EE616

Dr. Janusz Starzyk

Computer Aided Analysis of Electronic Circuits

• Innovations in numerical techniques had profound import on CAD:– Sparse matrix methods.– Multi-step methods for solution of differential

equation.– Adjoint techniques for sensitivity analysis.– Sequential quadratic programming in

optimization.

Fundamental Concepts• NETWORK ELEMENTS:

– One-portResistor voltage controlled.

or current controlled

Capacitor

Inductor

condition

)(vfi

)(ifv

+ V -i

dt

dqiandvfq )(

dt

dvandif

)(

0)0( f

.constv

.consti

Independence voltage source

Independence current source

Fundamental Concepts– Two-port:

Voltage to voltage transducer (VVT):

Voltage to current transducer (VCT):

Current to voltage transducer (CVT):

Current to current transducer (CCT):

Ideal transformer (IT):

Ideal gyrator (IG):

+

V1

-

+

V2

-

i1 i2

121 0 vvi

121 0 gvii

121 0 rivv

121 0 iiv

2121

1i

ninvv

2221 rivriv

Fundamental ConceptsPositive impedance converter (PIC)

Negative impedance converter (NIC)

Ideal operational amplifier (OPAMP)

OPAMP is equivalent to nullor constructed from two singular one-ports:

nullator

and norator

OPAMP nullor

211211 ikivkv

221211 ikivkv 00 11 iv

+ V -i 00 iv

+ V -i arbitraryiv,

+

V2

-

+

V1

- +

V1

-

+

V2

-

i1 i2

i1 i2

Network ScalingTypical design deals with network elements having resistivity from ohms

to MEG ohms, capacitance from fF to mF, inductance from mH to H within frequency range Hz. Consider

EXAMPLE:

Calculate derivative with 6 digits accuracy?

Let

but because of roundoff errors:

Which is 16% error.

910

)()()(

ooo xf

x

xfxxf

86.0)(

10

0000086.1)(

1)(

/

5

0

o

o

xf

x

xxf

xf

1)(

00001.1)(/

o

o

xfand

xxf

Scaling is used to bring network impedance close to unity

Impedance scaling: Design values have subscript d and scaled values

subscript s.

For scaling factor K we get:

Frequency scaling: has effect on reactive elements:

With:

K

LL

K

SL

K

ZZ

KCCKSCK

ZZ

K

RR

dS

dLdLS

dSd

cdCS

dS

1

os

SSdoSdL

SSdoSdC

LjLjLjZ

CjCjCjZ

111

doSdoS LLCC

For both impedance and frequency scaling we have:

WT, CCT, IT, PIC, NIC, OPAMP remain unchanged.

VCT the transcondactance g is multiplied by K.

CVT, IG the transresistance r is divided by K.

KCCK

LL

K

RR

odS

odS

dS

NODAL EQUATIONS

For (n+1) terminal network:

Y V = J

or:

V1

V2

V3

j1

j2

j3

Vn+1

Jn+1

1

2

1

1

2

1

121

122221

111211

nnnnnn

n

n

j

j

j

V

V

V

yyy

yyy

yyy

Y is called indefinite admittance matrix.

For network with R, L, C and VCT we can obtain Y directly from the network.

For VCT: k

m

i

jV1 gV1

gg

ggfrom i

to j

from k to m

when k=I and m=j we have one-port and g = Y:

Liner Equations and Gaussian Elimination:For liner network nodal equations are linear. Nonlinear networks can be

solved by linearization about operating point. Thus solution of linear equations is basic to many problems.

Consider the system of liner equations:

or:

i=Yv

K=i

m=j

Y

YY

YY

from k

from k

to m

to m

bxA

Solution can be obtained by inverting matrix

but this approach is not practical.

Gaussian elimination:

Rewrite equations in explicit from and denote bi by ai,n+1 to simplify notation :

bAx 1

nnnnnn

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

~A

How to start Gaussian elimination?

Divide the first equation by a11 obtaining:

Where

Multiply this equation by a21 and add it to the second. The coefficients of the new second equation are

with this transformation becomes zero. Similarly for the other equations, setting:

1,2211

1,22222121

1,11212111

nnnnnnn

nnn

nnn

axaxaxa

axaxaxa

axaxaxa

1,1

)1(

1

)1(

212

)1(

1 nnn axaxax

1,2,111

11

)1(

njaa

a jj

1,2,1)1(

12122

)1(

njaaaa jjj

)1(

21a

makes all coefficients of the first column zero with exception of .

