A small signal model of the design of a class D power
switching amplifier
Philippe Dondon
To cite this version:
Philippe Dondon. A small signal model of the design of a class D power switching amplifier.
JSF 03, Dec 2003, Tozeur, Tunisia. p 437, 2003. <hal-00347524>
HAL Id: hal-00347524
https://hal.archives-ouvertes.fr/hal-00347524
Submitted on 16 Dec 2008
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A SMALL SIGNAL MODEL OF THE DESIGN OF A CLASS D POWER
SWITCHING AMPLIFIER
Ph. DondonENSEIRB, 1 avenue A.Schweitzer- 33402 TALENCE-FRANCE
Abstract : The class D amplifier is not well known in audio applications. Its excellent power ratio (greater than 90%)
is the most important advantage. The MOS transistors switching power stage is able to drive a useful power up to
100W to the loud speaker. However, designing such an amplifier is more difficult than designing a classical class A
or AB power amplifier. The distortion level is also slightly higher than in AB structures. Despite the abundant
knowledge on class D, there was no electronic model available till now. Then, a small signal model is présentented
to make the design as simple as possible. The experimental results are given to illustrate the design method.
1. INTRODUCTION
The Class D amplification [1] uses MOS
power transistors in switching mode. It is able to
obtain a power ratio close to 95%. It is then possible to
reduce the size and thereby to miniaturize the audio
amplifier. The principle is given in figure 1. The audio
signal is compared to a high frequency triangular
signal. We then obtain a Pulse Width Modulated
intermediate signal. This two level signal drives a full
bridge MOS power transistor which works in switching
mode. Free running diodes are not shown in figure 1.
The power supply over 30V, enables it to carry a high
current in the load. A low pass ouput filter between the
MOS stage and the speaker restores the audio signal.
And the feed back active network set the voltage gain
of the circuit
R3
2.2 Spectrum analysis of the output PWM
signal
feed back network
V2
low pass filter
R4
C
audio
input
R5
+Va
V1
full bridge 4
Tr NMOS
+Vd
PWM
Modulator
triangle
generator
MOSFET
Driver
Filter
Filter
loud
speaker
overload
protection
Let "m" be the modulation index. It is given
by the ratio between the peak level Vbfe of the audio
signal and the peak level Vtri of the triangular
waweform :
m = Vbfe/Vtri
with 0<m<1
Letbf be the pulsation of the audio signal
and Ttri the period of the triangular signal. The
modulated pulse duration  can be written :
 (1+m sin (bf t) ) with  =Ttri /2
The "equivalent gain" at low frequencies (ratio
between the peak value of the output Low Frequency
contribution of the modulated signal and the input peak
value Vbfe) is constant. It is given by :
(1) Gpwm = Vd/(2.Vtri ), where Vd is the
peak value of PWM signal.
Shunt
Let Ftri be the frequency of the triangular
signal and Fbf the frequency of the audio input signal.
As a complete mathematical approach is very difficult,
we can simplify it, by assuming that the pwm signal is
periodic. Then, we can write : (2) pwm(t)=

Vd.
n.2.Ftri.0
1
 2Vd
( sin[(1 msinFbf .t)
].cos(n2Ftri .t))

1
n
Ttri
n
2

Figure 1 : class D amplifier principle
2. DESIGN METHODOLOGY
2.1 Principle of Pulse width modulation
An analog signal is compared to a high
frequency triangular waveform. The output is a two
level pulsed signal as shown in figure 2.
AudioEntrée
input
audio
Audio
entrée
inputBF
Horloge
triangular
triangulaire
waveform
Vtri
Vbf
t
+Vd
+
Sortie
PWM
modulée
PWM
output
0V
comparateur
comparator
Vd
t
Figure 2 : PWM modulation
The frequency spectrum of the PWM signal can
obviously be approximated with Bessel functions. But,
it is easily obtained by numerical simulation. An
example of the normalized spectrum is given in figure
3 for two values of m, 5% and 99%.
The spectrum of the output signal contains the
useful audio signal and also frequencies around n.Ftri 
k.Fbf. The peak value of these contributions depends on
the value of m. For its maximum value (m=1 ie Vbfe =
Vtri), the peak value of the audio frequency Vmax will
be obtained from equation (1), thus :
(3) Vmax=0,5.Vd
where Vd is the peak value of the PWM signal.
