Mathematical models for class-d amplifiers

Mathematical models for class-d amplifiers Stephen Cox School of Mathematical Sciences, University of Nottingham, UK 12 November 2012 Stephen Cox Mathematical models for class-d amplifiers 1/38

Background Stephen Cox Mathematical models for class-d amplifiers 2/38

Background Stephen Cox Mathematical models for class-d amplifiers 3/38

Background Stephen Cox Mathematical models for class-d amplifiers 4/38

Background Halcro manufacture reference quality, super-fidelity amplifiers La Revue du Son et du Home Cinéma Stephen Cox Mathematical models for class-d amplifiers 5/38

Background Halcro manufacture reference quality, super-fidelity amplifiers La Revue du Son et du Home Cinéma Their amplifiers are class D Class-D amplifiers are highly efficient, and (non-halcro examples) are used in mobile phones, laptops, hearing aids, In practical designs, there is negative feedback, but this introduces distortion We will quantify this distortion through a mathematical model for the amplifier design(s), and seek ways to eliminate it Bruce Candy, Tan Meng Tong (NTU, Singapore), Stephen Creagh Stephen Cox Mathematical models for class-d amplifiers 5/38

The ideal class-d amplifier Stephen Cox Mathematical models for class-d amplifiers 6/38

The ideal Class-D amplifier The ideal class-d amplifier s(t) + g(t) v(t) Audio signal s(t) compared with high-frequency triangular wave v(t) to generate high-frequency pulse-width-modulated (PWM) square wave g(t) +1 1 +1 1 v(t) g(t) s(t) n 1 n n+1 t The ideal class-d amplifier is distortion-free unfortunately negative feedback is desirable in practice, and this introduces distortion Stephen Cox Mathematical models for class-d amplifiers 7/38

A first-order class-d amplifier with negative feedback Stephen Cox Mathematical models for class-d amplifiers 8/38

Class-D amplifier with ve feedback s(t) h(t) + c dt + 0 v(t) + g(t) In the mathematical model: we scale voltages so that the supply voltage is ±1 the high-frequency carrier wave period is T Stephen Cox Mathematical models for class-d amplifiers 9/38

Negative-feedback model s(t) h(t) + c dt + 0 + g(t) v(t) Stephen Cox Mathematical models for class-d amplifiers 10/38

Negative-feedback model s(t) h(t) + c dt + 0 + g(t) v(t) carrier wave: v(t+t)=v(t) { 1 4t/T 0 t < 1 v(t) = 2 T 3 + 4t/T 1 2 T t < T Stephen Cox Mathematical models for class-d amplifiers 10/38

Negative-feedback model s(t) h(t) + c dt + 0 + g(t) v(t) carrier wave: v(t+t)=v(t) integrator output { 1 4t/T 0 t < 1 v(t) = 2 T 3 + 4t/T 1 2 T t < T dh dt = c(g(t) + s(t)) Stephen Cox Mathematical models for class-d amplifiers 10/38

Negative-feedback model s(t) h(t) + c dt + 0 + g(t) v(t) carrier wave: v(t+t)=v(t) integrator output { 1 4t/T 0 t < 1 v(t) = 2 T 3 + 4t/T 1 2 T t < T comparator output { 1 if h(t) + v(t) > 0 g(t) = 1 if h(t) + v(t) < 0 dh dt = c(g(t) + s(t)) Stephen Cox Mathematical models for class-d amplifiers 10/38

Negative-feedback model s(t) h(t) + c dt + 0 + g(t) v(t) carrier wave: v(t+t)=v(t) integrator output { 1 4t/T 0 t < 1 v(t) = 2 T 3 + 4t/T 1 2 T t < T dh dt = c(g(t) + s(t)) comparator output { 1 if h(t) + v(t) > 0 g(t) = 1 if h(t) + v(t) < 0 Aim: to find the Fourier spectrum of the output g(t), in particular its lowfrequency (audio) part Stephen Cox Mathematical models for class-d amplifiers 10/38

