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Motivation The Harmonic Balance Method Applications The Harmonic Balance Method 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 H.-D. Lang, X. Zhang The Harmonic Balance Method
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Page 1: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

The Harmonic Balance MethodFor Nonlinear Microwave Circuits

Hans-Dieter Lang, Xingqi Zhang

Thursday, April 25, 2013ECE 1254 – Modeling of Multiphysics Systems

Course Project Presentation

University of Toronto

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 2: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

The Harmonic Balance Method

Contents

Motivation

Balancing the harmonics

Applications

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 3: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Question: Why do we need another simulation method?

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 4: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Answer: MNA is great, but...

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 5: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Time-domain methods

, Transients

, Linear & nonlinear networks

/ Issues with stiff problems

/ Inefficient for steady-state

/ No dispersive effects

Frequency-domain methods

, Steady-state

, Fast (direct)

, Dispersive effects

/ Only linear networks

/ No transients

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 6: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Focus: RF & microwave circuits

Steady-state

Lumped elements to multiple-𝜆 TLs → stiff problems

Nonlinear elements

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 7: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Time-domain methods

, Transients

, Linear & nonlinear networks

/ Issues with stiff problems

/ Inefficient for steady-state

/ No dispersive effects

Frequency-domain methods

, Steady-state

, Fast (direct)

, Dispersive effects

/ Only linear networks

/ No transients

Hybrid method

Linear network: Frequency domain

Nonlinearities: Time domain

Combine solutions

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 8: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Time-domain methods

, Transients

, Linear & nonlinear networks

/ Issues with stiff problems

/ Inefficient for steady-state

/ No dispersive effects

Frequency-domain methods

, Steady-state

, Fast (direct)

, Dispersive effects

/ Only linear networks

/ No transients

Hybrid method

Linear network: Frequency domain

Nonlinearities: Time domain

Combine solutions

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 9: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Time-domain methods

, Transients

, Linear & nonlinear networks

/ Issues with stiff problems

/ Inefficient for steady-state

/ No dispersive effects

Frequency-domain methods

, Steady-state

, Fast (direct)

, Dispersive effects

/ Only linear networks

/ No transients

Hybrid method

Linear network: Frequency domain

Nonlinearities: Time domain

Combine solutions

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 10: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Motivation

Commercial use

..."everybody" uses harmonic balance:

ADS/Genesys

Microwave Office

Designer/Nexxim

Virtuoso Spectre

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 11: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method... 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

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 12: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

Linear subcircuit NonlinearitiesExcitation(s)

LTI 𝑔(𝑣)𝑣𝑠

𝑖1

𝑣1𝑣2

𝑖2𝑖𝑠

Harmonic balanceKCL: 𝑖1 + ��1 = 0 ∀ 𝑡, 𝜔

𝑖1 = 𝑌11𝑣1 + 𝑌12𝑣2→ Frequency-domain

��1(𝑣1) = �� · 𝑔(𝑣1)→ Time-domain

Cost function:

𝑓(𝑣1) = 𝑖1 + ��1 = 𝑌11𝑣1 + 𝑌12𝑣2 + ��(𝑣1)?≈ 0

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 13: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

Linear subcircuit NonlinearitiesExcitation(s)

LTI 𝑔(𝑣)𝑣𝑠

𝑖1 ��1

𝑣1𝑣2

𝑖2𝑖𝑠

Harmonic balanceKCL: 𝑖1 + ��1 = 0 ∀ 𝑡, 𝜔

𝑖1 = 𝑌11𝑣1 + 𝑌12𝑣2→ Frequency-domain

��1(𝑣1) = �� · 𝑔(𝑣1)→ Time-domain

Cost function:

𝑓(𝑣1) = 𝑖1 + ��1 = 𝑌11𝑣1 + 𝑌12𝑣2 + ��(𝑣1)?≈ 0

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 14: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

Linear subcircuit NonlinearitiesExcitation(s)

LTI 𝑔(𝑣)𝑣𝑠

𝑖1 ��1

𝑣1𝑣2

𝑖2𝑖𝑠

Harmonic balanceKCL: 𝑖1 + ��1 = 0 ∀ 𝑡, 𝜔

𝑖1 = 𝑌11𝑣1 + 𝑌12𝑣2→ Frequency-domain

��1(𝑣1) = �� · 𝑔(𝑣1)→ Time-domain

Cost function:

𝑓(𝑣1) = 𝑖1 + ��1 = 𝑌11𝑣1 + 𝑌12𝑣2 + ��(𝑣1)?≈ 0

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 15: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

Linear subcircuit NonlinearitiesExcitation(s)

