S-matrix calculation for 2-port, 3-port and 4-port waveguide components

1. Waveguide Components Analysis
using S-Matrix
Course Coordinator: Arpan Deyasi
3/6/2021 1
Arpan Deyasi, India

2. 3/6/2021 2
Arpan Deyasi, India
Isolator
Attenuator
Circulator
Gyrator

3. 3/6/2021 Arpan Deyasi, India 3
Circulator

4. 3/6/2021 4
Arpan Deyasi, India
Circulator
Circulator is a multiport device
in which RF signal travels only
from nth port to (n+1)th port
in one direction (either
clockwise or anti-clockwise)
only
1 2
3
For 3-port circulator, all ports have 120° difference
For 4-port circulator, all ports have 90° difference

5. 3/6/2021 5
Arpan Deyasi, India
Calculation of S-matrix of 3-port circulator
General S-matrix of a circulator is
[ ]
11 12 13
21 22 23
31 32 33
S S S
S S S S
S S S
 
 
=  
 
 

6. 3/6/2021 6
Arpan Deyasi, India
[ ]
11 12 13
21 22 23
31 32 33
S S S
S S S S
S S S
 
 
=  
 
 
Calculation of S-matrix of 3-port circulator
From properties of circulator, all ports are perfectly matched
Diagonal elements become zero
11 22 33 0
S S S
= = =

7. 3/6/2021 Arpan Deyasi, India 7
Calculation of S-matrix of 3-port circulator
For reciprocal ports
[ ]
12 13
21 23
31 32
0
0
0
S S
S S S
S S
 
 
=  
 
 
21 32 13
S S S
= =

8. 3/6/2021 Arpan Deyasi, India 8
Calculation of S-matrix of 3-port circulator
[ ]
12 21
21 23
31 21
0
0
0
S S
S S S
S S
 
 
=  
 
 
Rest of the s-matrix elements are zero
12 23 31 0
S S S
= = =

9. 3/6/2021 Arpan Deyasi, India 9
Calculation of S-matrix of 3-port circulator
[ ]
21
21
21
0 0
0 0
0 0
S
S S
S
 
 
=  
 
 
using Unitary property
[ ][ ]
*
S S I
=

10. 3/6/2021 Arpan Deyasi, India 10
Calculation of S-matrix of 3-port circulator
*
21 21
*
21 21
*
21 21
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
S S
S S
S S
 
   
 
   
=
 
   
 
   
   
 
*
21 21 1
S S =

11. 3/6/2021 Arpan Deyasi, India 11
Calculation of S-matrix of 3-port circulator
2
21 1
S =
Final S-matrix becomes
[ ]
0 0 1
1 0 0
0 1 0
S
 
 
=  
 
 

12. 3/6/2021 Arpan Deyasi, India 12
Isolator

13. 3/6/2021 13
Arpan Deyasi, India
RF Isolator is a 2-port non-reciprocal microwave device
which forwards the signal in one direction and blocks the
signal in the other direction
Isolator

14. 3/6/2021 Arpan Deyasi, India 14
Calculation of S-matrix of isolator
General S-matrix of an isolator is
[ ] 11 12
21 22
S S
S
S S
 
=  
 
From properties of isolator, all ports are perfectly matched
11 22 0
S S
= =

15. 3/6/2021 Arpan Deyasi, India 15
Calculation of S-matrix of isolator
[ ] 12
21
0
0
S
S
S
 
=  
 
From property of isolator
12 0
S =

16. 3/6/2021 16
Arpan Deyasi, India
Calculation of S-matrix of isolator
[ ]
21
0 0
0
S
S
 
=  
 
using Unitary property
[ ][ ]
*
S S I
=

17. 3/6/2021 17
Arpan Deyasi, India
Calculation of S-matrix of isolator
*
21
21
0 0 1 0
0
0 0 1
0 0
S
S
 
   
=
 
   
 
   
*
21 21 1
S S =

18. 3/6/2021 18
Arpan Deyasi, India
Calculation of S-matrix of isolator
2
21 1
S =
Final S-matrix becomes
[ ]
0 0
1 0
S
 
=  
 

19. 3/6/2021 Arpan Deyasi, India 19
Faraday Rotation Isolator

20. When an electromagnetic wave passes through ferrites,
plane of polarization continuously rotates to angle θ in
one particular direction (either clockwise or
anticlockwise).
This plane of polarization changes in the same direction
whatever may be the direction of propagation of wave.
This is called as Faraday Rotation.
What is Faraday Rotation?

