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

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

Tellegen’s Theorem I. 1 3 2 4

Tellegen’s Theorem (con’t) I. 1 3 2 4

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)

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

Outline Tellegen’s Theorem Resistive Network Dynamic System

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

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

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

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

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

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

Put it together… y

Outline Tellegen’s Theorem Resistive Network Dynamic System

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

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) + _

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

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

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

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

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

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