Demonstrative Video
Lumped equivalent circuit of a long line
\[\left[\begin{array}{c}
V_{S}\\
I_{S}
\end{array}\right]=\left[\begin{array}{cc}
\left(1+\dfrac{Y^{'}Z^{'}}{2}\right) & Z^{'}\\
Y^{'}\left(1+\dfrac{Y^{'}Z^{'}}{4}\right) &
\left(1+\dfrac{Y^{'}Z^{'}}{2}\right)
\end{array}\right]\left[\begin{array}{c}
V_{R}\\
I_{R}
\end{array}\right]\]
network,
For a
\[\left[\begin{array}{c}
V_{S}\\
I_{S}
\end{array}\right]=\left[\begin{array}{cc}
cosh\left(\gamma l\right) & Z_{c}sinh\left(\gamma l\right)\\
\dfrac{1}{Z_{c}}sinh\left(\gamma l\right) & cosh\left(\gamma
l\right)
\end{array}\right]\left[\begin{array}{c}
V_{R}\\
I_{R}
\end{array}\right]\]
\[\begin{aligned}
Z^{'} & =Z_{c}sinh\left(\gamma l\right)\\
1+\dfrac{1}{2} & Y^{'}Z^{'}=cosh\left(\gamma l\right)
\end{aligned}\]
According to exact solution of a long line
\[\begin{aligned}
Z^{'}&=\sqrt{\dfrac{z}{y}}sinh\left(\gamma l\right)\\
&=\dfrac{z.l}{\sqrt{yz}.l}sinh\left(\gamma l\right)\\
&\boxed{Z^{'}=Z.\left(\dfrac{sinh\left(\gamma l\right)}{\gamma
l}\right)}
\end{aligned}\]
\(Z^{'}\)\(\pi\)\(Z\)\(\Rightarrow
\left(\dfrac{sinh\left(\gamma l\right)}{\gamma l}\right)\)On simplifying
\[\begin{aligned}
1+\dfrac{1}{2}Y^{'}Z_{c}sinh\left(\gamma l\right) &
=cosh\left(\gamma l\right)\\
\Longrightarrow\dfrac{1}{2} &
Y^{'}=\dfrac{1}{Z_{c}}\left(\dfrac{cosh\left(\gamma
l\right)-1}{sinh\left(\gamma l\right)}\right)\\
& =\dfrac{1}{Z_{c}}tanh\left(\dfrac{\gamma
l}{2}\right)=\sqrt{\dfrac{y}{z}}tanh\left(\dfrac{\gamma l}{2}\right)\\
& =\dfrac{yl}{2}\left(\dfrac{tanh\left(\gamma l/2\right)}{\gamma
l/2}\right)=\dfrac{Y}{2}\left(\dfrac{tanh\left(\gamma l/2\right)}{\gamma
l/2}\right)
\end{aligned}\]
\((Y^{'}/2)\)\(\pi\)\(\Rightarrow
\left(\dfrac{tanh\left(\gamma l/2\right)}{\gamma l/2}\right)\)Further,
equivalent-T network can be obtained on similar lines
For load flow studies, equivalent-\(\pi\) or nominal-\(\pi\) is easily adopted, hence universally employed
\[\begin{aligned}
V_{s} & =\left(1+\frac{Y^{\prime} Z^{\prime}}{2}\right)
V_{r}+Z^{\prime}\left(1+\frac{Z^{\prime} Y^{\prime}}{4}\right) I_{r} \\
I_{s} & =Y^{\prime} V_{r}+Y^{\prime}\left(1+\frac{Z^{\prime}
Y^{\prime}}{2}\right) I_{r}
\end{aligned}\]
\[\begin{aligned}
1+\frac{Y^{\prime} Z^{\prime}}{2} & =\cosh \gamma l \\
Z^{\prime}\left(1+\frac{Z^{\prime} Y^{\prime}}{4}\right) &
=Z_{c} \sinh \gamma l
\end{aligned}\]
\[\begin{aligned}
& \boxed{Y^{\prime} =\frac{1}{Z_{c}} \sinh \gamma l }\\
1+\frac{Y^{\prime} Z^{\prime}}{2} & =\cosh \gamma l
\end{aligned}\]
On comparing
\[\begin{aligned}
1+\frac{Z^{\prime}}{2} \frac{1}{Z_{c}} \sinh \gamma l&=\cosh
\gamma l \\
\frac{Z^{\prime}}{2} \frac{1}{Z_{c}}&=\frac{\cosh \gamma
l-1}{\sinh \gamma l} \\
\frac{Z^{\prime}}{2}=\frac{2 \sinh ^{2} \frac{\gamma l}{2}}{2 \sinh
\frac{\gamma l}{2} \cosh \frac{\gamma l}{2}}&=Z_{c} \tanh
\frac{\gamma l}{2} \\
\frac{Z^{\prime}}{2}&=\sqrt{\frac{z}{y}} \frac{1}{2 \sqrt{y z}}
\sqrt{y z} \frac{\tanh \frac{\gamma l}{2}}{\frac{l}{2}} l \\
\frac{Z^{\prime}}{2}&=\frac{z l}{2} \frac{\tanh \frac{\gamma
l}{2}}{\frac{\gamma l}{2}} \\
&\boxed{\frac{Z^{\prime}}{2}=\frac{Z}{2} \frac{\tanh
\frac{\gamma l}{2}}{\frac{\gamma l}{2}}}
\end{aligned}\]