# How to Derive the Scattering Matrix of a Directional Coupler

### Key Takeaways

• A directional coupler is a four-port device that uses waveguides to distribute power.

• A directional coupler is characterized by its coupling factor, isolation, and directivity.

• The scattering matrix of a directional coupler is the most convenient representation of its behavior in complex systems.

Microwave circuits use directional couplers to distribute power

In microwave engineering, there are quite a few applications that require the division of power. One such device used to accomplish this is a directional coupler—a four-port device that uses waveguides to distribute power in microwave circuits.

In directional couplers, microwave power is scattered from the incident port to three other ports. Since there is a relationship between the power in various ports, using the scattering matrix of a directional coupler to study and analyze its characteristics and behavior is ideal.

## What Are Directional Couplers?

Directional couplers sample the incident power (typically, for measurement purposes) and supply small amounts of power to the other ports.

### Structure

Directional couplers consist of a primary waveguide, which forms ports 1 and 2 at either end. An auxiliary waveguide is connected to the primary waveguide and is associated with ports 3 and 4. The power incident (Pi) on port 1 travels towards port 2. The power received at port 2 is designated as received power, Pr. A fraction of the incident power reflects back to the incident side and is received at port 3 as back power, Pb. The power at port 4 is forward coupled power, Pf. Ports 1 and 3 are isolated from each other, as are ports 2 and 4.

### Characteristics

A directional coupler is characterized by its coupling factor, isolation, and directivity.

## The Scattering Matrix of a Directional Coupler

In microwave systems, a scattering matrix (S-matrix) is used to get a complete description of a multi-port network. The best thing about S-matrix theory is that the S-matrix of a multi-port device is enough to get information about the outputs at each port, even without the knowledge of the components inside the interior of the network.

### Deriving the Scattering Matrix

A directional coupler’s S-matrix is of order 4x4 and is given below:

4)

If all the ports are properly matched, then the matrix satisfies the following condition:

## 5) S11= S22= S33= S44

Since ports 1 and 3 are isolated from each other:

## 6) S13=0

Similarly, ports 2 and 4 are isolated from each other, therefore:

## 7) S24= 0

The S-matrix of a directional coupler is a symmetrical matrix, where i and j correspond to the row and column number, respectively:

## 8) Sij= Sji

Applying the symmetrical condition, we end up with the following equations:

## 9f) S41= S14

Substituting the above equations, the S-matrix given by equation 4 reduces to:

(10)

According to the unitary property of the S-matrix, the S-matrix of a directional coupler satisfies the following equation, where [S]* is the complex conjugate of the S-matrix of the directional coupler and [I] is the identity matrix:

## 11) [S][S]*= [ I ]

Equation 11 can be rewritten as:

(12)

From the multiplication of rows with columns, the equations obtained can be written as:

Row 1, column 1:

Row 2, column 2:

Row 3, column 3:

Row 4, column 4:

Row 1, column 3:

## 13e) S12S*23 + S14S*34= 0

From 13a, 13b, 13c, and 13d:

## 14a) S14= S23

Similarly, the other relationship obtained from the set of equations 13(a)-(e) is:

## 14b) S12 = S34

Assume S12is equal to a non-zero real number, P. From the relationship in equation 14b—and P being a real number—the following equations can be obtained:

## 15b) S*34= P

Substituting equation 15 in equation 13e gives you:

## 16a) PS*23 + S23P = 0

Equation 16b is obtained by rearranging equation 16a:

## 16b) S23=- S*23

Equation 16b is only possible if  S23is a purely imaginary number. Therefore, we can write:

## 17) S23= jq

And, if S23 is a purely imaginary number, so is S14(equation 14a).

From all the relationships obtained so far, the S-matrix of a directional coupler can be written as:

(18)

## Using the S-Matrix of a Directional Coupler

The scattering matrix of a directional coupler is the most convenient representation of a directional coupler’s behavior in complex systems, so knowing how to obtain one is helpful. If you are working with microwave applications that require the division of power, consider using an S-matrix. And, luckily, Cadence’s software offers tools to conduct S-parameter analysis of systems.