# Directional Coupler S Matrix Derivation

Directional Coupler is a 4 port device which has primary and secondary waveguides. The primary wave guide is from port 1 to port 2 and secondary waveguide is from port 3 to port 4.

• Directional coupler is used to couple microwave power, which is unidirectional in most of the cases. The properties of a directional coupler are,

1. All the ports are matched.

2. When the power moves from port 1 to port 2, little portion of it would gets coupled to port 4 and not to port 3.

3. When the power moves from port 2 to port 1, little portion of it would gets coupled to port 3 and not to port 4.

4. The coupling factor of a directional coupler is the ratio of incident power to forward power.

Coupling factor = 10 log(Pi/Pf)

• Directivity of the directional coupler is the ratio of forward power to back power.

Directivity = 10 log(Pf/Pb)

• Isolation of a directional coupler is the ratio of incident power to back power.
I = 10 log(Pi/Pb)

• Isolation = Coupling factor + Directivity.

• Two hole directional coupler is same as conventional directional coupler, but with two holes in common between primary and secondary waveguides.

The spacing between these two holes is given by,

L = (2n+1)λg/4

Where, n = an integer
λg = wavelength

• A fraction of energy entering into port 1 passes through holes and is radiated into port 2. The forward waves in port 4 are in the same phase and are added.

• The backward waves in port 3 are out of phase and are cancelled.

The general S matrix of a directional coupler is,

• Since all ports in a directional coupler are matched.

S11 = S22 = S33 = S44 = 0 ----------------- (2)

• Since there is no coupling between ports 1 & 3 and ports 2 & 4

S13 = S31 = S24 = S42 = 0 ------------- (3)

Apply equation (2) & (3) in (1)

By unitary property, [S][S]* = I

R1C1 => |S12|2 + |S14|2 = 1 ------------ (4)
R2C2 => |S12|2 + |S23|2 = 1 ------------ (5)
R3C3 => |S23|2 + |S34|2 = 1 ------------ (6)
R1C3 => S12 S23* + S14 S34* = 0 ----------- (7)

Comparing eq (4) and (5)

|S12|2 + |S14|2 = |S12|2 + |S23|2
S14 = S23 --------------- (8)

Comparing eq (5) and (6)

|S12|2 + |S23|2 = |S34|2 + |S23|2
S12 = S34 --------------- (9)

Let, S12 be real and positive,
ie, S12 = S34 = p --------------- (10)

applying equation (10) in (7)

Therefore, p S23* + S14 p = 0
p [S23* + S14] = 0
p [S23* + S23] = 0
S23* + S23 = 0

To satisfy the above condition, S23 should be a complex value.

Let S23 = jq

Therefore, the S matrix of directional coupler is,