EC6503 TRANSMISSION LINES AND WAVEGUIDES BY H.UMMA HABIBA, Professor SVCE,Chennai.

Overview of syllabus OBJECTIVES To introduce the various types of transmission lines and to discuss the losses associated. To give thorough understanding about impedance transformation and matching. To use the Smith chart in problem solving. To impart knowledge on filter theories and waveguide theories

Overview of syllabus OUTCOMES Upon completion of the course, students will be able to Discuss the propagation of signals through transmission lines. Analyze signal propagation at Radio frequencies. Explain radio propagation in guided systems. Utilize cavity resonators

Overview of syllabus TEXT BOOK 1. John D Ryder, “Networks lines and fields”, Prentice Hall of India, New Delhi, 2005 REFERENCES William H Hayt and Jr John A Buck, “Engineering Electromagnetics” Tata Mc Graw-Hill Publishing Company Ltd, New Delhi, 2008 David K Cheng, “Field and Wave Electromagnetics”, Pearson Education Inc, Delhi, 2004 John D Kraus and Daniel A Fleisch, “Electromagnetics with Applications”, Mc Graw Hill Book Co,2005 GSN Raju, “Electromagnetic Field Theory and Transmission Lines”, Pearson Education, 2005 Bhag Singh Guru and HR Hiziroglu, “Electromagnetic Field Theory Fundamentals”, Vikas Publishing House, New Delhi, 2001. 6. N. Narayana Rao, “ Elements of Engineering Electromagnetics” 6

Overview of syllabus UNIT I -TRANSMISSION LINE THEORY General theory of Transmission lines - the transmission line - general solution - The infinite line - Wavelength, velocity of propagation - Waveform distortion - the distortion-less line - Loading and different methods of loading - Line not terminated in Z0 - Reflection coefficient - calculation of current, voltage, power delivered and efficiency of transmission - Input and transfer impedance - Open and short circuited lines - reflection factor and reflection loss.

Overview of syllabus UNIT II -HIGH FREQUENCY TRANSMISSION LINES Transmission line equations at radio frequencies - Line of Zero dissipation - Voltage and current on the dissipation-less line, Standing Waves, Nodes, Standing Wave Ratio - Input impedance of the dissipation-less line - Open and short circuited lines - Power and impedance measurement on lines - Reflection losses - Measurement of VSWR and wavelength.

Overview of syllabus UNIT III-IMPEDANCE MATCHING IN HIGH FREQUENCY LINES Impedance matching: Quarter wave transformer - Impedance matching by stubs - Single stub and double stub matching - Smith chart - Solutions of problems using Smith chart - Single and double stub matching using Smith chart.

Overview of syllabus UNIT IV-PASSIVE FILTERS Characteristic impedance of symmetrical networks - filter fundamentals, Design of filters: Constant K - Low Pass, High Pass, Band Pass, Band Elimination, m- derived sections - low pass, high pass composite filters.

Overview of syllabus UNIT V -WAVE GUIDES AND CAVITY RESONATORS General Wave behaviours along uniform Guiding structures, Transverse Electromagnetic waves, Transverse Magnetic waves, Transverse Electric waves, TM and TE waves between parallel plates, TM and TE waves in Rectangular wave guides, Bessel‟s differential equation and Bessel function, TM and TE waves in Circular wave guides, Rectangular and circular cavity Resonators.

TRANSMISSION LINE THEORY UNIFORM PLANE WAVES UNIT I TRANSMISSION LINE THEORY UNIFORM PLANE WAVES

(shown connected to a 4:1 impedance-transforming balun) Transmission Line Properties Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz) Twin lead (shown connected to a 4:1 impedance-transforming balun) Coaxial cable (coax) 11

Transmission Line (cont.) CAT 5 cable (twisted pair) The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities. 12

Transmission Line (cont.) Transmission lines commonly met on printed-circuit boards Microstrip h w er er w Stripline h Coplanar strips h er w Coplanar waveguide (CPW) h er w 13

Transmission Line (cont.) Transmission lines are commonly met on printed-circuit boards. Microstrip line A microwave integrated circuit 14

Fiber-Optic Guide Properties Uses a dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields 15

Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-mode fiber Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength. 2) Multi-mode fiber Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect). 16

Fiber-Optic Guide (cont.) Higher index core region http://en.wikipedia.org/wiki/Optical_fiber 17