We repeat this process selecting diagonal elements as dividers and obtaining general formulas

where superscript shows how many changes were made. The resulting equations have the form:

ni

njaaaa jijj

,2

1,2,1)1(

1121

)1(

)1(

11a

kj

kk

ik

k

ijij

k

k

kk

k

kjkj

k

aaaa

aaa)()1()1()(

)1()1()(

/

1,,1

,,1

,,1

nkj

nki

nk

)(

1,

)2(

1,2

)2(

22

)1(

1,1

)1(

12

)1(

121

n

nnn

nnn

nnn

ax

axax

axaxax

Back substitution is used to obtain solution. Last variable is used to obtain xn-1 and so on.

In general:

Gaussian elimination requires operations.

EXAMPLE:

1,11

)()(

1,

nixaaxn

ij

i

jij

i

nii

33n

Example 2.5.b (p70)

While back substitutions requires .

Triangular decomposition:

Triangular decomposition has an advantage over Gaussian elimination as it can give simple solution to systems with different right-hand-side vectors and transpose systems required in sensitivity computations.

Assume we can factor matrix as follows:

where

22n

~A

~~~ULA

nnnn lll

ll

l

L

21

2221

11

~ 0

1

1

1

0

223

11312

~

n

n

uu

uuu

U

L stands for lower triangular and U for upper triangular. Replacing A by LU the system of equations takes a form:

L U X = b

Define an auxiliary vector Z as

U X = Z

then L X = b and Z can be found easily as:

so

Zn=b1/l11

and

nnnnnn bzlzlzl

bzlzl

bzl

2211

2222121

1111

nilZlbZ ii

i

jjijii ,,2

1

1

This is called forward elimination. Solution of UX=Z is called backward substitution. We have

so Xn=Zn

and

to find LU decomposition consider matrix.

Taking product of L and U we have :

4424421441432342134142124141

343324321431332332133132123131

242214212322132122122121

14111311121111

lulullulullull

ululullulullull

ulululullull

ululull

A

nn

nn

nn

zx

zxux

zxuxux

222

112121

1,,11

niZUZXn

ijjijii

44

From the first column we have

from the first row we find

from the second column we have

and so on…

In machine implementation L and U will overwrite A with L occupying the lower and U the upper triangle of A.

In general the algorithm of LU decomposition can be written as (Crout algorithm):

1. Set k=1 and go to step 3.

2. Compute column k of L using:

4,,111 ial ii

4,,21111 jlau jj

4,,212122 iulal iii

if k=n stop.

3. Compute row k of U using

4. Set k=k+1 and go to step 2.

This technique is represented in text by CROUT subroutine. Modification which is dealing with rows only by LUROW.

Modification of Gaussian elimination which give LU decompositions realized by LUG subroutine.

Features of LU decomposition:

1. Simple calculation of determinant

2.if only right-hand-side vector b is changed there is no need to recalculate the decomposition and only forward and backward substitution are performed, which takes n2 operations.

3.Transpose system AT X = C required for sensitivity calculation `can be solved easily as AT = UTLT.

kiulalk

mmkimikik

1

1

kjlulau kk

k

mmjkmkjkj

1

1

n

iiilA

1

det

4. Number of operation required for LU decomposition is (equivalent to Gaussian elimination.)

nnM 3

3

1

Example 2.5.1

2.6 PIVOTING:

the element by which we divide (must not be zero) in gaussian elimination is called pivot. To improve accuracy pivot element should have large absolute value.

Partial pivoting: search the largest element in the column.

Full pivoting: search the largest element in the matrix.

Example 2.6.1

SPARSE MATRIX PRINCIPLES

To reduce number of operations in case many coefficients in the matrix A are zero we use sparse matrix technique. This not only reduces time required to solve the system of equations but reduces memory requirements as zero coefficients are not stored at all.

(read section 2.7)

Pivot selection strategies are motivated mostly by the possibility to reduce the number of operations.