Negative-feedback model s(t) h(t) + c dt + 0 + g(t) v(t) +1 v(t) 1 h(t) +1 g(t) 1 (n 1)T nt (n+1)t t Stephen Cox Mathematical models for class-d amplifiers 11/38

Solving the negative-feedback model Aim: difference equations, to allow iteration from one carrier-wave period to the next +1 1 g(t) β n α n nt ( )T n+ α n n+ β n =A n =Bn ( )T (n+1)t t Stephen Cox Mathematical models for class-d amplifiers 12/38

Solving the negative-feedback model Aim: difference equations, to allow iteration from one carrier-wave period to the next +1 1 g(t) β n α n nt ( )T n+ α n n+ β n =A n =Bn Switching conditions are given by h(t) + v(t) = 0: h(b n ) = 3 4β n h(a n+1 ) = 1 + 4α n+1 ( )T (n+1)t Integrate ODE for h(t) from one down-switching to the next: h(b n ) = h(a n ) + ct (β n α n ) + c( f (B n ) + f (A n )) h(a n+1 ) = h(a n ) ct (α n+1 2β n + α n ) + c( f(a n+1 ) + f (A n )) where f (t) = s(t), so f(t) is the integrated input signal Stephen Cox Mathematical models for class-d amplifiers 12/38 t

Solving the negative-feedback model Eliminating h(b n ), we find the three equations h(a n+1 ) = 1 + 4α n+1 +1 1 g(t) β n α n nt ( )T n+ α n n+ β n =A n =Bn 3 4β n = h(a n ) + ct (β n α n ) + c( f (B n ) + f (A n )) ( )T (n+1)t h(a n+1 ) = h(a n ) ct (α n+1 2β n + α n ) + c( f (A n+1 ) + f (A n )) for the three quantities β n α n+1 h(a n+1 ) t Stephen Cox Mathematical models for class-d amplifiers 13/38

Solving the negative-feedback model Eliminating h(b n ), we find the three equations h(a n+1 ) = 1 + 4α n+1 +1 1 g(t) β n α n nt ( )T n+ α n n+ β n =A n =Bn 3 4β n = h(a n ) + ct (β n α n ) + c( f (B n ) + f (A n )) ( )T (n+1)t h(a n+1 ) = h(a n ) ct (α n+1 2β n + α n ) + c( f (A n+1 ) + f (A n )) for the three quantities β n α n+1 h(a n+1 ) We can iterate these equations to determine the behaviour of the amplifier - but this doesn t tell us the frequency spectrum! t Stephen Cox Mathematical models for class-d amplifiers 13/38

Solving the negative-feedback model How do we find the audio component of the output g(t) using the difference equations h(a n+1 ) = 1 + 4α n+1 3 4β n = h(a n ) + ct (β n α n ) c(f (B n ) f(a n )) h(a n+1 ) = h(a n ) ct (α n+1 2β n + α n ) c(f (A n+1 ) f(a n ))? Stephen Cox Mathematical models for class-d amplifiers 14/38

Solving the negative-feedback model +1 1 g(t) β n α n t βn 1 n n+1 n+2 n+3 A αn 1 αn αn+1 αn+2 αn+3 nt ( n+ α n )T ( n+ β n )T (n+1)t =A n =Bn ε(n 1) εn ε(n+1) ε(n+2) ε(n+3) τ β β β β B Introduce interpolating functions A(τ) and B(τ), where τ = ωt Here ω = typical signal frequency. Introduce ǫ = ωt 1 Stephen Cox Mathematical models for class-d amplifiers 15/38

Solving the negative-feedback model +1 1 g(t) β n α n t β β β βn 1 n n+1 n+2 n+3 A αn 1 αn αn+1 αn+2 αn+3 nt ( n+ α n )T ( n+ β n )T (n+1)t =A n =Bn ε(n 1) εn ε(n+1) ε(n+2) ε(n+3) τ β B Introduce interpolating functions A(τ) and B(τ), where τ = ωt Here ω = typical signal frequency. Introduce ǫ = ωt 1 We also introduce H(τ) with H(ǫn) = h(a n ) Only sampled values A(ǫn), B(ǫn) and H(ǫn) have meaning Stephen Cox Mathematical models for class-d amplifiers 15/38

