The telegraph equations for the local voltage V(x, t) and current I(x, t) along a transmission...

 

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The telegraph equations for the local voltage V(x, t) and current I(x, t) along a transmission line are Vx = -LI-RI & Ix = -CVt - GV, (1) where L, R and C denotes the impedance, resistance and capacitance of the line; G is the line conduc- tance. The transmission line has length l and there is no activity along it at t = 0 (so that V(x, 0) = I(x, 0) = 0). The voltage at the end x = l is suddenly increased to V (l,t) = Vo in order to send a signal to the other end at x = 0, which is held at zero voltage: V(0,t) = 0. The problem is to establish what signal, S(t) = I(0, t), is received. (a) (3 pts) First, eliminate I(x, t) to write a single PDE for V(x, t). Hint: differentiate in x the first PDE in (1), then use the second PDE in (1) to replace Ix. Show that this equation reduces to either the wave equation or the diffusion equation in certain limits of the parameters {L, R, C, G}. (b) (4 pts) Now (and from hereon) consider the case G = solution Vss(x). = R and CL. Find the steady state voltage (c) (11 pts) Next, attack the initial-value problem using separation of variables. (d) (4 pts) Last, use your solution for V(x,t) and the first PDE in (1), evaluated for x first-order ODE in time for S(t). Solve this ODE. To save you a little algebra, you may quote the integrals = 0, to write a L erz sin(yz)dz = Y - Yerz cos(yz) + rez sin yz 72 +12 erz cos(yz)dz rez cos(yz) - r+yerz sin yz 72 +12 The telegraph equations for the local voltage V(x, t) and current I(x, t) along a transmission line are Vx = -LI-RI & Ix = -CVt - GV, (1) where L, R and C denotes the impedance, resistance and capacitance of the line; G is the line conduc- tance. The transmission line has length l and there is no activity along it at t = 0 (so that V(x, 0) = I(x, 0) = 0). The voltage at the end x = l is suddenly increased to V (l,t) = Vo in order to send a signal to the other end at x = 0, which is held at zero voltage: V(0,t) = 0. The problem is to establish what signal, S(t) = I(0, t), is received. (a) (3 pts) First, eliminate I(x, t) to write a single PDE for V(x, t). Hint: differentiate in x the first PDE in (1), then use the second PDE in (1) to replace Ix. Show that this equation reduces to either the wave equation or the diffusion equation in certain limits of the parameters {L, R, C, G}. (b) (4 pts) Now (and from hereon) consider the case G = solution Vss(x). = R and CL. Find the steady state voltage (c) (11 pts) Next, attack the initial-value problem using separation of variables. (d) (4 pts) Last, use your solution for V(x,t) and the first PDE in (1), evaluated for x first-order ODE in time for S(t). Solve this ODE. To save you a little algebra, you may quote the integrals = 0, to write a L erz sin(yz)dz = Y - Yerz cos(yz) + rez sin yz 72 +12 erz cos(yz)dz rez cos(yz) - r+yerz sin yz 72 +12