lecture 12 s parameters

7/28/2019 Lecture 12 S parameters

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ENE 428

MicrowaveEngineering

Lecture 12 Power Dividers andDirectional Couplers


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Power dividers and directional couplers

Passive components that are used for power division orcombining.

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 90or a 180 phase shift between the outport ports.


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Types of power dividers and directionalcouplersT-junction power divider Resistive divider

Wilkinson power divider

Bethe Hole Coupler

Quadrature (90) hybrid and magic-T (180) hybrid

Coupled line directional coupler


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Basic properties of dividers and couplers

The simplest type is a T-junction or a three-port networkwith 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

couplerP1

P2aP1

P3(1-a)P1

Divider

or

coupler

P1=P2+P3P2

P3


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The scattering parameters losslessproperty

The unitary matrix:

This can be written in summation form as

where ij= 1 ifi= jand ij= 0 ifijthusifi= j,

while ifij,

1

tS S

-*

1, ,

N

ki kj ijk

S S for all i j*

1

1,N

ki kik

S S*

1

0.N

ki kj

k

S S*


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

If all ports are matched, then Sii= 0, and if the network isreciprocal 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


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It is impossible to construct a three-portlossless 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.

2 212 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

*

*

*

+

+

+


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Any matched lossless three-port networkmust be nonreciprocal. (1) The [S] matrix of a matched three-port network has thefollowing 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 SS S


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Any matched lossless three-port networkmust be nonreciprocal. (3) This results show that SijSji forij, therefore the device

must be nonreciprocal.

These S matrices represent two possible types of

circulators, forward and backward.


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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


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1212



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


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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=e

jq

S12=ejq

S33=ejf


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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

34

Input Through

CoupledIsolated

1 2

34

Input Through

CoupledIsolated


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Basic properties of directional couplers aredescribed 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.


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Conditions needed for a lossless reciprocalfour-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

+

+

+

+


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

1. The symmetrical coupler: q= f= /2.

2. The antisymmetrical coupler: q= 0, f= .

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

-

-


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Symmetrical and Antisymmetrical coupler (3) The two couplers differ only in the choice of the reference

planes. The amplitudes aand are not independent, eq

(6a) requires that

a2+ 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=aand

S12=S34=j, two possible solutions are given. FirstS14=S23=0, same as above.

The other solution is fora= =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 coupledport) 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 ora= = The quadrature hybrid coupler has a 90 phase shift

between ports 2 and 3 (q= f= /2) when fed at port 1.

21

Directional couplers characterization (1)2 2

13 .S

2 2 2

12 1 .S -a

1/ 2.


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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 couplers ability toisolate forward and backward waves, as is the isolation.

These quantities can be related as

I= D + CdB. 22

Directional couplers characterization(2)1

310log

P

P

3

4

10logP

P

1

4

10logP

P

14S


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

23

Ideal coupler


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Lossless divider (1)

A lumped susceptance, B, accounts for the storedenergy resulted from fringing fields and higher ordermodes associated with the discontinuity at the junction.

In order for the divider to be matched to the input lineimpedance Z0, and assume a TL to be lossless, we willhave

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 beselected 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 willhave


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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 poweris 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 allports, although the two output ports may not be isolated.

Z0/3

Z0

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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lecture 12 s parameters

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