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.

Read more about this topic: Impedance Parameters

Famous quotes containing the words relation to and/or relation:

“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)

“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (1623–1662)