# Scattering Matrix

## 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

I.e

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

I.e

If ai = incident wave

= incident wave ai gets divided into (n-1)

No reflections from 2^{nd} to nth line since that terminated into but there is mismatch in first line and hence,

=reflected wave going back to junction

Relation of b1 to a1

=(reflection coefficient) a1

= si1.a1

Where si1 = reflection coefficient of first line

1= reflection from 1^{st} 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 ()

Let all (n-1) lines be terminated in an impedance other than

Hence reflection from every line

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

Therefore we have,

. . . .

. . . .

In matrix form

a’s =inputs to particular ports

b’s =output to out of various port

Sij=scattering coefficient resulting due to input at i^{th} port and output taken as j^{th} poInput at i^{th} port and how much power reflected back from i^{th} port

## Properties of S-matrix

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