21. 3/6/2021 Arpan Deyasi, India 21
Common materials for use in 700-1100 nm range:
Terbium Doped Borosillicate Glass
and
Terbium Gallium Garnet Crystal
Material requirement
Linearly
Polarized
Light
Faraday active
crystal
Magnetic
Field

22. Relation of F.R with S- matrix
M
M
E
k
Mode 1
Mode 2

23. Isolator Based on Faraday Rotation
Polarizer at 0o
Polarizer at 45o
M
M
E
k
X
SMF
SMF
SMF
SMF

24. Faraday Rotation Isolator
comprises of components
i) rectangular waveguide with planar resistive card
ii) mechanical bend of 45° in anticlockwise direction.
It is reciprocal device
iii) circular waveguide with ferrite rod to give a
polarization rotation of 45° in clockwise
direction. It is nonreciprocal device
iv) rectangular waveguide with resistive card.

25. 3/6/2021 Arpan Deyasi, India 25
EM Wave Propagation inside FRI

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Principle of Operation
When light is reflected back to the output port of the
isolator with an unchanged polarization state, it can fully
transmit the output polarizer
Then, however, its polarization direction is rotated by
another 45° in the Faraday rotator, so that this light will be
blocked at the input polarizer, or can be sent to the
separate output port

27. 3/6/2021 Arpan Deyasi, India 27
Note that the output polarizer is important if light may
be reflected back with a modified polarization state.
Principle of Operation
If the rotation angle of the Faraday rotator somewhat
deviates from 45°, the orientation of the output polarizer
may still be adjusted for maximum transmission
In that case the degree of isolation is reduced.
It may be better to optimize that polarizer's orientation
for maximum isolation, while accepting a somewhat
higher insertion loss in forward direction.

28. 3/6/2021 Arpan Deyasi, India 28
Gyrator

29. 3/6/2021 29
Arpan Deyasi, India
RF Gyrator is a 2-port non-reciprocal microwave device
having 180° differential phase shift
Gyrator
180°
0°

30. 3/6/2021 Arpan Deyasi, India 30
Calculation of S-matrix of gyrator
General S-matrix of an gyrator is
[ ] 11 12
21 22
S S
S
S S
 
=  
 
From properties of gyrator, all ports are perfectly matched
11 22 0
S S
= =

31. 3/6/2021 Arpan Deyasi, India 31
Calculation of S-matrix of gyrator
[ ] 12
21
0
0
S
S
S
 
=  
 
From property of gyrator
12 21
S S
= −

32. 3/6/2021 32
Arpan Deyasi, India
Calculation of S-matrix of gyrator
[ ] 12
12
0
0
S
S
S
 
=  
−
 
using Unitary property
[ ][ ]
*
S S I
=

33. 3/6/2021 33
Arpan Deyasi, India
Calculation of S-matrix of gyrator
*
12 12
*
12 12
0 1 0
0
0 0 1
0
S S
S S
 
−
   
=
 
   
−  
   
*
12 12 1
S S =

34. 3/6/2021 34
Arpan Deyasi, India
Calculation of S-matrix of gyrator
2
12 1
S =
Final S-matrix becomes
[ ]
0 1
1 0
S
 
=  
−
 

35. 3/6/2021 Arpan Deyasi, India 35
Attenuator

36. 3/6/2021 36
Arpan Deyasi, India
Attenuator
RF attenuator is a 2-port reciprocal matched lossy
microwave device

37. 3/6/2021 37
Arpan Deyasi, India
Calculation of S-matrix of attenuator
General S-matrix of an attenuator is
[ ] 11 12
21 22
S S
S
S S
 
=  
 
From properties of attenuator, all ports are perfectly matched
11 22 0
S S
= =

38. 3/6/2021 Arpan Deyasi, India 38
Calculation of S-matrix of attenuator
[ ] 12
21
0
0
S
S
S
 
