An Analysis of THD in Class D Amplifiers

An Analysis of THD in Class D AmplifiersHuey Chian Foong and Meng Tong TanSchool of Electrical and Electronic EngineeringNanyang Technological University, SingaporeAbstract – In this paper, we investigate the parametersaffecting THD and PSRR in a PWM Class D amplifierswith feedback. From our mathematical analysis, we showthat the THD and PSRR of a PWM Class D Amplifierswith feedback can be improved by shaping the frequencyresponse of the 2nd-order integrator. We also show howthe THD can be further improved by means of dual-loopfeedback. Simulation shows that the THD of the single-loop and the dual-loop feedback Class D amplifiers can beimproved by as much as 3 times and 12 times respectively.I.I NTRODUCTIONClass D audio amplifiers have become more prevalent thanthe classical linear Class A, Class AB and Class B amplifiersin portable consumer products primarily due to their higherefficiency over a large modulation index range [1]. This isbecause the output transistors of the Class D amplifiersoperate in the triode and cut-off region thereby dissipatingvery little power.The block diagram of an open loop PWM Class D amplifier isdepicted in Figure 1. It consists of 3 blocks namely the PulseWidth Modulator (PWM), the output stage and the low passfilter. We have chosen the PWM approach because thistechnique is presently the most prevalent and features thelowest possible switching frequency. At the PWM, the inputsignal is compared against a high frequency carrier. Theresultant output of the comparator is a pulse width modulatedsignal where the width of the pulse is proportional to themagnitude of the input signal. The output stage comprising astring of inverters is used to provide sufficient current to drivea low impedance load at the low pass filter. The low passfilter is used to remove the high frequency carrier and recoverthe signal at the input.Figure 1 Schematic diagram of an open-loop Class Damplifier.The sound quality of an audio amplifier is largely determinedby its performance in term of distortion (e.g. THD) and noise(e.g. PSRR). One of the problems with open-loop Class Damplifier is the susceptibility of the PWM or output stage tonon-linearity and noise [2-4]. The non-linearity and noisewill as a result lead to poor THD and PSRR performance inan open-loop Class D amplifier. In a linear amplifier, theseproblems can be easily mitigated by feeding the non-linearities at the Class D output back to the input. However,the application of negative feedback in Class D amplifier isnot straight forward due to the high switching frequencycarrier and time delay in the feedback loop. To improve thestability and dynamic range of the input signal, a low passfilter is generally employed to provide sufficient attenuationto the amplitude of the high frequency carrier in the feedbackloop. As a result of low pass filtering, the THD of theclosed-loop Class D amplifier usually suffer from poorerTHD performance at higher frequency (e.g. 3kHz).In this paper, we investigate the mechanism for THD inclosed-loop Class D amplifiers on the basis of linear models.Our investigation include a commercial closed-loop Class Damplifier embodying a 2nd-order integrator and a dualfeedback loop Class D amplifier. Through our modeling andanalysis, we show how the THD and PSRR can be improvedby shaping the frequency response of the loop gain in theClass D amplifiers.In addition, we analyze and show how the THD can be furtherimproved by adding another loop to form a dual-loopfeedback. This gives us insight into the factors affecting THDand PSRR in a Class D amplifier.II. A 2nd-order Closed-Loop Class D AmplifierThe schematic diagram of a typical closed-loop class Damplifier embodying a 2nd-order integrator is depicted inFigure 2. This is a simple and yet elegant negative feedbackconfiguration and has been adopted in the design of somecommercial Class D amplifiers [5].The closed-loop Class D amplifier can be modeled as linearmodel if the gain of the PWM and output stage is assumed tobe constant. The model of this closed-loop Class D amplifieris depicted in Figure 3.7241-4244-0387-1/06/$20.00c 2006IEEEFigure 2 A 2nd-order closed-loop Class D amplifier.Figure 3 A linear model of the 2nd-order closed-loop ClassD amplifier.In this model, the non-linearities due to harmonic distortionand power supply noise is denoted as V N. For power supplynoise at the Class D output, the resultant noise will beattenuated by 50% after low pass filtering [6] and as a result,V N, is only half if the noise originates from the power supply.This should be reflected in the linear model of the Class Damplifier for PSRR simulation. The system transfer function(inoutVV) and noise transfer function (NoutVV) [4] isgiven asinfbPWMPWMVinoutRRHGGGGGVVN−≈+==1intint11(1)1int1int12/111HGGNNHGGVVPWMDPWMVNoutin+=×+==(2)where