Digital Control Of A PWM Switching Amplifier With Global .

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Digital Control of a PWM Switching Amplifierwith Global FeedbackToit Mouton 1 and Bruno Putzeys 21 Department2 Hypexof Electrical and Electronic Engineering, University of Stellenbosch, Private Bag X1, Matieland, 7602, South AfricaElectronics B.V., Groningen, The NetherlandsCorrespondence should be addressed to Toit Mouton (dtmouton@sun.ac.za)ABSTRACTA digitally controlled class-D amplifier using global feedback is presented. The output signal of the amplifieris sampled using a sigma-delta analogue-to-digital converter. A novel compensation strategy is used tominimize distortion resulting from ripple feedback of the output signal. An evaluation system, based on aField Programmable Gate Array, was developed and an experimental evaluation was performed. State-ofthe-art performance was achieved.1. INTRODUCTIONDigital pulsewidth modulated (PWM) signals can be synthesized with high accuracy from the PCM audio inputsignal at relatively modest switching frequencies in thedigital domain. However, the performance of open-loopdigital switching amplifiers is limited by nonlinearitiesand other imperfections introduced by the MOSFET output stage and output filter. These nonlinearities includetiming errors due to blanking time and current dependentdelays of the switching transitions as well as amplitudeerrors resulting from the non-linear on-state resistance ofthe MOSFET switches, current dependent overshoot andvariations in the power supply voltage. Another sourceof imperfection that is often overlooked is the low-passoutput filter. This filter modifies the frequency responseof the amplifier in a load dependent manner. It increasesthe amplifier’s output impedance and adds distortion dueto the non-linear nature of the filter inductor.Feedback provides an effective way to compensate all ofthe above-mentioned imperfections. To date digital feedback strategies for class-D amplifiers have mainly beenbased on local feedback loops [1]-[8], relying partly onthe use of analogue circuitry to reduce the required resolution of the analogue-to-digital converter that samplesthe error signal. While the local feedback approach effectively compensates for the non-idealities of the outputstage it provides no rejection of the imperfections associated with the output filter. The fact that the output filtermodifies the frequency response of this type of amplifieris often overlooked.This paper focuses on the digital closed-loop control ofa switching amplifier using global feedback. A sigmadelta analogue-to-digital converter is used to sample theoutput voltage of the amplifier. A novel ripple compensation strategy is used to eliminate the effect of aliasingof the PWM signal resulting from ripple feedback. Asimple way to characterise the load is presented.2. OVERVIEWOFTHECONTROLLED AMPLIFIERDIGITALLY-Fig. 1 shows an overall block diagram of the switchingamplifier with global feedback. The overall structure ofthe feedback loop is a classical control loop in its simplest form. The output signal of the amplifier is sampledby an analogue-to-digital converter and subtracted froman up sampled version of the 24-bit digital audio inputsignal. The resulting signal serves as input to a digitalloop filter which drives a digital pulse width modulator.The power stage can either be a half-bridge or full-bridgeconverter.Careful attention has to be paid to the choice of theanalogue-to-digital converter. The most obvious requirement is that the analogue-to-digital converter must havenoise and distortion specifications that exceed those putforward for the complete amplifier. Another requirementis the sampling rate of the analogue-to-digital converter.PWM results in groups of harmonics centred at integermultiples of the switching frequency. The magnitude ofAES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August 28–301

