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Investigation of switching frequency variations in self-oscillating class D amplifiers

Nielsen, Dennis; Knott, Arnold; Pfaffinger, Gerhard; Andersen, Michael A. E.

Published in:Proceedings of the AES 37TH INTERNATIONAL CONFERENCE

Publication date:2009

Document VersionEarly version, also known as pre-print

Link back to DTU Orbit

Citation (APA):Nielsen, D., Knott, A., Pfaffinger, G., & Andersen, M. A. E. (2009). Investigation of switching frequency variationsin self-oscillating class D amplifiers. In Proceedings of the AES 37TH INTERNATIONAL CONFERENCE

Investigation of switching frequencyvariations in self-oscillating class Damplifiers

Dennis Nielsen1, Arnold Knott1, Gerhard Pfaffinger2 and Michael A.E. Andersen1

1Technical University of Denmark, 2800 Kgs. Lyngby, Denmark

2Harman/Becker Automotive Systems GmbH, Schlesische Str. 135, 94315 Straubing, Germany

Correspondence should be addressed to Arnold Knott (akn@elektro.dtu.dk)

ABSTRACT

Class D audio amplifiers have gained significant influence in sound reproduction due to their high efficiency.One of the most commonly used control methods in class D amplifiers is known as self-oscillation. Anparameter of key interest in self-oscillating class D amplifiers is the switching frequency, which can bedirectly related to the performance of the amplifier. This paper will clearify the myth of the switchingfrequency through investigation of its dependency on modulation index and reference frequency. Validationis done using simulations and an 50 W amplifier providing 0.2 % of distortion. The switching frequencyis tracked through accurate spectrum measurements, and very good compliance with simulation results areobserved.

1. INTRODUCTIONAmplification of any signal is all about converting en-

ergy from one source to a some kind of consumer or load.Pulse Width Modulation (PWM) is a very well knownmethod of carrying out this task, and plays due to its highefficiency an important part in the design of todays powersupplies. However PWM is not limited to the applicationof power supplies, and can thus also be used in the ampli-fication of audio. Such amplifiers are known as class Damplifiers and have gained significant usage in commer-cial audio amplifiers. One of the most commonly usedcontrol scheme of class D audio amplifiers is known assliding mode control or self-oscillation. Self-oscillatingclass D audio amplifiers are characterized by having aopen loop bandwidth equal to the switching frequencyas oppose to traditional PWM amplifiers, where the loopbandwidth typically will be limited to one tenth of theswitching frequency. This increased loop bandwidth pro-vides valuable loop gain at low frequency, which are ben-eficial with respect to reduction of Total Harmonic Dis-tortion (THD). Many publications exists showing the ex-cellent performance of self oscillating class D audio am-plifiers. These are among others [1], [2], [3], [4] and [5].

The control scheme of self oscillating class D audio am-plifiers are characterized by having an open loop func-

∫1

τ

KFb

vRef (t)

vPSOP (t)

Fig. 1: Astable Integrating Modulator (AIM) withoutoutput filter.

tion shaped to obtain 180 of phase shift at the desiredswitching frequency [6]. This is typically achieved inone of two ways. The first is by use of extra phase shiftin the forward path as seen in [1], while the second usesa hysteresis window in order to obtain the desired phaseshift, [6]. This paper will only consider the later knownas hysteresis modulators.

One of the most basic class D audio amplifiers is theAstable Integrating Modulator (AIM) topology, which isdescribed in [11]. The AIM amplifier are shown in figure1, and consists of integrator in the forward path, feed-back taken before the output filter using a P regulator, anpower stage plus an hysteresis window. Note that in fig-ure 1 is the output omitted as this has no influence on the

AES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August28–30

1

switching frequency. The AIM amplifier will be consid-ered throughout this paper. Obviously does many differ-ent hysteresis modulators exists. The primary differenceof these modulator is however the feedback(s), and theway it is implemented. As feedback are used to reduceTHD within the audio band, and all self-oscillating am-plifiers need to fully the same requirements in order tooscillate, will the AIM amplifier provided an reasonablybasis for carrying out the desired investigation. The pa-per is further more limited to voltage mode controlledclass D audio amplifier.

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

x 10−4

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [Sec]

Am

plitu

de [V

]

Power stage output voltageCarrierReference signalHysteresis window

(a) Modulation index = 0.6

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

x 10−4

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [Sec]

Am

plitu

de [V

]

Power stage output voltageCarrierReference signalHysteresis window

(b) Modulation index = 0.9

Fig. 2: Power stage output voltage, carrier, reference sig-nal and hysteresis window.

