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,