Nonlinear Control Techniques for Low-Voltage, Low-Power Applications:

Class D Audio Amplifiers

Miguel Angel Rojas-Gonzlez and Edgar Snchez-Sinencio (sanchez@ece.tamu.edu)

Analog and Mixed Signal Center Texas A&M University

Outline
1. Introduction to audio amplifiers
a) b) a) b) a) b) c) a) b) Audio A dio amplifiers applications Linear vs. nonlinear audio amplifiers Typical architecture Parameters affecting amplifier performance Architecture description Design procedure Experimental results Architecture descriptions Preliminary results

2. Conventional class D audio amplifier p

3. 3 Proposed class D audio amplifier (single ended) (single-ended)

4. Class D audio amplifiers: two design approaches

1. Introduction to audio amplifiers

In a sound system, the power amplifier y , p p supplies power to the loudspeaker
The typical speaker input impedance is low, usually in the 4 to 8 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 by humans is ~ 0.3 % in average y g Telephone: THD ~ 10 % BW ~ 4 KHz

Main power amplifier classes

Linear
Class A, B, AB

Nonlinear
Class D

Linearity Efficiency

Power amplifier classes

Class A amplifier
Current flows continuously y High sound quality Poor efficiency (25 %)

Class D amplifier
Switching amplifier g p Ideal THD 0% Highest efficiency (100 %)
90 80 70

Class B amplifier
Current flows half of the period Linearity compromised by crossover Higher efficiency (78.5 %)
Effic ciency (%)

Power Efficiency of Class A, Class B and Class D Amplifiers

60 50 40 30 20 10 0 0 0.1 Class A Ideal Class B Ideal Class M Cl D Measured d 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Class AB amplifier
Hybrid between classes A and B sound quality Good sou d qua ty Higher efficiency (78.5 %)

Normalized Load Power (PL/PLmax)

2. Conventional class D audio amplifier

Continuously switch the output from one rail to another at supersonic frequency (Pulse Width Modulation -PWM-) 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 From 80 W to 1400 W Home theatre systems, car audio systems, etc. Keeps dissipation and heat sink size minimum

2. 2 High Power Outputs

2. Conventional class D audio amplifier

History
First proposed around 1950s 1950 s 5 % THD in 1976 state of the art amplifier

Efficiency y
100 % at all output levels ideally Between 80% and 90% due to parasitic losses in practice

Basic Principles
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 PWM is generated comparing the audio signal with a triangle wave carrier signal Triangle wave needs to be linear to prevent distortion
Carrier Signal Audio Signal

Power Stage L

PWM C Speaker Power Stage

Comparator

2. Conventional class D audio amplifier

Pulse Width Modulation (PWM)
Varies the duty cycle of the converter switches at a high switching frequency y y g g q y (supersonic) to achieve a target average low frequency output voltage.
Class D Audio Amplifier (THD = 0.00%)
0

fin = 1.0e+003 Hz

M Magnitude (dB)

-20 -40 -60 -80

Class D amplifier THD is calculated considering all harmonics below 20KHz (max. g ( audible frequency) Ideal class D amplifier THD is 0.0 % g y Performance is degraded due to system nonlinearities

10

20

30

40

50

60
Power Stage L

Frequency (KHz)

Nonlinearity sources
Passive components nonlinearities
MOS on-resistance Non-ideal triangle carrier signal
Carrier Signal Comparator Power Stage PWM Audio Signal C Speaker

2. Conventional class D audio amplifier

Passive components nonlinearities ()
The low saturation current of the load inductor causes frequency dependent nonnon linearity.
Saturation current of inductor is defined as the IDC current level where the effective inductance value is decreased to 90% of its value at zero DC current. An approximate expression of nonlinear inductance is obtained by series representation
Audio Signal Power Stage L

PWM C Speaker Power Stage

THD (% ) = 100

Vthird harmonic V fundamental

Carrier Signal

Comparator

3 L V 2 0.1 2 f in THD (% ) = 100 2 4 R 3 I sat

where L, V, fin, R and Isat are the inductance at zero DC current, the output voltage, the input signal frequency the speaker resistance and frequency, 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 b lifi because it output PWM i switch mode. its t t is it h d
The on-resistance is not negligible (~200m) and it will result in a slightly amplitude modulated signal at the PWM output. This modulation becomes more pronounced at higher modulation indexes.

