State-space average Modeling of DC-DC Converters with

State-space average Modeling of DC-DC Converters with
PROJECT REPORT
State-space average Modeling of DC-DC Converters with
parasitic in Discontinuous Conduction Mode (DCM).
A THESIS SUBMITTED IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
BACHELOR OF TECHNOLOGY
IN
ELECTRICAL ENGINEERING.
By
Antip Ghosh
Mayank Kandpal
Under the guidance of
Prof. S. Samanta
Department of Electrical Engineering
National Institute of Technology, Rourkela
1
PROJECT REPORT
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
CERTIFICATE
This is to certify that the thesis entitled” State-space average Modeling of DCDC Converters with parasitic in Discontinuous Conduction Mode” submitted by
Mr Antip Ghosh and Mr Mayank Kandpal in partial fulfilment of the
requirements for the award of Bachelor of Technology Degree in Electrical
Engineering at National Institute of Technology, Rourkela (Deemed University)
is an authentic work carried out by him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not
been submitted to any other University/ Institute for the award of any degree or
diploma.
Prof. S. Samanta
Department of Electrical Engineering
DATE -…./05/10
National Institute of Technology
Rourkela – 769008
2
PROJECT REPORT
ACKNOWLEDGEMENT
No thesis is created entirely by an individual, many people have helped to create
this thesis and each of their contribution has been valuable. My deepest
gratitude goes to my thesis supervisor, Prof. S. Samanta, Department of Electrical
Engineering, for his guidance, support, motivation and encouragement
throughout the period this work was carried out. His readiness for consultation
at all times, his educative comments, his concern and assistance even with
practical things have been invaluable. I would also like to thank all professors
and lecturers, and members of the department of Electrical Engineering for their
generous help in various ways for the completion of this thesis.
Antip Ghosh
Mayank Kandpal
Roll. No. 10602021
Roll. No. 10602056
Dept. of Electrical Engineering
Dept. of Electrical Engineering
NIT Rourkela
NIT Rourkela
3
PROJECT REPORT
Contents
List of Figures
05
Abstract
06
1 Introduction
07
1.1
Overview
08
1.2
Different topologies.
08
1.3
Different mode of operation.
10
1.4
About the thesis.
14
2
State Space Averaging
15
2.1
Procedure For State-Space Averaging.
16
3
Analysis In DCM
17
3.1
Overview.
18
3.2
Framework.
19
3.3 Reduced Order Model.
23
3.4
Full Order Model.
29
4
Inclusion Of Parasitics In Model
46
4.1
Overview
47
4.2
Transfer functions derivation considering parasitics
48
5
Results and Conclusion
57
References
60
4
PROJECT REPORT
List of Figures
S.l.No.
Name of Figure
1.1
Buck Converter Circuit Diagram
1.2
Boost Converter Circuit Diagram
1.3
Buck-Boost Converter Circuit Diagram
1.4
On-State Configuration for Boost Converter
1.5
Off-State Configuration for Boost Converter
1.6
On-State Configuration for Buck Converter
1.7
Off-State Configuration for Buck Converte
3.1
Waveform denoting Inductor Current in DC-DC Converter
3.2
Including Iload in Boost Converter(ON Position)
3.3
Including Iload in Boost Converter(OFF Position)
4.1
Boost Converter Circuit with parasitics of switched-inductor cell
5
PROJECT REPORT
ABSTRACT
Discontinuous Conduction Mode occurs due to low load current operation of
converters which employ current or voltage unidirectional switches. The
switching ripples in inductor current or capacitor voltage causes the polarity to
reverse of the applied switch current or voltage and thus a zero current mode is
reached. Nowadays, converters are so designed, to operate in DCM for all loads
due to its higher efficiency and lower losses. In this thesis, we have derived the
Reduced Order & Full Order Averaged Models for the Buck and Boost
configuration of converters. Also we calculated the output transfer function of
boost converter which can be further utilized for designing of controller. Then,
parasites effects have been taken into account for Boost Converter and
accordingly, its various transfer functions (Control, Output Impedance, etc.) and
various bode diagram have been plotted and compared with ideal cases.
6
PROJECT REPORT
Chapter 1
INTRODUCTION
1.1 Overview.
1.2 Different topologies.
1.3Different mode of operation.
1.4 About the thesis.
7
PROJECT REPORT
1.1 Overview
Over the years as the portable electronics industry progressed, different
requirements evolved such as increased battery lifetime, small and cheap
systems, brighter, full-color displays and a demand for increased talk-time in
cellular phones. An ever increasing demand from power systems has placed
power consumption at a premium. To keep up with these demands engineers
have worked towards developing efficient conversion techniques and also have
resulted in the subsequent formal growth of an interdisciplinary field of Power
Electronics. However it comes as no surprise that this new field has offered
challenges owing to the unique combination of three major disciplines of
electrical engineering: electronics, power and control.
DC-DC converters
These are electronic devices that are used whenever we want to change DC
electrical power efficiently from one voltage level to another. Generically
speaking the use of a switch or switches for the purpose of power conversion
can be regarded as an SMPS. A few applications of interest of DC-DC converters
are where 5V DC on a personal computer motherboard must be stepped down
to 3V, 2V or less for one of the latest CPU chips; where 1.5V from a single cell
must be stepped up to 5V or more, to operate electronic circuitry. Our main
focus is that in above mentioned applications is that to alter dc energy from a
particular level to other with minimum loss. The need for such converters have
risen due to the fact that transformers are unable to function on dc. A converter
is not manufacturing power. Whatever comes at the output has to come only
from input. Efficiency cannot be made equal to 100%, so input power is always
somewhat larger than output power.
1.2 Different topologies
 Buck converter
 Boost converter
 Buck –boost converter
Buck converter:8
PROJECT REPORT
A Buck converter is a step down DC-DC converter consisting mainly of inductor
and two switches (usually a transistor switch and a diode) for controlling
inductor. It fluctuates between connection of inductor to source voltage to
accumulate energy in inductor and then discharging the inductor’s energy to the
load.
Fig1.1
When the switch pictured above is closed (i.e. On-state), the voltage across the
inductor is VL = Vi − Vo. The current flowing through inductor linearly rises. The
diode doesn’t allow current to flow through it, since it is reverse-biased by
voltage V.
For Off Case (i.e. when switch pictured above is opened), diode is forward biased
and voltage is VL = − Vo (neglecting drop across diode) across inductor. The
inductor current which was rising in ON case, now decreases.
Boost converter:A boost converter (step-up converter), as its name suggests steps up the input
DC voltage value and provides at output. This converter contains basically a
diode, a transistor as switches and at least one energy storage element.
Capacitors are generally added to output so as to perform the function of
removing output voltage ripple and sometimes inductors are also combined
with.
Fig 1.2
9
PROJECT REPORT
Its operation is mainly of two distinct states:
 During the ON period, Switch is made to close its contacts which results in
increase of inductor current.
 During the OFF period, Switch is made to open and thus the only path for
inductor current to flow is through the fly-back diode ‘D’ and the parallel
combination of capacitor and load. This enables capacitor to transfer
energy gained by it during ON period.
Buck–boost converter:This type of converter gives output voltage which is having greater or lesser
magnitude than input value of voltage. Based on duty ratio of switching
transistor, output voltage is adjusted.
Fig 1.3
When switch is turned ON, then the inductor is connected to input voltage
source. This leads to accumulation of energy in the inductor and capacitor
performs the action of supplying energy to load.
When switch is turned OFF, the inductor is made to come in contact with
capacitor and load, so as to provide energy to load and discharged capacitor.
1.3 Different modes of operation:a) A dc-dc converter is said to be operating in CCM, if inductor current never
reaches to zero.
b) A dc-dc converter is said to be operating in DCM, if inductor current
reaches zero and remains there for certain period of time.
