CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification

Instructor:Prof. Chung-Kuan Cheng

Winter 2003

Lecture 1: Formulation

Jan. 24, 2003 Cheng & Zhu, UCSD @ 20032

Agenda RCL Network Sparse Tableau Analysis Modified Nodal Analysis


History of SPICE SPICE -- Simulation Program with Integrat

ed Circuit Emphasis 1969, CANCER developed by Laurence Nag

el on Prof. Ron Roher’s class 1970~1972, CANCER program May 1972, SPICE-I release July ’75, SPICE 2A, …, 2G Aug 1982, SPICE 3 (in C language) No new progress on software package sinc

e then


RCL circuit

Vs

R L

C

svi

v

ri

v

l

c 0

1

10

0

0

1

2

1

2

l

vi

v

l

r

l

ci

vs

0

1

10

1

2

1

2


RCL circuit (II) General Circuit Equation

Consider homogeneous form first

BUAYY

AYY

0YeY At

...!

...!2!1

22

k

tAtAAtIe

kkAt

Q: How to Compute Ak ?

and


Assume A has non-degenerate eigenvalues

and corresponding linearly independent eigenvectors , then A can be decomposed as

where and

Solving RCL Equation

1A

k ,...,, 21

k ,...,, 21

k

0

0

00

2

1

k ,...,, 21


What’s the implication then?

To compute the eigenvalues:

Solving RCL Equation (II)

1A

01

1 ...)det( ccAI nn

n

0)...(...))(( 212

0 ppp

realeigenvalue Conjugative

Complexeigenvalue

212A 212A

tAt ee 1 where

ke

e

e

e t

0

0

002

1


Solving RCL Equation (III)

In the previous example

)0(

)0(

1

2//1

/10

01

2

i

veXe

i

v tlrl

c

At

0

0

11

10 11A

2

31

2

31

j

j

where

1

2

31

12

31

3 j

jj

2

31

2

3111

1 jj

hence

e

ee At

0

01

Let c=r=l=1, we have


What if matrix A has degenerated eigenvalues? Jordan decomposition !

Solving RCL Equation (IV)

JA 1

J is in the Jordan Canonical form

And still JtAt ee 1


Jordan Decomposition

0

1J

t

ttJt

e

teete

00

1

10

01

00

10

01

J

t

tt

ttt

Jt

e

tee

et

tee

te

00

0!2

00

10

01

100

010

001

2

similarly




Equation Formulation KCL

Converge of node current KVL

Closure of loop voltage Brach equations

I, R relations


Types of elements Resistor Capacitor

Inductor

L is even dependent on frequency due to skin effect, etc…

Controlled Sources VCVS, VCCS, CCVS, CCCS

dt

dvvc

dt

dv

v

vQ

dt

dQi )(

)(

dt

diil

dt

di

i

i

dt

dv )(

)(


Cut-set analysis

1. Construct a spanning tree

2. Take as much capacitor branches as tree branches as possible

3. Derive the fundamental cut-set, in which each cut truncates exactly one tree branch

4. Write KCL equations for each cut

5. Write KVL equations for each tree link

6. Write the constitution equation for each branch

1

2

3

4

5

6


0

0

0

0

0

11000000

10110000

00011010

00000110

00000011

56

46

45

35

34

24

13

12

46

45

34

24

12

5646453534241312

i

i

i

i

i

i

i

i

c

c

c

c

ciiiiiiii

KCL Formulation

0

iA

1

2

3

4

5

6

#nodes-1 lines

#braches columns


KCL Formulation (II) Permute the columns to achieve a

systematic form

0~

|

iAI

0

0

0

0

0

100

110

011

001

001

10000

01000

00100

00010

00001

56

23

13

46

45

34

24

12

46

45

34

24

12

i

i

i

i

i

i

i

i

c

c

c

c

c5623134645342412 iiiiiiii

1

2

3

4

5

6


KVL Formulation

56

46

45

35

34

24

13

12

46

45

34

24

12

11000

10000

01000

01100

00100

00010

00111

00001

v

v

v

v

v

v

v

v

v

v

v

v

v

VeAT

0

10011000

01001100

00100111

56

35

13

46

45

34

24

12

56

35

13

v

v

v

v

v

v

v

v

l

l

l

0BV

Remove the equations for tree braches and systemize

IBB~

1

2

3

4

5

6


Cut & Loop relation

0

BV

VeAT

0TBA

0]~

][~[ TAIIB

0~

TAB

TAB~~

100

110

011

001

001

~A

11000

01100

00111~B

In the previous example


Sparse Tableau Analysis (STA) n=#nodes, b=#branches

e

v

i

KK

AI

A

vi

T

0

0

00(n-1) KCL

b KVL

b branch relations

b

b

n-1

S

0

0

Totally 2b+n-1 variables, 2b+n-1 equations

b b n-1

Due to independentsources


STA (II) Advantages

Covers any circuit Easy to assemble Very sparse

Ki, Kv, I each has exactly b non-zeros. A and AT each has at most 2b non-zeros.

