1. Notes 20 ECE 5317-6351 Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 20
Power Dividers and Couplers
Part 2

2. Directional Couplers Four-Port Networks If lossless  [S] is unitary.
Consider a reciprocal 4-port network with matched ports:
There are 6 degrees of freedom (independent S parameters).
If lossless  [S] is unitary.
The device is called a “directional coupler” if:

3. Directional Couplers Directional coupler:
There are now 4 degrees of freedom.
All ports are matched.
No power flows to port 4 from in the input port 1 (i.e., port 4 is “isolated”).
Ports 2 and 3 are isolated (power incident on port 2 does not get to port 3 and vice versa).

4. Directional Couplers Directional coupler symbol:
All ports are matched.
No power flows to port 4 from in the input port 1 (i.e., port 4 is “isolated”).
Ports 2 and 3 are isolated (power incident on port 2 does not get to port 3 and vice versa).

5. Directional Couplers
From the unitary property of the matrix, we have the following results:

6. Directional Couplers (cont.)
Choose
The first equation is really just a choice of reference planes on ports 2 and 3, which makes these parameters real.
(usually, n = 0)

7. Directional Couplers (cont.)
Two possible choices:
1) Symmetrical coupler
Example: 90o quadrature hybrid coupler
2) Anti-symmetrical coupler
Example: 180o rat-race hybrid coupler

8. Directional Couplers (cont.)
Assume a signal into port 1
Return loss
Ideally + dB
Directivity
Ideally + dB
Assume ports 2-4 are matched.
A “hybrid” coupler is one for which the coupling is 3 dB (equal power split at the output ports 2 and 3):
Isolation
Ideally + dB
Coupling

9. 90o (Quadrature) Hybrid Coupler
“A quadrature coupler is one in which the input is split into two signals (usually with a goal of equal magnitudes) that are 90 degrees apart in phase. Types of quadrature couplers include branchline couplers (also known as quadrature hybrid couplers), Lange couplers and overlay couplers.”
Taken from “Microwaves 101”
This is very useful for obtaining circular polarization:
There is a 90o phase difference between ports 2 and 3.

10. Quadrature Hybrid Coupler (cont.)
90o hybrid 1 2 4 3
The quadrature hybrid is a lossless 4-port (the S matrix is unitary ).
All four ports are matched.
The device is reciprocal (the S matrix is symmetric.)
Port 4 is isolated from port 1 and ports 2 and 3 are isolated from each other.

11. Quadrature Hybrid Coupler (cont.)
The quadrature hybrid is usually used as a splitter:
90o hybrid 1 2 4 3
+90o out of phase
-90o out of phase
The signal from port 1 splits evenly between ports 2 and 3, with a 90o phase difference.
The signal from port 4 splits evenly between ports 2 and 3, with a -90o phase difference.
Can be used to produce right-handed circular polarization.
Can be used to produce left-handed circular polarization.

12. Quadrature Hybrid Coupler (cont.)
(Branch-line coupler)
A microstrip realization is shown here.
Note: We only need to study what happens when we excite port 1, since the structure is symmetric.
We use even/odd mode analysis (exciting ports 1 and 4) to figure out what happens when we excite port 1.

13. Quadrature Hybrid Coupler (cont.)
Even Analysis 

14. Quadrature Hybrid Coupler (cont.)
Odd Analysis 

15. Quadrature Hybrid Coupler (cont.)
Consider the general case: .
In general:
Shunt load on line
Quarter-wave line

16. Quadrature Hybrid Coupler (cont.)
Hence we have

17. Quadrature Hybrid Coupler (cont.)
Continuing with the algebra, we have

18. Quadrature Hybrid Coupler (cont.)
Hence we have
Convert this to S parameters (use Table 4.2 in Pozar):

19. Quadrature Hybrid Coupler (cont.)
Hence
By symmetry:

20. Quadrature Hybrid Coupler (cont.)
By symmetry and reciprocity:

21. Quadrature Hybrid Coupler (cont.)
By symmetry and reciprocity:

22. Quadrature Hybrid Coupler (cont.)
By symmetry and reciprocity:

23. Quadrature Hybrid Coupler (cont.)
Summary
The input power to port 1 divides evenly between ports 2 and 3, with ports 2 and 3 being 90o out of phase.

24. 180o Hybrid Coupler (Rat-Race Ring Hybrid)
“Applications of rat-race couplers are numerous, and include mixers and phase shifters. The rat-race gets its name from its circular shape, shown below.”
Taken from “Microwaves 101”
Figure of Pozar Photograph of a microstrip ring hybrid. Courtesy of M. D. Abouzahra, MIT Lincoln Laboratory

25. Rat-Race Ring Hybrid (cont.)
180o hybrid 1 2 4 3
The rat race is a lossless 4-port (the S matrix is unitary).
All four ports are matched.
The device is reciprocal (the S matrix is symmetric).
Port 4 is isolated from port 1 and ports 2 and 3 are isolated from each other.

26. Rat-Race Ring Hybrid (cont.)
The rat race can be used as a splitter:
180o hybrid 1 2 4 3
In phase  
180o out of phase
The signal from the “sum port”  (port 1) splits evenly between ports 2 and 3, in phase.
The signal from the “difference port”  (port 4) splits evenly between ports 1 and 2, 180o out of phase.

27. Rat-Race Ring Hybrid (cont.)
The rat race can be used as a combiner:
Signal 1 (V1)  1 2
180o hybrid  4 3
Signal 2 (V2)
The signal from the sum port  (port 1) is the sum of the input signals 1 and 2.
The signal from the difference port  (port 4) is the difference of the input signals 1 and 2.

28. Rat-Race Ring Hybrid (cont.)
A microstrip realization is shown here.

29. 180o Hybrid Coupler (Magic T)
A waveguide realization of a 180o hybrid coupler is shown here, called a “Magic T.”
“Magic T”
Please see p. 361 of Pozar.
IEEE Microwave Theory and Techniques Society
Note the logo!

30. Rat-Race Ring Hybrid (cont.)
Layout
Schematic

31. Rat-Race Ring Hybrid (cont.)
Even Analysis 

32. Rat-Race Ring Hybrid (cont.)
Odd Analysis 

33. Rat-Race Ring Hybrid (cont.)
Proceeding as for the 90o coupler, we have:

34. Rat-Race Ring Hybrid (cont.)
Converting from the ABCD matrix to the S matrix, we have
Table 4.2 in Pozar

35. Rat-Race Ring Hybrid (cont.)
For the S parameters coming from port 1 excitation, we then have:

36. Rat-Race Ring Hybrid (cont.)
Similarly, exciting port 2, and using symmetry and reciprocity, we have the following results:

37. Rat-Race Ring Hybrid (cont.)
Summary