power dividers and directional couplers (7)juiching/microwave2.pdf · microwave circuits 1 power...

Microwave Circuits 1

Power Dividers and Directional Couplers(7)

The T-Junction Power Divider(7.2)

Lossless Divider

1. Lossless2. Match at the input port.3. Mismatch at the output ports.4. No isolation at the output ports.

Resistive Divider

1. Lossy.2. Match at all ports.3. No isolation.

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From the figure

The Wilkinson Power Divider(7.3)

1. Matched at all ports.

2. Isolation between output ports.

3. No power loss from input to output ports.

4. Half power loss from output to input ports.

Analysis

1. Excite port 1.

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Symmetry Y equal voltages at port 2 and 3 Y no current flowsthrough the resistor Y open. The circuit becomes

From the figure

To compute , let and denote the voltages of the

forward and backward propagating modes in one of the two

lines. Assume the reference plane is located at port 1. Let thevoltage of the incident wave at port 1 be and port 3 . We

have at port 1

At port 3,

2. Even and odd mode excitation at port 2 and 3

Rearranging the circuit as follow

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a. Even mode: Symmetry Y equal voltages at port 2and 3 Y no current flows through the resistor Y open.The circuit becomes

b. Odd mode: Anti-symmetry Y opposite voltages atport 2 and 3 Y short at the middle of the resistor. Thecircuit becomes

Since

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From , we have

Unequal Power Division

If power ration between ports 2 and 3 is ,

N-way, equal-split, Wilkinson power divider

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Basic Properties of a Three Port Device(7.1)

Impossible scenario: reciprocal, matching at all ports, lossless.

Reciprocal and matching at all ports give the following S matrix

If lossless, the matrix is unitary, that is,

Two of must be zero to satisfy the last 3 equations.

However, then, the first 3 equations will not be satisfied.

Possible scenario:1. Nonreciprocal, matching at all ports, lossless.

Lossless

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Two possible solutions

and

Example: Circulators

2. Reciprocal, lossless, matching only two ports.

Lossless

Possible solution

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3. Lossy, matching at all ports, reciprocal.

Basic Properties of a Four Port Device(7.1)

Reciprocal, matched at all ports.

If lossless and (directional coupler), the following

conditions are required.

where are real.

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1. Symmetrical:

2. Anti-symmetrical: ,

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Detailed Formulation1. Row 1* Row 2, Row 3 Row 4*

2. Multiply 1 with and , respectively and then subtract

3. Similarly, Row 1* Row 3, Row 2 Row 4*

4. Multiply 3 with and , respectively and then subtract

5. If ,

,

6. Without lose generality, choose and

, then Row 2* Row 3

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The Quadrature (90E) Hybrid (7.5)

Even-Odd Mode Analysis

Even Mode

Using ABCD matrix, we have

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Odd mode

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Coupled Line Directional Coupler (7.6)

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Assume and , we have

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where .

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If , and . Also

For , choose the mid-band frequency such that

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Example 7.7: Design a 20 dB single-section coupled line

coupler in stripline with a ground plane spacing of 0.32 cm, a

dielectric sonstant of 2.2, a characteristic impedance of 50 ,

and a center frequency 3 GHz.

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Design of Multi-section Coupled Line Coupler

If ,

If ,

where

Example 7.8: Design a three-section 20 dB coupled line couplerwith a binomial response, a system impedance of 50 , and acenter frequency of 3 GHz.

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180 Hybrid (7.8)

(a) Ring hybrid, rat-race.(b) Tapered coupled line hybrid.(c) Waveguide hybrid junction, magic-T.

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The Lange Coupler

4-wire even mode:

4-wire odd mode:

2-wire even mode:

2-wire odd mode:

Also approximate as follow

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Then we have

By applying coupled line theory

and

or in terms of and ,

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Analysis of the Tapered Coupled Line Hybrid

Consider the taper as an ideal lossless transformer, the evenmode ABCD matrix satisfy

Since

We have

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Therefore, the ABCD matrix of the transformers is

, for even mode.

Similarly,

, for odd mode.

Cascading all the ABCD matrices of the even mode, we have

Similarly, for odd mode

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Thus,

Let

The S matrix of the tapered coupled is

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Magic-T Junction