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Finite Element Method

- To be added later

Inductance

- Given a set of k conductors, compute the k?k

impedance matrix Z(?)

V1

V2

I1

I2

Partial Inductance

- For any two pieces of interconnect, the partial

inductance

k

l

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

Compute Inductance

- Send 1A current in one conductor and 0A current

through other conductors, then potential drop

gives impedance

V1

V2

1

0

Boundary Element Method

- Laplace integral equation
- where J(r) is current density, ? is

conductivity, and ??(r) is potential drop across

volume r

Discretization

- Partition conductors into n filaments

I1

I5

I2

I6

I3

I7

I1

I4

I6

Incident Matrix B

n2

f1

f5

f2

f6

n1

n3

f3

f7

f4

f8

n filaments

m nodes

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

Linear System (contd)

- L is the partial inductance matrix
- In addition, Kirchoffs Law must be satisfied
- where Id is the external current

Example

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

Problem

- The original system is hard to solve
- Some algorithms (FastHenry) solved it anyway
- We need a better formulation

Solenoidal Basis Method

- Linear system
- Solenoidal basis
- Basis for current that satisfies Kirchoffs law
- Reduced system

Intuition

- Any current vector I satisfying Kirchoffs 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

Example

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

Ip

1A

1A

New Formulation

- After some manipulation, the problem is changed

to the following - Solve Im from ZmImVm, 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(?)

What is Pre-conditioning?

- When matrix A is in bad shape, i.e., A has a

large condition number, then iterate methods to

solve Axb 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)xMb can be very fast - Ideally, if MA-1 then we are done

Preconditioning

- Reduced system
- Pre-conditioners

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

Avoiding Complex Numbers

- Reduced system
- Separate real and complex components ofthe

system - Solve this system by iterative method

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?

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 ACEFG and HMNLJ, give

the block entry matrix.

Assignment 3 (Due 3/13)

- 1. Use the solenoidal algorithm to perform

inductance extraction for a pair of conductors

x2y2?1, 0?z?10 and (x-10)2y2?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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