Solving an N-port in MATLAB

Question

I have an N-Port which I want to derive the equivalent Z matrix, where N = 2 is the number of ports in this instance. So, for example,

P = Z * U

Where P and U are 2x1 vectors and Z is a 2x2 matrix.

I've applied conditions for P and measured U for two different cases. Ex:

case 1: P = [1; -1], resulting U = [u11; u21]

case 2: P = [-1; 1], resulting U = [u12; u22]

Now I want to solve for what the Z matrix

Z = [Z11 Z12; Z21 Z22]

using my known values. In other words, I'd like to know how I can use MATLAB to solve for the components of Z in terms of the components of U and P. (i.e. a symbolic solution not a numeric one). I'd also like to know how to expand this since I may have a number of ports.

Thanks for your help!

Answer 1

Well in the N=2 case you have 2 equations in 4 unknowns, so you have 2 free variables, so say let Z12=s and Z22=t then

Z11=(P1-s*u2)/u1

and

Z21=(P2-t*u2)/u1.

This is easily generalised to arbitrary N. You will have N*(N-1) free variables, so Z12, Z13,...,Z1N,Z22,...,Z2N,...,ZN2,...,ZNN are all free, and then, for each i=1,2,...,N,