MOUTON AND PUTZEYSDigital Control of a PWM Switching Amplifier with Global FeedbackPowerSupplyDigitalAudioSourceUpsampler DigitalLoopFilterDigital PWMwithRipple CompensationLCPower StageSigma DeltaADConverterFig. 1: Overall schematic of the switching amplifier with global feedback.these harmonics shows a first order decrease with frequency. Since the LC low-pass filter is a second orderfilter a high sampling rate is required in order to avoidaliasing.State-of-the-art multi-bit sigma-delta analogue-to-digitalconverters satisfy the requirements in terms of signal-tonoise ratio and sampling frequency. However, the longdelays associated with the digital low-pass FIR filtersused to remove the shaped quantization noise in this typeof converter poses a significant challenge in terms of thedesign of the control loop. The solution is to use themodulator output of the sigma-delta AD converter andto substitute a minimum-phase IIR filter for the linearphase FIR filter. In a perfectly linear system the shapedhigh-frequency quantization noise would remain abovethe audio band. However, due to the sampling nature ofthe pulse width modulator some of the quantization noiseis aliased into the audio band [12] and additional digitalfiltering is required to lower the quantization noise to acceptable levels. The quality of the multi bit DA converterin the sigma-delta AD converter is critical, since its linearity determines the overall performance of the amplifier [9].3. RIPPLE COMPENSATIONIt is well-known that feedback of the ripple signal presenton the output voltage of a PWM amplifier causes a nonlinearity in the PWM process due to aliasing of the high-frequency carrier components [11], [13]. This causesdistortion of the output signal of the amplifier, even ifthe output stage is considered to be perfect. A detailedanalysis of the distortion caused by this phenomenon wascarried out in [11] under the assumption that the inputsignal to the amplifier is constant. Two distortion mechanisms were identified. The first mechanism is distortion of the pulse width which causes a DC non linearity.The second, more subtle distortion mechanism is relatedto a signal dependent non-linear time shift of the PWMpulses (phase modulation).In [11] a class of minimum aliasing error loop filters arepresented that obtains minimum distortion due to the useof quadrature sampling. A different approach to solving the ripple feedback problem (with a single integrator as loop filter) is presented in [13]. The carrier signalis modulated by a small-amplitude signal which is proportional to the derivative of the input signal. In [14] aremarkably simple solution to the ripple feedback problem was presented by cancelling the unmodulated edgein a single-sided modulator. This section further investigates this compensation strategy and shows how it canbe extended to the current system.First consider the simplified naturally-sampled singlesided PWM feedback loop, with loop filter G(s), shownin Fig. 2. The waveforms describing the operation of thiscircuit over a segment of the input signal i(t) are shownin Fig. 3. The feedback signal is taken directly from theAES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August 28–30Page 2 of 10

MOUTON AND PUTZEYSDigital Control of a PWM Switching Amplifier with Global FeedbackComparatori(t)x(t)G(s)Input Sawtooth Carrier1s(t) 1ponent of x(t) is independent of the average value of x(t),thereby significantly reducing the nonlinearity resultingfrom the interaction of the ripple component of the PWMinput signal and the pulse width modulation process.p(t) 1amplitude 1y(t) p(t)Fig. 2: PWM feedback loop with ripple compensation.average modulator output p(t)modulator output p(t). The sawtooth carrier s(t) is addedto p(t), thereby cancelling the unmodulated edge p(t).The resulting signal y(t) is a ‘sawtooth-like’ waveformof which the time average (calculated over one switching period) is equal to that of the modulator output p(t).The advantage of this ripple compensation technique liesin the observation that shape of the ripple component ofy(t) and hence x(t) is largely independent of the duty cycle p(t).unmodulated edgeamplitudemodulated edgey(t)average modulator input x(t)x(t) xs(t)r(t)p(t) xtimerminFig. 4: Ripple compensation with DC input.y(t)i(t)x(t)i(t)times(t)Fig. 3: Ripple compensation waveforms.Signal y(t) is subtracted from the audio input signal i(t)and passed through the loop filter G(s). The loop filtertypically consists of a chain of integrators with high gainthroughout the audio band and less than unity gain at theswitching frequency. Fig. 3 shows the output signal x(t)of a typical loop filter. Since the control loop accuratelytracks the input signal x(t), the crossings of x(t) and thesawtooth carrier s(t) coincide with those of i(t) and s(t).It can again be observed that the shape of the ripple com-In order to quantify the advantages of this ripple compensation strategy the DC-linearity of the pulsewidth modulator in the presence of the modified ripple signal is investigated. Consider the feedback loop of Fig. 2 in thecase where i(t) is constant (DC input signal). Fig. 4illustrates the operation of the pulsewidth modulator inthe presence of the modified ripple feedback signal. Thedifference x between the average modulator input x(t)and the average modulator output p(t) provides a quantitative measure of the non-linearity of the pulse widthmodulation process in the presence of the ripple signal.Now consider the ripple componentr(t) x(t) x(t)of x(t). This ripple component is the response of the loopfilter G(s) to a time-shifted replica of the sawtooth carriers(t). Furthermore, x is equal to the minimum value r minof r(t). Since changes in the average modulator inputonly affects the phase of r(t), x is independent of theaverage modulator input signal. This shows that the DCAES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August 28–30Page 3 of 10

