1. ECE 546 – Jose Schutt-Aine 1 ECE 546 Lecture -13 Latency Insertion Method Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu

2. ECE 546 – Jose Schutt-Aine 2 - Up to 16 layers - Hundreds of vias - Thousands of TLs - High density - Nonuniformity Vertical parallel-plate capacitance 0.05 fF/  m 2 Vertical parallel-plate capacitance (min width) 0.03 fF/  m Vertical fringing capacitance (each side) 0.01 fF/  m Horizontal coupling capacitance (each side) 0.03 Mitsumasa Koyanagi," High-Density Through Silicon Vias for 3-D LSIs" Proceedings of the IEEE, Vol. 97, No. 1, January 2009 TSV Density: 10/cm 2 - 10 8 /cm 2 Source: M. Bohr and Y. El-Mansy - IEEE TED Vol. 4, March 1998 Packaging Complexity Chip Complexity Challenges in Integration

3. ECE 546 – Jose Schutt-Aine 3 = Unit cell Typical workstation simulation time for a 1200-cell network is 2 h 40 min. Too time consuming! - Determine R,L,G,C parameters and define cell - Synthesize 2-D circuit model for ground plane - Use SPICE (MNA) to simulate transient Modeling PDN Modeling

4. ECE 546 – Jose Schutt-Aine 4 MNA has super-linear numerical complexity LIM has linear numerical complexity LIM has no matrix ill-conditioning problems Accuracy and stability in LIM are easily controlled LIM is much faster than MNA for large circuits Why LIM?

5. ECE 546 – Jose Schutt-Aine 5 Latency Insertion Method

6. ECE 546 – Jose Schutt-Aine 6 Each branch must have an inductor* Each node must have a shunt capacitor* Express branch current in terms of history of adjacent node voltages Express node voltage in terms of history of adjacent branch currents Latency Insertion Method * If branch or node has no inductor or capacitor, insert one with very small value

7. ECE 546 – Jose Schutt-Aine 7 Represents network as a grid of nodes and branches Discretizes Kirchhoff's current and voltage equations Uses ”leapfrog” scheme to solve for node voltages and branch currents Presence of reactive elements is required to generate latency Node structure Branch structure LIM Algorithm

8. ECE 546 – Jose Schutt-Aine 8 n: time LIM: Leapfrog Method Leapfrog method achieves second-order accuracy, i.e., error is proportional to  t 2

9. ECE 546 – Jose Schutt-Aine 9 For branch =1, N b Update current as per Equation: For time=1, N t Next branch; For node=1, N n Update voltage as per Equation Next node; Next time; time loop branch loop node loop LIM Code

10. ECE 546 – Jose Schutt-Aine 10 No of Nodes 20,00030,00040,00050,000 SPICE (sec) 1224293547417358 LIM (sec) 9131721 Speedup 136225278350 Simulation Times Standalone LIM vs SPICE

11. ECE 546 – Jose Schutt-Aine 11 LIM vs SPICE

12. ECE 546 – Jose Schutt-Aine 12  Mutual inductance  use reluctance (1/L)  Branch capacitors  special formulation  Shunt inductors  special formulation  Dependent sources  account for dependencies  Frequency dependence  use macromodels LIM: Problem Elements

13. ECE 546 – Jose Schutt-Aine 13 We wish to determine the current update equations for that branch. In order to handle the branch capacitor with LIM, we introduce a series inductor L. LIM Branch capacitor

14. ECE 546 – Jose Schutt-Aine 14 In the augmented circuit, we have: where V c is the voltage drop across the capacitor; V c =V o -V j. Solving for LIM Branch capacitor

15. ECE 546 – Jose Schutt-Aine 15 so that which leads to LIM Branch capacitor The voltage drop across the inductor is given by:

16. ECE 546 – Jose Schutt-Aine 16 In order to minimize the effect of L while maintaining stability, we see that - L must be very small - L must be A large branch capacitor helps as well as small time steps. LIM Branch capacitor After substitution and rearrangement, we get:

17. ECE 546 – Jose Schutt-Aine 17 Apply the difference equation directly to branch LIM Branch capacitor

18. ECE 546 – Jose Schutt-Aine 18 LIM Node Formulations Explicit Implicit Semi-Implicit

19. ECE 546 – Jose Schutt-Aine 19 LIM Branch Formulations Explicit Implicit Semi-Implicit

20. ECE 546 – Jose Schutt-Aine 20 Latency Insertion Method (LIM) Yee Algorithm Field Solution Circuit Solution FDTD & LIM Courant-Friedrichs-Lewy criteriastability condition

21. ECE 546 – Jose Schutt-Aine 21 LIM: Stability Analysis

22. ECE 546 – Jose Schutt-Aine 22 LIM and Dependent Sources

23. ECE 546 – Jose Schutt-Aine 23 LIM and Dependent Sources

24. ECE 546 – Jose Schutt-Aine 24 where M is the incidence matrix. M is defined as follows LIM and Dependent Sources

25. ECE 546 – Jose Schutt-Aine 25 Incidence Matrix

26. ECE 546 – Jose Schutt-Aine 26 LIM and Dependent Sources

27. ECE 546 – Jose Schutt-Aine 27 LIM and Dependent Sources

28. ECE 546 – Jose Schutt-Aine 28 LIM and Dependent Sources

29. ECE 546 – Jose Schutt-Aine 29 LIM and Dependent Sources

30. ECE 546 – Jose Schutt-Aine 30 DEFINE: so that Stability Analysis

31. ECE 546 – Jose Schutt-Aine 31 The 2 matrix equations can be combined in a single matrix equation that reads or Stability Analysis

32. ECE 546 – Jose Schutt-Aine 32 A is the amplification matrix. All the eigenvalues of A must be less than 1 in order to guarantee stability. Therefore to insure stability, we must choose  t such that all the eigenvalues of A are less than 1 Amplification Matrix

33. ECE 546 – Jose Schutt-Aine 33 Stability Methods for LIM LIM is conditionally stable  upper bound on the time step Δt. For uniform 1-D LC circuits [2] : For RLC/GLC circuits [3] : For general circuits [4] : [2] Z. Deng and J. E. Schutt-Ainé, IEEE EPEPS, Oct. 2004. [3] S. N. Lalgudi and M. Swaminathan, IEEE TCS II, vol. 55, no. 9, Sep. 2008. [4] J. E. Schutt-Ainé, IEEE EDAPS, Dec. 2008. Amplification matrix.

34. ECE 546 – Jose Schutt-Aine 34 Example – RLGC Grid * SPECTRE – A SPICE-like simulator from Cadence Design Systems Inc. * Current excitation with amplitude of 6A, rise and fall times of 10 ps and pulse width of 100 ps.

35. ECE 546 – Jose Schutt-Aine 35 Example – RLGC Grid Comparison of runtime for LIM and SPECTRE*. LIM exhibits linear numerical complexity! – Outperforms conventional SPICE-like simulators. Expand on LIM as a multi-purpose circuit simulator. Circuit Size # nodes (SPECTRE) SPECTRE (s) LIM (s) 20 × 2011600.151.53 40 × 4047202.146.25 60 × 601068025.7314.23 80 × 8019040140.5925.71 100 × 10029800445.7740.64 120 × 12042960945.0062.89 * Intel 3.16 GHz processor, 32 GB RAM.

36. ECE 546 – Jose Schutt-Aine 36 Computer simulations of distributed model (top) and experimental waveforms of coupled microstrip lines (bottom) for the transmission-line circuit shown at the near end (x=0) for line 1 (left) and line 2 (right). The pulse characteristics are magnitude = 4 V; width = 12 ns; rise and fall times = 1 ns. The photograph probe attenuation factors are 40 (left) and 10 (right). LIM Simulations

37. ECE 546 – Jose Schutt-Aine 37 Twisted-pair cable Simulation setup Generalized model [3] Ideal linelossy line Simulation resultsMeasured response LIM Simulation Examples