Utilization of non-linear converters for audio

Utilization of non-linear converters for audio
Utilization of non-linear converters for audio
amplification
Niels Iversen!, Thomas Birch!, and Arnold Knott!
! Technical University of Denmark, Lyngby, 2800, Denmark
Correspondence should be addressed to Thomas Birch ([email protected] .dtu.dk)
ABSTRACT
Class D amplifiers fits the automotive demands quite well. The traditional buck-based amplifier has reduced
both the cost and size of amplifiers. However the buck topology is not without its limitations. The maximum
peak AC output voltage produced by the power stage is only equal the supply voltage. The introduction of
non-linear converters for audio amplification defeats this limitation. A Cuk converter, designed to deliver
an AC peak output voltage twice the supply voltage, is presented in this paper. A 3V prototype has been
developed to prove the concept. The prototype shows that it is possible to achicve an peak AC output
voltage twice the supply voltage but also reveals some of the major obstacles and challenges which is also
discussed.
1. INTRODUCTION
Switch mode power audio amplifiers (Class D) are
known for their superior efficiency compared with lin-
ear amplifiers (Class A/AB/B). Therefore Class D am-
plifiers have become a conventional choice in systems
which demand high efficiency, such as mobile systems.
The traditional topology used for Class D amplifiers is
the Buck topology, typically in a full bridge configura-
tion as shown in fig. 1.
:
‘
Fig. 1: Full bridge Buck power stage
The transfer ratio of the full bridge Buck (1) is linear.
The linear property is desirable when amplifying audio
signals because it does not contribute to the Total Har-
monic Distortion (THD).
Vo
< = 2 - 1 1
у, (1)
Fig. 2 reveals that the DC gain of a full bridge Buck
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topology is limited and only achieves a gain of -1 to 1.
Feeding a class D amplifier with a sine wave, the maxi-
mum amplitude of the output sine wave will be Vg thus
obtaining a maximum output power of:
to
0 -
DCGan 94
Fig. 2: DC gain of full bridge Buck topology
(74)
R Load
tb
В, o,max — (2)
where Rzoad is the speaker impedance. This can become
a problem in systems where the voltage supply is low,
which is the case for many battery driven systems. Con-
ventional automotive audio systems have a voltage sup-
ply of Vg = 12V and speaker impedance of Ry = 4Q2
thus limiting the output power to:
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Iversen et al.
ae
P o.max —
The conventional way to achieve higher output power in
such systems is to add an additional DC-DC converter to
drive the amplifier power stage, boosting the voltage sup-
ply to a higher level thus obtaining more output power.
However the addition of a DC-DC converter will increase
the cost, increase the size and decrease the efficiency of
the amplifier.
Another way to achieve high output power in such sys-
tems 1s by using another topology for the power stage.
Non-linear topologies such as the Cuk topology can be
used. Unlike the Buck topology the Cuk topology DC
gain is not limited, making it able to deliver higher out-
put power thus avoiding the drawbacks of an additional
DC-DC converter. When using the Cuk topology a DC
gain of -2 to 2 is possible and therefore, when feeding an
amplifier with a sine wave using the Cuk topology as its
power stage, the maximum amplitude of the output sine
wave becomes 2Vg. In automotive audio system, with
V, = 12V, the output power becomes:
(2)
p= "7
¢ 40
= 12W (4)
Unfortunately the non-linear nature of the Cuk topol-
ogy will cause a high THD. However a sufficient neg-
ative feedback system can attenuate these problems, if
carefully designed, thereby obtaining good audio perfor-
mance.
2. THE CUK CONVERTER FOR AUDIO AMPLI-
FICATION
2.1. Basic operation of conventional Cuk con-
verter
The Cuk converter is a non-linear inverting converter
with a transfer ratio similar to Buck-boost and SEPIC
converters. Like the SEPIC converter the Cuk uses ca-
pacitive energy transfer when operating. Fig. 3 shows
the conceptual circuit outline of the Cuk converter. The
implementation of the Cuk converter is shown on fig. 4
58
Non-linear converters for audio amplification
Li F
с) h
—
+ Fr ” =
к „ей A
Cr == e > Aras
5 >
+
Li I;
5 ©, Mm h
— | ее
©: org Cr ==
Fig. 4: Implementation of Cuk converter
The Cuk converter is well described in the literature [1]
[2] [4] but the behavioral equations are repeated here for
convenience.
