Chapter 9 Guided Electromagnetic WavesSeveral wave guiding systems, Electromagneticwaves in rectangular and circular waveguidesCoaxial line,Cavityresonator1.TEMWave,TEWave,andTMWave2. Equations for Electromagnetic Waves in RectangularWaveguides3.Characterization of Electromagnetic Waves in RectangularWaveguides4.TE1oWave in RectangularWaveguides5. Group Velocity6.Circular Waveguides7.TransmittedPowerand LossinWaveguides8.ResonantCavity9. Coaxial Lines7
Chapter 9 Guided Electromagnetic Waves Several wave guiding systems, Electromagnetic waves in rectangular and circular waveguides Coaxial line，Cavity resonator 1. TEM Wave, TE Wave, and TM Wave 2. Equations for Electromagnetic Waves in Rectangular Waveguides 3. Characterization of Electromagnetic Waves in Rectangular Waveguides 4. TE10Wave in Rectangular Waveguides 5. Group Velocity 6. Circular Waveguides 7. Transmitted Power and Loss in Waveguides 8. Resonant Cavity 9. Coaxial Lines
The electromagneticwaves to be transmitted alongaconfined path are called guided electromagnetic wavesand the systems to transmitthe guided electromagneticwaves are called the wave guiding systemsTwo-wire line, coaxialline, strip line, microstrip, andmetal waveguides are often used in practice.we will discuss the metal waveguides and the coaxiallineonlyU
The electromagnetic waves to be transmitted along a confined path are called guided electromagnetic waves, and the systems to transmit the guided electromagnetic waves are called the wave guiding systems. Two-wire line, coaxial line, strip line, microstrip, and metal waveguides are often used in practice. we will discuss the metal waveguides and the coaxial line only
Two-wireCoaxial lineCircularRectangularlinewaveguidewaveguideDielectric waveguide,Strip lineMicrostriplineFiberoptic
Strip line Two-wire line Rectangular waveguide Microstrip line Dielectric waveguide, Fiber optic Coaxial line Circular waveguide
1.TEM Wave,TEWave,and TMWaveFLHTEM waveTEwaveTM waveThe wave guiding systems in which an electrostatic field can existmust beableto transmit TEM waveFrom Maxwell's equations we can prove that the metal waveguidecannot transmitTEM wave
1. TEM Wave, TE Wave, and TM Wave TEM wave E H es TE wave E H es TM wave E H es The wave guiding systems in which an electrostatic field can exist must be able to transmit TEM wave. From Maxwell’s equations we can prove that the metal waveguide cannot transmit TEM wave
The main properties of several wave guiding systemsEMWave typesWave bandSystemsshieldingTEM wavPoor>3mTwo-wirelineGood> 10cmCoaxiallineTEM waveTEM wavePoorCentimeterStrip linePoorCentimeterMicrostriplineQuasi-TEMwaveRectangularCentimeterGoodTE or TM waveMillimeterwaveguideCircularCentimeterGoodTEorTM waveMillimeterwaveguidePoorFiberopticTEorTM waveOptical waveu7
Systems Wave types EM shielding Wave band Two-wire line TEM wav Poor > 3m Coaxial line TEM wave Good > 10cm Strip line TEM wave Poor Centimeter Microstrip line Quasi-TEM wave Poor Centimeter Rectangular waveguide TE or TM wave Good Centimeter Millimeter Circular waveguide TE or TM wave Good Centimeter Millimeter Fiber optic TE or TM wave Poor Optical wave The main properties of several wave guiding systems
The general approach to study the wave guiding systemsSuppose the wave guiding system is infinitely long, and let it beplaced along the z-axis and the propagating direction be along thepositive z-direction. Then the electric and the magnetic fieldintensities can beexpressedasE(x, y,z)= E(x, y)e-k.:H(x, y,z)= H,(x, y)e-ik:where k,is the propagation constantin the z-direction, and they satisfythefollowing vectorHelmholtzequation:a?EaEa?E+kE=0ayO22ax?a?Hα?Ha"H+k"H=0dy202?ax?U
The general approach to study the wave guiding systems Suppose the wave guiding system is infinitely long, and let it be placed along the z-axis and the propagating direction be along the positive z-direction. Then the electric and the magnetic field intensities can be expressed as k zz x y z x y j 0 ( , , ) ( , )e − E = E k zz x y z x y j 0 ( , , ) ( , )e − H = H + = + + + = + + 0 0 2 2 2 2 2 2 2 2 H H H H E E E E 2 2 2 2 2 2 k x y z k x y z where kz is the propagation constant in the z-direction, and they satisfy the following vectorHelmholtz equation:
