1. Interconnect Network Modeling Motivation: Investigate the response of a complex interconnect network to external RF interference or internal coupling between different parts of a system On-chip interconnects on lossy substrate: capacitively and inductively coupled Characterized with S-parameter measurements Equivalent circuit models found by parameter-fitting

2. Modeling a Complex Interconnect Network Full-chip electromagnetic simulation: Too computation- intensive –On-chip scale changes; too many mesh points… –Simulation can not be repeated for all possible inputs Possible to do full wave simulation for small “unit cells”: Simple seed structures of single and coupled interconnects (Bo Yang) Create an equivalent circuit model from EM simulation results/S-parameters We have already developed a methodology to solve a large network composed of unit cells with random inputs while investigating chip heating (Akin Aktürk)

3. Modeling a Complex Interconnect Network Example Goal: Evaluate the sensitivity of different interconnect layouts to external pulses –Obtain unit cell equivalent circuits from full wave simulation/measurement –Set up a coupled network from unit cells, which emulates a certain type of interconnect network layout –Calculate the impulse responses {h i_j | i, j within chip} over time at each selected output point x i in the network, for input impulses at every possible coupling point x j –The interconnect network is a linear time invariant system: It is straightforward to calculate the output to any input distribution in space and time from the impulse responses. –Generate a random distribution of external input pulses to find the response and create a coupling map for this type of layout –Compare different layouts (i.e. different unit cell network configurations).

4. Calculating the response to a general input from impulse responses (One-dimensional input function, changing in time) Assume we can calculate the system’s response to an impulse at point x i :  (x-x i,t-t’)=  (x-x i )  (t-t’) h i (x,t) Take an input function applied only at one spatial point, time-dependent: f(x i,t)F i (x,t), where For a linear system, superposition holds:

5. The discrete case f[x i,t] F i [x,t], where  [x i ]= 1, x=x i 0, else Define the unit impulse:  [x-x i ]  [ t-t’]h i [x,t] Input function applied only at one spatial point: For a linear system, if

6. The discrete case The input values at discrete points in space and time can be selected randomly, depending on the characteristics of the interconnect network (coupling, etc.) and of the interference.

7. The Computational Advantage Choose a spatial mesh and a time period Calculate the impulse response over all the period to impulse inputs at possible input nodes (might be all of them) Then we can calculate the response to a random input by only summation and time shifting We can explore different random input distributions easily, more flexible than experimentation

8. A simple demonstration Simulate impulse responses at points F1..F6 to impulse inputs at points F1..F6, at time t=0 Simulate response at point F3 to a discrete-time input given by

9. A simple demonstration Theoretically, the response at point F3 to a discrete-time input should be Calculate this analytically from the simulated impulse responses and compare with simulation result

10. A simple demonstration