Hysteresis modulators as the AIM amplifier are known tochange the switching frequency with modulation index.This is among others shown in [6], [7] and [9], where itis found that

fSw(M) =VS

41−M2

τVHyst(1)

In (1) is VS the power supply voltage,M = VOutVS

themodulation index,VHyst the hight of the hysteresiswindow and τ the integration time constant. It isevident from (1), that the switching frequency will gotowards zero as M approaches 1. This is the reasonwhy the modulation index is normally limited to 0.8 inhysteresis controlled self oscillating class D amplifiers.Limiting the modulation index to 0.8 are done inorder to keep the switching frequency outside the audioband. Note that theτ of (1) is equal toτInt KFb of figure 1.

(1) can be extended to include the loop propagation de-lay, td . An example of this is found in [6], yielding

fSw(M) =VS

41−M2

τVHyst +12tdVS(1+M2)

(2)

Mikkel Høyerby has also derived an expression includ-ing the loop propagation delay, but using the duty cycle Dinstead of the modulation index. This is found in [8], andalso shows that the switching frequency travels towardszero in a parabolic fashion as the duty cycle is increasei.e. as the modulation index is increased.

(1) and (2) are plotted in figure 3 using an idle switchingfrequency of 300 kHz,τ = 99.58µs, VHyst = 500mV andtd = 7ns. It is seen, that the loop propagation delay has noinfluence on the switching frequency. Further more is itevident that the switching frequency falls for increasingmodulation index as expected.

The main drawback of (1) and (2) is that they relay onsmall signal models linearized around the modulation in-dex. This is unfortunate as the carrier signal for largemodulation indexes no longer is linear. Looking at figure2(a) and 2(b) is it evident that the carrier can be consid-ered to be linear for small modulation index. Howeverat high modulation indexes are the carrier degraded, andtakes the form of a exponential function. This is causedby the well-known response of an step function to an firstorder system, which will be shown in section 2. Small

signal models are thus not necessarily the best way to in-vestigate variations in the switching frequency. The fol-lowing section will illuminate the problems of deriving aswitching frequency model of hysteresis controlled self-oscillating class D audio amplifiers.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Modulation index

Nor

mal

ized

freq

uenc

y

(1)(2)

Fig. 3: Normalized switching frequency.

2. SWITCHING FREQUENCY MODEL

!"#

!$!&

%

!/$

!/&

%'()

0

1

Fig. 4: AIM audio amplifier without output filter.

The AIM topology has already been introduced in sec-tion 1. Using an passive integrator can the AIM ampli-fier be realized as presented in figure 4, where the outputfilter has been omitted. Note that the comparator can bemodeled to include the power stage if necessary. How-ever in order to investigate the switching nature of theAIM amplifier, feedback from the comparator can easilybe assumed.

An simple model of the AIM amplifier can be done byreplacing the comparator with a voltage source generat-ing pulses according to the PWM methodology. Furthermore is the comparator assumed to have a infinite input

!"#

!$

&

!%

&()*

Fig. 5: Proposed large signal model.

resistance. One thus obtain an model as seen in figure5. Notice thatRH1 andRH2 are omitted for simplicityas these just determines the hight of the hysteresis win-dow. The AIM amplifier will switch each time the volt-age across the capacitor becomes equal to the hight ofthe hysteresis window. Note that for this derivation willthe loop propagation delay be omitted. The followingequation can now be obtained

VHyst(s) =R2

R2+R1R1R2CR1+R2

s+1vRe f (s)+

R1R2+R1

R1R2CR1+R2

s+1vComp(s) (3)

The clear benefit of figure 5 is, that the AIM amplifiercan be considered as two first order systems. This isillustrated in (3), where each switching point is deter-mined by the superposition of the reference and com-parator response to the first order systems with time con-stantsτ = R1R2C

R1+R2.

Converting (3) to the time domain requires two well-known operations. These are the responses of a sinewave and a step function to an first order filter. The re-sponse of a step is a exponential function, while the re-sponse of an sine wave also will be a sine wave, [10].It is assumed, that the cut off frequency of the filter re-lated toVRe f is well above the reference frequency,fRe f .This allows for the phase and amplitude change of thesine wave be neglected. Grouping constants one obtainan expression of the form

k = sin(2π fRe f t)+ e−tτ (4)

In 4 is k an arbitrary constant. Note that the sin waveand exponential function might be weighted with respectto each by including yet another constant. However forsimplicity is this neglected.

