how to calculate DC resistance of transmission line from telegrapher's equation

Question

Can someone explain to me how to use the transmission line (TL) formula (from telegrapher's equations) to calculate the DC resistance of the TL?

Here's my work so far:

V(x,s) = A * exp(-gamma*x) + B * exp(gamma*x)

where A and B = complex amplitudes of forward going and backward going waves, respectively

gamma = sqrt((R+sL)(G+s*C))

Assumptions/model:

• Telegrapher's model (L,C,R,G constants)
• A DC source (Vs) is attached at one end of the transmission line, and the output of transmission line is a shorted.
• G = 0 (no shunt resistance)

I expect the DC resistance of the transmission line to be equal to R*(length of transmission line).

I should be able to calculate the constants A and B from the boundary conditions (voltage = Vs at one end, 0V at other).

However, when I plug s = 0 (DC) and G = 0 into formula for gamma, I get gamma = 0. When using gamma = 0 in transmission line equation, I get V(x,s) = A + B

This implies that V(x,s) is not a function of x at all, which contradicts the boundary conditions.

Please help me figure out where my reasoning went wrong. Do TL equations not apply at DC, and if so, why not?

Thank you.

Answer 1

I figured out the answer to my question. Yes, you can use the Telegrapher's equations to compute the DC resistance when a transmission line is terminated with a short and when G (shunt conductance) = 0. The key to using the equations is to keep G as a term but assume it to is very small at the end so that you can use the asymptotic behavior of the functions that is in. I'll explain below:

Using the boundary conditions for the short terminated wave equation, the constants A and B for the voltage TL equation can be determined. These are A = Vs/(2*sinh(gamma*Lwg)) and B = -A.

Input impedance is simply V(x,s) / I(x,s),

where

I(x,s) = A/Zc * exp(-gamma*x) - B/Zc * exp(gamma*x)

x = 0 at shorted end and x = -Lwg at source end.

Zc = sqrt((R + sL)/(G + sC)) (see Telegrapher's equations)

gamma = sqrt((R + sL)(G + sC)) (see Telegrapher's equations)

Plugging A and B into V and I and calculating V/I gives:

V(x,s)/I(x,s) = -tanh(gamma*x) * Zc

At DC, Zc = sqrt(R/G)

At DC, gamma = sqrt(RG)

With x = -Lwg, we get

V/I = tanh(sqrt(RG)*Lwg) * sqrt(R/G)

if G is very small, tanh(sqrt(RG)*Lwg) will be approximatley sqrt(RG)*Lwg

And thus V/I = sqrt(RG)*Lwg * sqrt(R/G) = R*Lwg

This is the result I was expecting.