# input impedance of Transmission Line

Consider the basic form of Transmission line with some impedance $Z_{R}$ at the Load end.

In order to find out the impedance at the input terminals of a Transmission line choose the Basic Transmission Line equations

$V=&space;V_{s}\cos&space;h\gamma&space;x-I_{s}Z_{o}\sin&space;h\gamma&space;x$.

$I=&space;I_{s}\cos&space;h\gamma&space;x-\frac{V_{s}}{Z_{o}}\sin&space;h\gamma&space;x$.

now at x= l , $V=V_{R}&space;\&space;and&space;\&space;I=I_{R}$ .

$V_{R}=&space;V_{s}\cos&space;h\gamma&space;l-I_{s}Z_{o}\sin&space;h\gamma&space;l-----EQN(1)$.

$I_{R}=&space;I_{s}\cos&space;h\gamma&space;l-\frac{V_{s}}{Z_{o}}\sin&space;h\gamma&space;l----EQN(2)$.

The load voltage is given by the equation $V_{R}=I_{R}Z_{R}$.

$V_{s}\cos&space;h\gamma&space;l-I_{s}Z_{o}\sin&space;h\gamma&space;l&space;=&space;Z_{R}(I_{s}\cos&space;h\gamma&space;l-\frac{V_{s}}{Z_{o}}\sin&space;h\gamma&space;l)$

$V_{s}Z_{O}\cos&space;h\gamma&space;l-I_{s}Z_{o}^{2}\sin&space;h\gamma&space;l&space;=&space;Z_{R}(I_{s}Z_{o}\cos&space;h\gamma&space;l-V_{S}\sin&space;h\gamma&space;l)$

$V_{s}(Z_{O}\cos&space;h\gamma&space;l+Z_{R}\sin&space;h\gamma&space;l)&space;=&space;I_{s}Z_{o}(Z_{o}\sin&space;h\gamma&space;l+Z_{R}\cos&space;h\gamma&space;l)$

$Z_{S}=\frac{V_{S}}{I_{S}}=Z_{o}&space;\frac{(Z_{R}\cos&space;h\gamma&space;l+Z_{O}\sin&space;h\gamma&space;l)}{(Z_{o}\cos&space;h\gamma&space;l+Z_{R}\sin&space;h\gamma&space;l)}$

$Z_{S}&space;\&space;(or)&space;\&space;Z_{in}=Z_{o}&space;\frac{(Z_{R}\cos&space;h\gamma&space;l+Z_{O}\sin&space;h\gamma&space;l)}{(Z_{o}\cos&space;h\gamma&space;l+Z_{R}\sin&space;h\gamma&space;l)}$.

$Z_{S}=Z_{o}&space;\frac{(Z_{R}+Z_{O}\tan&space;h\gamma&space;l)}{(Z_{o}+Z_{R}\tan&space;h\gamma&space;l)}$   represents the source (or) input impedance of a basic Transmission Line.

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## Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.