(4) might look like a fairly simple equation. The authorshave however not be able to find any general solution to

this equation. Obtaining an switching frequency modelof hysteresis controlled self-oscillating class D audio hasthus not be possible, and the attention will now be turnto results obtained through simulations.

3. SIMULINK SIMULATIONSSimulations have been preformed using the Simulink

model of figure 6. Further more an FFT-function hasbeen written in Matlab allowing for obtaining the switch-ing node spectrums, [14]. These spectrum will be used totrack the switching frequency. All simulated spectrumsas presented in this paper are plotted using an Hanningwindow. The simulation model of figure 6 are made ac-cording to figure 1, and is thus an AIM amplifier. No-tice that an output filter has been added in order to com-plete the simulations model. The plots of figure 2 areobtained using the simulation model of figure 6. All sim-ulations are performed with an idle switching frequencyof 300 kHz, an output filter cut off frequency of 49.5kHz,τInt = 12.45µs,VHyst = 500mV ,VS = 30V , td = 7ns,

KFb =18 andKhyst =

VHystVS

.

3.1. Spectrum simulationsThe switching node output spectrums obtained by simu-lations are shown in figure 7. Notice that the spectrum ofa self-oscillating class D audio amplifier deviates signif-icantly from the one of fixed frequency amplifiers. Thusare the sidebands no longer of equal magnitudes. Thisgives an indication of the complex mathematics, whichare need to describe such spectrums. Remember thatdouble fourier series are used to calculate the spectrumof fixed frequency amplifiers, [15]. In these derivationsbecomes the sidebands Bessel functions.

3.2. Switching frequencyStarting out with an relatively small modulation index of0.1 is the spectrum of figure 7(a) obtained. Here can theswitching frequency and its harmonics clearly be identi-fied. At modulation index 0.1 is the switching frequencyidentified to be 293 kHz, which is a reduction of 7 kHzcomparing with the idle switching frequency of 300 kHz.Increasing the modulation index to 0.3 causes an drop inswitching frequency of 18 kHz. Finally is the modulationindex increased to 0.6 resulting in the spectrum of Figure7(c). This shows the general problem of self-oscillatingclass D audio amplifiers, namely that at large modulationindex can the switching frequency not be identified. Theswitching frequency simply drowns and only an spec-tral distribution of peaks are observed. Is can thus bequestioned whether talking about a switching frequency

makes sense for large modulation indexes.

0.1 0.2 0.3 0.4 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Modulation indexN

orm

aliz

ed fr

eque

ncy

(2)Simulation

Fig. 8: Normalized simulated switching frequency com-pared with (2). For simulation is an reference signal of10 kHz used.

Using the model of figure 6 has a surface plot been pro-duced, which maps variations in switching frequency asfunction of modulation index and reference frequency.The plot can be seen in figure 9. Note that the plot onlyconsiders reference frequencies between 14 kHz and 20kHz, while the modulation index is limited to the interval0.1-0.5. The switching frequency is tracked by identify-ing the highest peak in the spectrum above the referencefrequency and below twice the first harmonic of the idleswitching frequency (600 kHz in this case). An clear andimportant conclusion of figure 9 is that the switching fre-quency is independent of the reference frequency. Thiscomply very well with the theory as presented in section1.

In order to investigate the switching frequency depen-dency on modulation index even closer, is the plot of fig-ure 8 produced. In this case is the reference kept fixedat 10 kHz, while modulation index is limited to the inter-val 0.1-0.5. Comparing is made with (2), and it is seen,that the simulated switching frequency actually falls a bitslower than predicted by the linearized model. As shownin section 2 must is be caused by the carrier, which nolonger is linear, but an exponential function. It can thusbe concluded, that (2) is to be considered as an worstcase approximation of the switching frequency depen-dency on modulation index.

Vbridge limit

TransportDelay

To Workspace5

v_Ref

To Workspace4

v_PSOV

To Workspace3

v_Out

To Workspace1

v_Carrier

SignalGenerator Output filter

with load

1

C_Out*L_Out.s +L_Out/R_Load.s+12

Integrator time cosntant

1/t_Int

Integrator

1s

Hyst window

K_Hyst

HF LPF forconvergence2

1

1e−8s+1

HF LPF forconvergence1

1

1e−8s+1

Feedback

K_Fb

Comparator gain

1e6

Fig. 6: Simulink simulation model.

