1. Sensitivity Analysis by Adjoint Network
CSE 245: Computer Aided Circuit Simulation and Verification
Sensitivity Analysis by Adjoint Network
CK Cheng
CSE Dept.
UC San Diego

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

4. Tellegen’s Theorem I. 1 3 2 4

5. Tellegen’s Theorem (con’t)I. 1 3 2 4

6. Tellegen’s Theorem (con’t)
II.
Example: Two circuits with the same topology
1 (2v)
3 (3v)
1 (0v)
3 (-1v) 1 -2 -1
4 (1v)
4 (3v) -1 1 2 -1 -1 -1 2
2 (4v)
2 (2v)

7. Tellegen’s Theorem (con’t)
II.
Example case: Two circuits with the same topology

8. Outline
Tellegen’s Theorem
Resistive Network
Dynamic System

9. Resistive Network Metric Sensitivity Calculation A: branch voltage
B: branch current
Sensitivity Calculation
D is the set of resistance ib R Vb

10. Example of Sensitivity Calculation
Given a circuit i2 v1
R’’’ i3
R’’ + _
R’=R+∆R
The objective function (e.g.) y=v1 + 2i2 + 4i3

11. Adjoint Network for Resistive Network
original network
adjoint network
-fk + _ Vk
R+∆R R gk ik + _

12. Adjoint Network for Resistive Network (con’t)
For set A (branch voltage),
original network
adjoint network
-fk + _ vk

13. Adjoint Network for Resistive Network (con’t)
For set B (branch current),
original network
adjoint network gk ik + _

14. Adjoint Network for Resistive Network (con’t)
For set D (resistance),
original network
adjoint network
R+∆R R

15. Put it together… y

16. Outline
Tellegen’s Theorem
Resistive Network
Dynamic System

17. Dynamic System Metric Sensitivity Calculation A: branch voltage
B: branch current
Sensitivity Calculation
D is the set of resistance
E is the set of capacitance
ib(t) R
Vb(t) C L
We omit L here which is similar to C

18. Adjoint Network for Dynamic System
original network
adjoint network
-fk(T- t) + _
vk(t)
R+∆R R
C+ ∆ C C
L+ ∆ L L
gk(T- t)
ik(t) + _

19. Adjoint Network for Dynamic System (con’t)
For set A (branch voltage),
original network
adjoint network
-fk(T- t) + _
vk(t)

20. Adjoint Network for Dynamic System (con’t)
For set B (branch current),
original network
adjoint network
gk(T- t)
ik(t) + _

21. Adjoint Network for Dynamic System (con’t)
For set D (resistance),
original network
adjoint network
R+∆R R

22. Adjoint Network for Dynamic System (con’t)
For set E (capacitance),
original network
adjoint network
C+ ∆ C C

23. Adjoint Network for Dynamic System (con’t)
For set E (capacitance),
cancellation
overall

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