Waveguides Properties Has a single hollow metal pipe Can propagate a signal only at high frequency:  > c The width must be at least one-half of a wavelength Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz) 18 http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)

Transmission-Line Theory Lumped circuits: resistors, capacitors, inductors neglect time delays (phase) Distributed circuit elements: transmission lines account for propagation and time delays (phase change) We need transmission-line theory whenever the length of a line is significant compared with a wavelength. 19

Transmission Line C = capacitance/length [F/m] 2 conductors 4 per-unit-length parameters: C = capacitance/length [F/m] L = inductance/length [H/m] R = resistance/length [/m] G = conductance/length [ /m or S/m] Dz  20

Transmission Line (cont.) + + + + + + + - - - - - - - - - - x B RDz LDz GDz CDz z v(z+z,t) + - v(z,t) i(z,t) i(z+z,t) 21

Transmission Line (cont.) RDz LDz GDz CDz z v(z+z,t) + - v(z,t) i(z,t) i(z+z,t) 22

TEM Transmission Line (cont.) Hence Now let Dz  0: “Telegrapher’sEquations” 23

TEM Transmission Line (cont.) To combine these, take the derivative of the first one with respect to z: Switch the order of the derivatives. 24

TEM Transmission Line (cont.) Hence, we have: The same equation also holds for i. 25

TEM Transmission Line (cont.) Time-Harmonic Waves: 26

TEM Transmission Line (cont.) Note that = series impedance/length = parallel admittance/length Then we can write: 27

TEM Transmission Line (cont.) Let Then Solution:  is called the "propagation constant." Convention: 28

TEM Transmission Line (cont.) Forward travelling wave (a wave traveling in the positive z direction): The wave “repeats” when: Hence: 29

Phase Velocity Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest. vp (phase velocity) 30

Phase Velocity (cont.) Set In expanded form: Hence 31

Characteristic Impedance Z0 + V+(z) - I+ (z) z A wave is traveling in the positive z direction. so (Z0 is a number, not a function of z.) 32

Characteristic Impedance Z0 (cont.) Use Telegrapher’s Equation: so Hence 33

Characteristic Impedance Z0 (cont.) From this we have: Using We have Note: The principal branch of the square root is chosen, so that Re (Z0) > 0. 34

General Case (Waves in Both Directions) wave in +z direction wave in -z direction Note: 35

Backward-Traveling Wave + V -(z) - I - (z) z A wave is traveling in the negative z direction. so Note: The reference directions for voltage and current are the same as for the forward wave. 36

General Case Most general case: I (z) + V (z) - z A general superposition of forward and backward traveling waves: Note: The reference directions for voltage and current are the same for forward and backward waves. 37

Summary of Basic TL formulas guided wavelength  g phase velocity  vp 38

Lossless Case so (real and indep. of freq.) (indep. of freq.) 39

Lossless Case (cont.) In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that (proof given later) The speed of light in a dielectric medium is Hence, we have that The phase velocity does not depend on the frequency, and it is always the speed of light (in the material). 40

Terminated Transmission Line Terminating impedance (load) Ampl. of voltage wave propagating in positive z direction at z = 0. Ampl. of voltage wave propagating in negative z direction at z = 0. Where do we assign z = 0? The usual choice is at the load. Note: The length l measures distance from the load: 41

Terminated Transmission Line (cont.) Terminating impedance (load) What if we know Can we use z = - l as a reference plane? Hence 42

Terminated Transmission Line (cont.) Terminating impedance (load) Compare: Note: This is simply a change of reference plane, from z = 0 to z = -l. 43

Terminated Transmission Line (cont.) Terminating impedance (load) What is V(-l )? propagating forwards propagating backwards The current at z = - l is then l  distance away from load 44

Terminated Transmission Line (cont.) Total volt. at distance l from the load Ampl. of volt. wave prop. towards load, at the load position (z = 0). Ampl. of volt. wave prop. away from load, at the load position (z = 0). L  Load reflection coefficient l  Reflection coefficient at z = - l Similarly, 45

Terminated Transmission Line (cont.) Input impedance seen “looking” towards load at z = -l . 46

Terminated Transmission Line (cont.) At the load (l = 0): Recall Thus, 47

Terminated Transmission Line (cont.) Simplifying, we have Hence, we have 48

Terminated Lossless Transmission Line Impedance is periodic with period g/2 tan repeats when Note: 49

