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= V_{s}\cos h\gamma x-I_{s}Z_{o}\sin h\gamma x.

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

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

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

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

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

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

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

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

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

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

Z_{S}=Z_{o} \frac{(Z_{R}+Z_{O}\tan h\gamma l)}{(Z_{o}+Z_{R}\tan h\gamma 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.