Passive Devices 3 port network Smatrix of 3

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, lossless, and only 2 ports matched Port 3 is isolated Reciprocal, matched, and lossy 3 port network 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 d. B 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 (2) -> some ports mismatched t or Lossless Power Divider: P Port #1 Input Admittance at Input Port for Input Port Matching, B=0 and Ex) 1: 2 divider (with B=0 & Port #1 matched: Reflection at Port #2: Reflection at Port #3: Po rt ) #3 #2

(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 • Symmetric • Not Unitary • All ports are matched • Reciprocal • Lossy • Output ports are isolated

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

4 -Port Network: Quadrature Hybrid 3 d. B 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 ground Even Mode Odd Mode

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