Zparameters

Zparameters, or impedance parameters, provide a linear two-port description of an electrical network in terms of port voltages and currents. For a two-port with currents I1 and I2 entering the ports and voltages V1 and V2 at the ports, the relationship is expressed as

V1 = Z11 I1 + Z12 I2

V2 = Z21 I1 + Z22 I2.

The Z-parameter matrix Z = [Z11 Z12; Z21 Z22] characterizes the network under the assumption of linear, bilateral behavior and fixed port reference directions.

The individual parameters are defined by open-circuit conditions: Z11 is the input impedance with the second port open (I2 = 0), Z22 is the output impedance with the first port open (I1 = 0), Z12 is the transfer impedance V1/I2 with I1 = 0, and Z21 is the transfer impedance V2/I1 with I2 = 0. These definitions reflect the standard convention that currents enter the ports.

Key properties include reciprocity and frequency dependence. A reciprocal network satisfies Z12 = Z21. Non-reciprocal networks (such as those containing certain active elements or non-reciprocal devices) can have Z12 ≠ Z21. Z-parameters are generally complex-valued functions of frequency and are defined for linear time-invariant systems.

Relation to other two-port models is straightforward: the Y-parameter matrix (I = YV) is the inverse of the Z matrix when Z is invertible, and S-parameters can be derived from Z or Y given a reference impedance. Z-parameters are particularly convenient for low-frequency or lumped-element analyses where open-circuit measurements are practical, and they are widely used in RF circuit modeling, impedance matching, and network characterization.

In practice, Z-parameters are obtained by measuring port voltages and currents under the specified open-circuit conditions and assembling the parameter matrix for the network.