10
PROJECT REPORT
Fig 1.4
In a Boost Converter, During ‘On’ Mode:From KVL
-L
=0
From KCL
+C
=0
In State Space form:-[
]=[
][
]+[ ]
;
=[
][
]
;
=[
][
]
During ‘OFF’ Mode:-
Fig 1.5
From KVL
From KCL
In State Space form:-[
=0
=0
]=[
][
During Discontinuous Conduction Mode:11
]+[ ]
PROJECT REPORT
From KVL
From KCL
=0
+C
=0
]=[
In State Space form:-[
][
]+* +
;
=[
][
]
;
=[
][
]
In a Buck Converter, During ‘ON’ Mode:-
Fig 1.6
From KVL
-L
-
From KCL
+C
-
In State Space form:-[
=0
=0
]=[
][
During ‘OFF’ Mode:-
12
]+[ ]
PROJECT REPORT
Fig 1.7
From KVL
=0
From KCL
=0
In State Space form:-[
]=[
][
]+* +
;
=[
][ ]
]+* +
;
=[
][
During Discontinuous Conduction Mode:From KVL
From KCL
=0
+C
In State Space form:-[
=0
]=[
][
13
]
PROJECT REPORT
1.4About the thesis
For modelling of converter two technique can be use
 Circuit averaging technique.
 State space averaging technique.
The latter approach has a number of advantages over circuit averaging
technique, these include:
 More compact representation of equations.
 Ability to obtain more transfer functions than was possible using
circuit averaging technique.
 Both DC and AC transfer functions are obtained with more ease
So in this thesis we are using State space averaging technique to derive Reduced
Order and Full Order Averaged Models for both the buck and boost converters
[1] and the output transfer function by taking a
as constant current source.
Then, parasites effects have been taken into account for Boost Converter as
parasitic are always present in system and accordingly, its various transfer
functions (Control, Output Impedance, etc.) and various bode diagram have
been plotted and compared with ideal cases.
14
PROJECT REPORT
CHAPTER 2
STATE SPACE AVERAGING
2.1 Procedure For State-Space Averaging.
15
PROJECT REPORT
2.1 Procedure For State-Space Averaging
 Draw the linear switched circuit model for each state of the switching
converter.
 Write state equations for each switched circuit model using Kirchhoff’s
voltage and current laws
 Averaging the State- space Equation using the duty ratio.
 Perturb the averaged state equation to yield steady-state (DC) and
dynamic (AC) terms and eliminate the product of any AC terms
 Transform the AC equations into S- domain to solve for Transfer Function.
16
PROJECT REPORT
CHAPTER 3
ANALYSIS IN DCM MODE
3.1 Overview.
3.2 Framework.
3.3 Reduced Order Model.
3.4 Full Order Model.
17
PROJECT REPORT
3.1 Overview
W
hen the implementation of ideal switches of a DC-DC converter are done
by using current unidirectional and /or voltage unidirectional semiconductor
switches, one or more new modes of operation known as DISCONTINUOUS
CONDUCTION MODE (DCM) can occur. It occurs when the load current
requirement is very low for certain operations like in the case of SMPS used in
computers, converters require a very low current during Hibernation or Sleep
Mode and switching ripples in inductor current or capacitor voltage causes the
polarity to reverse of the applied switch current or voltage and thus an zero
current mode is reached giving its name DCM. It is frequently observed in
inverters, DC-DC rectifiers and converters containing two quadrant switches, etc.
as it is usually required that it operates with their loads removed. Nowadays,
some converters are purposely designed to operate in DCM for all loads [2].
Various efforts had been done in the modelling of DCM PWM converters [3]–[6].
These models can be classified either analytically [3], [5] or equivalent circuit
form [4], [6], and can be grouped into two main divisions like: reduced-order model [3], [6];
 full-order model [5], [6].
The inductor current does not appear as a state variable in reduced order
model, which is undesirable for those applications where the paramount control
target is inductor current. In low frequency range, its prediction is accurately
defining the converter’s behaviour. But problem lies at large frequencies,
particularly in phase response, where large discrepancies do occur. Unlike in
reduced order, in full order model inductor current is retained and is much
accurate as compared to reduced order model.
18
PROJECT REPORT
3.2 Framework:In DCM, in addition to two modes as in CCM, there is a third mode of operation
in which capacitor voltage or inductor current is zero. For DCM operation, during
first interval (i.e. ON period) the switch is turned on and inductor current rises
and reached a peak when the switch is about to turn off, and resets to zero at
the end of the OFF period.
̅
Figure 3.1
Thus,
‘ON’ Mode
̇
=
x+
for t
[
‘OFF’ Mode
̇
=
x+
for t
[
‘DC’ Mode
̇
=
x+
for t
[
-(1)
]
]
]
-(2)
-(3)
Note: The duty ratio, ,is algebraically dependent on control and state variable.