=  
 
For reciprocal ports
21 12
S S
=

39. 3/6/2021 Arpan Deyasi, India 39
Calculation of S-matrix of attenuator
[ ] 12
12
0
0
S
S
S
 
=  
 
From property of attenuator
12
S α
=
1
α <

40. 3/6/2021 Arpan Deyasi, India 40
Calculation of S-matrix of attenuator
[ ]
0
0
S
α
α
 
=  
 

41. 3/6/2021 Arpan Deyasi, India 41
Waveguide Tees

42. 3/6/2021 Arpan Deyasi, India 42
Waveguide Tees
3 port component
E-plane tee connected in series
H-plane tee connected in shunt

43. E-plane Tee
Side arm
E
In E-plane tee, axis of the
side arm is parallel to the
E field

44. 3/6/2021 Arpan Deyasi, India 44
An E-Plane Tee junction is formed by attaching a simple
waveguide to the broader dimension of a rectangular
waveguide, which already has two ports
E-plane Tee
The arms of rectangular waveguides make two ports
called collinear ports i.e., Port1 and Port2, while the
new one, Port3 is called as Side arm or E-arm
E-plane Tee is also called as Series Tee

45. 3/6/2021 Arpan Deyasi, India 45
E-plane Tee
Port 1 Port 2
Port 3

46. 3/6/2021 Arpan Deyasi, India 46
Calculation of S-matrix of E-plane Tee
General S-matrix of an E-plane Tee is
[ ]
11 12 13
21 22 23
31 32 33
S S S
S S S S
S S S
 
 
=  
 
 

47. 3/6/2021 Arpan Deyasi, India 47
Calculation of S-matrix of E-plane Tee
With an input at port 3, Scattering coefficients S13 and S23
are out of phase by 180°
[ ]
11 12 13
21 22 13
31 32 33
S S S
S S S S
S S S
 
 
= −
 
 
 
13 23
S S
= −

48. 3/6/2021 Arpan Deyasi, India 48
Calculation of S-matrix of E-plane Tee
port 3 is perfectly matched
33 0
S =
[ ]
11 12 13
21 22 13
31 32 0
S S S
S S S S
S S
 
 
= −
 
 
 

49. 3/6/2021 Arpan Deyasi, India 49
Calculation of S-matrix of E-plane Tee
From symmetry property
12 21
S S
= 23 32
S S
= 13 31
S S
=
[ ]
11 12 13
12 22 13
13 13 0
S S S
S S S S
S S
 
 
= −
 
 
−
 

50. 3/6/2021 Arpan Deyasi, India 50
Calculation of S-matrix of E-plane Tee
using Unitary property
[ ][ ]
*
S S I
=
* * *
11 12 13 11 12 13
* * *
12 22 13 12 22 13
* *
13 13 13 13
1 0 0
0 1 0
0 0 0 0 1
S S S S S S
S S S S S S
S S S S
 
   
 
   
− − =
 
   
 
   
− −
   
 

51. 3/6/2021 Arpan Deyasi, India 51
Calculation of S-matrix of E-plane Tee
R3C3:
2 2
13 13 1
S S
+ =
13
1
2
S =

52. 3/6/2021 Arpan Deyasi, India 52
Calculation of S-matrix of E-plane Tee
R1C1:
2 2 2
11 12 13 1
S S S
+ + =
R2C2:
2 2 2
12 22 13 1
S S S
+ + =
11 22
S S
=

53. 3/6/2021 Arpan Deyasi, India 53
Calculation of S-matrix of E-plane Tee
R3C1:
13 11 13 12 0
S S S S
− =
11 12
S S
=
R1C1:
2 2
11 11
1
1
2
S S
+ + =
11
1
2
S =

54. 3/6/2021 Arpan Deyasi, India 54
Calculation of S-matrix of E-plane Tee
[ ]
1 1 1
2 2 2
1 1 1
2 2 2
1 1
0
2 2
S
 
 
 
 
−
 
 
 
−
 
 

55. 3/6/2021 Arpan Deyasi, India 55
H-plane Tee
H
Port 3
Collinear arms
Side arm
In H-plane tee, axis of the side arm is parallel
to the H field

