S-Parameters

Scattering parameters, popularly known as S-parameters deals with incident and reflected power waves in linear systems. S-parameters are complex numbers. Smith charts are used to represent s-parameters, even CAD tools and VNC also.

Consider a 2-port network shown in Figure 1.

2 port network with S-parameters

S-parameters for this 2-port network is defined as

(1) $\begin{equation*} \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}= \begin{pmatrix} S_{11} ~ S_{12} \\ S_{21} ~ S_{22} \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \end{equation*}$

where,
$a_1$ and $a_2$ are normalized incident voltages, and $b_1$ and $b_2$ are normalized reflected voltages

$|a_1|^2 \rightarrow$ power incident on the input port of the network

(2) $\begin{equation*} a_1 = { V_1 + I_1 Z_o \over 2\sqrt{Z_o} } = {V_{i1} \over \sqrt{Z_o} } \end{equation*}$

$|b_1|^2 \rightarrow$ power reflected from the input port of the network

(3) $\begin{equation*} b_1 = { V_1 - I_1 Z_o \over 2\sqrt{Z_o} } = {V_{r1} \over \sqrt{Z_o} } \end{equation*}$

$|a_2|^2 \rightarrow$ power incident on the output port of the network (or power reflected from load)

(4) $\begin{equation*} a_2 = { V_2 + I_2 Z_o \over 2\sqrt{Z_o} } = {V_{i2} \over \sqrt{Z_o} } \end{equation*}$

$|b_2|^2 \rightarrow$ power reflected from the output port of the network

(5) $\begin{equation*} b_2 = { V_2 - I_2 Z_o \over 2\sqrt{Z_o} } = {V_{r2} \over \sqrt{Z_o} } \end{equation*}$

Using these incident and reflected voltages, S-parameters of 2-port network are given below
Input reflection coefficient, $S_{11}$

(6) $\begin{eqnarray*} S_{11} &=& {b_1 \over a_1}\biggr\rvert_{a_2=0; {\mbox {output port terminated with matched load}}\\ &=& \frac{V_1 - I_1 Zo}{V_1+I_1Z_o} = {Z_{in} - Z_o \over Z_{in} + Z_o} = \Gamma_{in} \end{eqnarray*}$

$S_{11}$ should be as small as possible to ensure that very small power is reflected back.

Similarly output reflection coefficient, $S_{22}$

(7) $\begin{eqnarray*} S_{22} &=& {b_2 \over a_2}\biggr\rvert_{a_1=0; {\mbox {input port terminated with matched load}}\\ &=& \frac{V_2 - I_2 Zo}{V_2 + I_2 Z_o} = {Z_{L} - Z_o \over Z_{L} + Z_o} = \Gamma_{L} \end{eqnarray*}$

For a passive device the reflecting power is always lower than incident power. Therefore the reflection coefficients, $S_{11}$ and $S_{22}$ , are always smaller than unity for passive devices.