Z-parameters are also known as open-circuit impedance parameters as they are calculated under open circuit conditions. i.e., Ix=0, where x=1,2 refer to input and output currents flowing through the ports (of a two-port network in this case) respectively.
The Z-parameter matrix
A Z-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a black box with a number of ports. A port in this context is a pair of electrical terminals carrying equal and opposite currents into and out-of the network, and having a particular voltage between them. The Z-matrix gives no information about the behaviour of the network when the currents at any port are not balanced in this way (should this be possible), nor does it give any information about the voltage between terminals not belonging to the same port. Typically, it is intended that each external connection to the network is between the terminals of just one port, so that these limitations are appropriate.
For a generic multi-port network definition, it is assumed that each of the ports is allocated an integer n ranging from 1 to N, where N is the total number of ports. For port n, the associated Z-parameter definition is in terms of the port current and port voltage, and respectively.
For all ports the voltages may be defined in terms of the Z-parameter matrix and the currents by the following matrix equation:
where Z is an N × N matrix the elements of which can be indexed using conventional matrix notation. In general the elements of the Z-parameter matrix are complex numbers and functions of frequency. For a one-port network, the Z-matrix reduces to a single element, being the ordinary impedance measured between the two terminals. The Z-parameters are also known as the open circuit parameters because they are measured or calculated by applying current to one port and determining the resulting voltages at all the ports while the undriven ports are terminated into open circuits.
The equivalent circuit for Z-parameters of a two-port network.
The equivalent circuit for Z-parameters of a reciprocal two-port network.
The Z-parameter matrix for the two-port network is probably the most common. In this case the relationship between the port currents, port voltages and the Z-parameter matrix is given by:
For the general case of an N-port network,
The input impedance of a two-port network is given by:
where ZL is the impedance of the load connected to port two.
Similarly, the output impedance is given by:
where ZS is the impedance of the source connected to port one.
and is the corresponding diagonal matrix of square roots of characteristic admittances. In these expressions the matrices represented by the bracketed factors commute and so, as shown above, may be written in either order.[note 1]
In the special case of a two-port network, with the same characteristic impedance at each port, the above expressions reduce to
The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions
The above expressions will generally use complex numbers for and . Note that the value of can become 0 for specific values of so the division by in the calculations of may lead to a division by 0.
Relation to Y-parameters
Conversion from Y-parameters to Z-parameters is much simpler, as the Z-parameter matrix is just the inverse of the Y-parameter matrix. For a two-port: