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ECE 5317-6351 Microwave Engineering. Fall 2011. Prof. David R. Jackson Dept. of ECE. Notes 19. Power Dividers and Couplers Part 1. Three-port networks. Power Dividers and Directional Couplers. General 3-port:. Power Dividers and Directional Couplers (cont.).

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ECE 5317-6351

Microwave Engineering

Fall 2011

Prof. David R. Jackson

Dept. of ECE

Notes 19

Power Dividers and Couplers

Part 1

Three-port networks

## Power Dividers and Directional Couplers

General 3-port:

Power Dividers and Directional Couplers (cont.)

### For all ports matched, reciprocal, and lossless:

Not physically possible

(There are three distinct values.)

Lossless  [S] is unitary

These cannot all be satisfied.

(If only one is nonzero, we cannot satisfy all three.)

 At least 2 of S13, S12, S23must be zero.

(If only one is zero (or none is zero), we cannot satisfy all three.)

Power Dividers and Directional Couplers (cont.)

### Now consider a 3-port network that is non-reciprocal, all ports matched, and lossless:

“Circulator”

These equations will be satisfied if:

(There are six distinct values.)

1

2

Lossless

or

Note that Sij Sji.

Circulators

Clockwise (LH) circulator

1

2

Note: We have assumed here that the phases of all the S parameters are zero.

Circulators can be made using biased ferrite materials.

Counter clockwise (RH) circulator

### Power Dividers

T-Junction: lossless divider

If we match at port 1, we cannot match at the other ports!

### Power Dividers (cont.)

Assuming port 1 matched:

We can design the splitter to control the powers into the two output lines.

### Power Dividers (cont.)

For each port we have:

Also, we have

### Power Dividers (cont.)

If port 1 is matched:

The output ports are not isolated.

Powers:

Hence

### Power Dividers (cont.)

Summary

• The input port is matched, but not the output ports.

• The output ports are not isolated.

### Power Dividers (cont.)

Example: Microstrip T-junction power divider

### Resistive Power Divider

Same for Zin1 and Zin2

 All ports are matched.

### Resistive Power Divider (cont.)

By reciprocity and symmetry

### Resistive Power Divider (cont.)

Hence we have

All ports are matched, but 1/2 Pinis dissipated by resistors, and the output ports are not isolated.

### Even-Odd Mode Analysis

(This is needed for analyzing the Wilkenson.)

Obviously,

Example: We want to solve for V.

using even/odd mode analysis

“even” problem

“odd” problem

Even-Odd Mode Analysis (cont.)

### “Even” problem

Even-Odd Mode Analysis (cont.)

### “Odd” problem

short circuit (SC) plane of symmetry

Even-Odd Mode Analysis (cont.)

“odd” problem

“even” problem

By superposition:

Wilkenson Power Divider

Equal-split (3 dB) power divider

• All ports matched (S11 = S22 = S33 = 0)

• Output ports are isolated (S23 = S32 = 0)

Note: No power is lost in going from port 1 to ports 2 and 3.

Obviously not unitary

Wilkenson Power Divider (cont.)

Example: Microstrip Wilkenson power divider

Wilkenson Power Divider (cont.)

• Even and odd analysis is required to analyze structure when port 2 is excited.

 To determine

• Only even analysis is needed to analyze structure when port 1 is excited.

 To determine

The other components can be found by using symmetry and reciprocity.

Wilkenson Power Divider (cont.)

### Top view

A microstrip realization is shown.

Split structure along plane of symmetry (POS)

Even  voltage even about POS  place OC along POS

Odd  voltage odd about POS  place SC along POS

Wilkenson Power Divider (cont.)

How do you split a transmission line? (This is needed for the even case.)

top view

Voltage is the same for each half of line (V)

Current is halved for each half of line (I/2)

(magnetic wall)

Z0microstrip line

For each half

Wilkenson Power Divider (cont.)

### “Even” Problem

Note: The 2Z0 resistor has been split into two Z0 resistors in series.

Ports 2 and 3 are excited in phase.

Wilkenson Power Divider (cont.)

### “Odd” problem

Note: The 2Z0 resistor has been split into two Z0 resistors in series.

Ports 2 and 3 are excited 180o out of phase.

Wilkenson Power Divider (cont.)

### Even Problem

Port 2 excitation

Port 2

Wilkenson Power Divider (cont.)

### Odd Problem

Port 2 excitation

Port 2

Wilkenson Power Divider (cont.)

We add the results from the even and odd cases together:

Note: Since all ports have the same Z0, we ignore the normalizing factor Z0in the S parameter definition.

In summary,

Wilkenson Power Divider (cont.)

Port 1 excitation

Port 1

When port 1 is excited, the response, by symmetry, is even. (Hence, the total fields are the same as the even fields.)

Even Problem

Wilkenson Power Divider (cont.)

### Even Problem

Port 1 excitation

Hence

Wilkenson Power Divider (cont.)

### Even Problem

Port 1 excitation

Along g/4 wave transformer:

(reciprocal)

Wilkenson Power Divider (cont.)

For the other components:

By symmetry:

By reciprocity:

Wilkenson Power Divider (cont.)

All three ports are matched, and the output ports are isolated.

Wilkenson Power Divider (cont.)

• When a wave is incident from port 1, half of the total incident power gets transmitted to each output port(no loss of power).

• When a wave is incident from port 2 or port 3, half of the power gets transmitted to port 1 and half gets absorbed by the resistor, but nothing gets through to the other output port.

Wilkenson Power Divider (cont.)

Figure 7.15 of PozarPhotograph of a four-way corporate power divider network using three microstrip Wilkinson power dividers. Note the isolation chip resistors.Courtesy of M.D. Abouzahra, MIT Lincoln Laboratory.