Again, we will look at a transmission line circuit in Figure fig:FVTRLine to find the input impedance on a transmission line.
The equations for the voltage and current anywhere (any z) on a transmission line are
Using the equations from the previous section, we can replace the transmission line with its input impedance, Figure fig:FVTRLineEqCirc.
From Figure fig:FVTRLineEqCirc, we can find the input voltage on a transmission line using the voltage divider.
Using Equation eq:FVitlfin, we can also find the input voltage. The input voltage equation at the generator is:
Since these two equations represent the same input voltage we can make them equal.
Rearranging the equation, we find .
There is another way to find the input impedance as a function of the input reflection coefficient.
We write KVL for the circuit in Figure fig:FVTRLineEqCirc.
Substituting these two equations in Equation eq:FVKVL we get
We can re-write this equation as follows.
Using that is the input reflection coefficient, and multiplying through with .
Rearranging the equation, we get
is the input reflection coefficient.
Because the generator’s impedance is equal to the transmission line impedance, we will use the second equation. When we see that the denominator simplifies into , and we can further simplify the fraction to get the final value of .