Impedance parameters

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, I_n\, and V_n\, respectively. For all ports the voltages may be defined in terms of the Z-parameter matrix and the currents by the following matrix equation: :V = Z I\, 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. ==Two-port networks==

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: :\begin{pmatrix} V_1 \\ V_2\end{pmatrix} = \begin{pmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{pmatrix}\begin{pmatrix}I_1 \\ I_2\end{pmatrix} . where :Z_{11} = {V_1 \over I_1 } \bigg|_{I_2 = 0} \qquad Z_{12} = {V_1 \over I_2 } \bigg|_{I_1 = 0} :Z_{21} = {V_2 \over I_1 } \bigg|_{I_2 = 0} \qquad Z_{22} = {V_2 \over I_2 } \bigg|_{I_1 = 0} For the general case of an N-port network, :Z_{nm} = {V_n \over I_m } \bigg|_{I_k = 0 \text{ for } k \ne m} Impedance relations The input impedance of a two-port network is given by: :Z_\text{in} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22}+Z_L} where ZL is the impedance of the load connected to port two. Similarly, the output impedance is given by: :Z_\text{out} = Z_{22} - \frac{Z_{12}Z_{21}}{Z_{11}+Z_S} where ZS is the impedance of the source connected to port one. ==Relation to S-parameters==

The Z-parameters of a network are related to its S-parameters by : \begin{align} Z &= \sqrt{z} (1_{\!N} + S) (1_{\!N} - S)^{-1} \sqrt{z} \\ &= \sqrt{z} (1_{\!N} - S)^{-1} (1_{\!N} + S) \sqrt{z} \\ \end{align}   and Two port In the special case of a two-port network, with the same characteristic impedance z_{01} = z_{02} = Z_0 at each port, the above expressions reduce to :Z_{11} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \over \Delta_S} Z_0 \, :Z_{12} = {2 S_{12} \over \Delta_S} Z_0 \, :Z_{21} = {2 S_{21} \over \Delta_S} Z_0 \, :Z_{22} = {((1 - S_{11}) (1 + S_{22}) + S_{12} S_{21}) \over \Delta_S} Z_0 \, Where :\Delta_S = (1 - S_{11}) (1 - S_{22}) - S_{12} S_{21} \, The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions :S_{11} = {(Z_{11} - Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \over \Delta} :S_{12} = {2 Z_0 Z_{12} \over \Delta} \, :S_{21} = {2 Z_0 Z_{21} \over \Delta} \, :S_{22} = {(Z_{11} + Z_0) (Z_{22} - Z_0) - Z_{12} Z_{21} \over \Delta} where :\Delta = (Z_{11} + Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \, The above expressions will generally use complex numbers for S_{ij} \, and Z_{ij} \, . Note that the value of \Delta\, can become 0 for specific values of Z_{ij} \, so the division by \Delta \, in the calculations of S_{ij} \, 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: :Z_{11} = {Y_{22} \over \Delta_Y} \, :Z_{12} = {-Y_{12} \over \Delta_Y} \, :Z_{21} = {-Y_{21} \over \Delta_Y} \, :Z_{22} = {Y_{11} \over \Delta_Y} \, where :\Delta_Y = Y_{11} Y_{22} - Y_{12} Y_{21} \, is the determinant of the Y-parameter matrix. ==Notes==