the shorted line App K INSTALLED

This is the Title PhL 3.26.05 This is now an exercise in Appendix K. Reader Exercise: Consider the DC network transmission-line model shown below which is shorted at the right end, has length L, and z increases to the left as shown: Since this is for ω = 0, the inductors and capacitors shown in Fig K.1 have been removed. Each section has length dz, the horizontal resistors have resistance Ridz, and the vertical ones have conductance Gdz. The resistance looking in from the left at location z is R(z). (a) Show that R(z+dz) = || ( R(z) + R3dz ) = where R3 ≡ R1+ R2 Note that dim(Ri) = ohms/m, dim(G) = mhos/m, and dim[R(z)] = ohms. (b) for very small dz show that R(z+dz) ≈ R(z) + ( R3 - G [R(z)]2 )dz and therefore R(z) solves the following non-linear differential equation: + G[R(z)]2 = R3 . (c) Show that the general solution to this equation is R(z) = tanh ( z + C ) C = constant and, since R(z) = 0, the solution is R(z) = tanh ( z ) so R(L) = tanh ( L ) (d) Show that R(L) ≈ if L >> 1/ and that this agrees with (K.6) for an infinite line with ω= 0. Thus, for large L the fact that the line is shorted at the right end makes no difference. Interpret the distance 1/ . (e) Show that R(L) → R3L as G→ 0 which is the obviously correct limit for the figure shown above when G = 0. Here is some Maple code related to item (c):