of the stator current in
the rotor reference frame is obtained. Then, the instantaneous electromagnetic torque Te can
be obtained as stated in Eq. (2).
The necessary current references to the PWM inverter are obtained through two cascaded PI
controllers. The measured rotor speed ωr is compared with the given reference speed ωref
and the error is controlled to obtained the reference torque Teref. The calculated torque is
subtracted from the reference torque and the difference is controlled to obtain the modulus
of isref. The reference angle αsref is set equal to π/2, and the actual rotor angle is added to
(αsref − θr) to obtain the angle αsref of the stator current in the stationary D-Q frame. Theses
values are then transformed to the three-reference stator currents isAref, isBref and isCref and
used to drive the current controller.
The functions of the PI controllers (other controllers such as Fuzzy Logic, Adaptive, Slide
mode or combinations of such controller may be used) are to control both the speed and
torque to achieve predetermined setting values such as:
1. Zero study state error and minimum oscillation,
2. Wide range of regulated speed,
3. Short settling time,
4. Minimum torque ripples,
5. Limited starting current.
Based on the above description a FOC model was built in MatLab/Simulink as shown in
Fig. 2. The model responses for the data setting in Table 1 of SPMSM with ideal inverter
were displayed in Fig.3 to Fig.7. The PI controllers setting and reference values are:
Ts=1 μs, ωref =300, TL =5Nm, PI2: Kp=10, Ki=0.1 PI1: Kp=7, Ki=0.1.
Fig. 2. FOC model in Matlab/Simulink
Torque Control of PMSM and Associated Harmonic Ripples
Fig. 3. Torque response
Fig. 4. Speed response
Fig. 5. Line current response
159
160
Torque Control
Fig. 6. Vab switching pattern
Fig. 7. Regulated Speed range (0-450) rad/s
Vdc
120V
ΨF
0.1546 web.
Rs
1.4 ohm
Ld
0.0066 H
Lq
0.0066 H
J
0.00176 kGm2
B
0.000388 N/rad/s
Table 1. Motor parameters
The above figures show acceptable characteristics however, the torque pulsation cannot be
avoided and the line currents are almost sinusoidal with some harmonic values. The speed
can be regulated up to the rated value (300rad/s) with acceptable response. Bearing in mind
that sensors, analog/digital converters, switching elements of the inverter and algorithm
161
Torque Control of PMSM and Associated Harmonic Ripples
processing in DSP are time consuming, it is practically difficult to achieve such system with
small sampling period. Thus, in practice convenient sampling periods, such as 100 μs (or
larger) is normally selected for processing. In the following, simulation practical values will
be adopted to obtain reasonable results for comparison. So, PMSM with parameters shown
in Table 2 was simulated in the same model with the following setting values:
Ts=100 μs, TL =2Nm, ωref =70 rad/s
PI2: Kp=10, Ki=0.1 and PI1: Kp=7, Ki=0.1
Number of pole pairs
P
2
Stator leakageresistance Rs
5.8 Ohm
d-axis inductance
Lsq
102.7 mH
q-axis inductance
Lsd
44.8 mH
Permanent magnet flux ΨF
533 mWb
Inertia constant
J
0.000329Nms2
Friction constant
B
0.0
Reference speed
ω
70 rad/s
Load torque
TL
2 Nm
Table 2. IPMSM parameters
The simulation responses were shown below:
Fig. 8. FOC Torque response
Fig. 9. FOC Speed response
162
Torque Control
Fig. 10. FOC Line Voltage Switching
Fig. 11. FOC Current response
Fig. 12. Flux response
The responses showed that the torque pulsation is very high and line currents are full of
harmonic components which give rise to EMI noises, in addition flux and speed are not free
of ripples which result in unwanted phenomena such as machine vibration and acoustic
noise.
163
Torque Control of PMSM and Associated Harmonic Ripples
3. Algorithm 2: Hysteresis Direct Torque Control (HDTC)
This method which is also called Basic DTC can be explained by referring to Fig.13. In this
figure, the angle between the stator and rotor flux linkages δ is the load angle when the
stator resistance is neglected. In the study, state δ is constant corresponds to a load torque,
where stator and rotor flux rotate at synchronous speed. In transient operation, δ varies and
the stator and rotor flux rotate at different speeds. Since the electrical time constant is
normally much smaller than the mechanical time constant, the rotating speed of stator flux
with respect to rotor flux, can easily be changed also that the increase of torque can be
controlled by controlling the change of δ or the rotating speed of the stator flux (Zhong,
1997) as will be explained in the following analysis.
q
Q
is
β-axis
ψ sref
ψs
α-axix
isq
λ s λ sref
δ
θ
Ld isd
Lq isq
d Rotor direct axis
ΨF
D Stator direct axis
Fig. 13. Stator and rotor flux space phasors
3.1 Flux and torque criteria
Referring to Fig. 13 the flux equations in rotor dq axis frame can be rewritten as:
ψ sd = Lsd isd + ψ F = ψ s cos δ
(3)
ψ sq = Lsq isq = ψ s sin δ
(4)
Where, |ψs| represent the amplitude of the stator flux linkage calculated as:
(
ψ s = ( Lsd isd + ψ F ) + Lsq isd
2
2
)
2
(5)
In the general α-β reference frame the torque equation can be written as (Zhong, 1997):
Te =
3
P ψ s is β
2
(6)
Where; iβ is the component of the stator phasor space current perpendicular to the stator flux
axis α.
Equation (6) suggests that the torque is directly proportional to the β-axis component of the
stator current if the amplitude of the stator flux linkage is kept constant.
Now using Eq.(3) and Eq.(4) to rewrite the torque equation as:
164
Torque Control
Te =
3P ψ s
⎡ 2ψ F Lsq sin δ − ψ s Lsq − Lsd sin 2δ ⎤
⎦
4Lsd Lsq ⎣
(
)
(7)
For SPMSM Lsd = Lsq = Ls and this expression is reduced to
Te =
3P ψ s
2 Ls
ψ F sin δ =
3P ψ s
2 Ls
ψ F sin δ •t
(8)
Where δ• is the angular speed of the stator flux linkage relative to the permanent magnet
axis.
At constant flux values, Eq. (8) shows that Te-δ has sinusoidal relationship and the
derivative of this equation suggest that the increase of torque is proportional to the increase
of δ in the range of –π/2 to π/2. So the stator flux linkage should be kept constant and the
rotational speed δ• is controlled as fast as possible to obtain the maximum change in actual
torque.
For IPMSM, the torque expression contains in addition to the excitation torque, reluctance
torque and for each stator flux level value, there exist different Te-δ curve and different
maximum torque. Fig. 14 (Zhong, 1997) shows these relationship for different values of
|ψs|. Observe the crossing of curve |ψs|=2ψF where, the derivative of torque near zero
crossing has negative value, which implies that DTC can not be applied in this case.
Fig. 14. Different Te-δ curves for different stator flux values
Analytically this condition can be obtained from derivative of Eq. (7) as follows:
dTe 3 P ψ s ⎡
=
ψ F Lsqδ • − ψ s Lsq − Lsd δ ∗ ⎤
⎦
2Lsd Lsq ⎣
dt
(
)
(9)
And thus for positive torque derivative under positive δ•, |ψs| should be selected in such a
way that (Tang et al., 2002; Zhong et al. 1997):
ψs 〈
Lsq
Lsq − Lsd
ψF
(10)
165
Torque Control of PMSM and Associated Harmonic Ripples
That if fast dynamic response is required. Also that (Tang et al., 2002) for stable torque
control the following criteria should be satisfied.
δ < cos −1 (
Where, a =
Lsq
Lsq − Lsd
a /ψ s − ( a /ψ s )2 + 8
4
)
(11)
ψF
3.2 Control of stator flux strategy
The stator flux linkage of a PMSM in the stationary reference frame can be expressed as:
ψ s = ∫ (Vs − Ris )dt = Vst − R ∫ is dt + ψ s
t =0
(12)
During switching interval each voltage vector is constant, so if stator resistance is neglected
then, this equation implies that the stator flux will move in the direction of the applied
voltage vector.
To select the voltage vectors or controlling the amplitude of the stator flux linkage, the
voltage vector plane is divided into six sectors (FS1 to FS6) as shown in Fig. 15. In each
region two adjacent voltage vectors are selected to increase or decrease the amplitude
respectively of the flux within a hysteresis band. For example, the vectors V2 and V3 are
used to increase and decrease the flux amplitude when ψs is in region one and rotating in a
counter clockwise direction. If rotating in clockwise direction then V5 and V6 are used for
the same reason.
FS=3
FS=2
V2(110)
V3(010)
FS=4
V4(011)
V1(100)
FS=1
V6(101)
V5(001)
V0(000)
V7(111)
FS=6
FS=5
Fig. 15. Applied vectors position and flux sectors.
3.3 Implementation of Hysteresis DTC
The block diagram of a PMSM drive with HDTC may be as shown in Fig. 16, where the
measured current phase values and dc voltage are transferred to D-Q stationary axis values,
and the flux linkage components ψsD and ψsQ at the mth sampling instance are calculated
from the stator voltages as follows:
ψsD(m)= ψsD(m-1) +(VD(m-1)-RisD)Ts
(13)
166
Torque Control
ψsQ(m)= ψsQ(m-1) +(VQ(m-1)-RisQ)Ts
(14)
Where Ts is the sampling period and isD and isQ are calculated as average values of is(m-1)
and is(m) and thus, amplitude and flux angle position with respect to stationary D-Q axis
can be calculated as:
ψ s = ψ D2 ( m) + ψ Q2 ( m)
λs = tan −1
(15)
ψ Q ( m)
ψ D ( m)
The torque can be rewritten in the stationary reference frame as (Zhong et al., 1997):
Te ( m) =
3
P ψ sD (m)isQ (m) − ψ sQ (m)isD ( m)
2
(
)
(16)
However if the phase currents and the rotor speed and/or rotor position are monitored then
Eq. (3) and Eq. (4) can be used to calculate torque and flux values, where then the
transformation D-Q ↔ d-q is necessary to achieve the required values.
Fig. 16. HDTC of PMSM
The calculated Torque and Flux magnitude values are compared with their respective
reference values and the produced errors are inputs to their respective hysteresis
comparators. The flux linkage comparator is a two level comparator φ ε {1, 0} and the torque
comparator is a three level comparator τ ε{1, 0, -1}. The outputs of these comparators
together with stator position λs (or sector number) are inputs to optimum voltage switching
lookup table as the one shown in Table 3 (Luukko, 2000). The output of this table is
switching vector to the inverter driving the motor.
Based on the above description a HDTC of PMSM model was built in Matlab Simulink as
shown in Fig.17.
The torque and flux estimator is based on monitoring of phase currents and rotor angle. The
model responses for the Table 2 and controllers setting values as:
PI speed controller: Kp=0.04 and Ki=2,
Hysteresis logic: Flux band = ± 0.01; Torque Band = ±0.01; Sampling time: Ts= 0.0001s; has
been simulated with results displayed in Fig.18-Fig.22
167
Torque Control of PMSM and Associated Harmonic Ripples
FS
ф
1
0
τ
1
-30≤ λs <30
2
30≤ λs <90
3
90≤ λs <150
1
V2(110)
V3(010)
V4(011)
V5(001)
V6(101)
V1(100)
4
5
6
150≤ λs <210 210≤ λs <270 270≤ λs <330
0
V7(111)
V0(000)
V7(111)
V0(000)
V7(111)
V0(000)
-1
V6(101)
V1(100)
V2(110)
V3(010)
V4(011)
V5(001)
1
V3(010)
V4(011)
V5(001)
V6(101)
V1(100)
V2(110)
0
V0(000)
V7(111)
V0(000)
V7(111)
V0(000)
V7(111)
-1
V5(001)
V6(101)
V1(100)
V2(110)
V3(010)
V4(011)
Table 3. Optimum switching lookup table for HDTC inverter. Ф is the output of flux
hysteresis controller, τ is the output of the torque hysteresis controller, the entries Vi(…) is
the switching logic to the inverter and FS (Flux Sector) define the stator flux position sector
Fig. 17. HDTC of PMSM in Matlab/Simulink
Fig. 18. Torque Response
168
Fig. 19. Speed Response
Fig. 20. Voltage switching of line a-b
Fig. 21. HDTC Line current of phase-a
Torque Control
Torque Control of PMSM and Associated Harmonic Ripples
169
Fig. 22. HDTC Flux response
The responses showed that the torque pulsation is also high and line currents are full of
switching harmonics as compared to the FOC algorithm. In addition the flux and speed are
also not free of ripple which result in machine vibration and acoustic noise.
4. Algorithm 3: Space Vector Modulation Direct Torque Control (SVMDTC)
In this method, a mathematical model of PMSM and space vector modulation of inverter are
used to carry out system algorithm. Thus, instead of switching table and hysteresis
controller a space voltage modulation vectors depending on the flux positions are used to
compensate for errors in flux and torque (Dariusz et al, 2002; Tang et al., 2004). One of the
SVM-DTC block diagrams is shown in Fig. 23 (Dariusz et al., 2002).
Fig. 23. Direct Torque Control SV-Modulation
In Fig. 23, the torque error signal ΔTe and reference amplitude of the stator flux Ψsref are
delivered to predictive controller, which also uses information about the amplitude and
position of the actual stator flux vector and measured stator current vector. The predictive
controller determines the stator voltage command vector in polar co-ordinates Vsref = |Vsref|
∠φsref for space vector modulator (SVM) which finally generates the pulses SA, SB and SC to
control the PWM inverter.
Referring to Eq. (7), the electromagnetic torque produced by the motor is given by:
Te =
3 ψs
⎡ψ F Lq sin δ + 1 ψ s (Lsd − Lsq )sin 2δ ⎤
P
2
⎦
2 Ld Lq ⎣
(17)
170
Torque Control
From this equation, it can be seen that for constant stator flux amplitude and flux produced
by the permanent magnet, the electromagnetic torque can be changed by control of the
torque angle. The torque angle δ can be changed by changing position of the stator flux
vector with respect to the PM vector using the actual voltage vector supplied by the PWM
inverter (Dariusz, 2002). The flux and torque values can be calculated as in Section 3.1 or
may be estimated as in Section 3.3. The internal flux calculator is shown in Fig. 24.
isA
isB
isC
i
ABC sD
To
i
DQ sQ
DQ
isd
To
dq
isq
ΨF
Ld
Lq
Ψsd
ΨsD Cartesian
To
Polar
ΨsQ
dq
To
DQ
Ψsq
Ψs
λs
θr
Fig. 24. Flux Estimator Block Diagram
The internal structure of the predictive controller is in Fig. 25.
ψsref
VOLTAGES
PI
ΔTe
Vsref
M ODULTOR
Δδ
λsref
λs ψs
ϕsref
is
Fig. 25. Predictive Controller
Sampled torque error ΔTe and reference stator flux amplitude Ψsref are delivered to the
predictive controller. The error in the torque is passed to PI controller to generate the
increment in the load angle Δδ required to minimize the instantaneous error between
reference torque and actual torque value. The reference values of the stator voltage vector
are calculated as:
and ϕsref = tan −1
Vsref = VsD _ ref 2 + VsQ _ ref 2
VsQ _ ref
VsD _ ref
(18)
Where:
VsD _ ref =
VsQ _ ref =
ψ sref cos(λs + Δδ ) − ψ s cos λs
Ts
+ Rs isD .
ψ sref sin(λs + Δδ ) − ψ s sin λs
Ts
+ Rs isQ .
(19)
(20)
Where, Ts is the sampling period.
For constant flux operation region, the reference value of stator flux amplitude is equal to
the flux amplitude produced by the permanent magnet. So, normally the reference value of
the stator flux is considered to be equal to the permanent magnet flux.
Torque Control of PMSM and Associated Harmonic Ripples
171
4.1 Implementation of SVMDTC
The described system in Fig. 23 has been implemented in Matlab/Simulink, with the same
data and loading condition as in HDTC with PI controllers setting as:
Predictive Controller:
Ki=0.03,
Kp=1
Speed Controller: Ki= 1 Kp=0.04.
The simulation results are shown in Fig. 26 to Fig. 29. As evidence from the figures, the
SVM-DTC guarantee lower current pulsation, smooth speed as well as lower torque
pulsation. This is mainly due to the fact that the inverter switching in SVM-DTC is uni-polar
compared to that of FOC & HDTC (see Fig. 10, Fig. 20 and Fig. 28), in addition the
application of SVM reduces switching stress by avoiding direct transition from +Vdc to –
Vdc and thus avoiding instantaneous current reversal in dc link. However, the dynamic
response in Fig. 9, Fig. 19, and Fig. 27 show that HDTC has faster response compared to the
SVM-DTC and FOC.
Fig. 26. SVMDTC torque response
Fig. 27. SVMDTC rotor speed response
Fig. 28. SVMDTC Line voltage (Vab) waveform
172
Torque Control
Fig. 29. SVMDTC Line current response of phase a
Fig. 30. Stator flux response.
5. High Performance Direct Torque Control Algorithm (HP-DTC)
In this section, a new direct torque algorithm for IPMSM to improve the performance of
hysteresis direct torque control is described. The algorithm uses the output of two hysteresis
controllers used in the traditional HDTC to determine two adjacent active vectors. The
algorithm also uses the magnitude of the torque error and the stator flux linkage position to
select the switching time required for the two selected vectors. The selection of the switching
time utilizes suggested table structure which, reduce the complexity of calculation. Two
Matlab/Simulink models, one for the HDTC, and the other for the proposed model are
programmed to test the performance of the proposed algorithm. The simulation results of
the proposed algorithm show adequate dynamic torque performance and considerable
torque ripples reduction as well as lower flux ripples, lower harmonic current and lower
EMI noise reduction as compared to HDTC. Only one PI controller, two hysteresis
controllers, current sensors and speed sensor as well as initial rotor position and built-in
counters microcontroller are required to achieve this algorithm (Adam & Gulez, 2009).
5.1 Flux and torque bands limitations
In HDTC the motor torque control is achieved through two hysteresis controllers, one for
stator flux magnitude error control and the other for torque error control. The selection of
one active switching vector depends on the sign of these two errors without inspections of
their magnitude values with respect to the sampling time and level of the applied stator
voltage. In this section, short analysis concerning this issue will be discussed.
173
Torque Control of PMSM and Associated Harmonic Ripples
5.1.1 Flux band
Consider the motor stator voltage equation in space vector frame below:.
Vs = Rs is +
dΨ s
dt
(21)
Equation (21) can be written as:
dt =
dΨ s
Vs − Rs is
(22)
For small given flux band ΔΨso, the required fractional time to reach the limit of this value
from some reference flux Ψ* is given by:
Δt =
ΔΨ 0s
(23)
Vs − Rsis
And if the voltage drop in stator resistance is ignored, then the maximum time for the stator
flux to remain within the selected band starting from the reference value is given as:
Δtmax =
ΔΨ 0s
=
Vs
ΔΨ s0
2 / 3 Vdc
(24)
Thus if the selected sampling time Ts is large than Δtmax, then the stator flux linkage no
longer remains within the selected band causing higher flux and torque ripples.
According to (24) if the average voltage supplying the motor is reduced to follow the
magnitude of the flux linkage error, the problem can be solved, i.e. the required voltage
level to remain within the selected band is:
Vlevel =
Δtmax
Vkk
Ts
(25)
Where Vkk is the applied active vectors
Thus, by controlling the level of the applied voltage, the control of the flux error to remain
within the selected band can be achieved. For transient states, ΔΨs is most properly large
which, requires large voltage level to be applied in order to bring the machine into steady
state as quickly as possible.
5.1.2 Torque band
The maximum time Δttorque for the torque ripples to remain within selected hysteresis band
can be estimated as:
Δttorque =
Where, ΔT0; is the selected torque band
ΔT 0
Teref
* t0
(26)
174
Torque Control
Teref ; is the reference electromagnetic torque
t0; is the time required to accelerate the motor from standstill to some reference torque Teref.
The minimum of the values given in (24) and (26) can be considered as the maximum
switching time to achieve both flux and torque bands requirement. However, when the
torque ripples is the only matter of concern, as considered in this work, may be enough to
consider the maximum time as suggested by (26).
Now due to flux change by ΔΨs, the load angle δ will change by Δδ as shown in Fig. 31.
Under dynamic state, this change is normally small and can be approximated as:
Δδ ≈ sin −1
ΔΨ s ΔΨ s
≈
Ψs
Ψs
(27)
Δδ
q
Ψs
|ΔΨs|
ΨF
δ
d
θr
D
Fig. 31. Stator flux linkage variation under dynamic state
The corresponding change in torque due to change ΔΨs can be obtained by differentiation of
torque equation with respect to δ. Torque equation can be rewritten as:
Te =
Ψs
3
⎡ 2 Ψ F Lsq sin δ − Ψ s (Lsq − Lsd )sin 2δ ⎤
P
⎦
4 Lsd Lsq ⎣
(28)
∂Te
∂T Δψ s
⋅ Δδ ≈ e ⋅
∂δ
∂δ Ψ s
(29)
Where, then
ΔT =
Substitute (24) in (29) and evaluate to obtain:
ΔT =
3 Vs Δt
⎡ Ψ F Lsq cos δ − Ψ s (Lsq − Lsd )cos 2δ ⎤
P
⎦
2 Lsd Lsq ⎣
(30)
Where, Δt=minimum (Δtmax ,Δttorque)
Equation (30) suggests that ΔT can also be controlled by controlling the level of Vs. Thus
both ΔT and ΔΨs can be controlled to minimum when the average stator voltage level is
controlled to follow the magnitude of ΔT.
5.2 The HP-DTC Algorithm
The basic structure of the proposed algorithm is shown in Fig. 32.
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Torque Control of PMSM and Associated Harmonic Ripples
Fig. 32. The HPDTC system of PMSM
5.2.1 Vector selector
In Fig.32 the vector selector block contains algorithm to select two consecutive active vectors
Vk1, and Vk2 depending on the output of the hysteresis controllers of the flux error and the
torque error; φ and τ respectively as well as flux sector number; n. The vector selection table
is shown in Table 4., while vectors position and flux sectors is as shown in Fig.15
φ
τ
Vk1
Vk2
1
1
n+1
n+2
1
0
n-1
n-2
0
1
n+2
n+1
0
0
n-2
n-1
Table 4. Active vectors selection table
In the above table
if Vk>6 then Vk =Vk-6
if Vk<1 then Vk =Vk+6
5.2.2 Flux and torque estimator
In Fig. 32 the torque and flux estimator utilizes equation (21) to estimate flux and torque
values at m sampling period as follows:
ψ D (m) = ψ D (m − 1) + (VD (m − 1) − RsiD )Ts
(31)
ψ Q (m) = ψ Q (m − 1) + (VQ (m − 1) − RsiQ )Ts
(32)
ψ s = ψ D2 + ψ Q2
(33)
λs = Tan−1
ψQ
ψD
(34)
Where; the stationary D-Q axis voltage and current components are calculated as follows:
VD (m − 1) = (VDk 1t k 1 + VDk 2t k 2 ) / Ts
(35)
176
Torque Control
VQ (m − 1) = (VQk 1t k 1 + VQk 2t k 2 ) / Ts
(36)
iD = (iD (m − 1) + iD (m)) / 2
(37)
iQ = (iQ (m − 1) + iQ (m)) / 2
(38)
The torque value can be calculated using estimated flux values as:
Te =
3
P( Ψ D (m)iQ (m) − Ψ Q (m)iD ( m))
2
(39)
5.2.3 The timing selector structure
In Fig. 32 the timing selector block contains algorithm to select the timing period pairs of
vectors Vk1 and Vk2. The selection of timing pairs depends on two axes, one is the required
voltage level and the other is the reflected flux position in the sector contained between Vk1
and Vk2. The reflected flux position is given by:
ρs = λs mod 60 − π / 6
(40)
Where λs; is the stator flux linkage position in D-Q stationary reference frame.
