# Energy of a wave in a transmission line

1. Feb 16, 2010

### walking_edges

1. The problem statement, all variables and given/known data

An ideal transmission line has a characteristic impedance of Z=50 Ohms and
v=200,000km/s propagation velocity. A sinusoidal signal with frequency f=1GHz and
A=10mV amplitude is traveling down the line.Its total duration is 10s.What total
energy is it carrying?

2. Relevant equations

P = IV
w = 2[pi]f
k=w/v
V(x,t)=Re{A exp[i(wt-kx)]}
I(x,t)=Re{A/Z exp[i(wt-kx)]}

3. The attempt at a solution

I drew a picture of a transmission line that terminated into to a resistor with the same characteristic impedance as the line. The sinusoidal wave was coming in from the left only. The reflection amplitude of this wave is 0, so I think I should be able to say that all the energy in the wave was "burned" off by the resistor.

A resistor burns energy at the rate of V^2/R J/s, which I said was equal to (A)^2/(Z), the amplitude of the wave squared divided by the characteristic impedance. I then multiplied the power by 10 seconds to get the total energy dissipated by the resistor, which I calculated to be 2*10^-5 J.

The small answer and the fact that I didn't use all the information given pretty much yells to me that I did something wrong. I imagine that time and position dependence of the voltage and current might have something to do with it, but I'm not exactly sure.

Last edited: Feb 16, 2010