Post on 22-Jan-2021
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