Class D Audio Amplifiers

Nonlinear Control Techniques for Low-Voltage, Low-Power Applications: Class D Audio Amplifiers

Miguel Angel Rojas-González and Edgar Sánchez-Sinencio ([email protected], [email protected])

Analog and Mixed Signal Center Texas A&M University Spr ing 2008 Outline

1. Introduction to audio amplifiers a) Au dio amplifiers applications b) Linear vs. nonlinear audio amplifiers 2. Conventional class D audio amplifier a) Typical architecture b) Parameters affecting amplifier performance 3. Proposed class D audio amplifier (single-ended) a) Architecture description b) Design procedure c) Experimental results 4. Class D audio amplifiers: two design approaches a) Architecture descriptions b) Preliminary results 1. Introduction to audio amplifiers

In a sound syy,pstem, the power am plifier supplies power to the loudspeaker

The typical speaker input impedance is low, usually in the 4 Ω to 8 Ω range. Thus, the power amplifier must be able to supply the high peak currents required to drive the low impedance.

Standard audible frequency band is from 20 Hz to 20 KHz For high-fidelity sound system, THD must be < 0.1 %

Least detectable amount of harmonic distortion byyg humans is ~ 0.3 % in average

Telephone: THD ~ 10 % BW ~ 4 KHz Main power amplifier classes

Linear Nonlinear

Class A, B, AB Class D

Linearity Efficiency Power amplifier classes Class A amplifier Class D amplifier • Current flows continuously • Switchinggp amplifier • High sound quality • Ideal THD 0% • Poor efficiency (25 %) • Highest efficiency (100 %)

Power Efficiency of Class A, Class B and Class D Amplifiers Class B amplifier 90 • Current flows half of the period 80 70 • Linearity compromised by crossover 60

• Higher efficiency (78.5 %) 50 iency (%) iency Class AB amplifier cc 40 Effi 30

• Hybrid between classes A and B 20 Class A Ideal • Good soun d quali ty 10 Class B Ideal Class D M easured 0 • Higher efficiency (78.5 %) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Load Power (P /P ) L Lmax 2. Conventional class D audio amplifier Continuously switch the output from one rail to another at supersonic frequency (Pulse Width Modulation -PWM-)

There are two main areas of application for class D amplifiers: 1. Low Power Outputs • From few milliwats to around 5 W • Hearing aids, mobile phones, personal stereos, laptop computer audio, etc. • Portable products, battery driven • High efficiency required 2. High Power Outputs • From 80 W to 1400 W • Home theatre systems, car audio systems, etc. • Keeps dissipation and heat sink size minimum 2. Conventional class D audio amplifier History • First proposed around 1950’ s Efficiency • 5 % THD in 1976 state of the art amplifier • 100 % at all output levels ideally • Between 80% and 90% due to Basic Principles parasitic losses in practice • No output devices operating in the linear mode • Output stage switches at supersonic frequency • There is no inherent supply rejection

• Switching frequencies from 50KHz to 1MHz Power Stage L • PWM is generated comparing the audio

signal with a triangle wave carrier signal PWM Audio Signal C • Triangle wave needs to be Speaker

linear to prevent distortion Power Stage Comparator Carrier Signal 2. Conventional class D audio amplifier Pulse Width Modulation (PWM) Varies the duty cycle of the converter switches at a high switching freqqyuency (supersonic) to achieve a target average low frequency output voltage.

Class D Audio Amplifier (THD = 0.00%) • Class D amplifier THD is calculated f = 1.0e+003 Hz 0 in considering all harmonics below 20KHz (max. audible frequency) -20 • Ideal class D amplifier THD is 0.0 % -40

agnitude (dB) agnitude • Performance is degraded due to system MM -60 nonlinearities -80 0 10 20 30 40 50 60 Frequency (KHz) Power Stage

L Nonlinearity sources

• Passive components nonlinearities PWM Audio Signal C

• MOS on-resistance Speaker

Power Stage Comparator • Non-ideal triangle carrier signal Carrier Signal 2. Conventional class D audio amplifier Passive components nonlinearities (†) The low saturation current of the load inductor causes frequency dependent nonlinearity. Saturation current of inductor is defined as Power Stage the IDC current level where the effective inductance value is decreased to 90% of L its value at zero DC current.

PWM An approximate expression of Audio Signal C

nonlinear inductance is obtained by Speaker series representation Power Stage Comparator V Carrier Signal THD(%)=100 third harmonic V fundamental where L, V, fin, R and I are the inductance at 3⋅ L ⋅V 2 ⋅0.1⋅2π ⋅ f sat THD(%)=100 in zero DC current, the output voltage, the input 3 2 signal frequency, the speaker resistance and 4⋅ R ⋅ I sat the inductor saturation current respectively

(†) B. Kelleci, E. Sanchez-Sinencio, and A. Karsilayan, “THD+N Estimation in Class-D Amplifiers”, IEEE ISCAS, pp. 465-468, 2007 2. Conventional class D audio amplifier MOS on-resistance (†) The class D amplifier is more power efficient (ideally 100% efficiency) than linear amplifiers because itsoutttput PWM isswitch mode. The on-resistance is not negligible (~200mΩ)andit will result in a slightly “amplitude modulated” signal at the PWM output. This modulation becomes more Power Stage

pronounced at higher modulation indexes. L

PWM AdiSiAudio Signa l C

Speaker

Power Stage Comparator Carrier Signal

where fc,fs, M and Wp are the carrier signal frequency, the input signal frequency, the modulation index and the output transistor size respectively.

