1. Finite Element Method
To be added later
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2. Inductance
Given a set of k conductors, compute the kk impedance matrix Z() V1 V2 I1 I2
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3. Partial Inductance
For any two pieces of interconnect, the partial inductance k l
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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
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5. Compute Inductance
Send 1A current in one conductor and 0A current through other conductors, then potential drop gives impedance V1 V2 1
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6. Boundary Element Method
Laplace integral equation
where J(r) is current density,  is conductivity, and (r) is potential drop across volume r
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7. Discretization Partition conductors into n filaments I1 I1 I5 I2 I6 I3
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8. Incident Matrix B n filaments m nodes n2 f1 f5 f2 f6 n1 n3 f3 f7 f4 f8
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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:
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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
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11. Example I1 I2 I3 I4 I5 I6 I7 I8 n1 n2 n3
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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
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13. Problem The original system is hard to solve:
Some algorithms (FastHenry) solved it anyway
We need a better formulation
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14. Solenoidal Basis Method
Linear system
Solenoidal basis
Basis for current that satisfies Kirchoff’s law:
Reduced system
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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
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16. Example
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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 = + Ip 1A 1A
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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()
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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
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20. Preconditioning
Reduced system
Pre-conditioners
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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
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22. Avoiding Complex Numbers
Reduced system
Separate real and complex components of the system
Solve this system by iterative method
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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?
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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.
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25. 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 (x-10)2+y21, 0z10.
2. Download and compile FastHenry, and compare with the above results
. Hand in printout of input file and output
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