SISPAD ’06, Dilli, Goldsman, Akturk, Metze Model unit cells and combine them to form a network Model unit cells and combine them to form a network –Simplified lumped element model: Uses resistors and capacitors (Unit cells marked with red boxes in the figure). Pick critical points as output nodes of interest Pick critical points as output nodes of interest Solve for impulse responses to impulses at likely induction or injection points Solve for impulse responses to impulses at likely induction or injection points Use impulse responses to obtain outputs to general inputs Use impulse responses to obtain outputs to general inputsMethodology
SISPAD 06 A 3-D Time-Dependent Greens Function Approach to
Modeling Electromagnetic Noise in On-Chip Interconnect Networks
Zeynep Dilli, Neil Goldsman, Akn Aktrk, George Metze Dept. of
Electrical and Computer Eng. University of Maryland; University of
Maryland; Laboratory for Physical Sciences, Laboratory for Physical
Sciences, College Park, MD, USA SISPAD 06, Dilli, Goldsman, Akturk,
Metze Objective: Investigate the response of a complex on-chip
interconnect network to external RF interference, internal
parasitic signals, or coupling between different regions Full-chip
electromagnetic simulation: Too computationally- intensive, but
possible for small unit cells: Simple seed structures of single and
coupled interconnects We have developed a methodology to solve for
the response of such a unit cell network to random inputs. Sample
unit cells for a two-metal processIntroduction SISPAD 06, Dilli,
Goldsman, Akturk, Metze Model unit cells and combine them to form a
network Model unit cells and combine them to form a network
Simplified lumped element model: Uses resistors and capacitors
(Unit cells marked with red boxes in the figure). Pick critical
points as output nodes of interest Pick critical points as output
nodes of interest Solve for impulse responses to impulses at likely
induction or injection points Solve for impulse responses to
impulses at likely induction or injection points Use impulse
responses to obtain outputs to general inputs Use impulse responses
to obtain outputs to general inputsMethodology SISPAD 06, Dilli,
Goldsman, Akturk, Metze The interconnect network is a linear time
invariant system: Use Greens Function responses to calculate the
output to any input distribution in space and time. The
interconnect network is a linear time invariant system: Use Greens
Function responses to calculate the output to any input
distribution in space and time.Methodology SISPAD 06, Dilli,
Goldsman, Akturk, Metze We can write these input components f i [t]
as Writing f i [t] as the sum of a series of time-impulses marching
in time: [x-x i ]= 1, x=x i 0, else Define a unit impulse at point
x i : This yields a system impulse response: Let an input f[x,t] be
applied to the system. This input can be written as the
superposition of time-varying input components f i [t]=f[x i,t]
applied to each point x i : Numerical Modeling: Theory f[x,t] [x-x
i ] [t] h i [x,t] SISPAD 06, Dilli, Goldsman, Akturk, Metze f i
[t]F i [x,t] Let F i [x,t] be the systems response to this input
applied to x i : For a time-invariant system we can use the impulse
response to find F i [x,t] : Then, since Numerical Modeling: Theory
SISPAD 06, Dilli, Goldsman, Akturk, Metze Full-wave electromagnetic
solutions only possibly needed for small unit cells 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. Let Then we can
calculate the response to any such random input distribution ij by
only summation and time shifting We can explore different random
input distributions easily, more flexible than experimentation
Computational Advantages SISPAD 06, Dilli, Goldsman, Akturk, Metze
Knowing impulse responses h i [x,t] and input f i [x,t], the
response is calculated by adding the output contribution from each
input time step: For t in temporal and N in spatial input points
and impulse responses decaying in t h timesteps, this is done t in
N in t h times If t in