Solving the negative-feedback model The interpolating functions A(τ), B(τ) and H(τ) satisfy the functional equations H(τ + ǫ) = 1 + 4A(τ + ǫ) (4 + c)a(τ) = 1 + H(τ) (4 + c)b(τ) = 3 H(τ) + cǫ 1 [F(τ + ǫa(τ)) F(τ + ǫ + ǫa(τ + ǫ))] + cǫ 1 [F(τ + ǫb(τ)) F(τ + ǫa(τ))] where df(τ)/dτ = s(t) is the known audio input To solve these nonlinear equations we make perturbation expansions in the small parameter ǫ: A(τ) = ǫ m A m (τ) B(τ) = ǫ m B m (τ) H(τ) = ǫ m H m (τ) m=0 m=0 m=0 Stephen Cox Mathematical models for class-d amplifiers 16/38

Solving the negative-feedback model We collect terms at successive orders in ǫ, after writing Taylor series for the various functions (about τ) Details of calculation are messy (done by computer algebra) Special case: s(t) = s 0 constant input signal A = 1 16 (1 s 0)[4 c(1 + s 0 )] B = 1 2 + 1 16 (1 + s 0)[4 c(1 s 0 )] +1 1 g(t) B A nt (n+a)t t (n+ B)T (n+1)t Stephen Cox Mathematical models for class-d amplifiers 17/38

Solving the negative-feedback model We collect terms at successive orders in ǫ, after writing Taylor series for the various functions (about τ) Details of calculation are messy (done by computer algebra) Special case: s(t) = s 0 constant input signal A = 1 16 (1 s 0)[4 c(1 + s 0 )] B = 1 2 + 1 16 (1 + s 0)[4 c(1 s 0 )] +1 1 g(t) B A nt (n+a)t t (n+ B)T (n+1)t Time-average of the output (the only low-frequency part) is [ ] [ ] g(t) = +1 (1 B + A) + 1 (B A) = s 0 so there is no distortion (input reproduced, with change of sign) Stephen Cox Mathematical models for class-d amplifiers 17/38

Solving the negative-feedback model We collect terms at successive orders in ǫ, after writing Taylor series for the various functions (about τ) Details of calculation are messy (done by computer algebra) General case: s(t) = general input signal Details very messy, but can calculate A = A 0 + ǫa 1 +, etc: A 0 (τ) = 1 16 (1 s(t))[4 c(1 + s(t))] B 0 (τ) = 1 2 + 1 16 (1 + s(t))[4 c(1 s(t))] Stephen Cox Mathematical models for class-d amplifiers 17/38

Solving the negative-feedback model Once we have found the first few terms in A = A 0 + ǫa 1 + B = B 0 + ǫb 1 + H = H 0 + ǫh 1 + how do we find the audio part of the output g(t)? Stephen Cox Mathematical models for class-d amplifiers 18/38

Finding the audio output For general input s(t), we need to find the audio component of the output Notation: ψ(t; t 1, t 2 ) = top-hat function 1 0 t t 1 2 ψ( t) t Stephen Cox Mathematical models for class-d amplifiers 19/38

Finding the audio output For general input s(t), we need to find the audio component of the output Notation: ψ(t; t 1, t 2 ) = top-hat function 1 0 t t 1 2 ψ( t) t The amplifier output g(t) = n= { } ψ(t; B n, A n+1 ) ψ(t; A n, B n ) +1 g(t) 1 (n 1)T nt (n+1)t t Stephen Cox Mathematical models for class-d amplifiers 19/38

Finding the audio output The Fourier transform of the output is ĝ(ω) = e iωt 2 g(t) dt = (e iωt(n+αn) e iωt(n+βn)) iω Stephen Cox Mathematical models for class-d amplifiers 20/38

