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S-matrix analysis of waveguide components

This document discusses waveguide components and their S-matrix analysis. It begins with an introduction and overview of isolators, attenuators, circulators and gyrators. It then provides detailed calculations of the S-matrix for a 3-port circulator, 2-port isolator, 2-port gyrator and 2-port attenuator. The document also discusses Faraday rotation isolators, E-plane and H-plane waveguide tees, and includes the calculation of the S-matrix for an E-plane tee.

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Waveguide Components Analysis using S-Matrix Course Coordinator: Arpan Deyasi 3/6/2021 1 Arpan Deyasi, India
3/6/2021 2 Arpan Deyasi, India Isolator Attenuator Circulator Gyrator
3/6/2021 Arpan Deyasi, India 3 Circulator
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
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     =      
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 = = =
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 = =
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 = = =
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 =
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 =
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     =      
3/6/2021 Arpan Deyasi, India 12 Isolator
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
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 = =
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 =
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 =
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 =
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   =    
3/6/2021 Arpan Deyasi, India 19 Faraday Rotation Isolator
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?
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
Relation of F.R with S- matrix M M E k Mode 1 Mode 2
Isolator Based on Faraday Rotation Polarizer at 0o Polarizer at 45o M M E k X SMF SMF SMF SMF
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.
3/6/2021 Arpan Deyasi, India 25 EM Wave Propagation inside FRI
3/6/2021 Arpan Deyasi, India 26 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
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.
3/6/2021 Arpan Deyasi, India 28 Gyrator
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°
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 = =
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 = −
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 =
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 =
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   =   −  
3/6/2021 Arpan Deyasi, India 35 Attenuator
3/6/2021 36 Arpan Deyasi, India Attenuator RF attenuator is a 2-port reciprocal matched lossy microwave device
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 = =
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 =
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 <
3/6/2021 Arpan Deyasi, India 40 Calculation of S-matrix of attenuator [ ] 0 0 S α α   =    
3/6/2021 Arpan Deyasi, India 41 Waveguide Tees
3/6/2021 Arpan Deyasi, India 42 Waveguide Tees 3 port component E-plane tee connected in series H-plane tee connected in shunt
E-plane Tee Side arm E In E-plane tee, axis of the side arm is parallel to the E field
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
3/6/2021 Arpan Deyasi, India 45 E-plane Tee Port 1 Port 2 Port 3
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     =      
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 = −
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     = −      
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     = −     −  
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             − − =             − −      
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 =
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 =
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 =
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         −       −    
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
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
3/6/2021 Arpan Deyasi, India 57 H-plane Tee Port 3 Port 1 Port 2
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     =      
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 =
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     =      
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     =      
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             =                  
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 =
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 =
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 =
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   −       = −          
Hybrid Tee E-arm E In E-plane tee, axis of the side arm is parallel to the E field
3/6/2021 Arpan Deyasi, India 68 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
3/6/2021 Arpan Deyasi, India 69 Hybrid Tee Port 1 Port 2 Port 3 Port 4
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       =      
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       =      
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     −   =      
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     −   =      
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     −   =      
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 =
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     −   =     −  
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             − −       =             − −       
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 =
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 =
3/6/2021 Arpan Deyasi, India 80 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 =
3/6/2021 Arpan Deyasi, India 81 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         −     =         −    
3/6/2021 Arpan Deyasi, India 82 Directional Coupler
3/6/2021 Arpan Deyasi, India 83 terminated Directional Coupler
3/6/2021 Arpan Deyasi, India 84 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
3/6/2021 Arpan Deyasi, India 85 Directional Coupler Port 1 Port 2 Port 3 Port 4
3/6/2021 Arpan Deyasi, India 86 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.
3/6/2021 Arpan Deyasi, India 87 Performance parameters Directional Coupler Coupling Factor (C) ratio of incident power to the forward power 10log i f P C P   =      
3/6/2021 Arpan Deyasi, India 88 Performance parameters Directional Coupler Directivity (D) ratio of forward power to the back power 10log f b P D P   =    
3/6/2021 Arpan Deyasi, India 89 Isolation (I) Performance parameters Directional Coupler ratio of incident power to the back power 10log i b P I P   =    
3/6/2021 Arpan Deyasi, India 90 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       =      
3/6/2021 Arpan Deyasi, India 91 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       =      
3/6/2021 Arpan Deyasi, India 92 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 =
3/6/2021 Arpan Deyasi, India 93 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       =      
3/6/2021 Arpan Deyasi, India 94 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       =      
3/6/2021 Arpan Deyasi, India 95 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                   =                   
3/6/2021 Arpan Deyasi, India 96 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 =
3/6/2021 Arpan Deyasi, India 97 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 =
3/6/2021 Arpan Deyasi, India 98 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       =      
3/6/2021 Arpan Deyasi, India 99 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 + =
3/6/2021 Arpan Deyasi, India 100 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       =      
3/6/2021 Arpan Deyasi, India 101 Types of directional coupler: Two-Hole
3/6/2021 Arpan Deyasi, India 102 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
3/6/2021 Arpan Deyasi, India 103 Types of directional coupler: Bethe/Single-Hole
3/6/2021 Arpan Deyasi, India 104 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

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In this document
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Introduction to waveguide components analysis focusing on the S-Matrix.

