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

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.

For passive devices, |S_{11}| < 1 and |S_{22}| < 1

Forward transmission gain, S_{21}

(8)   \begin{equation*} S_{21} = {b_2 \over a_1}\biggr\rvert_{a_2=0; {\mbox {output port terminated with matched load}} \end{equation*}

Reverse transmission gain, S_{12}

(9)   \begin{equation*} S_{12} = {b_1 \over a_2}\biggr\rvert_{a_1=0; {\mbox {input port terminated with matched load}} \end{equation*}

Reflection coefficient S11 marked on Smith chart

Reflection coefficient, S11 marked on Smith chart

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