The Harmonic Balance Method - PDF Free Download

For Nonlinear Microwave Circuits Hans-Dieter Lang, Xingqi Zhang Thursday, April 25, 2013 ECE 1254 Modeling of Multiphysics Systems Course Project Presentation University of Toronto

Contents Balancing the harmonics

Question: Why do we need another simulation method?

Answer: MNA is great, but...

Time-domain methods Transients Linear & nonlinear networks Frequency-domain methods Steady-state Fast (direct) Dispersive effects Issues with stiff problems Inefficient for steady-state Only linear networks No transients No dispersive effects

Focus: RF & microwave circuits Steady-state Lumped elements to multiple-λ TLs stiff problems Nonlinear elements

Time-domain methods Frequency-domain methods Transients Steady-state Linear & nonlinear networks Hybrid method Fast (direct) Linear network: Frequency Dispersive domain effects Nonlinearities: Time domain Issues with stiff problems Only linear networks Combine solutions Inefficient for steady-state No transients No dispersive effects

Commercial use..."everybody" uses harmonic balance: ADS/Genesys Microwave Office Designer/Nexxim Virtuoso Spectre

Idea Derivation... since 1976 * * M. S. Nakhla, J. Vlach, A Piecewise Harmonic Balance Technique for Determination of Periodic Response of Nonlinear Systems, IEEE Transactions on Circuits and Systems, Vol. 23, No. 2, February 1976

Idea Derivation i s i 2 i 1 v s v 2 LTI v 1 g(v) Excitation(s) Linear subcircuit Nonlinearities

Idea Derivation Harmonic balance KCL: i 1 + ^i 1 = 0 t, ω i s i 2 i 1 ^i 1 v s v 2 LTI v 1 g(v) Excitation(s) Linear subcircuit Nonlinearities

Idea Derivation Harmonic balance KCL: i 1 + ^i 1 = 0 t, ω i s i 2 i 1 ^i 1 v s v 2 LTI v 1 g(v) Excitation(s) Linear subcircuit Nonlinearities i 1 = Y 11 v 1 + Y 12 v 2 Frequency-domain ^i 1 (v 1 ) = ^i g(v 1 ) Time-domain

Idea Derivation Harmonic balance KCL: i 1 + ^i 1 = 0 t, ω i s i 2 i 1 ^i 1 v s v 2 LTI v 1 g(v) Excitation(s) Linear subcircuit Nonlinearities i 1 = Y 11 v 1 + Y 12 v 2 ^i 1 (v 1 ) = ^i g(v 1 ) Frequency-domain Time-domain Cost function: f(v 1 ) = i 1 + ^i 1 = Y 11 v 1 + Y 12 v 2 + ^i(v 1 )? 0

Idea Derivation Notation R i s 2 i 2 1 i 1 ^i 1 v s v 2 = v s v 1 g(v 1 ) 0

Idea Derivation Notation R i s 2 i s 1 i ^i v s v s v g(v)

Idea Derivation R i s 2 i s 1 i ^i v s v s v g(v) Harmonic balance at node 1 in the frequency domain, k 0,..., K i(kω 0 ) + ^i(kω 0 ) = 0 k k i(ω) +^i(ω) = 0 Cost function with f(v) = i(ω) +^i(ω) 0 i(ω) = Y s v s (ω) +Yv(ω) i s (ω)

Idea Derivation R i s 2 i s 1 i ^i v s v s v g(v) Total linear current i(ω) = Y s v s (ω) + Yv(ω) consists of Y 12 (0) Y 12 (ω 0 )... i s (ω) = Y s v s (ω) = Y12(kω0)... Y12(Kω0) 0 1 0. 0 i(ω) = Diag[y 12 (ω)] v s (ω) + Diag[y 11 (ω)] v(ω)

Idea Derivation R i s 2 i s 1 i ^i v s v s v g(v) Nonlinear current ^i(ω) = F ^id (v(t)) = F ^i ( d F 1 ) v(ω) v(t) with nonlinear diode current function ^i d (v) = I s ( e v/v T 1 )