(†) M. Tan, et al, “An investigation Into The Parameters Affecting THD in Low-Voltage Low-Power Class-D Amplifiers”, IEEE TCAS I , Vol. 50, No. 10, pp. 1304-1315, 2003 2. Conventional class D audio amplifier Carrier signal nonlinearity Carrier signal non-idealities will affect directly the linearity performance in the class D audio amplifier.

There are three main carrier signal modulation Power Stage schemes: L • Triangle wave modulation

PWM • Sawtooth wave modulation Audio Signal C • Exponential wave modulation Speaker

Power Stage Comparator Carrier Signal • Harmonic frequency calculation of PWM is complex and is typically done using an FFT analysis of a simulated waveform but it usually leads to errors and t -T/2 T/2 miscalculations • Analytical calculations of harmonic components is usually done by using a Double Fourier Integral Analysis (DFIA). Mathematical expression is quite complex but accurate. 2. Conventional class D audio amplifier Carrier signal nonlinearity • A novel mathematical analysis method to model the carrier waveform has been ppproposed in (†) • Assume a exponential carrier signal instead of a triangle wave which may be generated by charging/discharging an RC integrator circuit with square pulses. • Shift the nonlinearity of the exponential carrier to the input modulating signal and then apply the Double Fourier Integral Analysis 1. Remove the nonlinearity of the trailing-edge exponential carrier by transform it to a linearized exponential carrier (linear sawtooth carrier) 2. Transform the initially-linear modulating signal (audio signal) to a transformed (nonlinear) modulating signal 3. Repeat (1) and (2) for the leading-edge exponential carrier. 4. Derive the Double Fourier coefficients of the double-sided PWM output by summing the Fourier coefficients of the trailing-edge and leading-edge PWM outputs. This mathematical analysis is accurate but its complexity and procedure are extensive

• We propose to use a simpler method to analyze carrier signal nonlinearity based on the PWM Analysis by Duty Cycle Variation for any kind of periodic carrier signal in the following presentation Overview of Double Fourier Integral Analysis of a PWM Waveform

• Fourier decomposition is based on the principle that any regular time-varying waveform f(t) can be expressed as an infinite series of sinusoidal harmonics: ∞ a0 f (t)= + ∑(am cos mωt + bm sin mωt) 2 m=1 where π 1 π ω a = f ()t cos m t d(ωt) m ∫ −π 1 π b = f (t)sin mωt d(ωt) m ∫ π −π 2. Conventional class D audio amplifier The analytical solution for the harmonic components of a PWM waveform assumes the existence of two time variables

x(t)= ωct

y()t = ωot where ωc and ωo are the carrier angular frequency and the input signal (audio wave) angular frequency respective lly. The objective is to find a function f(t) which describes the PWM signal as a periodic function of x and y by using Double Fourier Integral Analysis (†) Mcosy = Mcosω t 1 o The purpose of the Double Fourier Integral Analysis is to express the PWM waveform as a function of a x = ω t -ππc double variable controlled waveform.

-1

(†) H. S. Black, Modulation Theory, Princeton, NJ, Van Nostrand, 1953 ω

2. Conventional class ω D audio amplifier

In general, any double-variable time-varying function f(t) can be expressed, by using the Double Fourier Integral Analysis, in the following form (†) ω ∞ ∞ A00 f ()t = + ∑()()()A0n cos()n ot + B0n sin n ot ()+ ∑ Am0 cos m()ct + Bm0 sin mω0t 2 n=1 m=1 ∞∞ + ∑ ∑(Amn cos(mωct + nωot)+ Bmn sin(mωct + nωot)) m=1 n=−∞ ()n≠0 where m is the carrier index variable and n is the baseband index variable • The variables m and n define the angular frequency of each harmonic component

• The magnitudes of the harmonics components are the Amn and Bmn coefficients

This function f(t) will provide a exact solution to determine the harmonic components of a PWM opposed to the traditional method of computing an FFT of the waveform, which will always be sensitive to the time resolution of the simulation and the periodicity of the overall waveform.

(†) D. G. Holmes and T. A. Lipo, PWM For Power Converters, Wiley Inter-science, USA, 2003 2. Conventional class D audio amplifier

Analyzing the function f(t) we have that

A 00 where m=n=0,corresponds to the DC offset component of the PWM (if any) 2 ∞ where m=0, represents the fdfundamen tltal component ∑(A0n cos(nωot)+ B0n sin(nωot)) n=1 and baseband harmonics (if any)

∞ where n=0, dfidefines thecarrier wave hiharmonics (hig h- ∑(Am0 cos(mωct)+ Bm0 sin(mω0t)) m=1 frequency components)

∞∞ where m, n ≠ 0,represents the ∑ ∑(Amn cos(mωct + nωot)+ Bmn sin(mωct + nωot)) m=1 n=−∞ sideband harmonics ()n≠0

• The different summation terms in the PWM function f(t) will depend on the type of carrier wave • In some cases, it will be easier to express the summations using the Bessel function of the first kind (J) 2. Conventional class D audio amplifier Example 1: Sine-Sawtooth Modulation Harmonic components for sawtooth carrier when M = 0.9 and ω /ω = 21 Mcosy = Mcosω t c o 1 o 0

-20

x = ωct -40 nitude (dB) nitude

-T/2T/2 gg

Ma -60

-80 π -1 0 10 20 30 40 50 60 Harmonic Number π π The PWM function van(t) for a sawtooth carrier signal expressed in terms of its harmonics components is ω ∞ VDC 1 ω van (t)= VDCC +VDCC M cos(ωot)+ 2 ∑ [cos mπ − J 0 (mπMπ)]sin mωct π m=1 m ∞ ∞ ω VDC 1 ⎡ ⎤ + 2 ∑∑ J n ()m M ⎢sin n cos()m ct + n ot − cos n sin()m ct + nωot ⎥ mn=1 =−∞ m ⎣ 2 2 ⎦ (n≠0) As we expected, the THD of a class D audio amplifier will be 0.0 % since there are no baseband harmonics generated as we only have the fundamental tone. 2. Conventional class D audio amplifier Example 2: Sine-Triangle Modulation Harmonic components for triangle carrier when M = 0.9 and ω /ω = 21 Mcosy = Mcosω t c o 1 o 0