56. 3/6/2021 Arpan Deyasi, India 56
H-plane Tee
An H-Plane Tee junction is formed by cutting narrower
dimension of main waveguide and attaching a side arm
The arms of rectangular waveguides make two ports
called collinear ports i.e., Port1 and Port2, while the
new one, Port3 is called as Side arm or H-arm
H-plane Tee is also called as Shunt Tee

57. 3/6/2021 Arpan Deyasi, India 57
H-plane Tee
Port 3
Port 1 Port 2

58. 3/6/2021 Arpan Deyasi, India 58
Calculation of S-matrix of H-plane Tee
General S-matrix of an H-plane Tee is
[ ]
11 12 13
21 22 23
31 32 33
S S S
S S S S
S S S
 
 
=  
 
 

59. 3/6/2021 Arpan Deyasi, India 59
Calculation of S-matrix of H-plane Tee
With an input at port 3, Scattering coefficients S13 and S23
are perfectly matched
[ ]
11 12 13
21 22 13
31 32 33
S S S
S S S S
S S S
 
 
=  
 
 
13 23
S S
=

60. 3/6/2021 Arpan Deyasi, India 60
Calculation of S-matrix of H-plane Tee
port 3 is perfectly matched
33 0
S =
[ ]
11 12 13
21 22 13
31 32 0
S S S
S S S S
S S
 
 
=  
 
 

61. 3/6/2021 Arpan Deyasi, India 61
Calculation of S-matrix of H-plane Tee
From symmetry property
12 21
S S
= 23 32
S S
= 13 31
S S
=
[ ]
11 12 13
12 22 13
13 13 0
S S S
S S S S
S S
 
 
=  
 
 

62. 3/6/2021 Arpan Deyasi, India 62
Calculation of S-matrix of H-plane Tee
using Unitary property
[ ][ ]
*
S S I
=
* * *
11 12 13 11 12 13
* * *
12 22 13 12 22 13
* *
13 13 13 13
1 0 0
0 1 0
0 0 0 0 1
S S S S S S
S S S S S S
S S S S
 
   
 
   
=
 
   
 
   
   
 

63. 3/6/2021 Arpan Deyasi, India 63
Calculation of S-matrix of H-plane Tee
R3C3:
2 2
13 13 1
S S
+ =
13
1
2
S =

64. 3/6/2021 Arpan Deyasi, India 64
Calculation of S-matrix of H-plane Tee
R1C1:
2 2 2
11 12 13 1
S S S
+ + =
R2C2:
2 2 2
12 22 13 1
S S S
+ + =
11 22
S S
=

65. 3/6/2021 Arpan Deyasi, India 65
Calculation of S-matrix of H-plane Tee
R3C1:
13 11 13 12 0
S S S S
+ =
11 12
S S
= −
R1C1:
2 2
11 11
1
1
2
S S
+ + =
11
1
2
S =

66. 3/6/2021 Arpan Deyasi, India 66
Calculation of S-matrix of H-plane Tee
[ ]
1 1 1
2 2 2
1 1 1
2 2 2
1 1
0
2 2
S
 
−
 
 
 
= −
 
 
 
 
 

67. Hybrid Tee
E-arm
E
In E-plane tee, axis of the
side arm is parallel to the
E field

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Hybrid Tee
E_H plane Tee is formed by cutting width and breadth of
rectangular waveguide & attaching another waveguides
Four port hybrid junction gives power dividing property
of both E and H plane tee
It is also called Magic Tee

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Hybrid Tee
Port 1 Port 2
Port 3
Port 4

70. 3/6/2021 Arpan Deyasi, India 70
Calculation of S-matrix of Hybrid Tee
General S-matrix of an Hybrid Tee is
[ ]
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
S S S S
S S S S
S
S S S S
S S S S
 
 
 
=
 
 
 

71. 3/6/2021 Arpan Deyasi, India 71
Calculation of S-matrix of Hybrid Tee
Because of H-plane tee junction
13 23
S S
=
[ ]
11 12 13 14
21 22 13 24
31 32 33 34
41 42 43 44
S S S S
S S S S
S
S S S S
S S S S
 
 
 
=
 
 
 

72. 3/6/2021 Arpan Deyasi, India 72
Calculation of S-matrix of Hybrid Tee
Because of E-plane tee junction
14 24
S S
= −
[ ]
11 12 13 14
21 22 13 14
31 32 33 34
41 42 43 44
S S S S
S S S S
S
S S S S
S S S S
 