Finding the audio output The Fourier transform of the output is ĝ(ω) = e iωt 2 g(t) dt = (e iωt(n+αn) e iωt(n+βn)) iω 2 = (e iωt (n+a(ǫn)) e iωt(n+b(ǫn))) iω Stephen Cox Mathematical models for class-d amplifiers 20/38

Finding the audio output The Fourier transform of the output is ĝ(ω) = e iωt 2 g(t) dt = (e iωt(n+αn) e iωt(n+βn)) iω 2 = (e iωt (n+a(ǫn)) e iωt(n+b(ǫn))) iω 2 ( = iω e2πn iφ e iωt φ e iωt A(ǫφ) e iωt B(ǫφ)) dφ ( ) = 2 iωt e2πn iθ/t e iωθ ( e iωt A(ωθ) e iωt B(ωθ)) dθ where the step ( ) follows using Poisson resummation: f (n) = e 2πn iφ f(φ) dφ n= n= Stephen Cox Mathematical models for class-d amplifiers 20/38

Finding the audio output The low-frequency, audio part of the output is given by n = 0 terms: 2 ( ĝ audio (ω) = iωt e iωθ e iωt A(ωθ) e iωt B(ωθ)) dθ (ω 0) = = e iωθ 0 2( iωt ) n (n + 1)! e iωθ 2( T) n d n ( (n + 1)! dθ n 0 ( ) A n+1 (ωθ) B n+1 (ωθ) dθ ) A n+1 (ωθ) B n+1 (ωθ) and hence the audio part of the output is 2( T) n d n ( ) g audio (t) = 1 + (n + 1)! dt n A n+1 (ωt) B n+1 (ωt) 0 (1 + 2A(ωt) 2B(ωt)) T d ( ) A 2 (ωt) B 2 (ωt) + dt dθ Stephen Cox Mathematical models for class-d amplifiers 21/38

Finding the audio output Result for the audio output From much calculation (!), the amplifier output g audio (t) is s(t) + 1 c s (t) }{{} slight delay: s(t c 1 ) Stephen Cox Mathematical models for class-d amplifiers 22/38

Finding the audio output Result for the audio output From much calculation (!), the amplifier output g audio (t) is s(t) + 1 c s (t) }{{} slight delay: s(t c 1 ) [( + T 2 d2 1 dt 2 48 1 ) c 2 T 2 s(t) }{{} linear distortion + ] 1 48 s3 (t) }{{} nonlinear distortion + Remember, the input signal s(t) is slowly varying: T d dt 1 Stephen Cox Mathematical models for class-d amplifiers 22/38

Finding the audio output Result for the audio output From much calculation (!), the amplifier output g audio (t) is s(t) + 1 c s (t) }{{} slight delay: s(t c 1 ) [( + T 2 d2 1 dt 2 48 1 ) c 2 T 2 s(t) }{{} linear distortion + ] 1 48 s3 (t) }{{} nonlinear distortion + Remember, the input signal s(t) is slowly varying: T d dt 1 Distortion O(ǫ 2 ) = O (( ) typical signal freq. 2 ) ( ) 1kHz 2 switching freq. 500kHz Stephen Cox Mathematical models for class-d amplifiers 22/38

Removing the distortion How do we modify the amplifier design to retain the negative feedback but remove the distortion? Stephen Cox Mathematical models for class-d amplifiers 23/38

Removing the distortion: I One way to remove the distortion is to modulate the carrier wave Stephen Cox Mathematical models for class-d amplifiers 24/38

Removing the distortion: I One way to remove the distortion is to modulate the carrier wave Triangular carrier wave generator: v(t) = 4 T t w(τ) dτ 4 T dt w(t) v(t) Stephen Cox Mathematical models for class-d amplifiers 24/38