-20

x = ωct -40 nitude (dB) nitude

-T/2T/2 gg

Ma -60

-80 -1 0 10 20 30 40 50 60 Harmonic Number π The PWM function van(t) for a triangle carrier signal expressed in terms of its harmonics components is π π ∞ VDC 1 ⎛ ⎞ π van (t)= VDC +VDC M cos(ωot)+ 4 ∑ π J 0 ⎜m M ⎟sin m cos mωct π m=1 m ⎝ 2 ⎠ 2 ∞ ∞ ω VDC 1 ⎛ ⎞ ⎛ ⎞ + 4 ∑∑ J n ⎜m M ⎟sin⎜[]m + n ⎟cos()m ct + nωot mn=1 =−∞ m ⎝ 2 ⎠ ⎝ 2 ⎠ (n≠0) As in the previous case, the THD of a class D audio amplifier will be 0.0 % since there are no baseband harmonics generated as we only have the fundamental tone. 2. Conventional class D audio amplifier Carrier signal nonlinearity

• Previous examples demonstrated that class D audio amplifier provides 0.0% THD ideally • In reality, THD > 0.0% because the carrier signals are not ideal • In fact, for an ideal triangle wave carrier signal f(x) we would have (n−1) 8 ∞ ()−1 2 ⎛ 2nπx ⎞ f ()x sin = 2 ∑ 2 ⎜ ⎟ π n=1,3,5,... n ⎝ T ⎠ • And, for an ideal sawtooth wave carrier f(y) we would have Ideal Triangle Wave Spectrum 1 1π ∞ 1 ⎛ 2nπx ⎞ f ()y = − ∑ sin⎜ ⎟ 0 2 n=1 n ⎝ T ⎠

dB) -20 We would need an infinite bandwidth (( system to generate a perfect carrier signal! -40 Magnitude Magnitude Unfortunately, band-limited systems -60 degrade the performance of the -80 overall class D amplifier generating 0 10 20 30 40 50 undesired baseband components Harmonic Number 2. Conventional class D audio amplifier Carrier signal nonlinearity • Double Fourier Integral Analysis is a complex and tedious mathematical derivation • Instead, we can use the PWM Analysis by Duty Cycle variation if the input signal (audio wave) is assumed to be constant within each carrier cycle, i.e. ωc >> ωo, which is usually the case.

Example 1a: Sine-Sawtooth Modulation • Normalizing the period of the sawtooth to 2π and its amplitude to 1, we have

Mcosy = Mcosωot ∞ 1 a0 van (t)= + ∑(am cos mx + bm sin mx) 2 m=1 1 π x = ω t am = van (t)cos mx dx -ππc ∫ π π− 1 π b = v ()t sin mx dx 0 m ∫ an π −π

• We need to calculate the interval where van(t) switches from 0 to 1 π

2. Conventional π class D audio amplifier π Exampleπ 1a: Sine-Sawtooth Modulation (cont.) π π •For am coefficients,π when m ≠ 0 1 V πM cos y V a = v ()t cos mx dx = 2 πDC cos mx dx = 2 DC []sin()m M cos y + sin mπ m ∫ an ∫ − − π • For bm coefficients, when m ≠ 0 π π 1 π V πM cos y V b = v ()t sin mx dx = 2 DC sin mx dx = 2 DC []cos m − cos()mπM cos y m ∫ an ∫ −π −π • When m= 0 we have ω a = 2V ()1+ M cos y b = 0 0 DC 0 π • After some mathematical manipulation and applying theπ Jacobi-Anger expansions, we get the same expression obtained by using the Double Fourier Integral Analysisπ V ∞ 1 v ()t = V +V M cos t + (2 DC ) []cos m − J ()m M sin mω t an DC DC o ∑ m 0 c π m=1 ∞ ∞ ω ω VDC 1 ⎡ π π ⎤ + 2 ∑∑ J n ()mπM ⎢sin n cos()m ct + nωot − cos n sin()m ct + nωot ⎥ m=1 n=−∞ m ⎣ 2 2 ⎦ ()n≠0 2. Conventional class D audio amplifier Example 2a: Sine-Triangle Modulation • Normalizing the period of the sawtooth to 2π and its amplitude to 1, we have

∞ Mcosy = Mcosω t a 1 o 0 van ()t = + ∑()am cos mx + bm sin mx 2 m=1 1 π am = van (t)cos mx dx x = ωct ∫ -ππ π π− 1 π b = v (t)sin mx dx m ∫ an 0 π −π π

• We needπ to calculate the interval where van(t) switches from 0 to 1

•Fora coefficients, when m ≠ 0 m π π π ()1+M cos y 1 V 2 V ⎡ ⎛ π ⎞⎤ a = v (t)cos mx dx = 2 DC cos mx dx = 4 DC sin m (1+ M cos y) m ∫ an ∫ ⎢ ⎜ ⎟⎥ π π ⎣ ⎝ 2 ⎠⎦ − − ()1+M cos y 2 2. Conventional class D audio amplifier Exampleπ 2a: Sine-Triangle Modulation

