Sensitivity Analysis by Adjoint Network
24
Sensitivity Analysis by Adjoint Network CK Cheng CSE 245: Computer Aided Circuit Simulation and Verification 1
description
CSE 245: Computer Aided Circuit Simulation and Verification. Sensitivity Analysis by Adjoint Network. CK Cheng. Outline. Tellegen’s Theorem Resistive Network Dynamic System. Tellegen’s Theorem . Tellegen’s Theorem: For a vector of branch voltages and branch currents, we have . - PowerPoint PPT Presentation
Transcript of Sensitivity Analysis by Adjoint Network
1
Sensitivity Analysis by Adjoint Network
CK Cheng
CSE 245: Computer Aided Circuit Simulation and Verification
2
Outline
• Tellegen’s Theorem
• Resistive Network
• Dynamic System
3
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
4
Tellegen’s Theorem
0Tb bV I
1 3
2
4
12
131
202
303
344
40
1 1 0 01 0 1 00 1 0 00 0 1 00 0 1 10 0 0 1
vvvvvvvvvv
I.bn
T VVE
5
Tellegen’s Theorem (con’t)
1 3
2
4
0000
110000011010000101000011
40
34
30
20
13
12
iiiiii0bE I
0TT T T
b b n b n bV I E V I V EI
0Tb bV I I.
6
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
7
Tellegen’s Theorem (con’t)
0Tb bV I II.
Example case: Two circuits with the same topology
12
13
20
30
34
40
21
4321
vvvvvv
0011
11
~~~~~~
40
34
30
20
13
12
iiiiii 12
13
20
30
34
40
21214
3
vvvvvv
111
22
2
40
34
30
20
13
12
iiiiii
0T T T Tb b b b b b b bV I V I V I V I
8
Outline
• Tellegen’s Theorem
• Resistive Network
• Dynamic System
9
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
RDkRy
k
10
Example of Sensitivity Calculation
v1R’’’
R’=R+∆R
R’’i3
i2
+_
Given circuit
y=v1 + 2i2 + 4i3
11
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
12
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
13
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 ig i
where v 0k ki
ik
gk
+ _
14
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
15
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 ~
MRy
Ry
Ry
1 ki~ki
y
16
Outline
• Tellegen’s Theorem
• Resistive Network
• Dynamic System
17
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
EkCy
DkRy
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
18
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
19
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)
20
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
21
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
22
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
23
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
24
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
y i t i T t dt k DR
y v T t v t dt k EC