LTI 𝑔(𝑣)𝑣𝑠

𝑖1 ��1

𝑣1𝑣2

𝑖2𝑖𝑠

Harmonic balanceKCL: 𝑖1 + ��1 = 0 ∀ 𝑡, 𝜔

𝑖1 = 𝑌11𝑣1 + 𝑌12𝑣2→ Frequency-domain

��1(𝑣1) = �� · 𝑔(𝑣1)→ Time-domain

Cost function:

𝑓(𝑣1) = 𝑖1 + ��1 = 𝑌11𝑣1 + 𝑌12𝑣2 + ��(𝑣1)?≈ 0

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 16: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

Notation

0

𝑅2 1

𝑔(𝑣1)𝑣𝑠

𝑖1 ��1

𝑣1𝑣2 = 𝑣𝑠

𝑖2𝑖𝑠

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 17: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

Notation

0

𝑅2 1

𝑔(𝑣)𝑣𝑠

𝑖 ��

𝑣𝑣𝑠

𝑖𝑠𝑖𝑠

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 18: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

𝑅2 1

𝑔(𝑣)𝑣𝑠

𝑖 ��

𝑣𝑣𝑠

𝑖𝑠𝑖𝑠

Harmonic balance at node 1 in the frequency domain, ∀ 𝑘 ∈ 0, ...,𝐾

𝑖(𝑘𝜔0) + ��(𝑘𝜔0) = 0∀𝑘⇒k−−−−→ i(𝜔) + i(𝜔) = 0

Cost functionf(v) = i(𝜔) + i(𝜔) ≈ 0

with

i(𝜔) = Y𝑠v𝑠(𝜔)⏟ ⏞ i𝑠(𝜔)

+Yv(𝜔)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 19: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

𝑅2 1

𝑔(𝑣)𝑣𝑠

𝑖 ��

𝑣𝑣𝑠

𝑖𝑠𝑖𝑠

Total linear current i(𝜔) = Y𝑠v𝑠(𝜔) +Yv(𝜔) consists of

i𝑠(𝜔) = Y𝑠v𝑠(𝜔) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

𝑌12(0)𝑌12(𝜔0)

. . .𝑌12(𝑘𝜔0)

. . .𝑌12(𝐾𝜔0)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣010...0

⎤⎥⎥⎥⎥⎥⎦⇒ i(𝜔) = Diag[y12(𝜔)]v𝑠(𝜔) +Diag[y11(𝜔)]v(𝜔)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 20: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

𝑅2 1

𝑔(𝑣)𝑣𝑠

𝑖 ��

𝑣𝑣𝑠

𝑖𝑠𝑖𝑠

Nonlinear current

i(𝜔) = ℱ i𝑑(v(𝑡)) = ℱ i𝑑(ℱ−1v(𝜔)⏟ ⏞

v(𝑡)

)

with nonlinear diode current function

��𝑑(𝑣) = 𝐼𝑠(e𝑣/𝑣𝑇 − 1

)H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 21: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

𝑅2 1

𝑔(𝑣)𝑣𝑠

𝑖 ��

𝑣𝑣𝑠

𝑖𝑠𝑖𝑠

Cost function

f(v) = i(𝜔) + i(𝜔)

= Y𝑠v𝑠(𝜔) +Yv(𝜔) +ℱ i𝑑(ℱ−1v(𝜔))

= Diag[y12(𝜔)]v𝑠(𝜔) +Diag[y11(𝜔)]v(𝜔)⏟ ⏞ i(𝜔)

+ℱ i𝑑(ℱ−1v(𝜔))⏟ ⏞ i(𝜔)

Newton: as long as ‖f(v𝑚)‖ > 𝜀 and 𝑚 < 𝑚max

v𝑚+1 = v𝑚 − J−1f(v𝑚)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 22: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

𝑅2 1

𝑔(𝑣)𝑣𝑠

𝑖 ��

𝑣𝑣𝑠

𝑖𝑠𝑖𝑠

Main problem: finding the Jacobian

J =𝑑f(v)

𝑑v

v=v𝑚

⇔ 𝐽𝑖𝑗 =𝜕𝑓𝑖(v)

𝜕𝑣𝑗

of the cost function

f(v) = Y𝑠v𝑠(𝜔) +Yv(𝜔)⏟ ⏞ i(𝜔)

+ℱ i𝑑(ℱ−1v(𝜔))⏟ ⏞ i(𝜔)

ResultJ = Y +ℱ Diag

[i′𝑑(ℱ−1v(𝜔)

)]ℱ−1

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 23: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

IdeaDerivation

The Harmonic Balance Method

The algorithm

Initial guessv0(𝑡)

v𝑚(𝜔)

v𝑚(𝑡)

i𝑚(𝑡)

i𝑚(𝜔)

f(v)<𝜀?