Terminated Lossless Transmission Line For the remainder of our transmission line discussion we will assume that the transmission line is lossless. 50

Matched Load A Matched load: (ZL=Z0) No reflection from the load For any l 51

Short-Circuit Load B Short circuit load: (ZL = 0) Note: Always imaginary! S.C. can become an O.C. with a g/4 trans. line 52

Using Transmission Lines to Synthesize Loads This is very useful is microwave engineering. A microwave filter constructed from microstrip. 53

Example Find the voltage at any point on the line. 54

Example (cont.) Note: At l = d : Hence 55

Example (cont.) Some algebra: 56

Example (cont.) Hence, we have where Therefore, we have the following alternative form for the result: 57

Example (cont.) Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load). 58

Example (cont.) Wave-bounce method (illustrated for l = d ): 59

Example (cont.) Geometric series: 60

Example (cont.) Hence or This agrees with the previous result (setting l = d ). Note: This is a very tedious method – not recommended. 61

Time- Average Power Flow At a distance l from the load: Note: If Z0  real (low-loss transmission line) 62

Time- Average Power Flow Low-loss line Lossless line ( = 0) 63

Quarter-Wave Transformer ZL Z0 Z0T Zin This requires ZL to be real. so Hence 64

Voltage Standing Wave Ratio 65

Coaxial Cable a b Find C, L, G, R Here we present a “case study” of one particular transmission line, the coaxial cable. a b Find C, L, G, R For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics. We will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the per-unit-length parameters. 66

Coaxial Cable (cont.) h = 1 [m] Find C (capacitance / length) From Gauss’s law: -l0 l0 a b 67

Coaxial Cable (cont.) h = 1 [m] Coaxial cable Hence We then have -l0 68

Coaxial Cable (cont.) h Find L (inductance / length) h = 1 [m] I From Ampere’s law: Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency. S h I z center conductor Magnetic flux: 69

Coaxial Cable (cont.) Coaxial cable h = 1 [m] I Hence 70

Coaxial Cable (cont.) Observation: This result actually holds for any transmission line. 71

Coaxial Cable (cont.) For a lossless cable: 72

Coaxial Cable (cont.) h = 1 [m] Find G (conductance / length) From Gauss’s law: -l0 l0 a b 73

Coaxial Cable (cont.) -l0 l0 a b We then have or 74

Coaxial Cable (cont.) Observation: This result actually holds for any transmission line. 75

Coaxial Cable (cont.) As just derived, To be more general: This is the loss tangent that would arise from conductivity effects. The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral. Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies. 76

Coaxial Cable (cont.) General expression for loss tangent: Effective permittivity that accounts for conductivity Loss due to molecular friction Loss due to conductivity 77

Coaxial Cable (cont.) h = 1 [m] Find R (resistance / length) Rs = surface resistance of metal a b 78

General Transmission Line Formulas (1) (2) (4) (3) Equations (1) and (2) can be used to find L and C if we know the material properties and the characteristic impedance of the lossless line. Equation (3) can be used to find G if we know the material loss tangent. Equation (4) can be used to find R (discussed later). 79

General Transmission Line Formulas (cont.) Al four per-unit-length parameters can be found from 80

Common Transmission Lines Coax a b Twin-lead h a a 81

Common Transmission Lines (cont.) Microstrip h w er t 82

Common Transmission Lines (cont.) Microstrip h w er t 83

Limitations of Transmission-Line Theory At high frequency, discontinuity effects can become important. transmitted incident Bend reflected The simple TL model does not account for the bend. ZTH ZL Z0 + - 84

Limitations of Transmission-Line Theory (cont.) At high frequency, radiation effects can become important. We want energy to travel from the generator to the load, without radiating. ZTH ZL Z0 + - When will radiation occur? 85

Limitations of Transmission-Line Theory (cont.) The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, or under any circumstances. a b z The fields are confined to the region between the two conductors. 86

Limitations of Transmission-Line Theory (cont.) The twin lead is an open type of transmission line – the fields extend out to infinity. + - The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair.”) Having fields that extend to infinity is not the same thing as having radiation, however. 87

Limitations of Transmission-Line Theory (cont.) The infinite twin lead will not radiate by itself, regardless of how far apart the lines are. h incident reflected S No attenuation on an infinite lossless line The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency. 88