MOUTON AND PUTZEYSDigital Control of a PWM Switching Amplifier with Global Feedbacknon-linearity of the pulsewidth modulator has been reduced to a simple DC-offset. The value of this DC-offsetonly depends on the properties of the loop filter and isindependent of the average modulator input signal. ThisDC offset is easily compensated by the feedback loop.The next step is to show how the ripple compensationtechnique of Fig. 2 can be extended to the switchingamplifier that includes a low-pass LC filter, like that ofFig. 1. Fig. 5 shows three equivalent implementationsof the ripple compensation strategy. For the moment theMOSFET output stage is considered to be ideal and isrepresented by a gain, denoted by A. For a full-bridgeoutput stage this gain is equal to the DC-bus voltage V d ,while A V2d for a half-bridge output stage.Figs. 2 and 5(a) are essentially equivalent in terms ofripple feedback, the only difference being that the ripplecomponent only passes through G(s) in Fig. 2 while itpasses through the series combination of F(s) and G(s)in Fig. 5(a). A direct application of Fig. 5(a) is impractical since it would require adding an amplified versionof the sawtooth carrier to the output voltage of the powerstage before the low-pass LC filter.Figs. 5(b) and (c) are derived from Fig. 5(a) throughsimple block diagram manipulation. In Fig. 5(b) thesawtooth carrier is passed through a filter of which thetransfer function is identical to that of the low pass LCfilter before being subtracted from w(t). It should benoted that in a practical system the frequency responseof the LC filter at and above the switching frequency isrelatively independent of the load resistance. As a resultthe ripple compensation technique is relatively insensitive to the exact matching of LC filter‘s transfer function.Fig. 5(c) shows that the ripple compensation techniqueis equivalent to pre-distortion of the sawtooth carrier. Ina digital implementation this pre-distortion can be doneoff-line and the resulting carrier stored in a lookup table.The gain of the output stage is again represented byA, while any imperfection in the output stage are modelled by error source E(z). The sigma-delta analogue-todigital converter is represented by its closed-loop transferfunction H(z), which includes the transfer function of theanalogue-anti aliasing filter. For practical purposes it canbe assumed that the analogue-to-digital can be modelledby a single sample delay H(z) z 1 . Noise source N2 (z)represents the shaped quantization noise of the sigmadelta modulator. The ripple compensation strategy ofFig. 5(b) was used in the prototype design since thisimplementation doesn’t require recalculation of the ripple compensation lookup table when making changes toG2 (z) or G3 (z).The digital loop filter consists of three sections. The first,denoted by G 1 (z) in Fig. 6 provides digital cancellationof the poles of the low-pass LC filter by placing two zeros at the location of these poles. Exact cancellation ofthe LC filters poles isn‘t necessarily required. However,in order to ensure a stable control loop, two zeros wouldhave to be placed in the vicinity of these two poles. Italso contains two high-frequency poles to attenuate thequantization noise of the analogue-to-digital converter.The second section, denoted by G 2 (z) is a low pass filter. The aim of this filter is to attenuate the quantization noise of the analogue-to-digital converter. The thirdsection denoted G 3 (z) consists of a chain of integratorswith feed forward summation and local resonator feedback loops [16], p177. This section provides the requiredgain across the audio band.The ability of the control loop to reject imperfections inthe output stage and LC filter is described by the transferfunctionR(z) F(z)Vo (z) ,E(z)1 H(z)G(z)F(z)(1)whereG(z) G1 (z)G2 (z)G3 (z).4. DESIGN OF THE DIGITAL LOOP FILTERFig. 6 shows a z-domain block diagram of the feedbackloop. The low-pass LC filter has been transformed to thez-domain. The digital pulse width modulator is assumedto be a linear component, with noise source N 1 (z) representing the quantization noise associated the finite bitlength of the pulse width modulator. However, it shouldbe kept in mind that the pulse width modulation processsuffers from fold-back distortion. As a result high frequency noise in X(z) may fold back into the audio band.Furthermore, the transfer function describing the abilityof the control loop to attenuate the quantization noiseN1 (z) of the digital pulse width modulator is describedby:AF(z)Vo (z)NT F1 (z) (2)N1 (z) 1 H(z)G(z)F(z)Since both F(z) 1 and H(z) 1 throughout the audioband, the key to a successful design lies in ensuring thatG(z) has high gain throughout the audio band.AES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August 28–30Page 4 of 10

MOUTON AND PUTZEYSDigital Control of a PWM Switching Amplifier with Global FeedbackComparatori(t)x(t)G(s)InputOutput Stage 1 Sawtooth Carrier(a)p(t)A 1 LC low-pass Filtervo (t)F (s)1A 11AComparatori(t) w(t) G(s)x(t) 1p(t)Output Stage LC low-pass Filtervo (t)F (s)A 1Sawtooth Carrier(b)1F (s) 11AComparatori(t)G(s)x(t) 1p(t)Output Stage LC low-pass Filtervo (t)F (s)A 1Sawtooth Carrier(c)11 F (s)G(s) 11AFig. 5: Three equivalent implementations of the ripple compensation technique.The shaped quantization noise N 2 (z) of the analogue-todigital converter also requires some consideration. Theprototype design made use of a commercial sigma-deltaanalogue-to-digital converter [15]. This analogue-todigital-converter was operated at a sampling frequencyof 19.6608 MHz. According to [15], Fig. 41 the noisefloor of this sigma-delta AD converter is more or lessflat up to 1.5 MHz. It rises by approximately 60 dB athalf the clock frequency. As mentioned above, excessivelevels of high-frequency noise at the input of the pulsewidth modulator results in fold back distortion or mayeven cause instability of the feedback loop.To summarize, a number of factors have to be taken intoaccount when designing the loop filter: The feedback loop should have high enough openloop gain throughout the audio band to shape thequantization noise of the digital pulse width modulator and suppress errors resulting from imperfections in the output stage. The cut-off frequency ω c of the LC low-pass filteris usually selected to be significantly lower than theAES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August 28–30Page 5 of 10