ℱ−1

Nonlinearity

i𝑚

= v𝑚·𝑔(𝑣)

i𝑚(𝜔)

Update (Newton)

v𝑚+1 = v𝑚 − J−1f(v)

converged v(𝜔)

Time domain Frequency domain

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 24: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 25: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

"All electronic circuits are nonlinear:this is a fundamental truth ofelectronic engineering."

Stephen MaasDirector of Technology,

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 26: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Examples

Rectifiers

Diode mixer

Oscillator

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 27: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Half-wave rectifier

Diode nonlinearity + capacitor (dynamic)

Linear NonlinearExcitation

𝐶𝑅𝐿

𝑅3

2

1

𝑔(𝑣)𝑣𝑠

𝑖1 ��1

𝑖2 ��2

𝑣

𝑖3𝑖𝑠

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 28: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Half-wave rectifierMNA for steady-state: inefficient

0 5 10 15 20−3

−2

−1

0

1

2

3

Time (s)

Volta

ge (V

), C

urre

nt (A

)

vsv1v2id x 10

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 29: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Demo: Half-wave rectifier

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 30: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Half-wave rectifierError comparison

101 102 10310−4

10−3

10−2

10−1

100

Number of harmonics K+1

Abso

lute

erro

r

MNA L1HB L1MNA L2HB L2MNA L'HB L'

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 31: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Half-wave rectifierCPU time consumption

101 102 10310−2

10−1

100

101

102

Number of harmonics K+1

CPU

tim

e (s

)

MNA o=1MNA o=10MNA o=100HB o=1HB o=10HB o=100

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 32: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Delon bridge voltage doubler

Linear NonlinearExcitation

𝐶2

𝐶1

𝑅𝐿

𝑅

3

4

1

2

𝑔(𝑣)

𝑔(𝑣)

𝑣𝑠

𝑖3 ��3

𝑖2 ��2

𝑖1 ��1

𝑣𝑑2

𝑣𝑑1

𝑖4𝑖𝑠

Multiple nonlinearities

Different dynamics: 𝐶2 = 4𝐶1 = 1mF, 𝑅𝐿 = 10kΩ

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 33: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Delon bridge voltage doubler

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

Time

Volta

ge

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

Ampl

itude

Harmonic k

20 40 60 80 100 120 140 160 180 20010−6

10−4

10−2

100

Iteration m

Erro

r

vsvd1vd2vc2vc1

|Vd1||Vd2|

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

Time

Volta

ge

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

Ampl

itude

Harmonic k

20 40 60 80 100 120 140 160 180 20010−6

10−4

10−2

100

Iteration m

Erro

r

vsvd1vd2vc2vc1

|Vd1||Vd2|

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 34: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Rectifiers

HB more efficient than MNA for large 𝜏 and small 𝐾

Multiple nonlinearities and dynamics

Source stepping greatly improves convergence rate

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 35: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Diode mixer

Linear NonlinearExcitation

𝐶

𝑅3𝑅𝐿

𝑅1

5

4

2

1

3

𝑔(𝑣)

𝑣𝑠1

𝑅2

𝑣𝑠2

𝑖1 ��1

𝑖2 ��2

𝑣

𝑣2 = 𝑣𝑠

𝑖4𝑖𝑠1

𝑖5𝑖𝑠2

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 36: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Diode mixer

Nonlinear diode current

𝑖𝑑(𝑣) = 𝐼𝑠(e𝑣/𝑣𝑇 − 1) = 𝐼𝑠

(𝑣

𝑣𝑇+

𝑣2

𝑣2𝑇+

𝑣3

𝑣3𝑇+ ...

)Mixer products of 𝑣 = cos𝜔1 + cos𝜔2

𝑣2 = 1 +cos 2𝜔1𝑡+ cos 2𝜔2𝑡

2+ cos(𝜔1𝑡± 𝜔2𝑡)

𝑣3 =9

4(cos𝜔1𝑡+ cos𝜔2) +

1

4(cos 3𝜔1𝑡+ cos 3𝜔2𝑡)

+3

4

(cos(2𝜔1𝑡± 𝜔2𝑡) + cos(2𝜔2𝑡± 𝜔1𝑡)

)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 37: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Diode mixer

0 20 40 60 80 100 120

−2

0

2

Time step tn

Volta

ge

Input 1: v1(t)Input 2: v2(t)At diode: v3(t)Output: v4(t)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

Frequency kut0

Ampl

itude

At diode: v3(t)

Output: v4(t)

