Computer Aided Circuit Simulation and Verification Sensitivity Analysis
Explore the principles of sensitivity analysis in computer-aided circuit simulation, focusing on Tellegen's Theorem and resistive network dynamics. Understand how adjoint networks simplify sensitivity calculations for efficient verification. Dive into examples illustrating the application of Tellegen's Theorem for different circuits.
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CSE 245: Computer Aided Circuit Simulation and Verification Sensitivity Analysis by Adjoint Network CK Cheng CSE Dept. UC San Diego 1
Outline Introduction Tellegen s Theorem Resistive Network Dynamic System 2
Introduction Sensitivity Calculation Direct calculation: One simulation for each perturbation (Can work for multiple obj. functions). Adjoint network: One simulation for many possible perturbations (Work for one obj. function). 3
Tellegens Theorem Tellegen s Theorem: For a vector of branch voltages and branch currents, we have = = = = T T T T 0 V I V I V I V I b b b b b b b b Note that the branch voltages ??, ?? and branch currents ??, ?? obey the KVL and KCL. 4
Tellegens Theorem V I = T 0 I. b b = T E V V n b v v v v v v 1 1 0 0 0 0 1 0 0 0 0 0 12 = 0 1 0 0 0 1 v v v v 13 1 1 3 0 1 1 0 20 2 4 30 3 1 34 4 1 2 40 5
Tellegens Theorem (cont) V I = T 0 I. b b E I = 0 12 i b 1 3 1 1 0 0 0 0 0 13 i 0 1 0 1 0 0 0 0 20 i 4 = 0 1 0 1 1 0 0 30 i 0 0 0 1 1 0 34 i 2 40 i T = = = T T T 0 V I E V I V EI b b n b n b 6
Tellegens Theorem (cont) V I = T 0 II. b b Example: Two circuits with the same topology 1 (2v) 1 (0v) 3 (3v) 3 (-1v) 1 -2 0 -1 4 (1v) 4 (3v) -1 2 1 -1 0 -1 -1 2 2 (4v) 2 (2v) 7
Tellegens Theorem (cont) V I = T 0 II. b b Example case: Two circuits with the same topology 2 1 4 3 2 1 v 0 40 i ~ 12 i 1 v v v v v v v v v v v 1 2 2 12 i 12 12 ~ 13 i 1 2 2 13 i 13 13 ~ 20 i 1 1 2 20 i 20 20 = = = = ~ 30 i 1 4 1 30 i 30 30 ~ 34 i 0 1 34 i 34 34 ~ 3 1 40 i 40 40 = = = = T T T T 0 V I V I V I V I b b b b b b b b 8
Outline Tellegen s Theorem Resistive Network Dynamic System 9
Resistive Network Objective Function = + y k k f v g i k k k A k B Set A: branch voltages (? = 0) Set B: branch currents (? = 0) Sensitivity Calculation ib y k D R R k Set D: resistances Vb 10
Example of Sensitivity Calculation Given a circuit i2 ?1 R i3 R +_ R =R+ R The objective function (e.g.) ? = ?1 + 2?2 + 4?3 11
Adjoint Network for Resistive Network = + y k k f v g i k k k A k B original network adjoint network -fk _ + Vk R+ R R gk ik +_ = + + T T ( ) ( ) ( ) V I V I v i v i v i v i v i v i b b b b k k k k k k k k k k k k k A k B k D 12
Adjoint Network for Resistive Network (con t) For set A (branch voltage), original network adjoint network -fk _ + vk i = where 0 v ??= ?? where ?? ?? ??????? k k ( ) = = 0 f v v i v i v f k k k k k k k k 13
Adjoint Network for Resistive Network (con t) For set B (branch current), original network adjoint network gk ik +_ = where v 0 i = where is unknown k i v g k k k k = = 0 v i v i g i g i i k k k k k k k k k 14
Adjoint Network for Resistive Network (con t) For set D (resistance), original network adjoint network R+ R R ( ) = = + R v i v R R i k k k k k k k ?? ?? ?? ??= ?????? ????+ ?? ??= ?? ?? ?? 15