Power Stage L

PWM Audio Signal A di Si l C Speaker Power Stage

Carrier Signal

Comparator

where fc, fs, M and Wp are the carrier signal frequency, the input signal frequency, the modulation index and the output transistor size respectively 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 non idealities audio amplifier.
There are three main carrier signal modulation schemes: Triangle wave modulation Sawtooth wave modulation Exponential wave modulation
Carrier Signal Comparator Power Stage PWM Audio Signal C Speaker Power Stage L

-T/2

T/2

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 miscalculations y p 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 proposed in () y p p Assume a exponential carrier signal instead of a triangle wave which may be generated by charging/discharging an RC integrator circuit with square pulses. y p p g g 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
() 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
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 mt + bm sin mt ) i 2 m =1

where

am = bm =

f (t )cos mt d (t )

i f (t )sin mt d (t )

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 ) = c t

y (t ) = ot

where c and o are the carrier angular frequency and the input signal (audio wave) angular frequency respectively. f ti l 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 () 1 Mcosy = M M Mcosot The purpose of the Double Fourier Integral Analysis is to express the PWM waveform as a function of a double variable controlled waveform.

x = ct

-1
() H. S. Black, Modulation Theory, Princeton, NJ, Van Nostrand, 1953

2. Conventional class 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 ()
A00 f (t ) = + ( A0 n cos(no t ) + B0 n sin (no t )) + ( Am 0 cos(mc t ) + Bm 0 sin (m0t )) 2 n =1 m =1

m =1 n = (n 0 )

(A

mn

cos(mc t + not ) + Bmn sin (mc t + no t )) i

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

A00 2

where m = n = 0, corresponds to the DC offset component of the PWM (if any)

(A
n =1

0n

cos(no t ) + B0 n sin (not )) i cos(mc t ) + Bm 0 sin (m0t )) i

where m = 0 represents th f d h 0, t the fundamental component t l t and baseband harmonics (if any) where n = 0 d fi h 0, defines th carrier wave h the i harmonics (hi h i (highfrequency components) where m, n 0 represents th h 0, t the sideband harmonics

(A
m =1

m0

(A
m =1 n = (n 0 )

mn

cos(mc t + no t ) + Bmn sin (mc t + not )) i

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
1 Mcosy = Mcosot
Mag gnitude (dB) Harmonic components for sawtooth carrier when M = 0.9 and c/o = 21
0 -20 -40 -60 -80

-T/2

T/2

x = ct

-1

10

20

30

40

50

60

Harmonic Number

The PWM function van(t) for a sawtooth carrier signal expressed in terms of its harmonics components is

van (t ) = VDC + VDC M cos(ot ) + 2 C C +2 VDC

VDC

1 [cos m J 0 (mM )]sin mct m =1 m

1 J n (mM )sin n cos(mc t + no t ) cos n sin (mc t + not ) 1 n m 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
1 Mcosy = Mcosot
Mag gnitude (dB) Harmonic components for triangle carrier when M = 0.9 and c/o = 21
0 -20 -40 -60 -80

-T/2

T/2

x = ct

-1

10

20

30

40

50

60

Harmonic Number

The PWM function van(t) for a triangle carrier signal expressed in terms of its harmonics components is

van (t ) = VDC + VDC M cos(o t ) + 4 +4 VDC

VDC

1 J 0 m M sin m cos mc t m =1 m 2 2

1 J n m M sin [m + n] cos(mc t + no t ) m =1 n = 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 0.0% 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

f (x ) =

n =1, 3, 5,...

( 1)

( n 1)

n2

2nx sin T
Ideal Triangle Wave Spectrum
0

And, for an ideal sawtooth wave carrier f(y) we would have

f (y) =

1 1 1 2nx sin 2 n =1 n T

Magnitude (dB) (

We would need an infinite bandwidth system to generate a perfect carrier signal! Unfortunately, band-limited systems degrade the performance of the overall class D amplifier generating undesired baseband components

-20 -40 -60 -80

10

20

30

40

50

Harmonic Number

2. Conventional class D audio amplifier

Carrier signal nonlinearity
Double Fourier Integral Analysis is a complex and tedious mathematical derivation g y p 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. c >> o, which is usually the case.

