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
Modeling a Complex Interconnect Network
• Full-chip electromagnetic simulation: Too computationintensive
– 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)
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 {hi_j | i, j within chip} over
time at each selected output point xi in the network, for input
impulses at every possible coupling point xj
– 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).
Calculating the response to a general
input from impulse responses
(One-dimensional input
Assume we can calculate the system’s response to an
function, changing in time) impulse at point x :
i
(x-xi,t-t’)=(x-xi)(t-t’)
hi(x,t)
Take an input function applied only at one spatial point, time-dependent:
f ( xi , t )  f ( x, t ) ( x  xi )   f ( x,  ) ( x  xi ) (t   )d
f(x ,t)
i
Fi(x,t), where
Fi ( x, t )   f ( xi , )hi ( x, t   )d
For a linear system, superposition holds:
f ( x, t )   f ( xi , t )  F ( x, t )   Fi ( x, t )
i
i
 F ( x, t )    f ( xi , )hi ( x, t   )d
i
The discrete case
Define the unit impulse:
[x-xi][t-t’]
[xi]=
1, x=xi
0, else
hi[x,t]
Input function applied only at one spatial point:
f [ xi , t ]  f [ x, t ] [ x  xi ] 
  f [ x, t j ] [ x  xi ] [t  t j ]
j
Fi[x,t], where
f[x ,t]
i
Fi [ x, t ]   f [ xi , t j ]hi [ x, t  t j ]
For a linear system, if
j
f [ x, t ]   f [ xi , t ]  F[ x, t ]   Fi [ x, t ]
i
i
 F [ x, t ]   f i [ xi , t j ]hi [ x, t  t j ]
i
j
The discrete case
f [t ]   fi [t ]  F[t ]   Fi [t ]
 F [t ]   f [ xi , t j ]hi [ x, t  t j ]
i
j
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.
 ij : f [ xi , t j ]
 F [t ]    ij hi [ x, t  t j ]
i
j
The Computational Advantage
F [t ]    ij hi [ x, t  t j ]
i
j
•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
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
2 ( x  x2 , t )  3 ( x  x2 , t  200ns)
3 ( x  x5 , t  100ns)   ( x  x5 , t  400ns)
A simple demonstration
•Theoretically, the response at point F3 to a discrete-time input
2 ( x  x2 , t )  3 ( x  x2 , t  200ns)
3 ( x  x5 , t  100ns)   ( x  x5 , t  400ns)
should be
2h3_ 2 (t )  3h3_ 2 (t  200ns)  3h3_ 5 (t  100ns)  h3_ 5 (t  400ns)
•Calculate this analytically from the simulated impulse responses
and compare with simulation result
A simple demonstration
A simple demonstration