ElectromagneticFieldandWave電磁場與電磁波2011.5.211ElectromagneticFieldandWaveReview1.GeneralWaveBehaviorsalongUniformGuidingStructures2Review1.GeneralWaveBehavior33Maintopic1.RectangularWaveguides

4Maintopic1.RectangularWaveg1.RectangularWaveguides

Selecttherectangularcoordinatesystemandletthebroadsidebeplacedalongthe

x-axis,thenarrowsidealongthe

y-axis,andthepropagatingdirectionbealongthez-axis.azyxb,ForTMwaves,Hz=0

,andaccordingtothe

methodoflongitudinalfields,thecomponent

Ez

should

first

besolved,andfromwhichtheothercomponentscanbederived.

The

z-componentoftheelectricfieldintensitycanbewrittenas1.1TMwaves51.RectangularWaveguidesItsatisfiesthefollowingscalarHelmholtzequation,i.e.Inordertosolvetheaboveequation,themethodof

separationofvariables

isused.LetWeobtainwhereX"

denotes

thesecond

derivativeof

X

withrespectto

x,and

Y"

denotes

thesecond

derivativeof

Y

withrespectto

y.6ItsatisfiesthefollowingscaThe

onlyway

theequationcanbesatisfiedisthatbothtermsontheleftsideare

constants.

Nowlet

wherekx

and

ky

arecalledthe

separationconstants,andtheycanbefoundbyusingtheboundaryconditions.Obviously7Theonlywaytheequationcanwherealltheconstants

C1,C2,C3,C4,and

kx,ky,dependontheboundaryconditions.Thetwoequationsaresecondorderordinarydifferentialequations,andthegeneralsolutions,arerespectively

The

z-componentoftheelectricfieldintensitycanbewrittenasBoundaryconditions:azyxb,8wherealltheconstantsC1,C99Andallthefieldcomponentsarewhere10AndallthefieldcomponentsaThecutoffofaparticularmodeistheconditionthatmakesvanish.FortheTMmnmodethecutofffrequencyisAlternatively,wemaywriteWherecisthecutoffwavelength.

11Thecutoffofaparticularmod

1，電磁波的相位僅與變量z有關，而振幅與x,y有關。因此，在Z方向上為行波，在X及Y

方向上形成駐波。

2，z

等于常數的平面為波面。但振輻與x,y有關，因此上述TM波為非均勻的平面波；

3，當m

或n

為零時，上述各個分量均為零，因此m及n

應為非零的整數。m及n具有明顯的物理意義，m為寬壁上的半個駐波的數目，n為窄壁上半個駐波的數目。

4，由于m及n為多值，因此場結構均具有多種模式。m及n的每一種組合構成一種模式，以TMmn表示。例如TM11表示m=1,n=1的場結構，具有這種場結構的波稱為TM11波。5，數值大的m及n模式稱為高次模，數值小的稱為低次模。由于m及n均不為零，故矩形波導中TM波的最低模式是TM11波。121，電磁波的相位僅與變量z有關，而振幅與

Selecttherectangularcoordinatesystemandletthebroadsidebeplacedalongthe

x-axis,thenarrowsidealongthe

y-axis,andthepropagatingdirectionbealongthez-axis.azyxb,ForTEwaves,Ez=0

,andaccordingtothe

methodoflongitudinalfields,thecomponent

Hz

should

first

besolved,andfromwhichtheothercomponentscanbederived.

The

z-componentofthemagneticfieldintensitycanbewrittenas1.2TEwaves13SelecttherectangularItsatisfiesthefollowingscalarHelmholtzequation,i.e.Inordertosolvetheaboveequation,themethodof

separationofvariables

isused.LetWeobtainwhereX"

denotes

thesecond

derivativeof

X

withrespectto

x,and

Y"

denotes

thesecond

derivativeof

Y

withrespectto

y.14Itsatisfiesthefollowingsca

Similarly,wecanderiveallthecomponentsofa

TEwave

intherectangularwaveguide,asgivenbywhere,butbothshouldnotbezero

atthesametime.Boundaryconditions:15Similarly,wecanderivThecutoffofaparticularmodeistheconditionthatmakes

vanish.FortheTEmnmodethecutofffrequencyisAltermatively,wemaywriteWherecisthecutoffwavelength.