•For bπm coefficients, since the triangle wave is an even function π ()1+M cos y 1 V 2 b = v ()t sin mx dx = 2 DC sin mx dx = 0 m ∫ an ∫ −π π − (1+πM cos y) 2 • When m= 0 we have a = 2V ()1+ M cos y 0 DC ω

• After some mathematical manipulation, as wellπ as in the previousπ example, and applying the Jacobi-Anger expansions, we get the same expression obtained by using the Double Fourier Integral Analysis π ∞ VDC 1 ⎛ ⎞ van ()t = VDC +VDC M cos ot + (4 )∑ J 0 ⎜m M ⎟sin m cos mωct m=1 m ⎝ 2 ⎠ 2 ∞∞ VDC 1 ⎛ π ⎞ ⎛ π ⎞ + 4 ∑ ∑ J n ⎜m M ⎟sin⎜[m + n] ⎟cos(mωct + nωot) π m=1 n=−∞ m ⎝ 2 ⎠ ⎝ 2 ⎠ ()n≠0 2. Conventional class D audio amplifier Carrier signal nonlinearity • We pppropose to generalize the PWM Analysis by Duty Cycle variation and appl y ittoaany carrier waveform to calculate analytically the THD in a class D audio amplifier Example 3: Sine-Non-Ideal Triangle Modulation • Recall that a triangle wave carrier signal is constructed by an infinite sum of sinusoidal functions and since there are not unlimited bandwidth systems, the number of harmonics (n) in a triangle wave carrier signal is finite

Triangle Wave Carrier 1.5 Triangle Wave Carrier (Zoom-in) -1 1 -1.05 0.5 -1.1 e (V) 0 )) -1.15 Voltag -0.5 n = 1 -1.2 n = 5 (V Voltage n = 1 -1 n = 15 -1.25 n = 25 n = 5 n = 15 n = ∞ -1.3 -1.5 n = 25 0 0.2 0.4 0.6 0.8 1 n = ∞ Time (sec) -5 -1.35 x 10 4.5 5 5.5 -6 Time (sec) x 10 2. Conventional class D audio amplifier Sine-Non-Ideal Triangle Modulation • There are two trivial cases in a triangle wave shaped carrier signal 1. When the number of harmonics is infinite we have an ideal triangle wave carrier signal and the THD of the class D amplifier is 0.0% 2. When the number of harmonics is 1 then we have a sine-cosine modulation PWM and the THD of the class D amplifier will depend on the modulation index (M) • Lets examine the case where the non-ideal triangle wave carrier signal contains one single harmonic component Example 3: Sine-Cosine Modulation Mcosy = Mcosω t 1 o ∞ a0 van ()t = + ∑()am cos mx + bm sin mx 2 m=1 1 π x = ωct a = v ()t cos mx dx -ππ m ∫ an π − π 1 π b = v (t)sin mx dx 0 m ∫ an π −π 2. Conventional class D audio amplifier

Exampleπ 3: Sine-Cosine Modulation (cont.)

•For am coefficients, when m ≠ 0

⎛ π 2 ⎞ arccos⎜ − M cos y ⎟ ⎜ 8 ⎟ 1 V ⎝ ⎠ V ⎡ ⎛ ⎛ π 2 ⎞⎞⎤ a = v ()t cos mx dx = 2 DC cos mx dx = 4 DC sin⎜marccos⎜− M cos y⎟⎟ m ∫ an ∫ ⎢ ⎜ ⎜ ⎟⎟⎥ π −π π ⎛ 2 ⎞ π ⎢ ⎝ 8 ⎠ ⎥ ⎜ π ⎟ ⎣ ⎝ ⎠⎦ −arccos⎜ − M cos y ⎟ ⎝ 8 ⎠ π

•Forbm coefficients, since the cosine is an even function

⎛ π 2 ⎞ arccos⎜ − M cos y ⎟ ⎜ 8 ⎟ 1 V ⎝ ⎠ b = v ()t sin mx dx = 2 DC sin mx dx = 0 m ∫ an ∫ π −π π ⎛ 2 ⎞ ⎜ π ⎟ −arccos⎜ − M cos y ⎟ ⎝ 8 ⎠ • When m= 0 we have V ⎛ π 2 ⎞ DC ⎜ ⎟ a0 = 4 arccos⎜− M cos y⎟ π ⎝ 8 ⎠ 2. Conventional class D audio amplifier

Example π3: Sine-Cosine Modulation (cont.)

• Finally, the function van(()t) will be given by π V ⎛ ⎡ ∞ ⎛ ⎞ ⎛ π 2 ⎞ ⎤⎞ v ()t = 2 DC arccos⎜arcsin 2 sin k J ⎜− M ⎟cos()ky ⎟ an ⎜ ⎢ ∑ ⎜ ⎟ k ⎜ ⎟ ⎥⎟ ⎝ ⎣ k =1 ⎝ 2 ⎠ ⎝ 8 ⎠ ⎦⎠ π ∞ V ⎡ ⎛ ⎛ π 2 ⎞⎞⎤ DC ⎜ ⎜ ⎟⎟ + ∑ 4 ⎢sin⎜marccos⎜− M cos y⎟⎟⎥ cos mx m=1 ⎢⎣ ⎝ ⎝ 8 ⎠⎠⎦⎥

• We can see that the fundamental component is not alone in the expression but comes with baseband harmonics product of the cosine shaped-carrier waveform. Such baseband harmonics will produce the harmonic distortion in the class D audio amplifier • It can be appreciated that as we increment the modulation index M, the distortion increments exponentially. • Same procedure can be applied for a given number of harmonics components present in the triangle wave carrier, however, the only closed-form solution exists when n = 1 and n = ∞.The solution when 1 < n < ∞ must be calculated numerically. 2. Conventional class D audio amplifier Example 3: Sine-Cosine Modulation (cont.) •Inorder to verify the mathematical derivation and its results, we have created a simple SIMULINK model to simulate a class D audio amplifier and compare the traditional FFT method and the analytical solution to find the THD in the amplifier Class D Amplifier THD (n = 1) 14 Mathematical model 12 Simulink simulation 10

