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