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Sensitivity Analysis by Adjoint Network

CSE 245: Computer Aided Circuit Simulation and Verification. Sensitivity Analysis by Adjoint Network. CK Cheng, May 2010. 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.

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Sensitivity Analysis by Adjoint Network

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  1. CSE 245: Computer Aided Circuit Simulation and Verification Sensitivity Analysis by Adjoint Network CK Cheng, May 2010

  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 4 2

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

  6. Tellegen’s Theorem (con’t) II. 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. 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 • A: branch voltage • B: branch current • Sensitivity Calculation • D is the set of resistance ib R Vb

  10. Example of Sensitivity Calculation Given circuit i2 v1 R’’’ i3 R’’ + _ R’=R+∆R 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 • 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.

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