A
И, = —
|= т (5)
р
1%) = "1-5" (6)
] D I (7)
| = 1-D"”
V2
h = (8)
Road
The voltage and current waveforms is shown on fig. 5.
‘ UA A Vian, 1,
PDF (-DT DT f1-DIT
Ve- 14 1-15
fal), I Ia,
di Yo? Renag
DT (1-D)T DT (1-D)T
{>
Fig. 5: Cuk converter waveforms
Since the output voltage of the converter is Y, and the
output current is I» we can write
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Iversen et al.
D
Vo=V,= —T_p'e (9)
Le)
h=kh= (10)
RLoad
Using 10 the DC gain of the Cuk converter can be plotted
as a function of the duty cycle as shown in fig. 6
DC gain
Fig. 6: DC gain of the Cuk converter
2.2. Linearized Cuk for audio amplification
The Cuk converter has a non-linear transfer ratio which is
undesired in audio amplifiers. However combining two
Cuk converters, as shown in fig. 7, a full bridge con-
figuration is obtained, which linearizes the transfer ratio
enables the converter to produce both a positive and a
negative DC gain. This circuit is described by [4].
Fig. 7: Combined Cuk converters
In the full bridge configuration the two converters are
switching out of phase, meaning that Sl is at position
1 in the time interval DT while S2 is at position 4, thus
obtaining the two transfer ratios:
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Non-linear converters for audio amplification
——YVy (11)
Y, =-— Ve (12)
The combined transfer ratio for the whole converter can
be evaluated as follows:
у, = Vi — Vo |} (13)
р - (1 - р)
Vo = Р-р) * (14)
Neglecting the inverting property of this transfer function
one obtains:
_D-(1-D)
Y=> 1D)" (15)
The DC gain of the full bridge Cuk is shown on fig. 8.
DC Guin 0+
a y rá mi
D
Fig. 8: DC gain of full bridge Cuk
2.3. Note on bidirectional power flow
Even though the full bridge Cuk topology seems suited
for audio amplification it still lags a bidirectional power
flow which a conventional full bridge Buck topology pro-
vides. However this can be obtained by replacing the
diode with a MOSFET as shown i fig. 9.
The hardware implementation of the full bridge Cuk con-
verter for audio amplification is now obtained and shown
in fig. 10
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Iversen et al.
Bidirectional power flaw
47 oy
: -
Ci Ji Ge
7] —
no “Th Floss
>
+ «I
== |
+
—
La Las
=.
Fig. 10: Hardware implementation of full bridge Cuk
converter
3. MATLAB ANALYSIS OF CUK CONVERTER
FOR AUDIO AMPLIFICATION
To fully understand the consequences of using the Cuk
topology in class D this section presents a thorough in-
vestigation of THD and feedback using Matlab. All
investigations performed in this section is done with a
fixed voltage supply of V, = 12V and a load resistance of
Road = 492
3.1. THD investigation
In a DC-DC converter the duty cycle, D, of the PWM
signal switching the power stage is fixed. In a class D
amplifier the PWM signal varies with the audio input.
Assuming a class D amplifier is driven with a sine func-
tion, the duty cycle will become a sine function as well.
This means that the duty cycle can be modelled as fol-
lows:
D =0.5+A -sin(œt) (16)
A being the variation of the duty cycle, & = 27 f and ¢
the time. Substituting 16 into 15 yields:
__0.5+A -sin(wt) — (1 —0.5+A -sin(wt))
© 05+A-sin(wt) (1 —0.5+A -sin(œt))
и, (17)
60
Non-linear converters for audio amplification
Performing a FFT analysis of 17 one can visualise the
output of the amplifier. Since 17 is an odd function the
FFT analysis will contain the fundamental and odd num-
bered harmonic. Fig. 11 shows the output of the ampli-
fier where A = 0.25 and f = 6666kH:.
