6. 6
=
1
2π ππΈ
πΆ2 π2 π2
π4
[(
ππ
π
)
2
+ (
ππ
π
)
2
]
ππ
4
ImpossibilityofTEM mode in Rectangular waveguides
We have seenthat in a parallel plate waveguide,aTEMmode for whichboththe electricand
magneticfieldsare perpendiculartothe directionof propagation,exists.This,howeverisnot
true of rectangularwave guide,orforthat matter forany hollow conductorwave guide without
an innerconductor.
We knowthatlinesof H are closedloops.Since there isnozcomponentof the magneticfield,
such loopsmustlie inthe x-yplane.However,aloopinthe x-yplane,accordingtoAmpereβs
law,impliesanaxial current.If there isnoinnerconductor,there cannotbe a real current.The
onlyotherpossibilitythenisadisplacementcurrent. However,anaxial displacementcurrent
requiresanaxial componentof the electricfield,whichiszeroforthe TEM mode.ThusTEM
mode cannot existinahollowconductor.(forthe parallel plate waveguides,thisrestriction
doesnotapplyas the fieldlinesclose atinfinity.)
ResonatingCavity
In a rectangularwaveguide we hadahollow tube withfoursidesclosedand apropagation
directionwhichwasinfinitelylong.We wil now close the thirdside andconsider
electromagneticwave trappedinsidearectangularparallopipedof dimension π Γ π Γ π with
wallsbeingmade of perfectorconductor.(The thirddimensionistakenasd so as notto confuse
withthe speedof lightc).
Resonantingcavitiesare useful forstoringelectromagneticenergyjustasLCcircuit doesbutthe
formerhas an advantage inbeinglesslossyandhavingafrequencyrange muchhigher,above
100 MHz.
The Helmholtzequationforanyof the components πΈ πΌ of the electricfieldcanbe writtenas
β2 πΈ πΌ = βπ2 πππΈ πΌ
We write thisinCartesiananduse the technique of separationof variables,
πΈ πΌ( π₯, π¦, π§) = π πΌ( π₯) ππΌ( π¦) π πΌ(π§)
Introducingthisintothe equationanddividingby π πΌ( π₯) ππΌ( π¦) π πΌ(π§),we get,
1
π πΌ
π2 π πΌ
ππ₯2 +
1
ππΌ
π2 ππΌ
ππ¦2 +
1
π πΌ
π2 π πΌ
ππ§2 = βπ2 ππ
Since eachof the three termson the rightis a functionof an independentvariable,whilethe
righthand side isa constant,we musthave each of the three termsequalingaconstantsuch
that the three constantsadd up to the constanton the right. Let,
π2 π πΌ
ππ₯2 = βπ π₯
2 π πΌ
7. 7
π2 ππΌ
ππ¦2 = βπ π¦
2 ππΌ
π2 π πΌ
ππ§2 = βπ π§
2 π πΌ
such that π π₯
2 + π π¦
2 + π π§
2 = π2 ππ.
As eachof the equationshasa solutionintermsof linearcombinationof sine andcosine
functions,we write the solutionforthe electricfieldas
πΈ πΌ = (π΄ πΌ cos π π₯ π₯ + π΅ πΌ sin π π₯ π₯)(πΆ πΌ cos π π¦ π¦ + π· πΌ sin π π¦ π¦)(πΉcos π π§ π§ + πΊ πΌ sin π π§ π§)
Tha tangential componentof above mustvanishatthe metal boundary. Thisimplies,
1. At π₯ = 0 and at π₯ = π , πΈ π¦ and πΈ π§ = 0 for all valuesof π¦, π§,
2. At π¦ = 0 and π¦ = π, πΈ π₯ and πΈ π§ = 0 forall valuesof π₯, π§,
3. At π§ = 0 and z=d, πΈ π₯ and πΈ π¦ = 0 for all valuesof π₯, π¦
Let usconsider πΈ π₯ whichmustbe zero at π¦ = 0, π¦ = π, π§ = 0, π§ = π. Thisisposible if
πΈ π₯ = πΈ π₯0(π΄ π₯ cos π π₯ π₯ + π΅ π₯ sin π π₯ π₯)sin π π¦ π¦sin π π§ π§
with π π¦ =
ππ
π
and π π§ =
ππ
π
. Here m and n are non-zerointegers.
In a similarway,we have,
πΈ π¦ = πΈ π¦0sin π π₯ π₯ (πΆ π¦ cos π π¦ π¦ + π· π¦ sin π π¦ π¦)sin π π§ π§
πΈ π§ = πΈ π§0sin π π₯ π₯sin π π¦ π¦ (πΉπ§ cos π π§ π§ + πΊπ§ sin π π§ π§)
with π π₯ =
ππ
π
, with π beingnon-zerointeger.
We nowuse
β β πΈβ =
ππΈ π₯
ππ₯
+
ππΈ π¦
ππ¦
+
ππΈ π§
ππ§
= 0
Thisrelationmustbe satisfiedfor all valuesof π, π, π withinthe cavity. Thisrequires,
πΈ π₯0(βπ΄ π₯ π π₯ sin π π₯ π₯
+ π΅ π₯ π π₯cos π π₯ π₯)sin π π¦ π¦sin π π§ π§
+ πΈ π¦0sin π π₯ π₯(βπΆ π¦ π π¦ sin π π¦ π¦
+ π· π¦ π π¦ cos π π¦ π¦)sin π π§ π§ + πΈ π§0 sin π π₯ π₯π ππ π π¦ π¦(βπΉπ§ π π§ sin π π§ π§
+ πΊπ§ π π§ cos π π§ π§) = 0
Let uschoose some special pointsand try to satisfythisequation.Let π₯ = 0, π¦, π§ arbitrary,
Thisrequires π΅ π₯ = 0.Likewise,taking π¦ = 0,x,z arbitrary,we require π· π¦ = 0 andfinally, π§ = 0,
π₯, π¦ arbitrary gives πΊπ§ = 0.
Withthese our solutionsforthe componentsof the electricfieldbecomes,
πΈ π₯ = πΈ π₯0cos π π₯ π₯sin π π¦ π¦sin π π§ π§
πΈ π¦ = πΈ π¦0sin π π₯ π₯cos π π¦ π¦sin π π§ π§
πΈ π§ = πΈ π§0 sin π π₯ π₯sin π π¦ π¦cos π π§ π§
where we have redefinedourconstants πΈ π₯0,πΈ π¦0 and πΈ π§0.