Fig. 34 shows the proposed timing table. In this figure, the angle between the two vectors
Vk1 and Vk2 which is 600, is divided into 5 equal sections ρ-2, ρ-1, ρ0, ρ+1, and ρ+2. The required
voltage level is also divided into 5 levels.
The time pairs (tk1, tk2), expressed as points, (out of 20 points presenting the sampling
period) define the timing periods of Vk1 and Vk2 respectively. The remaining time points,
(t0=20-tk1 -tk2), is equally divided between zero vectors V0 and V7.
Fig. 34. Timing diagram for the suggested algorithm
The time structure shown in Fig.34 has the advantage of avoiding the complex mathematical
expressions used to calculate tk1 and tk2, as the case in space vector modulation used by
(Dariusz, 2002) and (Tan, 2004). In addition, it is more convenient to be programmed and
executed through the counter which controls the period tk1, tk2 and t0. The flow chart of the
algorithm is shown in Fig. 35.
177
Torque Control of PMSM and Associated Harmonic Ripples
Define timing table
Load initial & reference values
START
ADC &
Encoder
Read sensed values: currents, dc
link voltage and speed/position
Motor Sensed
values
Calculate iD, iQ, VD, VQ
Eq.s(35-38)
Calculate ΨD ,ΨQ ,λ s & Te
Eq.s (31, 34, 39)
Calculate ΔΨs , ΔT
Find Hysteresis controllers output values φ and τ
Find sector number n (Fig. 15)
Calculate torque error level ΔT ε {Level1...Level5}
Calculate reflected position Eq.18 ε {ρ-2 ,.. ρ+2 }
Determine tk1,tk2 & calculate t0
Get active vectors Vk1 , Vk2.
INVERTER SWITCHING
Send Vk1, Delay tk1/2
Send Vk2, Delay tk2/2
Send Vk2, Delay tk2/2
Send Vk1, Delay tk1/2
Send V7, Delay t0/2
Send V0, Delay t0/2
Fig. 35. A Flow chart of the proposed algorithm
5.3 Simulation and results
To examine the performance of the proposed DTC algorithm, two Matlab/Simulink models,
one for HDTC and the other for the HPDTC were programmed. The motor parameters are
shown in table 2. The inverter used in simulation is IGBT inverter with the following setting:
IGBT/Diode
Snubber Rs, Cs = (1e-3ohm,10e-6F)
Ron=1e-3ohm
Forward voltage (Vf Device,Vf Diode)= (0.6, 0.6)
Tf(s),Tt(s) = (1e-6, 2e-6)
DC link voltage= +132 to -132.
178
Torque Control
The simulation results with 100μs sampling time for the two algorithms under the same
operating conditions are shown in Fig. 36 -to- Fig. 41. The torque dynamic response is
simulated with open speed loop, while the steady state performance is simulated with
closed speed loop, 70rad/s as reference speed, and 2 Nm as load torque.
5.3.1 Torque dynamic response
The torque dynamic response with HDTC and the HPDTC are shown in Fig.36-a and Fig.36b respectively. The reference torque for both algorithms is changed from +2.0 to -2.0 and
then to 3.0 Nm. As shown in the figures, the dynamic response with the proposed algorithm
is adequately follows the reference torque with lower torque ripples. In the other hand,
the torque response with the proposed algorithm shows fast response as the HDTC
response.
(a)
(b)
Fig. 36. Motor dynamic torque with opened speed loop: (a) HDTC (b) HP-DTC
Fig. 37 demonstrates the idea of maximum time to remain within the proposed torque band
as suggested by equation (26). According to the shown simulated values, the time required
to accelerate the motor to 2 Nm is ≈ 0.8ms, so if the required limit torque ripple is not to
exceed 0.1 Nm, as suggested in this work, then, the maximum switching period according to
Eq. (26) is ≈0.05ms which is less than the sampling period (Ts=0.1 ms).
Fig. 37. Torque ripples and motor accelerating time
Although the torque ripple is brought under control, the flux ripples still high as shown in
Fig. 38 which, is mainly due to control of the voltage level according to the magnitude of
torque error only.
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Torque Control of PMSM and Associated Harmonic Ripples
Fig. 38. Flux response when only the torque error magnitude is used to approximate the
required voltage level
5.3.2 Motor steady state performance
The motor performance results under steady state are shown in Fig. 39 -to- Fig. 41. Fig. 39-a
and Fig. 39-b, show the phase currents of the motor windings under HDTC and the HPDTC
respectively, observe the change of the waveform under the proposed method, it is clear
that the phase currents approach sinusoidal waveform with almost free of current pulses
appear in Fig. 39-a. Better waveform can be obtained by increasing the partition of the
timing structure, however, when smoother waveform is not necessary, suitable division as
the one shown in Fig. 34 may be enough.
(a)
(b)
Fig. 39. Motor line currents: (a) HDTC (b) HPDTC
The torque response in Fig. 40 shows considerable reduction in torque ripples from 3.2Nm
(max. -to- max.) down to less than 0.15 Nm when the new method HP-DTC is used, which
in turn, will result in reduced motor mechanical vibration and acoustic noise, this reduction
also reflected in smoother speed response as shown in Fig. 41
(a)
Fig. 40. Motor steady state torque response: (a) HDTC (b) HPDTC
(b)
180
Torque Control
(a)
(b)
Fig. 41. Rotor speed response: (a) HDTC (b) HPDTC
6. Torque ripple and noise in PMSM algorithm
One of the major disadvantages of the PMSM drive is torque ripple that leads to mechanical
vibration and acoustic noise. The sensitivity of torque ripple depends on the application. If
the machine is used in a pump system, the torque ripple is of no importance. In other
applications, the amount of torque ripple is critical. For example, the quality of the surface
finish of a metal working machine is directly dependent on the smoothness of the delivered
torque (Jahns and Soong, 1996). Also in electrical or hybrid vehicle application, torque ripple
could result in vibration or noise producing source which in the worst case could affect the
active parts in the vehicle.
The different sources of torque ripples, harmoinc currents and noises in permanent magnet
machines can be abstracted in the following (Holtz and Springob 1996,1998):
•
Distortion of the stator flux linkage distribution
•
Stator slotting effects and cogging
•
Stator current offsets and scaling errors
•
Unbalanced magnetization
•
Inverter switching and EMI noise
However switching harmonics and voltage harmonics supplied by the power inverter
constitute the major source of harmonics in PMSM. In this section, the reduction of torque
ripple and harmonics generated due to inverter switching in PMSM control algorithms
using passive and active filter topology will be investigated.
Method1: Compound passive filter topology
6.1 The proposed passive filter topology
Fig. 42 shows a block diagram of basic structure of the proposed filter topology (Gulez et al.,
2007) with PMSM drive control system. It consists of compound dissipative filter cascaded
by RLC low pass filter. The compound filter has two tuning frequency points, one at
inverter switching frequency and the other at some average selected frequency.
181
Torque Control of PMSM and Associated Harmonic Ripples
Inverter
Trap
Compound
Filter
RLC
Filter
Currents
Control
System
PMSM
Speed
Fig. 42. Block diagram of the proposed filter topology with PMSM drive system
6.1.1 The compound trap filter
Fig. 43 shows the suggested compound trap filter. It consists of main three passes, one is low
frequency pass branch through R2 and L2, another is the high frequency pass through C2 and
R1 and the other is the average frequency pass through C1, L1 and R1 to the earth.
Fig. 43. The suggested compound trap filter
For some operating frequency ωo, the component of the low pass branch constitutes low
impedance path while at the same time shows high impedance for the high frequency
component, which forces the high frequency to pass through C2 to the earth. Some of the
average frequency components will find their way through the low pass branch. These
frequencies will be absorbed by tuning resonance of branch L1-C1 to some selected average
frequency such that.
ωo< ωav< ωsw
Where
ωo ; is the operating frequency
ωav = 1 / L1C 1 : is the selected average frequency
ωsw: is inverter switching frequency calculated as 1/(2Ts); Ts being the sampling period
The behavior of the Compound Trap filter can be explained by studying the behavior of the
impedances constitutes the Π equivalent circuit of the Compound Trap filter shown in Fig.
44.
In Fig. 44 the impedances Z1, Z2 and Z3 can be expressed as:
Z1 =
ωL1 − 1 / ωc1
1
1
+ j(ωL1 −
−
)
ω R1C 2
ωC 1 ωC 2
(41)
182
Torque Control
Fig. 44. Equivalent Π circuit of the compound trap filter.
Z2 = R1 − R1C 2ω (ωL1 −
Z3 = R1 −
1
ωC1
R1
) + j(ωL1 −
ω 2 L1C 2 − C 2 / C1
−j
1
ωC1
1
ωC 2
)
(42)
(43)
Figures 45, 46 and 47 show the frequency-magnitude characteristics of the impedances Z2, Z3
and Z12 respectively. Z12 is the equivalent impedance value between point 1 and 2 as shown
in Fig. 44.
Fig. 45. Z2 characteristics at C1=52.0e-6F and different L1 values.
Fig. 46. Z3 characteristics at C1=52.0e-6F and different L1 values.
183
Torque Control of PMSM and Associated Harmonic Ripples
Fig. 47. Z12 characteristics at C1=52.0e-6F and different L1 values.
As can be inferred from Equations (41 to 43) and Figures (Fig. 46 to Fig. 47) it is evidence
that both Z2 and Z3 show capacitive behavior at small values of L1. So if the value of L1 is
kept small (L1 < 1e-5 H), the tuning of high frequency current components is ensured
through the compound trap filter.
In the other hand the characteristics of Z12 shown in Fig. 47 demonstrates that at average
frequencies the impedance of Z12 is high while at low and high frequencies the impedance is
low thus Z12 constitute band stop filter to the average frequency current components. The
magnitude of Z12 can effectively be changed by changing the value of L1 at constant C1,
while the range of the average frequencies can effectively be changed by changing the value
of C1 at constant L1. Thus through proper tuning of L1 and C1 the desired average
frequency range can be selected.
6.1.2 RLC filter
Fig. 48 shows the suggested RLC filter, which play main role in reducing the high dv/dt of
line to line voltages at motor terminals. The transfer function of this circuit is given by:
V0
R3C 3s + 1
=
Vi C 3L3s 2 + (r3 + R3 )C 3s + 1
(44)
To obtain over damping response, the filter resistances are selected such that:
( R3 + r3 ) >
4L3
C3
(45)
With cutoff frequency ωc is given by:
ωc = 1 / L3C 3
(46)
To reduce ohmic losses the series resistance r3 is normally of small value, while the shunt
resistance R3 is selected high enough to limit the currents drawn by the filter. This current
can be expressed as:
iCR 3 =
zPMSM
zPMSM + R32 + (1 / sC 3 )2
iin
(47)
184
Torque Control
Where, ZPMSM is PMSM motor input impedance.
At the selected cutoff frequency, this current should be large compared to imotor drawn by
the motor; while at operating frequency this current should be very small compared to
imotor. Another point in selection of the RLC parameters is that, the filter inductors are
essentially shorted at line frequency while the capacitors are open circuit and for EMI
noise frequencies, the inductors are essentially open circuit while the capacitors are
essentially shorted, thus considerable amount of EMI noises will pass through the filter
resistors to the earth and cause frequency dependent voltage drop across the series branch
of the filter which, in turn, helps in smoothing of the voltage waveform supplying the
motor.
Fig. 48. RLC filter cascaded to the trap compound filter
To evaluate the performance of the suggested passive filter topology, it was applied to
HDTC algorithms under MatLab simulation. The following subsections show the results of
the simulations.
6.1.3 Torque ripples and noise reduction in HDTC using passive filter
Fig. 49 shows the basic structure of HDTC of PMSM with the proposed passive filter
topology. The switching table in Fig.49 is the same as that shown in Table 3. In this figure,
the switching of the inverter is updated only when the outputs of the hysteresis controllers
change states, which result in variable switching frequency and associated large harmonic
range and high current ripples.
Flux
Reference
Switching
Table
Ideal
Inverter
Trap
Compound
Filter
RLC
Filter
PMSM
Torque
Reference
Flux Estimator
Currents
Speed/Position
Fig. 49. The basic structure of HDTC of PMSM with the proposed filter topology
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Torque Control of PMSM and Associated Harmonic Ripples
6.1.4 Simulations and results
To simulate the performance of the proposed passive filter topology under HDTC
Matlab/Simulink was used.
Under base speed operation, the speed control was achieved through PI controller with Ki=
2.0 and Kp= 0.045. The flux reference is set equal to ΨF and hysteresis bands are set to 0.01 for
both the torque and flux hysteresis controllers. The motor parameters are shown in Table 2
and the passive filter parameters are in Table 5.
L1
C1
R1
20μH
52μF
56kΩ
L2
C2
R2
30mH
5.1μF
2.2Ω
L3
C3
R3
r3
30mH
12.5 μF
128 Ω
2Ω
Table 5. Passive Filter Parameters
The simulation results with 100μs sampling time are shown in Fig. 50 to Fig. 55. Fig. 50-a in
particular, shows the motor line voltage Vab without applying the proposed filter topology.
When the filter topology is connected, the switching frequency is reduced as depicted in Fig.
50-b compared to the one shown in Fig. 50-a. The line voltages provided to the motor
terminals approach sinusoidal waveform, observe the change of the waveform at the output
of the compound filter in Fig. 50-c and at the motor terminal in Fig. 50-d. Better waveform as
mentioned before can be obtained by increasing the series inductance L3 and decreasing the
resistance r3.
(a)
(c)
(b)
(d)
Fig. 50. Motor line voltage (a) before applying the filter topology (b) at inverter terminals
after applying the filter topology (c) at the compound Filter output terminals (d) at the
motor terminals as output of the RLC filter
186
Torque Control
The motor performance before and after applying the filter topology are shown in Fig. 51 to
Fig. 54. In Fig. 51, the motor line currents show considerable reduction in noise and
harmonic components after applying the filter which reflects in smoother current waveform.
(a)
(b)
Fig. 51. Motor line currents: (a) before (b) after applying the filter topology
The torque response in Fig. 52 shows considerable drop in torque ripples from 1.4Nm
(ripples to ripples) down to 0.6 Nm after applying the Filter topology, which will result in
reduced motor mechanical vibration and acoustic noise. The speed response in Fig. 52,
shows slight smoothness after applying the passive filter topology.
The status of the line current harmonics and EMI noise before and after connecting the filter
topology are shown in Fig. 53 to Fig. 54.
(a)
(b)
Fig. 52. Motor torque: (a) before (b) after applying the filter topology (load torque is 2Nm)
(a)
(b)
Fig. 53. Rotor speed: (a) before (b) after applying the filter topology
In Fig. 54-a the spectrum of the line current without connecting the filter shows that
harmonics currents with THD of ~3% have widely distributed with a dominant harmonics
187
Torque Control of PMSM and Associated Harmonic Ripples
concentration in the range around 2 kHz.. After connecting the filter topology, the THD is
effectively reduced to less than 1.7% with dominant harmonics concentration in the low
frequency range (less than 0.5 kHz.), while the high frequency range is almost free of
parasitic harmonics as shown in Fig. 54-b.
(a)
(b)
Fig. 54. Phase-a current spectrum: (a) before applying the filter topology (b) after applying
the filter topology
The EMI noise level near zero crossing before applying the filter topology in Fig. 55-a shows
a noise level of ~ -5dB at operating frequency , ~-10dB at switching frequency (5KHz) and
~-47dB at the most high frequencies (greater than 0.2 MHz.). When the filter is connected,
the EMI noise level is damped down to ~-20dB at operating frequency , ~-30dB at switching
frequency and ~-67dBs at the most high frequencies as shown in Fig. 55-b.
(a)
(b)
Fig. 55. EMI level: (a) before applying the filter topology (b) after applying the filter
topology
6.2 Method 2: active filter topology
In this section an active filter topology will be proposed to reduce torque ripples and
harmonic noises in PMSM when controlled by FOC or HDTC equipped with hysteresis
188
Torque Control
controllers. The filter topology consists of IGBT active filter (AF) and two RLC filters, one in
the primary circuit and the other in the secondary circuit of a coupling transformer. The AF
is characterized by detecting the harmonics in the motor phase voltages and uses hysteresis
voltage control method to provide almost sinusoidal voltage to the motor windings.
6.2.1 The proposed active filter topology
When the PMSM is controlled by HDTC, the motor line currents and/or torque are
controlled to oscillate within a predefined hysteresis band. Fig. 56, for example, shows
typical current waveform and the associated inverter output voltage switching.
In the shown figure the inverter changes state at the end of a sampling period only when the
actual line current increases or decrease beyond the hysteresis band which result in high
ripple current full of harmonic components.
Motor
current
Hysteresis
Band
Inverter
voltage
Required
voltage
Fig. 56. Current waveform and associated inverter voltage switching equipped with
hysteresis controllers
To reduce the severe of these ripples two methods can be mentioned, the first one is to
reduce the sampling period which implies very fast switching elements, and the second one
is to affect the voltage provided to the motor terminals in such a way to almost follow
sinusoidal reference guide. The last method will be adopted here so, active filter topology is
used to affect inverter voltage waveform to follow the required signal voltage.
Series active power filters were introduced by the end of the 1980s and operate mainly as a
voltage regulator and as a harmonic isolator between the nonlinear load and the utility
system (Hugh et al, 2003). Since series active power filter injects a voltage component in
series with the supply voltage, they can be regarded as a controlled voltage source. Thus
this type of filters is adopted here to compensate the harmonic voltages from the inverter
supplying the motor.
Fig. 57 shows a schematic diagram of basic structure of the proposed filter topology;
including the active filter, coupling transformer, RLC filters and block diagram of the active
filter control circuit
In Fig. 57 Vsig is the desired voltage to be injected in order to obtain sinusoidal voltage at
motor terminals and VAF is the measured output voltage of the active filter. VAF is subtracted
from Vsig and passed to hysteresis controller in order to generate the required switching
signal to the active filter. The active filter storage capacitor CF which operates as voltage
source should carefully be selected to hold up to the motor line voltage. The smoothing
inductance LF should be large enough to obtain almost sinusoidal voltage at the motor
Torque Control of PMSM and Associated Harmonic Ripples
189
terminals. The reference sinusoidal voltage V* which should be in phase with the main
inverter output voltage Vinv, is calculated using information of the motor variables.
In the following sections firstly, the operating principle of voltage reference control circuit
will be explained then the two other parts will follow.
Fig. 57. Basic Structure of the Proposed Filter Topology
6.2.2 Voltage reference signal generator
The effectiveness of the active filter is mainly defined by the algorithm used to generate the
reference signals required by the control system. These reference signals must allow current
and voltage compensation with minimum time delay. In this study the method used to
generate the voltage reference signals is related to FOC algorithm, which use motor model
in rotor d-q reference frame and rotor field oriented control principles with monitored rotor
position/speed and monitored phase currents. The motor model in this synchronously
rotating reference frame is given by:
⎡ vsd ⎤ ⎡ R + pLsd
⎢ ⎥=⎢
⎢⎣ vsq ⎥⎦ ⎢⎣ ωr Lsd
Te =
−ωr Lsq ⎤ ⎡isd ⎤ ⎡ 0 ⎤
⎥⎢ ⎥ +
R + pLsq ⎥⎦ ⎣ isq ⎦ ⎢⎣ωrψ F ⎥⎦
3
P(ψ F isq + (Lsd − Lsq )isd isq ))
2
(48)
(49)
Under base speed operation, the speed or torque control can be achieved by forcing the
stator current component isd to be zero while controlling the isq component to directly
proportional to the motor torque Te as in (50):
190
Torque Control
Te =
3
Pψ F isq
2
(50)
The instantaneous q-axis current can be extracted from (50) and hence by setting isd to zero,
the instantaneous d and q axis voltages can be calculated from (48) as:
Vsd = −ωr Lsq isq
(51)
Vsq = Risq + pLsq isq + ωrψ F
(52)
Once the values of d-axis and q-axis voltage components are obtained, Park and Clarke
transformation can be used to obtain the reference sinusoidal voltages as:
⎡ va* ⎤
0 ⎤
⎡ 1
⎢ *⎥
⎢
⎥ ⎡ cosθ
3 /2 ⎥⎢
⎢ vb ⎥ = K ⎢ −1 / 2
⎢ *⎥
⎢
⎥ ⎣ sin θ
⎣ −1 / 2 − 3 / 2 ⎦
⎣⎢ vc ⎦⎥
− sin θ ⎤ ⎡Vsd ⎤
⎢ ⎥
cosθ ⎥⎦ ⎣Vsq ⎦
(53)
Where, K is the transformation constant and θ is rotor position
6.2.3 Active filter compensation circuit
Fig. 58 shows simplified power circuit of the proposed topology (the passive RCL filters are
±
is equivalent
not shown). In this circuit Vdc is the voltage of the main inverter circuit, VCF
compensated voltage source of the active filter. In order to generate the required
compensation voltages that follow the voltage signal vsig; bearing in the mind that the main
inverter change switching state only when the line current violates the condition of the hysteresis
band and that the capacitor voltage polarity can not change abruptly, the switches sw1 and sw2 are
controlled within each consecutive voltage switching of the main inverter to keep the motor
winding voltages with acceptable hysteresis band.
The motor line current im is controlled within the motor main control circuit with hysteresis
current controller to provide the required load torque; therefore, two hysteresis controller
systems, one for voltage and the other for current are working independently to supply the
motor with almost sinusoidal voltage
In Fig. 58, when switching signal (eg.100) is send to the main inverter, i.e. phase a is active
high while phase b and c are active low, then, following the path of the current im in Fig.58
the voltage provided to the motor terminal can be expressed as:
2
3 di
±
Vs = (Vdc − VCF
− LF m )
dt
3
2
(54)
The limit values of inductor LF and the capacitor CF can be determined as follows:
During a sampling period Ts, the change in the capacitor voltage can be calculated as:
ΔVCF =
1
CF
Ts
∫ imdt
(55)
0
So if maximum capacitor voltage change is determined as Vdc, the minimum capacitor
value can be calculated as:
Torque Control of PMSM and Associated Harmonic Ripples
191
Fig. 58. Simplified power circuit of the proposed active filter topology.
( n + 1)Ts
CF ≥
∫nTs
im dt
Vdc
=
Ts • imav
Vdc
(56)
Where, imav is the maximum of the average current change which can be occurred per
sample periods.
