Transmission line modelling

Question

folks

I've been lately struggling with modelling a transmission line. What I would like to do is to take a signal on one side of a cable, pass it through the cable and look how it has changed on the output.

My initial thought was to numerically solve the telegrapher's equation but as I suck at computational physics I dug around some more and found that there is a way to compute the influence of transmission line by the means of transfer function.

The transfer function I found on wikipedia as well as in sources cited by wikipedia is

$H(ω)= \frac{Z_LZ_C}{Z_C(Z_L+Z_S)\mathrm{cosh}(γx)+(Z_LZ_S+Z_C^2)\mathrm{sinh}(γx)}$

Here $Z_L$ is the load impedance, $Z_S$ is the source impedance and $Z_C$ is the characteristic impedance.

$γ=\sqrt{(R+jωL)(G+jωC)}$

$Z_C=\sqrt{\frac{R+jωL}{G+jωC}}$

You can see the modelled situation in the first picture.

What I did in my code was load the signal, compute the spectrum via scipy fft, used the frequency range to calculate the transfer function and multiplied the spectrum with it to compute the new spectrum after the transfer and preformed an inverse fft. The result is weird. With a cable of lenght 1 meter the amplitude drop is too big. The parameteres I used should be those of a coaxial cable RG-58, that is

R = 50 $\Omega$/meter

L = 25 nH/meter

C = 100 pF/meter

G = 0.02 S/meter

x = 1 meter

I included the results in the second picture.

Answer 1

My initial thought was to numerically solve the telegrapher's equation but as I suck at computational physics I dug around some more and found that there is a way to compute the influence of transmission line by the means of transfer function.

Unfortunately you will have to go back to the telegrapher's equation in this particular case as you are dealing with the wave-like behavior of the electromagnetic signals traveling through the transmission line.

signal_hat = sfft.fft(signal) 
freq = sfft.fftfreq(len(signal), (time[1]-time[0])) 
gamma = np.sqrt((R+1j*freq*L)*(G+1j*freq*C)) 
Z_c = np.sqrt((R + 1j * freq * L) / (G + 1j * freq * C)) 
Z_L = Z_c 
Z_s = Z_c 
signal_hat_new = signal_hat*(Z_L*Z_c)/(Z_c*(Z_L+Z_s)*np.cosh(gamma*x) + (Z_L*Z_s + Z_c**2)*np.sinh(gamma*x)) 
signal_new = sfft.ifft(signal_hat_new)

You can see that the approach is not new and there are many attempts in the literature to convert the time-domain problem into the frequency domain:

https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=48710115f87c77fd04134aa3824d62cec0e97edd

Nonetheless, you should be aware that even though the treatment in the frequency domain is indeed easier, you still need to account for the wave-like behavior of the signal in the time domain which in turn will distort the signal and show the values found in the literature.

For this specific issue I think you will find much better feedback here: https://electronics.stackexchange.com/ as there are plenty of very well informed users in this particular topic in that forum.