How to Derive the Scattering Matrix of a Directional Coupler
Key Takeaways

A directional coupler is a fourport 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 fourport 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 (P_{i}) on port 1 travels towards port 2. The power received at port 2 is designated as received power, P_{r}. 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, P_{f}. 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 (Smatrix) is used to get a complete description of a multiport network. The best thing about Smatrix theory is that the Smatrix of a multiport 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 Smatrix is of order 4x4 and is given below:
4)
If all the ports are properly matched, then the matrix satisfies the following condition:
5) S_{11}= S_{22}= S_{33}= S_{44}
Since ports 1 and 3 are isolated from each other:
6) S_{13}=0
Similarly, ports 2 and 4 are isolated from each other, therefore:
7) S_{24}= 0
The Smatrix of a directional coupler is a symmetrical matrix, where i and j correspond to the row and column number, respectively:
8) S_{ij}= S_{ji}
Applying the symmetrical condition, we end up with the following equations:
9a) S_{21}= S_{12}
9b) S_{23}= S_{32}
9c) S_{24}= S_{42}
9d) S_{31}= S_{13}
9e) S_{34}= S_{43}
9f) S_{41}= S_{14}
Substituting the above equations, the Smatrix given by equation 4 reduces to:
(10)
According to the unitary property of the Smatrix, the Smatrix of a directional coupler satisfies the following equation, where [S]* is the complex conjugate of the Smatrix 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:
13a) S_{12}^{2}+S_{12}^{2}= 1
Row 2, column 2:
13b) S_{12}^{2}+S_{23}^{2}=1
Row 3, column 3:
13c) S_{23}^{2}+S_{34}^{2 }= 1
Row 4, column 4:
13d) S_{14}^{2}+S_{34}^{2 }= 1
Row 1, column 3:
13e) S_{12}S^{*}_{23} + S_{14}S^{*}_{34}= 0
From 13a, 13b, 13c, and 13d:
14a) S_{14}= S_{23}
Similarly, the other relationship obtained from the set of equations 13(a)(e) is:
14b) S_{12} = S_{34}
Assume S12is equal to a nonzero real number, P. From the relationship in equation 14b—and P being a real number—the following equations can be obtained:
15a) S_{12}= S_{34}= P
15b) S^{*}_{34}= P
Substituting equation 15 in equation 13e gives you:
16a) PS^{*}_{23} + S_{23}P = 0
Equation 16b is obtained by rearranging equation 16a:
16b) S_{23}= S^{*}_{23}
Equation 16b is only possible if S_{23}is a purely imaginary number. Therefore, we can write:
17) S_{23}= jq
And, if S_{23} is a purely imaginary number, so is S_{14}(equation 14a).
From all the relationships obtained so far, the Smatrix of a directional coupler can be written as:
(18)
Using the SMatrix 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 Smatrix. And, luckily, Cadence’s software offers tools to conduct Sparameter analysis of systems.
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