The limit values of the smoothing inductance LF can be expressed as:
1
2
(2π f sw ) CF
< LF ≤
VLF max
di
3
max( m )
2
dt
(57)
Where, the lower limit is determined by selecting the resonance frequency of the
combination CFLF to be less than the inverter switching frequency fsw to guarantee reduced
switching frequency harmonics. The upper limit is calculated by determining the maximum
voltage drop across the inductors VLFmax, and the maximum current change per sampling
period dim/dt.
6.2.4 The Coupling
The coupling between the main inverter circuit and the active filter circuit is achieved
through 1:1 transformer, and to attenuate the higher frequency EMI noises, LCR filters are
used at the transformer primary and secondary windings as suggested by Fig. 59
Fig. 59. Coupling between AF and main inverter from one side, and PMSM in the other side.
192
Torque Control
The important point here is that, the resonance which may arise between capacitor C1 and
transformer primary winding and between capacitor C2 and motor inductance winding
should be avoided when selecting capacitor values.
At selected cutoff frequency, the currents iCR1 and iCR2 derived by the RLC filters are given
by
iCR 1 =
iCR 2 =
zT
zT +
R12
+ (1 / sC 1 )2
im 1
and
(58)
zPMSM
zPMSM + R22 + (1 / sC 2 )2
im 2
Where, ZT and ZPMSM are as defined in Fig. 59.
Bearing in the mind the conditions required in the selection of RLC, these currents should be
large compared to im1, drawn by the transformer, and/or im, drawn by the motor at selected
cutoff frequency; while at operating frequency these currents should be very small
compared to im1 and im.
6.2.5 Simulation and results
In order to verify that the proposed filter topology does actually improve the performance
of the conventional HDTC methods, the HDTC is implemented in Matlab/Simulink to
compare the performance of the PMSM with and without the filter topology under the same
operating and loading conditions
The motor parameters are in Table 2 and the filters parameters are in Table 6. The AF
capacitor used is 200μF and its inductors are 200mH. The drive is IGBT inverter.
L1
C1
R1
1μH
2μF
250Ω
L2
C2
R2
1.5μH
2μF
750Ω
Table 6. Active Filter Topology parameters
The simulation results with 100μs sampling time and ±0.1 Nm hysteresis torque band are
shown in Fig. 60 to Fig. 66. The torque dynamic response is simulated with open speed loop,
while the steady state performance is simulated with closed speed loop at 70rad/s as
reference speed, and 2 Nm as load torque.
The torque dynamic responses before and after connecting the AF are shown in Fig. 60-a
and Fig. 60-b respectively. The reference torque for both figures is changed from +2.0 to -2.0
and then to 3.0 Nm. As shown in the figures, the dynamic response with the proposed filter
topology is adequately follows the reference torque with lower torque ripples and settles
down within ±0.1 Nm band of the reference torque; while the torque dynamic under HDTC
without filter topology can not settle down within the specified torque bands due to
presence of high torque ripples (± 1.0 Nm). On the other hand, the torque response time
without filter topology is shorter (~1.2ms) than the torque response time with the proposed
filter topology (2.5ms). This delay in the torque response with the proposed filter topology is
mainly due to delay of current propagation through the LFCF loop of the active filter
however; this is not significant if compared with the results provided by Tang et al (2004).
193
Torque Control of PMSM and Associated Harmonic Ripples
(a)
(b)
Fig. 60. Motor torque dynamic under basic HDTC: (a) before (b) after connecting the AF
The motor steady state performance before and after applying the AF are shown in Fig. 61 to
Fig. 64. Fig. 61-a and Fig. 61-b, show the phase voltage provided to the motor terminals
before and after applying the filter topology respectively, observe the change of the
waveform after applying the AF, it is clear that the phase voltage approaches sinusoidal
waveform with almost free of voltage pulses appear in Fig. 61-a due to inverter switching.
Better waveform can be obtained by increasing the active filter inductance LF however, the
cost and size of the AF will increase, and therefore suitable inductance value can be selected
to achieve acceptable performance. Similar results have been provided by Yilmaz, (Yilmaz et
al. 2000), however as compared to above result, their sinusoidal voltage waveform provided
to the motor terminals is full of harmonic components.
(a)
(b)
Fig. 61. Starting motor phase voltage: (a) before (b) after connecting the AF topology
Fig. 62-a and Fig. 62-b show the response of the motor line currents under HDTC without
and with the proposed filter topology respectively. In Fig. 62-a high distortion in line
current can be observed, however the current waveform is smoother after applying the
proposed filter topology. The reason of the high current distortion (ripples) is mainly due to
the fact that switching of the inverter is only updated once at the sampling instances
when the hysteresis controllers change state so, with existence of the proposed active filter
a proper voltage is provided to the motor terminal which, in turn decreases current
ripples.
194
Torque Control
(a)
(b)
Fig. 62. Motor line currents: (a) before (b) after applying the AF topology.
The torque response in Fig. 63 shows considerable reduction in torque ripples around the
load torque when the proposed active filter is connected. The higher ripples of ±1.62Nm
around the load torque in Fig. 63-a is mainly due to the existence of harmonic voltages
provided to the motor terminals, so when the harmonics are reduced after insertion of the
proposed filter topology the torque ripples is decreased down to ±0.1 Nm as shown in Fig.
63-b. The reduction in the torque ripples normally reflected in reduced motor mechanical
vibration and hence reduced acoustic noise as well as smoother speed response as shown in
Fig. 64.
(a)
(b)
Fig. 63. Steady state motor torque response under basic HDTC with 2.0 Nm as load torque
(a) before (b) after connecting the AF topology
(a)
(b)
Fig. 64. Rotor speed under basic HDTC (a)before (b)after applying the AF topology
195
Torque Control of PMSM and Associated Harmonic Ripples
The status of the phase voltage harmonics and EMI noise in the line currents before and
after connecting the AF are shown in Fig. 65 to Fig. 66.
In Fig. 65-a the spectrum of the phase voltage before connecting the AF shows that
disastrous harmonic voltages with THD of ~79% have widely scattered in the shown
frequency range. These harmonic voltages if not cleared or reduced, it will result in parasitic
ripples in motor developed torque and contribute to electromagnetic interference noise, so
after connecting the AF, the THD is effectively reduced to less than 5% as in Fig. 65-b.
(a)
(b)
Fig. 65. Phase-a voltage (upper) and it is spectrum (lower):
(a) Before connecting the AF topology (b) after connecting the AF topology
The EMI noise level before connecting the AF in Fig. 66-a shows a noise level of ~ 20dB at
operating frequency, ~18dB at switching frequency (5KHz), and almost -40dBs for the most
high frequencies (>0.2 MHz). These noise component frequencies have bad effect on the
control system if not filtered. When the AF is connected the EMI noise level is tuned down
to ~-18dB at operating frequency, ~-25dB at switching frequency and less than ~-60dBs for
the most high frequencies as shown in Fig. 66-b.
From the results presented it can be seen that the steady state performance of the HDTC
with the proposed filter topology is much better than the performance presented by
Zhong(1997). This result can also be compared with experimental result presented by Tang
et al (2004) though the effective average switching sampling time in that method is much
less than the selected sampling period (150μs) and that due to the fact space vector
modulation was used to drive the inverter.
The motor voltage waveform is better than that provided by Yilmaz, et al(2000), beside the
filter topology presented by Yilmaz, et al (2000) is continuously required to be tuned when
the switching frequency is changed. In addition in order to obtain acceptable sinusoidal
196
Torque Control
waveform, the resistor value used in the RLC loop is small, which involves larger current to
flow through the loop composed of the RLC and the inverter which in turn causes over
loading to the inverter elements.
(a)
(b)
Fig. 66. EMI noise level: (a) before (b) after connecting the AF topology
7. References
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Algorithm for PMSM with Minimum Torque Ripples”, COMPEL, Vol.28, No.2, p.p.
437-453, April 2009.
Dariusz, S., Martin, P. K. And Frede, B., (2002),”DSP Based Direct Torque Control of
Permanent Magnet Synchronous Motor Using Space Vector Modulation”
Proceeding of the 2002 IEEE International Symposium on Industrial Electronics,
ISIE 2002 , Vol. 3, 26-29 May , pp. 723-727.
Darwin, R., Morán, L., Dixon, W. J., and Espinoza, J. R,. (2003), “Improving Passive Filter
Compensation Performance With Active Techniques,” IEEE Transaction on
Industrial Electronics, Vol. 50( 1), pp. 161-170, Feb. 2003.
Depenbrock, M., (1984), “Direct Self-Control”, U.S. Patent, No: 4678248, Oct. 1984.
Depenbrock, M., (1988), “Direct Self-Control of inverter-fed machine”, IEEE Transactions on.
Power Electronics Vol. 3, No.4, pp. 420-429, Oct. 1988.
Dirk, D., Jacobs, J., De Doncker, R. W. and Mall, H.G.,(2001), “ A new Hybrid Filter to
Dampen Resonances and Compensate Harmonic Currents in Industrial Power
System With Power Factor Correction Equipment,” IEEE Transaction on Industrial
Electronics, Vol. 16(6), pp. 821-827, Nov. 2001.
Erik, P.,(1992), “ Transient Effects in Application of PWM Inverters to Induction Motors”,
IEEE Transactions on Industry Application, Vol. 28, No. 5, pp. 1095-1101 Sept./
Oct. 1992.
French, C. and Acarnley, P., (1996), “Direct torque control of permanent magnet drives,”
IEEE Transactions on Industrial Applications., Vol. 32 Issue: 5, pp.1080–1088,
Sept./Oct. 1996.
Gulez K., Adam A. A., Pastacı H. (2007), “Passive Filter Topology to Minimize Torque
Ripples and Harmonic Noises in IPMSM Derived with HDTC”, IJE-International
Journal of Electronics, Vol. 94, No:1, p.p.23-33, Jan. (2007).
Torque Control of PMSM and Associated Harmonic Ripples
197
Gulez K., Adam A. A., Pastacı H. (2008) “Torque Ripples and EMI Noise Minimization in
PMSM Using Active Filter Topology and Field Oriented Control”, IEEETransactions on Industrial Electronics, Vol. 55, No. 1, Jan. (2008).
Hideaki, F., Takahiro, Y., and Hirofumi, A.,(2000), “A Hybrid active Filter For Damping of
Harmonic Resonance in Industrial Power Systems,’’ IEEE Transaction on Power
Electronics, Vol. 15 ( 2) , pp. 215-222, Mar. 2000.
Holtz, J. and Springob, L.,(1996), “Identification and Compensation of Torque Ripple in
High- Precision Permanent Magnet Motor Drives”, IEEE Transactions on Industrial
Electronics, Vol. 43, No. 2, April 1996, pp.309-320
Hugh, R., Juan, D. and Morán, L., (2003), “Active power filters as a solution to power quality
problems in distribution networks”, IEEE power & energy magazine Sept./Oct.
2003 pp. 32-40
Jahns, T. M. and Soong, W. L., (1996), “Pulsating torque minimization techniques for
permanent magnet AC motor drives – a review,” IEEE Transactions on Industrial
Electronics, vol. 43, no. 2, pp. 321-330, Feb. 1996.
Jeong-seong, K., Shinji, D. and Muneaki, I.,(2002), “Improvement of IPMSM Sensor less
control performance Using Fourier Transform and Repetitive control”, IECON 02
Industrial Electronic Society Conference,5-7 Nov. 2002, IEEE, vol. 1 pp. 597-602
Luukko, J.,(2000), Direct Torque Control of Permanent Magnet Synchronous Machines Analysis and Implementation, Diss. Lappeenranta University of Technology,
Lappeenranta, Stockholm, 2000.
Satomi, H., Muneaki, I. and Takamasa, H., (2001), “Vibration Suppression Control Method
for PMSM Utilizing Repetitive Control With Auto-tuning Function and Fourier
Transform” IECON’01: The 27th Annual Conference of IEEE Industrial Electronics
Society, 2001, pp 1673-1679.
Se-Kyo, C., Hyun-Soo, K. and Myun-Joong, Y., (1998),”A new Instantaneous Torque Control
of PM Synchronous Motor for High-Performance Direct-Drive Applications”, IEEE
Transactions on Power Electronics Vol. 13, No. 3.
Springob, L. and Holtz, J., (1998),“High-Bandwidth Current Control for Torque-Ripple
Compensation in PM Synchronous Machines”, IEEE Transactions on Industrial
Electronics, Vol. 45, NO. 5, October 1998, pp.713-721
Takahashi, I. and Naguchi, T.(1998), “A new quick-response and high efficiency control
strategy of an induction motor,” IEEE Transactions on Industrial Applications, vol.
34, No. 6 pp. 1246-1253, Nov./Dec. 1998.
Tan, Z. Y. and Li, M., (2001), ”A Direct Torque Control of Induction Motor Based on Three
Level Inverter” , IEEE, PESC’200, Vol. 2 pp. 1435-1439
Tang, L., Zhong, L., Rahman, M. F. and Hu, Y., (2004), “A Novel Direct Torque Controlled
Interior Permanent Magnet Synchronous Machines Drive with Low Ripple in Flux
and Torque and Fixed Switching Frequency”, IEEE Transactions on Power
Electronics Vol. 19, No. 2, Mar. 2004
Tang, L., Zhong, L, Rahman, M. F., and Hu, Y., (2001), “ A novel Direct Torque Control for
Interior Permanent Magnet Synchronous Machine Drive System with Low Ripple
In torque and Flux-A Speed Sensor less Approach” IEEE, IAS, 13-18 Oct. 2002, vol.
1, pp.104-111.
Vas, P.,(1996), Electrical Machines and Drives- A Space-Vector Theory Approach, Oxford,
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Torque Control
Yilmaz, S., David, A. T., and Suhan, R., (2000), “New Inverter Output Filter Topology for
PWM
Motor Drives”, IEEE Transaction on Power Electronics Vol. 15 No 6, pp.
1007-1017, Nov. 2000
Zhong, L., Rahman, M. F., Hu, W.Y. and Lim, K.W., (1997), “Analysis of direct torque
control in permanent magnet synchronous motor drives,” IEEE Transactions. on
Power Electronics, vol. 12 Issue: 3, pp. 528 –536, May 1997.
Part 3
Special Controller Design and Torque Control
of Switched Reluctance Machine
8
8
Switched Reluctance Motor
Jin-Woo Ahn, Ph.D
Kyungsung University
Korea
1. Introduction
Switched Reluctance Motors (SRM) have inherent advantages such as simple structure with
non winding construction in rotor side, fail safe because of its characteristic which has a
high tolerances, robustness, low cost with no permanent magnet in the structure, and
possible operation in high temperatures or in intense temperature variations. The torque
production in switched reluctance motor comes from the tendency of the rotor poles to align
with the excited stator poles. The operation principle is based on the difference in magnetic
reluctance for magnetic field lines between aligned and unaligned rotor position when a
stator coil is excited, the rotor experiences a force which will pull the rotor to the aligned
position. However, because SRM construction with doubly salient poles and its non-linear
magnetic characteristics, the problems of acoustic noise and torque ripple are more severe
than these of other traditional motors. The torque ripple is an inherent drawback of
switched reluctance motor drives. The causes of the torque ripple include the geometric
structure including doubly salient motor, excitation windings concentrated around the
stator poles and the working modes which are necessity of magnetic saturation in order to
maximize the torque per mass ratio and pulsed magnetic field obtained by feeding
successively the different stator windings. The phase current commutation is the main cause
of the torque ripple.
The torque ripple can be minimized through magnetic circuit design in a motor design stage
or by using torque control techniques. In contrast to rotating field machines, torque control of
switched reluctance machines is not based on model reference control theory, such as fieldoriented control, but is achieved by setting control variables according to calculated or
measured functions. By controlling the torque of the SRM, low torque ripple, noise reduction
or even increasing of the efficiency can be achieved. There are many different types of control
strategy from simple methods to complicated methods. In this book, motor design factors are
not considered and detailed characteristics of each control method are introduced in order to
give the advanced knowledge about torque control method in SRM drive.
1.1 Characteristic of Switched Reluctance Motor
The SRM is an electric machine that converts the reluctance torque into mechanical power.
In the SRM, both the stator and rotor have a structure of salient-pole, which contributes to
produce a high output torque. The torque is produced by the alignment tendency of poles.
The rotor will shift to a position where reluctance is to be minimized and thus the
inductance of the excited winding is maximized. The SRM has a doubly salient structure,
202
202
Torque Control
Torque Controlo
but there are no windings or permanent magnets on the rotor [Lawrenson, 1980]. The rotor
is basically a piece of steel (and laminations) shaped to form salient poles. So it is the only
motor type with salient poles in both the rotor and stator. As a result of its inherent
simplicity, the SRM promises a reliable and a low-cost variable-speed drive and will
undoubtedly take the place of many drives now using the cage induction, PM and DC
machines in the short future. The number of poles on the SRM’s stator is usually unequal to
the number of the rotor to avoid the possibility of the rotor being in a state where it cannot
produce initial torque, which occurs when all the rotor poles are aligned with the stator
poles. Fig.1 shows a 8/6 SRM with one phase asymmetric inverter. This 4-phase SRM has 8
stator and 6 rotor poles, each phase comprises two coils wound on opposite poles and
connected in series or parallel consisting of a number of electrically separated circuit or
phases. These phase windings can be excited separately or together depending on the
control scheme or converter. Due to the simple motor construction, an SRM requires a
simple converter and it is simple to control.
Inverter
Speed Controller
SRM
Encoder
Fig. 1. SRM with one phase asymmetric inverter
The aligned position of a phase is defined to be the situation when the stator and rotor poles
of the phase are perfectly aligned with each other (ߠଵ െ ߠଶ ), attaining the minimum
reluctance position and at this position phase inductance is maximum (‫ܮ‬௔ ). The phase
inductance decreases gradually as the rotor poles move away from the aligned position in
either direction. When the rotor poles are symmetrically misaligned with the stator poles of
a phase (ߠଷ െ ߠ௦ ), the position is said to be the unaligned position and at this position the
phase has minimum inductance (‫ܮ‬௨ ). Although the concept of inductance is not valid for a
highly saturated machine like SR motor, the unsaturated aligned and unaligned incremental
inductances are the two key reference positions for the controller. The relationship between
inductance and torque production according to rotor position is shown in Fig. 2.
There are some advantages of an SRM compared with the other motor type. The SRM has a
low rotor inertia and high torque/inertia ratio; the winding losses only appear in the stator
because there is no winding in the rotor side; SRM has rigid structure and absence of
permanent magnets and rotor windings; SRM can be used in extremely high speed
application and the maximum permissible rotor temperature is high, since there are no
permanent magnets and rotor windings [Miller, 1988].
203
203
Switched Reluctance Motor
Switched Reluctance Motor
(a)
(b)
Fig. 2. (a) Inductance and (b) torque in SRM
Constructions of SRM with no magnets or windings on the rotor also bring some
disadvantage in SRM. Since there is only a single excitation source and because of magnetic
saturation, the power density of reluctance motor is lower than PM motor. The construction
of SRM is shown in Fig. 3. The dependence on magnetic saturation for torque production,
coupled with the effects of fringing fields, and the classical fundamental square wave
excitation result in nonlinear control characteristics for the reluctance motor. The double
saliency construction and the discrete nature of torque production by the independent
phases lead to higher torque ripple compared with other machines. The higher torque
ripple, and the need to recover some energy from the magnetic flux, also cause the ripple
current in the DC supply to be quite large, necessitating a large filter capacitor. The doubly
salient structure of the SRM also causes higher acoustic noise compared with other
machines. The main source of acoustic noise is the radial magnetic force induced. So higher
torque ripple and acoustic noise are the most critical disadvantages of the SRM.
The absence of permanent magnets imposes the burden of excitation on the stator windings
and converter, which increases the converter kVA requirement. Compared with PM
brushless machines, the per unit stator copper losses will be higher, reducing the efficiency
and torque per ampere. However, the maximum speed at constant power is not limited by
the fixed magnet flux as in the PM machine, and, hence, an extended constant power region
of operation is possible in SRM.
The torque-speed characteristics of an SRM are shown in Fig. 4. Based on different speed
ranges, the motor torque generation has been divided into three different regions: constant
torque, constant power and falling power region.
204
204
Torque
e Control
Torque Controlo
Fig
g. 3. The construcction of SRM
Zb
Zp
Fig
g. 4. The torque-speed of SRM
Th
he base speed ǚb is
i the maximum speed at which maximum
m
curren
nt and rated torqu
ue can
be achieved at rated
d voltage. Below ǚb, the torque ca
an be maintained constant or control the
fatt-top phase curreent. At lower sp
peed, the phase current
c
rises alm
most instantly aftter the
ph
hase switches turrn-on since the back
b
EMF is sma
all at this time. S
So it can be set at any
desired level by means of regulaators (hysteresiss or PWM conttroller). Thereforre, the
adjjustment of firing
g angle and phase current can red
duce noise and im
mprove torque rip
pple or
effficiency.
Wiith speed increasse, the back-EMF
F is increased. An
n advance turn-on angle is necesssary to
reaach the desired current
c
level befo
ore rotor and sta
ator poles start to
o overlap. The desired
d
currrent level depen
nds on the speeed and the load condition. At th
he same time, sin
nce no
currrent chopping appears
a
during th
he dwell angle, on
nly the angle con
ntrol can be used at this
staage. So the torquee cannot be kept constant and is falling
f
linearly w
with the speed inccrease,
ressulting in a constaant power produ
uction.
In the falling poweer region, as the speed increases, the turn-on ang
gle cannot be adv
vanced
furrther. Because torrque falls off morre rapidly, the co
onstant power can
nnot be maintain
ned. As
thee speed grows, th
he tail current of the
t phase windin
ng extends to the n
negative torque region.
r
205
205
Switched Reluctance Motor
Switched Reluctance Motor
The tail current may not even drop to zero. In the high speed operation, the continued
conduction of current in the phase winding can increase magnitude of phase current and the
power density can be increased.
1.2 Equivalent circuit of Switched Reluctance Motor
The equivalent circuit for SRM can be consisting of resistance and inductance with some
condition. The effects of magnetic saturation, fringing flux around the pole corners, leakage
flux, and the mutual coupling of phases are not considered. The linear analytical model of
the SRM can be described by three differential equations, which can be classified as the
voltage equation, the motional equation and the electromagnetic torque equation. The
voltage equation is:
ൌ Ǥ ‹ ൅
ୢ஛ሺ஘ǡ୧ሻ
(1)
ୢ୲
An equivalent circuit of the SRM is shown in Fig. 5. Where V is the applied phase voltage to
phase, R is the phase resistance, and e is back-EMF. Ordinarily, e is the function of phase
current and rotor position, and nj can be expressed as the product of inductance and winding
current:
ɉሺɅǡ ‹ሻ ൌ ሺɅǡ ‹ሻǤ ‹
(2)
And from (1) and (2), the function can be rewritten as:
ൌ Ǥ ‹ ൅ ୢ஛ሺ஘ǡ୧ሻ
ୢ୧
ୢ୧
. ୢ୲ ൅ ୢ஛ሺ஘ǡ୧ሻ
ୢ஘
ୢ஘
Ǥ ୢ୲
(3)
R
V
Fig. 5. Equivalent circuit of SR motor
For the electromechanical energy conversion, a nonlinear analysis takes account of the
saturation of the magnetic circuit. Generally, the stored magnetic energy is defined as Wf
and the co-energy is defined as Wc :
୤ ൌ ‫†‹ ׬‬ɗ
(4)
ୡ ൌ ‫ ׬‬ɗ †‹
(5)
The relationship between energy (Wf) and co-energy (Wc) as a function of flux and current
shows in Fig. 6.
