Transcript I. Overview - Texas A&M University

Finite Element Method

To be added later
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1
Inductance

Given a set of k conductors, compute the
kk impedance matrix Z()
V1
V2
I1
I2
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 Z11(ω)

 Z21(ω)
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Z12(ω)  I1   V1 
     
Z22(ω) I2   V2 
2
Partial Inductance
For any two pieces of interconnect, the
partial inductance
μ 1
uk  ul
Lkl 
dVldVk


4π ak al rk Vk rlVl rk  rl

k
l
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Application

Partial inductance assumes




Unit current
Current return at infinity
It works OK for thin conductors and
known current distribution
It does not work for large plate or if
current distribution is unknown
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Compute Inductance
Send 1A current in one conductor and
0A current through other conductors,
then potential drop gives impedance
V1
V2

1
0
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 Z11(ω)

 Z21(ω)
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Z12(ω)  1   V1 
     
Z22(ω)  0   V2 
5
Boundary Element Method

Laplace integral equation
Jr 
μ (r  r )Jr 
 jω
dV  Φr 

4π
r  r
V
where J(r) is current density,  is
conductivity, and (r) is potential
drop across volume r
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Discretization

Partition conductors into n filaments
I1
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I1
I2
I3
I4
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I5
I6
I7
I6
7
Incident Matrix B
n1
f1
f2
f3
f4
n2
f5
f6
f7
f8
n3
n filaments
 1 1 1 1 0 0 0 0 


T
B   1 1 1 1  1  1  1  1

m nodes  0
0
0
0
1
1
1
1


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Linear Systems




Linear system for current and potential
R  jω LI  
I is filament current vector
 is filament potential drop vector
R is a diagonal matrix of filament DC
resistance:
length i
Rii 
  areai
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Linear System (cont’d)
L is the partial inductance matrix
μ 1
uk  ul
Lkl 
dVldVk


4π ak al rk Vk rlVl rk  rl
 In addition, Kirchoff’s Law must be
satisfied

B I  Id
T
where Id is the external current
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Example
n1
I1
I2
I3
I4
n2
I5
I6
I7
I8
n3
 1 1 1 1 0 0 0 0   1 

  
T
B I  Id   1 1 1 1  1  1  1  1 I   0 
 0 0 0 0 1 1 1 1    1

  
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Rewrite Linear System
 R  jω L  B   I   0 

     
T
0   V   Id 
 B



Note that =BV, where V is the node
potential
Large system; R, B: sparse; L: dense
Solution methodology
Iterative methods
 Pre-conditioners are critical

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Problem

The original system is hard to solve:
 R  jω L  B   I   0 

     
T
0   V   Id 
 B


Some algorithms (FastHenry) solved it
anyway
We need a better formulation
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Solenoidal Basis Method

Linear system

T
B
P0
Solenoidal basis
 R  jω L

T
B



 B  I  F
     
0   V  0
Basis for current that satisfies Kirchoff’s
law: I  Px  BTI  0
Reduced system PT R  jω M Px  PTF
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Intuition

Any current vector I satisfying
Kirchoff’s law and boundary condition
B I  Id
T
can be written as the sum of two
parts:


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A unit current from external node to
external node
A linear combination of loop currents
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Example
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Mesh Currents

1A
Filament current vector I can be written
as the sum of a particular current Ip and
a linear combination of mesh currents
1A
+
=
1A
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Ip
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1A
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New Formulation

After some manipulation, the problem is
changed to the following:





Solve Im from ZmIm=Vm, where
Zm is mesh-to-mesh impedance matrix
Im is mesh current vector, and
Vm is a vector of voltage drop on the Ip path, due
to unit current at each mesh
Solution of Im gives potential drop between
external nodes, which is one row of Z()
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What is Pre-conditioning?



When matrix A is in “bad” shape, i.e., A
has a large condition number, then iterate
methods to solve Ax=b take a long time to
converge
If we can find a matrix M, called the preconditioner, such that (MA) is in “good”
shape, then solving (MA)x=Mb can be very
fast
Ideally, if M=A-1 then we are done
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Preconditioning

Reduced system

Pre-conditioners
PT R  jω L Px  PTF


 
1
dVldVk
rk  rl
~~
~ -1 ~
M  L R  jω L L
~
ωμ 1
Lkl 
4π ak al
Mlow
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~ ~ -1~
 LR L
rk Vk rl Vl
Mhigh
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~
  jω L
20
Hierarchical Approximations


Both L and M are dense and large
Hierarchical method used to compute
matrix-vector products with both L and


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Used for fast decaying Greens functions,
such as 1/r (r : distance from origin)
Reduced accuracy at lower cost
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Avoiding Complex Numbers

Reduced system
PT R  jω L Px  PTF

Separate real and complex components of
the system
 PTRP - ωPTLP   xr   br 







 ωPTLP PTRP   x   b 

 j  j

Solve this system by iterative method
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Extract R, C and L together




Existence of C affects the accuracy of
above method
Most accurate approach is to extract R,
C and L all in one equation
Introduce current variables normal to
the conductor surface and relate it to
charge
Expensive. Necessary in the future?
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Assignment #2 (Due 3/6)


1. Use FEM to solve the capacitance problem.
2. For the hierarchical algorithm discussed on
1/28, assume the two panels (A and H) are of
size 2x4, and the distance between them is 1.
Assuming the partition is A=C+E+F+G and
H=M+N+L+J, give the block entry matrix.
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Assignment #3 (Due 3/13)


1. Use the solenoidal algorithm to
perform inductance extraction for a pair
of conductors: x2+y21, 0z10 and (x10)2+y21, 0z10.
2. Download and compile FastHenry,
and compare with the above results
http://rleweb.mit.edu/vlsi/codes.htm .
Hand in printout of input file and output
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