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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
MotivationThe Harmonic Balance Method
Applications
The Harmonic Balance Method
Contents
Motivation
Balancing the harmonics
Applications
H.-D. Lang, X. Zhang The Harmonic Balance Method
MotivationThe Harmonic Balance Method
Applications
Motivation
Question: Why do we need another simulation method?
H.-D. Lang, X. Zhang The Harmonic Balance Method
MotivationThe Harmonic Balance Method
Applications
Motivation
Answer: MNA is great, but...
H.-D. Lang, X. Zhang The Harmonic Balance Method
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
ℱ
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
MotivationThe Harmonic Balance Method
Applications
RectifiersDiode mixerOscillator
Applications
H.-D. Lang, X. Zhang The Harmonic Balance Method
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
MotivationThe Harmonic Balance Method
Applications
RectifiersDiode mixerOscillator
Applications
Examples
Rectifiers
Diode mixer
Oscillator
H.-D. Lang, X. Zhang The Harmonic Balance Method
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
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
MotivationThe Harmonic Balance Method
Applications
RectifiersDiode mixerOscillator
Applications
Demo: Half-wave rectifier
H.-D. Lang, X. Zhang The Harmonic Balance Method
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
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
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
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
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
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
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
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
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
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
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
MotivationThe Harmonic Balance Method
Applications
RectifiersDiode mixerOscillator
Applications
Van der Pol oscillator
-3 -2 -1 1 2 3v
-3
-2
-1
1
2
3
v
𝜖 = 0 (linear)
H.-D. Lang, X. Zhang The Harmonic Balance Method
MotivationThe Harmonic Balance Method
Applications
RectifiersDiode mixerOscillator
Applications
Van der Pol oscillator
-3 -2 -1 1 2 3v
-3
-2
-1
1
2
3
v
𝜖 = 1 (nonlinear)
H.-D. Lang, X. Zhang The Harmonic Balance Method
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
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
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
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
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
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
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
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
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
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
MotivationThe Harmonic Balance Method
Applications
Harmonic Balance
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NonlinearLinearsubcircuitsubcircuit
H.-D. Lang, X. Zhang The Harmonic Balance Method