1. 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

2. Matched, lossless, and nonreciprocal 3 port network
Circulator

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

4. 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

6. Hybrid Coupler(3 dB Coupler)

7. 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

8. 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:

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

10. 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

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

12. Wilkinson Power Divider: even mode
Voltage on transmission line

13. 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

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

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

16. 4-Port Network: Quadrature Hybrid
3 dB Directional Coupler with Quadrature Phase Difference between two output ports
Reciprocal, Lossless, and all ports are matched.

17. Even-mode

18. Odd-mode

20. 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

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

22. –port Analysis
Even mode excitation

23. –port Analysis
Odd mode excitation

24. –port Analysis
Even mode excitation

25. –port Analysis
Odd mode excitation

27. Propagating Modes on Symmetric Coupled Line
Coupled Line Theory
Propagating Modes on Symmetric Coupled Line
Even Mode
Odd Mode
ground

28. 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:

29. 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:

31. 4-Port Network: Coupled Line Coupler
Boundary Condition at each Port

37. Weakly Coupled Line Coupler
Symmetric coupler

38. 2-Port Network Using Symmetric Coupled Lines

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

41. Impedance Matrix Z11 = Z22=Z33= Z44

42. Impedance Matrix Z12=Z21=Z34=Z43

43. Impedance Matrix Z13 = Z31 = Z24 = Z42

44. Impedance Matrix Z14 = Z41 = Z32 = Z23

45. Image Impedance and Propagation Constant of 2-Port Network

47. Port 1
Port 2

48. Port 4
Port 1

49. Port 1
Port 4

50. Coupled Line Filter
Admittance
Inverter

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

52. Coupled Line Filter