1. 1 Steady-State Methods UCB EE219A Oct 31 2002 Joel Phillips, Cadence Berkeley Labs Thanks to: K. Kundert

2. 2 Steady-State Methods: Goals Understand alternative way of analyzing differential equations ‑ Faster ‑ Application-Specific “Tie together” several numerical themes ‑ Circuit theory ‑ Solution of ODEs/DAEs ‑ Newton methods ‑ Iterative solvers & preconditioning

3. 3 Today Review material from last time In-depth look at time-domain methods for periodic steady-state problem ‑ Matrix-implicit implementation ‑ Analysis Quick survey of more advanced techniques ‑ Small-signal, RF noise analysis ‑ Oscillators ‑ Multi-frequency steady-state, envelope

4. 4 Review: AC Small-Signal Find DC operating point Linearize around operating point Solve the AC analysis equation

5. 5 Simple Nonlinear Steady-State Problems Compute harmonic distortion in the amplifier Compute conversion gain in the mixer Compute noise with large-signal bias *    

6. 6 IVP Approach Apply source Solve IVP (Trap, Euler, etc.) Wait till steady state is reached Fourier-transform the output

7. 7 Problems with IVP Speed -- Especially with multiple periodic inputs Accuracy ‑ Steady-state not reached, a-periodicity errors, aliasing errors, interpolation errors Sim Time ~ Fmax / Fmin

8. 8 Periodic Steady-State Computation Apply a sinusoid or other periodic input signal Directly solve for the periodic response ‑ Time-domain solution over one “fundamental” period ‑ Or spectrum: Fourier coefficients at fundamental + harmonics Need to solve a boundary value problem (BVP)

9. 9 Note on PBCs If solution to DAE is unique, then solution on one period determines solution for all time ‑ Both the shooting method and spectral interval methods (harmonic balance) use this fact From knowledge of solution at one timepoint, can easily construction solution over entire period by solving IVP ‑ We will exploit this in the shooting method

10. 10 Enforcing PBCs Approach 1: Build BCs in basis function ‑ Example: Fourier series satisfy periodic boundary conditions Approach 2: Write extra equations ‑ PBC

11. 11 PSS Algorithm #1: Harmonic Balance Periodic solution can be expressed in terms of Fourier series with fundamental frequency Pick Spectral derivatives (real solutions please!)

12. 12 Harmonic Balance: Equation Formation Enforce Pseudo-spectral approach: force at selected timepoints ‑ Uniformly spaced to compute derivatives via FFT

13. 13 Equation Structure BVP becomes Jacobian with

14. 14 Equation Solution We need to solve These matrices are dense in either Fourier- or real- space  LU factorization is bad news They are potentially very large Yet a matrix-vector product can be done fast Ideal candidate for iterative solution methods (GMRES!) Good preconditioners are necessary, but hard to construct

15. 15 Algorithm #2: “Finite Difference” Given the same derivative discretization for IVP (Gear, trapezoidal, etc.) Step 1: Write all the discretized circuit equations for the whole period

16. 16 Algorithm #2: “Finite Difference” Step 2: Write the PBC equation

17. 17 Algorithm #2: “Finite Difference” Step 2: Write the PBC equation

18. 18 Algorithm #2: “Finite Difference” Step 3: Solve them ‑ With N circuit equations, M timepoints, system has O(MN) unknowns ‑ Big system! Recall: Newton method for IVP at timestep n, iteration k

19. 19 Newton for PSS-FD Step 2: Write the equations w/ PBC PBC

20. 20 Algorithm #2: “Finite Difference” Jacobian structure PBC

21. 21 Similar Concept, Distinctive Computations AC PSS-HB PSS-FD

22. 22 Algorithm #2: “Finite Difference” Linear system solution ‑ With N circuit equations, M timepoints, system has O(MN) unknowns ‑ Big system! Will use iterative methods ‑ GMRES in particular Efficient? ‑ How many iterations? ‑ Cost of each iteration? (matrix-vector product?)

