(G- signal generator; Z_a- output impedance; I- signal receiver Z_b- input impedance; L- transmission line length; Z₀- Line impedance.

When line is tuned and without losses then input voltage:

U_b(t) = Z_bE(t-t)/(Z_a + Z_b)

E(t)- generators signal amplitude; t- signal delay in line.

t =l/v ;

l- line lenght; v- signal speed.

If line is not tuned up, then there are distortions in line because of reflections in line:

If signal E(t) is step function:

Then signal in line exit in discrete time moments will be:

U(0) = p;

U(t) = p*(1+p_b)

U(2t) = p*(1+p_b+p_a*p_b)

U(3t) = p*(1+p_b+p_a*p_b+p_a*p_b^2);

U(4t) = p*(1+p_b+p_a*p_b+p_a*p_b^2+p_a^2*p_b^2);

U(5t) = p*(1+p_b+p_a*p_b+p_a*p_b^2+p_a^2*p_b^2+p_a^2*p_b^3) …

where

Depending on reflectance coefficients and their signs distortions can differentiate or integrate:

Real model using MathCAD was implemented to see how signal looks like on exit depending on parameters. Below you see used algorithm structure used in modeling:

Algorithm Part 1
Algorithm Part 2

One of results using trapezoid signal:

In exit we get distorted signal.

Download your MathCAD Transmition Line (13, 12 and 2001i versions ) file to test with various parameters and different signal shapes.