Medium Transmission Lines
Demonstrative Video
Medium Transmission line
TL length of more than 80 kms but less than 250 kms
The parameters (Resistance, Inductance, and Capacitance) are distributed uniformly along the line.
Unlike a short transmission line, the line \(I_c\) of a medium transmission line is appreciable and hence the shunt capacitance must be considered (this is also the case for long transmission lines).
\(Y\) and \(Z\) are considered as a lumped parameter
Lumped parameters can be represented using two different models, namely:
Nominal \(\Pi\) representation (nominal pi model)
Nominal \(\mathrm{T}\) representation (nominal T model)
Nominal \(\mathrm{T}\) representation
Let, \(V_{c} =\) voltage across the capacitor and \(Y = j\omega C\) \[\begin{aligned} V_{c} & =V_{r}+I_{r}\left(Z/2\right)\\ I_{S} & =I_{r}+V_{c}Y\\ &=I_{r}+Y \cdot V_{r}+I_{r}\left(Z/2\right)\cdot Y\\ V_{s} & =V_{c}+I_{s}\left(Z/2\right) \end{aligned}\]
Substituting \(V_{c}\) and \(I_{s}\) in \(V_{s}\) \[V_{s}=V_{r}\left(1+\dfrac{YZ}{2}\right)+I_{r}Z\left(1+\dfrac{YZ}{4}\right)\]
$$\boxed{ \left[\begin{array}{c} V_s \\ I_s \end{array}\right]=\left[\begin{array}{cc} \left(1+\frac{Y Z}{2}\right) & Z\left(1+\frac{Y Z}{4}\right) \\ Y & \left(1+\frac{Y Z}{2}\right) \end{array}\right]\left[\begin{array}{c} V_r \\ I_r \end{array}\right] }$$
\[\begin{aligned} A & = \left(1+\dfrac{YZ}{2}\right) & B & = Z\left(1+\dfrac{YZ}{4}\right)\\ C & = Y & D & = \left(1+\dfrac{YZ}{2}\right) \end{aligned}\]
Nominal \(\mathrm{Pi}\) representation
\[\begin{aligned}
I_{s} & =I_{r}+V_{r}\dfrac{Y}{2}+V_{s}\dfrac{Y}{2}\\
V_{s} & =V_{r}+\left(I_{r}+V_{r}\dfrac{Y}{2}\right)Z\\
&=V_{r}\left(1+\dfrac{YZ}{2}\right)+I_{r}Z
\end{aligned}\]
\(\therefore I_{s}=V_{r}Y\left(1+\dfrac{YZ}{4}\right)+I_{r}\left(1+\dfrac{YZ}{2}\right)\)
Hence,
$$\boxed{ \left[\begin{array}{c} V_s \\ I_s \end{array}\right]=\left[\begin{array}{cc} \left(1+\frac{Y Z}{2}\right) & Z \\ Y\left(1+\frac{Y Z}{4}\right) & \left(1+\frac{Y Z}{2}\right) \end{array}\right]\left[\begin{array}{c} V_r \\ I_r \end{array}\right] }$$
\[\begin{aligned} A& = \left(1+\dfrac{YZ}{2}\right) & B& = Z\\ C & =Y\left(1+\dfrac{YZ}{4}\right) & D & = \left(1+\dfrac{YZ}{2}\right)\\ \end{aligned}\]