Sensitivity Analysis by Adjoint Network

CK Cheng, May 2010

CSE 245: Computer Aided Circuit Simulation and Verification

1

Outline

• Tellegen’s Theorem

• Resistive Network

• Dynamic System

2

Tellegen’s Theorem

Tellegen’s Theorem: For a vector of branch voltages and branch currents, we have

0T T T Tb b b b b b b bV I V I V I V I

0Tb bV I

0Tb bV I

3

Tellegen’s Theorem

0Tb bV I

1 3

2

4

12

131

202

303

344

40

1 1 0 0

1 0 1 0

0 1 0 0

0 0 1 0

0 0 1 1

0 0 0 1

v

vv

vv

vv

vv

v

I.bn

T VVE

4

Tellegen’s Theorem (con’t)

1 3

2

4

0

0

0

0

110000

011010

000101

000011

40

34

30

20

13

12

i

i

i

i

i

i0bE I

0TT T T

b b n b n bV I E V I V EI

0Tb bV I I.

5

Tellegen’s Theorem (con’t)

0Tb bV I II.

Example: Two circuits with the same topology

1 (2v) 3 (3v)

2 (4v)

4 (1v)

0

0

1-1

1

-1

1 (0v) 3 (-1v)

2 (2v)

4 (3v)

-1

-1

-12

-2

2

6

Tellegen’s Theorem (con’t)

0Tb bV I II.

Example case: Two circuits with the same topology

12

13

20

30

34

40

2

1

4

3

2

1

v

v

v

v

v

v

0

0

1

1

1

1

~

~

~

~

~

~

40

34

30

20

13

12

i

i

i

i

i

i 12

13

20

30

34

40

2

1

2

1

4

3

v

v

v

v

v

v

1

1

1

2

2

2

40

34

30

20

13

12

i

i

i

i

i

i

0T T T Tb b b b b b b bV I V I V I V I

7

Outline

• Tellegen’s Theorem

• Resistive Network

• Dynamic System

8

Resistive Network

• Metric

– A: branch voltage– B: branch current

• Sensitivity Calculation

– D is the set of resistance

0T Tb b b bV I V I k k k k

k A k B

y f v g i

ib

Vb

RDk

R

y

k

9

Example of Sensitivity Calculation

v1R’’’

R’=R+∆R

R’’i3

i2

+_

Given circuit

y=v1 + 2i2 + 4i3

10

Adjoint Network for Resistive Network

Vk

ik

-fk

gk

+ _

+ _

original network adjoint network

( ) ( ) ( )T Tb b b b k k k k k k k k k k k k

k A k B k D

V I V I v i v i v i v i v i v i

RR+∆R

11

Adjoint Network for Resistive Network (con’t)

For set A (branch voltage),

vk

-fk

+ _

original network adjoint network

where k k kv i f

0k k k k k k

k k

v i v i v f

f v

where 0k kv i

12

Adjoint Network for Resistive Network (con’t)

For set B (branch current),

original network adjoint network

where is unknownk k kv g i

0k k k k k k k

k k

v i v i g i i

g i

where v 0k ki

ik

gk

+ _13

Adjoint Network for Resistive Network (con’t)

For set D (resistance),

original network adjoint network

k k k kv R R i

k k k k k k k k k k k

k k k

v i v i i R i i R R i

i R i

kRk kv i

RR+∆R

14

Put it together…

( ) ( ) ( ) 0T Tb b b b k k k k k k k k k k k k

k A k B k D

V I V I v i v i v i v i v i v i

Dk

kkkBk

kkAk

kk iRiigVf~

MR

y

R

y

R

y

1 ki~ki

y

15

Outline

• Tellegen’s Theorem

• Resistive Network

• Dynamic System

16

Dynamic System • Metric

– A: branch voltage– B: branch current

• Sensitivity Calculation

– D is the set of resistance– E is the set of capacitance

0 0

( ) ( ) ( ) ( )T T

k k k kk A k B

y f t v t dt g t i t dt

R

EkC

y

DkR

y

k

k

C

L

0

( ) ( ) ( ) ( ) 0T

b b b bv T t i t v t i T t dt

Vb(t)

ib(t)

We omit L here which is similar to C 17

Adjoint Network for Dynamic System

vk(t)

ik(t)

-fk(T- t)

gk(T- t)

+ _

+ _

original network adjoint network

RR+∆R

C

L

C+ ∆ C

L+ ∆ L

, , ,0 0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0T T

b b b b k k k kk A B D E

v T t i t v t i T t dt v T t i t v t i T t dt

18

Adjoint Network for Dynamic System (con’t)

For set A (branch voltage),

+ _

original network adjoint network

where ( ) ( )k k kv (t) i t f t

0 0

( ) ( ) ( ) ( ) ( )T T

k k k k k kv T t i t v t i T t dt f t v t dt

where ( ) 0k kv (t) i t

vk(t)

-fk(T- t)

19

Adjoint Network for Dynamic System (con’t)

For set B (branch current),

original network adjoint network

t where t is unknownk k kv T t g i t where v 0k ki t

ik(t)gk(T- t)

+ _

0 0

( ) ( ) ( ) ( ) ( )T T

k k k k k kv T t i t v t i T t dt g t i t dt

20

Adjoint Network for Dynamic System (con’t)

For set D (resistance),

original network adjoint network

k k k kv t R R i t kt Rk kv t i

RR+∆R

0 0

0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )T T

k k k k k k k k k k k

T

k k k

v T t i t v t i T t dt i T t R i t i t R R i T t dt

i t R i T t dt

21

Adjoint Network for Dynamic System (con’t)

For set E (capacitance),

original network adjoint network

where ( )k k k k kv t i t dt C C dv t where t ( )k k k kv t i dt C dv t

CC+ ∆ C

22

Adjoint Network for Dynamic System (con’t)

For set E (capacitance),

0

( ) ( ) ( ) ( )T

k k k kv T t i t v t i T t dt

0 0

0 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

T T

k k k k k k

T T

k k k k k k

v T t i t v T t C C dv t

v T t C dv t v T t C dv t

00 0

0

( ) ( ) ( ) ( ) ( ) ( ( ) )

(0) ( ) ( ) (0) ( ) ( ( ) )

T TT

k k k k k k k k kt

T

k k k k k k k k k

v T t C dv t v T t C v t v t d v T t C

v C v T v T C v v t d v T t C

cancellation

0 0

( ) ( ) ( ) ( ) ( ) (0) (0) ( ) ( )T T

k k k k k k k k k kv T t i t v t i T t dt v T C v v T t C v t dt �overall

Set v (0) 0k

23

Put it together…

, , ,0 0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0T T

b b b b k k k kb k A B D E

v T t i t v t i T t dt v T t i t v t i T t dt

0 0

( ) ( ) ( ) (0) ( ) ( ) ( )T T

k k k k k k k k kk D k E k E

y i t i T t R dt v T C v v T t C t v t dt

�

The initial voltage of the capacitor remains fixed when other element parameters vary. Thus, delta vk(0)=0.

0

0

( ) ( )

( ) ( )

T

k kk

T

k kk

yi t i T t dt k D

R

yv T t v t dt k E

C

�

24