Example 1a: S Sine-Sawtooth Modulation S

Normalizing the period of the sawtooth to 2 and its amplitude to 1, we have
1 Mcosy = Mcosot

x = ct

a0 van (t ) = + (am cos mx + bm sin mx ) 2 m =1 1 am = van (t ) cos mx dx

bm =

v (t )sin mx dx
an

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

2. Conventional class D audio amplifier

Example 1a: Sine-Sawtooth Modulation (cont.)
For am coefficients, when m 0
M cos y

am =

van (t )cos mx dx = 2

VDC

cos mx dx = 2

VDC

[sin (mM cos y ) + sin m ]

For bm coefficients when m 0 coefficients,

bm =

van (t )sin mx dx = 2

VDC

M cos y

sin mx dx = 2

VDC

[cos m cos(mM cos y )]

a0 = 2VDC (1+ M cos y )

When m= 0 we have

b0 = 0
VDC 1 [cos m J 0 (mM )]sin mct m =1 m

After some mathematical manipulation and applying the Jacobi-Anger expansions, we get the same expression obtained by using the Double Fourier Integral Analysis

van (t ) = VDC + VDC M cos(ot ) + 2 +2 VDC

1 J n (mM )sin n cos(mc t + not ) cos n sin (mc t + not ) 1 n m m = = 2 2

(n 0 )

2. Conventional class D audio amplifier

Example 2a: Sine-Triangle Modulation
Normalizing the period of the sawtooth to 2 and its amplitude to 1, we have
1 Mcosy = Mcosot

x = ct

a0 van (t ) = + (am cos mx + bm sin mx ) 2 m =1 1 am = van (t ) cos mx dx

bm =

i dx v (t )sin mx d
an

We need to calculate the interval where van(t) switches from 0 to 1 For am coefficients, when m 0

am =

dx van (t )cos mx d = 2

VDC

(1+ M cos y )

(1+ M cos y )

dx cos mx d = 4

VDC i sin m (1 + M cos y ) 2

2. Conventional class D audio amplifier

Example 2a: Sine-Triangle Modulation
For bm coefficients, since the triangle wave is an even function g

bm =

van (t )sin mx dx = 2

VDC

(1+ M cos y )

(1+ M cos y )

sin mx dx = 0

When m= 0 we have

a0 = 2VDC (1+ M cos y )

After some mathematical manipulation, as well as in the previous example, and applying the Jacobi-Anger expansions, we get the same expression obtained by using the Double Fourier Integral Analysis

van (t ) = VDC + VDC M cos(ot ) + 4 +4 VDC

VDC

1 J 0 m M sin m cos mc t m =1 m 2 2

1 J n m M sin [m + n] cos(mc t + not ) 2 m =1 n = m 2

(n 0 )

2. Conventional class D audio amplifier

Carrier signal nonlinearity
We propose to generalize the PWM Analysis by Duty Cycle variation and apply it to a any p p g y y y y pp y y 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
1.5 1 -1.05 0.5

Triangle Wave Carrier

-1

Triangle Wave Carrier (Zoom-in)

Voltage (V)

-1.1 0 -0.5 -1 -1.5

Voltage (V) )

-1.15 -1.2 -1.25 -1.3 n=1 n=5 n = 15 n = 25 n= 5 5.5 x 10

-6

n=1 n=5 n = 15 n = 25 n= 0 0.2 0.4 0.6 0.8 x 10 1

-5

Time (sec)

-1.35 4.5

Time (sec)

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

1 Mcosy = Mcosot

x = ct

a0 van (t ) = + (am cos mx + bm sin mx ) 2 m =1 1 am = van (t ) cos mx dx

bm =

v (t )sin mx dx
an

2. Conventional class D audio amplifier

Example 3: Sine-Cosine Modulation (cont.)
For am coefficients, when m 0

2 VDC m arccos am = van (t ) cos mx dx = 2 2cos mx dx = 4 sin 8 M cos y arccos M cos y 1

VDC

2 arccos M cos y 8

For bm coefficients, since the cosine is an even function

bm =

van (t )sin mx dx = 2

VDC

2 arccos M cos y 8

2 arccos M cos y 8

sin mx dx = 0

When m= 0 we have

2 a0 = 4 arccos M cos y 8 VDC

2. Conventional class D audio amplifier

Example 3: Sine-Cosine Modulation (cont.)
Finally, the function van(t) will be given by y () g y
2 van (t ) = 2 arccos arcsin 2 sin k J k 8 M cos(ky ) k =1 2 2 VDC m arccos + 4 sin 8 M cos y cos mx m =1

