Ok so I'm analyzing a problem with a very simple lossless transmission line:
In my case $$R=20 \Omega$$ and I have to find the evolution of uL(t)
So I had no problem doing the first iterations.
I think I don't need to give much details but basically
$$u_L=u_i + u_r$$
where i and r means incident wave and reflected wave
I will constantly have:
$$u_i=U_0-u_r$$
$$u_r=\frac{R - R_{\omega}}{R + R_{\omega}}u_i$$
So I will have the following sequence
$$u_i=U_0=120V$$ $$u_r=-60V$$ $$u_L=60V$$
$$u_i=U_0=180V$$ $$u_r=-90V$$ $$u_L=90V$$
$$u_i=U_0=210V$$ $$u_r=-105V$$ $$u_L=105V$$
and so on and so forth.
I have no problem computing this iterations and I totally understand the phenomena of wave propagation in the transmission lines.
What I don't understand is when my book says that in infinite time u_L will become 120 V. Is it a general property? I mean, I think it makes sense: as time goes to infinity the output voltage will tend to be equal to the input voltage, like we would have in traditional circuit analysis. But I'm not sure if I can jump to this conclusion.
I also did the analysis with the limit cases of R=0 and R=infinity: with R=0 I would of course always have a zero voltage as it should be. While in the case of an open-circuit I constantly have the output voltage switching from 240 V and 0 V as time goes by (but the mean value is indeed 120 V).
Can someone explain me mathematically why the output voltage will tend to be equal to the input voltage when we have a finite value resistor (different than zero)? Thanks!