TEAM PROJECT. Transmission Line and Related PDEs. Consider a long cable or telephone

Chapter 12, Problem 12.1.200

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TEAM PROJECT. Transmission Line and Related PDEs. Consider a long cable or telephone wire (Fig. 312) 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. 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 (a) Show that ("first transmission line equation")au ai - - =Ri + Laxatwhere u(x, t) is the potential in the cable. Hint: ApplyKirchhoff's voltage law to a small portion of the cablebetween x and x + !n (difference of the potentials atx and x + !n = resistive drop + inductive drop).(b) Show that for the cable in (a) ("secondtransmission line equation"),ai au - - = Gu + CaxatHint: Use Kirchhoff's current law (difference of thecurrents at x and x + LlX = loss due to leakage toground + capacitive loss).(c) Second-order PDEs. Show that elimination ofi or u from the transmission line equations leads touxx = LCutt + (RC + GLhlt + RGu.ixx = LCitt + (RC + GL)it + RGi.(d) Telegraph equations. For a submarine cable,G is negligible and the frequencies are low. Show thatthis leads to the so-called submarine cable equationsor telegraph equationsFind the potential in a submarine cable with ends(x = 0, x = l) grounded and initial voltage distributionVa = const.(e) High-frequency line equations. Show that in thecase of alternating currents of high frequencies theequations in (c) can be approximated by the so-calledhigh-frequency line equationsSolve the first of them, assuming that the initialpotential isVo sin (7fx/l).and ut(x. 0) = 0 and u = 0 at the ends x = 0 andx = I for all t.