6 THD (%) 4 Class D Amplifier THD (n = 5) 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.5 Modulation index 2 ))

Non-ideal triangle wave carrier signal with n = 1 1.5

THD (% 1

0.5 Mathematical model Simulink simulation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Modulation index Non-ideal triangle wave carrier signal with n = 5 2. Conventional class D audio amplifier Example 3: Sine-Cosine Modulation (cont.) Class D Amplifier THD (n = 15) Class D Amplifier THD (n = 25) 1 0.7 Mathematical model Mathematical model 0.6 Simulink simulation Simulink simulation 0.8 0.5

0.4 0.6 0.3 THD (%) THD THD (%)

0.4 0.2 0.1

0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Modulation index Modulation index Non-ideal triangle wave carrier signal with n = 15 Non-ideal triangle wave carrier signal with n = 25 Class D Amplifier THD (n = 99) Class D Amplifier THD for M = 0.8 0.04 10 Mathematical model Mathematical model Simulink simulation Simulink simulation 0.03 1 ))

0.02

THD (% 0.1

0.01 THD (%)

0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Modulation index 0.001 Non-ideal triangle wave carrier signal with n = 99 0 20 40 60 80 100 Number of Harmonics in Triangle Wave Carrier 2. Conventional class D audio amplifier Example 3: Sine-Cosine Modulation (cont.) Class D (THD = 3.59352% (-28.890dB)) Triangle Wave Carrier (n = 3) Triangg()le Wave Carrier (n = 3): Mathematical Model 0 3 2.5 -20 2 -40 1.5 agnitude (dB) tage (V) MM -60 ll 1 Vo 0.5 -80 0 10 20 30 40 50 60 0 Frequency (Hz) -0.5 Class D output spectrum when n = 3 -0.5 0 0.5 Time (sec) (Ma thema tical Mo de l)

Class D Audio Amplifier (THD = 3.60424% (-28.864dB) Triangle Wave) The mathematical model predicts with high 0 f = 1.0e+000 Hz in HD1 = -3.3 dB accuracy the result in the SIMULINK HD2 = -138.9 dB dB)

(( -20 HD3 = -32 .4 dB simulation! HD4 = -144.4 dB -40 HD5 = -45.4 dB Magnitude Magnitude -60 Exercise: Can you provide an analytical -80 0 10 20 30 40 50 60 expression for the THD output spectrum when Frequency (Hz) a sawtooth carrier signal with only one Class D output spectrum when n = 3 harmonic is used to PWM an audio signal? (SIMULINK Simulation) 2. Conventional class D audio amplifier Carrier signal nonlinearity • Lets now analyze the case when an exponential waveform is used as a carrier signal • An exponential waveform is usually employed as a carrier signal due to its simple implementation Example 4: Sine-Exponential Modulation • The exponential waveform is generated by charging/discharging a simple RC integrator with square pulses. Exponential Carrier Signals 1.5 Defining a set of normalized exponential waves A B ⎛ ⎛ T ⎞ ⎞ 1 ⎧ ⎧ ⎜ −⎜ x+ ⎟ ⎟ ⎫ ⎫ C ⎪ ⎜ ⎝ 2 ⎠ ⎟ ⎪ ⎪ ⎜ t ⎟ ⎪ D ⎪ ⎜ 0 ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ E ⎪ ⎪ e − error ⎪ T ⎪ 0.5 VDC ⎨2 −1⎬, − < x < 0

⎪ ⎪ (V) ⎪ 1− error ⎪ 2 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f ()x = ⎨ ⎩ ⎭ ⎬ Voltage ⎧ ⎛ −x ⎞ ⎫ -0.5 ⎪ ⎛ ⎜ ⎟ ⎞ ⎪ ⎪ ⎪ ⎜1− e⎝ t0 ⎠ ⎟ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎪ ⎝ ⎠ ⎪ T ⎪ -1 ⎪VDC ⎨2 −1⎬, 0 < x < ⎪ ⎪ ⎪ 1− error ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ -1.5 ⎪ ⎪ 0.5 1 1.5 ⎩⎪ ⎭⎪ -5 ⎩ ⎭ Time (sec) x 10 2. Conventional class D audio amplifier Example 4: Sine-Exponential Modulation (cont.)

Normalizing the exponential carrier to a period of 2π we can calculate the Fourier coefficients based on the PWM Analysis by Duty Cycle Variation

Mcosy = Mcosω t 1 o ∞ a0 van ()t = + ∑()am cos mx + bm sin mx 2 m=1 x = ω t -ππc 1 π a = v ()t cos mx dx 0 m ∫ an π −π 1 π b = v ()t sin mx dx m ∫ an π −π 2. Conventional class D audio amplifier Example 4: Sine-Exponential Modulation (cont.)

