MAXWELL’S EQUATIONS AND TRANSMISSION MEDIA CHARACTERISTICS ENEE 482 Spring 2002 DR. KAWTHAR ZAKI

MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS Two conductor wire Coaxial line Shielded Strip line Dielectric ENEE482

Common Hollow-pipe waveguides Rectangular guide Ridge guide Circular guide ENEE482

STRIP LINE CONFIGURATIONS W SINGLE STRIP LINE COUPLED LINES COUPLED STRIPS TOP & BOTTOM COUPLED ROUND BARS ENEE482

MICROSTRIP LINE CONFIGURATIONS SINGLE MICROSTRIP TWO COUPLED MICROSTRIPS TWO SUSPENDED SUBSTRATE LINES SUSPENDED SUBSTRATE LINE ENEE482

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 ENEE482

Electromagnetic Theory Maxwell’s Equations ENEE482

Auxiliary Relations: ENEE482

Maxwell’s Equations in Large Scale Form ENEE482

Maxwell’s Equations for the Time - Harmonic Case ENEE482

Boundary Conditions at a General Material Interface h E1t E2t m1,e1 m2,e2 D1n D2n h Ds ENEE482

Fields at a Dielectric Interface ENEE482

+ + + n rs Js Ht ENEE482

The magnetic wall boundary condition ENEE482

Wave Equation ENEE482

Plane Waves ENEE482

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H is perpendicular to E and to n. (TEM waves) z y E n H x H is perpendicular to E and to n. (TEM waves) ENEE482

Plane Wave in a Good Conductor ENEE482

Boundary Conditions at the Surface of a Good Conductor 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 ENEE482

Reflection From A Dielectric Interface Parallel Polarization e x Er n2 e0 Et n3 q2 q3 q1 z n1 Ei ENEE482

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Energy and Power Under steady-state sinusoidal time-varying Conditions, the time-average energy stored in the Electric field is ENEE482

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Poynting Theorem ENEE482

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Circuit Analogy L R I V C ENEE482

Potential Theory ENEE482

Solution For Vector Potential J (x’,y’, z’) R (x,y,z) r’ r ENEE482

Waves on An Ideal 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 ENEE482

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Steady State Sinusoidal Waves ENEE482

Transmission Line Parameters C2 C1 S ENEE482

Terminated Transmission Line ZL Zc Z To generator ENEE482

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Transmission Lines & Waveguides Wave Propagation in the Positive z-Direction is Represented By:e-jbz ENEE482

Modes Classification: 1. Transverse Electromagnetic (TEM) Waves 2. Transverse Electric (TE), or H Modes 3. Transverse Magnetic (TM), or E Modes 4. Hybrid Modes ENEE482

TEM WAVES ENEE482

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TE WAVES ENEE482

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TM WAVES ENEE482

TEM TRANSMISSION LINES Coaxial Two-wire Parallel -plate a b e ENEE482

COAXIAL LINES a b e ENEE482

THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0 ENEE482

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 ENEE482

Transmission line with small losses ENEE482

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Qc OF COAXIAL LINE AS A FUNCTION OF Zo er Zc ENEE482

Parallel Plate Waveguide TEM Modes y d x w ENEE482

TM modes ENEE482

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TE Modes ENEE482

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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 ENEE482

WAVEGUIDES HOLLOW CONDUCTORS RECTANGULAR OR CIRCULAR. PROPAGATE ELECTROMAGNETIC ENERGY ABOVE A CERTAIN FREQUENCY (CUT OFF) INFINITE NUMBER OF MODES CAN PROPAGATE, EITHER TE OR TM MODES WHEN OPERATING IN A SINGLE MODE, WAVEGUIDE CAN BE DESCRIBED AS A TRANSMISSION LINE WITH C/C IMPEDANCE Zc & PROPAGATION CONSTANT g ENEE482

WAVEGUIDE PROPERTIES FOR A W/G FILLED WITH DIELECTRIC er : ENEE482

PROPAGATION PHASE CONSTANT: FOR RECTANGULAR GUIDE a X b, CUTOFF WAVELENGTH OF TE10 MODES ARE: : CUT OFF FREQUENCY IN GHz (lc INCHES): FOR CIRCULAR WAVEGUIDE OF DIAMETER D CUTOFF WAVE LENGTH OF TE11 MODE IS: lc = 1.706 D DOMINANT MODES ARE TE10 AND TE11 MODE FOR RECTANGULAR & CIRCULAR WAVEGUIDES ENEE482