VDC

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.)
In order to verify the mathematical derivation and its results, we have created a simple y p 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
14 12 10

Class D Amplifier THD (n = 1)

Mathematical model Simulink simulation

THD (%)

8 6 4 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 2.5

Class D Amplifier THD (n = 5)

Modulation index

Non-ideal triangle wave carrier signal with n = 1

THD (%) )

2 1.5 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 Mathematical model Simulink simulation 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)
1 0.8 Mathematical model Simulink simulation
Class D Amplifier THD (n = 25)

0.7 0.6 0.5

THD (%)

Mathematical model Simulink simulation

THD (%)

0.6

0.4 0.3 0.2 0.1

0.4 0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Modulation index

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Modulation index

Non-ideal Non ideal triangle wave carrier signal with n = 15

0.04 0.03

Non-ideal Non ideal triangle wave carrier signal with n = 25

Class D Amplifier THD for M = 0.8
10 Mathematical model Simulink simulation 1

Class D Amplifier THD (n = 99)

Mathematical model Simulink simulation

THD (%) )

THD (%)

0.02 0.01

0.1

0 0.1

0.01

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9
0.001

Modulation index

Non-ideal triangle wave carrier signal with n = 99

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)
0

3 2.5 Vol ltage (V) 2 1.5 1 0.5 0 -0.5 -0.5

Triangle Wave Carrier ( = 3): Mathematical Model g (n )

M Magnitude (dB)

-20 -40 -60 60 -80

10

20

30

40

50

60

Frequency (Hz)

Class D output spectrum when n = 3 (Mathematical M d l) (M th ti l Model)

Class D Audio Amplifier (THD = 3.60424% (-28.864dB) Triangle Wave)
0

0 Time (sec)

0.5

fin = 1.0e+000 Hz

Magnitude (dB) (

-20 -40 -60 -80

HD1 = -3.3 dB HD2 = -138.9 dB HD3 = -32.4 dB 32 4 HD4 = -144.4 dB HD5 = -45.4 dB

The mathematical model predicts with high accuracy the result in the SIMULINK simulation! Exercise: Can you provide an analytical expression for the THD output spectrum when a sawtooth carrier signal with only one harmonic is used to PWM an audio signal?

10

20

30

40

50

60

Frequency (Hz)

Class D output spectrum when n = 3 (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 y p g An exponential waveform is usually employed as a carrier signal due to its simple implementation

Example 4: Sine-Exponential Modulation p p

The exponential waveform is generated by charging/discharging a simple RC integrator with square pulses. Defining a set of normalized exponential waves
x + T t0 2 error V 2 e 1, DC 1 error f (x ) = x 1 e t 0 1, VDC 2 1 error T < x < 0 2 T 0< x< 2
1.5 1 0.5

Exponential Carrier Signals

A B C D E

Voltage (V)

0 -0.5 -1 -1.5 0.5

1.5 x 10
-5

Time (sec)

2. Conventional class D audio amplifier

Example 4: Sine-Exponential Modulation (cont.)
Normalizing the exponential carrier to a period of 2 we can calculate the Fourier coefficients based on the PWM Analysis by Duty Cycle Variation
Mcosy = Mcosot

a0 van (t ) = + (am cos mx + bm sin mx ) 2 m =1

x = ct

am = bm =

v (t )cos mx dx
an an

v (t )sin mx dx

2. Conventional class D audio amplifier

Example 4: Sine-Exponential Modulation (cont.)
For am coefficients, when m 0
am = 2 VDC
1 M t 0 ln 1 (1 error ) V cos y +1 2 DC M (1+ error ) e o t 0 ln 2V cos y (1 error )+ 2 DC

cos mx dx

VDC mt0 ln1 1 (1 error ) M cos y + 1 sin m t0 ln M cos y (1 error ) + (1 + error ) am = 2 sin 2V V 2 m 2 DC DC

