chapter8 (2)

8/20/2019 Chapter8 (2)

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*Note: The information below can be referenced to: Carr, J., Practical Antenna

Handbook, Tab Books, Blue Ridge ummit, PA, !"#", $B%: &'#(&)'"*+&'(.

dminister, J., lectromagnetics -chaums /utline0, 1c2raw'Hill, %ew 3ork, %3,!""(, $B%: &'&+'&!#""('4.

Chapter 8 Waveguides

The microwa5e 6ortion of the radio s6ectrum co5ers fre7uencies from about "&& 1H8 to

(&& 2H8, with wa5elengths in free's6ace ranging from (( cm down to ! mm.Transmission lines are used at fre7uencies from dc to about 4& or )& 2H8, but an9thing

abo5e 4 2H8 onl9 short runs are 6ractical, because attenuation increases dramaticall9 as

fre7uenc9 increases. There are three t96es of losses in con5entional transmission lines:ohmic, dielectric, and radiation. The ohmic losses are caused b9 the current flowing in

the resistance of the conductors making u6 the transmission lines. The skin effect will

increase the resistance at higher fre7uencies therefore the losses tend to increase in the

microwa5e region. ;ielectric losses are caused b9 the electric field acting on the

molecules of the insulator and thus, will cause heating through molecular agitation.Radiation losses are the loss of energ9 as the electromagnetic wa5e 6ro6agates awa9 from

the surface of the transmission line conductor.<osses on long runs of commonl9 used coa=ial transmission line causes concern as low

as >&& 1H8. Because of the increased losses the 6ower handling ca6abilit9 decreases at

higher fre7uencies, therefore, at higher microwa5e fre7uencies, or where long runs makecoa= attenuation losses unacce6table, or where high 6ower le5els causes the coa= to

o5erheat, wa5eguides are used instead of the transmission lines.

This cha6ter will describe the 6ro6agation characteristics in a single conductortransmission lines referred to as wa5eguides. ?hat is a wa5eguide@ Consider the light

6i6e analog9 illustrated in igure #.&A. A flashlight ser5es as our r'f source, which

gi5en that light is also an electromagnetic wa5e is not all that unreasonable. The sourceradiates into free's6ace, and s6reads out as a function of distance. The intensit9 6er unitarea at the destination D a wall falls off as a function of distance -;0 according to the

in5erse s7uare law -!E;*0. %ow consider the transmission scheme in igure #.&B, the

light wa5e still 6ro6agates o5er a distance ;, but is now confined to the interior of amirrored 6i6e. Almost all of the energ9 cou6led to the in6ut end is deli5ered to the out6ut

end, where the intensit9 is 6racticall9 undiminished. The light 6i6e analog9 ma9 not be

the best wa9 to e=6lain the o6eration of wa5eguides, but rather a neat summar9 on asim6le le5el.

The internal walls of the wa5eguide are not mirrored surfaces, but instead electrical

conductors. 1ost wa5eguides are made of aluminum, brass, or co66er. ome

wa5eguides internal surfaces are electro6lated with either gold or sil5er to reduce ohmiclosses. The gold or sil5er ha5e lower resisti5ities than most other metals.

?a5eguides are hollow 6i6es, and ma9 ha5e either circular or rectangular cross sections.

Rectangular are, b9 far, the most common. These wa5eguides are used for highfre7uenc9 transmission in the gigahert8 -microwa5e0 range. The T1 mode cannot

6ro6agate in these single conductor transmission lines. /nl9 higher modes in the form of

trans5erse electric -T0 and trans5erse magnetic -T10 modes can 6ro6agate in thewa5eguide.

%otes b9: ;ebbie Prestridge !

8/20/2019 Chapter8 (2)


8/20/2019 Chapter8 (2)


The dimensions for the cross section are inside dimensions. igure #.!-a0 is a rectangular

wa5eguide shown in Cartesian coordinate s9stem igure #.!-b0 shows a circular or

c9lindrical wa5eguide of radius a in a c9lindrical coordinate s9stem.

The time de6endence e j t ω will be assumed for the electromagnetic field in the dielectric

core. The following e=6ressions for the field 5ector F -which stands for either E or H0,assuming the wa5e is 6ro6agating in the F8 direction.

