Reflection coefficient equations

Γ → Z tool

Given a wave propagating through a medium with characteristic impedance Z0, the reflection coefficient indicates how much of the power is reflected back because of the change in the charateristic impedance. So, if the load impedance is ZL, the reflection coefficient is calculated as:

$$\Gamma = {Z_L - Z_0 \over Z_L + Z_0} \tag{1}$$

The load impedance can be calculated from the reflection coefficient as follows:

$$Z_L = Z_0 · {(1 + \Gamma) \over (1 - \Gamma)} \tag{2}$$

On the other hand, the return loss is calculated as:

$$S_{11} = {20 · log_{10}(\vert \Gamma \vert)} \tag{3}$$

$${\vert \Gamma \vert} = 10^{{S_{11} } \over 20} \tag{4}$$

The standing wave ratio (SWR) is given by:

$$SWR = {1+ \vert \Gamma| \over 1 - \vert \Gamma \vert} \tag{5}$$

Then:

$${SWR - SWR · \vert \Gamma \vert} = {1 + \vert \Gamma \vert} \Rightarrow {SWR - 1} = {\vert \Gamma \vert + SWR · \vert \Gamma \vert} \tag{6}$$

so,

$${\vert \Gamma \vert} = { {SWR - 1} \over {SWR + 1} } \tag{7}$$

Finally,

$${S_{11}} = 20 · log_{10} \left( {SWR - 1} \over {SWR + 1}\right) \tag{8}$$

Γ → Z tool