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MathematicalAnalysisofWaveguides(Continued)

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Onthepreviouspage,thezcomponentoftheelectricvectorpotentialfortheTEmodewasderived:

Usingthefieldrelationships:

WecanwritetheallowablefieldconfigurationsfortheTE(transverseelectric)modeswithinawaveguide:

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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

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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

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Mode

TE10 6.56GHzTE20 13.1GHzTE01 14.8GHzTE11 16.2GHzTE30 19.7GHzTE21 19.8GHzTE02 29.5GHz

Ifwearerunningasignalthatiscenteredaround15GHzwitha1GHzbandwidth,thentheonlyTEmodesthatwouldpropagatewouldbeTE10,TE20andTE01.Inthenextsection,we'lllookattheTMmode.

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