Impedance and Admittance Matrices

Impedance and Admittance Matrices
As in low frequency electrical circuits, a matrix description for portions of microwave circuits can prove useful in simulations and for understanding the behavior of the subcircuit, among other reasons. Matrix descriptions are a very convenient way to integrate the effects of the subcircuit into a circuit without having to concern yourself with the specific details of the subcircuit. We will primarily be interested in ABCD and S matrices in this course, though Z and Y matrices will also prove useful. The ABCD and S parameters are probably new to you. As we’ll see, using these matrix descriptions is very similar to other two-port models for circuits you’ve seen before, such as Z and Y matrices.
Z Matrices
As an example of Z matrices, consider this two-port network:

The Z-matrix description of this two-port is defined as

As an example, let’s determine the Z matrix for this T-network

Applying (1) repeatedly to all four Z parameters, we find:

Collecting these calculations, then for this T-network:

Notice that this matrix is symmetric. That is, Zij = Zji for i ¹ j . It can be shown that [Z] will be symmetric for all “reciprocal” networks. What’s the usefulness of an impedance matrix description? For one thing, if a complicated circuit exists between the ports, one can conveniently amalgamate the electrical characteristics into this one matrix. Second, if one has networks connected in series, it’s very easy to combine the Z matrices. For example:

By definition

From the figure we see that I1' = I1', I2' = I2', and that V1 =V1' +V 1 ', V2 = V2 ' + V2'. So, summing the above two matrix equations gives

Also from the figure, note that I1 = I1 ' and I2 = I2 ' . Therefore,

From this result, we see that for a series connection of two-port networks, we can simply add the Z matrices to form a single “super” Z matrix
[Z] = [Z']+[Z'']
that incorporates the electrical characteristics of both networks and their mutual interaction.
Y Matrices
A closely related characterization is the Y-matrix description of a network:

By definition:


Comparing (4) and (1) we see that
The Y-parameter description is useful when connecting networks in parallel:

From this diagram, we see that

[Y] = [ Y] + [y ‘’ ]
Z and Y Matrices for Microwave Networks
We can easily generalize these Z and Y parameter descriptions for microwave networks and multiport networks. Consider an N-port network connected to transmission lines

The locations tn , n =1,…,N , are the terminal planes for each port. These are the positions on that TL where the phase is arbitrarily chosen equal to zero. At these terminal planes (which are also called the phase planes), z0 =0 so that the voltage on the nth TL
Vn(zn) = V n + e -jb n zn + Vn - e + jb nzn
Vn ( zn = 0 ) = Vn + + V n -
In ( z n - 0 ) = I n + + I n -
Since the telegrapher’s equations are linear, any N linearly independent combinations of the 2N quantities Vn and In may be chosen as the independent variables. For an impedance description, we choose In as the independent variables. Then,

[V] = [Z] .[I ]
For an admittance description, we choose Vn as the independent variables:

[I] = [Y ] . [V]
Global Characteristics of Z and Y Matrices
Finally, these are two extremely important properties of Z and Y matrices:
1. For a reciprocal network
Zij = Z ji and Yij = Yji
That is, the matrices are symmetrical about the main diagonal. (We observed this characteristic in the Z matrix of an impedance T-network earlier in this lecture.) A reciprocal network is one where a source instrument and a measurement instrument can be exchanged between two ports and the measured quantity remains unchanged. All passive (and some active) circuits you’ve encountered in circuits and electronics courses are reciprocal networks.
2. For a lossless network
Re {Zij} = 0 " I,j
From (5), this implies that
Re {Yij} = 0 " I,j
In other words, for a lossless network the Z and Y matrices are purely imaginary