When rotor position matches the turn-on position, the phase switches are turned on; the
phase voltage starts to build up phase current. At this time, one part of the input energy will
206
206
Torque Control
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Fig. 6. Relationship between energy (Wf) and co-energy (Wc)
be stored in magnetic field. With the increasing inductance, the magnetic field energy will
increase until turn-off angle. The other parts of input energy will be converted to mechanical
work and loss. In Fig. 7, the flux of the SR motor operation is not a constant; nevertheless,
uniform variation of the flux is the key point to obtain smoothing torque. W1 is the
mechanical work produced during the magnetization process, in other words, W1 is coenergy in energy conversion. F+W2 is magnetic field energy between turn-on and turn-off.
During the derivation of the energy curve and the energy balance, constant supply voltage
Vs and rotor speed ǚ are assumed.
When rotor position matches the turn-off position, phase switches are turned off. So the
power source will stop to input energy. But magnetic field energy is F+W2 at that moment.
The magnetic field energy needs to be released, and then the phase current starts to
feedback energy to power source. At this time, some of magnetic field energy, which is W2,
is converted into mechanical work and loss. The surplus of field energy F is feedback to the
power source.
Fig. 7. Graphical interpretation of energy and co-energy for SR motor
The analytical answer of the current can be obtained from (3). The electromagnetic torque
equation is:
207
207
Switched Reluctance Motor
Switched Reluctance Motor
ୣ ൌ ப୛ᇱ
ப஘
ൌ
ப୛ᇲ ሺ஘ାο஘ሻିப୛ᇲ ሺ஘ሻ
ο஘
(6)
From (6), an analytical solution for the torque can be obtained. W' is the co-energy, which
can be expressed as:
୧
ᇱ ൌ ‫׬‬଴ ɉ†‹
(7)
And the motion equation is:
ୣ ൌ ப୛
ப୲
൅ ɘ ൅ ୐
(8)
ୢ஘
ɘ ൌ ୢ୲
(9)
Whereܶ௅ , ܶ௘ , J, ǚ and D are load the electromagnetic torque, the rotor speed, the rotor
inertia and the friction coefficient respectively.
The equations which have been mentioned above, can be combined together to build the
simulation model for a SRM system. However, the function of inductance needs to be
obtained by using a finite element method or by doing experiments with a prototype motor.
1.3 Torque control in Switch Reluctance Motor
The torque in SRM is generated toward the direction that the reluctance being to minimized.
The magnitude of torque generated in each phase is proportional to the square of the phase
current which controlled by the converter or drive circuit, and the torque control scheme.
The drive circuit and torque control scheme directly affected to the performance and
characteristic of the SRM. Many different topologies have emerged with a reduced number
of power switch, faster excitation, faster demagnetization, high efficiency, high power factor
and high power through continued research. Conventionally, there has always been a tradeoff between gaining some of the advantages and losing some with each new topology.
The torque is proportional to the square of current and the slope of inductance. Since the
torque is proportional to the square of current, it can be generated regardless of the direction
of the current. And also because the polarity of torque is changed due to the slope of
inductance, a negative torque zone is formed according to the rotor position. To have a
motoring torque, switching excitation must be synchronized with the rotor position angle.
As shown in Fig. 8, an inductance profile is classified into three regions,
increasingሺߠ௠௜௡ଵ ̱ߠ௠௔௫ଵ ሻ, constant ሺߠ௠௔௫ଵ ̱ߠ௠௔௫ଶ ሻ and decreasing ሺߠ௠௔௫ଶ ̱ߠ௠௜௡ଶ ሻ period.
If a constant exciting current flows through the phase winding, a positive torque is
generated. When that is operated in inductance increasing period ሺߠ௠௜௡ଵ ̱ߠ௠௔௫ଶ ሻ and viceversa in inductance decreasingሺߠ௠௔௫ଶ ̱ߠ௠௜௡ଶ ሻ.
In the case of a constant excitation, it cannot be generated any torque, because a positive
torque and negative one are canceled out, and the shaft torque becomes zero. As a result, to
achieve an effective rotating power, switching excitation must be synchronized with the
inductance profile. In order to derive the phase current from (3), exact information about the
inductance profile of the SRM is essential. In (10), the first term of the right side is voltage
drops of winding resistance, the second term is the voltage drop of reactance and the last
term is both the emf (electromotive magnetic force) and the mechanical output.
ൌ ‹ ൅ ‹ሺ–ሻ
ୢࣦሺ஘ǡ୧ሻ
ୢ஘
ɘ ൅ ࣦሺɅǡ ‹ሻ
ୢ୧ሺ୲ሻ
ୢ୲
(10)
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T
T
T
T
T
T
T
Fig. 8. (a)Inductance profile and (b) Torque zone
where, ɘ is the angular speed of the rotor.
In (10), the second in the right side can be considered as the back-emf; therefore, this term is
expressed as:
‡ൌ
ୢࣦሺ஘ǡ୧ሻ
ୢ஘
ɘ‹ሺ–ሻ ൌ ɘ‹ሺ–ሻ
where, ൌ
ୢࣦሺ஘ǡ୧ሻ
ୢ஘
(11)
(12)
As shown in (11), the back-emf equals to that of the DC motor. And also torque equation in
(12) is equivalent with that of the DC series motor; therefore, the speed-torque of the
magnetic energy in SRM is different from that of a mutual torque machine. And it operates
more saturated level. The field energy in the magnetization curve is shown in Fig. 9.
Fig. 9. Magnetizing curve and flux-linkage curve of SRM
209
209
Switched Reluctance Motor
Switched Reluctance Motor
It shows the magnetization curves from an aligned to an unaligned position. In SRM design,
when poles of a rotor and a stator are aligned, the other phases are unaligned. In an aligned
position, it has a maximum inductance with magnetically saturated easily. On the other
hand, in an unaligned position it has a minimum inductance. As magnetic saturation is
proportional to a rotor position, the magnetization curve according to the rotor position is
an important factor to investigate the motor characteristics and to calculate the output
power. The torque produced by a motor can be obtained by considering the energy
variation. The generated torque is as:
ୢ୛ᇲ
ൌ ሾ ୢ஘ ሿ୧ୀୡ୭୬ୱ୲Ǥ
(13)
where, w' means the co-energy, and it is given as:
୧
ᇱ ൌ ‫׬‬଴ ɉ†‹
(14)
Under a constant phase current as shown in Fig. 10, when the rotor and total flux linkage are
shifted from A to B, the SRM exchanges energy with the power source; thus, the stored field
energy is also changed. The limitation to a constant current is that mechanical work done
during the shifting region is exactly equal to the variation of co-energy. At a constant
current, if the displacement between A and B is AB, the variation of energy received from
the source can be expressed as:
ȟୣ ൌ (15)
ȟୡ ൌ െ (16)
Then the mechanical work can be written as:
ȟ୫ ൌ οɅ ൌ οୣ െ οୡ ൌ (17)
Fig. 10. Calculation of instant torque by the variation of co-energy at constant current
The above equation just shows the instantaneous mechanical output; therefore, in order to
understand the characteristics of the motor, the average torque generated during an energy
conversion cycle may be considered. The mechanical output is expressed as an area in an
energy conversion curve (i-ɉ graph), the processes are separated with two stages as shown
in Fig. 11.
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Fig. 11. Average torque (Energy conversion loop)
The total flux linkage is increased with phase current and inductance. Its operating area (i, Ȝ)
follows the curve between 0 and C as shown in Fig. 11(a). When the total flux linkage exists
at point C, the mechanical work and stored energy between 0 and C becomes ܹ௠ଵ and ܹ௙ ,
respectively. Therefore, the total energy received from the source is summed up the
mechanical work and the stored energy. On the other hand, when the demagnetizing
voltage is applied at the point C, terminal voltage becomes negative; then current flows to
the source through the diode. Its area follows the curve between C and 0 in Fig. 11(b).
During process, some of the stored energy in SRM are appeared as a mechanical power;.
211
211
Switched Reluctance Motor
Switched Reluctance Motor
During the energy conversion, the ratio of supply and recovered energy considerably affects
to the efficiency of energy conversion. To augment the conversion efficiency, the motor must
be controlled toward to increase the ratio. Lawrenson [Lawrenson,1980]] proposed the
energy ratio E that explains the usage ability of the intrinsic energy.
ൌ ୫ଵ ൅ ୫ଶ
(18)
ൌ ୢ ൌ ୤ െ ୫ଶ (19)
The energy ratio is similar to the power factor in AC machines. However, because this is
more general concept, it is not sufficient to investigate the energy flowing in AC machines.
The larger energy conversion ratio resulted in decreasing a reactive power, which improves
efficiency of the motor. In a general SRM control method, the energy conversion ratio is
approximately 0.6 - 0.7.
୛
ൌ ୛ାୖ
(20)
In conventional switching angle control for an SRM, the switching frequency is determined
by the number of stator and rotor poles.
ଵ
ˆୣ ൌ ଶ ’ୱ ’୰ ሾœሿ
(21)
The general switching angle control has three modes, i.e., flat-topped current build-up,
excitation or magnetizing, and demagnetizing. Each equivalent circuit is illustrated in Fig. 12.
+
i
+
R
v
v
+
v
s
L
+
R
L
v
i
+
R
v
v
+
s
L
i
R
L
(a)
(b)
v
v
R
+
v
s
L
+
e
+
R
L
+
e
+
(c)
Fig. 12. Equivalent circuits when general switching angle control
(a) build-up mode (b) excitation mode (c) demagnetizing mode
Fig. 12(a) is a build-up mode for flat-topped current before inductance increasing. This
mode starts at minimum inductance region. During this mode, there is no inductance
variation; therefore, it can be considered as a simple RL circuit that has no back-emf. Fig.
12(b) shows an equivalent circuit at a magnetizing mode. In this mode, torque is generated
from the built-up current. Most of mechanical torque is generated during this mode. A
demagnetizing mode is shown in Fig. 12(c). During this mode, a negative voltage is applied
to demagnetize the magnetic circuit not to generate a negative torque.
An additional freewheeling mode shown in Fig.13 is added to achieve a near unity energy
conversion ratio. This is very effective under a light-load. By employing this mode, the
energy stored is not returned to the source but converted to a mechanical power that is
multiplication of phase current and back-emf. This means that the phase current is
decreased by the back-emf.
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Fig. 13. Equivalent circuit of additional wheeling mode supplemented to conventional
If the increasing period of inductance is sufficiently large compared with the additional
mode, the stored field energy in inductance can be entirely converted into a mechanical
energy; then the energy conversion ratio becomes near unity.
1.4 Power converter for Switched Reluctance Motor
The selection of converter topology for a certain application is an important issue. Basically,
the SRM converter has some requirements, such as:
x
Each phase of the SR motor should be able to conduct independently of the other
phases. It means that one phase has at least one switch for motor operation.
x
The converter should be able to demagnetize the phase before it steps into the
regenerating region. If the machine is operating as a motor, it should be able to excite
the phase before it enters the generating region.
In order to improve the performance, such as higher efficiency, faster excitation time, fast
demagnetization, high power, fault tolerance etc., the converter must satisfy some
additional requirements. Some of these requirements are listed below.
Additional Requirements:
x
The converter should be able to allow phase overlap control.
x
The converter should be able to utilize the demagnetization energy from the outgoing
phase in a useful way by either feeding it back to the source (DC-link capacitor) or
using it in the incoming phase.
x
In order to make the commutation period small the converter should generate a
sufficiently high negative voltage for the outgoing phase to reduce demagnetization
time.
x
The converter should be able freewheel during the chopping period to reduce the
switching frequency. So the switching loss and hysteresis loss may be reduced.
x
The converter should be able to support high positive excitation voltage for building up
a higher phase current, which may improve the output power of motor.
x
The converter should have resonant circuit to apply zero-voltage or zero-current
switching for reducing switching loss.
1.4.1 Basic Components of SR Converter
The block diagram of a conventional SRM converter is shown in Fig. 14. It can be divided
into: utility, AC/DC converter, capacitor network, DC/DC power converter and SR motor.
213
213
Switched Reluctance Motor
Switched Reluctance Motor
Fig. 14. Component block diagram of conventional SR drive
The converter for SRM drive is regarded as three parts: the utility interface, the front-end
circuit and the power converter as shown in Fig. 15. The front-end and the power converter
are called as SR converter.
Fig. 15. Modules of SR Drive
D1
is
Vdc1_ ripple
Vs
D1
D3
Vs
Vdc _ ripple
VG
D2
D2
Vdc 2 _ ripple
D4
(a) Voltage doubler rectifier
Q1
Q3
(b) 1-phase diode bridge rectifier
Q1
Q3
is
is
Vdc _ ripple
Vdc _ ripple
Vs
D2
D4
(c) Half controlled rectifier
Fig. 16. Utility interface
Vs
Q2
Q4
(d) Full controlled rectifier
214
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Torque Control
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A. Utility Interface
The main function of utility interface is to rectify AC to DC voltage. The line current input
from the source needs to be sinusoidal and in phase with the AC source voltage. The
AC/DC rectifier provides the DC bus for DC/DC converter. The basic, the voltage doubler
and the diode bridge rectifier are popular for use in SR drives.
B. Front-end circuit
Due to the high voltage ripple of rectifier output, a large capacitor is connected as a filter on
the DC-link side in the voltage source power converter. This capacitor gets charged to a
value close to the peak of the AC input voltage. As a result, the voltage ripple is reduced to
an acceptable valve, if the smoothing capacitor is big enough. However, during heavy load
conditions, a higher voltage ripple appears with two times the line frequency. For the SR
drive, another important function is that the capacitor should store the circulating energy
when the phase winding returned to.
Single Capacitor
Two Capacitor in series
Pure Capacitor
Two Capacitor in Parallel
Split dc-link
Doubler dc-link voltage
Passive type
Capacitor with diode
Series type
Parallel type
Series - Parallel type
Connected dc-link
Series - Parallel active type 2
`
Active type
Series - Parallel active type 1
Series type
Separated dc-link
Parallel type
Series - Parallel active type 3
Fig. 17. Classification of capacitive type front-end topology
To improve performance of the SR drive, one or more power components are added. In this
discussion, two capacitors networks are considered and no inductance in the front-end for
reasonable implementation. Two types of capacitor network are introduced below: a two
capacitors network with diodes and two capacitors with an active switch. The maximum
boost voltage reaches two times the DC-link voltage.
The two capacitors network with diodes, which is a passive type circuit, is shown in Fig. 19.
The output voltages of the series and parallel type front-ends are not controlled. Detailed
characteristics are analyzed in Table 1.
215
215
Switched Reluctance Motor
Switched Reluctance Motor
(a) Single cap.
(b) Two cap. in series
(d) Split dc-link
(c) Two cap. in parallel
(e) Doublers dc-link voltage
Fig. 18. Pure capacitor network
(a) Series type
(b) Parallel type
c) Series-parallel type
Fig. 19. Two capacitors network with diodes
Type
Series
Parallel
Series-parallel
No. of Capacitor
2
2
2
No. of Diode
1
1
3
VC1+VC2
VC2
VC1+VC2
Vdc
VDC
VDC
VDC
Spec. Boost Capacitor
VDC
Vboost
VDC
Spec. Diode
VDC
VDC
VDC
Vboost
Table 1. Characteristics of two capacitor network with diodes
The active type of the two capacitors network connected to the DC-link, which is a two
output terminal active boost circuit, is shown in Fig. 20 and Table 2.
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Torque Control
Torque Controlo
(a) Series-parallel active type 1
(b) Series-parallel active type 2
Fig. 20. Active type of two capacitors network connected to DC-link
Type
Series-parallel 1
Series-parallel 2
No. of Capacitor
2
2
No. of Switch
1
1
No. of Diode
2
3
Vboost
VC1+VC2
VC2
Vdemag
- (VC1+VC2)
- (VC1+VC2)
Dc-link
VDC
VDC
Spec. Boost Capacitor
VDC
Vboost
Spec. Diode
VDC
VDC
Table 2. Characteristics of active type of two capacitors connected to DC-link
The active type of two capacitors network separated to DC-link is shown in Fig. 21 and
Table 3.
(a) Series type
(b) Parallel type
(c) Series-parallel active type3
Fig. 21. Active type of two capacitors network separated to DC-link
217
217
Switched Reluctance Motor
Switched Reluctance Motor
Type
Series
Parallel
Series-parallel type 3
No. of Capacitor
2
2
2
No. of Switch
1
1
1
No. of Diode
1
1
3
Vboost
VC1+VC2
VC2
VC2
Vdemag
- ( VC1+VC2)
- VC2
- ( VC1+VC2)
Vdc
VDC
VDC
VDC
Spec. Capacitor
VDC
Vboost
VC2
Spec. Diode
VDC
VDC
VC2
Table 3. Characteristics of active type of two capacitors separated to DC-link
C. Power converter
The power circuit topology is shown in Fig. 22 and Table 4. In this figure, five types of DCDC converter are shown.
(a) One switch
(b) Asymmetric
(d) Full bridge
Fig. 22. Active type of two capacitors network separated to DC-link
(c) Bidirectional
(e) Shared switch
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Torque
e Control
Torque Controlo
Type
One switch
No. Switch
1
2
2
4
3
No. Diode.
1
2
0
0
3
A
Asymetric
Bi-diirectional Full b
bridge
Shared sw
witch
No. Phase
1
1
1
1
2
VExcitation
Vdc
Vdc
Vdc/2
Vddc
Vdc
VDemagnetitation
Vdc
Vdc
Vdc/2
Vddc
Vdc
C
Current
Direction
n
Uni.
Uni.
Bi.
Un
ni.
Uni.
Taable 4. Compariso
on of 5 types of DC/DC converter topology
1.4
4.2 Classification
n of SR converte
er
On
ne of the well-kn
nown classificatio
ons of SRM conv
verters only conssidering the num
mber of
po
ower switches an
nd diodes is intro
oduced [miller,19
990]. Different fro
om the classification, a
no
ovel classification
n, which focusess on the charactteristics of conv
verters, is propossed in
[Krrishnan,2001].
A. SR converter by
y phase switch
Th
he classification of power converteer focuses on the number of poweer switches and diodes.
d
Th
hese options havee given way to power
p
converter topologies with
h q, (q+1), 1.5q, and
a
2q
sw
witch topologies, where q is thee number of mo
otor phases. Theese configuration
ns are
claassified and listed
d in Fig. 23 for easy reference. A tw
wo-stage power cconverter configu
uration
wh
hich does not fitt this categorizaation based on th
he number of m
machine phases is
i also
inccluded.
Fig
g. 23. SR converteer classification by
y phase switch
Sw
witched Reluctance Motor
Switched Reluctance Motor
219
219
All the converter topologies, excep
pt the two-stagee power convertter, assume that a DC
vailable for theirr inputs. This DC
C source may be from batteries orr most
voltage source is av
usu
ually a rectified AC supply with a filter to provid
de a stable DC in
nput voltage to the
t SR
con
nverters.
Ev
ven though it is easy with the classsification to find the number of semiconductors and the
cosst by counting the number of
o active compo
onents, it does not show imp
portant
chaaracteristics of a power converteer, and the voltage ratings for th
he power switchees and
dio
odes are difficult to consider.
B. SR converter by commutation
ber of switches may obtain diffferent
Diffferent converterrs which have the same numb
performance and ch
haracteristics. Fro
om this point of view, such a classsification is not useful
forr finding the charracteristic of an SR
RM converter.
Th
he three types in the classification
n were presented as: extra commu
utation, half bridg
ge and
sellf commutation. In the extra co
ommutation circuit, the capacitiive, the magnetiic and
disssipative circuit is included. However,
H
the disstinction betweeen three types in
i the
claassification is not
n
clearly defin
ned. Convention
nally, the half bridge and th
he self
com
mmutation circu
uit also need a large capacitor in the front-end
d. They could also
a
be
claassified as capacitive circuit. Moreeover, the characcteristic of circuits which contain one or
mo
ore inductances iss not shown in th
he classification.
Fig
g. 24. SR converteer configuration by
b commutation type
t
An
n SR converter configuration
c
by commutation ty
ype is shows in Fig. 24. Based on
o the
com
mmutation type of
o the most of thee returned or disssipated stored maagnetic energy, th
he four
maajor sorts are classsified: dissipativ
ve, magnetic, resonant and capaccitive type. Becau
use the
220
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Torque Control
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capacitive type is focused in this discussion, the capacitive converter category is split into
several subclasses. The concepts for passive and active converters are introduced. The
distinction between active and passive is determined by whether they include a controllable
power switch or not.
1. Dissipative converter
The dissipative type dissipates some or all of the stored magnetic energy using a phase
resistor, an external resistor or both of them. The remaining energy is transformed to
mechanical energy. Therefore, none of the stored magnetic energy in the phase winding is
returned to DC-link capacitor or source. The advantage of this type of converter is that it is
simple; a low cost and has a low count of semiconductor components.
(a) R-dump
(b) Zener-dump
Fig. 25. Two types of dissipative SR converter
2. Magnetic converter
The magnetic type is where the stored magnetic energy is transferred to a closely coupled
second winding. Of course, that energy could be stored in DC-link capacitor or used to
energize the incoming phase for multi-phase motors or use special auxiliary winding. The
major advantage is a simple topology. The one switch per phase power circuit can be used.
However, the potential rate of change of current is very high due to the stored magnetic
energy is recovered by a magnetic manner. And the coupled magnetic phase winding which
should be manufactured increases the weight of copper and cost of motor. Moreover, the
power density of the motor is lower than that of the conventional ones.
(a) Bifilar
(b) Single controllable switch
Fig. 26. Two types of magnetic SR converter
3. Resonant converter
The resonant type has one or more external inductances for buck, boost or resonant
purposes. Conventionally, the inductance, the diode and the power switch are designed as a
snubber circuit. So, the dump voltage can be easily controlled, and the low voltage is easy to
boost. In a special case, an inductance is used to construct a resonant converter. The major
advantage is that the voltage of phase winding can be regulated by a snubber circuit.
However, adding an inductance increases the size and cost of converter. The other
221
221
Switched Reluctance Motor
Switched Reluctance Motor
additional components also increase the cost of converter. Three types of resonant type are
shown in Fig. 27. All of them use a snubber circuit, which is composed by a power switch, a
diode and an inductance.
Qr
CCD
Lr
QAH
CDC
DAL
(a) C-Dump
(b) Boost
DAH
A
Dr
QAL
(c) High
demagnetization
Fig. 27. Three types of resonant SR converter
4. Capacitive converter
The magnetic energy in the capacitive converters is fed directly back to the boost capacitor,
the DC-link capacitor or both of the capacitors. Compared to the dissipative, magnetic, and
resonant converters, one component is added in the main circuit. So, this component will
increase the loss of the converter. Different from the other converters, the stored magnetic
energy can easily be fed back using only the inductance of phase winding. Although the
capacitor has an equivalent series resistance (ESR), the loss of ESR is lower than that of other
converters. Therefore, the capacitive converter is more effective for use in SR drive.
