# Scattering Matrix

Updated on 2017/08/10 00:00

## Syllabus

• Scattering Matrix: Significance, formulation and properties.

## Introduction and Significance

Low frequency ckt can be described by two port network and their parameter such as z,y,h,abcd etc. as per network theory

Here network parameter relate the total voltage and total current as shown in fig.

## Formulation

-it is a square matrix which gives all the combination of power relationship between the various input and output ports of a microwave junction

- the element of the matrix are called scatter coeffients or scattering (s) parameter

To obtain the relationship between scattering matrix and input output powers at different ports, consider a junction of ‘n’ number of transmission line where i th line ( i can any line from 1 to n) is terminated in source as shown in fig.

### Case 1: First line terminated in impedance other than $Z_{0}$

I.e $Z_{L}\neq Z_{0}$

and remaining lines( from 2nd to nth line) terminated in

I.e $Z_{L}\doteq Z_{0}$

If ai = incident wave

$a_{1},a_{2},a_{3},....a_{n}\doteq a_{i}$ = incident wave ai gets divided into (n-1)

No reflections from 2nd to nth line since that terminated into $Z_{L}\doteq Z_{0}$  but there is mismatch in first line and hence,

$b_{1}$=reflected wave going back to junction

Relation of b1 to a1

$b_{1}$=(reflection coefficient) a1

$b_{1}$ = si1.a1

Where si1 = reflection coefficient of first line

1= reflection from 1st line

i= source connected to ith line

Hence contribution to outward traveling wave in the ith line is given by

bi= si1.a1 (since b2=b3=….bn= 0)

### Case 2: All (n-1) lines terminated in impedance other than $Z_{0}$($Z_{L}\neq Z_{0}$)

Let all (n-1) lines be terminated in an impedance other than $Z_{L}\neq Z_{0}$

Hence reflection from every line

$b_{i}=S_{i1}a_{1}+S_{i2}a_{2}+..........S_{in}a_{n}$

Here i= 1 to n since i can be any line from 1 to n

Therefore we have,

$b_{1}=S_{11}a_{1}+S_{12}a_{2}+..........S_{1n}a_{n}$
$b_{2}=S_{21}a_{1}+S_{22}a_{2}+..........S_{2n}a_{n}$
.          .              .                      .
.          .              .                      .
$b_{n}=S_{n1}a_{1}+S_{n2}a_{2}+..........S_{nn}a_{n}$

$\begin{bmatrix} b \end{bmatrix}=\begin{bmatrix} S \end{bmatrix}\begin{bmatrix} a \end{bmatrix}$

In matrix form

$\begin{bmatrix} b1\\ b2\\ b3\\ .\\ .\\ b_{n} \end{bmatrix}=\begin{bmatrix} S_{11} &S_{12} &S_{13} & . & . &S_{1n} \\ S_{11}&S_{11} & S_{11} &. & . &S_{2n} \\ S_{11} &S_{11} &S_{11} & . &. &S_{3n} \\ . &. &. &. & . &. \\ .& . &. & . &. &. \\ S_{n1}&S_{n2} & S_{n3} &. &. & S_{nn} \end{bmatrix}\begin{bmatrix} a1\\ a2\\ a3\\ .\\ .\\ an\\ \end{bmatrix}$

a’s =inputs to particular ports

b’s =output to out of various port

Sij=scattering coefficient resulting due to input at ith port and output taken as jth poInput at ith port and how much power reflected back from ith port

## Properties of S-matrix

[S] is always a square matrix of order (n*n)
[S] is symmetric matrix i.e., Sij = Sji
[S] is a unitary matrix i.e. [S][S]*=[I]
Where [S]* = complex conjugate of [S]
[I] = unit matrix
The sum of the products of each term of any row (or column) multiplied by the complex conjugate of the corresponding terms of any other row(or column) is zero.
$\sum_{i=1}^{n}S_{ik}S^{*}_{ik}=0 k\neq j$

if any of the terminal or reference plane (say the kth port) are moved away from the junction by an electric distance βl, each of the coefficients Sij involving k will be multiplied by the factor .

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Created by Sujit Wagh on 2017/07/26 23:09

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