Idea Derivation R i s 2 i s 1 i ^i v s v s v g(v) Cost function f(v) = i(ω) +^i(ω) = Y s v s (ω) + Yv(ω) + F ^i d (F 1 v(ω)) = Diag[y 12 (ω)] v s (ω) + Diag[y 11 (ω)] v(ω) + F ^i d (F 1 v(ω)) i(ω) ^i(ω) Newton: as long as f(v m ) > ε and m < m max v m+1 = v m J 1 f(v m )

Idea Derivation R i s 2 i s 1 i ^i v s v s v g(v) Main problem: finding the Jacobian J = df(v) dv v=v m of the cost function Result J ij = f i(v) v j f(v) = Y s v s (ω) + Yv(ω) + F ^i d (F 1 v(ω)) i(ω) ^i(ω) J = Y + F Diag [ i ( d F 1 v(ω) ) ] F 1

Idea Derivation The algorithm Time domain Frequency domain F 1 v m (ω) Initial guess v 0 (t) v m (t) Update (Newton) v m+1 = v m J 1 f(v) Nonlinearity ^i m = vm g(v) i m (ω) f(v) <ε? converged v(ω) ^i m (t) F ^i m (ω)

Rectifiers Oscillator

Rectifiers Oscillator "All electronic circuits are nonlinear: this is a fundamental truth of electronic engineering." Stephen Maas Director of Technology,

Rectifiers Oscillator Examples Rectifiers Oscillator

Rectifiers Oscillator Half-wave rectifier Diode nonlinearity + capacitor (dynamic) i s 3 i 3 R i 1 1 ^i 1 v s v i 2 ^i 2 g(v) R L C 2 Excitation Linear Nonlinear

Rectifiers Oscillator Half-wave rectifier MNA for steady-state: inefficient 3 Voltage (V), Current (A) 2 1 0 1 2 3 0 5 10 15 20 Time (s) v s v 1 v 2 i d x 10

Rectifiers Oscillator Demo: Half-wave rectifier

Rectifiers Oscillator Half-wave rectifier Error comparison 10 0 10 1 Absolute error 10 2 10 3 10 4 MNA L1 HB L1 MNA L2 HB L2 MNA L HB L 10 1 10 2 10 3 Number of harmonics K+1

Rectifiers Oscillator Half-wave rectifier CPU time consumption 10 2 10 1 MNA =1 MNA =10 MNA =100 HB =1 HB =10 HB =100 CPU time (s) 10 0 10 1 10 2 10 1 10 2 10 3 Number of harmonics K+1

Rectifiers Oscillator Delon bridge voltage doubler i 3 3 ^i 3 i s 4 i 4 R C 1 v d1 i 2 2 ^i 2 g(v) v s R L C 2 v d2 i 1 ^i 1 g(v) 1 Excitation Linear Nonlinear Multiple nonlinearities Different dynamics: C 2 = 4C 1 = 1 mf, R L = 10 kω

Rectifiers Oscillator Delon bridge voltage doubler 4 Voltage Voltage 42 20 2 0 2 4 4 6 0 0.2 0.4 0.6 0.8 1 6 Time 0 0.2 0.4 0.6 0.8 1 Time v s v d1 s v d2 d1 v c2 d2 v c1 c2 v c1 2.5 Amplitude Amplitude 2.52 1.5 2 1.51 0.5 1 0.50 0 5 10 15 20 25 30 0 Harmonic k 0 5 10 15 20 25 30 H.-D. Lang, Harmonic X. Zhang k V d1 V V d2 d1 V d2

Rectifiers Oscillator Rectifiers HB more efficient than MNA for large τ and small K Multiple nonlinearities and dynamics Source stepping greatly improves convergence rate

Rectifiers Oscillator i s1 4 i 4 R 1 v s1 v 2 = v s i s2 5 i 5 R 2 i 1 1 ^i 1 v s2 3 C v i 2 ^i 2 g(v) 2 R L R 3 Excitation Linear Nonlinear