Sy Y 0" ter TC ATA Y
T 1 1
OT À
+
Fig. 11: FFT analysis of audio output
Only considering the 3 harmonic at 20KHz, the THD of
the output can be calculated using:
( 7
THD= —_""? . 100
(18)
Vind
The RMS voltage of sine waves can be evaluated know-
ing the magnitude using:
Y
Vrms = М (19)
V2
Using fig. 11 as an example the RMS voltage can
roughly be estimated:
1.9
Vharm = — ~ 1.34V 20
j 7 (20)
3
Vid = —= ~21.21V 21
fund 77 (21)
(22)
Thus obtaining a THD of:
v1.34V?
THD = ——. ~ 6.
ЭТИ 100 ~ 6.3% (23)
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Iversen et al.
THO 36 1 fanegor: £1 Ea ampiCude AO? Ma estr cycle 411 3 9) «avo put of HE = 5005
T T T 1 h
£ 1 } 1 | Ll. 1 L
es 21 0.15 92 925 03 9.75 04 45
Arius A Ide uy Cyde
Fig. 12: THD vs. A
Keeping the input frequency at f = 6666kHz one can
calculate the THD for a range of different values of A
and obtain a plot of the THD vs. A as shown in fig. 12.
3.2. Investigation of feedback gain
The needed K feedback at 20kHz to compensate for the
non-linear behavior of the Cuk topology can be derived
knowing the actual THD for a given value of A.
Viarm
K feedback = 20-108 (e) ‚(а В] (24)
crit harm
Where Verit harm 18 the critical RMS value of the harmonic
to obtain a wanted THD. The critical RMS value of the
harmonic can be determined as follows:
THDyaned Vfund
Verit,harm — - 100 Jun (25)
Applying 25 in 24 yields:
Vi
K feedback — 20 - log ( TH Bo Vrs ) (26)
100
A typical value for desired THD in class D amplifiers is
0.1%. Using fig. 11 as an example one obtain:
1.34
K feedback = 20-log (ot) ~ 36dB (27)
100
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Non-linear converters for audio amplification
Fuectad gan at HI
Fig. 13: Feedback gain vs. A
Using a range of different values for A one can plot the
feedback gain as a function of A as shown in fig. 13
3.3. Summary of Matlab analysis
Section 3.1 predict that the THD will increase with duty
cycle variation A. The feedback needed to significantly
reduce the THD is also very depedent on duty cycle
variations as seen in 3.2. A feedback gain of approxi-
mately 35-40dB will be sufficient to obtain good audio
performance while delivering an output sine wave with
Vsine = 2Vp amplitude.
3.4. Note on paracitic resistances in the induc-
tors
According to [4] the paracitic resistances in the induc-
tors tend to linearize the full bridge Cuk’s transfer ratio
further. This will result in a better audio performance of
the amplifier reducing the needed feedback gain. An in-
vestigation performed in [9] shows that in order to get
low power loss the paracitic resistances in the inductors
should be as low as possible, since the power loss will be
very dependent on this value when aiming for an output
sine wave with Vsine = 2V, amplitude.
4. IMPLEMENTATION
A prototype of an amplifier using full bridge Cuk topol-
ogy has been build with У, = ЗИ. A small signal AC
model have been constructed in order to properly dimen-
sion the component values. The final transfer function
for a single Cuk power stage, is shown in eq. 28
The differential transfer function, e.g the complete trans-
fer function for the whole power stage, can be written
as
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Iversen et al.
Non-linear converters for audio amplification
2C,d°L Ld?
Gu = Sra + Saran t+] (28)
MET 4 C,d*LyL,C; C,d*L, L3 Cid*L; | Lid?C, | LiC, 2 ?
