Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can not be lossless, reciprocal, and match at all port

Matched, lossless, and nonreciprocal 3 port network Circulator

Reciprocal, matched, and lossy 3 port network Reciprocal, lossless, and only 2 ports matched Port 3 is isolated Lossy power divider

Passive Devices (4 port network) S-matrix of reciprocal and matched network For lossless, S-matrix must be unitary matrix

Hybrid Coupler(3 dB Coupler)

Even-Odd Mode Circuit Analysis If circuit can be decomposed into the superposition of an even-mode excitation and odd- mode excitation Adding two sets( even & odd) of excitations produces the original excitation

3-Port Network: Power Divider (1)Reciprocal and Lossless -> some ports mismatched Lossless Power Divider: Input Admittance at Input Port for Input Port Matching, B=0 and Port #1 Port #2 Port #3 Ex) 1:2 divider (with B=0 & Port #1 matched: ) Reflection at Port #2 : Reflection at Port #3:

(2)Resistive Divider – All ports are matched - output ports are not isolated

Wilkinson Power Divider( equal power division) Resistive divider -all ports are matched -out-ports are isolated Microstrip Wilkinson Power Divider Equivalent Circuit

Wilkinson Power Divider even mode odd mode Equivalent Circuit in normalized and symmetric form

Wilkinson Power Divider: even mode Voltage on transmission line

Wilkinson Power Divider: odd mode All power from Port #2 is delivered to r/2 resister, and all dissipated No power is delivered to Port #1

Wilkinson Power Divider: input port Wilkinson Power Divider: S-matrix All ports are matched Reciprocal Lossy Output ports are isolated Symmetric Not Unitary

Wilkinson Power Divider: unequal power division Wilkinson Power Divider: N-way Divider

4-Port Network: Quadrature Hybrid 3 dB Directional Coupler with Quadrature Phase Difference between two output ports Reciprocal, Lossless, and all ports are matched.

Even-mode

Odd-mode

Coupled Line Theory Propagating Modes on Symmetric Coupled Line Even ModeOdd Mode ground

Even Mode Line Characteristics for even mode propagation Capacitance per unit length w/o dielectrics: Inductance per unit length: Capacitance per unit length with dielectrics: Characteristic Impedance: Propagation Constant:

Odd Mode Line Characteristics for even mode propagation Capacitance per unit length w/o dielectrics: Inductance per unit length: Capacitance per unit length with dielectrics: Characteristic Impedance: Propagation Constant:

(1)Voltage and Current Waves of Even Mode Excitation

(1)Voltage and Current Waves of Odd Mode Excitation

Voltage and Current Waves on Coupled Lines (1)Voltage and Current Waves on Line #1(2)Voltage and Current Waves on Line #2

Impedance Matrix Z 11 = Z 22 =Z 33 = Z 44

Impedance Matrix Z 12 =Z 21 =Z 34 =Z 43

Impedance Matrix Z 13 = Z 31 = Z 24 = Z 42

Impedance Matrix Z 14 = Z 41 = Z 32 = Z 23

4-Port Network: Coupled Line Coupler

Single Section Coupled Line Coupler Boundary Condition at each Port