Limitations of Transmission-Line Theory (cont.) A discontinuity on the twin lead will cause radiation to occur. h Incident wave pipe Obstacle Reflected wave Note: Radiation effects increase as the frequency increases. Bend h Incident wave bend Reflected wave 89

Limitations of Transmission-Line Theory (cont.) To reduce radiation effects of the twin lead at discontinuities: Reduce the separation distance h (keep h << ). Twist the lines (twisted pair). h CAT 5 cable (twisted pair) 90

Two conductor wire Coaxial line Shielded Strip line Dielectric 91

Two conductor wire Coaxial line Shielded Strip line Dielectric 92

Common Hollow-pipe waveguides Rectangular guide Ridge guide Circular guide 93

STRIP LINE CONFIGURATIONS W SINGLE STRIP LINE COUPLED LINES COUPLED STRIPS TOP & BOTTOM COUPLED ROUND BARS 94

MICROSTRIP LINE CONFIGURATIONS SINGLE MICROSTRIP TWO COUPLED MICROSTRIPS TWO SUSPENDED SUBSTRATE LINES SUSPENDED SUBSTRATE LINE 95

TRANSVERSE ELECTROMAGNETIC (TEM): COAXIAL LINES TRANSMISSION MEDIA TRANSVERSE ELECTROMAGNETIC (TEM): COAXIAL LINES MICROSTRIP LINES (Quasi TEM) STRIP LINES AND SUSPENDED SUBSTRATE METALLIC WAVEGUIDES: RECTANGULAR WAVEGUIDES CIRCULAR WAVEGUIDES DIELECTRIC LOADED WAVEGUIDES ANALYSIS OF WAVE PROPAGATION ON THESE TRANSMISSION MEDIA THROUGH MAXWELL’S EQUATIONS 96

Auxiliary Relations: 97

Maxwell’s Equations in Large Scale Form 98

Maxwell’s Equations for the Time - Harmonic Case ENEE482 99

Boundary Conditions at a General Material Interface h E1t E2t m1,e1 m2,e2 D1n D2n h Ds 100

Fields at a Dielectric Interface 101

+ + + n rs Js Ht 102

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Wave Equation 104

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The field amplitude decays exponentially from its surface According to e-u/ds where u is the normal distance into the Conductor, ds is the skin depth 108

Parallel Polarization x Er n2 e0 Et n3 q2 q3 q1 z n1 Ei 109

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Under steady-state sinusoidal time-varying Conditions, the time-average energy stored in the Electric field is 111

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L R I V C 116

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Solution For Vector Potential J (x’,y’, z’) R (x,y,z) r’ r 118

Lumped element circuit model for a transmission line Rg z Lumped element circuit model for a transmission line Ldz I(z,t)+dI/dz dz I(z,t) V(z,t) Cdz V(z,t)+dv/dz dz 119

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ZL Zc Z To generator 122

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Wave Propagation in the Positive z-Direction is Represented By:e-jbz Transmission Lines & Waveguides Wave Propagation in the Positive z-Direction is Represented By:e-jbz 124

Modes Classification: 1. Transverse Electromagnetic (TEM) Waves 2. Transverse Electric (TE), or H Modes 3. Transverse Magnetic (TM), or E Modes 4. Hybrid Modes 125

TEM WAVES 126

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TE WAVES 128

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TM WAVES 130

TEM TRANSMISSION LINES Coaxial Two-wire Parallel -plate a b e 131

COAXIAL LINES a b e 132

THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0 133

Zc OF COAXIAL LINE AS A FUNCTION OF b/a 1 10 100 20 40 60 80 120 140 160 180 200 220 240 260 = er Zo 134

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Qc OF COAXIAL LINE AS A FUNCTION OF Zo er Zc 137

TEM Modes y d x w 138

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COUPLED LINES EVEN & ODD MODES OF EXCITATIONS AXIS OF EVEN SYMMETRY AXIS OF ODD SYMMETRY P.M.C. P.E.C. ODD MODE ELECTRIC FIELD DISTRIBUTION EVEN MODE ELECTRIC FIELD DISTRUBUTION =ODD MODE CHAR. IMPEDANCE =EVEN MODE CHAR. IMPEDANCE Equal currents are flowing in the two lines Equal &opposite currents are flowing in the two lines 144