TEwavehasthe

multi-mode

characteristicsasthe

TM

wave.

The

lowest

ordermodeofTEwaveisthe

TE01

orTE10

wave.16ThecutoffofaparticularmodExample.Theinsideofarectangularmetalwaveguideisvacuum,andthecross-sectionis

25mm10mm.Whatmodes

canbetransmittedifanelectromagneticwaveoffrequencyentersthewaveguide?Willthemodesbechangedifthewaveguideisfilledwithaperfect

dielectric

ofrelativepermittivity?

Solution:

Duetotheinsideisvacuum,theoperatingwavelengthis

andthecutoffwavelengthis

Thenthecutoffwavelengthof

TE10

waveis,thatof

TE20

waveis,andthatof

TE01

waveis.Thecutoffwavelengthofthehighermodeswillbeevenshorter.Inviewofthis,only

TE10

wavecanbetransmittedinthiswaveguide.17Example.Theinsideof

Ifthewaveguideisfilledwitha

perfectdielectric

of,thentheoperatingwavelengthisHence,

TE10

and

TE20

wavescanbetransmitted,andsomeothermodes

TE01，TE30，TE11，TM11，TE21，TM21

canexist.18Ifthewaveguideisfill1.3TE10WaveinRectangularWaveguides

Let,wefindThecorresponding

instantaneous

valuesareAnd.191.3TE10WaveinRectangularLet

m=1,n=0,wefindthecutoffwavelengthof

TE10

modeas

Itmeansthatthecutoffwavelengthofthe

TE10

waveisindependentofthe

narrow

side.The

phasevelocity

andthe

guidewavelength

canbefoundasthe

energyvelocity

as20Letm=1,n=0,wefindth

Example.

Arectangularwave-guideisfilledwithdielectric(perfect)medium(r=1,r=9),.andoperatesatafrequency3GHz.Ifthedimensionsofthewave-guideis

a=2cmandb=1cmSolution：Showthatthecanpropagateatthisfrequency

m/s

Sothecanpropagateatthisfrequencyinthewaveguide.21Example.Arectangularwave-g(2)Determinephaseconstant:

rad/m

(3)Determinewaveimpedance22(2)Determinephaseconstant:r(4)Determinethephasevelocity(5)Determinethegroupvelocity23(4)Determinethephaseveloci1.RectangularWaveguides

themethodof

separationofvariablesazyxb,the

method

oflongitudinalfieldssummary241.RectangularWaveguidestheTMwaveTEwave25TMwaveTEwave252626homework10-14;10-16;10-18Thankyou!Bye-bye!答疑安排時間：地點：27homework10-14;10-16;10-18ThanElectromagneticFieldandWave電磁場與電磁波2011.5.2128ElectromagneticFieldandWaveReview1.GeneralWaveBehaviorsalongUniformGuidingStructures29Review1.GeneralWaveBehavior303Maintopic1.RectangularWaveguides

31Maintopic1.RectangularWaveg1.RectangularWaveguides

Selecttherectangularcoordinatesystemandletthebroadsidebeplacedalongthe

x-axis,thenarrowsidealongthe

y-axis,andthepropagatingdirectionbealongthez-axis.azyxb,ForTMwaves,Hz=0

,andaccordingtothe

methodoflongitudinalfields,thecomponent

Ez

should

first

besolved,andfromwhichtheothercomponentscanbederived.

The

z-componentoftheelectricfieldintensitycanbewrittenas1.1TMwaves321.RectangularWaveguidesItsatisfiesthefollowingscalarHelmholtzequation,i.e.Inordertosolvetheaboveequation,themethodof

separationofvariables

isused.LetWeobtainwhereX"

denotes

thesecond

derivativeof

X

withrespectto

x,and

Y"

denotes

thesecond

derivativeof

Y

withrespectto

y.33ItsatisfiesthefollowingscaThe

onlyway

theequationcanbesatisfiedisthatbothtermsontheleftsideare

constants.