23. 23 GMRES Convergence Analysis Phillips’ rule of preconditioner analysis: ‑ Real life is always worse than an example you can solve analytically Simplified PSS-FD matrices ‑ Consider linear case ‑ Consider one variable ‑ Constant timestep

24. 24 Eigenvalues of PSS-FD Matrix Eigenvalues tend to unit circle as h  0 ‑ Very bad for GMRES!!!! ‑ Need a preconditioner!

25. 25 PSS-FD Solution Procedure Viewpoint 1: Preconditioning ‑ Hard to invert/factor entire matrix ‑ Can invert lower-triangular piece fairly easily

26. 26 GMRES with FD System Need to solve system with We will use iterative method (GMRES). ‑ Can combined preconditioner and matrix product! ‑ Must exploit structure for efficiency! Must compute ‑ Products with ‑ Solves with

27. 27 Exploiting Structure in Step 1: Compute Step 2: Compute

28. 28 PSS-FD Solution Procedure Viewpoint 2: Compressed System ‑ If we knew the last point/first point, we can easily obtain all the rest of the points by a forward- substitution

29. 29 Compressed FD System Structure Need to solve system with

30. 30 Compressed FD System Structure Need to solve system with Apply GMRES to !!

31. 31 Compressed GMRES Products Step 1: Compute Step 2: Compute

32. 32 Final PSS-FD Procedure Form equations For each Newton iteration ‑ Compute residual ‑ Compute Jacobian matrices ‑ Solve PSS-FD Jacobian equation Preconditioned by L: Using GMRES, solve compressed matrix equation Find update on whole interval by forward substitution (application of ) ‑ Update solution waveform on interval

33. 33 Compressed System Analysis GMRES convergence? ‑ With N equations, M timesteps, have reduced system size from MN to N. Nice! ‑ What is eigenvalue distribution? Must consider for a meaningful analysis. What about the nonlinear part of the solution? Can we apply a similar trick?

34. 34 Recall: Note on PBCs If solution to DAE is unique, then solution on one period determines solution for all time From knowledge of solution at one timepoint, can easily construction solution over entire period by solving IVP ‑ We will exploit this in the shooting method

35. 35 Shooting Procedure Guess x(0) Integrate to get x(T) Update x(0) to correct residual ‑ How? Newton! T 0

36. 36 Transition Function Define transition function s.t Physical interpretation: integrate forward in time with as initial condition Periodicity condition is T 0

37. 37 Transition Function Example Consider linear system Recall from theory of (homogeneous) linear ODEs

38. 38 Shooting Method Updates Periodicity condition is Want to solve this equation via Newton Need: Sensitivity matrix How to compute? Depends on method used for shooting….

39. 39 Sensitivity Computation Given equation for final timepoint, take partial derivatives Use chain rule

40. 40 Sensitivity Computation Apply this formula recursively Starting condition is

41. 41 Sensitivity Matrix-Vector Product GMRES needs matrix-vector product for some Compute, starting at step 1, proceed to step M ‑ Convenient: At step k, solve

42. 42 Remarkable Fact: Connection Shooting/FD We have the *same equations* to compute the sensitivity update. Yet not the same method.

43. 43 Final PSS-Shooting Procedure Form equations For each Newton iteration ‑ Compute residual in periodicity condition By integrating IVP forward to find x(T) ‑ Solve shooting update equation Using GMRES, solve sensitivity matrix equation ‑ Update first time point x(0)

44. 44 Shooting vs. PSS-FD Procedures Same linear system is solved ‑ Same GMRES convergence properties Different nonlinear updates ‑ Different Newton convergence properties Shooting Updates end point only Satisfies ODE at every iteration Tries to converge boundary condition PSS-FD Updates entire interval Satisfies BC at every iteration Tries to converge ODE solution

45. 45 GMRES Convergence Analysis Consider linear model problem Transition function / matrix: Eigenvalues of exponential matrix are exponentials of eigenvalues Stability analysis: circuits tend to be designed to be stable  eigenvalues have negative real parts  all eigevalues of are inside unit circle

46. 46 GMRES convergence “Stiff” modes go away immediately “Neutrally stable” modes slow convergence problems With DAEs: ‑ Algebraic equations, conservations (e.g. KCL) ‑ Only dynamic modes show up in matrix ‑ Resistors do not hold “state”, only capacitors, inductors, other energy storage elements

47. 47 Irony Consider transient matrix: ‑ Ill-conditioned if stiff  bad for iterative methods PSS-FD ‑ Stiff modes: eigenvalues with large negative real part ‑ Map to unity in this procedure What made transient hard (esp. iterative methods) makes PSS problem easy!!