the transfer function of each sub-blocks are given as:)R/(RRG fb1fb1+=(3))R/(RRH fb111+=(4)223//3_//333_int]1))2/(1/([)2(1)12))[2/(1/(sCRRfsACsRRCsRfsAGdBDCopdBDCop+++++++−=ππ---------------------------------------------- (5)where R = R1//R fbFrom equation (2), it can be observed that the PSRR isdetermined primarily by 3 parameters [4]: (a) the gain of thePWM stage, G PWM, (b) the feedback factor, H1 and (c) thegain of the integrator, G INT. These 3 parameters can also beoptimized to improve the THD of the Class D amplifiers.However, as the harmonics at the Class D amp output is afunction of the input frequency, the THD is as a consequencea function of the input frequency. In this paper, we propose toincrease the bandwidth of the integrator G INT to furtherimprove the THD of the Class D amplifier. We shall nowdescribe how these 3 parameters affect the PSRR and THD.The 1st parameter the gain of the PWM, G PWM, is determinedby the ratio of the power supply voltage over the amplitude ofthe carrier. In a typical PWM, the peak-to-peak amplitude ofthe high frequency carrier is typically in the range of 30%-70% of the supply voltage, V DD. In general, G PWM is less than5 for a larger input dynamic range and signal-to-noise ratio.For the 2nd parameter the feedback factor, H1, it is determinedby the closed-loop gain at audio frequency (the frequencyband of interest) according to equation (1) and isapproximately equal to1RRfb−. For a larger input dynamicrange and a higher loop gain as well as a better THD andPSRR performance, we recommend H1 to be as high aspossible for a higher loop gain.The 3rd parameter, the gain of the integrator is the dominantfactor that determines the THD and PSRR of the Class Damplifiers. This is because the power supply noise, typically100Hz – 120Hz (if noise source is from a full wave rectifier),can be attenuated by a large open loop gain of the op-amp ofthe integrator. The open-loop gain of the op-amp is typically80dB to 100dB. For simplicity, we assume the open-loopgain of the op-amp to be 80dB. The THD, on the other hand,is largely determined by the open-loop gain of the op-amp aswell as the bandwidth of the 2nd-order integrator. This isbecause as the gain of the integrator is much lower than thegain at higher frequency due to the 40dB/dec roll off at thecut-off frequency. Put differently, the THD will increasewhen the input frequency increases. In summary, the gainand bandwidth of the integrator has to be optimized toimprove the PSRR and THD.The transfer function of the integrator in equation (5) can besimplified as follows3//2233////12)12(1RRCssCRRsCRRsCVVino+−=+−=(6)where R = R fb//R1Due to the large time constant required for the 2nd-orderintegrator, the resistor and capacitors (R3, C1 and C2) areusually realized externally as discrete surface mountcomponents. Since the frequency response of the integratoris determined by the values of these 3 external components,an engineer designing the Class D amplifier will have theflexibility to improve the THD by changing the values ofthese 3 components.3APCCAS2006725We depict in Figure 4 the frequency responses of a 2nd-order integrator based on the RC values given in the datasheet of a commercial Class D amplifier (R3=68kΩ and C=470pF). This frequency response is depicted in solid line.In Figure 4, we note that the bandwidth of the 2nd-order integrator is 60Hz. The gain of the integrator starts to roll-off at 40dB/dec until 2 kHz when a zero is introduced.Figure 4 Frequency response of the 2nd-order integrator. From the frequency response in Figure 4, we note that the bandwidth can be increased without affecting signal to carrier attenuation ratio (carrier frequency is ~200kHz). We achieved this in 2 steps. First, we shift the zero of the integrator transfer function, G int, to the right such that it is greater than the switching frequency of the carrier. The amplitude of the carrier, as a result, will have a larger attenuation than the former case. This can be achieved by decreasing the value of R3 in equation (6) to 3kΩ. With a higher carrier attenuation, we can increase the bandwidth of the integrator in the second step by shifting the double poles of the 2nd-order integrator to the right. The frequency response of the proposed design is depicted as dotted line in Figure 4. From the proposed frequency response, we note that the gain at the audio frequency range is improved by as much as 20dB. The phase margin at unity gain is also improved by as much as 30° thereby improving the stability of the amplifier. It may be worthwhile to remark here that the bandwidth can also be increased by increasing the value of the capacitor C. However, this approach will decrease the amount of attenuation on the carrier and as a result a larger carrier amplitude will appear at the output of the