How Can We Verify the Telegraph Equation for an Electrical Cable with Leakage?

[cbarker1]

Dear Everyone,

I am confused about how to start with the following problem:

Consider an electrical cable running along the x-axis that is not well insulated from the ground, so that leakage occurs along its entire length. Let V(x,t) and I(x,t) denote the voltage and current at point x in the wire at time t. These functions are related to each other by the system

${V}_{x}=-L{I}_{t}-RI$ and ${I}_{x}=-C{V}_{t}-GV$ where L is the inductance, R is the resistance, C is the capacitance, and G is the leakage to ground. Show that V and I each satisfy:

${u}_{xx}=LC{u}_{tt}+(RC+L){u}_{t}+RGu$.

Thanks,
Cbarker1

 

[GJA]

Hi Cbarker1,

There appears to be an error in the second order equation; it should be $$u_{xx}= LCu_{tt}+(RC+GL)u_{t}+RGu.$$ From here you want to work with the two given equations for $V_{x}$ and $I_{x}$ to see if you can get them to satisfy $$V_{xx} = LCV_{tt}+(RC+GL)V_{t}+RGV$$ and $$I_{xx} = LCI_{tt}+(RC+GL)I_{t}+RGI,$$ respectively. For example, you could start by using the $V_{x}$ equation to compute $V_{xx}$ obtaining $$V_{xx} = -LI_{tx}-RI_{x} = -LI_{tx}+RCV_{t}+RGV,$$ where the second equality comes from using the equation for $I_{x}$. You can see that this is a promising start because the terms $RCV_{t}$ and $RGV$ are already present, which is good because they should be there according to the formula we are trying to establish. See if you can fill in the rest of the details for $V_{xx}$, then try moving on to $I_{xx}.$