
In this section, we will derive the equation for the reflection coefficient. The reflection coefficient relates the forward-going voltage with reflected voltage.

### Reflection coefficient at the load

Equations eq:RCvtl-eq:RCctl represent the voltage and current on a lossless transmission line shown in Figure fig:RCTLCircuit.

We set up the z-axis so that the $z=0$ is at the load, and the generator is at $z=-l$. At $z=0$, the load impedance is connected. The definition of impedance is $Z=V/I$, therefore at the z=0 end of the transmission line, the voltage and current on the transmission line at that point have to obey boundary condition that the load impedance imposes.

Substituting z=0, the boundary condition, in Equations eq:RCvtl-eq:RCctl, we get Equations eq:RCvtl1-eq:RCctl1.

Dividing the two above equations, we get the impedance at the load.

We can now solve the above equation for $\tilde {V}_0^-$

Lowercase $z_L$ is called the ”normalized load impedance”. It is the actual impedance divided by the transmission line impedance $z_L=\frac {Z_L}{Z_0}$. For example, if the load impedance is $Z_L=100 \Omega$, and the transmission-line impedance is $Z)=50 \Omega$, then the normalized impedance is $z_L=\frac {100 \Omega }{50 \Omega }=2$. Normalized impedance is a unitless quantity.

#### Example

(a)
100 $\Omega$ transmission line is terminated in a series connection of a 50 $\Omega$ resistor and 10 pF capacitor. The frequency of operation is 100 MHz. Find the voltage reflection coefficient.
(b)
For purely reactive load $Z_L=j 50 \Omega$, find the reflection coefficient.

### Voltage and Current on a transmission line

Now that we related forward and reflected voltage on a transmission line with the reflection coefficient at the load, we can re-write the equations for the current and voltage on a transmission line as:

We see that if we know the length of the line, line type, the load impedance, and the transmission line impedance, we can calculate all variables above, except for $\tilde {V}_0^+$. In the following chapters, we will derive the equation for the forward going voltage at the load, but first, we will look at little more at the various reflection coefficients on a transmission line.

### Reflection coefficient anywhere on the line

Equations eq:RCvtl-eq:RCctl can be concisely written as

Where $\tilde {V}(z)^+$ is the forward voltage anywhere on the line, $\tilde {V}(z)^-$ is reflected voltage anywhere on the line, $\tilde {I}(z)^+$ is the forward current anywhere on the line, and $\tilde {I}(z)^-$ is the reflected current anywhere on the line.

We can then define a reflection coefficient anywhere on the line as

Since we already defined the reflection coefficient at the load, the reflection at any point on the line $z=-l$ is

### Reflection coefficient at the input of the transmission line

Using the reasoning above, the reflection coefficient at the input of the line whose length is $l$ is