Taking Ftri around 200kHz, (i.e greater than
ten times Fbf), the reproduction of the audio signal will
be quite easy with a simple ouptut second order
lowpass filter.
Ftri
1
0,05
normalized peak value Fbf
(audio)
1
3.4 Feed back network
Ftri
0,5
Fbf
(audio)
2.Ftri
Frequency
0
15kHz
0
15kHz
Frequency
m = 5%
m = 99%
Figure 3 : PWM signal spectrum
2.3 Starting the design
Let Va to be the bridge power supply and Pu
the maximum power transmitted to the load. Assuming
that the load is fully resistive in audio bandwidth, (loud
speaker 8 or 4 ), we first calculate the peak value of
the useful low frequency audio lines of the modulated
signal by Pu = Vbfs2/R. We initially consider a perfect
ouput filter without losses and we assume that the
MOS "on state" drop voltage is negligible . As the
"equivalent voltage gain" of the full bridge is : 2Va/Vd,
and taking m=1 (best case of modulation), we can
determine the minimum supply voltage Va of the
bridge ; from (3), it yields :
Vbfs max = 0,5.Vd x (2Va/Vd)
and :
Vbfs max = Va
Knowing the normalized input voltage level,
the value of the required voltage gain is easily
calculated. The power ratio of the amplifier obviously
depends on the switching and the conduction losses of
the MOS transistors and also of the output filter losses.
Thus, a small "Ron" MOS transistor should be used.
3. OPEN LOOP DESCRIPTION
3.1 Driver stage
The driver circuit (in our example HIP4080
[3]), located between the PWM modulator and the full
bridge, allows the MOS transistors (here IRF 530) to
switch under the best conditions by :
-delivering a 12V pulse and a peak current of
3A to improve the switching times.
-generating a dead time to avoid the
simultaneous conduction of two transistors at the
switching times.
-inhibiting the gates command signal in case
of current overload.
3.2 Overload protection circuit
The current overload protection consists of
one "shunt" resistor, a low pass filter, a threshold
comparator, and a MOS transistor driver inhibition
circuit (inside the driver circuit).
3.3 Triangle generator
The triangle generator is a commercial circuit
MAX 038. The output level is ± 1V. A voltage level
shift is added to center the signal around 6V. Its
frequency can be adjusted between 100kHz and 1Mhz.
The diagram of this active lowpass filter is given in
figure 4.
The first order transfer function H(p) is :
Vs
R3
1

(4)
C1. R1. R2
V2  V1 (R1  R2)
(1 
p)
R1  R2
The gain in the audio bandwidth is:
R3
GBP 
(R1  R2 )
R3
R1
R2
R4
to the integrator
input
R1
R2
V2
V1
C1
R3
Load
filter
filter
Figure 4 : feedback network
4. CLOSED LOOP MODEL
4.1 Small signal model
From a low frequencies point of view, The
PWM modulator and the bridge can be considered as a
simple gain K, proportionnal to Gpwm, Va and 1/Vd
(see equation (3) and (4)).
K = Gpwm .2Va/Vd soit : K = Va/Vtri
Moreover, at the audio frequency, the
feedback network may be considered as a real gain.
The circuit can be simplified as indicated in figure 5.
A classical small signal analysis is then possible.
R4
GBP=R3/(R1+R2)
V1
V2
R5
audio
input
C
gain K
Load
filter
filter
V1-V2
Figure 5 : amplifier small signal model
Simplifiying the circuit, it comes :
Req = (R1+R2).(R4/R3)
C/K
audio
input
R5
to the
output
filter
V1-V2
Gain (dB)
30
20
10
0
-10
-20
-30
1,E+01
Figure 6 : final LF small signal equivalent diagram
4.2 Stability study
The feed back loop should be taken just before
the loud speaker for an optimum effect. Including this
filter into the loop would generate a too great phase
rotation. And it would be impossible to compensate the
system. So, the feed back is done just before the
output filter.
5. OUTPUT FILTER
It is a second order low pass filter with a cutoff frequency of 12KHz. The order choice comes from
a compromise between complexity and filtering
efficiency. The R,C serial network value depends on
the speaker characteristics.
L=30uH
V1-V2
C=1uF
+Va
-Va
t
C=1uF
VHP
R=3,9
Speaker
C=0,47uF
L=30uH
1,E+03 1,E+04 1,E+05
Fréquency (Hz)
1,E+06
Figure 8 : voltage gain response
-170
Phase
(degres)
-220
-270
-320
-370
1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06
Fréquency (Hz)
Figure 9 : phase response
The gain and phase margin are large enough to make
the amplifier unconditionally stable. The output filter
transfer function VHP/(V1-V2), including the speaker
load is given in figure 10.
