## Two Port Matrices

A handy way of manipulating multiple things is to consider them as generalized two port networks, represented by a 2x2 matrix. There are 3 forms I have found useful:

Z (impedance) matrix, which gives the voltages at each port given the currents: [V]=[Z][I]

Y (admittance) matrix, which gives the currents at each port given the voltages: [I]=[Y][V]. The admittance matrix is the foundation of nodal network analysis programs (like SPICE), which build a row of the matrix for each node, making use of the fact that the sum of all currents into a given node must be zero.

ABCD (chain or transmission line) matrix, which gives voltage and current at one port, given voltage and current at the other.

### Conversion among forms

One often needs to convert from one form to another. Z and Y matrices are fairly straightforward to convert, one being the matrix inverse of the other. ABCD matrices are a bit trickier. In any case, for the limited 2x2 case, explicit equations can be written.

Z
Y
ABCD
Z
 Z11 Z12 Z21 Z22
 Y22/det(Y) -Y12/det(Y) -Y21/det(Y) Y11/det(Y)
 A/C det(T)/C 1/C D/C
Y
 Z22/det(Z) -Z12/det(Z) -Z21/det(Z) Z11/det(Z)
 Y11 Y12 Y21 Y22
 D/B -det(T)/B -1/B A/B
ABCD
 Z11/Z21 det(Z)/Z21 1/Z21 Z22/Z21
 -Y22/Y21 -1/Y21 -det(Y)/Y21 -Y11/Y12
 A B C D

This Excel spreadsheet contains conversions among Z, Y, and ABCD matrices. You can copy and paste the relevant cells as needed. Two forms are provided: the first has the real and imaginary components in separate cells, and should work in almost any form of spreadsheet; the second uses the complex math functions provided in the Analysis ToolPak.

### Z matrix for a T network

To generate the Z matrix for a T network, here are the equations. Note that it degenerates in some cases.

 z11 = Z1 + Z3 z22 = Z2 + Z3 z21,z12 = Z3

### Y matrix for a Pi network

To generate the Y matrix for a Pi network, here are the equations.

 y11 = Ya + Yc = 1/Za + 1/Zc y22 = Yb + Yc = 1/Zb + 1/Zc y21,y12 = Yc = 1/Zc

### More reference information

Most network analysis textbooks cover this in some detail. The reference I've used the most is:
M.E. Van Valkenburg, Network Analysis, 2nd ed., Prentice-Hall, 1964 (There's probably a newer edition)

radio/math/twoport.htm - 31 Dec 2002 - Jim Lux