Overview of key waveguide components: Isolator, Attenuator, Circulator, Gyrator.

Description of a circulator device with signal directionality in multi-port configurations.

Hierarchical S-matrix calculation for a 3-port circulator with reciprocal ports.

Introduction to RF Isolator as a non-reciprocal device that blocks signals in one direction.

Step-by-step S-matrix calculation for isolator highlighting matching conditions.

Explanation of Faraday rotation in light due to electromagnetic wave interactions.

Discussion on the use of Faraday rotation in isolator design and components.

Overview of electromagnetic wave propagation principles in Faraday rotation isolators.

Introduction to RF Gyrator as a 2-port non-reciprocal device that introduces 180° phase shift.

Detailed S-matrix calculations for gyrators considering reciprocity and matching.

Defining RF attenuator as a matched lossy device for signal power reduction.

Stepwise S-matrix calculations for an attenuator detailing port relations.

Explanation of types of waveguide tees, with focus on E-plane and H-plane designs.

Detailed functions of E-plane tee and calculations of its S-matrix.

H-plane tee device description and its S-matrix calculations.

Defining Hybrid Tee, a combination of E and H plane concepts in waveguide technology.

S-matrix calculations for Hybrid Tee, focusing on junction properties.

Overview of directional coupler, a 4-port device for microwave power sampling.

Key performance parameters of directional couplers: Coupling Factor, Directivity, Isolation.

Stepwise S-matrix calculations for directional couplers emphasizing port relationships.

Describing two types of directional couplers: Two-Hole and Bethe/Single-Hole.

S-matrix analysis of waveguide components

  • 1.
    Waveguide Components Analysis usingS-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 electromagneticwave 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.Rwith S- matrix M M E k Mode 1 Mode 2
  • 23.
    Isolator Based onFaraday Rotation Polarizer at 0o Polarizer at 45o M M E k X SMF SMF SMF SMF
  • 24.
    Faraday Rotation Isolator comprisesof 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
  • 26.
    3/6/2021 Arpan Deyasi,India 26 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 InE-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-planetee, axis of the side arm is parallel to the E field
  • 68.
    3/6/2021 Arpan Deyasi,India 68 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
  • 69.
    3/6/2021 Arpan Deyasi,India 69 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 =
  • 80.
    3/6/2021 Arpan Deyasi,India 80 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 =
  • 81.
    3/6/2021 Arpan Deyasi,India 81 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         −     =         −    
  • 82.
    3/6/2021 Arpan Deyasi,India 82 Directional Coupler
  • 83.
    3/6/2021 Arpan Deyasi,India 83 terminated Directional Coupler
  • 84.
    3/6/2021 Arpan Deyasi,India 84 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
  • 85.
    3/6/2021 Arpan Deyasi,India 85 Directional Coupler Port 1 Port 2 Port 3 Port 4
  • 86.
    3/6/2021 Arpan Deyasi,India 86 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.
  • 87.
    3/6/2021 Arpan Deyasi,India 87 Performance parameters Directional Coupler Coupling Factor (C) ratio of incident power to the forward power 10log i f P C P   =      
  • 88.
    3/6/2021 Arpan Deyasi,India 88 Performance parameters Directional Coupler Directivity (D) ratio of forward power to the back power 10log f b P D P   =    
  • 89.
    3/6/2021 Arpan Deyasi,India 89 Isolation (I) Performance parameters Directional Coupler ratio of incident power to the back power 10log i b P I P   =    
  • 90.
    3/6/2021 Arpan Deyasi,India 90 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       =      
  • 91.
    3/6/2021 Arpan Deyasi,India 91 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       =      
  • 92.
    3/6/2021 Arpan Deyasi,India 92 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 =
  • 93.
    3/6/2021 Arpan Deyasi,India 93 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       =      
  • 94.
    3/6/2021 Arpan Deyasi,India 94 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       =      
  • 95.
    3/6/2021 Arpan Deyasi,India 95 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                   =                   
  • 96.
    3/6/2021 Arpan Deyasi,India 96 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 =
  • 97.
    3/6/2021 Arpan Deyasi,India 97 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 =
  • 98.
    3/6/2021 Arpan Deyasi,India 98 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       =      
  • 99.
    3/6/2021 Arpan Deyasi,India 99 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 + =
  • 100.
    3/6/2021 Arpan Deyasi,India 100 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       =      
  • 101.
    3/6/2021 Arpan Deyasi,India 101 Types of directional coupler: Two-Hole
  • 102.
    3/6/2021 Arpan Deyasi,India 102 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
  • 103.
    3/6/2021 Arpan Deyasi,India 103 Types of directional coupler: Bethe/Single-Hole
  • 104.
    3/6/2021 Arpan Deyasi,India 104 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