This dependency is defined in terms of average values of current and voltage.
That way we can eliminate from state variables and get a model which can be
expressed in averaged state variables. The function which is describing this
dependency is normally termed as the ‘duty-ratio-constraint’.
19
PROJECT REPORT
The modelling method for DCM operation comprises of three steps:
a) Averaging;
b) Inductor current analysis;
c) Duty-ratio constraint.
State space averaging techniques are employed to get a set of equations that
describe the system over one switching period. After applying averaging
technique to equations (1)-(3), we get the following expression:̅̇ = [
] ̅+[
]u –(4)
The above equation can be written as ̅̇ = A ̅ + B u, where,
A=[
] and B=
.
In state space averaging technique in DCM, we are averaging only the matrix
parameter and not the state variables. Equation (4) will hold good when we use
true average of every state variable.
From figure 3.1, it can be deduced that average is:̅
–(5)
Consider when switch is ‘on’, the current which is delivered to capacitor is not
necessarily having the same value as average inductor current. As inductor
current charges rapidly with time, it is quite easy to derive the capacitor
equation with the help of ‘conservation of charge’ principle, and after that
averaging step is performed. The total amount of charge which capacitor obtains
from the inductor during switching cycle is:-
Thus average charging current would be of value:(6)
When a capacitor is connected to resistive load, then the net charge which is
delivered to the capacitor is given by :
20
PROJECT REPORT
(
)-
.
On the average
Note here that the above expression differs from the Kirchhoff Current Law
expression of capacitor which is obtained through state-space averaging. From
model (4), we can define state-space-averaged (SSA) charging current as the
inductor current’s average multiplied with duty ratio for which inductor is
charging the capacitor. From (5), the SSA charging current can be expressed as:̅
– (7)
This expression is different from actual charging current in (6). It can be implied
that a ‘charge conservation’ law is violated in unmodified SSA as the averaging
step is done on complete model thus leading to un matching of responses with
averaged response of dc-dc converters. Thus (4) is modified by dividing by factor
of
the inductor current. The basic method is to rearrange the x, thus x
=[
] , where all inductor currents ( ) are contained in and define a matrix
K, as below:
[
]
With this correction vector, the averaged modified model becomes
̅=[
[
]K ̅ +
]
- (8)
For a Buck Converter,
=[
=[ ] ,
],
=* + ,
=[
],
],
=[
=* +.
21
PROJECT REPORT
The state space averaged model for the above equation is
[
̅
]=[
̅̅̅
][
̅
]
̅̅̅
[ ]vin .
Since in this model only one inductor is involved plus with x’s dimension is two,
the modification matrix denoted by K is simply given as:
].
K =[
Thus the resulted averaged model after modification will be given by:
[
̅
]=[
̅̅̅
]
=[
][
[
̅
]
̅̅̅
[ ]vin
̅
] + [ ]vin.
̅̅̅
– (9)
For a Boost Converter,
=[
=[ ] ,
],
=[ ] ,
],
=[
],
=[
=* +.
The state space averaged model for the above equation is
[
̅
]=[
̅̅̅
][
̅
]
̅̅̅
[
]vin .
As we can see only one inductor is there and the x is having dimension two, the
modification matrix K is simply.
K =[
].
Thus modified averaged model of boost converter in DCM would be like below:-
22
PROJECT REPORT
[
̅
]=[
̅̅̅
=[
]
[
̅
]
̅̅̅
][
[
]vin
̅
]+[
̅̅̅
]vin. – (10)
3.3 Reduced Order Model:To complete the averaged model (8), a duty ratio constraint is defined showing
the dependency of on other variables. Usually in conventional state-space
averaging technique [3], inductor’s voltage balance eqn. is used in defining dutyratio constraint.