Fig. 28. Classification of capacitive SR converter
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222
Torque Control
Torque Controlo
(a) Asymmetric
(c) H-bridge
(b) Shared switch
(d) Modified C-dump
Fig. 29. Single capacitor type in capacitive SR converter
The capacitive converter can be divided two sorts: single capacitor and multi-capacitor type.
i. Single-capacitor converter
Single-capacitor converters have simple structure, which makes them very popular.
Four single capacitor types are shown in Fig. 29. One capacitive converter has as a
simple front-end as shown in Fig. 29(a)-(c). This capacitor should be large enough to
remove the voltage ripple of the rectifier and store the magnetic energy. Since the DClink capacitor voltage is uncontrollable during charging and discharging, this type of
converter is defined as a passive converter. The modified C-dump converter is shown
in Fig. 29(d). In this converter, the boost capacitor only stores the recovered energy to
build up a boost voltage. Unfortunately, one power switch should be placed in front of
the boost capacitor to control the voltage. Because the boost capacitor does not reduce
the DC-link voltage from the rectifier, the fluctuating DC-link voltage is input directly
to the phase winding. The boost capacitor has only to be big enough for the stored
magnetic energy, so the size of this capacitor is smaller than that of conventional DClink capacitor. The Single capacitor in capacitive converters simplifies the construction
of the converter. However, the input voltage for the phase winding is kept fixed by the
DC-link capacitor. If only a boost capacitor is used, the DC-link voltage is fluctuating,
and one power switch is added to control the boost voltage. This extra switch may
increase the cost of converter.
ii. Multi-capacitor converter
Multi-capacitor converters include two or more capacitors in the converter topology to
obtain boost voltage. Extra capacitors may make the topology of converter more
complex. In this discussion, different converter topologies, which include two
capacitors, are considered. The different types of passive type front-ends are shown in
Fig. 30. The passive converter with two capacitors in parallel type is in Fig. 30(a). Due to
the direction of diode, the stored magnetic energy is only feed back to the boost
capacitor. The maximum boost voltage can be obtained by a suitable size of the
capacitor. Because the discharge of the boost capacitor is not controllable in the passive
converter, the voltage of the boost capacitor is changed by the stored magnetic energy
223
223
Switched Reluctance Motor
Switched Reluctance Motor
during different operating condition. When the phase switch is turned on, the voltage
of the boost capacitor may fall very fast until the voltage reaches the DC-link voltage.
Due to the non-linear characteristic of the SR motor, it is difficult to estimate advance
angle or turn-on angle.
A passive converter with two capacitors in series is shown in Fig. 30(b). The stored
magnetic energy charges the two capacitors in series. So, a part of the energy is stored
in the boost capacitor to build up a boost voltage. It has the same advantage as for the
parallel passive converter. However, the voltage rating of the boost capacitor is less
than that of the parallel converter.
(a) Parallel type
(b) Series type
(c) Series-parallel type
Fig. 30. Passive boost converter with two capacitors
Another passive converter of two capacitors in series-parallel type is in Fig. 30(c). This
converter is made of rectifier, the passive boost circuit and an asymmetric converter. The
excitation voltage is the DC-link voltage, but the demagnetization voltage is twice of DClink voltage. The high demagnetization voltage can reduce the tail current and negative
torque; it could also extend the dwell angle to increase the output.
(a) Split dc-link type
(b) Doublers dc-link voltage type
Fig. 31. other passive SR converter with series capacitor type
Other passive SR converter with two series capacitors is shown in Fig. 31. The front-end and
DC-DC converter are same, but the bridge rectifier and the voltage doubling rectifier are
224
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Torque Control
Torque Controlo
connected. The split DC-link converter is shown in Fig. 31(a). The phase voltage of this
converter is a half of DC-link voltage. The double dc-link voltage converter is shown in Fig.
31(b). The phase voltage is same to DC-link voltage. The main advantage of these two
converters is that one switch and one diode per phase is used. However, the voltage rating
of power switch and diode is the twice the input excitation voltage.
The active boost converter with two capacitors connected in parallel is shown in Fig. 32. The
four active boost converters with two capacitors connected in parallel are introduced. To
handle the charging of the capacitor in the beginning of the conduction period, one diode is
needed to series or parallel with the power switch to protect the power switch. When
parallel type 1 and 2 are used with the asymmetric converter, the maximum voltage rating
of the power diode and the switch is the same as the desired boost voltage. While the diode
is connected to the power switch, the boost capacitor is only charged by the stored magnetic
energy. In the beginning, the voltage of the boost capacitor is increased from 0 to the desired
value. For the parallel converter of type 2, a diode in parallel with the power switch is used,
so the boost capacitor can be charged by the DC-link capacitor. Parallel converters of type 3
and 4 which belong to capacitor dump converters are shown in Fig. 32(c) and (d). If the
demagnetization voltage is required to be the same to DC-link, the voltage rating of power
diode and switch is at least twice of DC-link voltage.
(a) Parallel type 1
(c) Parallel type 3
(b) Parallel type 2
(d) Parallel types 4
Fig. 32. Active boost converter with two capacitors connected in parallel
An active boost converter with two series connected capacitors is in Fig. 33(a). The stored
magnetic energy charges the two series connected capacitors, so the boost voltage can be
built up in the boost capacitor. The power switch Qcd is used to control the boost voltage of
the boost capacitor.
225
225
Switched Reluctance Motor
Switched Reluctance Motor
CBoost
QCD
DCD
QAH
DAH
A
CDC
DAL
QAL
(a) Series capacitor type
(b) Series-parallel capacitor type
Fig. 33. Active boost converter
An active boost converter with a series-parallel connection of the two capacitors is shown in
Fig. 33(b). The active capacitor circuit added to the front-end consists of three diodes and
one capacitor. This circuit combines a series-connected and a parallel-connected structure of
two capacitors. Based on this active boost capacitor network, the two capacitors can be
connected in series or parallel during different modes of operation. The operation mode of
whole converter is presented in [Khrishnan,2001]. The fast excitation and demagnetization is
easily obtained from the two series-connected capacitors. The stable voltage achieved with
the two parallel-connected capacitors.
4 types of converter are compared in Table. 5. The converter with two capacitors connected
in series or the converter with two capacitors connected in parallel may obtain a higher
boost voltage than the series-parallel converter. However, an increased boost voltage may
increase the cost of the converter. Since the series-parallel converter can limit the maximum
voltage to twice the DC-link voltage, it is more stable and controllable.
Asymmetric
Vmax
Vcontrol
VC1_rate
VC2_rate
No.Switch
No. Diode
Stability
Vdc
No
Vdc
Vdc
2
2
Good
2-capacitor in 2-capacitor in
series type
parallel type
’/2Vdc
’/2Vdc
Yes
Yes
Vdc
Vdc
’/Vdc
’/2Vdc
3
3
3
3
Normal
Normal
2-capacitor in
series-parallel
2Vdc
optional
Vdc
Vdc
3
4
Good
Table 5. Comparison of 2-capacitor types
2. Torque control strategy
2.1 Angle control method
The switched reluctance drive is known to provide good adjustable speed characteristics
with high efficiency. However, higher torque ripple and lack of the precise speed control are
drawbacks of this machine. These problems lie in the fact that SR drive is not operated with
an mmf current specified for dwell angle and input voltage. To have precise speed control
with a high efficiency drive, SR drive has to control the dwell angle and input voltage
instantaneously. The advance angle in the dwell angle control is adjusted to have high
efficiency drive through efficiency test.
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2.1.1 Switching angle control method
In SRM drive, it is important to synchronize the stator phase excitation with the rotor
position; therefore, the information about rotor position is an essential for the proper
switching operation. By synchronizing the appropriate rotor position with the exiting
current in one phase; the optimal efficiency of SRM can be achieved. In this part various
types of switching angle control method to achieve the optimal efficiency will be discussed.
A. Fixed angle switching method
Current source is a proper type to excite an SRM for its good feature of electromagnetic
characteristics because it produces rectangular or flat-topped current and it is easy to control
the torque production period. Therefore, it is considered as an ideal excitation method for
switched reluctance machine but difficult and expensive to realize it.
To produce similar current shapes in voltage source, it is needed to regulate the supply
voltage in the variable reluctance conditions. Usually PWM or chopper technique is used for
this propose. But it is complex in its control circuit and increases loss. The other technique
which is more simply in control is excitation voltage to form a flat-topped current by using
fixed switching angle at various operation conditions. Fig. 34 shows excitation scheme with
fixed switching angle control method.
Fig. 34. Excitation scheme with fixed switching angle control method
In the fixed angle switching method, the turn-on angle and the turn-off angle of the main
switches in the power converter are fixed; the triggering signals of the main switches are
modulated by the PWM signal. The average voltage of phase winding could be adjusted by
regulating the duty ratio of the PWM signal. So the output torque and the rotor speed of the
motor are adjustable by regulating the phase winding average voltage.
Constant voltage source with current controller is substituted with variable voltage source
to make the current flat-topped. Voltage equation of SRM for a phase is shown in (3). If
winding resistance and magnetic saturation are ignored, an applied voltage to form a flattopped current in the torque developed region is
ܸ௖ ൌ ‫ܫܭ‬௖ ߱
(22)
Where ܸ௖ is amplitude of voltage, K is݀‫ܮ‬Τ݀ߠ, ‫ܫ‬௖ is required current to balance load torque a,
and ߱ is angular velocity. If magnetic saturation is considered, this equation is to be
modified as
227
227
Switched Reluctance Motor
Switched Reluctance Motor
ᇱ ൌ ɐ‫ܫ‬௖ ߱
(23)
Where ߪ is saturation factor. To calculate proper excitation voltage and switching angle for
flat-topped current, let consider phase voltage and current as shown in Fig. 35. ߠ௢௡ and
ߠ௢௙௙ are switching-on and switching-off angle, respectively. Phase current reaches to the
desired value of current, ‫ܫ‬௖ at ߠ௦ , and become flat-topped current by this scheme, and the
current decrease rapidly by reversing the applied voltage. ߠ௢௙௙ is to be set in order to
prevent the generation of negative torque. It can be divided into 3 regions to calculate the
angles and voltages. In Region I and III, switching-on and switching-off angles are
determined respectively. And in Region II, proper excitation voltage is calculated.
Fig. 35. Flat-topped phase current
x
Region I : ߠ௢௡ ൑ ߠ ൑ ߠ௦ ( switching-on angle determination )
ߠ௢௡ is determined in this region. It is to ensure that current is to be settled to the desired
value at ߠ௦ . In this region, voltage equation becomes (24).
ୢ୧
ୡୱ ൌ ‹ ൅ ୳ ୢ୲
(24)
Where ‫ܮ‬௨ , is the minimum value of the inductance.
Required time, ‫ݐ‬௦ to build up a phase current from 0 to‫ܫ‬௖ , which is the current to balance
load torque, is derived from (23) and (24).
–ୱ ൌ ஘౩ ି஘౥౤
ன
ൌെ
୐౫
ୖ
ୖ
Žሺͳ െ ஢୏னሻ
(25)
Therefore, ߠ௢௡ is
Ʌ୭୬ ൌ Ʌୱ െ ன୐౫
ୖ
ୖ
Žሺͳ െ ஢୏னሻ
(26)
ߠ௢௡ is affected merely by saturation factor and not by speed variation except the range
where speed is very low. Therefore, it can be fixed at the center of variation range of
switching-on and compensate current build-up via applied voltage regulation for simple
control.
x
Region III : ߠ௢௙௙ ൑ Ͳ ൑ ߠ௧ ( Switching-off angle determination )
228
228
Torque Control
Torque Controlo
In this region, applied voltage must be reversed to accelerate current decay. It is divided
into two sub-regions:
x
Sub-region III-1
Voltage and current equation are as follows.
ୢ୧
െୡୱ ൌ ‹ ൅ ୢ୲ ൅ ɐɘ‹
‹ ൌ ୡ ሺʹ‡ି
x
ಚేಡ
୲
ై
െ ͳሻ
(27)
(28)
These equations are effective only during ߠ௢௙௙ ൑ ߠ ൑ ߠଵ
Sub-region III-2
In this region, the inductance has its maximum value ‫ܮ‬௔௝ and is constant. So,
current is
‹ ൌെ
୚ౙ౩
୐౗ౠ
– ൅ ଴
(29)
Where ‫ܫ‬଴ is the current value at ߠଵ . This equation is effective during ߠଵ ൑ ߠ ൑ ߠଶ .
B. Advance angle control method
The SRM is controlled by input voltage, switch-on and switch-off angle. Switch-on and
switch-off angle regulate the magnitude and shape of the current waveform. Also it results
in affecting the magnitude and shape of the torque developed. To build up the current
effectively with a voltage source, an advance switching before the poles meet is needed. The
switch-on angle is one of the main factors to control the build-up currents. Therefore, this
angle is controlled precisely to get optimal driving characteristics.
Fig. 36. Block diagram of advance angle control with feedback signal
In the real control system, control of advance angle which is controlled by variable load
condition can be realized by simple feedback circuit using detecting load current. The block
diagram of the advance angle control with a feedback signal shows in Fig.36.
The regulation of speed-torque characteristics of SRM drive is achieved by controlling
advance angle and applied voltage. The advance angle is regulated to come up with the load
variation in cooperation with the applied voltage.
The signal from the control loops is translated into individual current reference signal for
each phase. The torque is controlled by regulating these currents. The feedback signal which
is proportional to the phase detector is used to regulate the instantaneous applied voltage.
229
229
Sw
witched Reluctance Motor
Switched Reluctance Motor
Vaariation relationsship of torque with
w
current or torque with ro
otor position mu
ust be
com
mpensated in the feed forward torque
t
control allgorithm. The rellation between to
orques
and current of a phase is shown in Fig.
F 37.
g. 37. Advance an
ngle control
Fig
C. Switching-off an
ngle control meth
hod
Co
ontrol method off switch-off angle is introduced for variable load
d. The switching
g angle
con
ntrol method is based
b
on two com
mmand signals fo
or switching–on aand switching–offf angle
ind
dependently. Acccording to the motor
m
speed and load condition, a proper switch
hing-on
angle T on ¬ is set at th
he cross point of negative slope off the sensor signaal and the switchiing–on
com
mmand signal Von
o is
T on
§
V ·
¬¨ 1 on ¸ ¬T0 ¬T a ¬T a
V
max ¹
©
(30)
he maximum swittching-on angle is
i in the minimum
m inductance reg
gion. So, a fast bu
uild up
Th
of current is possiblle at the rated loaad. The minimum
m switching–on an
ngle is in the incrreasing
reg
gion of inductancce. Therefore a sm
mooth build up of
o current is possiible at a light load with
T off is seet at the cross po
a smooth
s
torque production.
p
Simillarly, the delay angle
a
oint of
po
ositive slope of thee signal and the switching
s
off com
mmand signal Vofff as
T offff
Voff
Vmax
¬T d ¬T0 ¬T 0
(31)
230
230
Torque
e Control
Torque Controlo
In addition, the dw
well angle is the in
nterval of switch
hing-on and switcching–off angles, which
tak
kes the form
T dwell
T off ¬T on
(32)
here are two typees of control swittch-off angle, onee is constant torq
que angle ( TTQ ) control
c
Th
and the other is con
nstant dwell anglee ( T Dw )¬ control.
ue angle control
1. Constant torqu
orque angle is thee angle between the increasing off inductance to th
he switching-off angle.
To
Th
his control method
d is fixed the turn
n-off angle and tu
urn-on angle is tu
uned for a fluctuaation of
speeed and load by
y constant torquee angle control method.
m
The fluctuation of efficieency is
sm
mall until rated po
ower, but if the tu
urn-on angle moves toward for an
n increase torquee, even
in the region of deecreasing inductaance, the currentt will flow and n
negative torque will
w be
pro
oduced. Thus, th
he efficiency beco
omes reduced. Therefore,
T
it is neeeded to find a proper
p
po
osition of turn-on angle and the ph
hase current whicch determined by
y constant torquee angle
con
ntrol method.
g. 38. Constant to
orque angle contro
ol
Fig
2.
Constant dwelll angle control
Th
he constant dwelll angle method controls
c
the turn
n-on or turn-off aangle by keep co
onstant
dw
well angle ( T Dw ) for speed or outtput control. Wh
hen turn-on anglee is moved to keeep the
con
nstant speed, effeect of negative to
orque is regardlesss of speed and lo
oad. But becausee of the
lim
mits of rated pow
wer, it can be unsttable to drive on overload. This m
method makes a control
c
sysstem simple and easy to avoid neegative torque in the switching-offf region. Fig. 39 shows
thee relation between
n current and rottor position in con
nstant dwell anglle ( T Dw ) control.
2.1
1.2 Single pulse
e control method
d
To
orque production
n in SRM is not constant
c
and it must
m
be establish
hed from zero att every
strroke. Each phase must be energizzed at the turn-on
n angle and switcched off at the tu
urn-off
angle. In the low sp
peed range, the torque
t
is limited only by the current, which is reg
gulated
231
231
Sw
witched Reluctance Motor
Switched Reluctance Motor
eitther by voltage-P
PWM or by instan
ntaneous currentt. As the speed in
ncreases the back
k-EMF
inccreases too, and there is insufficieent voltage available to regulate the current; the torque
can
n be controlled on
nly by the timing
g of the current pulse.
p
This contro
ol mode is called singles
pu
ulse mode.
Fig
g. 39. Constant dw
well angle ( T Dw ) control
In single pulse operation the powerr supply is kept switched
s
on durin
ng the dwell ang
gle and
s
off at th
he phase commu
utation angle. As there is no contrrol of the currentt and a
is switched
shaarp increase of current, the amou
unt of time availa
able to get the d
desired current is short.
Ty
ypically, single pu
ulse operation is used at high mecchanical speed w
with respect to thee turnon
n angle determineed as a function of speed. Fig.40 shows the phasee current in high speed
reg
gion using an assymmetric conveerter. As shown in Fig. 40, SR drrive is excited at T on
Zrm
LTre T adv
a
Desired Phase Current
ias
i*as
Actual Phase Current
At High Sp
peed
LTre T on T 1
T offf
Positive torq
que
region
T2
T re
Negative
N
torque
region
Fig
g. 40. Build-up of phase current in
n high speed regio
on
232
232
Torque Control
Torque Controlo
position advanced as ߠ௔ௗ௩ , than the start point of positive torque region ߠଵ in order to
establish the sufficient torque current. The desired phase current shown as dash line in Fig.
40 is demagnetized at ߠ௢௙௙ , and decreased as zero before the starting point of negative
torque region ߠଶ to avoid negative torque.
‫כ‬
, the advance angle
In order to secure enough time to build-up the desire phase current ݅௔௦
ߠ௔ௗ௩ can be adjusted according to motor speed ߱௠ . From the voltage equations of SRM, the
proper advance angle can be calculated by the current rising time as follows regardless of
phase resistance at the turn-on position.
ο‫ ݐ‬ൌ ‫ܮ‬ሺߠଵ ሻǤ
‫כ‬
௜ೌ್೎ೞ
௏ೌ್೎ೞ
(33)
‫כ‬
Where, ݅௔௕௖௦
denotes the desired phase current of current controller and ܸ௔௕௖௦ is the terminal
voltage of each phase windings. And the advance angle is determined by motor speed and
(33) as follow
ߠ௔ௗ௩ ൌ ߱௠ Ǥ ο‫ݐ‬
(34)
As speed increase, the advance angle is to be larger and turn-on position may be advanced
not to develop a negative torque. At the fixed turn-on position, the actual phase current
denoted as solid line could not reach the desire value in high speed region as shown in Fig.
40. Consequently, the SRM cannot produce sufficient output torque. At the high speed
region, turn-on and turn-off position are fixed and driving speed is changed. To overcome
this problem, high excitation terminal voltage is required during turn-on region from ߠ௢௡
toߠଵ .
2.1.3 Dynamic angle control method
The dynamic angle control scheme is similar to power angle control in synchronous
machine. When an SRM is driven in a steady-state condition, traces such as shown in Fig.
41(a) are produced. The switch-off instant is fixed at a preset rotor position. This may
readily be done by a shaft mounted encoder. If the load is decreased, the motor is
accelerated almost instantaneously. The pulse signal from a rotor encoder is advanced by
this acceleration. This effect will reduce switch-off interval until the load torque and the
developed torque balances [Ahn,1995]. Fig. 41(b) shows this action. On the contrary, if load
is increased, the rotor will be decelerated and the switch-off instant will be delayed. The
effect results in increasing the developed torque. Fig. 41(c) shows the regulating process of
the dwell angle at this moment.
The principle of dynamic dwell angle is similar to PLL control. The function of the PLL in
this control is to adjust the dwell angle for precise speed control. The phase detector in the
PLL loop detects load variation and regulates the dwell angle by compares a reference
signal (input) with a feedback signal (output) and locks its phase difference to be constant.
Fig. 42 shows the block diagram of PLL in SR drive. It has a phase comparator, loop filter,
and SRM drive.
The reference signal is a speed command and used for the switch-on signal. The output of
the phase detector is used to control voltage through the loop filter. The switching inverter
regulates switching angles. The output of phase detector is made by phase difference
between reference signal and the signal of rotor encoder. It is affected by load variations.
The dwell angle is similar to phase difference in a phase detector. To apply dynamic angle
Switched Reluctance Motor
Switched Reluctance Motor
233
233
control in an SR drive system, a reference frequency signals are used to switch-on, and the
rotor encoder signal is used to switch-off similar to the function of a phase detector. The
switch-off angle is fixed by the position of the rotor encoder. Therefore, the rotor encoder
signal is delayed as load torque increased. This result is an increase of advance angle and
initial phase current.
Fig. 41. Regulation of dwell angle according to load variation.
(a) steady-state. (b) load decreased. (c) load increased.
Fig. 42. Block diagram of PLL in SR drive.
2.2 Current control method
Control of the switched reluctance motor can be done in different ways. One of them is by
using current control method. The current control method is normally used to control the
torque efficiently. Voltage control has no limitation of the current as the current sensor is
avoided, which makes it applicable in low-cost systems. Due to the development of
234
234
Torque Control
Torque Controlo
microcontrollers, the different control loops have changed from analog to digital
implementation, which allows more advanced control features. However, problems are still
raised when designing high-performance current loop [miller,1990].
The main idea of current control method is timing and width of the voltage pulses. Two
methods are too used in the current control, one is voltage chopping control method, and
the other is hysteresis control method.