7 11

4 18

14 22DC

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 38: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Diode mixer

Multi-tone + nonlinearities = large spectrum

1st harmonic 𝜔0:

{common factor of source frequencies

special techniques (more efficient)

Source stepping greatly improves convergence rate

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 39: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

Linear Nonlinear

𝐿 𝐶 𝑔(𝑣)

1𝑖𝐶 + 𝑖𝐿 = 𝑖 �� = ��𝑅

𝑣

KCL: 𝑖𝐿 + 𝑖𝐶 + ��𝑅 = 0

Nonlinear resistor: 𝑔(𝑣) =𝑣2

3− 1 → ��𝑅 = 𝑣 · 𝑔(𝑣) = 𝑣3

3− 𝑣

Capacitor: 𝑖𝐶 = 𝐶��,

Inductor: 𝑣 = 𝐿��𝐿 and 𝑖𝐿 = −(𝑖𝐶 + ��𝑅)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 40: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

Linear Nonlinear

𝐿 𝐶 𝑔(𝑣)

1𝑖𝐶 + 𝑖𝐿 = 𝑖 �� = ��𝑅

𝑣

𝑣 = −𝐿𝑑

𝑑𝑡(𝑖𝐶 + ��𝑅) = −𝐿𝐶𝑣 − �� 𝑔(𝑣)− 𝑣

𝜕𝑔(𝑣)

𝜕𝑣��

Van der Pol equation:

𝐿𝐶𝑣 + 𝐿(𝑣2 − 1)�� + 𝑣 = 0

𝑣 + 𝜖(𝑣2 − 1)�� + 𝑣 = 0

with 𝜖 = 1/𝐶 = 𝐿

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 41: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

-3 -2 -1 1 2 3v

-3

-2

-1

1

2

3

𝜖 = 0 (linear)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 42: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

-3 -2 -1 1 2 3v

-3

-2

-1

1

2

3

𝜖 = 1 (nonlinear)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 43: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

Dependence on 𝜖 or L,C

-4 -2 2 4

-4

-2

2

4

𝜖 = 0

10 20 30 40

-4

-2

2

4

𝑡

𝑣(𝑡), ��(𝑡)

𝑣(𝑡)

��(𝑡)𝑣(𝑡)

��(𝑡)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 44: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

Dependence on 𝜖 or L,C

-4 -2 2 4

-4

-2

2

4

𝜖 = 1

10 20 30 40

-4

-2

2

4

𝑡

𝑣(𝑡), ��(𝑡)

𝑣(𝑡)

��(𝑡)𝑣(𝑡)

��(𝑡)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 45: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

Dependence on 𝜖 or L,C

-4 -2 2 4

-4

-2

2

4

𝜖 = 2

10 20 30 40

-4

-2

2

4

𝑡

𝑣(𝑡), ��(𝑡)

𝑣(𝑡)

��(𝑡)𝑣(𝑡)

��(𝑡)

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 46: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Van der Pol oscillator

Results for 𝜖 = 2

MNA HB

392 394 396 398−3

−2

−1

0

1

2

3

Normalized time (period)

Ampl

itude

0 2 4 6 8 1010−20

10−10

100

Frequency

Ampl

itude

vdv/dt

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

Normalized time

Ampl

itude

0 10 20 30 40 50 6010−20

10−10

100

Normalized frequency

Ampl

itude

vdv/dt

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 47: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

RectifiersDiode mixerOscillator

Applications

Oscillators & Harmonic balance

No source stepping

Frequency unknown

{additional variables

guess & choose 𝐾 large enough

Various issues → special techniques for autonomous circuits

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 48: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

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.)

Fast convergence: Source stepping

Special techniques for multi-tone simulations and oscillators

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 49: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

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.)

Fast convergence: Source stepping

Special techniques for multi-tone simulations and oscillators

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 50: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

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.)

Fast convergence: Source stepping

Special techniques for multi-tone simulations and oscillators

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 51: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

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.)

Fast convergence: Source stepping

Special techniques for multi-tone simulations and oscillators

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 52: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

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.)

Fast convergence: Source stepping

Special techniques for multi-tone simulations and oscillators

H.-D. Lang, X. Zhang The Harmonic Balance Method

Page 53: The Harmonic Balance Method - hade.ch › docs › presentation_HB.pdf · The Harmonic Balance Method Applications The Harmonic Balance Method For Nonlinear Microwave Circuits Hans-Dieter

MotivationThe Harmonic Balance Method

Applications

Harmonic Balance

i

i

NonlinearLinearsubcircuitsubcircuit

H.-D. Lang, X. Zhang The Harmonic Balance Method


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