•For am coefficients, when m ≠ 0 π ⎛ 1 ⎛ M ⎞ ⎞ −t ln⎜1− ()1−error ⎜ cos y+1⎟ ⎟ 0 ⎜ 2 ⎜ V ⎟ ⎟ V ⎝ ⎝ DC ⎠ ⎠ a = 2 DC cos mx dx = m ∫ π ⎛ M ()1+error ⎞ −π −t0 ln⎜ cos y(1−eeorror )+ ⎟ ⎝ 2VDC 2 ⎠ V ⎡ ⎛ ⎛ 1 ⎛ M ⎞⎞⎞ ⎛ ⎛ ⎛ M 1+ error ⎞⎞⎞⎤ a = 2 DC ⎢sin⎜− mt ln⎜1− ()1− error ⎜ cos y +1⎟⎟⎟ − sin⎜m⎜−π − t ln⎜ cos y()1− error + ()⎟⎟⎟⎥ m m ⎜ 0 ⎜ 2 ⎜V ⎟⎟⎟ ⎜ ⎜ 0 ⎜ 2V 2 ⎟⎟⎟ ⎣⎢ ⎝ ⎝ ⎝ DC ⎠⎠⎠ ⎝ ⎝ ⎝ DC ⎠⎠⎠⎦⎥

•Forbm coefficients

⎛ 1 ⎛ M ⎞ ⎞ −t ln⎜1− ()1−error ⎜ cos y+1⎟ ⎟ 0 ⎜ 2 ⎜ V ⎟ ⎟ V ⎝ ⎝ DC ⎠ ⎠ b = 2 DC sin mx dx = m π ∫ ⎛ M 1+error()⎞ −π −t0 ln⎜ cos y()1−error + ⎟ ⎝ 2VDC 2 ⎠ V ⎡ ⎛ ⎛ 1 ⎛ M ⎞⎞⎞ ⎛ ⎛ ⎛ M 1+ error ⎞⎞⎞⎤ b = −2 DC ⎢cos⎜− mt ln⎜1− ()1− error ⎜ cos y +1⎟⎟⎟ − cos⎜m⎜−π − t ln⎜ cos y()1− error + ()⎟⎟⎟⎥ m mπ ⎜ 0 ⎜ 2 ⎜V ⎟⎟⎟ ⎜ ⎜ 0 ⎜ 2V 2 ⎟⎟⎟ ⎣⎢ ⎝ ⎝ ⎝ DC ⎠⎠⎠ ⎝ ⎝ ⎝ DC ⎠⎠⎠⎦⎥ 2. Conventional class D audio amplifier Example 4: Sine-Exponential Modulation (cont.)

•For am coefficients, when m ≠ 0

⎛ 1 ⎛ M ⎞ ⎞ −t ln⎜1− ()1−error ⎜ cos y+1⎟ ⎟ 0 ⎜ 2 ⎜ V ⎟ ⎟ V ⎝ ⎝ DC ⎠ ⎠ a = 2 DC dx 0 ∫ π ⎛ M (1+error )⎞ −π −t0 ln⎜ cos y()1−error + ⎟ ⎝ 2VDC 2 ⎠ V ⎡ ⎛ 1 ⎛ M ⎞⎞ ⎛ ⎛ M 1+ error()⎞⎞⎤ a = 2 DC ⎢− t ln⎜1− ()1− error ⎜ cos y +1⎟⎟ + ⎜π + t ln⎜ cos y()1− error + ⎟⎟⎥ 0 π 0 ⎜ 2 ⎜V ⎟⎟ ⎜ 0 ⎜ 2V 2 ⎟⎟ ⎢⎣ ⎝ ⎝ DC ⎠⎠ ⎝ ⎝ DC ⎠⎠⎦⎥

• As in the previous example, the coefficient a0 will have the fundamental tone as well as baseband harmonics that will degrade the class D audio amplifier THD • As the ‘error’ parameter increases, the exponential wave behaves in a quasi-triangular way and the baseband harmonics magnitude decrease. • Like in the previous example, a SIMULINK model was created and simulated in order to compare the results of both procedures. 2. Conventional class D audio amplifier Example 4: Sine-Exponential Modulation (cont.) Exponential Carrier Signals 1.5 A B Class D Amplifier THD (Exponential waves 'B' and 'A') 1 C 10 D 0.5 E 1 ) e (V) gg 0 %% Volta -0.5 THD ( 0.1 Mathematical model (wave 'B') Mathematical model (wave 'A') Simulink simulation (wave 'B') -1 Simulink simulation (wave 'A') 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1.5 0.5 1 1.5 Modulation index -5 Time (sec) x 10 Class D Amplifier THD (Exponential wave 'E') Class D Amplifier THD (Exponential waves 'D' and 'C') 0.02 0.8 Mathematical model (wave 'D') Mathematical model (wave 'C') 0.015 0.6 Simulink simulation (wave 'D') Simulink simulation (wave 'C')

0.01 0.4 THD (%) THD (%)

0.005 0.2 Mathematical model Simulink simulation 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Modulation index Modulation index 2. Conventional class D audio amplifier Example 4: Sine-Exponential Modulation (cont.) Class D (THD = 0.23470% (-52.590dB)) Exponential Carrier Exponential Carrier: Mathematical Model 3 0 2.5 2 -20 1.5 -40 1 oltage (V) gnitude (dB) gnitude VV 0.5 Ma -60 0 -80 -0.5 0 10 20 30 40 50 60 -0.5 0 0.5 Frequency (Hz) Time (sec) Class D ou tpu t spec trum (ma thema tica l mo de l)

Class D Audio Amplifier (THD = 0.23494% (-52.5dB) Exponential Wave) The mathematical model predicts with high 0 fin = 1.0e+000 Hz HD1 = -3.7 dB accuracy the result in the SIMULINK )

BB HD2 =- 144.8 dB -20 simulation! HD3 = -56.3 dB HD4 = -146.4 dB -40 HD5 = -103.7 dB

Magnitude (d Magnitude -60 This analysis method can be further applied to any carrier wave or even multiphase systems -80 0 10 20 30 40 50 60 where multilevel PWM is generated. Frequency (Hz) Class D output spectrum (SIMULINK simulation) 3. Proposed class D audio amplifier (†) (single ended architecture)