0.5 1 1.5 2

x 106

0

20

40

60

80

100

Frequency [Hz]

|vP

SO

V| [

dBµV

]

(a) Modulation index = 0.1

0.5 1 1.5 2

x 106

0

20

40

60

80

100

Frequency [Hz]

|vP

SO

V| [

dBµV

]

(b) Modulation index = 0.3

0.5 1 1.5 2

x 106

0

20

40

60

80

100

Frequency [Hz]

|vP

SO

V| [

dBµV

]

(c) Modulation index = 0.6

Fig. 7: Simulated spectrums using different modulation indexes. All measurements are performed with an 10 kHreference signal.

Fig. 9: Surface plot of normalized switching frequency.

4. VERIFICATION BY PROTOTYPINGExperimental measurements are performed on an 50 Wself-oscillating class D audio amplifier with an rated To-tal Harmonic Distortion of 0.2 %. The prototype can beseen in figure 10, while table 4 collects the key parame-ters of the prototype. THD over power are obtained usingan AP2 from Audio Precission, and the result are shownin figure 11. Notice that all simulations and calculationsas presented earlier in this paper are performed with re-spect to table 4.

4.1. Spectrum measurementsAll spectrum measurements are performed with a EMI

Test Receiver (ESI7 from Rohde & Schwarz). Again isthe switch node selected as the measuring point. Fig-ure 12 and 13 is an selection of the measured spectrums.As observed in section 3 is the sidebands not of equalmagnitude which is the case in fixed frequency class D

Switching frequency (idle) 300 kHzSupply ±30V

Output power 50WLoad 4 ΩGain 8

Time delay 7 nsOutput filter cut off 49.5kHz

Table 1: Key parameters of prototype.

amplifiers.

4.2. Switching frequency measurementsFigure 12 shows the spectrum of modulation index 0.1,0.3 and 0.6. Remembering that the idle switching fre-quency is 300 kHz one observe an drop in switchingfrequency of 2.6 kHz at modulation index 0.1. As ex-pected does the reduction in switching frequency con-tinue yielding a drop of 18.4 kHz at modulation index0.3. Notice that the simulation predicted an drop of 18kHz with modulation index 0.3. This complies very wellwith the measurement. At modulation index 0.6 is theproblem of identifying the switching frequency again ob-served.

Investigation of the switching frequency dependency onreference frequency are done by use of figure 13. Themeasurements are all performed using an modulation in-dex of 0.2. For the three measurements are switching fre-quencies of 291.2 kHz, 290.8 kHz and 290.4 kHz found.These variation are so small, that they can not be caused

Fig. 10: Developed prototype.

Fig. 11: THD vs power (blue 1 kHz and red 6.65kHz).

by the change of reference frequency. This is as acceptedfrom the simulations.

Measured variations in switching frequency are mappedand shown in figure 14. All measurements are performedwith an reference of 10 kHz and an modulation limitedto the range 0.1-0.5. Again is the switching frequencyidentified as the highest peak found in the spectrum be-tween the reference frequency and the firs harmonic ofthe switching frequency.

Figure 14 compares measurements with (2) and the sim-ulation results of figure 8. It is seen, that the simulationresults and measurements following each other nicely formodulation index less than 0.3. At modulation index0.3 is an jump in the measured switching frequency ob-served. This jump is caused by the problems of trackingthe switching frequency as the modulation index is in-creased. Finally is it observed, that the measurementsdoes not decay as rapidly as predicted by (2). The mea-surements thus emphasizes the conclusion of section 3,where (2) are characterized as an worst case approxima-tion of the switching frequency dependency on modula-tion index.

5. CONCLUSIONIt has been shown that the switching frequency of self-

oscillating class D audio amplifiers is independent of thereference frequency. Further more is it concluded, thatthe spectrum of self-oscillating class D audio amplifierdeviates significantly from the one of fixed frequencyamplifier. This is seen by the sidebands, which is notof equal magnitudes.

(a) Modulation index = 0.1 (b) Modulation index = 0.3 (c) Modulation index = 0.6

Fig. 12: Spectrum using different modulation indexes. All measurements are performed with an 10 kHz referencesignal.

(a) Reference signal of 10 kH. (b) Reference signal of 17 kH. (c) Reference signal of 20 kH.

