A Mixed Time Frequency Algorithm for Circuit Simulations€¦ · A Mixed Time Frequency Algorithm for Circuit Simulations Outline 1 Introduction 2 Formulation of Algorithm Selection

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


Outline

1 Introduction

2 Formulation of AlgorithmSelection of Initial Values

3 Test ExamplesNon-Autonomous CircuitsAutonomous Circuits



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.



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)




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)




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)




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




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



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



Test Examples


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




Test Examples


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) →




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



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



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



Test Examples

Autonomous Circuits

Thank you


Documents

A Mixed Time Frequency Algorithm for Circuit Simulations€¦ · A Mixed Time Frequency Algorithm for Circuit Simulations Outline 1 Introduction 2 Formulation of Algorithm Selection