Chapter 7Power Dividers and Couplers

• Three Port Network S Parameter Matrix • Review of Power Dividers and Couplers• Directional Coupler• Hybrid Couplers• The Lossless T Junction• Junction T Example• The Resistive Divider• Scattering Matrix of the Resistive Divider

Three Port Network S Parameter

The scattering matrix of an arbitrary three port network has nine independent variables:

333231

232221

131211

SSS

SSS

SSS

S

If all the components of the network are passive and it does not contain any active component, then its S parameter matrix must be reciprocal. The [S] matrix will also be symmetric. Usually to avoid power loss, we would like to have a network that is matched at all ports and is lossless. However, it is impossible to construct a three port lossless reciprocal network that is matched at all ports.

jiij SS

Three Port Network S Parameter

If all the three ports are matched the [S] matrix can be written as:

0

0

0

2313

2312

1312

SS

SS

SS

S

If the three port network is not reciprocal then and its [S] matrix will not be symmetric.

jiij SS

Power Dividers and Couplers

In low frequency design, it is common to sum signals by adding their currents or voltages. In high frequency design direct summation is not used. Instead there is a class of circuits used to sum or split power.

Power SplitterPower Combiner

Power Dividers and Couplers

Two basic types of these circuits are: Power Divider and Power Combiner.

1

Three port networks are shown here, but both dividers and combiners can be multi-port networks. The most common value for in splitter is –3 dB (when P2 = P3 ). The power ratio in splitter can range up to –40 dB for one path.

Directional Coupler

One specific class of power divider is the directional coupler. This is a four port device that samples the power flowing into port 1 coupled in to port 3 (the coupled port) with the remainder of the power delivered to port 2 (the through port) and no power delivered to the isolated port.

Directional CouplerInput

Isolated

Through

Coupled

Usually the isolated port is terminated within the coupler casing. In such case, the coupler appears to be a three port device. In ideal case no power is delivered to port 4 (the isolated port).

Directional Coupler

Directional CouplerInput

Isolated

Through

Coupled

Directional couplers are described by three specifications:• Coupling (C) - The ratio of input power to the couple power.• Directivity (D)- The ratio of coupled power to the power at the isolated port.• Isolation (I) – The ratio of input power to power out of the isolated port.

3

1log10P

PC

4

3log10P

PD

4

1log10P

PI dBCDI

Hybrid Couplers

• Hybrid couplers are special cases of directional couplers, where the coupling factor is 3 dB.

• There are two types of hybrids. 1) The quadrature hybrid has a 90 degree phase shift between port 2 and 3 when fed from port 1, with the following [S] matrix.

2) The magic-T hybrid or rat-race hybrid has a 180 degree phase shift between port 2 and 3 when fed from port 4, with the following [S] matrix:

010

100

001

010

2

1

j

j

j

j

S

0110

1001

1001

0110

2

1S

The Lossless T Junction

The simplest type of power divider is the T junction. T junction can be implemented using virtually any type of transmission line. However, the T junction is very simple to implement, it must be treated with care because it does not offer any isolation between its ports.

Matching requirement for the T junction input:132

111

ZZZ

The Lossless T Junction

In order for the input port to be matched, the output lines must be matched (terminated in their characteristic impedance). The power dividing ratio can be selected by using different values of characteristic impedance for ports 2 and 3.

11321 PPPPP 1

2

1

1

2

12

2

2 2

1

2

1

Z

Z

Z

VP

Z

VP oo

3

1

1

2

13

2

3 2

1

2

1

Z

Z

Z

VP

Z

VP oo

The Lossless T Junction

Input matching requirement: 0

||

||

312132

312132

132

132

1

11

ZZZZZZ

ZZZZZZ

ZZZ

ZZZ

ZZ

ZZ

L

L

32

32132132 )(

ZZ

ZZZZZZZZ

Therefore, the input to the T junction can be matched through the correct choice of impedances in port 2 and 3. What about matching of port 2 and 3?

The Lossless T Junction

Now lets consider matching of port 2.

322132

322132

231

231

2

22 ||

||

ZZZZZZ

ZZZZZZ

ZZZ

ZZZ

ZZ

ZZ

L

L

If port 2 is matched, then 2 = 0 and we will have:

31

31231231 0)(

ZZ

ZZZZZZZZ

Substitution of yields Z2 = 0.

Which shows that the lossless T junction cannot be matched at all three ports simultaneously. (Problem 7.1 Pozar)

32

321 ZZ

ZZZ

Junction T Example

If the T junction contains lossy components then it is possible to match all the three ports. In this case the signal power will be reduced due to loss in the junction.

The resistive power divider for an equal power split.

The Resistive Divider

Assuming that all the lumped-element resistors are terminated in the characteristic impedance Zo, the input impedance looking into any port is:

ooo

inoo

ooo

in ZZZ

ZZZ

ZZZ

Z

3

2

3333

Since the network is symmetric from all three ports, the output ports are also matched. Therefore, S11=S22=S33=0

The Resistive Divider

The voltage at the center of the junction is:

11 3

2

32

3

32

VZZ

Z

VVoo

o

The output voltages V2 and V3 are equal to:

10

0

32 2

1

4

3

3

VVZZ

ZVVV o

Scattering Matrix of the Resistive Divider

Since the network is symmetric and reciprocal, S21=S31=S23=1/2.

Thus, the output power is –6 dB below the input power level (lossy).The power delivered to the input and outputs of the divider are:

011

101

110

2

1S

oin Z

VP

21

2

1

in

oo

PZ

V

Z

VPP

4

1

8

121

2

1 21

2

1

32

Half of the supplied power is dissipated in the resistors.