2.2.1 Voltage chopping control method
The voltage chopping control method compares a control signal ܸ௖௢௡௧௥௢௟ (constant or slowly
varying in time) with a repetitive switching-frequency triangular waveform or Pulse Width
Modulation (PWM) in order to generate the switching signals. Controlling the switch duty
ratios in this way allowed the average dc voltage output to be controlled. In order to have a
fast built-up of the excitation current, high switching voltage is required. Fig. 43 shows an
asymmetric bridge converter for SR drive. The asymmetric bridge converter is very popular
for SR drives, consists of two power switches and two diodes per phase. This type of the SR
drive can support independent control of each phase and handle phase overlap. The
asymmetric converter has three modes, which are defined as magnetization mode,
freewheeling mode, and demagnetization mode as shown in Fig. 44.
ia
ib
ic
Fig. 43. Asymmetric bridge converter for SR drive
(a) Magnetization
(b) Freewheeling
(c) Demagnetization
Fig. 44. Operation modes of asymmetric converter
From Fig. 44 (a) and (c), it is clear that amplitudes of the excitation and demagnetization
voltage are close to terminal voltage of the filter capacitor. The fixed DC-link voltage limits
the performance of the SR drive in the high speed application. On the other hand, the
235
235
Switched Reluctance Motor
Switched Reluctance Motor
voltage chopping method is useful for controlling the current at low speeds. This PWM
strategy works with a fixed chopping frequency. The chopping voltage method can be
separated into two modes: the hard chopping and the soft chopping method. In the hard
chopping method both phase transistors are driven by the same pulsed signal: the two
transistors are switched on and switched off at the same time. The power electronics board
is then easier to design and is relatively cheap as it handles only three pulsed signals. A
disadvantage of the hard chopping operation is that it increases the current ripple by a large
factor. The soft chopping strategy allows not only control of the current but a minimization
of the current ripple as well. In this soft chopping mode the low side transistor is left on
during the dwell angle and the high side transistor switches according to the pulsed signal.
In this case, the power electronics board has to handle six PWM signals [Liang,2006].
2.2.2 Hysteresis control method
Due to the hysteresis control, the current is flat, but if boost voltage is applied, the switching
is higher than in the conventional case. The voltage of the boost capacitor is higher in the
two capacitor parallel connected converter. The hysteresis control schemes for outgoing and
incoming phases are shown on the right side of Fig. 45.
Solid and dash lines denote the rising and falling rules, respectively. The y axis denotes
phase state and the x axis denotes torque error ሺοܶ௘௥௥ ሻ, which is defined as,
οୣ୰୰ ൌ ୰ୣ୤ െ ୣୱ୲
(35)
The threshold values of torque error are used to control state variation in hysteresis
controller. Compared to previous research, this method only has 3 threshold values (ο‫ܧ‬, 0
and -ο‫)ܧ‬, which simplifies the control scheme. In order to reduce switching frequency, only
one switch opens or closes at a time. In region 1, the incoming phase must remain in state 1
to build up phase current, and outgoing phase state changes to maintain constant torque.
For example, assume that the starting point is (-1, 1), and the torque error is greater than 0.
The switching states for the two phases will change to (0, 1). At the next evaluation period,
the switching state will change to (1, 1) if torque error is more than ο‫ ܧ‬and (-1, 1) if torque
error is less than -ο‫ܧ‬. So the combinatorial states of (-1, 1), (0, 0) and (1, 1) are selected by the
control scheme. The control schemes for region 2 and region 3 are shown in Fig. 45(b) and
(c), respectively.
3. Advanced torque control strategy
There are some various strategies of torque control: one method is direct torque control,
which uses the simple control scheme and the torque hysteresis controller to reduce the
torque ripple. Based on a simple algorithm, the short control period can be used to improve
control precision. The direct instantaneous torque control (DITC) and advanced DITC
(ADITC), torque sharing function (TSF) method are introduced in this section.
3.1 Direct Instantaneous Torque Control (DITC)
The asymmetric converter is very popular in SRM drive system. The operating modes of
asymmetric converter are shown in Fig. 46. The asymmetric converter has three states,
which are defined as state 1, state 0 and state -1 in DITC method, respectively.
236
236
Torque Control
Torque Controlo
(a) Region 1
(b) Region 2
(c) Region 3
Fig. 45. The hysteresis control schemes for outgoing and incoming phases
237
237
Switched Reluctance Motor
Switched Reluctance Motor
ia
ia
(a) state 1
(b) state 0
ia
(c) state -1
Fig. 46. 3 states in the asymmetric converter
In order to reduce a torque ripple, DITC method is introduced. By the given hysteresis
control scheme, appropriate torque of each phase can be produced, and constant total
torque can be obtained. The phase inductance has been divided into 3 regions shown as Fig.
47. The regions depend on the structure geometry and load. The boundaries of 3 regions are
ߠ௢௡ଵ , ߠଵ , ߠଶ and ߠ௢௡ଶ in Fig. 47. ߠ௢௡ଵ and ߠ௢௡ଶ are turn-on angle in the incoming phase and
the next incoming phase, respectively, which depend on load and speed. The ߠଵ is a rotor
position which is initial overlap of stator and rotor. And ߠଶ is aligned position of inductance
in outgoing phase. Total length of these regions is 120 electrical degrees in 3 phases SRM.
Here, let outgoing phase is phase A and incoming phase is phase B in Fig. 47. When the first
region 3 is over, outgoing phase will be replaced by phase B in next 3 regions.
The DITC schemes of asymmetric converter are shown in Fig. 48. The combinatorial states of
outgoing and incoming phase are shown as a square mesh. x and y axis denote state of
outgoing and incoming phase, respectively. Each phase has 3 states, so the square mesh has
9 combinatorial states. However, only the black points are used in DITC scheme.
Z
Fig. 47. Three regions of phase inductance in DITC method
238
238
Torque Control
Torque Controlo
Incoming phase
(-1,1)
'Terr ! 0
(0,1) 'Terr
Incoming phase
!'E
(1,1)
'Terr !'E
1
Outgoing
phase
'Terr ! 0
'Terr 'E
Outgoing
phase
-1
(a) region 1
'Terr 0
'Terr 0
'Terr 'E
Incoming phase
Outgoing
phase
(b) region 2
(c) region 3
Fig. 48. DITC scheme of asymmetric converter
Control diagram of DITC SR motor drive is shown in Fig. 49. The torque estimation block is
generally implemented by 3-D lookup table according to the phase currents and rotor
position. And the digital torque hysteresis controller which carries out DITC scheme
generates the state signals for all activated machine phases according to torque error
between the reference torque and estimated torque. The state signal is converted as
switching signals by switching table block to control converter.
Through estimation of instantaneous torque and a simple hysteresis control, the average of
total torque can be kept in a bandwidth. And the major benefits of this control method are
its high robustness and fast toque response. The switching of power switches can be
reduced.
However, based on its typical hysteresis control strategy, switching frequency is not
constant. At the same time, the instantaneous torque cannot be controlled within a given
bandwidth of hysteresis controller. The torque ripple is limited by the controller sampling
time, so torque ripple will increase with speed increased.
Tref*
Test
T
Fig. 49. Control diagram of DITC
239
239
Switched Reluctance Motor
Switched Reluctance Motor
3.2 Advanced Direct Instantaneous Torque Control (ADITC)
The conventional DITC method uses a simple hysteresis switch rules, so only one phase
state is applied according to torque error at every sampling period. The toque variation with
sampling time and speed under full dc-link voltage is shown in Fig. 50. In order to
guarantee the torque ripple within a range, it has two methods: one is that reduces sampling
time, which will increase the cost of hardware. Another is that control average voltage of
phase winding in sampling time. PWM method can be used.
'Tm [%]
10
5
0
100
1000
50
Sampling time [ Ps]
500
0
0
Speed [rpm]
Fig. 50. Torque variation with sampling time and speed
ADITC combines the conventional DITC and PWM method. The duty ratio of the phase
switch is regulated according to the torque error and simple control rules of DITC.
Therefore, the sampling time of control can be extended, which allows implementation on
low cost microcontrollers.
ADITC is improved from the conventional DITC, so the divided region of phase inductance
is similar to DITC method. The control scheme of ADITC is shown in Fig. 51, ‫ܦ‬௧ሺ௞ሻ means
incoming phase, ‫ܦ‬௧ሺ௞ିଵሻ means outgoing phase. X-axis denotes torque error, and y-axis
denotes switching state of ‫ܦ‬௧ሺ௞ሻ and ‫ܦ‬௧ሺ௞ିଵሻ .
Terr
'TH
(a) Region 1
'TH
(b) Region 2
Terr
'TH
Terr
(c) Region
Fig. 51. ADITC scheme of asymmetric converter
Profit from the effect of PWM, the average voltage of phase winding can be adjusted from 0
to ܸௗ௖ in one sampling time. And the hysteresis rule is removed from the control scheme.
Now, the current state can select the phase state between state 0 and 1 by duty ratio of
PWM.
240
240
Torque Control
Torque Controlo
(1 Dt ) ˜ TS
Dt ˜ TS
TS
TS
(a) Incoming phase
(b) outgoing phase
Fig. 52. Switching modes of incoming and outgoing phase
The duty ratio of switching modes is decided by the torque error as shown in Fig. 52, and
‫ܦ‬௧ is expressed as follows:
୲ ൌ „•ሺୣ୰୰ ሻȀοୌ
(36)
Where, ܶ௘௥௥ is torque error, Ʀܶு is torque error bandwidth. The control block diagram of
ADITC is similar to Fig. 53. The hysteresis controller is replaced by Advanced DITC
controller, and the PWM generator is added.
Tref*
Test
T
Fig. 53. Control diagram of ADITC
ADITC method can adjust average phase voltage to control variety of phase current in one
sampling time, which can extend the sampling time and obtain smaller torque ripple than
conventional DITC. However, PWM generator is added, and the switching frequency of
241
241
Switched Reluctance Motor
Switched Reluctance Motor
ADITC is double of DITC’s with uniform sampling time in the worst case. So the switching
loss and EMC noise are increased in ADITC method.
3.3 Torque sharing control
Another control method to produce continuous and constant torque is indirect torque
control, which uses the complicated algorithms or distribution function to distribute each
phase torque and obtain current command. And then, the current controller is used to
control phase torque by given current command. The linear, cosine and non linear logical
torque sharing function (TSF) are introduced.
Among them, the simple but powerful method is torque sharing function (TSF). The TSF
method uses the pre-measured non-linear torque characteristic, and simply divided torque
sharing curve is used for constant torque generation. Besides the direct torque control
method, another method is indirect torque control. TSF is simple but powerful and popular
method among the indirect torque control method. It simply divided by torque sharing
curve that is used for constant torque generation. And the phase torque can be assigned to
each phase current to control smoothing torque. But phase torque has relationship of square
current. So the current ripple should keep small enough to generate smooth torque. So the
frequency of current controller should be increased.
Fig. 54 shows the torque control block diagram with TSF method. The input torque
reference is divided into three-phase torque command according to rotor position. Torque
references of each phase are changed to current command signal in the “Torque-to-Current”
block according to rotor position. Since the output torque is determined by the inductance
slope and phase current, and the inductance slope is changed by rotor position, so the
reference currents of each phase is determined by the target torque and rotor position. The
switching rule generates an active switching signal of asymmetric converter according to
current error and hysteresis switching tables.
Vdc
Switching
Rule
Torque-to-Current
Tm*( A)
Tm*
TSF
Tm*( B )
Tm*(C )
I m* ( A)
jœ™™Œ•›G
yŒŒ™Œ•ŠŒG
nŒ•Œ™ˆ›–™
+
-
I m* ( B )
+
I m* (C ) + -
Sm( k )
1
S m ( A)
Sm ( B )
0
'I m ( k )
-1
S m (C )
-
Trm
Trm
ias
ibs
ics
Encoder
Trm
Fig. 54. The torque control block diagram with TSF method
In the over-lap region of inductances, the two-phase currents generate the output torque
together. A simple torque sharing curves are studied for constant torque generation in the
commutation region such as linear and cosine function.
Fig. 55 shows the inductance profiles of three-phase SRM, cosine and linear TSF curves. As
shown in Fig. 55, region 2 denotes the one phase activation area. Region 1 and region 3 are
242
242
Torque Control
Torque Controlo
two phases activation area explained as the commutation region. In one phase activation
region, TSF is constant in every torque sharing functions. But TSF is different in the
commutation regions. The linear TSF has constant slope of torque in commutation region.
This method is simple, but it is very difficult to generate the linear torque slope in the
commutation region due to the non-linear inductance characteristics.
fT ( k 1)
T overlap
fT ( k )
Ton(k1) T off ( k )
T on ( k ) T off ( k 1)
fT ( k )
fT ( k 1)
Ton(k 2)
T rm
Ton(k 2)
T rm
fT ( k 1)
Ton(k1) T off ( k )
T on ( k ) T off ( k 1)
T rm
fT ( k 1)
Fig. 55. Phase inductances and cosine, and linear TSF curves
The cosine TSF uses the cosine function in commutation region as shown in Fig. 55. The
cosine function is relatively simple and it is similar to the non-linear inductance
characteristics. But the non-linear characteristic of SRM is very complex, so cosine torque
function can not be satisfied in the aspect of torque ripple and efficiency.
In the cosine TSF, the TSF of each phase in the commutation region are defined as follow
஘౨ౣ ି஘౥౤ሺౡሻ
ଵ
ˆ୘ሺ୩ሻ ൌ ଶ ൤ͳ െ …‘• ൬
஘౥౬౛౨ౢ౗౦
Ɏ൰൨
(37)
ˆ୘ሺ୩ିଵሻ ൌ ͳ െ ˆ୘ሺ୩ሻ
(38)
ˆ୘ሺ୩ାଵሻ ൌ Ͳ
(39)
And the linear TSF method, the TSF of each phase can be obtained as follow
ˆ୘ሺ୩ሻ ൌ ஘౨ౣ ି஘౥౤ሺౡሻ
஘౥౬౛౨ౢ౗౦
(40)
243
243
Switched Reluctance Motor
Switched Reluctance Motor
ˆ୘ሺ୩ିଵሻ ൌ ͳ െ ˆ୘ሺ୩ሻ
(41)
ˆ୘ሺ୩ାଵሻ ൌ Ͳ
(42)
These two TSFs are very simple, but they can not consider nonlinear phenomena of the SRM
and torque dip is much serious according to rotor speed. For the high performance torque
control, a novel non-linear torque sharing function is suitable to use. In order to reduce
torque ripple and to improve efficiency in commutation region, the TSF uses a non-linear
current distribution technique at every rotor position. And the torque sharing function can
be easily obtained by the current coordinates of each rotor position. In the commutation
region, the total torque reference is divided by two-phase torque reference.
‫כ‬
‫כ‬
‫כ‬
ൌ ୫ሺ୩ሻ
൅ ୫ሺ୩ାଵሻ
୫
(43)
In the equation, the subscripts k+1 denotes the incoming phase and k denotes outgoing
phase. The actual torque can be obtained by inductance slope and phase current. So the
torque equation can be derived as follows.
‫כ‬
୫
=
where,
ƒ ൌ ඨப୐ሺౡሻ
୍‫כ‬మ
ౣሺౡሻ
ୟమ
ଶ
൘ப୐
ሺౣሻ
+
‫כ‬మ
୍ౣሺౡశభሻ
(44)
ୠమ
ଶ
, „ ൌ ඨப୐ሺౡశభሻ
൘ப୐
ሺౣሻ
(45)
This equation is same as ellipse equation. In order to generate a constant torque reference,
current references of the outgoing and incoming phases is placed on the ellipse trajectory in
the commutation region. And the aspect of the ellipse and its trajectory is changed according
to rotor position, inductance shape and the reference torque. Since the TSFs uses a fixed
torque curve such as linear and cosine, the outgoing phase current should keep up the
reference. And the actual current should remain higher level around rotor aligned position.
Fig. 56 shows each phase current reference and actual phase torque for constant torque
production according to rotor position. As shown in Fig. 56, the actual torque profile has
non-linear characteristics around match position of rotor and stator position. So the current
reference of each phase for constant torque generation is changed according to the rotor
position and the amplitude of the torque reference. However, the actual phase current is
limited by the performance of a motor and a drive. And the actual torque can not be
satisfied the torque reference around the aligned position due to the non-linear torque
characteristics shown as Fig. 56. If the current of outgoing phase is increased as a limit value
of the motor, the actual torque is decreased after‫ܦ‬௞ position. And the actual torque of
incoming phase can not be satisfied at the start position of the commutation due to the same
reason. In order to generate the constant torque from ‫ܣ‬௞ –‘‫ܩ‬௞ାଵ , the outgoing and incoming
current reference should be properly selected so that the total torque of each phase is
remained as constant value ofܶ௠‫ כ‬.
In order to reduce the commutation region, the outgoing phase current should be decreased
fast, and the incoming phase current should be increased fast with a constant torque
generation. At the starting point of commutation, the incoming phase current should be
increased from zero to ‫ܣ‬௞ାଵ point, and the end of the commutation, the outgoing phase
current should be decreased from ‫ܩ‬௞ point to zero as soon as possible shown in Fig. 56.
244
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Torque Control
Torque Controlo
T overlap
Ak+1
Bk+1
Ck+1
Dk+1
Tm(Trm)
Tm*
Dk
Tm ( k )
I m* ( k )
Ek
T1
Gk+1
T2
T3
Current
Limit
Tm ( k 1)
Fk+1
Bk
Ak
Gk
Ek+1
Ck
T on ( k 1)
Fk
T4
T5
I m* ( k 1)
T off ( k )
T rm
Fig. 56. Phase current and actual torque trajectory for constant torque production during
phase commutation
Im* (k1)
Im* (k1)
In order to reduce the torque ripple and increase the operating efficiency, a non-linear TSF is
based on minimum changing method. One phase current reference is fixed, and the other
phase current reference is changed to generate constant torque during commutation. Fig. 57
shows the basic principle of the non-linear TSF commutation method.
I m* ( k )
I m* ( k )
(a) In case of ܶ௠ ൏ ܶ௠‫כ‬
(b) In case of ܶ௠ ൐ ܶ௠‫כ‬
Fig. 57. Basic principle of the commutation method based on minimum changing
In this method, the incoming phase current is changed to a remaining or an increasing
direction to produce the primary torque. And the outgoing phase current is changed to a
remaining or a decreasing direction to produce the auxiliary torque. In case of ܶ௠ ൏ ܶ௠‫ כ‬, the
outgoing phase current is fixed, and the incoming phase current is increased to reach the
constant torque line from ܲଵ to ܺଵ shown as Fig. 57(a). If the incoming phase current is
245
245
Switched Reluctance Motor
Switched Reluctance Motor
limited by the current limit and the actual torque is under the reference value, the auxiliary
torque is generated by the outgoing phase current from ܲଶ to ܺଶ shown as Fig. 57(a). In case
of ܶ௠ ൐ ܶ௠‫ כ‬, the incoming phase current is fixed, and the outgoing phase current is decreased
to reach the constant torque line fromܳଵ to ܻଵ shown as Fig. 57(b), because the incoming
phase current is sufficient to generate the reference torque. If the outgoing phase current is
reached to zero, and the actual torque is over to reference value, the incoming phase current
is decreased from ܳଶ to ܻଶ shown as Fig. 57(b). This method is very simple, but the
switching number for torque control can be reduced due to the minimum number changing
of phase. As the other phase is fixed as the previous state, the torque ripple is dominated by
the one phase switching. Especially, the outgoing phase current is naturally decreased when
the incoming phase current is sufficient to produce the torque reference. The
demagnetization can be decreased fast, and the tail current which generates negative torque
can be suppressed.
Table 6 shows the logical TSF, and the Fig. 58 is the ideal current trajectory during
commutation region. In Fig. 58, the ellipse curves are current trajectory for constant torque
at each rotor position under commutation.
‫כ‬
In case of ୫ ൏ ୫
‫כ‬
୫ሺ୩ାଵሻ
‫כ‬
୫
‫כ‬
In case of ୫ ൐ ୫
when
‫כ‬
୫ሺ୩ାଵሻ
െ ୫ሺ୩ሻ
୫ ሺ୩ାଵሻ‫כ‬
൏ ୫ୟ୶
‫כ‬
୫ሺ୩ାଵሻ
൐ ୫ୟ୶
*At current limit
‫כ‬
୫ሺ୩ሻ
‫כ‬
୫
െ ୫ሺ୩ାଵሻ
0
୫ሺ୩ାଵሻ
‫כ‬
୫ሺ୩ሻ
‫כ‬
‫כ‬
୫
െ ୫ሺ୩ାଵሻ
൐Ͳ
‫כ‬
୫ሺ୩ሻ
൏Ͳ
‫כ‬
൐Ͳ
୫ሺ୩ሻ
‫כ‬
୫ሺ୩ାଵሻ
‫כ‬୫ െ ୫ሺ୩ሻ
Table 6. The logical TSF in commutation region.
I m* ( k )
when
‫כ‬
୫ሺ୩ሻ
T2
Ton(k) T1
T3
T4
Ton(k) T1 T2 T3 T4 T5 Toff (k1)
T5
Toff (k 1)
I m* ( k 1)
Fig. 58. The ideal current trajectory at commutation region
‫כ‬
୫ሺ୩ሻ
൏Ͳ
246
246
Torque Control
Torque Controlo
Tm*
Tm
Tm* (k 1)
Tm(k 1)
Im(k 1)
Im* (k 1)
(a) Linear TSF
Tm*
Tm
Tm* (k 1)
Im* (k 1)
Tm(k 1)
Im(k 1)
(b) Cosine TSF
Fig. 59. Simulation result at 500 rpm with rated torque
247
247
Switched Reluctance Motor
Switched Reluctance Motor
Tm*
Tm
Tm* (k 1)
*
Im
(k 1)
Tm(k 1)
Im(k 1)
(c) non-linear Logical TSF
Fig. 59. Simulation results at 500rpm with rated torque (continued)
In order to verify the non-linear TSF control scheme, computer simulations are executed and
compared with conventional methods. Matlab and simulink are used for simulation. Fig. 59
shows the simulation comparison results at 500[rpm] with rated torque reference. The
simulation results show the total reference torque, actual total torque, reference phase
torque, actual phase torque, reference phase current, actual phase current and phase
voltage, respectively. As shown in Fig. 59, torque ripple is linear TSF > cosine TSF > the
logical TSF.
Fig. 60 shows the actual current trajectory in the commutation region. In the conventional
case, the cross over of the outgoing and incoming phase is serious and two-phase current
are changed at each rotor position. But the cross over is very small and one-phase current is
changed at each rotor position in the logical TSF method.
I m( k 1)
I m( k 1)
I m( k 1)
Im(k )
(a) Linear TSF
Im(k )
I m(k )
(b) Cosine TSF
(c) logical TSF
Fig. 60. The current trajectory for constant torque production in commutation region
248
248
Torque Control
Torque Controlo
Fig. 61 shows the experimental setup. The main controller is designed by TMS320F2812
from TI(Texas Instruments) and phase current and voltage signals are feedback to 12bit
ADC embedded by DSP. The rotor position and speed is obtained by 512ppr optical
encoder. At every 1.6[ms], the rotor speed is calculated from captured encoder pulse by QEP
function of DSP.
Fig. 61. The experimental configuration
Fig. 62, 63 and 64 show the experimental results in case of linear TSF, cosine TSF and the
non-linear logical TSF at 500rpm, respectively. Torque ripple can be reduced in case of the
TSF method due to the minimum phase changing.
Tm* ( A)
Tm( A)
Tm
Tm*
ias
vas
(a)
ibs
ias
(b)
Fig. 62. Experimental results in linear TSF(at 500[rpm])
(a) Reference torque, actual torque, phase current and terminal voltage
(b) Total reference torque, actual torque and phase currents
Fig. 65 shows experimental results at 1200rpm. As speed increase, torque ripple is increased
due to the reduction of the commutation time. However, the control performance is much
improved in this case.