Put it together = + + = T T ( ) ( ) ( ) 0 V I V I v i v i v i v i v i v i b b b b k k k k k k k k k k k k k A k k B k D ~ k k + = f V g i i R i k k k k k k k A B D y y R 1 y where ?? = ???= ???? R R y M 16
Outline Tellegen s Theorem Resistive Network Dynamic System 17
Dynamic System Objective Function ib(t) T T = + ( ) ( ) f t v t dt ( ) ( ) k g t i t dt y k k k R k A k B 0 0 Set A: branch voltages Set B: branch currents Sensitivity Calculation Vb(t) C R y L k D k C y k E T = ( ) ( ) b t i t ( ) ( v t i T ) 0 v T t dt k b b b 0 Set D: resistances Set E: capacitances (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 ??(0) = 0 L+ L L gk(T- t) ik(t) +_ T T = = ( ) ( ) b t i t ( ) ( v t i T ) ( ) ( ) k t i t ( ) ( v t i T ) 0 v T t dt v T t dt b b b k k k k A B D E , , , 0 0 19
Adjoint Network for Dynamic System (con t) For set A (branch voltage), original network adjoint network -fk(T- t) _ + vk(t) i t = where ( ) v (t) 0 ??? ? = ??? ? ??? ??(?) ?? ??????? k k T T ( ) t v t dt = ( ) ( ) k t i t ( ) ( v t i T ) ( ) v T t dt f k k k k k 0 0 20
Adjoint Network for Dynamic System (con t) For set B (branch current), original network adjoint network gk(T- t) ik(t) +_ ( ) t where v ( ) t = ( ) ( ) t where ( ) t is unknown = 0 i v T t g i k k k k k T T ( ) t i t dt = ( ) ( ) k t i t ( ) ( v t i T ) ( ) v T t dt g k k k k k 0 0 21
Adjoint Network for Dynamic System (con t) For set D (resistance), original network adjoint network R+ R R ( ) ( ) t R = ( ) ( v t ) ( ) v t i = + R R i t k k k k k k k T T ( ) = R i T + ( ) ( ) k t i t ( ) ( v t i T ) ( ) ( ) ( ) ( ) v T t dt i T t R i t i t R t dt k k k k k k k k k k 0 0 T ( ) ( ) = R i T i t t dt k k k 0 22
Adjoint Network for Dynamic System (con t) For set E (capacitance), original network adjoint network C+ C C ( ) ( ) t ( ) ( ) ( ) = = + where ( ) v t i dt C dv t where ( ) v t i t dt C C dv t k k k k k k k k k 23
Adjoint Network for Dynamic System (con t) For set E (capacitance), T ( ) ( ) k t i t ( ) ( v t i T ) v T t dt k k k 0 T T = C dv t + ( ) ( ) k t i t ( ) ( ) v T v T t C k k k k k 0 0 T T cancellation = + ( ) ( ) ( ) ( ) v T t C dv t v T t C dv t k k k k k k 0 0 T T T = ( ) ( ) ( ) ( ) ( ) ( ( v t d v T ) ) v T t C dv t v T t C v t t C k k k k k k k k k = 0 t 0 0 ??0 = 0 ?? ??????? ??????? T = (0) ( ) ( ) (0) ( ) ( ( v t d v T ) ) v C v T v T C v t C k k k k k k k k k 0 T T overall v T C = + ( ) ( ) k t i t ( ) ( v t i T ) ( ) (0) (0) v ( ) ( ) v T t dt v T t C v t dt k k k k k k k k k 0 0 24
Put it together T T = = ( ) ( ) b t i t ( ) ( v t i T ) ( ) ( ) k t i t ( ) ( v t i T ) 0 v T t dt v T t dt b b b k k k k A B D E , , , b 0 0 T T = + ( ) ( ) C t v t dt ( ) ( i t i T ) ( ) (0) ( ) y t R dt v T C v v T t k k k k k k k k k k D k E k E 0 0 Initial voltage, ??0 , of capacitor ? remains fixed when parameters ??,?? vary. Thus, this term is not relevant to the sensitivities. D T y = ( ) ( i t i T ) t dt k k k R k 0 T y = ( ) ( ) k t v t dt v T k E k C k 0 25
Conclusion Adjoint network can derive sensitivities of all parameters for one objective function. The integration traces backward on the time domain for the dynamic adjoint network. The derivation uses Tellegen s theorm which depends upon the circuit topology only. 26
