Impedance Parameters - Relation To S-parameters

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

$\begin{align} S &= (\sqrt{y}Z\sqrt{y} \,- 1_{\!N}) (\sqrt{y}Z\sqrt{y} \,+ 1_{\!N})^{-1} \\ &= (\sqrt{y}Z\sqrt{y} \,+ 1_{\!N})^{-1} (\sqrt{y}Z\sqrt{y} \,- 1_{\!N}) \\ \end{align}$

where is the identity matrix, is a diagonal matrix having the square root of the characteristic impedance at each port as its non-zero elements,

$\sqrt{z} = \begin{pmatrix} \sqrt{z_{01}} & \\ & \sqrt{z_{02}} \\ & & \ddots \\ & & & \sqrt{z_{0N}} \end{pmatrix}$

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.

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