Removing the distortion: I One way to remove the distortion is to modulate the carrier wave Triangular carrier wave generator: v(t) = 4 T t w(τ) dτ 4 T dt w(t) v(t) Modulated carrier wave generator: v(t) = 4 T t { } w(τ) + r(τ) dτ + r(t) 4 T dt w(t) v(t) Here r(t) is some slowly varying signal to be found Stephen Cox Mathematical models for class-d amplifiers 24/38

Removing the distortion: I Now the period of the carrier wave is slowly modulated by r(t). We find g audio (t) to be s + 1 c ds dt [ ( ( )) T d T + s 2 ds 4 dt 4 dt r + d ([ T 2 dt 48 1 ] ds c 2 dt T ) ] 4 r + Stephen Cox Mathematical models for class-d amplifiers 25/38

Removing the distortion: I Now the period of the carrier wave is slowly modulated by r(t). We find g audio (t) to be s + 1 c ds dt [ ( ( )) T d T + s 2 ds 4 dt 4 dt r + d ([ T 2 dt 48 1 ] ds c 2 dt T ) ] 4 r + If we choose the modulating function to be r(t) = Ts (t)/4 then the nonlinear distortion term is eliminated This is easily implemented in the circuit Stephen Cox Mathematical models for class-d amplifiers 25/38

Removing the distortion: I Now the period of the carrier wave is slowly modulated by r(t). We find g audio (t) to be s + 1 c ds dt [ ( ( )) T d T + s 2 ds 4 dt 4 dt r + d ([ T 2 dt 48 1 ] ds c 2 dt T ) ] 4 r + If we choose the modulating function to be r(t) = Ts (t)/4 then the nonlinear distortion term is eliminated This is easily implemented in the circuit In fact, this is Bruce Candy s method for removing the nonlinear distortion: remember the Halcro dm88? Stephen Cox Mathematical models for class-d amplifiers 25/38

Removing the distortion: II Another way to remove the distortion arises from the mathematical model Stephen Cox Mathematical models for class-d amplifiers 26/38

Removing the distortion: II Another way to remove the distortion arises from the mathematical model It involves a sample-and-hold element and a multiplier in the circuit S/H s(t) + c dt h(t) v(t) + 0 + g(t) Again g audio (t) has only linear distortion Stephen Cox Mathematical models for class-d amplifiers 26/38

Removing the distortion: II Another way to remove the distortion arises from the mathematical model It involves a sample-and-hold element and a multiplier in the circuit S/H s(t) + c dt h(t) v(t) + 0 + g(t) Again g audio (t) has only linear distortion it works! Stephen Cox Mathematical models for class-d amplifiers 26/38

Pulse skipping in a second-order class-d amplifier with negative feedback Stephen Cox Mathematical models for class-d amplifiers 27/38

A second-order class-d amplifier s(t) m(t) + c dt + 1 p(t) c 2 dt 0 + v(t) g(t) Stephen Cox Mathematical models for class-d amplifiers 28/38

A second-order class-d amplifier s(t) m(t) + c dt + 1 p(t) c 2 dt 0 + v(t) g(t) For an audio input s(t) = 0.7 sin(ωt) freq. 5000Hz (ω = 10000π) the audio output of the amplifier is 0.702 sinωt + 5.08 10 4 sin 3ωt mathematical model 0.701 sinωt + 5.11 10 4 sin 3ωt engineering simulations for v(t) with frequency T 1 250kHz; thus ǫ = ωt 0.126 Stephen Cox Mathematical models for class-d amplifiers 28/38

A second-order class-d amplifier So agreement is excellent between the mathematical model and engineering simulations Stephen Cox Mathematical models for class-d amplifiers 29/38

Pulse-skipping in a class-d amplifier s(t) = 0.7 sin(200πt) [100Hz] v(t) [250kHz] Yu Jun, Tan Meng Tong (NTU, Singapore) Stephen Cox Mathematical models for class-d amplifiers 30/38

Pulse-skipping in a class-d amplifier s(t) = 0.7 sin(200πt) [100Hz] v(t) [250kHz] Pulse skipping occurs when the calculated switching times fall outside the ranges 0 < α n < 1 2 1 2 < β n < 1 +1 1 g(t) β n α n nt ( n+ α n )T ( n+ β n )T (n+1)t =A n =Bn t Stephen Cox Mathematical models for class-d amplifiers 31/38