AA+ LA + La + 52) + (5h + 2) +1
15. The conventional way to implement this circuit will
be to use a n-channel MOSFET for high- and a p-channel
Ga = Gad + Ga (29) MOSEET for low side. However, when switching at
The following component values have been selected,
based on the AC model:
e La = 5.4uH and R; + 50:mQ.
e Energy transfer capacitor: 1uF
e Output capacitor: 4.7uF
The prototype utilizes a self oscillating circuit known as
"Astable Integrating Modulator’ or AIM, described by
[5], for PWM generation. The switching frequency is
set to fu, = IMHz. The prototype serves as a proof of
concept for increased output power compared with con-
ventional class D amplifiers as predicted by 4. No global
feedback has been implemented.
4.1. Driving the MOSFETS
The gate drivers uses individual voltage supplies to drive
the low side MOSFETs. This 1s due to the unconven-
tional half bridge configuration needed in the Cuk topol-
ogy. In a conventional half bridge the high side MOS-
FET's source 15 referred to the low side MOSFET's drain
while the low side MOSFET*s source referred to ground.
Fig. 14 shows the conventional half bridge with a boot-
strap circuit on the gate driver.
VDH I
=
JUL
=
GNDH
Fig. 14: Conventional half bridge
The half bridge in the Cuk topology is actually two low
side MOSFETs on each side of ground as shown in fig.
high frequencies, sufficient p-channel MOSFETs are not
available. Instead the use of an individual battery power
supplies for the gate drivers driving the low side MOS-
FETs is a solution.
Fig. 15: Half bridge using the Cuk topology
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5. MEASUREMENTS
A series of measurements are conducted on the proto-
type with the specific goal to verify the gain of the power
stage. À sine wave Of fsine = 2000Hz with an amplitude
of Vsine = 0.5V 1s applied to the input in order to see the
gain delivered by the power stage. The power stage sup-
ply is set to V, = 2V in the first measurement. The output
wave is shown in fig 16.
Level СТ ГУ
Fig. 16: Output sine of the power stage with 0.5V sine
input and 2V power supply
The amplitude of the signal shown in 16 is Vp ~4V |
This yields a power output of:
Po = rel 30
°= "10 (30)
P, =2W (31)
The measurement is repeated for a power supply voltage
of V, = 3V. The result is shown in figure 17.
The amplitude of the signal shown in 17 is Vj; ~7.5V
and the power delivered is:
P,=1W (32)
À frequency response measurement with a sine sweep
input from 20Hz to 20KHz with the amplitude Vj, =
0.5V is conducted. The resultant frequency response is
shown in 18
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Non-linear converters for audio amplification
Level Cht/V
9% 000 «eu sec 100 To wou sony 1m
Son 8 gy Time ls
Fig. 17: Output sine of the power stage with 0.5V sine
input and 3V power supply
Funstion Ch da
9
a м om ome
Sean | 0 at?
x x 209 0 th x A
Frequeney! Hz
* Tn 1% =.
Fig. 18: Frequency response with a sine sweep from
20Hz to 20KHz and 0.5V amplitude
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Iversen et al.
Function Chi 4%
El: E
“ x su 70 1
Soni vais
00 00 sm TE в mm Tx lig 20
Frequency 1 Hz
Fig. 19: THD sweep with 0.1V input sine
The frequency response shows a relatively [lat response
until the roll off at around /,, = 12KHz with minor res-
onant peaks around f,,; = 10Khz ~ and fres = 17Khz.
THD measurements are also conducted with different in-
put amplitudes. Fig 19 shows a THD measurement with
an input sine amplitude of Vsine = 0.1V and a power stage
voltage of 3V The next measurement is conducted with
an amplitude of Vyine = 0.3V shown in fig 20 and the last
THD measurement is conducted with Vsine = 0.5V sine
input is shown in fig. 21
Funeden Chi / %
"i
a a =m Te сн x =
Frequeney / Ha
we th am
Fig. 20: THD sweep with 0.3V input sine
It is seen that the THD increases along with the input and
thereby the duty cycle variation A.
