antenna tutorials -analysis of waveguides (continued) - te fields
DESCRIPTION
jshjsjTRANSCRIPT
-
4/20/2015 AntennaTutorialsAnalysisofWaveguides(Continued)TEFields
http://www.antennatheory.com/tutorial/waveguides/waveguides3.php 1/4
MathematicalAnalysisofWaveguides(Continued)
Back:MathematicalAnalysis1 Waveguides Antennas(Home)
Onthepreviouspage,thezcomponentoftheelectricvectorpotentialfortheTEmodewasderived:
Usingthefieldrelationships:
WecanwritetheallowablefieldconfigurationsfortheTE(transverseelectric)modeswithinawaveguide:
-
4/20/2015 AntennaTutorialsAnalysisofWaveguides(Continued)TEFields
http://www.antennatheory.com/tutorial/waveguides/waveguides3.php 2/4
Intheabove,theconstantsarewrittenasAmnthisimpliesthattheamplitudeforeachmodecanbeindependentoftheothershowever,thefieldcomponentsforasinglemodemustallberelated(thatis,ExandHydonothaveindependentcoefficients).
CutoffFrequency(fc)
Atthispointintheanalysis,weareabletosaysomethingintelligent.Recallthatthecomponentsofthewavenumbermustsatisfytherelationship:
[3]
Sincekxandkyarerestrainedtoonlytakeoncertainvalues,wecanplugthisfactin:
[4]
Aninterestingquestionarisesatthispoint:WhatisthelowestfrequencyinwhichthewaveguidewillpropagatetheTEmode?
Forpropagationtooccur, .Ifthisistrue,thenkzisarealnumber,sothatthefieldcomponents(equations[1]and[2])willcontaincomplexexponentials,whichrepresent
-
4/20/2015 AntennaTutorialsAnalysisofWaveguides(Continued)TEFields
http://www.antennatheory.com/tutorial/waveguides/waveguides3.php 3/4
propagatingwaves.Ifontheotherhand, ,thenkzwillbeanimaginarynumber,inwhichcasethecomplexexponentialaboveinequations[12]becomesadecayingrealexponential.Inthiscase,thefieldswillnotpropagatebutinsteadquicklydieoutwithinthewaveguide.Electromagneticfieldsthatdieoffinsteadofpropagatearereferredtoasevanescentwaves.
Tofindthelowestfrequencyinwhichpropagationcanoccur,wesetkz=0.Thisisthetransitionbetweenthecutoffregion(evanescent)andthepropagationregion.Settingkz=0inequation[4],weobtain:
[5]
Ifmandnarebothzero,thenallofthefieldcomponentsin[12]becomezero,sowecannothavethiscondition.Thelowestvaluethelefthandsideofequation[5]cantakeoccurswhenm=1andn=0.Thesolutiontoequation[5]whenm=1andn=0,givesthecutofffrequencyforthiswaveguide:
Anyfrequencybelowthecutofffrequency(fc)willonlyresultinevanescentordecayingmodes.Thewaveguidewillnottransportenergyatthesefrequencies.Inaddition,ifthewaveguideisoperatingatafrequencyjustabovefc,thentheonlymodethatisapropagatingmodewillbetheTE10mode.Allothermodeswillbedecaying.Hence,theTE10mode,sinceithasthelowestcutofffrequency,isreferredtoasthedominantmode.
Everymodethatcanexistwithinthewaveguidehasitsowncutofffrequency.Thatis,foragivenmodetopropagate,theoperatingfrequencymustbeabovethecutofffrequencyforthatmode.Bysolving[5]inamoregeneralform,thecutofffrequencyfortheTEmnmodeisgivenby:
Althoughwehaven'tdiscussedtheTM(transversemagnetic)mode,itwillturnoutthatthedominantTMmodehasahighercutofffrequencythanthedominantTEmode.Thiswillbediscussedinthenextsection.
Togiveanexampleofthecutofffrequenciesofvariousmodes,let'sconsiderastandardxbandwaveguide,withdimensionsofa=0.9"(2.286cm)andb=0.4"(1.016cm).Assumingthewaveguideisfilledwithair(oravacuum),thenthecutofffrequenciesforvariousmodesaregiveninthefollowingtable:
TableI.CutoffFrequencyforTEmnModesinanXbandRectangularWaveguide
-
4/20/2015 AntennaTutorialsAnalysisofWaveguides(Continued)TEFields
http://www.antennatheory.com/tutorial/waveguides/waveguides3.php 4/4
Mode
TE10 6.56GHzTE20 13.1GHzTE01 14.8GHzTE11 16.2GHzTE30 19.7GHzTE21 19.8GHzTE02 29.5GHz
Ifwearerunningasignalthatiscenteredaround15GHzwitha1GHzbandwidth,thentheonlyTEmodesthatwouldpropagatewouldbeTE10,TE20andTE01.Inthenextsection,we'lllookattheTMmode.
Next:TheTransverseMagnetic(TM)ModesinWaveguides
Previous:MathematicalAnalysisofWaveguides1
Waveguides(TableofContents)
AntennaTheory(Home)