MOUTON AND PUTZEYSDigital Control of a PWM Switching Amplifier with Global FeedbackDigitalAudioSourceUpsamplerLow-pass Chain ofintegratorsN1 (z)filterX(z) P(z)W (z)G1 (z)G2 (z)G3 (z) PWMLC-filterRippleCompensationPole cancellationSawtooth CarrierI(z) Output StageA E(z)LC FilterF(z)Vo (z)1F(z)H(z)G1 (z) 1Sigma-Delta AD Converter N2 (z)H(z)1AFig. 6: Block diagram of the feedback loop.switching frequency. The two zeros of G 1 (z) thatcancel these poles causes a second order rise in themagnitude of the open-loop transfer functionNT F2 (z) X(z)N2 (z)G(z)(3)above ωc . This gives rise to the amplification of thequantization noise N2 (z) of the analogue-to-digitalconverter, resulting in fold-back distortion. A possible solution to this problem is to place a lead compensator in the analogue domain. In the implementation that is described in this paper it is possible toovercome this problem through careful design of thedigital loop filter without additional analogue circuitry. However, a commercial system would usean analogue to digital converter with much moreout-of-band noise in which case an analogue leadcompensator would be required. Additional digital low-pass filtering is required toattenuate the quantization noise of the analogue-todigital converter. However, placing the poles associated with this filter near the audio band leads toinstability of the control loop. The stability of the control loop is also dependent onthe proper implementation of the ripple compensation method described in the previous section. Theexperimental system proved to be unstable withoutproper ripple compensation.Finding the optimal trade-off between these requirements requires careful sculpting of the transfer functionof the digital loop filter.A switching frequency of 768 kHz was selected for theexperimental system and the low-pass LC-filter has acut-off frequency of 70 kHz. The digital feedback loopis clocked at 19.6608 MHz, expect for the 7-bit digitalpulse width modulator which operates at a clock frequency of 98.304 MHz. The input signal X(z) of thismodulator is thus updated at every fifth clock cycle of the98.304 MHz clock. During the experimental evaluationit was found that this 5:1 clock ratio didn‘t cause measurable distortion in the audio band as long as sufficientanalog low-pass filtering (which the LC filter provides)was included before the analog-to-digital converter.AES 37TH INTERNATIONAL CONFERENCE, Hillerød, Denmark, 2009 August 28–30Page 6 of 10

MOUTON AND PUTZEYSDigital Control of a PWM Switching Amplifier with Global Feedback1503Magnitude of NTF2(z) (dB)Magnitude (dB)100Poles of G (z)500Zeros of G (z)3LC pole cancellationPoles of G1(z) 50100Poles of G1(z)500Poles of G2(z) 100 110210310410Frequency (Hz)Poles of G (z)Zeros of G (z)21510610710 50 110210310410Frequency (Hz)510610107Fig. 7: Open loop Bode plot of G 1 (z)G2 (z)G3 (z)F(z).Fig. 8: Bode plot of NT F2 (z).As second order noise shaping loop with noise transferfunctionNT F3 (z) (1 z 1)2(4)Fig. 8 shows a bode plot of transfer function NT F2 (z)as defined in equation 3. The second order rise in thistransfer function above 70 kHz is a result of the zerosof G1 (z), as described earlier. The poles of G 1 (z) andthe low-pass filter G2 (z) provide sufficient attenuation ofthe high-frequency quantization noise of the analogueto-digital converter to prevent fold back distortion.(as described in [17]) was placed around the pulsewidthmodulator. The intersection between the sawtooth carrier and the input signal of the pulse width modulator isdetected and the noise shaper is only clocked at this intersection. In this way the noise shaper is clocked at theswitching

CONTROLLED AMPLIFIER Fig. 1 shows an overall block diagram of the switching amplifier with global feedback. The overall structure of the feedback loop is a classical control loop in its sim-plest .

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