Controller Comparator Power Stage Output Filter

VA Sliding U VIN iL Mode VOUT VOUT L Controller VDD vc CR MDP

MDN

VSS

Class D amplifier with sliding surface

(†) M. Rojas-Gonzalez, E. Sanchez-Sinencio, “Design of a Class D Audio Amplifier IC Using Sliding Mode Control and Negative Feedback”, IEEE Transactions on Consumer Electronics, Vol. 53, No. 2, May 2007. 3. Proposed class D audio amplifier: PWM generation Traditional architecture

• Ideally fixed frequency • Jitter ((gdegrades linearit y) • Non-linear circuit • Carrier generator adds complexity • Non-ideal triangle wave

Proposed architecture

Controller Comparator Power Stage Output Filter • No dedicated triangle wave

VA Sliding U VIN iL Mode VOUT • Linearity no compromised VOUT L Controller VDD vc C R • Good transient response MDP • Variable frequency (450KHz-600KHz) MDN VSS 3. Proposed class D audio amplifier

Error function

V1 = e1 = ()VA −VOUT

Switching function

V3 = s(e1,e2 )= e1 + αe2 = e1 + αe&1 = (1+ αs)e1(s)

Phase portraits in a class D audio amplifier Class D Amplifier with sliding surface 3. Proposed class D audio amplifier

Error function

V1 = e1 = (VA −VOUT )

Switching function

V3 = s(e1 )= e1 + αe&1 = (1+ αs)e1(s)

IC Microphotograph 3. Proposed class D audio amplifier

Efficiency vs. input signal

Sliding mode phases A – Initial condition B – RhidReaching mode C – Sliding surface D – Sliding equilibrium point

Output spectrum (300 mVpp) 3. Proposed class D audio amplifier

THD versus audio frequency input THD versus audio voltage input

SNR versus audio frequency input PSRR versus audio frequency input 3. Proposed class D audio amplifier

Design THD η Supply Load I0 [1] 0.28% 92% 2.5 V 8 Ω 25.2 μA [2] 0.11% 70% 5.0 V 8 Ω - [3] 0.03% 76% 4.2 V 8 Ω 4.7 mA [4] 0.20% 90% 5.0 V 4 Ω - [5]* 0.08% 85% 5.0 V 4 Ω 8.0 mA [6]* 0.40% 87% 2.7 V 4 Ω 2.8 mA [7] 0.04% 79% 3.6 V 8 Ω 2.5 mA [8] 0.10% 92% 12 V 8 Ω - This work 0.08% 91% 2.7 V 8 Ω 2.0 mA

[1] S. C. Li, V. C. Lin, K. Nandhasri and J. Ngarmnil, “New high-efficiency 2.5V/0.45W RWDM class D audio amplifier for portable consumer electronics”, IEEE Trans. on Circuits and Systems I, Vol. 52, No. 9, pp. 1767-1774, September 2005. [2] K. Philips, J. Van Der Homber and C. Dijkmas, “Power DAC: a single-chip audio DAC with 70% efficient power stage in 0.5um CMOS”, IEEE International Solid-State Circuits Conference, pp. 154-155, February 1999. [3] B. Fore jt, V. Ren ta la, J. D. Ar teaga an d G. Burra, “A 700+-mW cl ass D des ign w ith direct b att ery hoo kup in a 90nm process ”, IEEE Journa l o f So lid-Stat e Ci rcuits it , Vol. 40, No. 9, pp. 1880-1887, September 2005. [4] J. Lee, J. Lee, G. Lee and S. Kim, “A 2W BTL single-chip class D power amplifier with very high efficiency for audio applications”, IEEE International Symposium on Circuits and Systems, Vol. 5, pp. 493-496, May 2000. [5] TPA2000D2 2W Filterless Stereo class D Audio Power Amplifier Datasheet, Texas Instruments Inc., Publication Number SLOS291E, May 2003. [6] MAX4295 Mono, 2W Switch-Mode (class D) Audio Power Amplifier Datasheet, Maxim Integrated Products Inc., January 2001. [7] P. Muggler, W. Chen, C. Jones, P. Dagli and N. Yazdi, “A filter free class D audio amplifier with 86% power efficiency”, Proceedings of the 2004 International Symposium on Circuits and Systems, Vol. 1, pp. I-1036-1039, May 2004. [8] S. Choi, J. Lee, W. Jin and J. So, “A design of a 10W single-chip class D audio amplifier with very high efficiency using CMOS technology”, IEEE Trans. on Consumer Electronics, Vol. 45, No. 3, pp. 465-473, August 1999. 4. Class D audio amplifiers: two design approaches Motivation Single-ended class D amplifier • Single ended architecture generates even-order distortion tones which degrades linearity • Single ended version generates “quasi” differential output by adding an extra inverter (causes delay and distortion)

Advantages of fully-differential version • Even-order cancellation enhances linearity • No delay in signal paths Fully-differential class D amplifier

Improvements from single-ended version • Reduction of building blocks (operations are done in a single OPAMP) • Compara tor d es ign is done w ith in terna l pos itive feedback instead of poly-resistors 4. Class D audio amplifiers: two design approaches

Output waveforms

Output spectrum 4. Class D audio amplifiers: two design approaches Motivation • MltilMultilevel convert ers present tbttli better linear ity as num ber o f level increases • High frequency components are pushed to higher frequencies (for three level modulation, carrier fs is pushed to 2xfs) • Possibility of adding an extra-level by using H-bridge already present in class D output stage

