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- 1. Quiescent Steady State (DC) Analysis The Newton-Raphson MethodJ. Roychowdhury, University of California at Berkeley Slide 1
- 2. Solving the Systems DAEs d ~ (~ (t)) + f (~ (t)) + ~ = ~ q x ~ x b(t) 0 dt ● DAEs: many types of solutions useful ● DC steady state: no time variations state ● transient: ckt. waveforms changing with time transient ● periodic steady state: changes periodic w time ➔ linear(ized): all sinusoidal waveforms: AC analysis ➔ nonlinear steady state: shooting, harmonic balance shooting ● noise analysis: random/stochastic waveforms analysis ● sensitivity analysis: effects of changes in circuit analysis parametersJ. Roychowdhury, University of California at Berkeley Slide 2
- 3. QSS: Quiescent Steady State (“DC”) Analysis d ~ (~ (t)) + f (~ (t)) + ~ = ~ q x ~ x b(t) 0 dt ● Assumption: nothing changes with time ● x, b are constant vectors; d/dt term vanishes ~ (~ ) g x z }| { f (~ ) + ~ = ~ ~ x b 0 ● Why do QSS? ➔ quiescent operation: first step in verifying functionality ➔ stepping stone to other analyses: AC, transient, noise, ... ● Nonlinear system of equations ➔ the problem: solving them numerically ➔ most common/useful technique: Newton-Raphson methodJ. Roychowdhury, University of California at Berkeley Slide 3
- 4. The Newton Raphson Method ● Iterative numerical algorithm to solve ~ (~ ) = ~ g x 0 1 start with some guess for the solution 2 repeat a check if current guess solves equation i if yes: done! ii if no: do something to update/improve the guess ● Newton-Raphson algorithm ● start with initial guess ~ 0 ; i=0 x ● repeat until “convergence” (or max #iterations) d~ (~ i ) g x ➔ compute Jacobian matrix: Ji = d~x ➔ solve for update ±~ : Ji ±~ = ¡~ (~ i ) x x g x ➔ ➔ update guess: ~ i+1 = ~ i + ±~ x x x ➔ i++;J. Roychowdhury, University of California at Berkeley Slide 4
- 5. Newton-Raphson Graphically g(x) ● Scalar case above ● Key property: generalizes to vector caseJ. Roychowdhury, University of California at Berkeley Slide 5
- 6. Newton Raphson (contd.) ● Does it always work? No. ● Conditions for NR to converge reliably ➔ g(x) must be “smooth”: continuous, differentiable ➔ starting guess “close enough” to solution ● practical NR: needs application-specific heuristicsJ. Roychowdhury, University of California at Berkeley Slide 6
- 7. NR: Convergence Rate ● Key property of NR: quadratic convergence ¤ ● Suppose x is the exact solution of g(x) = 0 ● At the i th NR iteration, define the error ²i = xi ¡ x¤ ● meaning of quadratic convergence: ²i+1 < c²2 i ● (where c is a constant) ● NRs quadratic convergence properties ➔ if g(x) is smooth (at least continuous 1st and 2nd derivatives) ➔ and g 0 (x¤ ) 6= 0 ➔ and kxi ¡ x¤ k is small enough, then: ➔ NR features quadratic convergenceJ. Roychowdhury, University of California at Berkeley Slide 7
- 8. Convergence Rate in Digits of Accuracy Quadratic convergence Linear convergenceJ. Roychowdhury, University of California at Berkeley Slide 8
- 9. NR: Convergence Strategies ● reltol-abstol on deltax ● stop if norm(deltax) <= tolerance ➔ tolerance = abstol + reltol*x ● reltol ~ 1e-3 to 1e-6 ● abstol ~ 1e-9 to 1e-12 ● better ➔ apply to individual vector entries (and AND) ➔ organize x in variable groups: e.g., voltages, currents, … ➔ (scale DAE equations/unknowns first) ● more sophisticated possible ➔ e.g., use sequence of x values to estimate conv. rate ● residual convergence criterion ● stop if k~ (~ )k < ²residual g x ● Combinations of deltax and residual ● ultimately: heuristics, tuned to applicationJ. Roychowdhury, University of California at Berkeley Slide 9
- 10. Newton Raphson Update Step ● Need to solve linear matrix equation ● J ¢~ = ¡~ (~ ) : Ax = b problem x g x d~ (~ ) g x ● J= : Jacobian matrix d~x ● Derivatives of vector functions 2 3 2 3 x1 g1 (x1 ; ¢ ¢ ¢ ; xn ) 6 . 7 6 7 ● If ~ (t) = 4 . 5 ; x . ~ (~ ) = 4 g x . . . 5 xn g1 (x1 ; ¢ ¢ ¢ ; xn ) 2 dg1 dg1 dg1 dg1 3 dx1 dx2 ¢¢¢ dxn¡1 dxn 6 dg2 dg2 dg2 dg2 7 6 ¢¢¢ 7 d~ g 6 dx1 dx2 dxn¡1 dxn 7 6 . . . . 7 ● … then ,6 . . ¢¢¢ . . 7 x 6 dg . d~ . . . 7 6 n¡1 dgn¡1 ¢¢¢ dgn¡1 dgn¡1 7 4 dx1 dx2 dxn¡1 dxn 5 dgn dgn dgn dgn dx1 dx2 ¢¢¢ dxn¡1 dxnJ. Roychowdhury, University of California at Berkeley Slide 10
- 11. DAE Jacobian Matrices 1 ° 2 ° d ● Ckt DAE: ~ (~ (t)) + f (~ (t)) + ~ = ~ q x ~ x b(t) 0 iE dt iL 2 3 2 2 3 3 2 3 e1 (t) 0 ¡diode(¡e1 ; IS ; Vt ) ¡ iE 0 6 e2 (t) 7 6 Ce2 7 6 0 7 ~ (t) = 6 7 ~(~ ) = 6 7 f (~ ) = 6 iE + iL + e2 7~ 7 b(t) = 6 7 x 4 iL (t) 5 q x 4 0 5 ~x 6 4 e2 ¡ e1 R 5 4¡E(t)5 iE (t) ¡LiL e2 0 2 3 2 ddiode 3 0 0 0 0 dv (¡e1 ) 0 0 ¡1 d~ q 60 C 0 07 ~ 6 df 0 1 1 17 Jq , =6 40 7 Jf , =6 R 7 d~ x 0 0 05 d~ x 4 ¡1 1 0 05 0 0 ¡L 0 0 1 0 0J. Roychowdhury, University of California at Berkeley Slide 11
- 12. Newton Raphson: Computation ● Need to solve linear matrix equation ● J ¢~ = ¡~ (~ ) : Ax = b problem x g x ● Ax=b: where much of the computation lies ● large circuits (many nodes): large DAE systems, large Jacobian matrices ● in general (for arbitrary matrices of size n) ➔ solving Ax = b requires ● O(n2) memory ● O(n3) computation! ● (using, e.g., Gaussian Elimination) ➔ but for most circuit Jacobian matrices ● O(n) memory, ~O(n1.4) computation ● … because circuit Jacobians are typically sparseJ. Roychowdhury, University of California at Berkeley Slide 12
- 13. Dense vs Sparse Matrices ● Sparse Jacobians: typically 3N-4N non-zeros ● compare against N2 for denseJ. Roychowdhury, University of California at Berkeley Slide 13

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