Question

Consider a long cable or telephone wire, shown below, that is imperfectly insulated, so that leaks occur along the entire length of the cable. The source S of the current i(x,t) in the cable is at x-0, the receiving end T at x = I. The current flows from S to T and through the load, and returns to the ground. Let the constants R, L, C, and G denote the resistance, inductance, capacitance to ground, and conductance to ground, respectively, of the cable per unit length. Transmission Line tl ai (a) Show that ("first transmission line equation" Ri+ L at where u(x, t) is the potential in the cable. Hint: Apply Kirchhoffs voltage law to a small portion of the cable between x and x Δχ (difference of the potentials at x and x + Δx-resistive drop +inductive drop) ai (b) Show that for the cable in (a) ("second transmission line equation"Gut C at Hint: Use Kirchhoff's current law (difference of the currents at x + ΔΧ-loss due to leakage to ground capacitive loss) (c) Second-Order PDEs. Show that elimination of i and u from the transmission equations leads to (d) Telegraph Equations. For a submarine cable, G is negligible, and the frequencies are low. Shovw 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 Uoconstant. (e) High-frequency Line Equations. 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: ra rax (f) Solve the first of them, assuming that the initial potential is Uosin ('ー) and ut (x, 0-0 and u 0 at the ends x = 0 and x 1 for all t.

Answer 1