 
−
 
=
 
 
 

73. 3/6/2021 Arpan Deyasi, India 73
Calculation of S-matrix of Hybrid Tee
Because of geometry of junctions, port 3 and port 4 are isolated
34 43 0
S S
= =
[ ]
11 12 13 14
21 22 13 14
31 32 33
41 42 44
0
0
S S S S
S S S S
S
S S S
S S S
 
 
−
 
=
 
 
 

74. 3/6/2021 Arpan Deyasi, India 74
Calculation of S-matrix of Hybrid Tee
port 3 and port 4 are perfectly matched
33 44 0
S S
= =
[ ]
11 12 13 14
21 22 13 14
31 32
41 42
0 0
0 0
S S S S
S S S S
S
S S
S S
 
 
−
 
=
 
 
 

75. 3/6/2021 Arpan Deyasi, India 75
Calculation of S-matrix of Hybrid Tee
From symmetry property
12 21
S S
= 23 32
S S
= 13 31
S S
=
14 41
S S
= 24 42
S S
=

76. 3/6/2021 Arpan Deyasi, India 76
Calculation of S-matrix of Hybrid Tee
[ ]
11 12 13 14
12 22 13 14
13 13
14 14
0 0
0 0
S S S S
S S S S
S
S S
S S
 
 
−
 
=
 
 
−
 

77. 3/6/2021 Arpan Deyasi, India 77
Calculation of S-matrix of Hybrid Tee
using Unitary property
[ ][ ]
*
S S I
=
* * * *
11 12 13 14 11 12 13 14
* * * *
12 22 13 14 12 22 13 14
* *
13 13 13 13
* *
14 14 14 14
1 0 0 0
0 1 0 0
0 0 0 0 1 0
0 0
0 0 0 0 0 1
0 0
S S S S S S S S
S S S S S S S S
S S S S
S S S S
 
   
 
   
− −
 
   
=
 
   
 
   
− −
   
  

78. 3/6/2021 Arpan Deyasi, India 78
Calculation of S-matrix of Hybrid Tee
R3C3:
2 2
13 13 1
S S
+ =
13
1
2
S =
R4C4:
2 2
14 14 1
S S
+ =
14
1
2
S =

79. 3/6/2021 Arpan Deyasi, India 79
Calculation of S-matrix of Hybrid Tee
R1C1:
2 2 2 2
11 12 13 14 1
S S S S
+ + + =
R2C2:
2 2 2 2
12 22 13 14 1
S S S S
+ + + =
11 22
S S
=

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Calculation of S-matrix of Hybrid Tee
2 2
11 12
1 1
1
2 2
S S
+ + + =
11 12 0
S S
= =
22 0
S =

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Calculation of S-matrix of Hybrid Tee
[ ]
1 1
0 0
2 2
1 1
0 0
2 2
1 1
0 0
2 2
1 1
0 0
2 2
S
 
 
 
 
−
 
 
=
 
 
 
 
−
 
 

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Directional Coupler

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terminated
Directional Coupler

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Directional Coupler
It is a device that samples a small amount of Microwave
power for measurement purposes
It is a 4-port waveguide junction consisting of a
primary main waveguide and a secondary auxiliary
waveguide

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Directional Coupler
Port 1 Port 2
Port 3
Port 4

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Properties of Directional Coupler
 All the terminations are matched to the ports.
 When the power travels from Port 1 to Port 2, some portion
of it gets coupled to Port 4 but not to Port 3.
 As it is also a bi-directional coupler, when the power travels
from Port 2 to Port 1, some portion of it gets coupled to Port 3
but not to Port 4.
 If the power is incident through Port 3, a portion of it is
coupled to Port 2, but not to Port 1.
 If the power is incident through Port 4, a portion of it is
coupled to Port 1, but not to Port 2.
 Port 1 and 3 are decoupled as are Port 2 and Port 4.