Rectangular coordinates -=, 90 e'Ik8 where:

F x y F x y a F x y a F x y a F x y F x y a x x y y z z z z - , 0 - , 0 - , 0 - , 0 - , 0 - , 0= + + ≡ +Τ

C9lindrical coordinates F f r e jkz = −- , 0φ where:

F r F r a F r a F r a F r F r ar r z z r z - , 0 - , 0 - , 0 - , 0 - , 0 - , 0φ φ φ φ φ φ φ φ = + + ≡ +Τ

The wa5e will 6ro6agate without attenuation, because the dielectric is lossless - &0.

<et k *π λ

-in radEm0 be the wa5e number and is constrained to be real and 6ositi5e.

The reason for se6arating the field 5ector into a trans5erse 5ector com6onent FT and an

a=ial 5ector com6onent 8a8 is two'fold. The com6lete E K H fields in the wa5eguide

are known once either cartesian com6onent ;8 or H8 is known.

Transverse Components rom Axial Components

Assume a rectangular coordinate s9stem. 1a=wells e7uation 9ields three scalare7uations:

-!a0 ( ) ( ) ( )− = + j j k y x y

xω µ

∂

∂ Η Ε Ε

-!b0 ( ) ( ) ( )− = − − j j k x y x

z ω µ ∂

∂ Η Ε

Ε

-!c0 ( ) ( )− = − j x y z

y xω µ

∂

∂

∂

∂ Η

Ε Ε

1a=wells e7uation 9ields three additional scalar e7uations with &:

-*a0 ( ) ( ) ( )− = + j j k y

x y

xω ε ∂

∂ Ε Η

Η

-*b0 ( ) ( ) ( )− = − − j j k x y x

z ω ε

∂

∂ Ε Η

Η

-*c0 ( ) ( )− = − j x y z

y xω ε ∂

∂

∂

∂ Ε

Η Η

liminate H= between -!a0 and -*b0 and H9 between -!b0 and -*a0:

%otes b9: ;ebbie Prestridge (

8/20/2019 Chapter8 (2)


-(a0( ) ( )

Ε Η Η

y

c

z

c

z jk

k y

j

k x= − −* *

∂

∂

ω µ ∂

∂

-(b0( ) ( )

Ε Ε Η

x

c

z

c

z jk

k x

j

k y

= − −* *

∂

∂

ω µ ∂

∂ L- ( ) ( )k k

c

* * *≡ −ω µ ε 0 The 6arameter k c -also radEm0 functions as a critical wa5e number.

Example or ! ":

?hat is critical about the number k c@

or 6ro6agation through a lossless dielectric, the wa5e number k must be real, but

( )k k k k c o c= ∗ − = −ω µ ε * * * *

The wa5e number k o is of a uniform 6lane wa5e in the unbounded dielectric at the

gi5en M. Thus k c is a critical wa5e number in the sense that a guided wa5es same D fre7uenc9 twin must ha5e a wa5e number e=ceeding k c. tated otherwise, the

fre7uenc9 f of the guided wa5e must e=ceed the 7uantit9

( ) ( )u k where uo c o* !π µ ε , = is the wa5e 5elocit9 in the unbounded dielectric.

inall9, take -(b0 and -(a0 substitute into -*a0 and -*b0:

-(c0( ) ( )

Η Η Ε

y

c

z

c

z jk

k y

j

k x= − −* *

∂

∂

ω ε ∂

∂

-(d0( ) ( )

Η Η Ε

x

c

z

c

z jk

k x

j

k y= − −* *

∂

∂

ω ε ∂

∂

$t is 6ossible to force either 8 or H8 -but not both0 to 5anish identicall9. The non'

5anishing a=ial com6onent will determine all other com6onents 5ia e7uations -(0.

Example 8.#:

=6ress 1a=wells e7uations -!0 and -*0 in scalar form in c9lindrical coordinate

s9stem.

-!0 ( ) ( )( )∇ ∧ = +H Eσ ω ε j

-*0 ( ) ( )( )∇ ∧ = −E Hσ ω µ j

L%ote: or the curl in c9lindrical coordinates refer to D

( )∇∧ = −

+ −

+ ∗ −

A

! !

r z a

z r a

r r r a

z

r

r z r

z

∂

∂ φ

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ φ

φ

φ φ

Α Α Α ΑΑ

Α

%otes b9: ;ebbie Prestridge >

8/20/2019 Chapter8 (2)


7uation -!0 9ields - &0:

( ) ( ) jr

j k r

z

ω ε ∂

∂ φ φ Ε Η

Η = + ∗!

