Question

5. (Telegraph equations) Using basic electrical theory, it can be shown that the current I(r, t) and voltage V(z, t) in an electrical transmission (eg., a power line or telephone line) at position in the lne andimt obey the equations Math 121 Applied DiffEqns II Homework 1 Due: Febuary 7, 2019 where C is the capacitance per u length, Gis the leakage per unit length, R is the resistance per unit length, and L is the inductance per unit length (a) Assume C,G, R, L are ll constants. Eliminate Vi the equations above to obtain the PDE ar Hint: Differentiate the first equation with respect to z, the second with respect to t, and combine to eliminate the mired derivative Vt and Vz terms. Use the second equation again to eliminate the remaining V term. (b) Similar to (a), determine the PDE for the voltage if I is eliminated. (c) If R 0 and G0, then this idealized case is called a loseless transmission line. Show for lossless transmission lines, (2) reduces to the two wave equations where c := /

Answer 1

Because Given equations are partial differential I/partial differential x + c partial differential V/partial differential t + Gv = 0 --------------(I) partial differential V/partial differential x + L partial differential I/partial differential t + RI = 0-----------(II) }--------(2) where V(x, t) is the voltage, I(x, t) is the current and c is the leakage per unit length , R is the resistance per unit length , L is the inductance per unit length (a) Assuming C, G, R, L all are constant , eliminate V in the equations above to obtain the partial differential equation partial differential^2 I/partial differential x^2 = CL partial differential^2 I/partial differential t^+(CR + GL) partial differential I/partial differential t + GRI differentiating equation (I) partially with respect to x we get partial differential^2 I/partial differential x^2 + C partial differential^2 V/partial differential x partial differential t + G partial differential V/partial differential x = 0