For the buck topology, utilizing the volt second balance over the switching cycle,
For time, T1 = d1TS,
For Time,
By removing
;
,
from above equations,
Similarly for Boost Converter,
23
PROJECT REPORT
For time,
,
;
For Time,
By removing
from above equations,
d2 =
For Buck Converter, Substituting
from (11) in equation (9), we get
̅
̅̅̅
̅
̅̅̅
Similarly for Boost Converter, Substituting
in equation (10), we get
̅
̅̅̅
̅̅̅
̅
From these calculations for buck and boost converters, it can be seen that
inductor current dynamics disappear thus resulting into degenerate model.
Since inductor current is not present in state variable in this reduced order
model, it must be replaced by expressing it as an algebraic function of other
variables, so that inductor dynamics is removed.
For a buck converter, peak of inductor current is given by,
24
PROJECT REPORT
Average of inductor current is given by,
̅
=
Substituting (17), the above relation (14) can be written as
̅
(
)(
)
For a boost converter, peak of inductor current is given by,
Average of inductor current is given by,
̅
Substituting (20), the above relation can be written as
̅̅̅
̅
̅̅̅
̅ from (18) can be replaced into (14) to give CONVENTIONAL AVERAGED
MODEL for BUCK CONVERTER in DCM [3], to remove dependencyon ̅ .
̅̅̅
̅̅̅
̅̅̅
̅̅̅
Similarly for Boost Converter, the model will be obtained by replacing (21) into
(16)
̅̅̅
̅̅̅
̅̅̅
25
PROJECT REPORT
Reduced Order Averaged Model for Buck
Converter
Now apply standard linearization technique and apply perturbations as follows
to (22):̅
̃
̅̅̅
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃̇
]
̃̇
[
][
̃
]
̃
[
][
Which is in the form of:[
Then,[
̃
]=̃
̃
Where, (s I - A)-1 = {*
= .[
̃
̃ , and converting it to state space form,
Separating terms of ̃ ̃ ̃
[
̃
̃
̃
̃
]=A[ ]+B[ ]
̃
̃
̃
(s I - A)-1 B U(s)
]}-1
+- [
]Where, =
26
̃
]
̃
PROJECT REPORT
[
̃
]
̃
.
[
[
̃
][
]
̃
][
̃
]=[
̃
].[
].
Thus, two transfer functions are as follows:̃
̃
And,
̃
̃
Reduced Order Averaged Model for Boost
Converter
Similarly, apply standard linearization technique and apply perturbations as
follows to (23):̅
̃
̅̅̅
̃
̃
̃
̃
̃
̃
̃
27
̃
PROJECT REPORT
̃
̃)
(
̃
̃̇
]
̃̇
[
][
̃
]
̃
[
][
[
Which is in the form of:̃
Then,[
]=̃
̃
Where, (s I - A)-1 = {*
= .[
̃
[
]
̃
[
̃
̃ , and converting it to state space form,
Separating terms of ̃ ̃ ̃
[
̃
̃
]
̃
̃
̃
̃
]=A[ ]+B[ ]
̃
̃
̃
(s I - A)-1 B U(s)
]}-1
+- [
]Where, =
.[
[
(
)]
̃
][
]
̃
][
̃
]=[
̃
].[
Thus, two transfer functions are as follows:̃
̃
And,
28
].
PROJECT REPORT
̃
̃
Control transfer function:
3.4 NEW FULL ORDER AVERAGED
MODELS:Limitations of Reduced order model is that although it can correctly predict dc
and low frequency behaviour of PWM converters, at high frequencies, it is
unable to capture the dynamics of boost and buck converter. Full-order models
however can very well predict the high-frequency responses and are therefore
desired.
MODEL DERIVATION :-
29
PROJECT REPORT
For Buck Converter:The New Full order derivation starts from modified averaged model (8). This
model differs from reduced one in terms of duty ratio constraint. From (16), we
can get this relation:From (5)
̅
Substituting
into this, we get duty constraint
̅
This constraint is different from the earlier one which is derived for reduced
order model showing that it enforces correct average charging of output
capacitor. Putting into (9), these relations are derived:̅
̅ ̅̅̅̅
=
̅̅̅
̅̅̅̅ ̅
=
̅̅̅̅
DC analysis:
The dc operating point can be determined by
̅
And
̅̅̅
Let G=
̅̅̅
From (24) and (25)
̅
(26)
30
PROJECT REPORT
√
(27)
Now apply standard linearization technique and apply perturbations as follows
to (24) and (25):̅
̃
̅̅̅
̃
̃
̃
Thus,
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
Also,
̃
̃
̃
̃̇
̃
̃
Small Signal Model can be derived to following equation
[
̃
̃
̃
]=A[ ]+B[ ]
̃
̃
̃
Where
] , B =[
A =[
̃
Then, [
]=̃
̃
(s I - A)-1 B U(s)
31
].