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Performance parameters Directional Coupler
Coupling Factor (C)
ratio of incident power to the forward power
10log i
f
P
C
P
 
=  
 
 

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Performance parameters Directional Coupler
Directivity (D)
ratio of forward power to the back power
10log
f
b
P
D
P
 
=  
 

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Isolation (I)
Performance parameters Directional Coupler
ratio of incident power to the back power
10log i
b
P
I
P
 
=  
 

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Calculation of S-matrix of Direction Coupler
General S-matrix of a Directional Coupler is
[ ]
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
S S S S
S S S S
S
S S S S
S S S S
 
 
 
=
 
 
 

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Calculation of S-matrix of Direction Coupler
All four ports are perfectly matched
11 22 33 44 0
S S S S
= = = =
[ ]
12 13 14
21 23 24
31 32 34
41 42 43
0
0
0
0
S S S
S S S
S
S S S
S S S
 
 
 
=
 
 
 

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Calculation of S-matrix of Direction Coupler
From symmetry property
12 21
S S
= 23 32
S S
= 34 43
S S
=
14 41
S S
= 24 42
S S
= 13 31
S S
=

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Calculation of S-matrix of Direction Coupler
No coupling between port 1 and port 3
13 31 0
S S
= =
[ ]
12 14
21 23 24
32 34
41 42 43
0 0
0
0 0
0
S S
S S S
S
S S
S S S
 
 
 
=
 
 
 

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Calculation of S-matrix of Direction Coupler
No coupling between port 2 and port 4
24 42 0
S S
= =
[ ]
12 14
21 23
32 34
41 43
0 0
0 0
0 0
0 0
S S
S S
S
S S
S S
 
 
 
=
 
 
 

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Calculation of S-matrix of Direction Coupler
using Unitary property
[ ][ ]
*
S S I
=
* *
12 14 12 14
* *
21 23 21 23
* *
32 34 32 34
* *
41 43 41 43
0 0 1 0 0 0
0 0
0 0 0 1 0 0
0 0
0 0 0 0 1 0
0 0
0 0 0 0 0 1
0 0
S S S S
S S S S
S S S S
S S S S
 
   
 
   
 
   
=
 
   
 
   
   
  

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Calculation of S-matrix of Direction Coupler
R2C2:
2 2
12 14 1
S S
+ =
R1C1:
2 2
21 23 1
S S
+ =
14 23
S S
=

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Calculation of S-matrix of Direction Coupler
R2C2:
2 2
21 23 1
S S
+ =
R3C3:
2 2
32 34 1
S S
+ =
21 34
S S
=

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Calculation of S-matrix of Direction Coupler
[ ]
12 14
12 14
14 12
14 12
0 0
0 0
0 0
0 0
S S
S S
S
S S
S S
 
 
 
=
 
 
 

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Calculation of S-matrix of Direction Coupler
Let S12 is real and positive
R1C3:
12
S p
=
* *
12 23 14 43 0
S S S S
+ =
*
23 14
. . 0
p S S p
+ =
*
23 23
[ ] 0
p S S
+ =

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Calculation of S-matrix of Direction Coupler
To satisfy the condition, S23 should be complex quantity
23 14
S S jq
= =
[ ]
0 0
0 0
0 0
0 0
p jq
p jq
S
jq p
jq p
 
 
 
=
 
 
 

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Types of directional coupler: Two-Hole

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Directional coupler with same main and auxiliary waveguides, but
with two small holes that are common between them
Types of directional coupler: Two-Hole
Designed to avoid back power
Some of the power while travelling between Port 1 and Port 2, escapes
through the holes 1 and 2.The magnitude of the power depends upon the
dimensions of the holes. This leakage power at both the holes are in phase
at hole 2, adding up the power contributing to the forward power.
However, it is out of phase at hole 1, cancelling each other and preventing
the back power to occur. Hence, the directivity of a directional coupler
improves
Holes are λg/4 distance apart

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Types of directional coupler: Bethe/Single-Hole

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It consists of only one hole.
Power entering in port1 is coupled to co-axial probe.
Output entering in port2 is absorbed by matched load.
The auxiliary guide is placed at such angle that the magnitude
of magnetically excited wave is made equal to that of
electrically excited wave. This improves directivity.
The wave in auxiliary wave guide is generated through single
hole with both electric and magnetic field.
Because of phase relationship, single generated by two types
of coupling cancel in forward direction and reinforce in
reverse direction.
Types of directional coupler: Bethe/Single-Hole