Fig. 13: Spectrum using different reference frequencies. All measurements are performed with M = 0.2.

Comparing with the results obtained by linearized mod-els is it to be concluded, that such models provide anworst case approximation of the switching frequency de-pendency on modulation index. However these modelsshould note be used to describe the switching frequency ahigh modulation indexes. This is due to the fact, that theswitching frequency simply is not defined for such mod-ulation indexes. For high modulation indexes does theswitching node spectrum consists of spectral distributedpeaks, where no switching frequency are to be identified.

The results of this paper has been verify throughSimulink simulations and prototyping an 0.2 % THDclass D audio amplifier.

Future work includes investigating whether the audioband is disturbed at high modulation indexes. Furthermore would it be desirable to have an model of self-

oscillating class D audio amplifiers spectrum in order toimprove EMI design.

6. REFERENCES

[1] Simple Self-Oscillating Class D Amplifier withFull Output Filter Control, Bruno Putzeys, PhilipsApplied Technologies, BE3001 Leuven, Belgium.

[2] Derivation and analysis of a low-cost, high-performance analogue BPCM control schemefor class-D audio power amplifiers, MikkelC. W. Høyerby, Michael A. E. Andersen,Ørsted·DTU/Automation, Technical University ofDenmark, Denmark, AES 27th International Con-ference, Copenhagen.

[3] Søren Poulsen, Michael A. E. Andersen, ”Sim-ple PWM modulator with excellent dynamic be-

0.1 0.2 0.3 0.4 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Modulation index

Nor

mal

ized

sw

itchi

ng fr

eque

ncy

(2)MeasurementSimulation

Fig. 14: Comparison of switching frequency obtained by(2) and measuring. The measurement are performed with10 kHz reference signal.

haviour”, proc. of APEC 2004, February 2004,Anaheim, USA, pp 486-492.

[4] Paul van der Hulst, Andre Veltman, Rene Groen-berg, ”An asynchronous switching high-end poweramplifier”, proc of 112th AES Convention, May2002, Munich, Germany, preprint no. 5503

[5] Søren Poulsen, Michael A. E. Andersen, “Self os-cillating PWM modulators, a topological compari-son”, proc. of IEEE Power Modulators Conference2004, July 2004, San Francisco, USA, pp 403-407.

[6] Søren Poulsen, “Towards Active Transducers,”Ørsted DTU, Kgs. Lyngby, 2004, ISBN 87-91184-39-8.

[7] J. Vanderkoo, Comments on ”Design parametersimportant for the optimization of very high fi-delity PWM (class D) audio amplifiers”, Letters tothe editor, Audio Research Group, Department ofPhysics, University of Waterloo.

[8] Mikkel C. W. Høyerby and Michael A. E. An-dersen, A small-signal model of the hystereticcomparator in linear-carrier self-oscillating switch-mode controllers, NORPIE 2006 paper #052.

[9] Gael Pillonnet, Remy Cellier, Nacer Abouchi andMonique Chiollaz, An Integrated Class D Au-dio Amplifier based on Sliding Mode Control,

Advanced Audio Research Laboratory at CPELyon/INL, Grenoble/Lyon, France, 978-1-4244-1811 2008 IEEE.

[10] Electrical engineering, Principles and applications,page 150-152 and 1266-268, Third edition, AllanR. Hambley, Pearson Prentice Hall, ISBN: 0-13-127764-2.

[11] ELBO GmbH, 47509 Rheurdt: Selbstschwingen-der Digitalverstarker, DE 198 38 765 A1, Germanpatent, May 1998.

[12] Discrete-time modeling of continuous-time pulsewidth modulator loops, Lars Risbo, Digital AudioVideo Division, Texas Instruments Denmark, Lyn-gby, Denmark.

[13] Practical considerations for integrating switchmode audio amplifiers and loudspeakers fora higher power efficiency, Søren Poulsen andMichael A. E. Andersen, Ørsted·DTU, Automa-tion, Technical University of Denmark, DK-2800,Denmark.

[14] MATLAB R© documentation for the FFT function.The Mathworks Inc.

[15] Audio Power Amplifier Techniques With EnergyEfficient Power Conversion, Volume I, Ph.D. The-sis, Karsten Nielsen, Department of Applied Elec-tronics, Building 451, Technical University of Den-mark, DK-2800 Lyngby, April 30, 1998.