Nowlet

wherekx

and

ky

arecalledthe

separationconstants,andtheycanbefoundbyusingtheboundaryconditions.Obviously34Theonlywaytheequationcanwherealltheconstants

C1,C2,C3,C4,and

kx,ky,dependontheboundaryconditions.Thetwoequationsaresecondorderordinarydifferentialequations,andthegeneralsolutions,arerespectively

The

z-componentoftheelectricfieldintensitycanbewrittenasBoundaryconditions:azyxb,35wherealltheconstantsC1,C369Andallthefieldcomponentsarewhere37AndallthefieldcomponentsaThecutoffofaparticularmodeistheconditionthatmakesvanish.FortheTMmnmodethecutofffrequencyisAlternatively,wemaywriteWherecisthecutoffwavelength.

38Thecutoffofaparticularmod

1，電磁波的相位僅與變量z有關，而振幅與x,y有關。因此，在Z方向上為行波，在X及Y

方向上形成駐波。

2，z

等于常數的平面為波面。但振輻與x,y有關，因此上述TM波為非均勻的平面波；

3，當m

或n

為零時，上述各個分量均為零，因此m及n

應為非零的整數。m及n具有明顯的物理意義，m為寬壁上的半個駐波的數目，n為窄壁上半個駐波的數目。

4，由于m及n為多值，因此場結構均具有多種模式。m及n的每一種組合構成一種模式，以TMmn表示。例如TM11表示m=1,n=1的場結構，具有這種場結構的波稱為TM11波。5，數值大的m及n模式稱為高次模，數值小的稱為低次模。由于m及n均不為零，故矩形波導中TM波的最低模式是TM11波。391，電磁波的相位僅與變量z有關，而振幅與

Selecttherectangularcoordinatesystemandletthebroadsidebeplacedalongthe

x-axis,thenarrowsidealongthe

y-axis,andthepropagatingdirectionbealongthez-axis.azyxb,ForTEwaves,Ez=0

,andaccordingtothe

methodoflongitudinalfields,thecomponent

Hz

should

first

besolved,andfromwhichtheothercomponentscanbederived.

The

z-componentofthemagneticfieldintensitycanbewrittenas1.2TEwaves40SelecttherectangularItsatisfiesthefollowingscalarHelmholtzequation,i.e.Inordertosolvetheaboveequation,themethodof

separationofvariables

isused.LetWeobtainwhereX"

denotes

thesecond

derivativeof

X

withrespectto

x,and

Y"

denotes

thesecond

derivativeof

Y

withrespectto

y.41Itsatisfiesthefollowingsca

Similarly,wecanderiveallthecomponentsofa

TEwave

intherectangularwaveguide,asgivenbywhere,butbothshouldnotbezero

atthesametime.Boundaryconditions:42Similarly,wecanderivThecutoffofaparticularmodeistheconditionthatmakes

vanish.FortheTEmnmodethecutofffrequencyisAltermatively,wemaywriteWherecisthecutoffwavelength.

TEwavehasthe

multi-mode

characteristicsasthe

TM

wave.

The

lowest

ordermodeofTEwaveisthe

TE01

orTE10

wave.43ThecutoffofaparticularmodExample.Theinsideofarectangularmetalwaveguideisvacuum,andthecross-sectionis

25mm10mm.Whatmodes

canbetransmittedifanelectromagneticwaveoffrequencyentersthewaveguide?Willthemodesbechangedifthewaveguideisfilledwithaperfect

dielectric

ofrelativepermittivity?

Solution:

Duetotheinsideisvacuum,theoperatingwavelengthis

andthecutoffwavelengthis

Thenthecutoffwavelengthof

TE10

waveis,thatof

TE20

waveis,andthatof

TE01

waveis.Thecutoffwavelengthofthehighermodeswillbeevenshorter.Inviewofthis,only

TE10

wavecanbetransmittedinthiswaveguide.44Example.Theinsideof

Ifthewaveguideisfilledwitha

perfectdielectric

of,thentheoperatingwavelengthisHence,

TE10

and

TE20

wavescanbetransmitted,andsomeothermodes

TE01，TE30，TE11，TM11，TE21，TM21

canexist.45Ifthewaveguideisfill1.3TE10WaveinRectangularWaveguides

Let