**Subject :**Microwave Engineering

## S-Matrix of a Directional Coupler

In a directional coupler all four ports are completely matched. Thus the diagonal elements of the S matrix are zeros and

...(1)

As noted, there is no coupling between port 1 and port 3 and between port 2 and port 4. Thus

..(2)

Consequently, the S matrix of a directional coupler becomes

..(3)

Equation (4-5-6) can be further reduced by means of the zero property of the S matrix,so we have

..(4)

**Figure: Two hole directional coupler
**

...(5)

Also from the unity property of the S matrix, we can write

.....(6)

Equations (4) and (5) can also be written as

...(7)

Since S_{12} = S_{21} , S_{14} = S_{41} , S_{23} = S_{32}, and S_{34} = S_{43}, then

...(8)

Let

...(9)

where p is positive and real. Then from Eq. (6)

..(10)

Let

...(11)

where q is positive and real. Then from Eq. (6)