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Research Projects

Research Projects. Account for 40% of total grade Type of projects Research Implementation Schedule Proposal due no later than 3/20 Weekly meeting Presentation 4/22, 4/24, 4/29. Suggested Topics 1. Capacitance extraction Extend FastCap/HiCap to multi-layer dielectric

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Research Projects

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  1. Research Projects • Account for 40% of total grade • Type of projects • Research • Implementation • Schedule • Proposal due no later than 3/20 • Weekly meeting • Presentation 4/22, 4/24, 4/29 ELEN689

  2. Suggested Topics 1 • Capacitance extraction • Extend FastCap/HiCap to multi-layer dielectric • Extend FastCap/HiCap to MEMS • Integrate FastCap with HiCap • Solve Pq=v for different v’s quickly • Compute capacitance under the new definition without infinity ground • Use computer graphics techniques ELEN689

  3. Suggested Topics 2 • Inductance extraction • Properties and extraction of (R+jL)-1 • Divide-and-conquer method • Power/ground network • Synthesis of clock tree under inductance ELEN689

  4. Model Order Reduction • 1. Motivation • 2. Basic concepts • 3. Pade approximation • 4. Computing moments • 5. Driving point impedance ELEN689

  5. 1. Motivation • Extracted circuits are too complex ELEN689

  6. Problem Description f(x(t)) u(t) y(t) Model Order Reduction u(t) y(t) fr(z(t)) q << n ELEN689

  7. Applications • Digital circuits • Delay and waveform evaluation • Cross talk analysis • Easy • Analog circuits • Reduce simulation time • Hard ELEN689

  8. 2. Basic Concepts • Simplest interconnect interconnect When R small Ctotal Uniform RC When R large ELEN689

  9. Uniform RC Segment • Transfer function of URC can not be obtained in closed rational form (R,C) ELEN689

  10. Approximation of URC • Approximate by a finite RC tree (R,C) R/(2n) R/n R/n R/(2n) … C/n C/n C/n ELEN689

  11. How Many Segments? • A segment of URC can be lumped if its physical length is much shorter than the wavelength • Depends on input frequency and RC r/2 r/2 c short long ELEN689

  12. Extract RC(L) • For each segment, extract RC(L) within a window • Repeat for each segment R/2 R/2 C ELEN689

  13. Extracted RC Circuit • Example extracted circuit • We want to obtain transfer function H(s)=Vout(s)/Vin(s) 1 1 1 1 1 2 3 4 5 Vin 1F 1F 1F 1F Vout ELEN689

  14. 1 1 1 1 1 2 3 4 5 Vin 1F 1F 1F 1F Vout Derive Transfer Function 1 2 3 4 5 6 ELEN689

  15. Transfer Function • We obtain • In general, transfer functions are in the form ELEN689

  16. How to Approximate H(s)? • Taylor series • Pade approximation ELEN689

  17. Taylor v.s. Pade Taylor Pade H(s) ELEN689

  18. 3. Pade Approximation • An [n,m] Pade approximation is in the form • Normally, set m=n+1, and call it m-th order Pade approximation • Any other reason why in this form? ELEN689

  19. Poles • Given a Pade approximation • Write the denominator as • Then ELEN689

  20. Inverse L-Transform • Frequency domain • Time domain ELEN689

  21. Example • The above example ELEN689

  22. Example • First order approximation • We have ELEN689

  23. Example (cont’d) • Second order approximation ELEN689

  24. Quality of Approximation Step Response Something is wrong with DC solution of 1st order Time (sec) ELEN689

  25. Moment • Definition of H(s) • The name moment comes from probability theory ELEN689

  26. Moment Matching • Show the idea for m=4 • Assume we have H(s) in this form where m’s are called moments • We want to equate H(s) with ELEN689

  27. Moment Matching … • Therefore • First three powers of s: ELEN689

  28. Moment Matching … • Next four moments ELEN689

  29. Linear Systems • Given the first 8 moments, we have • Solve it to find b’s Hankel Matrix ELEN689

  30. Another Linear System • Once b’s are obtained, we can compute a’s: • This completes the approximation ELEN689

  31. Inverse L-Transform • Obtained Pade approximation • Solve non-linear equation • The 4 roots give the poles ELEN689

  32. Inverse L-Transform • Solve for k’s: • We will get a Vandermond Matrix. Solve it to get k’s. ELEN689

  33. 4. Computing Moments • Solve linear system • Path tracing for RC trees 4 1 2 3 R1 R2 R3 R4 C1 C2 C3 C4 m0_1=m0_2=m0_3=m0_4=1 ELEN689

