Chapter 11, Problem 6P

To determine

Show that the expressions $-\frac{\partial I\left(z,t\right)}{\partial z}=GV\left(z,t\right)+C\frac{\partial V\left(z,t\right)}{\partial t}$ and $-\frac{\partial V\left(z,t\right)}{\partial z}=RI\left(z,t\right)+L\frac{\partial I\left(z,t\right)}{\partial t}$ are same for both $\Pi \text{-type}$ and $\text{T-type}$.

Explanation of Solution

Calculation:

Consider $\Pi \text{-type}$ network.

The KVL equations are,

$V\left(z+\Delta z,t\right)=V\left(z,t\right)-R\Delta z{I}_1-L\Delta z\frac{\partial I_1}{\partial t}$        (1)

$I_1=I\left(z,t\right)-\frac{C}{2}\Delta z\frac{\partial V\left(z,t\right)}{\partial t}-\frac{G}{2}\Delta zV\left(z,t\right)$        (2)

Substitute Equation (2) in Equation (1).

$\begin{array}{l}V\left(z+\Delta z,t\right)=\left\{\begin{array}{l}V\left(z,t\right)-R\Delta z\left[I\left(z,t\right)-\frac{C}{2}\Delta z\frac{\partial V\left(z,t\right)}{\partial t}-\frac{G}{2}\Delta zV\left(z,t\right)\right]-\\ L\Delta z\left[\frac{\partial I}{\partial t}-\frac{C}{2}\Delta z\frac{{\partial }^{2}V\left(z,t\right)}{\partial t^2}-\frac{G}{2}\Delta z\frac{\partial V\left(z,t\right)}{\partial t}\right]\end{array}\right\}\\ \frac{V\left(z+\Delta z,t\right)-V\left(z,t\right)}{\Delta z}=\left\{\begin{array}{l}-R\left[I\left(z,t\right)-\frac{C}{2}\frac{\partial V\left(z,t\right)}{\partial t}-\frac{G}{2}V\left(z,t\right)\right]-\\ L\left[\frac{\partial I}{\partial t}-\frac{C}{2}\frac{{\partial }^{2}V\left(z,t\right)}{\partial t^2}-\frac{G}{2}\frac{\partial V\left(z,t\right)}{\partial t}\right]\end{array}\right\}\end{array}$

Take limits $\underset{\Delta z\to 0}{\lim }$.

$\begin{array}{l}\underset{\Delta z\to 0}{\lim }\frac{V\left(z+\Delta z,t\right)-V\left(z,t\right)}{\Delta z}=\underset{\Delta z\to 0}{\lim }\left\{\begin{array}{l}-R\left[I-\frac{C}{2}\Delta z\frac{\partial V}{\partial t}-\frac{G}{2}\Delta zV\right]-\\ L\left[\frac{\partial I}{\partial t}-\frac{C}{2}\Delta z\frac{{\partial }^{2}V}{\partial t^2}-\frac{G}{2}\Delta z\frac{\partial V}{\partial t}\right]\end{array}\right\}\\ \frac{\partial V}{\partial z}=-RI-L\frac{\partial I}{\partial t}\end{array}$

$-\frac{\partial V\left(z,t\right)}{\partial z}=RI\left(z,t\right)+L\frac{\partial I\left(z,t\right)}{\partial t}$        (3)

The KCL equation is,

$I\left(z+\Delta z,t\right)-I\left(z,t\right)=-G\Delta z\left(\frac{V\left(z,t\right)+V\left(z+\Delta z\right)}{2}\right)-C\Delta z\frac{\partial }{\partial t}\left(\frac{V\left(z,t\right)+V\left(z+\Delta z,t\right)}{2}\right)$

Take limits $\underset{\Delta z\to 0}{\lim }$.

$\underset{\Delta z\to 0}{\lim }\left[\frac{I\left(z+\Delta z,t\right)-I\left(z,t\right)}{\Delta z}\right]=\underset{\Delta z\to 0}{\lim }\left[\begin{array}{l}-G\left(\frac{V\left(z,t\right)+V\left(z+\Delta z\right)}{2}\right)-\\ C\frac{\partial }{\partial t}\left(\frac{V\left(z,t\right)+V\left(z+\Delta z,t\right)}{2}\right)\end{array}\right]$

$-\frac{\partial I\left(z,t\right)}{\partial z}=GV\left(z,t\right)+C\frac{\partial V\left(z,t\right)}{\partial t}$        (4)

Consider $\text{T-type}$ network.

The KVL equation is,

$\begin{array}{l}V\left(z,t\right)=R\frac{\Delta l}{2}I\left(z,t\right)+L\frac{\Delta l}{2}\frac{\partial I\left(z,t\right)}{\partial t}+V\left(z+\Delta l/2,t\right)\\ -\frac{V\left(z+\Delta l/2,t\right)-V\left(z,t\right)}{\Delta l/2}=RI\left(z,t\right)+L\frac{\Delta l}{2}\frac{\partial I\left(z,t\right)}{\partial t}\end{array}$

Take limits $\underset{\Delta z\to 0}{\lim }$.

$\underset{\Delta z\to 0}{\lim }\left[-\frac{V\left(z+\Delta l/2,t\right)-V\left(z,t\right)}{\Delta l/2}\right]=\underset{\Delta z\to 0}{\lim }\left[RI\left(z,t\right)+L\frac{\Delta l}{2}\frac{\partial I\left(z,t\right)}{\partial t}\right]$

$-\frac{\partial V\left(z,t\right)}{\partial z}=RI\left(z,t\right)+L\frac{\partial I\left(z,t\right)}{\partial t}$        (5)

The KCL equation is,

$\begin{array}{l}I\left(z,t\right)=I\left(z+\Delta l,t\right)+G\Delta lV\left(z+\Delta l/2,t\right)+C\Delta l\frac{\partial V\left(z+\Delta l/2,t\right)}{\partial t}\\ -\frac{I\left(z+\Delta l,t\right)-I\left(z,t\right)}{\Delta l}=GV\left(z+\Delta l/2,t\right)+C\frac{\partial V\left(z+\Delta l/2,t\right)}{\partial t}\end{array}$

Take limits $\underset{\Delta z\to 0}{\lim }$.

$\underset{\Delta z\to 0}{\lim }\left[-\frac{I\left(z+\Delta l,t\right)-I\left(z,t\right)}{\Delta l}\right]=\underset{\Delta z\to 0}{\lim }\left[GV\left(z+\Delta l/2,t\right)+C\frac{\partial V\left(z+\Delta l/2,t\right)}{\partial t}\right]$

$-\frac{\partial I\left(z,t\right)}{\partial z}=GV\left(z,t\right)+C\frac{\partial V\left(z,t\right)}{\partial t}$        (6)

The Equations (3) and (5), (4) and (6) are equal.

Conclusion:

Hence, the expressions $-\frac{\partial I\left(z,t\right)}{\partial z}=GV\left(z,t\right)+C\frac{\partial V\left(z,t\right)}{\partial t}$ and $-\frac{\partial V\left(z,t\right)}{\partial z}=RI\left(z,t\right)+L\frac{\partial I\left(z,t\right)}{\partial t}$ are same for both $\Pi \text{-type}$ and $\text{T-type}$.