The above equation includes six components, E, E,E, andH.,H.,H,in rectangular coordinate system,and they satisfythescalarHelmhotzequationBased on the boundary conditions of the wave guiding systemand by using the method of separation ofvariables, we can findthesolutions fortheseequationsFrom Maxwell's eguations, we can find the relationshipsbetweenthe x-componentor they-componentand the z-componentasaEaH.aH.aEHjk-jkHj0sjouaxayyah.oH.OE.-jkE1jkH+ joμj0skeayayk2axaxWhere k? = k?-k?. These relationships are called the representationof the transverse components by the longitudinal components.u
The above equation includes six components, and , in rectangular coordinate system, and they satisfy the scalar Helmhotz equation. Ex Ey Ez , , H x H y H z , , From Maxwell’s equations, we can find the relationships between the x-component or the y-component and the z-component as − = − y H x E k k E z z x j z j 1 2 c + = − x H y E k k E z z y j z j 1 2 c − = x H k y E k H z z z x j j 1 2 c − = − y H k x E k H z z z y j j 1 2 c Where . 2 2 2 c z k = k − k Based on the boundary conditions of the wave guiding system and by using the method of separation of variables, we can find the solutions for these equations. These relationships are called the representation of the transverse components by the longitudinal components
We onlyneed to solvethe scalarHelmholtzequationforthelongitudinal components, and then from the relationships between thetransverse components and the longitudinalcomponents all transversecomponentscanbederived.In the same way,in cylindricalcoordinatesthe z-componentcanbe expressed in terms of the r-component and -component asE.ouaEjkkearadaH.aE合jopadaroH.aE08jkHapk.arOEaHk.HJOek2OradU
We only need to solve the scalar Helmholtz equation for the longitudinal components, and then from the relationships between the transverse components and the longitudinal components all transverse components can be derived. In the same way, in cylindrical coordinates the z-component can be expressed in terms of the r-component and –component as + = − z z r z H r r E k k E j j 1 2 c + = − r E H r k k E z z z j j 1 2 c − = r H k E k r H z z z r j j 1 2 c + = − z z Hz r k r E k H j j 1 2 c
2.Equationsfor ElectromagneticWavesin RectangularWaveguidesSelect the rectangularcoordinate system and let the broad sidebe placed along the x-axis, the narrow side along the y-axis,and thepropagating directionbe along the z-axis.For TM waves, H, = o , andaccordingto the method oflongitudinal fields,the componentE,shouldfirstbe solved,and fromhe,uwhichthe other components can0be derived.The z-component of the electric field intensity can be written asE. = E.o(x, y)e-ik:UV
2. Equations for Electromagnetic Waves in Rectangular Waveguides Select the rectangular coordinate system and let the broad side be placed along the x-axis, the narrow side along the y-axis, and the propagating direction be along the z-axis. a z y x b , For TM waves, Hz = 0 , and according to the method of longitudinal fields, the component Ez should first be solved, and from which the other components can be derived. The z-component of the electric field intensity can be written as k z z z z E E x y j 0 ( , )e − =
It satisfies the following scalarHelmholtzeguation,i.eo'E."E.+kE.=0oy?Ox?And the amplitude is found to satisfy the same scalar Helmholtzequation, given byOE00'E.0+kE.-0ax?OyIn order to solve the above equation, the method of separation ofvariablesis used. LetE.0 (x、y)= X(x)Y(y)XV--kWeobtainXVwhere X" denotes the second derivative of X with respect to x, and y"denotes the second derivativeof Y with respectto y.UV
It satisfiesthe following scalarHelmholtz equation, i.e. 0 2 2 c 2 2 2 + = + z z z k E y E x E And the amplitude is found to satisfy the same scalar Helmholtz equation, given by 0 0 2 2 c 0 2 2 0 2 + = + z z z k E y E x E In order to solve the above equation, the method of separation of variablesis used. Let ( ) ( ) ( ) 0 E x y X x Y y z 、 = We obtain 2 c k Y Y X X = − + where X" denotes the second derivative of X with respect to x, and Y" denotes the second derivative of Y with respect to y