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

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It is possible to show, using basic laws of electrical circuit theory, that the electrical current i ( x , t ) satisfies the PDE

2 i x 2 = L C 2 i t 2 + ( R C + G L ) i t + R G i (5)

where the constants R , L , C and G are, for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage v ( x , t ) also satisfies (5). Special cases of (5) arise in particular situations. For a submarine cable G is negligible and frequencies are low so inductive effects can also be neglected. In this case (5) becomes

2 i x 2 = R C i t (6)

which is called the submarine equation or telegraph equation . For high frequency alternating currents, again with negligible leakage, (5) can be approximated by

2 i x 2 = L C 2 i t 2 (7)

which is called the high frequency line equation .


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.