Rectifiers Oscillator Nonlinear diode current i d (v) = I s (e v/v T 1) = I s ( v v T + v2 v 2 T Mixer products of v = cos ω 1 + cos ω 2 v 2 = 1 + cos 2ω 1t + cos 2ω 2 t 2 + v3 v 3 T + cos(ω 1 t ± ω 2 t) ) +... v 3 = 9 4 (cos ω 1t + cos ω 2 ) + 1 4 (cos 3ω 1t + cos 3ω 2 t) + 3 ( ) cos(2ω 1 t ± ω 2 t) + cos(2ω 2 t ± ω 1 t) 4

Rectifiers Oscillator Voltage 2 0 Input 1: v 1 (t) Input 2: v 2 (t) At diode: v 3 (t) Output: v 4 (t) 2 Amplitude 0.5 0.4 0.3 0.2 0 20 40 60 80 100 120 Time step t 7 11 n DC 0.1 4 18 14 22 At diode: v 3 ( ) Output: v 4 ( ) 0 0 10 20 30 40 50 60 Frequency k 0

Rectifiers Oscillator Multi-tone + nonlinearities = large spectrum { common factor of source frequencies 1st harmonic ω 0 : special techniques (more efficient) Source stepping greatly improves convergence rate

Rectifiers Oscillator Van der Pol oscillator i C + i L = i 1 ^i = ^i R L C v g(v) Linear Nonlinear KCL: i L + i C + ^i R = 0 Nonlinear resistor: g(v) = v2 3 1 ^i R = v g(v) = v3 3 v Capacitor: i C = C v, Inductor: v = L i L and i L = (i C + ^i R )

Rectifiers Oscillator Van der Pol oscillator i C + i L = i 1 ^i = ^i R L C v g(v) Linear Nonlinear v = L d dt (i C + ^i R ) = LC v v g(v) v g(v) v Van der Pol equation: v with ε = 1/C = L LC v + L(v 2 1) v + v = 0 v + ε(v 2 1) v + v = 0

Rectifiers Oscillator Van der Pol oscillator ε = 0 (linear) 3 2 1 3 2 1 1 2 3 v 1 2 3

Rectifiers Oscillator Van der Pol oscillator ε = 1 (nonlinear) 3 2 1 3 2 1 1 2 3 v 1 2 3

Rectifiers Oscillator Van der Pol oscillator Dependence on ε or L,C ε = 0 v(t) 4 v(t), v(t) 4 v(t) v(t) 2 2 4 2 2 4 v(t) 10 20 30 40 t 2 4 2 4

Rectifiers Oscillator Van der Pol oscillator Dependence on ε or L,C ε = 1 v(t) 4 v(t), v(t) 4 v(t) v(t) 2 2 4 2 2 4 v(t) 10 20 30 40 t 2 4 2 4

Rectifiers Oscillator Van der Pol oscillator Dependence on ε or L,C ε = 2 v(t) 4 v(t), v(t) 4 v(t) v(t) 2 2 4 2 2 4 v(t) 10 20 30 40 t 2 4 2 4

Rectifiers Oscillator Van der Pol oscillator Results for ε = 2 Amplitude 3 2 1 0 1 2 3 10 0 MNA v dv/dt 392 394 396 398 Normalized time (period) Amplitude 3 2 1 0 1 2 HB 3 0 0.2 0.4 0.6 0.8 1 Normalized time 10 0 v dv/dt Amplitude 10 10 Amplitude 10 10 10 20 0 2 4 6 8 10 Frequency 10 20 0 10 20 30 40 50 60 Normalized frequency

Rectifiers Oscillator Oscillators & Harmonic balance No source stepping { additional variables Frequency unknown guess & choose K large enough Various issues special techniques for autonomous circuits

Summary & Conclusions Time- vs. frequency-domain: Hybrid is the answer

Summary & Conclusions Time- vs. frequency-domain: Hybrid is the answer The harmonic balance method: Linear subcircuits frequency domain Nonlinear subcircuits time domain Balance currents at interfaces Advantages for Steady-state simulations Stiff problems Others (dispersion, optimization, etc.)

Harmonic Balance i ^i Linear subcircuit Nonlinear subcircuit