1. ENE 428 Microwave Engineering
Lecture 12 Power Dividers and Directional Couplers 1

2. Power dividers and directional couplers
Passive components that are used for power division or combining.
The coupler may be a three-port or a four-port component
Three-port networks take the form of T-junctions
Four-port networks take the form of directional couplers and hybrids.
Hybrid junctions have equal power division and either 90 or a 180 phase shift between the outport ports. 2

3. Types of power dividers and directional couplers
T-junction power divider
Resistive divider
Wilkinson power divider
Bethe Hole Coupler
Quadrature (90) hybrid and magic-T (180) hybrid
Coupled line directional coupler 3

4. Basic properties of dividers and couplers
The simplest type is a T-junction or a three-port network with two inputs and one output.
The scattering matrix of an arbitrary three-port network has nine independent elements 4

5. The scattering parameters’ lossless property
The unitary matrix:
This can be written in summation form as
where ij = 1 if i = j and ij = 0 if i  j thus
if i = j,
while if i  j , 5

6. It is impossible to construct a three-port lossless reciprocal network
If all ports are matched, then Sii = 0, and if the network is reciprocal the scattering matrix reduces to
If the network is lossless, the scattering matrix must be unitary that leads to 6

7. It is impossible to construct a three-port lossless reciprocal network
Two of the three parameters (S12, S13, S23) must be zeros but this will be inconsistent with one of eq. (1a-c), implying that a three-port network cannot be lossless, reciprocal, and matched at all ports. 7

8. Any matched lossless three-port network must be nonreciprocal. (1)
The [S] matrix of a matched three-port network has the following form:
If the network is lossless, [S] must be unitary, which implies the following: 8

9. Any matched lossless three-port network must be nonreciprocal. (2)
Either of these followings can satisfy above equations, or 9

10. Any matched lossless three-port network must be nonreciprocal. (3)
This results show that Sij  Sji for i  j, therefore the device must be nonreciprocal.
These S matrices represent two possible types of circulators, forward and backward. 10

11. A lossless and reciprocal three-port network can be physically realized if only two of its ports are matched. (1)
If ports 1 and 2 are matched ports, then
To be lossless, the following unitary conditions must be satisfied: 11

12. A lossless and reciprocal three-port network can be physically realized if only two of its ports are matched. (2)
From (3a-b), , so (3d) shows that S13 = S23 = 0. Then |S12|=|S33|=1. 12

13. A lossless and reciprocal three-port network can be physically realized if only two of its ports are matched. (3)
The scattering matrix and signal flow graph are shown below.
If a three-port network is lossy, it can be reciprocal and matched at all ports. 13

14. Four-port networks (Directional Couplers)
Power supplied to port 1 is coupled to port 3 (the coupled port), while the remainder of the input power is delivered to port 2 (the through port)
In an ideal directional coupler, no power is delivered to port 4 (the isolated port). 14

15. Basic properties of directional couplers are described by four-port networks.(1)
The [ S ] matrix of a reciprocal four-port network matched at all ports has the above form.
If the network is lossless, there will be 10 equations result from the unitary condition.
Unitary applies to row as well
S13*S23+S14*S24=0
Multiply by S24*
S14*S13+S24*S23=0
Multiply by S13*
Subtract both eqs 15

16. Conditions needed for a lossless reciprocal four-port network (1)
The multiplication of row 1 and row 2, and the multiplication of row 4 and row 3 can be arranged so that
(4)
The multiplication of row 1 and row 3, and the multiplication of row 2 and row 4 can be arranged so that
(5)
If S14 = S23 = 0, a directional coupler can be obtained.
Eq5
S12*S23+S14*S34=0
S14*S12+S34*S23=0
Multiply by S12 and S34, then subtract 16

17. Conditions needed for a lossless reciprocal four-port network (2)
Then the self-products of the rows of the unitary [S] matrix yield the following equations:
which imply that |S13|=|S24|and that |S12|=|S24|. 17

18. Symmetrical and Antisymmetrical coupler (1)
The phase references of three of the four ports are chosen as S12 = S34 = , S13 = ej, and S24 = ej, where  and  are real, and  and  are phase constants to be determined.
The dot products or rows 2 and 3 gives
which yields a relation between the remaining phase constant as
 +  =  2n. 18

19. Symmetrical and Antisymmetrical coupler (2)
If 2 is ignored, we yield
1. The symmetrical coupler:  =  = /2.
2. The antisymmetrical coupler:  = 0,  = . 19

20. Symmetrical and Antisymmetrical coupler (3)
The two couplers differ only in the choice of the reference planes. The amplitudes  and  are not independent, eq (6a) requires that
2 + 2 =1.
Another way for eq. (4) and (5) to be satisfied is if |S13|=|S24| and |S12|=|S34|.
If phase references are chosen such that S13=S24= and S12=S34=j, two possible solutions are given. First S14=S23=0, same as above.
The other solution is for  =  =0, which implies S12=S13=S24=S34=0, the case of two decoupled two-port network. 20

21. Directional coupler’s characterization (1)
Power supplied to port 1 is coupled to port 3 (the coupled port) with the coupling factor
The remainder of the input power is delivered to port 2 (the through port) with the coefficient
In an ideal coupler, no power is delivered to port 4 (the isolated port).
Hybrid couplers have the coupling factor of 3 dB or  =  = The quadrature hybrid coupler has a 90 phase shift between ports 2 and 3 ( =  = /2) when fed at port 1. 21

22. Directional coupler’s characterization (2)
Coupling = C = = -20log dB,
Directivity = D = = 20log dB,
Isolation = I = = -20log|S14| dB.
The coupling factor indicates the fraction of the input power coupled to the output port.
The directivity is a measure of the coupler’s ability to isolate forward and backward waves, as is the isolation. These quantities can be related as
I = D + C dB. 22

23. Ideal coupler
The ideal coupler would have infinite directivity and isolation (S14 = 0). 23

24. The T-junction power divider
The T-junction power divider can be implemented in any type of transmission line medium.

25. Lossless divider (1)
A lumped susceptance, B, accounts for the stored energy resulted from fringing fields and higher order modes associated with the discontinuity at the junction.
In order for the divider to be matched to the input line impedance Z0, and assume a TL to be lossless, we will have

26. Lossless divider (2)
The output line impedances Z1 and Z2 can then be selected to provide various power division ratios.
In order for the divider to be matched to the input line impedance Z0, and assume a TL to be lossless, we will have

27. Ex1 A lossless T-junction power divider has a source impedance of 50 
Ex1 A lossless T-junction power divider has a source impedance of 50 . Find the output characteristic impedances so that the input power is divided in a 3:1 ratio. Compute the reflection coefficients seen looking into the output ports.

28. Resistive divider
A lossy three-port divider can be made to matched at all ports, although the two output ports may not be isolated.

29. The Wilkinson power divider
The lossless T-junction divider cannot be matched at all ports and does not have any isolation between output ports.
The resistive divider can be matched at all ports but the isolation is still not achieved.
The Wilkinson power divider can be matched at all ports and isolation can be achieved between the output ports.