lossless, reciprocal, and matched at all port 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 matched at all port

Matched, lossless, and nonreciprocal 3 port network Circulator

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

S-matrix of reciprocal and matched network 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 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 Wilkinson Power Divider( equal power division) Resistive divider -all ports are matched -out-ports are isolated Microstrip Wilkinson Power Divider Equivalent Circuit

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

Wilkinson Power Divider: even mode Voltage on transmission line

No power is delivered to Port #1 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

4-Port Network: 180o Hybrid Coupler 3 dB Directional Coupler with 180o Phase Difference between two output ports Reciprocal, Lossless, and all ports are matched. Ring hybrid Tapered coupled line hybrid Magic-T

Circuit for –port Analysis Circuit for –port Analysis Ring hybrid symmetric Circuit for –port Analysis symmetric Circuit for –port Analysis

–port Analysis Even mode excitation

–port Analysis Odd mode excitation

–port Analysis Even mode excitation

–port Analysis Odd mode excitation

Propagating Modes on Symmetric Coupled Line Coupled Line Theory Propagating Modes on Symmetric Coupled Line Even Mode Odd 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:

4-Port Network: Coupled Line Coupler Boundary Condition at each Port

Weakly Coupled Line Coupler Symmetric coupler

2-Port Network Using Symmetric Coupled Lines

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 Z11 = Z22=Z33= Z44

Impedance Matrix Z12=Z21=Z34=Z43

Impedance Matrix Z13 = Z31 = Z24 = Z42

Impedance Matrix Z14 = Z41 = Z32 = Z23

Image Impedance and Propagation Constant of 2-Port Network

Port 1 Port 2

Port 4 Port 1

Port 1 Port 4

Coupled Line Filter Admittance Inverter

Coupled Line Filter N+1 Section Coupled Line Filter Equivalent Circuit using Admittance Inverters and TL’s

Coupled Line Filter