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A Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for CircuitSimulations
Ashish Awasthi & H.G. BrachtendorfResearch Center
Upper Austria University of Applied Sciences, HagenbergAustria
October 2009
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Outline
1 Introduction
2 Formulation of AlgorithmSelection of Initial Values
3 Test ExamplesNon-Autonomous CircuitsAutonomous Circuits
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Introduction
Introduction
At the high data rates requires huge signal bandwidths and highcenter frequency of several GHz led CAD tools to their limits.
A novel method has been developed to circumvent Nyquist rateproblem.
The method is based on reformulating the ordinary DAE to asystem of PDEs, also known as multirate PDEs (MPDEs).
Formulation of PDE depends on the circuit class under investigation.
The formulation of PDE also depends on number of fundamentaltones or frequencies, therefore autonomous and non-autonomouscircuits can be treated.
Autonomous circuits comprise mainly oscillators, the frequency ofoscillations is not known a-priori.
The PDE formulation differs significantly from autonomous tonon-autonomous case.
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Formulation of Algorithm
Cont...
Consider the system of ordinary DAEs
d
dtq(x)(t) = f(b(t), x(t)) (1)
x0 = x(t0) (2)
where, x : R→ RN , b : R→ RN and f : RN ×RN → RN
Introduce x̂ : Rm → RN for the state variables and b̂ : Rm → RN of theinput signals.For simplicity, we consider the function f, b, b̂ ∈ C0 and q, x, x̂ ∈ C1. In1996, Brachtendorf et al. (Numerical steady state analysis of circuitsdriven by multi-tone signals, published in Elect. Eng.) have introducedthe corresponding multirate partial differential algebraic equation(MPDAE)
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Formulation of Algorithm
Formulation(∂
∂τ+∂(τω1(τ))
∂τ
∂
∂t1+ . . .+
∂(τωm−1(τ))
∂τ
∂
∂tm−1
)q(x̂)
= f(b̂(τ, t1, . . . , tm−1), x̂(τ, t1, . . . , tm−1)).
Important Result
A given solution x̂ of the MPDAE (3) coincides with a solution x ofDAE (1) along the curve i .e. characteristic curve
x(t) = x̂(t, ω1t, . . . , ωm−1t). (3)
In Radio Frequency (RF) applications, many system comprises exactlytwo different time scales :
MPDAE (∂
∂τ+∂(ω(τ)τ)
∂τ
∂
∂t1
)q(x̂(τ, t1)) = f(b̂(τ, t1), x̂(τ, t1)) (4)
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Formulation of Algorithm
Formulation(∂
∂τ+∂(τω1(τ))
∂τ
∂
∂t1+ . . .+
∂(τωm−1(τ))
∂τ
∂
∂tm−1
)q(x̂)
= f(b̂(τ, t1, . . . , tm−1), x̂(τ, t1, . . . , tm−1)).
Important Result
A given solution x̂ of the MPDAE (3) coincides with a solution x ofDAE (1) along the curve i .e. characteristic curve
x(t) = x̂(t, ω1t, . . . , ωm−1t). (3)
In Radio Frequency (RF) applications, many system comprises exactlytwo different time scales :
MPDAE (∂
∂τ+∂(ω(τ)τ)
∂τ
∂
∂t1
)q(x̂(τ, t1)) = f(b̂(τ, t1), x̂(τ, t1)) (4)
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Formulation of Algorithm
Selection of Initial Values
Cont...x̂(0, t1) = h(t1) ∀t1 ∈ R, (5)
x̂(τ, t1) = x̂(τ, t1 + T ) ∀τ, t1 ∈ R,
x̂(τ, t1) ≈K∑
k=−K
Xk(τ) exp (iω(τ)kt1) (6)
The Fourier coefficient Xk : R→ CN with k = −K , . . . ,K .