Gain (db)
The closed loop function B(p) = (V1-V2)/Ve is :
R4 R1  R2
1
(5) B(p) 
C R4
R5 R3
1
(R1  R2)p
K R3
For a given voltage gain Gv, an audio bandwidth BP
and a choosen gain K, it comes :
1 C R4
R4 R1  R2
(R1  R2)
et BP 
(6) Gv 
2p K R3
R5 R3
From these formulas, we can calculate the required
capacitor and resistor values.
1,E+02
5
0
-5
-10
-15
-20
-25
-30
-35
1,E+01
BP 10kHz
1,E+02
1,E+03
1,E+04
Fréquency (Hz)
Figure 10 : output filter gain response
The loss of 1dB in the audio bandwidth,
comes from the parasitic serial resistor of the coils
which are not negligible compared to the speaker
resistor value.
6.2 Time response
Figure 11 shows the PWM signal modulated at 90%.
Figure 7 : output filter
6. EXPERIMENTAL RESULTS
For safety reasons, the test was performed on
the amplifier limited to an output power of 15W.
6.1 Frequency response
The experimental open loop transfer function
is given in figure 8 and figure 9.
Figure 11 : PWM signal modulated at 90%
6.3 Spectrum mesurements
Entrée
audio
input
audio
triangular
Horloge
triangulaire
signal
t
PWM
Sortie
modulée
output
t
Figure 13 : overmodulation
Peak
amplitude
Va
Va
Va
Figure 12 : speaker output signal spectrum
Figure 12 shows the output spectrum for a 500
Hz sinewave input signal. The second harmonic
attenuation is better than 40db. Thus, the distortion
rate is smaller than 1% as indicated before.
7. DISCUSSION
7.1 Sample frequency variation effects
The power ratio obviously depends on the
ouput level. In our example it reaches 83% for an
output power of 15W and a "sample" frequency Ftri
around 200 kHz. Increasing the frequency Ftri does not
improve the distortion level. Moreover, it increases the
switching losses and lowers the power ratio. Thus Ftri
should not exceed ten times the maximum audio
frequency.
7.2 Bridge power supply variations effects
Assuming that nothing else changes :
-if the supply voltage increases, then the gain
K also increases. The PWM modulator input signal
decreases making the index m smaller. The HF
spectrum lines increase. The distortion and the
bandwidth in closed loop increase. The noise to signal
ratio is then worse and the power ratio is smaller.
-if the supply voltage decreases, m
increases.In the worst case, the LF signal is always
greater than the triangular signal. We obtain then, the
modulation indicated in figure 13. The amplifier
saturates and the output signal is then a squared
waveform with a fundamental frequency of Fbf. The
harmonic contents 3, 5, 7 can appear in the bandwidth
and give high harmonic distortion (figure 14).
Fbf 3Fbf 5Fbf
Bandwidth
Frequency
Ftri-nFbf Ftri Ftri+nFbf
Figure 14 : power supply setup effect
Theses effects must be taken into account to
choose carefully the power supply voltage.
8.CONCLUSION
In this study, we presented a method to design
a power switching class D amplifier as simply as
possible. Modelling the closed loop circuit at the audio
frequency for small signals was the most difficult step
in this method. The experimentals results show a good
matching with our theoretical approach. However, this
method only enables us to find the most important
parameters of the amplifier. The power ratio must be
taken into account when the electronic design is
started. In our example, it could be easily improved by
reducing the power comsumption of the triangle
generator and the active feedback filter.
The optimisation of the power ratio will allow
the reduction of the size of the heatsinks and probably
their elimination for a medium power audio amplifier
(i.e 15W). Thus, the full integration of our amplifier
will be possible.
REFERENCES
[1] M Girard , Amplificateur de puissance
Mc GrawHill 1988
[2] Audio and HIFI Handbook, second edition
ed. Ian Sinclair Butterworth-Heinemann 1993
[3] Harris power seminar handbook 1993
[4] A tutorial of Class D operation TPA005D02 Texas
instrument
http:/www.ti.com/sc/docs/msp/pran/tutorial.html
[5] Philips TDA8920 2x50W SOI Class D amplifier
data sheet 1998