3 Transmission line equations
In a long electrical cable or a telephone wire both the current and voltage depend upon position along the wire as well as the time (see Figure 6).
Figure 6
It is possible to show, using basic laws of electrical circuit theory, that the electrical current [maths rendering] satisfies the PDE
[maths rendering] (5)
where the constants [maths rendering] and [maths rendering] are, for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage [maths rendering] also satisfies (5). Special cases of (5) arise in particular situations. For a submarine cable [maths rendering] is negligible and frequencies are low so inductive effects can also be neglected. In this case (5) becomes
[maths rendering] (6)
which is called the submarine equation or telegraph equation . For high frequency alternating currents, again with negligible leakage, (5) can be approximated by
[maths rendering] (7)
which is called the high frequency line equation .
Task!
What PDEs, already discussed, have the same form as equations (6) or (7)?
(6) has the same form as the one-dimensional heat conduction equation.
(7) has the same form as the one-dimensional wave equation.