Consider a long cable or telephone wire (Fig. 315) that is imperfectly insulated, so that leaks occur
Question:
(a) Show that (first transmission line equation)
where u (x, t) is the potential in the cable. Apply Kirchhoffs voltage law to a small portion of the cable between x and x+ Îx (difference of the potentials at x and x + Îx = resistive drop + inductive drop).
(b) Show that elimination of i or u from the transmission line equations leads to
(c) For a submarine cable, G is negligible and the frequencies are low. Show that this leads to the so-called submarine cable equations or telegraph equations
Find the potential in a submarine cable with ends (x = 0, x = l) grounded and initial voltage distribution U0 = const.
(d) Show that in the case of alternating currents of high frequencies the equations in (c) can be approximated by the so-called high-frequency line equations
Solve the first of them, assuming that the initial potential is
and ut (x, 0) = 0 and u = 0 at the ends x = 0 and x = l for all t.
Step by Step Answer:
Modeling And Analysis Of Dynamic Systems
ISBN: 9781138726420
3rd Edition
Authors: Ramin S. Esfandiari, Bei Lu