# Directional Coupler S-Matrix derivation | Directional Coupler scattering matrix

This page describes **Directional Coupler S-Matrix** (**scattering matrix**).
It also covers basics of Directional Coupler along with its S-matrix derivation.

## What is Directional Coupler?

RF Directional coupler consists of four ports, input port (P1) , transmitted port (P2) , isolated port (P3) and coupled port (P4) . The directional coupler is used in RF comformance tests of many wireless devices. It is also used to couple the power to other RF circuits from the main transmission path. As mentioned above directional coupler is generally specified by the two factors viz. coupling factor and directivity.

The figure-1 depicts Directional Coupler. The equations are also mentioned.

## Directional Coupler S-Matrix derivation

Let us understand directional coupler S-matrix derivation which is described below.

➤Step-1: As we know directional coupler is a 4-port network.
Hence [S] matrix is a 4 x 4 matrix and is expressed as follows

[s] =

|S11 S12 S13 S14|

|S21 S22 S23 S24|

|S31 S32 S33 S34|

|S41 S42 S43 S44| ...Equation-1

➤Step-2: Since all the four ports are completely matched. All the diagonal elements of S-matrix are zeros.

Hence, S11 = S22 = S33 = S44 = 0

➤Step-3: There is no coupling between port-1 and port-3 and between port-2 and port-4.
Hence S31 = 0 and S42 = 0.

➤Step-4: Due to symmetry property Sij = Sji,

Hence, S24 = S42 = 0 and S13 = S31 = 0

➤Putting above values in Equation-1 we get S-matrix as follows.

## S-Matrix of other RF devices

3-port and 4-port circulator S-matrix

RF Isolator S-matrix

Magic Tee S-matrix

Rat Race Coupler S-matrix

Branch Line Coupler S-matrix

Directional Coupler S-matrix

Fixed Attenuator S-matrix

Hybrid Ring S-matrix

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