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Notes 19 ECE Microwave Engineering Fall 2011 Power Dividers and Couplers Part 1 Prof. David R. Jackson Dept. of ECE 1

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Power Dividers and Directional Couplers Three-port networks General 3-port: 2

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For all ports matched, reciprocal, and lossless: Lossless [S] is unitary Not physically possible These cannot all be satisfied. At least 2 of S 13, S 12, S 23 must be zero. 3 Power Dividers and Directional Couplers (cont.) (There are three distinct values.) (If only one is zero (or none is zero), we cannot satisfy all three.) (If only one is nonzero, we cannot satisfy all three.)

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Now consider a 3-port network that is non-reciprocal, all ports matched, and lossless: Lossless These equations will be satisfied if: “Circulator” 4 Power Dividers and Directional Couplers (cont.) (There are six distinct values.) or 12 Note that S ij S ji.

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Clockwise (LH) circulatorCirculators Counter clockwise (RH) circulator Circulators can be made using biased ferrite materials. Note: We have assumed here that the phases of all the S parameters are zero.

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Power Dividers T-Junction: lossless divider 6 If we match at port 1, we cannot match at the other ports!

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Assuming port 1 matched: 7 Power Dividers (cont.) We can design the splitter to control the powers into the two output lines.

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For each port we have: 8 Power Dividers (cont.)

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9 Also, we have

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If port 1 is matched: The output ports are not isolated. 10 Power Dividers (cont.)

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Powers: 11 Power Dividers (cont.) Hence

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Summary 12 Power Dividers (cont.) The input port is matched, but not the output ports. The output ports are not isolated.

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Example: Microstrip T-junction power divider 13 Power Dividers (cont.)

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Resistive Power Divider Same for Z in1 and Z in2 All ports are matched. 14

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By reciprocity and symmetry 15 Resistive Power Divider (cont.)

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All ports are matched, but 1/2 P in is dissipated by resistors, and the output ports are not isolated. 16 Resistive Power Divider (cont.) Hence we have

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Even-Odd Mode Analysis Obviously, Example: We want to solve for V. using even/odd mode analysis “odd” problem“even” problem 17 (This is needed for analyzing the Wilkenson.)

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“Even” problem 18 Even-Odd Mode Analysis (cont.)

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“Odd” problem short circuit (SC) plane of symmetry 19 Even-Odd Mode Analysis (cont.)

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“even” problem “odd” problem By superposition: 20 Even-Odd Mode Analysis (cont.)

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Wilkenson Power Divider Equal-split (3 dB) power divider All ports matched ( S 11 = S 22 = S 33 = 0 ) Output ports are isolated ( S 23 = S 32 = 0 ) Obviously not unitary 21 Note: No power is lost in going from port 1 to ports 2 and 3.

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Example: Microstrip Wilkenson power divider 22 Wilkenson Power Divider (cont.)

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Even and odd analysis is required to analyze structure when port 2 is excited. Only even analysis is needed to analyze structure when port 1 is excited. To determine 23 Wilkenson Power Divider (cont.) The other components can be found by using symmetry and reciprocity.

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Top view 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 24 Wilkenson Power Divider (cont.) A microstrip realization is shown.

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Z 0 microstrip line 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 ) For each half 25 Wilkenson Power Divider (cont.) (magnetic wall)

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“Even” Problem 26 Wilkenson Power Divider (cont.) Ports 2 and 3 are excited in phase. Note: The 2Z 0 resistor has been split into two Z 0 resistors in series.

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27 Wilkenson Power Divider (cont.) “Odd” problem Ports 2 and 3 are excited 180 o out of phase. Note: The 2Z 0 resistor has been split into two Z 0 resistors in series.

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28 Wilkenson Power Divider (cont.) Even Problem Port 2 excitation Port 2

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29 Wilkenson Power Divider (cont.) Port 2 Odd Problem Port 2 excitation

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30 Wilkenson Power Divider (cont.) In summary, We add the results from the even and odd cases together: Note: Since all ports have the same Z 0, we ignore the normalizing factor Z 0 in the S parameter definition.

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When port 1 is excited, the response, by symmetry, is even. (Hence, the total fields are the same as the even fields.) Even Problem 31 Wilkenson Power Divider (cont.) Port 1 Port 1 excitation

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32 Wilkenson Power Divider (cont.) Hence Even Problem Port 1 excitation

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(reciprocal) 33 Wilkenson Power Divider (cont.) Along g /4 wave transformer: Even Problem Port 1 excitation

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By symmetry: 34 Wilkenson Power Divider (cont.) By reciprocity: For the other components:

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35 Wilkenson Power Divider (cont.) All three ports are matched, and the output ports are isolated.

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36 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.

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37 Wilkenson Power Divider (cont.) Figure 7.15 of Pozar Photograph 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.

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