64
Non-linear converters for audio amplification
Funesdon Chi 4%
3 в =” зы 200 el 6 fe ak к =
Fraquenoy! Hz
un 16 хе
Fig. 21: THD sweep with 0.5V input sine
6. FUTURE WORK
6.1. Driving the power stage
As described in section 4.1 the MOSFETs in the power
stage 1s driven using battery power supplies. This not an
optimal solution if the class D amplifier using Cuk topol-
ogy is to be a serious product in audio systems. There-
fore a further investigation of the driver circuit, driving
the MOSFETs, will be relevant in order to find an opti-
mal solution to this problem. The development of high
speed p-channel MOSFETs might be a solution.
6.2. 12V prototype
Based on the results presented in this report a natural next
step will be an implementation of a 12V prototype with
a global feedback system. A 12V prototype with good
audio performance and good efficiency would be of great
interest to automotive audio systems.
7. CONCLUSION
The following can be concluded from the investigation:
e The optimal choice of paracitic resistances is a trade
off between good DC gain, feedback gain needed to
obtain good THD and power loss. In order to get
low power loss the paracitic resistances in the induc-
tors should be as low as possible, since the power
loss will be very dependent on this value when aim-
ing for an output sine wave with Vsine = 2V, ampli-
tude.
e Section 3 show that a feedback gain of 35-40dB is
sufficient to reduce the THD and obtain a good au-
dio performance.
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e Measurements show that the prototype is capable
of delivering a sine wave with an amplitude Vsine >
2V,, e.g greater than two times higher than the
power stage voltage. This proves the superiority of
the Cuk topology over the conventional Buck topol-
ogy when it comes to gain and output power.
e Âs expected, the THD increases dramatically when
increasing the duty cycle variation. This is caused
by the inherent non-linearity of the converter when
reaching the outer most regions of duty cycle varia-
tion. This coincides with the theoretical predictions
shown in section 3
e The prototype also exposed another set of issues,
including driving of the MOSFETs, that deserves a
more extensive investigation.
7.1. Perspective
The concept of driving a class D amplifier with a Cuk
(or any non-linear converter) is very attractive if the in-
herent non-linearities can be controlled and the THD can
be reduced enough to ensure a proper audio quality. The
subject contains many challenges but also shows great
potential, judged by the results of work presented in this
report. The potential reduction in size and cost is es-
pecially interesting to the automotive industry, as earlier
stated, and deserves a further examination.
8. REFERENCES
[1] Robert W. Erickson and Dragan Maksimovic, ’Fun-
dementals of power eletronics’, Kluwer Academic
Publishers, 2000, ISBN 0-7923-7270-0.
[2] K.I. Hwu and K.W. Huang, "Simple Modeling of
DC-DC converter’, IEEE, 2011.
[3] Ole Jannerup and Paul Haase Srensen, ’Regulering-
steknik’, Polyteknisk forlag, 2009, ISBN 87-502-
0982-5.
[4] Robert W. Erickson and Slobodan Cuk, 'A con-
ceptually new high frequency switch mode power
amplifier technique eliminates current ripple’, Pro-
ceedings of Powercon5, 1978.
[5] Soeren Poulsen, Towards Active Transducers’,
DTU, 2004, Phd. Thesis.
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[6]
[7]
[8]
[9]
Non-linear converters for audio amplification
Amold Knott and Gerhard R. Pfaffinger and
Michael A.E. Andersen, A Self-Oscillating Con-
trol Scheme for a Boost Converter Providing a Con-
trolled Output Current’, IEEE, 2011.
Amir Hassanzadeh and Mohammad Monfared and
Saeed Golestan and Reza Dowlatabadi, ’Small Sig-
nal Averaged Model of DC Choppers for Control
Studies’, Internation Conference on electrical En-
gineering and Informatics, 2011.
PR.K Chetty, "Modelling and Analysis of Cuk
Converter Using Current Injected Equivalent Cir-
cuit Approach’, IEEE, 1983.
Niels Iversen and Thomas Haagen Birch, *Utiliza-
tion of non-linear converters for audio amplifica-
tion’, Bachelor thesis at DTU, 2012.
65
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