Directional Coupler S-Matrix Derivation Explained

This page explains the S-Matrix (scattering matrix) of a directional coupler. We’ll cover the basics of a directional coupler and then walk through the derivation of its S-matrix.

What is a Directional Coupler?

An RF Directional Coupler is a four-port device consisting of:

• Port 1 (P1): Input port
• Port 2 (P2): Transmitted port
• Port 3 (P3): Isolated port
• Port 4 (P4): Coupled port

Directional couplers are commonly used in RF conformance tests of wireless devices. They’re also employed to couple power to other RF circuits from the main transmission path.

Two key specifications of a directional coupler are its coupling factor and directivity.

Figure 1 depicts a directional coupler and its related equations.

Directional Coupler S-Matrix Derivation

Let’s break down the derivation of the directional coupler’s S-matrix step by step:

** Step-1:** A directional coupler is a four-port network. Therefore, its S-matrix is a 4x4 matrix, which can be represented as:

[s] = |S11 S12 S13 S14| |S21 S22 S23 S24| |S31 S32 S33 S34| |S41 S42 S43 S44| …Equation-1

** Step-2:** Assuming all four ports are perfectly matched, all diagonal elements of the S-matrix are zero. This means:

S11 = S22 = S33 = S44 = 0

** Step-3:** There's no coupling between Port 1 and Port 3, nor between Port 2 and Port 4. Thus:

S31 = 0 and S42 = 0

** Step-4:** Due to the symmetry property (Sij = Sji), we can infer:

S24 = S42 = 0 and S13 = S31 = 0

By substituting the values obtained in the previous steps into Equation-1, we derive the S-matrix for the directional coupler as follows:

![Directional Coupler S-Matrix](../../assets/Directional-Coupler-S-Matrix.jpg)