Microwave Circuits 1 Power Dividers and Directional Couplers (7) The T-Junction Power Divider(7.2) Lossless Divider 1. Lossless 2. 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.
Microwave Circuits 2 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.
Microwave Circuits 3 Symmetry Y equal voltages at port 2 and 3 Y no current flows through 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 the voltage 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
Microwave Circuits 4 a. Even mode: Symmetry Y equal voltages at port 2 and 3 Y no current flows through the resistor Y open. The circuit becomes b. Odd mode: Anti-symmetry Y opposite voltages at port 2 and 3 Y short at the middle of the resistor. The circuit becomes Since
Microwave Circuits 5 From, we have Unequal Power Division If power ration between ports 2 and 3 is, N-way, equal-split, Wilkinson power divider
Microwave Circuits 6 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
Microwave Circuits 7 Two possible solutions and Example: Circulators 2. Reciprocal, lossless, matching only two ports. Lossless Possible solution
Microwave Circuits 8 3. Lossy, matching at all ports, reciprocal. Basic Properties of a Four Port Device(7.1) Reciprocal, matched at all ports. If lossless and conditions are required. (directional coupler), the following where are real.
Microwave Circuits 9 1. Symmetrical: 2. Anti-symmetrical:,
Microwave Circuits 10 Detailed Formulation 1. 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
Microwave Circuits 11 The Quadrature (90E) Hybrid (7.5) Even-Odd Mode Analysis Even Mode Using ABCD matrix, we have
Odd mode Microwave Circuits 12
Coupled Line Directional Coupler (7.6) Microwave Circuits 13
Assume and, we have Microwave Circuits 14
where. Microwave Circuits 15
Microwave Circuits 16 If, and. Also For, choose the mid-band frequency such that
Microwave Circuits 17
Microwave Circuits 18 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.
Microwave Circuits 19 Design of Multi-section Coupled Line Coupler If, If, where Example 7.8: Design a three-section 20 db coupled line coupler with a binomial response, a system impedance of 50, and a center frequency of 3 GHz.
Microwave Circuits 20
Microwave Circuits 21 180 Hybrid (7.8) (a) Ring hybrid, rat-race. (b) Tapered coupled line hybrid. (c) Waveguide hybrid junction, magic-t.
Microwave Circuits 22 The Lange Coupler 4-wire even mode: 4-wire odd mode: 2-wire even mode: 2-wire odd mode: Also approximate as follow
Microwave Circuits 23 Then we have By applying coupled line theory and or in terms of and,
Microwave Circuits 24 Analysis of the Tapered Coupled Line Hybrid Consider the taper as an ideal lossless transformer, the even mode ABCD matrix satisfy Since We have
Microwave Circuits 25 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
Microwave Circuits 26 Thus, Let The S matrix of the tapered coupled is
Magic-T Junction Microwave Circuits 27