Fig. 66 shows efficiency of the logical control schemes. In the low speed range, the TSF
control scheme has about 5% higher efficiency than that of the conventional ones with low
torque ripple. In high speed range, the actual efficiency is similar to all other control method
due to the short commutation time. But the practical torque ripple can be reduced than other
two control schemes shown in simulation and experimental results.
249
249
Switched Reluctance Motor
Switched Reluctance Motor
Tm( A)
Tm*
Tm
Tm* ( A)
ias
vas
ibs
(a)
ias
(b)
Fig. 63. Experimental results in cosine TSF(at 500[rpm])
(a) Reference, actual torque, phase current and terminal voltage
(b) Total reference torque, actual torque and phase currents
(a)
(b)
Fig. 64. Experimental results in case of the non-linear logical TSF(at 500[rpm])
(a) Reference, actual torque, phase current and terminal voltage
(b) Total reference torque, actual torque and phase currents
4. Conclusion
The torque production in switched reluctance motor structures comes from the tendency of
the rotor poles to align with the excited stator poles. However, because SRM has doubly
salient poles and non-linear magnetic characteristics, the torque ripple is more severe than
these of other traditional motors. The torque ripple can be minimized through magnetic
circuit design or drive control. By controlling the torque of the SRM, low torque ripple,
noise reduction or even increasing of the efficiency can be achieved. There are many
different types of control methods. In this chapter, detailed characteristics of each control
method are introduced in order to give the advanced knowledge about torque control
method in SRM drive.
250
250
Torque Control
Torque Controlo
(a) Reference torque, total torque and phase currents in linear TSF
(b) Reference torque, total torque and phase currents in cosine TSF
(c) Reference torque, total torque and phase currents in non-linear logical TSF
Fig. 65. Experimental results at 1200rpm with rated torque
͖ͩ͡
Efficiency
͖ͨ͡
͖ͧ͡
͖ͦ͡
͖ͥ͡
΁ΣΠΡΠΤΖΕ͑΅΄ͷ
͖ͤ͡
ͥ͡͡
ͧ͡͡
ͩ͡͡
͢͡͡͡
ʹΠΤΚΟΖ͑΅΄ͷ
ͣ͢͡͡
ͥ͢͡͡
Speed [rpm]
Fig. 66. Efficiency comparison
ͧ͢͡͡
ͽΚΟΖΒΣ͑΅΄ͷ
ͩ͢͡͡
ͣ͡͡͡
Switched Reluctance Motor
Switched Reluctance Motor
251
251
5. References
A. Chiba, K. Chida and T. Fukao, "Principles and Characteristics of a Reluctance Motor with
Windings of Magnetic Bearing," in Proc. PEC Tokyo, pp.919-926, 1990.
Bass, J. T., Ehsani, M. and Miller, T. J. E ; "Robust torque control of a switched reluctance
motor without a shaft position sensor," IEEE Transactions, Vol.IE-33, No.33, 1986,
212-216.
Bausch, H. and Rieke, B.; “Speed and torque control of thyristorfed reluctance motors."
Proceedings ICEM, Vienna Pt.I, 1978, 128.1-128.10. Also : "Performance of
thyristorfed electric car reluctance machines." Proceedings ICEM, Brussels E4/2.12.10
Byrne, J. V. and Lacy, J.G.; "Characteristics of saturable stepper and reluctance motors." IEE
Conf. Publ. No.136,Small Electrical Machines, 1976, 93-96.
Corda, J. and Stephenson, J. M., "Speed control of switched reluctance motors," International
Conference on Electrical Machines, Budapest, 1982.
Cossar, C. and miller, T.J.E., "Electromagnetic testing of switched reluctance motors,"
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9
Controller Design for Synchronous
Reluctance Motor Drive Systems
with Direct Torque Control
Tian-Hua Liu
Department of Electrical Engineering,
National Taiwan University of Science and Technolog
Taiwan
1. Introduction
A. Background
The synchronous reluctance motor (SynRM) has many advantages over other ac motors. For
example, its structure is simple and rugged. In addition, its rotor does not have any winding
or magnetic material. Prior to twenty years ago, the SynRM was regarded as inferior to
other types of ac motors due to its lower average torque and larger torque pulsation.
Recently, many researchers have proposed several methods to improve the performance of
the motor and drive system [1]-[3]. In fact, the SynRM has been shown to be suitable for ac
drive systems for several reasons. For example, it is not necessary to compute the slip of the
SynRM as it is with the induction motor. As a result, there is no parameter sensitivity
problem. In addition, it does not require any permanent magnetic material as the permanent
synchronous motor does.
The sensorless drive is becoming more and more popular for synchronous reluctance
motors. The major reason is that the sensorless drive can save space and reduce cost.
Generally speaking, there are two major methods to achieve a sensorless drive system:
vector control and direct torque control. Although most researchers focus on vector control
for a sensorless synchronous reluctance drive [4]-[12], direct torque control is simpler. By
using direct torque control, the plane of the voltage vectors is divided into six or twelve
sectors. Then, an optimal switching strategy is defined for each sector. The purpose of the
direct torque control is to restrict the torque error and the stator flux error within given
hysteresis bands. After executing hysteresis control, a switching pattern is selected to
generate the required torque and flux of the motor. A closed-loop drive system is thus
obtained.
Although many papers discuss the direct torque control of induction motors [13]-[15], only a
few papers study the direct torque control for synchronous reluctance motors. For example,
Consoli et al. proposed a sensorless torque control for synchronous reluctance motor drives
[16]. In this published paper, however, only a PI controller was used. As a result, the
transient responses and load disturbance responses were not satisfactory. To solve the
problem, in this chapter, an adaptive backstepping controller and a model-reference
adaptive controller are proposed for a SynRM direct torque control system. By using the
254
Torque Control
proposed controllers, the transient responses and load disturbance rejection capability are
obviously improved. In addition, the proposed system has excellent tracking ability. As to
the authors best knowledge, this is the first time that the adaptive backstepping controller
and model reference adaptive controller have been used in the direct torque control of
synchronous reluctance motor drives. Several experimental results validate the theoretical
analysis.
B. Literature Review
Several researchers have studied synchronous reluctance motors. These researchers use
different methods to improve the performance of the synchronous reluctance motor drive
system. The major categories include the following five methods:
1. Design and manufacture of the synchronous reluctance motor
The most effective way to improve the performance of the synchronous reluctance motor is
to design the structure of the motor, which includes the rotor configuration, the windings,
and the material. Miller et al. proposed a new configuration to design the rotor
configuration. By using the proposed method, a maximum Ld / Lq ratio to reach high power
factor, high torque, and low torque pulsations was achieved [17]. In addition, Vagati et al.
used the optimization technique to design a rotor of the synchronous reluctance motor. By
applying the finite element method, a high performance, low torque pulsation synchronous
reluctance motor has been designed [18]. Generally speaking, the design and manufacture of
the synchronous reluctance motor require a lot of experience and knowledge.
2. Development of Mathematical Model for the synchronous reluctance motor
The mathematical model description is required for analyzing the characteristics of the
motor and for designing controllers for the closed-loop drive system. Generally speaking,
the core loss and saturation effect are not included in the mathematical model. However,
recently, several researchers have considered the influence of the core loss and saturation.
For example, Uezato et al. derived a mathematical model for a synchronous reluctance
motor including stator iron loss [19]. Sturtzer et al. proposed a torque equation for
synchronous reluctance motors considering saturation effect [2]. Stumberger discussed a
parameter measuring method of linear synchronous reluctance motors by using current,
rotor position, flux linkages, and friction force [20]. Ichikawa et al. proposed a rotor
estimating technique using an on-line parameter identification method taking into account
magnetic saturation [5].
3. Controller Design
As we know, the controller design can effectively improve the transient responses, load
disturbance responses, and tracking responses for a closed-loop drive system. The PI
controller is a very popular controller, which is easy to design and implement.
Unfortunately, it is impossible to obtain fast transient responses and good load disturbance
responses by using a PI controller. To solve the difficulty, several advanced controllers have
been developed. For example, Chiang et al. proposed a sliding mode speed controller with a
grey prediction compensator to eliminate chattering and reduce steady-state error [21]. Lin
et al. used an adaptive recurrent fuzzy neural network controller for synchronous reluctance
motor drives [22]. Morimoto proposed a low resolution encoder to achieve a high
performance closed-loop drive system [7].
4. Rotor estimating technique
The sensorless synchronous reluctance drive system provides several advantages. For
example, sensorless drive systems do not require an encoder, which increases cost,
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
255
generates noise, and requires space. As a result, the sensorless drive systems can reduce
costs and improve reliability. Several researchers have studied the rotor estimating
technique to realize a sensorless drive. For example, Lin et al. used a current-slope to
estimate the rotor position and rotor speed [4]. Platt et al. implemented a sensorless vector
controller for a synchronous reluctance motor [9]. Kang et al. combined the flux-linkage
estimating method and the high-frequency injecting current method to achieve a sensorless
rotor position/speed drive system [23]. Ichikawa presented an extended EMF model and
initial position estimation for synchronous motors [10].
5. Switching strategy of the inverter for synchronous reluctance motor
Some researchers proposed the switching strategies of the inverter for synchronous
reluctance motors. For example, Shi and Toliyat proposed a vector control of a five-phase
synchronous reluctance motor with space vector pulse width modulation for minimum
switching losses [24].
Recently, many researchers have created new research topics for synchronous reluctance
motor drives. For example, Gao and Chau present the occurrence of Hopf bifurcation and
chaos in practical synchronous reluctance motor drive systems [25]. Bianchi, Bolognani, Bon,
and Pre propose a torque harmonic compensation method for a synchronous reluctance
motor [26]. Iqbal analyzes dynamic performance of a vector-controlled five-phase
synchronous reluctance motor drive by using an experimental investigation [27]. Morales
and Pacas design an encoderless predictive direct torque control for synchronous reluctance
machines at very low and zero speed [28]. Park, Kalev, and Hofmann propose a control
algorithm of high-speed solid-rotor synchronous reluctance motor/generator for flywheelbased uniterruptible power supplies [29]. Liu, Lin, and Yang propose a nonlinear controller
for a synchronous reluctance drive with reduced switching frequency [30]. Ichikawa,
Tomita, Doki, and Okuma present sensorless control of synchronous reluctance motors
based on extended EMF models considering magnetic saturation with online parameter
identification [31].
2. The synchronous reluctance motor
In the section, the synchronous reluctance motor is described. The details are discussed as
follows.
2.1 Structure and characteristics
Synchronous reluctance motors have been used as a viable alternative to induction and
switched reluctance motors in medium-performance drive applications, such as: pumps,
high-efficiency fans, and light road vehicles. Recently, axially laminated rotor motors have
been developed to reach high power factor and high torque density. The synchronous
reluctance motor has many advantages. For example, the synchronous reluctance motor
does not have any rotor copper loss like the induction motor has. In addition, the
synchronous reluctance motor has a smaller torque pulsation as compared to the switched
reluctance motor.
2.2 Dynamic mathematical model
In synchronous d-q reference frame, the voltage equations of the synchronous reluctance
motor can be described as
256
Torque Control
vqs = rsiqs + pλqs + ωr λds
(1)
vds = rsids + pλds − ωr λqs
(2)
where vqs and vds are the q-axis and the d-axis voltages, rs is the stator resistance, iqs is the
q-axis equivalent current, ids is the d-axis equivalent current, p is the differential operator,
λqs and λds are the q-axis and d-axis flux linkages, and ωr is the motor speed. The flux
linkage equations are
λqs = (Lls + Lmq ) iqs
(3)
λds = (Lls + Lmd ) ids
(4)
where Lls is the leakage inductance, and Lmq and Lmd are the q- axis and d-axis mutual
inductances. The electro-magnetic torque can be expressed as
Te =
3 P0
( Lmd − Lmq ) ids iqs
2 2
(5)
where Te is the electro-magnetic torque of the motor, and P0 is the number of poles of the
motor. The rotor speed and position of the motor can be expressed as
p ωrm =
1
( Te - Tl - B ωrm )
J
(6)
and
p θ rm = ωrm
(7)
where J is the inertia constant of the motor and load, Tl is the external load torque, B is the
viscous frictional coefficient of the motor and load, θ rm is the mechanical rotor position, and
ωrm is the mechanical rotor speed. The electrical rotor speed and position are
ωr =
P0
ωrm
2
(8)
θr =
P0
θrm
2
(9)
where ωr is the electrical rotor speed, and θr is the electrical rotor position of the motor.
2.3 Steady-state analysis
When the synchronous reluctance motor is operated in the steady-state condition, the d-q
axis currents, id and iq , become constant values. We can then assume xq = ωe Lqs and
xd = ωe Lds , and derive the steady-state d-q axis voltages as follows:
vd = rs id − xq iq
(10)
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
257
vq = rsiq + xd id
(11)
The stator voltage can be expressed as a vector Vs and shown as follows
Vs = vq − jvd
(12)
Now, from equations (10) and (11), we can solve the d-axis current and q-axis current as
id =
rs vd + xq vq
rs 2 + xd xq
(13)
and
iq =
rs vq − xd vd
rs 2 + xd xq
(14)
By substituting equations (13)-(14) into (5), we can obtain the steady-state torque equation as
Te =
x d − xq
3P 1
[( rs xq vq 2 ) − (rs xd vd 2 ) + (rs 2 − xd xq )vd vq ]
2
2 2 ωe (rs + xd xq )2
(15)
According to (15), when the stator resistance rs is very small and can be neglected, the
torque equation (15) can be simplified as
Te =
3 P 1 x d − xq 2
Vs sin(2δ )
2 2 ω e 2 x d xq
(16)
The output power is
P = Te
=
ωe
P
( )
2
3 x d − xq
2 2 x d xq
(17)
Vs 2 sin(2δ )
where P is the output power, and δ is the load angle.
3. Direct torque control
3.1 Basic principle
Fig. 1 shows the block diagram of the direct torque control system. The system includes two
major loops: the torque-control loop and the flux-control loop. As you can observe, the flux
and torque are directly controlled individually. In addition, the current-control loop is not
required here. The basic principle of the direct torque control is to bound the torque error
and the flux error in hysteresis bands by properly choosing the switching states of the
inverter. To achieve this goal, the plan of the voltage vector is divided into six operating
258
Torque Control
sectors and a suitable switching state is associated with each sector. As a result, when the
voltage vector rotates, the switching state can be automatically changed. For practical
implementation, the switching procedure is determined by a state selector based on precalculated look up tables. The actual stator flux position is obtained by sensing the stator
voltages and currents of the motor. Then, the operating sector is selected. The resolution of
the sector is 60 degrees for every sector. Although the direct torque is very simple, it shows
good dynamic performance in torque regulation and flux regulation. In fact, the two loops
on torque and flux can compensate the imperfect field orientation caused by the parameter
variations. The disadvantage of the direct torque control is the high frequency ripples of the
torque and flux, which may deteriorate the performance of the drive system. In addition, an
advanced controller is not easy to apply due to the large torque pulsation of the motor.
In Fig.1, the estimating torque and flux can be obtained by measuring the a-phase and the bphase voltages and currents. Next, the speed command is compared with the estimating
speed to compute the speed error. Then, the speed error is processed by the speed controller
to obtain the torque command. On the other hand, the flux command is compared to the
estimated flux. Finally, the errors ΔTe and Δλs go through the hysteresis controllers and the
switching table to generate the required switching states. The synchronous reluctance motor
rotates and a closed-loop drive system is thus achieved. Due to the limitation of the scope of
this paper, the details are not discussed here.
Fig. 1. The block diagram of the direct torque control system
3.2 Controller design
The SynRM is easily saturated due to its lack of permanent magnet material. As a result, it
has nonlinear characteristics under a heavy load. To solve the problem, adaptive control
algorithms are required. In this paper, two different adaptive controllers are proposed.
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
259
A. Adaptive Backstepping Controller
From equation (6), it is not difficult to derive
and
d
1
ωr = [Te − TL − Bmωr ]
dt
Jm
= A1Te + A2TL + A3ωr
A1 =
1
Jm
(18)
(19)
A2 = −
1
Jm
(20)
A3 = −
Bm
Jm
(21)
Where A1 , A2 , A3 are constant parameters which are related to the motor parameters. In
the real world, unfortunately, the parameters of the SynRM can not be precisely measured
and are varied by saturated effect or temperature. As a result, a controller designer should
consider the problem. In this paper, we proposed two control methods. The first one is an
adaptive backstepping controller. In this method, we consider the parameter variations and
external load together. Then
d
ωr = A1Te + A3ωr +( A2TL + ΔA1Te + ΔA2 TL + ΔA3ωr ) = A1Te + A3ωr +d
dt
(22)
d= ( A2TL + ΔA1Te + ΔA2 TL + ΔA3ωrm )
(23)
and
where ΔA1 , ΔA2 , ΔA3 are the variations of the parameters, and d is the uncertainty
including the effects of the parameter variations and the external load.
Define the speed error e2 as
*
e2 = ωrm
− ωrm
(24)
Taking the derivation of both sides, it is easy to obtain
*
e2 = ω rm
− ω rm
(25)
In this paper, we select a Lyapunov function as
1 2 1 1 2
e2 +
d
2
2γ
1
11
d − dˆ
= e2 2 +
2
2γ
V=
(
)
2
(26)
260
Torque Control
Taking the derivation of equation (26), it is easy to obtain
= e e + 1 dd
V
2 2
γ
= e2 e2 +
= e2 e2 −
( )
1 ˆ
d d−d
γ
(27)
1 ˆ
dd
γ
By substituting (25) into (27) and doing some arrangement, we can obtain
(
)
ˆ
= e ω * − A T − A ω − d − 1 dd
V
rm
2
1 e
3 rm
(
γ
)
1 ˆ
*
= e2 ω rm
− A 1Te − A 3ωrm − d − dˆ − dd
(28)
γ
Assume the torque can satisfy the following equation
Te =
(
1
*
ω rm
− A 3ωrm − dˆ + Me2
A1
)
(29)
Substituting (29) into (28), we can obtain
− 1 dd
ˆ
= −Me 2 − de
V
2
2
γ
(30)
From equation (30), it is possible to cancel the last two terms by selecting the following
adaptive law
d̂ = −γ e2
(31)
∧
In equation (31), the convergence rate of the d is related to the parameter γ . By submitting
(31) into (30), we can obtain
= − Me 2 ≤ 0
V
2
(32)
From equation (32), we can conclude that the system is stable; however, we are required to
use Barbalet Lemma to show the system is asymptotical stable [32]-[34].
By integrating equation (32), we can obtain
∞
∫0
dτ = V(∞ ) − V(0) < ∞
V
(33)
From equation (33), the integrating of parameter e2 2 of the equation (32) is less than infinite.
Then, e2 (t ) ∈ L ∞ ∩ L 2 , and e2 (t ) is bounded. According to Barbalet Lemma, we can
conclude [32]-[34]
lim e2 (t ) = 0
t →∞
(34)
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
261
The block diagram of the proposed adaptive backstepping control system is shown in Fig. 2,
which is obtained from equations (29) and (31).
Fig. 2. The adaptive backstepping controller.
B. Model-Reference Adaptive Controller
Generally speaking, after the torque is applied, the speed of the motor incurs a delay of
several micro seconds. As a result, the transfer function between the speed and the torque of
a motor can be expressed as:
ωrm
Te
1
=
Jm
e -τ s
⎛ s + Bm ⎞
⎜
J m ⎟⎠
⎝
(35)
Where τ is the delay time of the speed response. In addition, the last term of equation (35)
can be described as
e −τ s ≅
1 /τ
1
≅
1 + τ s s + 1 /τ
(36)
Substituting (36) into (35), one can obtain
ωrm
Te
1
=
(s +
Jm
Bm
1
τ
Jm
1
) (s + )
=
τ
b0
s 2 + a1s + a0
(37)
where
a1 = (
Bm 1
+ )
Jm τ
(38a)
262
Torque Control
a0 =
b0 =
Bm
J mτ
(38b)
1
(38c)
J mτ
Equation (37) can be described as a state-space representation:
⎡ x 1 ⎤ ⎡ 0
⎢ x ⎥ = ⎢ - a
⎣ 2⎦ ⎣ 0
1 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤
+
u
- a1 ⎥⎦ ⎢⎣ x2 ⎥⎦ ⎢⎣b0 ⎥⎦
⎡x ⎤
y = [1 0] ⎢ 1 ⎥
⎣ x2 ⎦
(39a)
(39b)
Where x1 = ωrm = y p , x2 = ω rm , and u = Te . Next, the equations (39a) and (39b) can be
rewritten as :
X p = Ap X p +Bp u
(40a)
y p = C Tp X p
(40b)
⎡ x1 ⎤
Xp = ⎢ ⎥
⎣ x2 ⎦
(41a)
and
where
⎡ 0
Ap = ⎢
⎣ − a0
1 ⎤
− a1 ⎥⎦
(41b)
⎡0⎤
Bp = ⎢ ⎥
⎣b0 ⎦
(41c)
C pT = [ 1 0 ]
(41d)
After that, we define two state variables w1 and w2 as:
1 = - hw1 + u
w
(42)
2 = - hw2 + y p
w
(43)
and
The control input u can be described as
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
263
u = Kr + Q1 w1 + Q2 w2 + Q0 y p
(44)
= θ Tφ
where
θ T = [ K Q1 Q2 Q0 ]
and
φ = ⎡⎣r w1 w2
y p ⎤⎦
T
where γ is the reference command. Combining (40a),(42), and (43), we can obtain a new
dynamic equation as
⎡ X p ⎤ ⎡ Ap
⎢ ⎥ ⎢
1⎥ = ⎢ 0
⎢w
⎢w
⎥ ⎢ T
⎣⎢ 2 ⎦⎥ ⎢⎣C p
0 ⎤ ⎡ X p ⎤ ⎡ Bp ⎤
⎥⎢ ⎥ ⎢ ⎥
- h 0 ⎥ ⎢ w1 ⎥ + ⎢ 1 ⎥ u
⎥
0 - h ⎥⎦ ⎢⎣ w2 ⎥⎦ ⎢⎣ 0 ⎥⎦
0
(45)
Substituting (44) into (45), we can obtain
T
⎡ X p ⎤ ⎡ Ap + BpQ0C p
⎢
⎢ ⎥
1⎥ = ⎢
Q0C Tp
⎢w
⎢
⎢w
⎥
C Tp
⎣⎢ 2 ⎦⎥ ⎣⎢
BpQ1
- h + Q1
0
BpQ2 ⎤ ⎡ X p ⎤ ⎡ BpK ⎤
⎥⎢ ⎥ ⎢
⎥
Q2 ⎥ ⎢ w1 ⎥ + ⎢ K ⎥ r
⎥⎢ ⎥ ⎢
⎥
- h ⎦⎥ ⎣ w2 ⎦ ⎣ 0 ⎦
(46)
Define K = K − K * , Q 1 = Q1 − Q1* , Q 2 = Q2 − Q2* , Q 0 = Q0 − Q0*
Then, equation (46) can be rearranged as
* T
⎡ X p ⎤ ⎡ Ap + BpQ0C p
⎢ ⎥ ⎢
1⎥ = ⎢
Q0*C Tp
⎢w
⎢
⎢w
⎥
C Tp
⎣⎢ 2 ⎦⎥ ⎢⎣
Where θT = ⎡⎣K Q 1 Q 2
(47) as a simplified form
BpQ1*
- h + Q1*
0
BpQ2* ⎤ ⎡ Bp K * ⎤
⎡ Bp ⎤
⎥ ⎢
⎥
⎢ ⎥
Q2* ⎥ + ⎢ K * ⎥ r + ⎢ 1 ⎥ θTφ
⎥ ⎢
⎥
⎢0⎥
- h ⎥⎦ ⎢⎣ 0 ⎥⎦
⎣ ⎦
(47)
Q 0 ⎤⎦ is the parameter errors. It is possible to rearrange equation
⎡ Bp K * ⎤
⎢
⎥
X c = Am Xc + ⎢ K * ⎥ r + BmθTφ
⎢
⎥
⎢ 0 ⎥
⎣
⎦
(48)
T
Yc = C m
Xc
(49)
and
264
Torque Control
where
⎡ Ap + BpQ0*C Tp
⎡X p ⎤
⎢
⎢ ⎥
Xc = ⎢ w1 ⎥ , Am = ⎢
Q0*C Tp
⎢
⎢w ⎥
⎢
C Tp
⎣ 2⎦
⎣
T
Bm
= ⎡⎣ Bp
T
1 0 ⎤⎦ , C m
= ⎡⎣C Tp
BpQ1*
- h + Q1*
0
BpQ2* ⎤
⎥
Q2* ⎥ ,
⎥
- h ⎥⎦
0 0 ⎤⎦ .