Initial values estimation, Stephanie Knorr, Wavelet-Based Simulation of MPDAS in RF Applications
Ph.D. Dissertation, Univ. of Wup, 2007
T 2T
T
τ
t1
Figure : Initial values for simulations
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Formulation of Algorithm
Selection of Initial Values
Cont...x̂(0, t1) = h(t1) ∀t1 ∈ R, (5)
x̂(τ, t1) = x̂(τ, t1 + T ) ∀τ, t1 ∈ R,
x̂(τ, t1) ≈K∑
k=−K
Xk(τ) exp (iω(τ)kt1) (6)
The Fourier coefficient Xk : R→ CN with k = −K , . . . ,K .
Initial values estimation, Stephanie Knorr, Wavelet-Based Simulation of MPDAS in RF Applications
Ph.D. Dissertation, Univ. of Wup, 2007
T 2T
T
τ
t1
Figure : Initial values for simulations
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Non-Autonomous Circuits
Amplifier Circuit
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
* [V
] →
t [s] →
ODE solutions
vinvout
00.002
0.0040.006
0.0080.01
00.2
0.40.6
0.81
x 10−3
−1
−0.5
0
0.5
1
τ →
Envelope solutions
← t1
x(τ,
t 1) →
(a) (b)Figure : (a) Solutions along characteristic curve, (b) Waveforms foroutput voltage
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Non-Autonomous Circuits
Differential Flip-Flop Circuit
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10−5
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
t [s] →
* [V
] →
Solution along characteristic line
d.vnclk.vnq.vn
(a) (b)Figure : (a) Solutions along characteristic curve, (b) Waveforms foroutput voltage
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Non-Autonomous Circuits
Differential to Single Circuit
0 0.2 0.4 0.6 0.8 1 1.2
x 10−5
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
t [s] →
* [V
] →
ODE solutions
vnnvnpvq
01
23
45
x 10−6
0
0.5
1
1.5
x 10−5
−1
0
1
2
3
4
t1 [s] →
pde solution
τ [s] →
x(τ,
t 1) →
(a) (b)Figure : (a) Solutions along characteristic curve, (b) Waveforms foroutput voltage
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Autonomous Circuits
Colpitts Oscillators
0 1 2 3 4 5 6 7 8
x 10−6
0.975
0.98
0.985
0.99
0.995
1
1.005 time dependent variable angular frequency
τ →
ω →
Angular frequency
(a) (b)
0 1 2 3 4 5 6 7 8
x 10−6
−5
0
5
10
15
20
25
t→
x →
solution along charcteristics
vn1vn2ibn1
(c)Figure :(a)Angular Frequency variations, (b) Dominating Waveforms (c)Solution along characteristic
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Autonomous Circuits
Voltage Controlled Oscillators
0 0.5 1 1.5 2 2.5
x 10−5
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14 time dependent variable angular frequency
τ →
ω →
Angular frequency
00.5
11.5
22.5
x 10−5
0
0.5
1
1.5
2
x 10−6
−1
0
1
2
3
4
τ →
waveforms
← t1
x(τ,
t 1) →
(a) (b)
0 0.5 1 1.5 2 2.5
x 10−5
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
t→
x →
solution along charcteristic line
outn vnoutp vnIn vn
(c)Figure :(a) Angular Frequency variations, (b) Dominating Waveforms (c)Solution along characteristic curve
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Autonomous Circuits
Pierce Quartz Crystal Oscillators
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.081
1
1
1
1
1 time dependent variable angular frequency
τ →
ω →
Angular frequency
00.02
0.040.06
0.08
0
1
2
3
x 10−3
−4
−3
−2
−1
0
1
2
3
4
x 104
τ →
waveforms
← t1
x(τ,
t 1) →
(a) (b)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08−30
−20
−10
0
10
20
30
t→
x →
solution along charcteristics
vn1vn2vn3vn4vn5
(c)Figure :(a) Angular Frequency variations, (b) Dominating Waveforms (c)
Solution along characteristic curve
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations
A Mixed Time Frequency Algorithm for Circuit Simulations
Test Examples
Autonomous Circuits
Thank you
Ashish Awasthi & H.G. Brachtendorf Research Center Upper Austria University of Applied Sciences, Hagenberg AustriaA Mixed Time Frequency Algorithm for Circuit Simulations