11

ENE 428Microwave Engineering

Lecture 12 Power Dividers and Directional Couplers

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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.

33

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

44

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

11 12 13

21 22 23

31 32 33

S S S

S S S S

S S S

Divider or

coupler P1

P2=aP1

P3=(1- a)P1

Divider or

coupler

P1=P2+P3P2

P3

55

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 thusif i = j,

while if i j ,

1tS S

-

1, ,

N

ki kj ijkS S for all i j

=

11,

N

ki kikS S

=

10.

N

ki kjkS S

=

66

It is impossible to construct a three-port lossless reciprocal network. (1)

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

12 13

12 23

13 23

0

0 .

0

S S

S S S

S S

77

It is impossible to construct a three-port lossless reciprocal network. (2)

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.

2 2

12 13

2 2

12 23

2 2

13 23

13 23

23 12

12 13

1, (1 )

1, (1 )

1, (1 )

0, (1 )

0, (1 )

0. (1 )

S S a

S S b

S S c

S S d

S S e

S S f

88

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:

12 13

21 23

31 32

0

0 .

0

S S

S S S

S S

99

Any matched lossless three-port network must be nonreciprocal. (2)

Either of these followings can satisfy above equations,

or

2 2

12 13

2 2

21 23

2 2

31 32

31 32

21 23

12 13

1, (2 )

1, (2 )

1, (2 )

0, (2 )

0, (2 )

0. (2 )

S S a

S S b

S S c

S S d

S S e

S S f

12 23 31 21 32 13

21 32 13 12 23 31

0, 1, (3 )

0, 1. (3 )

S S S S S S a

S S S S S S b

1010

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.

1111

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:

12 13

12 23

13 23 33

0

0 .

S S

S S S

S S S

1212

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.

2 2

12 13

2 2

12 23

2 2 2

13 23 33

13 23

12 13 23 33

23 12 33 13

1, (3 )

1, (3 )

1, (3 )

0, (3 )

0, (3 )

0. (3 )

S S a

S S b

S S S c

S S d

S S S S e

S S S S f

13 23S S

1313

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.

21

3

S21=ejq

S12=ejq

S33=ejf

1414

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).

1 2

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Input Through

CoupledIsolated

1 2

34

Input Through

CoupledIsolated

1515

Basic properties of directional couplers are described by four-port networks.(1)

12 13 14

12 23 24

13 23 34

14 24 34

0

0.

0

0

S S S

S S SS

S S S

S S S

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.

1616

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.

2 2

14 13 24( ) 0.S S S

2 2

23 12 34( ) 0.S S S

1717

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|.

2 2

12 13

2 2

12 24

2 2

13 34

2 2

24 34

1, (6 )

1, (6 )

1, (6 )

1, (6 )

S S a

S S b

S S c

S S d

1818

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.

12 13 24 34 0S S S S

1919

Symmetrical and Antisymmetrical coupler (2) If 2 is ignored, we yield

1. The symmetrical coupler: = = /2.

2. The antisymmetrical coupler: = 0, = .

0 0

0 0.

0 0

0 0

j

jS

j

j

a a a

a

=

0 0

0 0.

0 0

0 0

S

a a a

a

- = -

2020

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.

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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.

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Directional coupler’s characterization (1)

2 213 .S

2 2 212 1 .S a

1/ 2.

22

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.

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Directional coupler’s characterization (2)

1

3

10logPP

3

4

10logPP

1

4

10logPP

14S

23

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

23

Ideal coupler

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The T-junction power divider The T-junction power divider can be implemented

in any type of transmission line medium.

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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

1 2 0

1 1 1.inY Z Z Z

= =

jB

Z1

Z2

Z0

+V0

-

Yin

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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

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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.

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Resistive divider A lossy three-port divider can be made to

matched at all ports, although the two output ports may not be isolated.

Z0/3

Z0

+V-

Zin

Z

Z0+V2

-

+V3

-

Z0/3

Z0/3+V1-

P1

Port 1

Port 2

P2

P3

Port 3

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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.