integrator.III. Class D Amplifier with Dual Feedback LoopWe have mentioned in the previous section that the THD of a 2nd-order closed-loop Class D amplifier at higher frequency will be poorer due to insufficient gain at higher frequency as a result of the 40dB/dec roll-off. To further improve the THD and PSRR, we can increase the gain and bandwidth of the loop gain by cascading another 2nd-order integrator to form a dual loop feedback Class D amplifier as depicted in Figure 5.Figure 5 A dual-loop Class D amplifier based on 2nd-order integrator.With a dual 2nd-order feedback loop, the noise transfer function is given as))(1)()(1(intint iPWMoPWMnino HGGHGGVV++= (7)where()oPWMHGGintis the loop gain of the outer loopand ()iPWMHGGintis the loop gain of the inner loop.From equation (7), we note that the undesired signal at various frequencies is attenuated by a factor of the product of the loop gain of the first integrator and the second integrator. With an additional feedback loop, the 4th-order loop gain response of the dual loop feedback will have a much steeper roll-off of 80dB/dec.Assuming R1 = R1’, R2 = R2’ and R3 = R3’, the close loop transfer function for the dual loop topology can be derived as follow:2321123212133112322322232222132122131232)214414444()22(RRRRCsRsCRRRRRRCsRRRCsRsCRRRCsRsCRRRCsRsCRRRRCsRRsCRRRsCRVVinO++++++++++++++=(7)The transfer function of the dual loop can also be derived as follows:)12()22(.)()12('33132232123212213121323222+++++++++=sCRRRCsRRCsRRsCRRsCRRRRCsRRsCRRRsCRsCRRRCsVVin(8)Figure 6 depicted the loop-gain frequency response of the dual-loop feedback Class D amplifier before and after increasing the bandwidth of the two 2nd-order integrators.New zero locationOriginal zero location726APCCAS2006In Figure 6, we note that by increasing the bandwidth of the 2 integrators, the gain is increased by as much as 55dB. Since the harmonics of the Class D amplifier are usually dominated by the 2nd and 3rd harmonics, the THD of the dual loop Class D amplifier will be greatly improved if the input frequency is less than 6kHz.Figure 6Loop-gain frequency response with dual loopfeedback.IV. Analysis and Comparison of Simulation ResultsWe have simulated the 2nd-oder closed-loop Class D amplifier in Figure 2 for resistor R3= 68kΩ, 10kΩ and 3kΩ. The THD based on HSPICE simulation is depicted in Figure 5. In Figure 5, we note that THD improves by as much as 3 times when the zero of the integrator transfer function is shifted to the right by decreasing R3 from 68kΩ to 3kΩ. As expected, the THD increases at higher frequency because the loop gain of the Class D amplifier decreases at higher frequency due to the 40dB/dec roll-off at the cut-off frequency.Figure 7 Simulation Results of a single-loop feedback ClassD amplifier.We depict in Figure 8 the THD of a single-loop and a dual-loop feedback Class D amplifiers when resistors, R3=R’3=3kΩand C=C’=470nF. In Figure 8, we note that the THD of the dual-loop feedback Class D amplifier improved tremendously in comparison to that of the single-loop Class D amplifier. The THD for input frequency between 1kHz to 7kHz is less than 0.2% - an improvement of more than 12 times. This method can be applied easily since the resistor and capacitorsD amplifier.IV. CONCLUSIONSWe have investigated the parameters affecting THD and PSRR in a PWM Class D Amplifiers with feedback. We have shown through mathematical analysis and simulation how the THD of a commercial single loop feedback Class D amplifier can be optimized by shaping the frequency response of the 2nd-order integrator. We have also shown how the THD can be further improved by means of dual-loop feedback. Simulation shows that the THD of the single loop feedback commercial Class D amplifier can be improved by as much as 3 times by shaping the frequency response of the 2nd-order integrator. The THD of a dual loop Class D amplifier is improved by as much as 12 times.REFERENCES[1] “Digital Audio’s Final Frontier”, IEEE spectrum, Mar 2003.[2] M.T. Tan, J.S. Chang, H.C. Chua and G.B. Gwee., “AnInvestigation into the Parameters Affecting Total Harmonic Distortion in Low Voltage Low Power Class D Amplifiers”, IEEE Trans on Circuits and Systems I, Vol. 50, No. 10, pp.1304–1315, Oct 2003.[3] T. Ge, J.S. Chang, W. Shu and M.T. Tan, “Modeling andAnalysis of PSRR in PWM Class D Amplifiers”, ISCAS ’06, Kos, Greece, pp. 1386–1389, May 2006.[4] W. Shu, J.S. Chang, T. Ge and M.T. Tan, “Fourier SeriesAnalysis of the Nonlinearities in Analog Closed-Loop PWM Class D Amplifiers”, ISCAS ’06, Kos, Greece, pp. 1382– 1385, May 2006.[5] “25W + 25W Stereo Class D Amplifier”, TDA7490Datasheet, ST Microelectronics, 1999.[6] M. Berkhout, “Integrated Class D Amplifier,” 112thConvention of Audio Engineering Society, Germany, May 2002.Loop gain curve after shapingLoop gain curve before shapingAPCCAS2006727。