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
(radio math index) (radio index) (Jim's home page)