1 adjoint method in network analysis dr. janusz a. starzyk

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Adjoint Method in Network Analysis

Dr. Janusz A. Starzyk

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Outline

-- Definition of Sensitivities

-- Derivatives of Linear Algebraic Systems

-- Adjoint Method

-- Adjoint Analysis in Electrical Networks

-- Consideration of Parasitic Elements

-- Solution of Linear Systems using the Adjoint Vector

-- Noise Analysis Using the Adjoint Vector

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Sensitivity

Normalized sensitivity of a function F w.r.t. parameter

Two semi-normalized sensitivities are discussed when either F or h is zero

and

F can be a network function, its pole or zero, quality factor, resonant frequency, etc., while

h can be component value, frequency s, operating temperature humidity, etc.

h

F

F

h

hln

FlnS F

h

h

Fh

h

FS Fh

ln h

F

Fh

FS Fh

1ln

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Derivatives of Linear Algebraic Systems

Consider a linear system

(i) TX = W

where T and W are, in general case, functions of parameters h. Differentiate (i) with respect to a single parameter hi

We are interested in derivatives of the response vector, so we can get

(ii)

hi

WX

hi

T

hi

XT

hi

WX

hi

TT

hi

X 1

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Adjoint Method

Very often, the output function is a linear combination of the components of X

(iii)

where d is a constant (selector) vector. We will compute using the so called adjoint method.

From (ii) and (iii) we will get

Let us define an adjoint vector to get

(iv)

Xd T

hi /

hi

WX

hi

TTd

hi1T

1TTa TdX

hi

WX

hi

TX

hi

Ta

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Adjoint Method

From its definition, the adjoint vector can be obtained by solving

(v)

Note that solution of this system can be obtained based on LU factorization of the original system - thus saving computations, since

1TTa TdX dXT aT

TTTT LULUT

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Adjoint Method - example

Find sensitivity of Vout with respect to G4.

From KCL:

System equations TX = W are

C2=1

E=1G3=1

G1=1

G4=4

Vout

+

-

+

-

343

221

GGG

sCsCG

out

4

v

v

0

EG1

=

0

1

15

12 4

outv

v

v4 0

0

4344

4241

out

out

vvGvG

vvsCEvG

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If we use s = 1 then the solution for X is

calculate

therefore

3/5

3/1

0

1

25

11

3

10

1

15

121

4

outv

v

0

0,

01

00

4343

221

44G

Wand

GGG

sCsCG

GG

T

3

1

0

3

53

1

01

00

44 G

wX

G

T

9


Since Vout =[0 1] X, we get d = [0 1]T, and compute the adjoint vector from

so

and the output derivative is obtained from equation (iv)

dXT aT

1

0

11

52

2

1a

a

x

x

3/2

3/5

1

0

21

51

3

1

2

1

a

a

x

x

9

2

3/1

03/23/5

444

G

WX

G

TX

G

v Taout

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Adjoint Analysis in Electrical Networks

Adjoint analysis is extremely simple in electrical networks and have the following features:

1. Derivative to a source is simple, since in this case

and

where eK is defined as a unit vector:

and the output derivative w.r.t. source is

0h

T

ji eeh

W

0...001..00K

T

Ke

ai

ajji

Ta xxeeXh

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2. Derivative to a component is also simple, since each component value appears in at most 4 locations in matrix T

so

and the derivative of the output function is found as

hh

hh

j

T

iLKtionertin sec

10 orisvwhereeeeesh

T TLKji

v

LKaj

ai

vTLKji

Tav xxxxsXeeeeXsh

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In the previously analyzed network we had:

and

Thus to find the derivative we need to calculate

- only a single multiplication

Adjoint Analysis in Electrical Networks - example

4G

vout

9

2

3

1

3

212

0

4

xxsG

v aout

3

53

1

X

3

23

5

aX

343

221

GGG

sCsCG

out

4

v

v

0

EG1

=

13


3. Derivative to parasitic elements can be calculated without additional analysis. We can use the same vectors X and Xa,

since the nominal value of a parasitic is zero. Suppose that we want to find a derivative with respect to a

parasitic capacitance CP shown in the same system, then

343

221

GGG

sCsCG

out

4

v

v

0

EG1

=

C2=1

E=1G3=1

G1=1

G4=4

Vout

+

-

+

-Cp

9

5

3

1

3

511

ssxxs

Ca

P

considering parasitic location

and there is no need to repeat the circuit analysis

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Finding a response of a network with different right hand side vectors is easy using the adjoint vectors.

– Consider a system with different r.h.s. vectors:

– (vi)

– we have– (vii)

– so all i can be obtained with a single analysis of the adjoint system

– this is a significant improvement comparing to repeating forward

and backward substitutions for each vector Wi.

Solution of Linear Systems using the Adjoint Vector

miXd

WTX

i

T

i

ii ...,,2,1

i

Tai

1Ti

1Ti WXWTdWTd

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Noise analysis is always performed with the use of linearized network model because amplitudes involved are extremely small.

– To illustrate how the adjoint analysis can be used in estimation of the noise signal let us consider thermal noise of a resistive element described by an independent current

source in parallel with noiseless resistor.

Noise Analysis Using the Adjoint Vector

R

RfkTin 4

where k Boltzmann's constantT temperature in Kelvinsf frequency bandwidth

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We assume that noise sources are random and uncorrelated.

The mean-square value of the output noise energy is

– where is the output signal due to the i-th noise source.

Since the noise sources are uncorrelated, we cannot use superposition.

Instead the linear circuit has to be analyzed with different noise sources as excitations (different r.h.s. vectors in system equations).


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2

2

1

2... n

mnnn VVVV

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We can use equation (vi) to perform noise analysis very efficiently. We will get

(viii)

– where is the output signal due to the i-th noise source.

Since contains at most two entries

then only one subtraction and one multiplication are needed for each noise source.


ni

Tani WXv

niW

nii

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Noise Analysis Using Adjoint Vectors - example

Example:

Calculate the signal-to-noise ratio for the output voltage. Ignore noise due to op-amp.

C2=1

E=1G3=1

G1=1

G4=4

Vout

+

-

+

-

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The adjoint vector was found in the previous example.

Using (viii) we have the nominal output

The same equation is used to obtain noise outputs:


3/2

3/5

2

1a

a

x

x

0

1

3

2

3

500 WxV

Ta

nn

nTan ii

WXV 11

11 3

5

03

2

3

5

20

and

Thus the total noise signal is:


nn

n

nn

n

ii

V

ii

V

44

4

33

3

3

20

3

2

3

5

3

20

3

2

3

5

2

4

2

3

2

1

2

9

4

9

4

9

25 nnnn iiiV

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We can replace by with to obtain

and the signal to noise ratio is computed from:


nii iGfTk 4

ii R

G1

fkTfkT

GGGfkTV n

20)16425(49

1

)4425(49

1431

2

203

50

fkTV

Vn

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Adjoint method is an efficient numerical technique Adjoint vector can be used used to calculate output

derivatives to various circuit parameters Adjoint vector can be used to find a response of a

network with different right hand side vectors Sensitivity analysis, circuit optimization and noise

analysis can benefit from this approach

Summary

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Questions?

1 adjoint method in network analysis dr. janusz a. starzyk

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