  34. Path Tracing 4 1 2 3 R1 R2 R3 R4 C1 C2 C3 C4 m1_4= –[R1(C1+C2+C3+C4)+R2(C2+C3+C4)+R4C4] m2_4= –[R1(C1•m1_1+C2•m1_2+C3•m1_3+C4•m1_4) +R2(C2•m1_2+C3•m1_3+C4•m1_4) +R4C4•m1_4] … Why? ELEN689

  35. Example 4 1 1 1 1 1 2 3 1 1 1 1 m1_1= –4, m1_2= –7, m1_3= –8, m1_4= –8 m2_1= 27, m2_2= 50, m2_3= 58, m2_4= 58 The i-th moments of all nodes can be computed in O(n) time, where n is the number of nodes ELEN689

  36. Drawback • Moments increase drastically • When m is large (>8), the following matrix cause numerical error ELEN689

  37. What to do? • For order less than 5, do not worry • For order greater than 8, use PVL or PRIMA • Also performs moment matching, but does not compute moment explicitly ELEN689

  38. 5. Driving Point Admittance • Approximate load by a simple model • Admittance Y(s)=I(s)/V(s) • For RC trees, Y(s) is better than Z(s) • Moment matching Y(s) Y’(s) ELEN689

  39. Compute Admittance 1 • Since Iu=sCV+Id, Yu(s)=sC+Yd(s) Iu Id Yu(s) Yd(s) C Up stream Down stream ELEN689

  40. Compute Admittance 2 • Since IR=Vu–Vd, Yu(s)=Yd(s)/(1+R•Yd(s)) =Yd(s)(1–R•Yd(s)+(R•Yd(s))2–…) R Vu Vd Yu(s) Yd(s) Up stream Down stream ELEN689

  41. Compute Admittance 3 • Since Iu=I1+…+Iks, Yu(s)=Yd1(s)+…+Ydk(s) Iu Id1 Yu(s) Yd1(s) … Idk Ydk(s) ELEN689

  42. Equivalent Load • Use moment matching to find equivalent load that match first, second, third and fourth moments ELEN689

  43. Moment Matching • Now we have Y(s) • We want to equate Y(s) with a circuit model ELEN689

  44. Lumped C Load • Let Y(s)=m0+m1*s+… • For lumped C load: Y=Cs • C=m1 Y(s) C ELEN689

  45. RC Loads • Let Y(s)=m0+m1*s+… • For RC load: Y=sC/(1+sRC) • R=–m2/(m1)2 • C=m1 Y(s) R C ELEN689

  46.  (CRC) Loads • Let Y(s)=m0+m1*s+… • For CRC load: Y=sC2+sC1/(1+sRC1) • R=–(m3)2/(m2)3 • C1=(m2)2/(m3) • C2=m1–(m2)2/m3 Y(s) R C1 C2 ELEN689

  47. Assignment #3 (Due 2/25) • For the RC circuit in the next slide, compute 2 pole approximation and 3 pole approximation, and compare it with SPICE. • For the same RC circuit, perform driving-point admittance approximation using C, RC and CRC models, and compare with SPICE. • Prove the path tracing algorithm gives the moments. ELEN689

  48. RC Circuit 1 2 Vout 2 2 1 Vin 1 1 1 2 1 ELEN689

  49. Readings on MOR • P. R.O’brien and T. L. Savarino, “Modeling driving-point characteristic of resistive interconnect for accurate delay estimation,” Proc. ICCAD, 1989. • L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. CAD, Vol. 9, No. 4, 1990. • C. L. Ratzlaff and L. T. Pillage, “RICE: rapid interconnect circuit evaluation using AWE,” IEEE Trans. CAD, Vol. 13, No. 6, 1994. • P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Pade approximation via the Lanczos process,” IEEE Trans. CAD, Vol. 14, No. 5, 1995. ELEN689

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