1. Wei Yao weiyao@ee.ucla.edu
Homework 3
Wei Yao

2. Modified Nodal Analysis (MNA) Equations
Any RLC circuit can be represented by first order differential equations
Gx(t) + C = Bu(t) (G+ sC)x(s) = Bu(s)
(Laplace(s) domain)
i(t) = LT x(t)
dx(t) dt

3. (G+ sC)x(s) = Bu(s) (Laplace(s) domain)
PRIMA review
G x(t) C = B u(t)
(G+ sC)x(s) = Bu(s) (Laplace(s) domain)
Only interested in inputs/outputs, and
Dynamics is important
Can we reduce the equation size?
Reduce the number of variables (column # of G and C)
Reduce the number of equations (row # of G and C)
dx(t) dt

4. Projection Framework
Check q equations
qxn G
qxq
nxn
nxq

5. Intuitive view of Krylov subspace choice for Uq
Taylor series expansion: A=-G-1C, R=G-1B  x=(I-sA)-1Ru
change base and use only the first few vectors of the Taylor series expansion: equivalent to match first q derivatives around expansion point

6. Orthonormalization of Uq:The Arnoldi Algorithm
For i = 1 to q-1
Generates k+1 vectors!
Orthogonalize new vector:
Remove the projection on other normalized vectors
For j = 1 to i
end
Normalize new vector
end 6

7. We know how to select Uq now…
but how about Vq?

8. PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98)
Select Vq=Uq with Arnoldi Krylov Projection Framework:
Use Arnoldi: Numerically very stable

9. Moment Matching Theorem
PRIMA preserves the moments of the transfer function up to the q-th order, i.e., 9

10. Combine point and moment matching: multipoint moment matching
Multiple expansion points give larger band
Moment (derivates) matching gives more accurate
behavior in between expansion points 10

11. Homework 3 [due ? ]
[3] Modify the PRIMA code with single frequency expansion to multiple points expansion. You should use a vector fspan to pass the frequency expansion points. Compare the waveforms of the reduced model between the following two cases: 1. Single point expansion at s=1e4. 2. Four-point expansion at s=1e3, 1e5, 1e7, 1e9.
Elfadel, I.M.; Ling, D.D.; , "A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks," Computer-Aided Design, Digest of Technical Papers., 1997 IEEE/ACM International Conference on , vol., no., pp.66-71, 9-13 Nov 1997
Odabasioglu, A.; Celik, M.; Pileggi, L.T.; , "PRIMA: passive reduced-order interconnect macromodeling algorithm ," Computer-Aided Design, Digest of Technical Papers., 1997 IEEE/ACM International Conference on , vol., no., pp.58-65, 9-13 Nov 1997 11

12. Matlab Files We provide two matlab files: prima.m
PRIMA function on single freq point moment matching
demo2_11.m
perform MOR (function call “prima”), calculate and compare corresponding time and frequency domain response between original matrix and MATLAB reduced matrix.
prima.m GC
demo2_11.m

13. Format of the input matrices for test
e-15
e-15
e-15
e-13
e-13
e-15 e
e-13
e-13
…..
The input files GC8 and GC9 each has 4 columns. They are: row number m, column number n, (m,n) entry in G matrix - G(m,n), (m,n) entry in C matrix - C(m,n) .
If both G(m,n) and C(m,n) are zero, that entry is omitted in input file. 13

16. Frequency and time domain response for single point expansion

17. Impulse response for single point expansion