Advantages of three-level architecture • Multi-level modulation = linearity improvement • No additional hardware cost Characteristics • Two identical switching surfaces are created (Two binary comparators are used) • Each switching surface is fed by the audio signal shifted 180 degrees from each other • Each output stage operates at two different levels • Differential output becomes multi-level!!! 4. Class D audio amplifiers: two design approaches

Output waveform

Three-level PWM

Output spectrum 4. Class D audio amplifiers: two design

Class D Amplifiers SNR approaches 80 75 Class D Amplifiers Efficiency 90 70 80 65 60 70 SNR (dB) 55

60 50

50 45 Class D Fully Differential Class D Multilevel

40 2 3 4 40 10 10 10 fficiency (%) Frequency (Hz) EE 30 Class D Amplifiers PSRR 80 20 75 10 Class D Fully Differential Class D Multilevel 70 0 0 0.05 0.1 0.15 0.2 0.25 Output Power (W) 65 PSRR (dB) Class D fully-differential and multilevel have 60

55 similar efficiencies and linearity but multilevel Class D Fully Differential modulation gives better SNR and PSRR due to Class D Multilevel 50 2 3 4 the extra level of quantization. 10 10 10 Frequency (Hz) η FM = Table of comparison I 0 ×THD ×100e3 SNR PSRR f P Design THD η Supply Load I0 s O, max Area Process FM 81dB - 1.5MHz 381 mW 0.35um [3] 0.20% - 3.0V 8Ω - 1.20 mm2 - CMOS 80dB 85dB 200KHz 330mW 0.50um [4] 0.07% 92% 2.5V 8Ω 25.2uA 0.60 mm2 5 CMOS [5]0.50%85%---85dB40dB------90dB 1.0MHz 250mW 0.50um [6] 0.11% 70% 5.0V 8Ω - 12.5 mm2 - CMOS 98dB 70dB 410KHz 700mW 90nm [7] 0.03% 76% 4.2V 8Ω 4.7mA 0.44 mm2 6 DCMOS - - 450KHz 1250mW 0.65um [8] 0.20% 90% 5.0V 4Ω - 12.3 mm2 - CMOS [9]* 0.08% 70% 5.0V 4Ω 4.0mA 87dB 77dB 250KHz 1000mW --2 [10]* 0.40% 87% 2.7V 4Ω 2.8mA - - 125KHz 700mW --1 - 84dB 250KHz 500mW 12um1.2um [11] 0.04% 79% 3.6V 8Ω 2.5mA 2.25 mm2 8 BiCMOS - - 180KHz 10.0W 4.00um [12] 0.10% 92% 12.0V 8Ω - 25.0 mm2 - CMOS 95dB - 700KHz 400mW 0.35um [13] 0.19% - 3.0V 8Ω - 2.25 mm2 - CMOS - - 20MHz - 0.35um [14] 0.04% 80% 3.3V 4Ω - - - CMOS 65dB 70dB 500KHz 200mW 0.50um SE 0.08% 91% 2.7V 8Ω 2.0mA 4.70 mm2 6 CMOS 75dB 62dB 450KHz 250mW 0.50um FD 0.04% 89% 2.7V 8Ω 1.3mA 1.88 mm2 17 CMOS 78dB 75dB 450KHz 250mW 0.50um ML 0.10% 85% 2.7V 8Ω 836uA 2.48 mm2 10 CMOS

* Commercial product. Thank you References

[3] A. Yasuda, T. Kimura, K. Ochiai and T. Hamasaki, “A class D amplifier using a spectrum shaping technique”, Proceedings of the IEEE 2004 Custom Integrated Circuits Conference, pp. 173-176, October 2004. [4] S. C. Li, V. C. Lin, K. Nandhasri and J. Ngarmnil, “New high-efficiency 2.5V/0.45W RWDM class D audio amplifier for portable consumer electronics”, IEEE Trans. on Circuits and Systems I, Vol. 52, No. 9, pp. 1767-1774, September 2005. [5] M. Score and D. Dapkus, “Optimized modulation scheme eliminates output filter”, Proceedings of the 109th AES Convention , pp. 22-25, September 2000 . [6] K. Philips, J. Van Der Homber and C. Dijkmas, “Power DAC: a single-chip audio DAC with 70% efficient power stage in 0.5um CMOS”, IEEE International Solid-State Circuits Conference, pp. 154-155, February 1999. [7] B. Forejt, V. Rentala, J. D. Arteaga and G. Burra, “A 700+-mW class D design with direct battery hookup in a 90nm process”, IEEE Journal of Solid-State Circuits, Vol. 40, No. 9, pp. 1880-1887, September 2005. [8] J. Lee, J. Lee, G. Lee an d S. Kim, “A 2W BTL s ing le-chip cl ass D power amplifi er with very hig h e ffic iency for audio applications”, IEEE International Symposium on Circuits and Systems, Vol. 5, pp. 493-496, May 2000. [9] TPA2000D2 2W Filterless Stereo class D Audio Power Amplifier Datasheet, Texas Instruments Inc., Publication Number SLOS291E, May 2003. [[]10] MAX4295 Mono, 2W Switch-Mode (()class D) Audio Power Am plifier Datasheet , Maxim Inte grated Products Inc., January 2001. [11] P. Muggler, W. Chen, C. Jones, P. Dagli and N. Yazdi, “A filter free class D audio amplifier with 86% power efficiency”, Proceedings of the 2004 International Symposium on Circuits and Systems, Vol. 1, pp. I-1036-1039, May 2004. [12] S . Choi , J. Lee, W. Jin and J. So, “A design of a 10W single-chip class D audio amplifier with very high efficiency using CMOS technology”, IEEE Trans. on Consumer Electronics, Vol. 45, No. 3, pp. 465-473, August 1999.