48. 48 Multi-Interval Decomposition Low-order where needed, high-order where possible Refinement strategy: ‑ order increase, in smooth region, III, ‑ interval decomposition, in sharp-transition region, II. I II III IV V Subintervals Multi-Interval Chebyshev Discretization

49. 49 10 -6 Relative error of output Total # of time steps or collocation points 1000 uniform 2nd-order MIC Uniform high-order 100 500 10 0 10 -5 10 -1 10 -3 High order is more efficient and MIC (2-16) is more efficient than uniform high order! Example Convergence Result

50. 50 Linear Periodically Varying Circuits Model Frequency-Translating Devices Assume devices is near-linear with respect to data inputs May be very nonlinear with respect to other inputs

51. 51 Small-Signal Analysis Assume second signal is “small” sinusoid Linearize around time-varying operating point Solve for transfer functions

52. 52 Calculations Let v = v L + v s and u = u L + u s Let u s = 0, find v L, linearize about v L, then apply u s Taylor series expansion about v L Let u v

53. 53 Cyclostationary Noise Cyclostationary noise is periodically modulated noise ‑ Results when circuits have periodic operating points Noise is cyclostationary if its autocorrelation is periodic in t ‑ Implies variance is periodic in t ‑ Implies noise is correlated in frequency Cyclostationarity generalizes to non-periodic variations ‑ In particular, multiple periodicities

54. 54 Origins of Cyclostationary Noise Modulated (time-varying) noise sources ‑ Periodic bias current generating shot noise ‑ Periodic variation in resistance of channel generating thermal noise Modulated (time-varying) signal path ‑ Modulation of gain by nonlinear devices and periodic operating point Modulated noise source Modulated signal path

55. 55 Cyclostationary Noise vs. Time Noise transmitted only when switch is closed Noise is shaped in time Noisy Resistor & Clocked Switch vovo NoisyNoiseless t n

56. 56 Modulated Noise Spectrum Time shaping  Frequency correlation Stationary Noise Source Periodic Modulation Noise Folding Terms Cyclostationary Noise Replicate & Translate Sum -2012-33 -2012-33 f f f f Convolve

57. 57 Adjoint Analysis: Circuit Interpretation Linearized Circuit Output 1 Output 2 Output 3 Output 4 Input 1 Input 2 Input 3 For one output configuration, compute TF from all possible inputs

58. 58 Exotic Simulation Creatures Oscillator Analyses Quasi-periodic analyses ‑ Steady-State (intermodulation distortion) ‑ AC, XF (transfer function) ‑ Noise (in mixer with blocker) Envelope methods

59. 59 Two-Frequency Analysis Main issue: frequency resolution Sim Time ~ Fmax / Fmin Multi-Freq Harmonic Balance Shooting methods (MFT)

60. 60 Sample quasi-periodic waveform at clock frequency Represent “envelope” with Fourier series Use time-domain method to resolve “fast” behavior in each cycle Only a few cycles/samples are needed Mixed Frequency-Time Algorithm

61. 61 Envelope Analyses Track non-periodic carrier envelope without tracing all carrier cycles RF analog of transient analysis

62. 62 Summary Steady-state methods are powerful techniques for analyzing complicated – yet structured – phenomena in circuits Essential for RF design ‑ Useful in many other contexts!!! Numerical techniques ‑ Shooting, HB, Chebyshev-Interval, MFT ‑ Iterative solutions methods ‑ Must exploit problem structure to get efficient algorithms!