- 0 -i0

( ) ( ) j j k r r

z ω ε ∂ ∂ φ Ε Η Η = − ∗ −- 0 -ii0

( ) ( ) jr r r

r z ω ε

∂

∂

∂

∂ φ Ε

Η = −

! !-iii0

7uation -*0 9ields:

( ) ( )− = + ∗ jr

j k r

z

ω µ ∂

∂ φ φ Η Ε

Ε !

- 0 -i50

( ) ( )− = − ∗ −

j j k r r

z ω µ

∂

∂ φ Η Ε

Ε - 0

-50

( ) ( )− = − jr r r

r z ω µ

∂

∂

∂

∂ φ Η

Ε ! !-5i0

Example 8.$:

Nsing the e7uations of e=am6le #.!, find all c9lindrical field com6onents in termsof 8 and H8.

rom -i0 and -50, with k c as 6re5iousl9 defined,

( ) ( ) ( )Ε

Η Ε r

c

z

c

z j

k r

j k

k r

= − −ω µ ∂

∂ φ

∂

∂

* *

!-!0

rom -ii0 and -i50,

( ) ( ) ( )Η

Ε Η r

c

z

c

z j

k r

j k

k r = −

ω ε ∂

∂ φ

∂

∂ * *

!-*0

rom -!0 and -i0,

( ) ( ) ( )Η

Ε Η φ

ω ε ∂

∂

∂

∂ φ = − −

j

k r r

j k

k r c

z

c

z

* *

! !-(0

rom -!0 and -ii0,

( ) ( ) ( ) ( )Ε Ε Η φ

ω ε ∂ ∂ φ

ω µ ∂ ∂

= + j

k r

j

k r r c

z

c

z

* *

! ! ->0

%ropagation &odes in Waveguide

$n a wa5eguide a signal will 6ro6agate as an electromagnetic wa5e. 5en in a

transmission line the signal 6ro6agates as a wa5e because the current in motion down the

line gi5es rise to the electric and magnetic fields that beha5es as an electromagnetic field.

%otes b9: ;ebbie Prestridge 4

8/20/2019 Chapter8 (2)


The trans5erse electromagnetic -T10 field is the s6ecific t96e of field found in

transmission lines. ?e also know that the term trans5erse im6lies to things at right

angles to each other, so the electric and magnetic fields are 6er6endicular to the directionof tra5el. These right angle wa5es are said to be normal or orthogonal to the

direction of tra5el.

The boundar9 conditions that a66l9 to wa5eguides will not allow a T1 wa5e to 6ro6agate. Howe5er, the wa5e in the wa5eguide will 6ro6agate through air or inert gas

dielectric in a manner similar to free s6ace 6ro6agation, the 6henomena is bounded b9 the

walls of the wa5eguide and that im6lies certain conditions that must be met. The boundar9 conditions for wa5eguides are:

!. The electric field must be orthogonal to the conductor in order to e=ist at the

surface of that conductor.*. The magnetic field must not be orthogonal to the surface of the wa5eguide.

The wa5eguide has two different t96es of 6ro6agation modes to satisf9 these boundar9

conditions:

!. T D trans5erse electric -8 &0*. T1 D trans5erse magnetic -H8 &0

The trans5erse electric field re7uirement means that the 'field must be 6er6endicular tothe conductor wall of the wa5eguide. This re7uirement can be met with 6ro6er cou6ling

at the in6ut end of the wa5eguide. A 5erticall9 6olari8ed cou6ling radiator will 6ro5ide

the necessar9 trans5erse field.

/ne boundar9 condition will re7uire the magnetic -H0 field not to be orthogonal to theconductor surface. ince it is at right angles to the 'field, the re7uirement will be met.

The 6lanes that are formed b9 the H'field will be 6arallel to the direction of 6ro6agation

and to the surface.

Waveguide 'mpedan"es

or an9 trans5erse electromagnetic wa5e , the wa5e im6edance -in ohms0 is defined as

being a66ro=imatel9 e7ual to the ratio of the electric and magnetic fields, and con5erges

as a function of fre7uenc9 to the intrinsic im6edance of the dielectric:

η ≡E

H

Τ

Τ

->0

or a T mode wa5eguide, -!a0 K -!b0 im6l9:

E HΤΤΕ ΤΕ

ΤΕ Ε Η Η * * **

* *

*

*

= + = ∗

+

=

x y y x

k k

ω µ ω µ

%otes b9: ;ebbie Prestridge )

8/20/2019 Chapter8 (2)


/r η ω µ

ΤΕ ΤΕ

= ∗k

-40

7uation ->0 in5ol5es onl9 lengths of two'dimensional 5ectors, so O must be inde6endent

of the coordinate s9stem. =am6le #.( will confirm the 5alue of OT b9 recalculating it in

c9lindrical coordinates. =am6le #.> shows -using rectangular coordinates0 that:

( ) ( )η

ω ε ΤΜΤΜ=

k -)0

Example 8.(:

Calculate OT from the field com6onents in c9lindrical coordinates.