PROJECT REPORT
(s I - A)-1 = {*
]}-1
+- [
]
= .[
Where, =
[
(
)
̃
]
̃
]
.[
[
̃
]=[
̃
[
]*[
] *[
̃
]
̃
̃
].
̃
Thus, Transfer functions can be formulated from small signal model to below
equations
̃
̃
̃
̃
̃
̃
̃
̃
=*
+. ,
=*
+
=*
=*
+
,
+
32
PROJECT REPORT
By putting value of R,L,C ,Ts,Vg and using equation (26) and (27)
L=5µH, C=40µF ,fs=100kHz (Ts=10µs) , Vg=5V ,R=20Ω , D= 0.7
We Get
(a)Inductor current to input voltage ratio
̃
̃
=
(a)
(b)Inductor current to duty ratio:
̃
̃=
33
PROJECT REPORT
(b)
(c)Audio susceptibility:
̃
̃
=
34
PROJECT REPORT
(c)
(d)Control transfer function:
̃
̃
=
35
PROJECT REPORT
(d)
For Boost Converter:From (19), we can get this relation:-
Substituting this into (5), we get duty constraint
̅
Putting
into (10), these relations are derived:̅
=
̅
̅̅̅̅ ̅
(
= -
̅̅̅̅
)+
-
36
̅̅̅̅
̅̅̅̅
PROJECT REPORT
Equating (26) and (27) to zero and finding solution for and , we obtain dcoperating point. Let the scalar value of G be the output to input voltage ratio.
Thus,
G=
̅̅̅̅
and ̅ =
= + √1 +
Now apply standard linearization technique and apply perturbations as follows
to (18) and (19):̅
̃
̅̅̅
̃
̃
̃
Thus,
̃
̃
̃)
(
̃
̃
̃
)
̃
(
̃
̃
̃
̃
̃
̃
̃
Also,
(
̃)
̃
̃ ̇ (
̃)
̃
̃
̃
̃
Small Signal Model can be derived to following equation
[
̃
̃
̃
]=A[ ]+B[ ]
̃
̃
̃
Where
37
̃
̃
PROJECT REPORT
] , B =[
A =[
̃
Then, [
]=̃
̃
(s I - A)-1 = {*
+- [
̃
]
̃
(s I - A)-1 B U(s)
]}-1
]Where, =
= .[
[
].
.[
][
(
)
][
̃
]
̃
̃
[
]=
̃
[
[
].
̃
].
̃
38
PROJECT REPORT
[
[
].
̃
].
̃
Thus, Transfer functions can be formulated from small signal model to below
equations
̃
̃
̃
̃
̃
̃
̃
̃
=*
=*
+. ,
+
=*
+
=*
,
+
Bode Plots :-
By putting L=5µH, C=40µF ,fs=100kHz (Ts=10µs) , Vg=5V ,R=20Ω , D= 0.7
39
PROJECT REPORT
Inductor current to input voltage ratio
Bode Plot for
Inductor current to input voltage
Bode Plot for
40
PROJECT REPORT
Audio susceptibility:
Bode Plot for
Control transfer function:
Bode plot for
41
PROJECT REPORT
Taking ILoad in Boost Converter:
Figure 3.2
During ON Period:
=[
],
=[
],
During OFF Period:
Figure 3.3
=[
],
=[
],
During DCM Period
=[
],
=[
].
According to (4), the model’s expression would be:42
PROJECT REPORT
[
̅
]=[
̅̅̅
][
̅
]
̅̅̅
[
][
]
We will introduce correction matrix as, where
].
K =[
From (8)
[
̅
]=[
̅̅̅
][
̅
]
̅̅̅
[
][
] –(30)
By applying duty ratio constraint i.e.