Disadvantages Sophisticated data structures & programmin

g techniques




Nodal Analysis Derivation

SvKiK

eAv

iA

vi

T

0

SAKeAKAK

SAKvKAKiA

SKvKKi

iT

vi

ivi

ivi

11

11

11

From STA:

(1)

(2)

(3)

(3) x Ki-1

(4) x A

(4)

Using (a)

(5)

(6)

Tree trunk voltages

Substitute with node voltages (to a given reference), we get the nodal analysis equations.

e


Nodal Analysis (II)

SAKeAKAK iT

vi

11

1iK

12y13y

24y34y

35y45y

46y56y

IKv

11000000

10110000

00011010

00000110

00000011

A

46

45

34

24

12

564656

5656453535

353534131313

13241313

13131312

1

v

v

v

v

v

yyy

yyyyy

yyyyyy

yyyy

yyyy

eAKAK Tvi

1

2

3

4

5

6


2

2

2222

2

W

J

I

V

ZAY

AY nT

n

Modified Nodal Analysis General Form

Node Conductance matrix

KCL

Independent current source

Independent voltage source

Due to non-conductive elements

Yn can be easily derived

Add extra rows/columns for each non-conductive elements using templates


11000000

10110000

01101000

00011010

00000101

00000011

MNA (II) Fill Yn matrix according to incidence matrix

5645354535

45464534243424

353435341313

24241212

13121312

00

0

0

00

00

yyyyy

yyyyyyy

yyyyyy

yyyy

yyyy

Yn

5646453534241312 iiiiiiii

6

5

4

3

2

1

n

n

n

n

n

n

A

Choose n6 as reference node

1

2

3

4

5

6

Ten AAYY


MNA Templates

j

j

i

i'j

j

j

'j

ji

11

1

1

1

'

m

j

j

'jj vv

gv

Independent current source

Independent voltage source

j

'j

gv

Add to the right-hand side of the equation

jiji


MNA Templates (II)

CCVS

'j

ji

j k

'k

ji

CCCS

11

1

1

jcc iNNNN

1

m

N

N

N

N

c

c

r11

11

1

1

1

1kjkkjj iivvvv ''

2

1

'

'

m

m

k

k

j

j

'j

ji

j k

'k

jri

ki


MNA Templates (III)

'j

j k

k

jv

VCVS

'j

jv

j k

'k

jmvg

VCCS

11

1

1

kkkjj ivvvv ''

1

'

'

m

k

k

j

j+

-

jv

ki

+

-

mm

mm

gg

gg'k

k

'jj vv


2

1

11

11

1

1

1

1

LjMj

MjLj

MNA Templates (IV)

j

'j

k

'k

+ +

- -

jv kv

1L 2L

1i 2i

j

'j

k

'k

i

'

'

k

k

j

j

M'' kkjj vvvv 2i

'

'

k

k

j

j

2

1

m

m

Mutual inductance

Operational Amplifier

11

1

1

'' kkjj vvvv i

1m

1i


MNA Example

n 1n 2

n 3 n 4G 2 G 3

C 4

C 5v g

n 0

+

_

v 6 v 7= v 6

Circuit Topology

0

0

0

0

0

00100

000001

1000

0000

00

0100

4

3

2

1

44

533

434322

22

g

vcvs

g

n

n

n

n

v

i

i

v

v

v

v

CjCj

CjGG

CjGCjGGG

GG

n 1 n 2 n 3 n 4

n 0

2 3

4

5 6 71i g

i vcvs

MNA

Equations


MNA Summary Advantages

Covers any circuits Can be assembled directly from input data.

Matrix form is close to Yn

Disadvantages We may have zeros on the main diagonal. Principle minors could be singular

Documents

CSE 245: Computer Aided Circuit Simulation and Verification