After that, the referencing model of the closed-loop system can be described as :
X m = Am Xm + BmK *r
(50)
T
Ym = C m
Xm
(51)
and
T
where Xm
= ⎡⎣ X p* w1* w2* ⎤⎦ is the vector of the state variables, and Ym is the output of the
referencing model. Now, we define the derivation of the state variable error and the output
error as:
e = X c − X m
(52)
e1 = Yc − Ym
(53)
and
Substituting (50)-(51) into (52) and (53), one can obtain
e = Am e + BmθTφ
(54a)
T
e1 = C m
e
(54b)
and
By letting Bm = BmK * , it is not difficult to rearrange (54a) as
e = Am e + Bm
1 T
θ φ
K*
(55a)
Combining (54b) and (55a), one can obtain
e1 =
1 T
-1
C m ( sI − Am ) BmθTφ
K*
(56)
It is essential that the degree of the referencing model equal the uncontrolled plant. As a
result, equation (55a) has to be revised as [12]:
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
265
e = Am e + Bm1
1 T
θ φ
K*
(57a)
where Bm1 = BmL( s ) , φ = L(-1s )φ , L( s ) = s + F ; F > 0 ’
After that, we can obtain
e1 =
1 T
-1
C m ( sI − Am ) Bm1θTφ
K*
(58)
Now, selecting a Lyapunov function as
V=
1 T
1
1
e Pm e + θT Γ -1θ *
2
2
K
(59)
where Pm is a symmetry positive real matrix, and Γ is a positive real vector.
The matrix Pm satisfies the following two equations:
T
Am
Pm + Pm Am = −Q
(60)
T
Pm Bm1 = C m
(61)
and
where Q is a symmetry positive real matrix . Taking the derivation of equation (59) and
substituting (60), (61) into the derivation equation, we can obtain
= -1 eT Qe + eT P B 1 θTφ + θT Γ -1θ 1
V
m m1 *
2
K
K*
-1
1
1
= eT Qe + e1 * θTφ + θT Γ -1θ *
2
K
K
(62)
It is possible to select the adaptive law as
θ = - sgn (
where sgn(
1
)Γe1φ
K*
(63)
K*
1
)
=
, substituting (63) into (62), we can obtain :
K*
K*
= -1 eT Qe ≤ 0
V
2
(64)
Next, by using Barbalet Lemma, we can obtain that the system is asymmetrical and
lim
t→∞
Finally, we can obtain
e1 (t ) =0
(65)
266
Torque Control
up = L( s )θ T L( s ) -1φ = L( s )θ Tφ
= θTφ + θ Tφ + Fθ Tφ = θTφ + θ Tφ
(66)
The block diagram of the model-reference control system is shown in Fig. 3, which includes
referencing model, adaptive controller, and adaptive law.
ym
b
s + as + b
2
r
−
e1
+
up = θ φ + θ φ
T
φ
1
s+F
T
φ
J mτ
B
B
1
s + ( m + )s + m
Jm τ
J mτ
yp
2
w1
1
s+h
w2
1
s+h
T
φ = ⎡⎣r w1 w2 yp ⎤⎦
up
1
1
θ = − sgn( * )Γe1φ
K
T
θ
T
1
s
θ
T
Fig. 3. The block diagram of the model reference adaptive controller.
4. Implementation
The implemented system is shown in Fig. 4. The system includes two major parts: the
hardware circuits and the software programs. The hardware circuits include: the
synchronous reluctance motor, the driver and inverter, the current and voltage sensors, and
the A/D converters. The software programs consist of the torque estimator, the flux
estimator, the speed estimator, the adaptive speed controller, and the direct torque control
algorithm. As you can observe, the most important jobs are executed by the digital signal
processor; as a result, the hardware is quite simple. The rotor position can be obtained by
stator flux, which is computed from the stator voltages and the stator currents. The digital
signal processor outputs triggering signals every 50 μ s ; as a result, the switching frequency
of the inverter is 20 kHz. In addition, the sampling interval of the speed control loop is 1 ms
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
267
although the adaptive controllers are quite complicated. The whole drive system, therefore,
is a multi-rate fully digital control system.
T1 , T1'
DSP
T2 , T2'
'
3
T3 , T
Driver
and
Inverter
SynRM
va
Voltage
and
Current
Sensors
vb
ia
ib
Fig. 4. The implemented system.
A. Hardware Circuits
The hardware circuits of the synchronous reluctance drive system includes the major parts.
The details are discussed as follows.
a. The delay circuit of the IGBT triggering signals.
Fig. 5 shows the proposed delay circuit of the IGBT triggering signals. The delay circuit is
designed to avoid the overlapping period of the turn-on interval of the upper IGBT and the
lower IGBT for the inverter. Then, the inverter can avoid a short circuit. In this paper, the
delay time of the delay circuit is set as 2 μ s . To achieve the goal, two integrated circuit chips
are used: 74LS174 and 74LS193. The basic idea is described as follows. First, the digital
signal processor sends a clock signal to 74LS193. The time period of the clock is 62.5 μ s . The
74LS193 executes the dividing frequency function and finally generates a clock signal with a
0.5 μ s period. After that, the 74LS193 sends it into the CLK pin of 74LS174. The 74LS174
provides 6 series D-type flip-flop to generate a 3 μ s delay. Finally, an AND gate is used to
make a 3 μ s for a rising-edge triggering signal but not a falling-edge triggering signal.
b. The driver of the IGBTs
The power switch modules used in the paper are IGBT modules, type 2MBI50-120. Each
module includes two IGBTs and two power diodes. The driver of the IGBT is type EX-B840,
made by Fuji company. The detailed circuit of the driver for an IGBT is shown in Fig. 6. In
Fig. 6, the EX-B840, which is a driver, uses photo-couple to convert the control signal into a
268
Torque Control
U1
2
5
7
10
12
15
1Q
2Q
3Q
4Q
5Q
6Q
1D
2D
3D
4D
5D
6D
CLK
CLR
3
4
6
11
13
14
9
1
DSP trigger
U7
H1
3
2
6
7
QA
QB
QC
QD
A
B
C
D
15
1
10
9
U8A
Driver
2
3
74LS174
+5V
1
7408
12
13
UP
CO
BO DOWN
LOAD
CLR
5
4
11
14
DSP H1
74LS193
Fig. 5. The delay circuit of the IGBT triggering signals.
triggering signal for an IGBT. In addition, the EX-B840 provides the isolation and overcurrent protection as well.
When the control signal is “High”, the photo transistor is turned on. Then, the photo diode
is conducted. A 15V can across the gate and emitter of the IGBT to turn on the IGBT. On the
other hand, when the control signal is “Low”, the photo transistor is turned off. As a result,
the photo diode is cut off. A -5V can across the gate and emitter of the IGBT to make IGBT
turn off immediately.
The protection of the IGBT is included in Fig. 6. When the IGBT has over-current, the
voltage across the collector and emitter of the IGBT is obviously dropped. After the 6-pin of
EX-B840 detects the dropped voltage, the 5-pin of the EX-B840 sends a “Low” voltage to the
photo diode. After that, the photo diode is opened, and a -5V across the gate and emitter of
the IGBT is sent to turn off the IGBT.
c. The snubber circuit
The snubber circuit is used to absorb spike voltages when the IGBT is turned off. As we
know, the synchronous reluctance motor is a kind of inductive load. In Fig. 7, when the
upper leg IGBT T is turned off, the low leg IGBT T ' cannot be turned on immediately due to
the required dead-time, which can avoid short circuits. A new current path to keep the
current continuous flow is required. The new current path includes the fast diode D and the
snubber capacitor C s . So, the current can flow through the fast diode D and the
capacitor CS , and then stores its energy into the capacitor CS . On the other hand, when the
IGBT is turned on in next time interval, the stored energy in the capacitor CS can flow
through the resistance Rs and the IGBT T . Finally, the energy dissipates in the resistance
Rs . By suitably selecting the parameter C s and Rs , a snubber circuit with satisfactory
performance can be obtained.
d. The current detecting circuit
The current detecting circuit is used to measure the stator current of the synchronous
reluctance motor, and can be shown in Fig. 8. The Hall current sensor, typed LP-100, is used
to sense the stator current of the motor and to provide the isolation between the power stage
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
269
and the control circuit. The primary side of the LT-100P can measure 0 to 100 A with a
bandwidth of 100 kHz. The basic principle is discussed as follows. The primary of the LT100P has 5 turns. As a result, when 1A flows into the primary side, the secondary side of the
LT-100P can generate 5mA. The current flows from the M pin of the LP-100P to the 0.1 KΩ
resistance, and then provides 0.5V voltage drop. As a result, in this chapter, for every 1A
primary current, the circuit can output 0.5V. A low-pass filter is designed to eliminate the
high-frequency noise.
Fig. 6. The circuit of the driver for IGBTs.
Fig. 7. The snubber circuit.
270
Torque Control
Fig. 8. The current detecting circuit.
e. The voltage detecting circuit
The voltage detecting circuit is used to sense the stator voltage of the synchronous
reluctance motor, which is an important item for computing the estimated flux of the motor.
A voltage isolation amplifier, AD210, is selected to isolate the input side and output side. In
the chaper, R0 and R1 are used to attenuate the input voltage to be 0.05 vab . As a result,
the input of AD 210 is limited under ±10V .
Fig. 9. The voltage detecting circuit.
f. The A/D conversion circuit
The measured voltages and currents from Hall current sensor and AD210 are analog signals.
In order to be read by a digital signal processor, the A/D conversion is required. In this
chapter, the 12 bit A/D converter with a 3 μ s conversion time is used. The A/D converter is
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
271
typed AD578. The detailed circuit is shown in Fig. 10. There are two sets: one for voltage
conversion, and the other for current conversion.
When the analog signal is ready, the digital signal processor outputs a triggering signal to
the A/D converter. Then, each AD578 converter starts to convert the analog signal into a
digital signal. When the conversion process finishes, an EOC signal is sent from the AD578
to latch the 74LS373. Next, the digital signal processor reads the data. In this chapter, a timer
with a fixed clock is used to start the conversion of the AD578 and then the digital signal
processor can read the data. By using the method, we can simplify the software program of
the digital signal processor.
START
2
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
1
74LS04
CURRENT
100
100
+5V
CLDADJ
CLKOUT DIG GND
SHT CYC
CLK IN
BIT 1(N)
EOC
BIT 1
START
SER OUT(N) BIT 2
BIT 3
SER OUT
BIT 4
REF OUT
BIT 5
GAIN
BIT 6
OFFSET
BIT 7
10V SPAN
BIT 8
20V SPAN
BIT 9
ZERO ADJ
BIT 10
ANA GND
BIT 11
+15V
BIT 12
-15V
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
18
17
14
13
8
7
4
3
11
1
8D
7D
6D
5D
4D
3D
2D
1D
C
OC
8Q
7Q
6Q
5Q
4Q
3Q
2Q
1Q
19
16
15
12
9
6
5
2
18
17
14
13
8
7
4
3
11
1
74LS373
8D
7D
6D
5D
4D
3D
2D
1D
C
OC
8Q
7Q
6Q
5Q
4Q
3Q
2Q
1Q
19
16
15
12
9
6
5
2
74LS373
READ
AD578
6.8uf
C1
DATA BUS
Fig. 10. A/D converter circuit.
SUB1 15
14
SUB2 13
12
SUB3 11
10
9
7
Y0
A
Y1
B
Y2
C
Y3
Y4
Y5
G1
Y6 G2A
Y7 G2B
1
2
3
18
16
14
12
9
7
5
3
6
4
5
1Y1
1Y2
1Y3
1Y4
2Y1
2Y2
2Y3
2Y4
1A1
1A2
1A3
1A4
2A1
2A2
2A3
2A4
74LS138
1G
2G
2
4
6
8
11
13
15
17
ICLK1
CLKX0
DX0
FSX0
CLKX1
DX1
FSX1
XF1
1
19
74LS244
CURRENT
VOLTAGE
VOLINE
POSITION
OUTPUT
15
14
13
12
11
10
9
7
Y0
A
Y1
B
Y2
C
Y3
Y4
G1
Y5
Y6 G2A
Y7 G2B
1
2
3
18
16
14
12
9
7
5
3
6
4
5
74LS138
1Y1
1Y2
1Y3
1Y4
2Y1
2Y2
2Y3
2Y4
1A1
1A2
1A3
1A4
2A1
2A2
2A3
2A4
1G
2G
2
4
6
8
11
13
15
17
RESET
IOSTRB
H1
1
3
5
7
9
11
13
15
17
19
21
A11 23
A9 25
A7 27
A5 29
31
A3 33
A1 35
37
39
41
43
45
47
49
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
ICLK0
CLKR0
DR0
FSR0
CLKR1
DR1
FSR1
XF0
A12
A10
A8
A6
A4
A2
A0
TODRY
IOR/N
H3
HEADER
1
19
IGBT TRIGGER
74LS244
1
2
74LS04
1
2
74LS04
1
2
74LS04
1
2
1
74LS04 2
3
74LS32
Fig. 11. The interfacing circuit of the DSP.
272
Torque Control
g. The interfacing circuit of the digital signal processor
In the chapter, the digital signal processor, type TMS320-C30, is manufactured by Texas
Instruments. The digital signal processor is a floating-point operating processor. The
application board, developed by Texas Instruments, is used as the major module. In
addition, the expansion bus in the application board is used to interface to the hardware
circuit. The voltage, current, speed, and rotor position of the drive system are obtained by
using the expansion bus. As a result, the address decoding technique can be used to provide
different address for data transfer. In addition, the triggering signals of the IGBTs are sent
by the following pins: CLKX1, DX1, and FSX1. The details are shown in Fig. 11.
A. Software Development
a. The Main Program
Fig. 12 shows the flowchart of the initialization of the main program. First, the DSP enables
the interrupt service routine. Then, the DSP initializes the peripheral devices. Next, the DSP
sets up parameters of the controller, inverter, A/D converter, and counter. After that, the
DSP enables the counter, and clear the register. Finally, the DSP checks if the main program
is ended. If it is ended, the main program stops; if it is not, the main program goes back to
the initializing peripheral devices and carries out the following processes mentioned.
Fig. 12. The flowchart of the initialization of the main program.
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
273
b. The interrupt service routines
The interrupt service routines include: the backstepping adaptive controller, the reference
model adaptive controller, and the switching method of the inverter. The detailed
flowcharts are shown in Fig. 13, Fig. 14, and Fig. 15.
d̂
Fig. 13. The subroutine of the backstepping adaptive controller.
274
Torque Control
{
K ,Q 1,Q 2,Q 0}
*
Te = uP =θ T φ +θT φ
Fig. 14. The subroutine of the reference model adaptive controller.
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
275
ΔTe = T e −Tˆe
*
Δλ ss = λ ss −λˆss
*
Fig. 15. The subroutine of the switching method of the inverter.
276
Torque Control
5. Experimental results
Several experimental results are shown here. The input dc voltage of the inverter is 150V.
The switching frequency of the inverter is 20 kHz. In addition, the sampling interval of the
minor loop is 50 μ s , and the sampling interval of the speed loop is 1 ms. The parameters of
the PI controller are K P =0.006 and K I =0.001. The parameters of the adaptive backstepping
controller are M=3 and γ =0.8. The parameters of the model referencing controller are Γ =
[ -0.0002 -0.004 -0.004 -0.0006]. Fig. 16(a)(b) show the measured steady-state waveforms.
Fig. 16(a) is the measured a-phase current and Fig. 16(b) is the measured line-line voltage,
vab . Fig. 17(a) is the simulated fluxes at 1000 r/min. Fig. 17(b) is the simulated flux
trajectory at 1000 r/min. Fig. 17(c) is the measured fluxes at 1000 r/min. Fig. 17(d) is the
measured flux trajectory at 1000 r/min. As you can observe, the trajectories are both near
circles in both simulation and measurement. Fig. 18(a) shows the comparison of the
measured estimating rotor angle and the measured real rotor angle at 50 r/min. As we
know, when the motor is operated at a lower speed, the flux becomes smaller. As a result,
the motor cannot be operated well at lower speeds due to its small back emf. The estimating
error, shown in Fig. 18(b) is obvious. Fig. 19(a)(b) show the measured estimating rotor angle
at 1000 r/min. Fig. 19(a) shows the comparison of the measured estimating rotor angle and
the measured real rotor angle at 1000 r/min. Fig. 19(b) shows the estimating error, which is
around 2 degrees. As a result, the estimating error is reduced when the motor speed is
increased. In addition, Fig. 19(b) is varied more smoothly than the Fig. 18(b) is. The major
reason is that the back emf has a better signal/noise ratio when the motor speed increases.
Fig. 20(a) shows the measured transient responses at 50 r/min. Fig. 20(b) shows the
measured load disturbance responses under 2 N.m external load. The model reference
control performs the best. The steady-state errors of Fig. 20(a)(b) are: 2.7 r/min for PI
controller, 0.5 r/min for ABSC controller, and 0.1 r/min for MRAC controller, respectively.
According to the measured results, the MRAC controller performs the best and the PI
controller performs the worst in steady-state. Fig. 21(a)(b) show the measured speed
responses at 1000 r/min. Fig. 21(a) is the measured transient responses. Fig. 21(b) is the load
disturbance responses under 2 N.m. According to the measured results, the model-reference
controller performs better than the other two controllers in both transient response and load
disturbance response again. The steady-state errors of Fig. 21(a)(b) are: 7.3 r/min for PI
controller, 1.9 r/min for ABSC controller, and 0.1 r/min for MRAC controller, respectively.
As you can observe, the conclusions are similar to the results of Fig. 20(a)(b). Fig. 22(a)
shows the measured external - d̂ of the adaptive backstepping control. Fig. 22(b) shows the
measured speed error of the adaptive backstepping control by selecting different
parameters. Fig. 23(a)(b)(c)(d) show the relative measured parameters K, K , Q1 , Q2 , Q0 of the
model-reference controller. All the parameters converge to constant values. Fig. 24(a)(b)(c)
show the measured speed responses of a triangular speed command. The PI controller has a
larger steady-state error than the adaptive controllers have. Fig. 25(a)(b)(c) show the
measured speed responses of a sinusoidal speed command. As you can observe, the modelreference controller performs the best. The model- reference controller has a smaller steadystate error and performs a better tracking ability than the other controllers.
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
277
(a)
(b)
Fig. 16. The measured steady-state waveforms. (a) phase current (b) line voltage.
278
Torque Control
(a)
(b)
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
279
(c)
(d)
Fig. 17. The stator flux trajectories at 1000 r/min. simulated fluxes (b) simulated trajectory
(c) measured fluxes (d) measured trajectory.
280
Torque Control
(a)
(b)
Fig. 18. The measured estimating rotor angle at 50 r/min. (a) comparison (b) estimating
error.
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
281
(a)
(b)
Fig. 19. The measured estimating rotor angle at 1000 r/min. (a) comparison (b) estimating
error.
282
Torque Control
(a)
(b)
Fig. 20. The measured speed responses at 50 r/min, (a) transient responses (b) load
disturbance responses
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
283
(a)
(b)
Fig. 21. The measured speed responses at 1000 r/min. (a) transient responses (b) load
disturbance responses.
284
Torque Control
(a)
(b)
Fig. 22. The measured responses of adaptive backstepping control. (a) - dˆ (b) speed error.
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
285
(a)
(b)
286
Torque Control
(c)
(d)
Fig. 23. The measured responses of model-reference control. (a) K (b) Q1 (c) Q2 (d) Q0 .
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
287
(a)
(b)
288
Torque Control
(c)
Fig. 24 The measured speed responses of a triangular speed command.
(a) PI (b) backstepping (c) model-reference.
(a)
Controller Design for Synchronous Reluctance
Motor Drive Systems with Direct Torque Control
289
(b)
(c)
Fig. 25. The speed responses of a sinusoidal speed command. (a) PI (b) backstepping (c)
model-reference.
290
Torque Control
6. Future trends
In this chapter, by using the torque control, a closed-loop sensorless speed drive system has
been implemented. The proposed system can be operated from 30 r/min to 2000 r/min with
satisfactory performance. Unfortunately, the proposed system cannot be operated from
standstill to 30 r/min. As a result, it is necessary in the future to continuously improve the
controller design, hardware design, and software design to reduce the torque pulsations and
then provide better performance in low-speed operating range. In addition, it is another aim
to realize a closed-loop high performance position control system by using a torque control
method.
7. Conclusions
In this chapter, two different adaptive controllers have been proposed for a synchronous
reluctance motor drive system. The parameters of the controllers are on-line tuned. The
adaptive backstepping controller has simple control algorithm. It is more easily
implemented than the model reference adaptive controller is. On the other hand, the model
reference adaptive controller performs better in transient responses and steady-state
characteristics. A digital signal process is used to execute the control algorithm. As a result,
the hardware circuit is very simple. The implemented system shows good transient
responses, load disturbance responses, and tracking ability in triangular and sinusoidal
commands. This paper provides a new direction in the application of adaptive controller
design for a synchronous reluctance motor drive system.
8. References
[1] Park, J. M., Kim, S., Hong, J. P., and Lee, J. H.: ‘Rotor design on torque ripple reduction
for a synchronous reluctance motor with concentrated winding using response
surface methodology’, IEEE Trans. Magnet., vol. 42, no. 10, pp. 3479-3481, 2006.
[2] G. Sturtzer, D. Flieller, and J. P. Louis, “Mathematical and experi- mental method to
obtain the inverse modeling of nonsinusoidal and saturated synchronous
reluctance motors,” IEEE Trans. Energy Conversion, vol. 18, no. 4, pp. 494-500, Dec.
2003.
[3] Hofmann, H. F., Sanders, S. R., and Antably, A.: ‘Stator-flux-oriented vector control of
synchronous reluctance machines with maximized efficiency’, IEEE Trans. Ind.
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