open circuit line and Short Circuit line

open circuit line:-

In order to observe the properties of an open circuited Transmission Line, chose a Transmission line of length l and is open circuited at the load end.

from the figure

V_{S}  –  Source voltage (at sending end (or) source end).

I_{S} –  Source current (at sending end (or) source end).

V_{R} –  Load voltage (at receiving end (or) Load end).

I_{R} –  Load current (at receiving end (or) Load end).

at Load end I_{R} =0  ,  V_{R} -  maximum voltage  and Z_{R}=\infty .

We knew  that  the input impedance of a Transmission line is given by  Z_{S} = Z_{O} \frac{(Z_{R}+Z_{O} \tan h\gamma l)}{(Z_{O}+Z_{R} \tan h\gamma l)}.

after substituting the conditions of a open circuit conditions  that is Z_{R}=\infty

Z_{S} =\frac{Z_{O}}{\tan h\gamma l}.

Z_{S} =Z_{O}\cot h\gamma l.

if the Transmission line is a loss-less line then \alpha =0   and  \gamma = j\beta .

Z_{S} =Z_{O}\cot (h j\beta l) .

Z_{S} =-jZ_{O}\cot \beta l.     since \cot (hj\beta l)=-j\cot \beta l .

alternative method:- 

The alternative way to derive the input impedance is by using voltage and current equations of a basic Transmission line

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.

at x=l   ,  V=V_{R}    and  I=I_{R} then the equations changes to 

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).

by substituting I_{R} =0  in equation(2)  and also choosing the line as loss-less line

0= I_{s}\cos (j\beta l)-\frac{V_{s}}{Z_{o}}\sin (j\beta l) .    

I_{s}\cos \beta l=\frac{V_{s}}{Z_{o}} j\sin \beta l .     since  \cos (hj\beta l) = \cos \beta l   and  \sin (hj\beta l) = j\sin \beta l .

Z_{S} =\frac{V_{s}}{I_{s}}=Z_{o} \frac{\cos \beta l}{j\sin \beta l}.

Z_{S}=Z_{OC} =-jZ_{o} \cot \beta l.

short circuit line:-

In order to observe the properties of an short circuited Transmission Line, chose a Transmission line of length l and is short circuited at the load end.

from the figure

V_{S}  –  Source voltage (at sending end (or) source end).

I_{S} –  Source current (at sending end (or) source end).

V_{R} –  Load voltage (at receiving end (or) Load end).

I_{R} –  Load current (at receiving end (or) Load end).

at Load end V_{R} =0  ,  I_{R} -  maximum current and Z_{R}=0 .

We knew  that  the input impedance of a Transmission line is given by  Z_{S} = Z_{O} \frac{(Z_{R}+Z_{O} \tan h\gamma l)}{(Z_{O}+Z_{R} \tan h\gamma l)}.

after substituting the conditions of a short circuit conditions  that is Z_{R}=0

Z_{S} =Z_{O}\tan h\gamma l.

if the Transmission line is a loss-less line then \alpha =0   and  \gamma = j\beta .

Z_{S} =Z_{O}\tan (h j\beta l) .

Z_{S} =jZ_{O}\tan \beta l.     since \tan (hj\beta l)=j\tan \beta l .

alternative method:- 

The alternative way to derive the input impedance is by using voltage and current equations of a basic Transmission line

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.

at x=l   ,  V=V_{R}    and  I=I_{R} then the equations changes to 

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).

by substituting V_{R} =0  in equation(1)  and also choosing the line as loss-less line

0= V_{s}\cos (j\beta l)-I_{s}Z_{o}\sin (j\beta l) .    

V_{s}\cos \beta l=I_{s}Z_{o} j\sin \beta l .     since  \cos (hj\beta l) = \cos \beta l   and  \sin (hj\beta l) = j\sin \beta l .

Z_{S} =\frac{V_{s}}{I_{s}}=Z_{o} \frac{j\sin \beta l}{\cos \beta l}.

Z_{S}=Z_{SC} =jZ_{o} \tan \beta l.

 

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Author: Lakshmi Prasanna Ponnala

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