For bm coefficients
VDC
1 M t 0 ln 1 (1 error ) V cos y +1 2 DC M (1+ error ) t 0 ln 2V cos y (1 error )+ 2 DC

bm = 2

sin mx dx

VDC mt0 ln1 1 (1 error ) M cos y + 1 cos m t0 ln M cos y (1 error ) + (1 + error ) bm = 2 cos V 2V 2 m 2 DC DC

2. Conventional class D audio amplifier

Example 4: Sine-Exponential Modulation (cont.)
For am coefficients, when m 0
1 M t 0 ln 1 (1 error ) V cos y +1 2 DC M (1+ error ) t0 ln 2V cos y (1 error )+ 2 DC

a0 = 2

VDC

dx

a0 = 2

1 M M (1 + error ) VDC cos y + 1 + + t0 ln cos y (1 error ) + t0 ln1 (1 error ) V 2V 2 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.)
1.5 1 0.5

Exponential Carrier Signals

A B C D E

10

Class D Amplifier THD (Exponential waves 'B' and 'A')

Voltag (V) ge

0 -0.5 -1 -1.5 0.5

THD (% %)

0.1

0.01 0.1
1 1.5 x 10
-5

Mathematical model (wave 'B') Mathematical model (wave 'A') Simulink simulation (wave 'B') Simulink simulation (wave 'A') 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Modulation index Class D Amplifier THD (Exponential waves 'D' and 'C')
Mathematical model (wave 'D') Mathematical model (wave 'C') Simulink simulation (wave 'D') Simulink simulation (wave 'C')

Time (sec)
0.02 0.015

Class D Amplifier THD (Exponential wave 'E')

0.8 0.6

THD (%)

0.01 0.005 Mathematical model Simulink simulation 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

THD (%)

0.4 0.2

0 0.1

0 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
0

3 2.5 V Voltage (V) 2 1.5 1 0.5 0

Exponential Carrier: Mathematical Model

Magnitude (dB)

-20 -40 -60 -80

10

20

30

40

50

60

Frequency (Hz)

-0.5 -0.5

0 Time (sec)

0.5

Class D output spectrum ( th Cl t t t (mathematical model) ti l d l)

Class D Audio Amplifier (THD = 0.23494% (-52.5dB) Exponential Wave)
0

fin = 1.0e+000 Hz

Magnitude (dB B)

-20 -40 -60 -80

HD1 = -3.7 dB HD2 = -144.8 dB 144.8 HD3 = -56.3 dB HD4 = -146.4 dB HD5 = -103.7 dB

The mathematical model predicts with high accuracy the result in the SIMULINK simulation! This analysis method can be further applied to any carrier wave or even multiphase systems where multilevel PWM is generated.

10

20

30

40

50

60

Frequency (Hz)

Class D output spectrum (SIMULINK simulation)

3. Proposed class D audio amplifier () (single ended architecture)

Controller VA VOUT Comparator Power Stage Output Filter Sliding Mode Controller

U
VDD MDP MDN VSS

VIN L

iL vc

VOUT C R

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 ( g y) Jitter (degrades linearity) Non-linear circuit Carrier generator adds complexity Non ideal triangle wave Non-ideal

Proposed architecture
No dedicated triangle wave
VA Controller Comparator Power Stage Output Filter

Linearity no compromised Good transient response Variable frequency (450KHz-600KHz)

VOUT

Sliding Mode Controller

U
VDD MDP MDN VSS

VIN L

iL vc

VOUT C R

3. Proposed class D audio amplifier

V1 = e1 = (V A VOUT )

Error function

& V3 = s(e1 , e2 ) = e1 + e2 = e1 + e1 = (1 + s )e1 (s )

Switching function

Phase portraits in a class D audio amplifier

Class D Amplifier with sliding surface

3. Proposed class D audio amplifier

V1 = e1 = (V A VOUT )
Error function

& V3 = s(e1 ) = e1 + e1 = (1+ s )e1 (s )

Switching function

IC Microphotograph

3. Proposed class D audio amplifier

Efficiency vs. input signal

Sliding mode phases A Initial condition BR Reaching mode hi d 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 [1] [2] [3] [4] [5]* [6]* [7] [8] This work THD 0.28% 0.11% 0.03% 0.20% 0.08% 0.40% 0.04% 0.10% 0.08% 92% 70% 76% 90% 85% 87% 79% 92% 91% Supply 2.5 V 5.0 V 4.2 V 5.0 V 5.0 V 2.7 V 3.6 V 12 V 2.7 V Load 8 8 8 4 4 4 8 8 8 I0 25.2 A 4.7 mA 8.0 mA 2.8 mA 2.5 mA 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 F jt V R t l J D A t B. Forejt, V. Rentala, J. D. Arteaga and G B d G. Burra, A 700+ W class D d i with di t battery h k i a 90 700+-mW l design ith direct b tt hookup in 90nm process, IEEE J Journal of S lid St t Circuits, l f Solid-State Ci 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 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)

Single-ended class D amplifier g p

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) C Comparator design i d t d i is done with i t ith internal positive l iti 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
M ltil Multilevel converters present better linearity as number of l t t b tt li it b 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 approaches

80 75 70 SNR (dB) 65 60 55 50 45 40 10
2

Class D Amplifiers SNR

90 80 70 E Efficiency (%) 60 50 40 30 20 10 0 0

Class D Amplifiers Efficiency

Class D Fully Differential Class D Multilevel 10 Frequency (Hz) Class D Amplifiers PSRR
3

10

80

Class D Fully Differential Class D Multilevel 0.05 0.1 0.15 Output Power (W) 0.2 0.25
PSRR (dB)

75 70 65 60 55 50 Class D Fully Differential Class D Multilevel 10

2

Class D fully-differential and multilevel have similar efficiencies and linearity but multilevel modulation gives better SNR and PSRR due to the extra level of quantization.

10 Frequency (Hz)

10

Table of comparison
Design [3] [ ] [4] [5] [6] [7] [8] [9]* [10]* [11] [12] [13] [14] SE FD ML THD 0.20% 0.07% 0.50% 0.11% 0.03% 0.20% 0.08% 0.40% 0.04% 0.10% 0.19% 0.04% 0.08% 0.04% 0.10% 92% 85% 70% 76% 90% 70% 87% 79% 92% 80% 91% 89% 85% Supply 3.0V 2.5V 5.0V 4.2V 5.0V 5.0V 2.7V 3.6V 12.0V 3.0V 3.3V 2.7V 2.7V 2.7V Load 8 8 8 8 4 4 4 8 8 8 4 8 8 8 I0 25.2uA 4.7mA 4.0mA 2.8mA 2.5mA 2.0mA 1.3mA 836uA SNR 81dB 80dB 85dB 98dB 87dB 95dB 65dB 75dB 78dB PSRR fs 1.5MHz PO, max 381 mW 330mW 250mW 700mW 1250mW 1000mW 700mW 500mW 10.0W 400mW 200mW 250mW 250mW 85dB 200KHz 40dB 90dB 70dB 77dB 84dB 70dB 62dB 75dB 1.0MHz 410KHz 450KHz 250KHz 125KHz 250KHz 180KHz 700KHz 20MHz 500KHz 450KHz 450KHz

FM =
Area 1.20 mm2 0.60 mm2 12.5 mm2 0.44 mm2 12.3 mm2 2.25 mm2 25.0 mm2 2.25 mm2 4.70 mm2 1.88 mm2 2.48 mm2

I 0 THD 100e3 100e

Process 0.35um CMOS 0.50um CMOS 0.50um CMOS 90nm DCMOS 0.65um CMOS 1.2um 1 2um BiCMOS 4.00um CMOS 0.35um CMOS 0.35um CMOS 0.50um CMOS 0.50um CMOS 0.50um CMOS FM 5 6 2 1 8 6 17 10

* Commercial product.

References

[1] J. Torres, A. Colli-Menchi, M.A. Rojas-Gonzlez and E. Snchez-Sinencio, A Low-Power High-PSRR Clock Free Current-Controlled Class-D Audio Amplifier, IEEE J. Solid-State g S C oc ee Cu e t Co t o ed C ass ud o p e , J So d State Circuits, July 2011.

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. Convention pp. 22 25, 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 L J. Lee, J L J. Lee, G L and S Ki A 2W BTL single-chip class D power amplifier with very hi h efficiency f G. Lee d S. Kim, i l hi l lifi ith high ffi i 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 Amplifier Datasheet, Maxim Integrated Products , ( ) p , g 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 S Choi J. Lee, W. J. So, A single-chip efficiency using CMOS technology, IEEE Trans. on Consumer Electronics, Vol. 45, No. 3, pp. 465-473, August 1999.