ZParameter

Z-parameter, short for impedance parameter, is a set of parameters used to describe a linear two-port network in terms of port voltages and currents at a given frequency. For a network with port voltages V1 and V2 and currents I1 and I2 entering the ports, the Z-parameters are defined by the equations

V1 = Z11 I1 + Z12 I2

V2 = Z21 I1 + Z22 I2.

The four constants Z11, Z12, Z21, and Z22 form the impedance matrix Z = [[Z11, Z12], [Z21, Z22]], which is generally complex and depends on frequency.

Z-parameters can be determined from open-circuit measurements. Specifically:

Z11 = V1 / I1 with I2 = 0

Z22 = V2 / I2 with I1 = 0

Z12 = V1 / I2 with I1 = 0

Z21 = V2 / I1 with I2 = 0.

The usual convention is that currents are defined as entering the network at both ports; some texts use opposite conventions, which affects sign.

Reciprocity is a key property: for reciprocal networks (those without unilateral elements), Z12 = Z21. In general, Z-parameters are valid for linear, time-invariant networks at a fixed frequency; they are not applicable to nonlinear or broadband behavior without treating the parameters as frequency-dependent.

Relation to other parameter sets: Z-parameters can be converted to Y-parameters (admittance), H-parameters (hybrid), or ABCD-parameters through standard formulas. They are widely used in RF and microwave engineering to model and analyze two-port networks, especially when the ports are exposed to fixed source and load impedances or when cascaded networks are involved.

Limitations include dependence on frequency, strict linearity, and careful attention to sign conventions and terminating conditions.