̅
Into (30), we get
̅
̅
̅̅̅
And,
̅̅̅
̅
̅
Now using linearization technique and applying perturbations to (31) and (32),
̅
̅̅̅
̃
̃
̃
̃
We get small signal model as:-
[
̃
̃
̃
]=A[ ]+B[
]
̃
̃
̃
43
PROJECT REPORT
Where
[
]
] , B =[
A =[
̃
Then, [
]=̃
̃
].
(s I - A)-1 B U(s)
̃
][
]
̃
][
.[
Where,
(
[
)
̃
]=
̃
[
[
]
[
[
.[
](
]
[
[
)
]
̃
].
̃
Thus, Transfer functions can be formulated from small signal model to below
equations
̃
̃
̃
̃
=*
=*
[
[
[
]
]
+
44
]
+. ,
]
]
PROJECT REPORT
̃
̃
̃
̃
=[
=*
(
)
(
+
̃
̃
45
)[
]
]
,
PROJECT REPORT
Chapter 4
Inclusion of Parasitic In Model
4.1
Overview
4.2
Transfer functions derivation considering parasitics
46
PROJECT REPORT
4.1 Overview:In the modelling of converter systems, due to the various difficulties faced in the
complexities and modelling procedure, the parasitic such as switch conduction
voltages, conduction resistances, diode drop and resistances, switching times
and ESR’s of capacitors are commonly ignored[8]. The idea of considering
ideal/lossless components and leaving parasitic like we have derived model
earlier, significantly simplifies model development and is of high importance at it
contributes to the understanding of the main features of a switching system[9].
Most conventional modelling (like reduced order and Full Order Model) are
adequate for this purpose. So it is no doubt that these modelling are successful
in the primary stage design of a switching system. However the effects of
parasitic and losses are important for improving model accuracy, study
efficiency, dynamic performance, and robustness of system poles. The problem
with including the parasitic leads to nonlinear current/voltage waveforms and
further complicates the analytical derivations.
47
PROJECT REPORT
4.2Parasitic Realization in DC-DC
Converters:For a Boost Converter, circuit with parasitic will look like
Fig 4.1
During ON State:
=
[
]
=
[
]= [
Where, A1=[
][
] +[ ][
] and
=[ ]
48
]
PROJECT REPORT
During OFF State:-
=
[
[
]
(
=
]
)
(
)
=
=(
[
)/C
]=
] + [ ][
[
[
[
]
]
B2=[ ]
Where, A2=
[
]
[
]
]
During DCM period:
=0
=
49
PROJECT REPORT
Thus, [
]= [
Where, A3=[
][
]
] + * +*
and
+
B3= * +
Now applying averaging technique, we get:A =A1d1 + A2d2 + A3(1-d1-d2)
B=B1d1 + B2d2 + B3(1-d1-d2)
Thus,
(
A= [
*
)
]
+
And,
]
B=[
Let correction Matrix be K and be defined as :-
]
K=[
State space equation will look like:
[
̅
]=[
̅̅̅
(
*
+
50
)
][
̅
][ ]
̅̅̅
PROJECT REPORT
][
+[
]
By using duty constraint d2, i.e.
̅
In the above state space equation, replace
̅
̅
̅
(
by duty constraint, we get
)
[
̅
]
[
̅
(
̅
)
]̅
̅̅̅
̅
̅
And,
̅
̅̅̅
̅̅̅
̅
̅
Now, apply perturbations we can get small signal model as:-
[
̃
̃
̃
]=A[ ]+B[ ]
̃
̃
̃
Where
] , B =[
A =[
(
]
)
(
51
)
;
PROJECT REPORT
(
)
(
)
̃
Then, [
]=̃
̃
(s I - A)-1 = {*
(s I - A)-1 B U(s)
]}-1
+- [
]
= .[
Where,
[
(
)
̃
]
̃
.[
]
[
]*[
52
̃
]
̃
PROJECT REPORT
̃
[
]=
̃
(
)
[
]
̃
*[
].
̃
Thus, Transfer functions can be formulated from small signal model to below
equations
̃
=*
̃
̃
̃
+. ,
+
=*
̃
̃
)
=*
̃
̃
(
+
=*
,
+
⁄
⁄
[
⁄
⁄
⁄
][
This is in the form of:Y(s) = C(s) X(s)
We know that,
B(s) U(s)
Thus,
Y(s) = C(s)
B(s) U(s)
53
]
PROJECT REPORT
Y(s) =
[
⁄
⁄
(
]*
)
[
[
]
̃
]
̃
[
⁄
⁄
][
][
̃
]
̃
Thus, two transfer functions are:̃
̃
⁄
̃
̃
⁄
Taking :
L=5µH, C=40µF ,fs=100kHz (Ts=10µs) , Vg=5V ,R=20Ω , D= 0.7 Rc= 30m
Rd =.15 ,RL=.176 ,Rsw=.17
54
PROJECT REPORT
̃
̃
g1= With parasitic
g2= Without parasitic
55
PROJECT REPORT
̃
̃
g1=Without parasitic.
g2=With parasitic.
Conclusion for Bode plot:-From the bode plot we can derive the
inference that in the low frequency range the gain magnitude
decreases by 5 dB as we take into account the effect of parasitic in the
model.
56
PROJECT REPORT
Chapter 5
Conclusion
57
PROJECT REPORT
F
irstly we have studied the various aspects of averaged modelling of DC-DC
converter(buck and boost) operating in discontinuous conduction mode.
The modelling procedure consists of basic three steps:
 Averaging the matrix parameters and selection of the correction matrix
(K) depending on the number of inductor currents of the converter.
 Representation of state space equations into the differential equations of
inductor current and capacitor voltage.
 Defining an duty ratio constraint so that the expression consists of only
one duty ratio.
We have plotted various bode diagrams for reduced averaged model and new
full order model and found out that, reduced order can estimate the behavior in
low frequency range but in new order model, since dynamics of inductor is
present, it is more precise.
We have also modeled the load into a constant current source and parallel
resistance so as to obtain the output impedance transfer function which can be
utilized for designing of Controller.
58
PROJECT REPORT
After that, various components parasitic are taken into consideration and a full
order model is developed. On comparing IDEAL and NON-IDEAL model behavior
through bode plot of control transfer function, we have found that the model
verifies the fact that dc gain decreases in case of parasitic.
59
PROJECT REPORT
REFERENCES:[1] J. Sun, Daniel M. Mitchell, Matthew F. Greuel, Philip T. Krein and Richard M. Bass, “Averaged
Modelling of PWM Converters Operating in Discontinuous Mode,” IEEE Transaction on Power
Electronics, Volume 16, NO. 4, Jul 2001.
[2] Robert W. Erickson and Dragon Maksimonic, Fundamentals of Power Electronics, Second Edition,
Springer International Edition, Chapter-5.
[3] S. Cuk and R. D. Middle brook, “A General Unified Approach to modeling switching DC-to-DC
converters in Discontinuous Conduction Mode(DCM),” in Proc. IEEE PESC’77, pp. 36–57, 1977.
[4] R. Tymerski and V. Vorperian, “Generation, Classification and Analysis of Switched-mode DC-to-DC
converters by the use of Converter cells,” in Proc. INTELEC’86, pp. 181–195, Oct. 1986.
[5] D. Maksimovic and S. Cuk, “A Unified Analysis of PWM converters in Discontinuous mode,” IEEE
Transaction On Power Electronics, Volume 6, pp. 476–490, May 1991.
[6] V. Vorperian, “Simplified analysis of PWM converters using model of PWM switch, Part II:
Discontinuous conduction mode,” IEEE Transaction on Aerospace Electron. Systems, Volume 26, pp.
497–505, May 1990.
[7] Xin Cheng and Guang-jun Xie,“ Full Order Models and Simulation of Boost Converters Operating in
DCM,” 2009 International Conference on Electronic Computer Technology.
[8] Chun T. Rim, Gyu B. Joung, and Gyu H. Cho,” Practical Switch Based State-Space Modeling of DC-DC
Converters with All Parasitics,” IEEE Trans. on power electronics, vol. 6 No. 4 October 1991
[9] Ali Davoudi and Juri Jatskevich,” Parasitics Realization in State-Space Average-Value
Modeling of PWM DC–DC Converters Using an Equal Area Method,” IEEE Tran. On circuits and
systems-I: regular papers, vol. 54, No. 9, September.
60
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