Pulse-skipping in a class-d amplifier s(t) = 0.7 sin(200πt) [100Hz] v(t) [250kHz] But the mathematical model predicts switching times with α n 1 16 (1 s(ωn))[4 c 1(1 + s(ωn))] β n 1 2 + 1 16 (1 + s(ωn))[4 c 1(1 s(ωn))] and hence (since 1 < s(t) < 1 and 0 < c 1 < 2) 0 < α n < 1 1 2 2 < β n < 1 So the mathematical model predicts no pulse skipping Stephen Cox Mathematical models for class-d amplifiers 31/38

Pulse-skipping in a class-d amplifier The mathematical model predicts no pulse skipping, yet the engineering simulations clearly observe it Stephen Cox Mathematical models for class-d amplifiers 32/38

Pulse-skipping in a class-d amplifier The mathematical model predicts no pulse skipping, yet the engineering simulations clearly observe it In fact, the mathematical model and the simulations do agree The analytical solution turns out to be unstable for large enough signal amplitudes: s(t) > 0.7 Λ 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 s 0 This can lead to pulse skipping, as observed in engineering simulations Stephen Cox Mathematical models for class-d amplifiers 32/38

Pulse-skipping in a class-d amplifier The mathematical model predicts no pulse skipping, yet the engineering simulations clearly observe it In fact, the mathematical model and the simulations do agree The analytical solution turns out to be unstable for large enough signal amplitudes: s(t) > 0.7 Λ 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 s 0 This can lead to pulse skipping, as observed in engineering simulations 1.2 1 0.8 0.6 0.4 0.2 0 β n Λ α n 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 t Need to decrease the precision in Maple to see pulse skipping Stephen Cox Mathematical models for class-d amplifiers 33/38

Pulse-skipping in a class-d amplifier The mathematical model predicts no pulse skipping, yet the engineering simulations clearly observe it In fact, the mathematical model and the simulations do agree The analytical solution turns out to be unstable for large enough signal amplitudes: s(t) > 0.7 This can lead to pulse skipping, as observed in engineering simulations Λ g(t) 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 s 0 1 0.5 0-0.5-1 0.0038 0.00385 0.0039 Need to decrease the precision in Maple to see pulse skipping Stephen Cox Mathematical models for class-d amplifiers 34/38 t

Pulse-skipping in a class-d amplifier In summary, for this second-order amplifier The mathematical model accurately predicts the amplifier s audio output A stability analysis of the clean analytical solution shows why pulse skipping is observed in practice (quantisation noise) Next we need to find a way to eliminate the nonlinear (third-harmonic) distortion and eliminate pulse skipping! Stephen Cox Mathematical models for class-d amplifiers 35/38

Proof of the pudding The analytical result for the audio output can be compared with engineering simulations and experiments Stephen Cox Mathematical models for class-d amplifiers 36/38

Proof of the pudding Fund. 3rd harm. 0.74 0.73 0.72 0.71 0.70 0.69 0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 Maple fund. Matlab fund. 100 1000 10000 frequency Maple 3rd harm. Matlab 3rd harm. 100 1000 10000 frequency Stephen Cox Mathematical models for class-d amplifiers 37/38

Summary Negative-feedback class-d amplifiers are all around us, in part because they are so efficient Stephen Cox Mathematical models for class-d amplifiers 38/38

Summary Negative-feedback class-d amplifiers are all around us, in part because they are so efficient Mathematical models for these amplifiers provide useful information to engineers (on distortion, pulse skipping, etc) Mathematical models can be developed to help improve amplifier designs Current focus: How to eliminate distortion in the second-order amplifier How to stabilise the behaviour (to eliminate pulse skipping) Better ways to analyse the amplifier? Stephen Cox Mathematical models for class-d amplifiers 38/38