04/19/23 ELEN 689 1

Finite Element Method To be added later

04/19/23 ELEN 689 2

Inductance Given a set of k conductors, compute

the kk impedance matrix Z()

I1

I2

V1V2

2

1

2

1

2221

1211

V

V

I

I

ω)Zω)Z

ω)Z ω)Z

((

((

04/19/23 ELEN 689 3

Partial Inductance For any two pieces of interconnect,

the partial inductance

klVr Vrlk

kl dVdVaa

1

4π

μL

kk ll

lk

lk

rr

uu

k

l

04/19/23 ELEN 689 4

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

04/19/23 ELEN 689 5

Compute Inductance Send 1A current in one conductor

and 0A current through other conductors, then potential drop gives impedance

1

0

V1V2

2

1

2221

1211

V

V

0ω)Zω)Z

ω)Z ω)Z 1

((

((

04/19/23 ELEN 689 6

Boundary Element Method Laplace integral equation

where J(r) is current density, is conductivity, and (r) is potential drop across volume r

rrr

r)Jr(rrJΦVd

4π

μjω

V

04/19/23 ELEN 689 7

Discretization Partition conductors into n

filaments

I1I2I3I4

I5I6I7I6I1

04/19/23 ELEN 689 8

Incident Matrix Bf1f2f3f4

f5f6f7f8

n1

n2

n3

11110000

11111111

00001111TB

n filaments

m nodes

04/19/23 ELEN 689 9

Linear Systems Linear system for current and

potential

I is filament current vector is filament potential drop vector R is a diagonal matrix of filament

DC resistance:

IL jωR

i

i

ii area

lengthR

04/19/23 ELEN 689 10

Linear System (cont’d) L is the partial inductance matrix

In addition, Kirchoff’s Law must be satisfied

where Id is the external current

klVr Vrlk

kl dVdVaa

1

4π

μL

kk ll

lk

lk

rr

uu

dT IIB

04/19/23 ELEN 689 11

ExampleI1I2I3I4

I5I6I7I8

n1

n2

n3

1

0

1

11110000

11111111

00001111

I IIB dT

04/19/23 ELEN 689 12

Rewrite Linear System

Note that =BV, where V is the node potential

Large system; R, B: sparse; L: dense Solution methodology

Iterative methods Pre-conditioners are critical

dT I

0

V

I

0B

BL jωR

04/19/23 ELEN 689 13

Problem The original system is hard to solve:

Some algorithms (FastHenry) solved it anyway

We need a better formulation

dT I

0

V

I

0B

BL jωR

04/19/23 ELEN 689 14

Solenoidal Basis Method Linear system

Solenoidal basis Basis for current that satisfies

Kirchoff’s law: Reduced system

0

F

V

I

0B

BL jωRT

0PBT

0IBPxI T

FPPxM jωRP TT

04/19/23 ELEN 689 15

Intuition Any current vector I satisfying

Kirchoff’s law and boundary condition

can be written as the sum of two

parts: A unit current from external node to

external node A linear combination of loop currents

dT IIB

04/19/23 ELEN 689 16

Example

04/19/23 ELEN 689 17

Mesh Currents Filament current vector I can be

written as the sum of a particular current Ip and a linear combination of mesh currents

1A

1A

1A

1A

= +

Ip

04/19/23 ELEN 689 18

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()

04/19/23 ELEN 689 19

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 pre-conditioner, 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

04/19/23 ELEN 689 20

Preconditioning Reduced system

Pre-conditioners

FPPxL jωRP TT

LL jωRL-1~~~~

M

klVr Vrlk

kl dVdVaa

1

4π

ωμL

kk ll

lk rr

1~

L jωLRL -1 ~~~~ highlow MM

04/19/23 ELEN 689 21

Hierarchical Approximations Both L and M are dense and large Hierarchical method used to

compute matrix-vector products with both L and Used for fast decaying Greens

functions, such as 1/r (r : distance from origin)

Reduced accuracy at lower cost

04/19/23 ELEN 689 22

Avoiding Complex Numbers Reduced system

Separate real and complex components ofthe system

Solve this system by iterative method

j

r

j

r

TT

TT

b

b

x

x

RPPLPωP

LPωP-RPP

FPPxL jωRP TT

04/19/23 ELEN 689 23

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?

04/19/23 ELEN 689 24

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.

04/19/23 ELEN 689 25

Assignment #3 (Due 3/13) 1. Use the solenoidal algorithm to

perform inductance extraction for a pair of conductors: x2+y21, 0z10 and (x-10)2+y21, 0z10.

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