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 3

Tellegen’s Theorem I. 4

Tellegen’s Theorem (con’t) I. 5

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

Tellegen’s Theorem (con’t) II. Example case: Two circuits with the same topology 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 ibib VbVb R 9

Example of Sensitivity Calculation v1v1 R’’’ R’=R+∆R R’’i3i3 i2i2 +_+_ Given circuit y=v 1 + 2i 2 + 4i 3 10

Adjoint Network for Resistive Network VkVk ikik -f k gkgk +_+_ + _ original networkadjoint network RR+∆R 11

Adjoint Network for Resistive Network (con’t) For set A (branch voltage), vkvk -f k + _ original networkadjoint network 12

Adjoint Network for Resistive Network (con’t) For set B (branch current), original networkadjoint network ikik gkgk +_+_ 13

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

Put it together… 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 R C L V b (t) i b (t) We omit L here which is similar to C 17

Adjoint Network for Dynamic System v k (t) i k (t) -f k (T- t) g k (T- t) +_+_ + _ original networkadjoint network RR+∆R C L C+ ∆ C L+ ∆ L 18

Adjoint Network for Dynamic System (con’t) For set A (branch voltage), + _ original networkadjoint network v k (t) -f k (T- t) 19

Adjoint Network for Dynamic System (con’t) For set B (branch current), original networkadjoint network i k (t) g k (T- t) +_+_ 20

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

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

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

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