RECTANGULAR WAVEGUIDE MODE FIELDS y b z x a CONFIGURATION ENEE482

TE modes ENEE482

TEmn MODES ENEE482

The dominant mode is TE10 ENEE482

TMmn MODES ENEE482

TE Modes of a Partially Loaded Waveguide x ENEE482

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CIRCULAR WAVEGUIDE MODES y r a f x z ENEE482

TE Modes ENEE482

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TEnm MODES ENEE482

TMnmMODES ENEE482

Cutoff frequencies of the first few TE And TM modes in circular waveguide TE11 TE21 TE01 TE31 1 fc/fcTE11 TM01 TM11 TM21 ENEE482

ATTENUATION IN WAVEGUIDES ATTENUATION OF THE DOMINANT MODES (TEm0) IN A COPPER RECTANGULAR WAVEGUIDE DIM. a X b, AND (TE11) CIRCULAR WAVEGUIDE, DIA. D ARE: WHERE f IS THE FREQUENCY IN GHz ENEE482

ATTENUATION IN COPPER WAVEGUIDES DUE TO CONDUCTOR LOSS ENEE482

Higher Order Modes in Coaxial Line TE Modes: ENEE482

Grounded Dielectric Slab x z Dielectric Ground plane TM Modes ENEE482

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Stripline y w b x z Approximate Electrostatic Solution: y b/2 a/2 -a/2 a/2 -a/2 ENEE482

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Microstrip y w d -a/2 x a/2 ENEE482

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The Transverse Resonance Technique TM Modes for the parallel plate waveguide y y d d w x ENEE482

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MODES IN DIELTECTRIC LOADED WAVEGUIDE b er1 a er2 CATEGORIES OF FIELD SOLUTIONS: TE0m MODES TM0m MODES HYBRID HEnm MODES ENEE482

BOUNDARY CONDITIONS FIELDS SATISFY THE WAVE EQUATION, SUBJECT TO THE BOUNDARY CONDITIONS Ez , Ef , Hz , Hf ARE CONTINUOUS AT r=b Ez , Ef VANISH AT r=a ENEE482

WHERE A IS AN ARBITRARY CONSTANT ENEE482

Characteristic equation Where z=x1a is the radial wave number in er ENEE482

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For n = 0, the Characteristic Equation Degenerates in two Separate Independent Equations for TE and TM Modes: For TE Modes And: For TM Modes ENEE482

COMPLEX MODES COMPLEX PROPAGATION CONSTANT : g = a +jb ONLY HE MODE CAN SUPPORT COMPLEX WAVES PROPAGATION CONSTANT OF COMPLEX MODES ARE CONJUGATE : COMPLEX MODES DON’T CARRY REAL POWER COMPLEX MODES CONSTITUTE PART OF THE COMPLETE SET OF ELECTROMAGNETIC FIELD SPACE COMPLEX MODES HAVE TO BE INCLUDED IN THE FIELD EXPANSIONS FOR CONVERGENCE TO CORRECT SOLUTIONS IN MODE MATCHING TECHNIQUES. ENEE482

OPTICAL FIBER 2a e1 IN CIRCULAR CYLINDRICAL COORDINATES: Step-index fiber ENEE482

For the symmetric case n=0, the solution break into Separate TE and TM sets. The continuity condition for Ez1= Ez2 and Hf1= Hf2 at r=a gives for the TM set: The continuity condition for Hz1= Hz2 and Ef1= Ef2 at r=a gives for the TE set: If n is different from 0, the fields do not separate into TM and TE types, but all the fields become coupled through continuity conditions. ENEE482

Parallel Plate Transmission Line b c er Partially loaded parallel Plate waveguide y x ENEE482

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Low Frequency Solution When the frequency is low, ENEE482

y c b er a x -W W ENEE482

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High Frequency Solution: ENEE482

Microstrip Transmission Line w y H x ENEE482

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Boundary conditions: ENEE482

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