8 &, -i50 and -50 of =am6le #.! gi5es:

( )

( )− = + ∗ j r j k r

z

ω µ

∂

∂ φ φ Η

Ε

Ε

!

- 0 -i50

( ) ( )− = − ∗ − j j k r

r

z ω µ

∂

∂ φ Η Ε Ε

- 0 -50

( ) ( ) ( )H Τ

ΤΕ ΤΕ ΤΕ ΤΗ Η Ε Ε Ε = + =

+

=r r

k k k * *

*

*

*

* *

φ φ ω µ ω µ ω µ

( )η

ω µ ΤΕ

Τ

Τ ΤΕ ≡ =E

H k

Example 8.)

Calculate OT1 from the field of com6onents in rectangular coordinates.H8 &, -*a0 and -*b0:

-*a0 ( ) ( ) ( )− = + j j k y

x y

xω ε

∂

∂ Ε Η

Η

-*b0 ( ) ( ) ( )− = − − j j k x y x

z ω ε

∂

∂ Ε Η

Η

%otes b9: ;ebbie Prestridge +

8/20/2019 Chapter8 (2)


olution:( ) ( )

Ε Ε Η Η Ε Η ΤΜΤ

ΤΜΤ x y y x

k or

k * **

* * *+ =

+

=ω ε ω ε

( )η

ω ε ΤΕ Τ

Τ

ΤΜ≡ =E

H

k

+etermination o the Axial Fields

All that remains for a com6lete descri6tion of T and T1 modes is the

determination of the res6ecti5e a=ial fields:

8 H8 T

8 8 T1

The cartesian coordinate F e z

jkz − of F -in either rectangular or c9lindrical coordinates0,

must satisf9 the scalar wa5e e7uation∂

∂

∂

∂

*

* *

*

*

!F

F

z u t = ,

( ) ( ) ( ) ( )∇ = −− −* * F e F e z

jkz

z

jkz ω µ ε -+0

And the a66ro6riate boundar9 conditions which are inferred from the boundar9

conditions on the com6onents of FT. L%ote: Trans5erse com6onents such as Η φ e jkz −

are not cartesian com6onents and do not obe9 a scalar wa5e e7uation.

Expli"it olutions or TE &odes o a ,e"tangular -uide

The wa5e e7uation -+0 becomes:

∂

∂

∂

∂

*

*

*

*

* &H H

H z z

c z x y k + + =ΤΕ

This was 6re5iousl9 defined as ( ) ( )k k cΤΕ ΤΕ * * *= −ω µ ε . ol5e b9 using se6aration of

5ariables:

( ) ( )Η Α Β Α Β z x x x x y y y y x y k x k x k y k y- , 0 cos sin cos sin= + + -#0

where k k k x y c

* * *+ = ΤΕ . The se6aration constants k = and k 9 are determined b9 the boundar9

conditions. Consider first the ='conditionsΕ Ε y y y a y- , 0 - , 0& &= = in 5iew of -(a0 '

( ) ( )Ε

Η Η y

c

z

c

z jk

k y

j

k x= − −* *

∂

∂

ω µ ∂

∂ and 8 & these translate into:

∂

∂

∂

∂

Η Η z z

x x x x a= =

= =&

&

%otes b9: ;ebbie Prestridge #

8/20/2019 Chapter8 (2)


A66l9 these conditions to: -#0 '

( ) ( )Η Α Β Α Β z x x x x y y y y x y k x k x k y k y- , 0 cos sin cos sin= + +

This will result in B= & and ( )sin , , ,...k a or k m

a m x x= = =& & ! *

π and b9 s9mmetr9, the

boundar9 conditions in 9 force B9

& and ( )k

n

b n

y = =

π & ! *, , ,...

ach 6air of nonnegati5e integers -m, n0 Dwith the e=ce6tion of -&, &0 which will

result in a tri5ial solution'identifies a distinct T mode, indicated as Tmn. This mode has

the a=ial field

Η Η zmn mn x ym x

a

n y

b- , 0 cos cos=

π π -"0

And the trans5erse field is obtained through -(0 D -refer to 6ages *'( of these notes0. The

critical wa5e number for Tmn is:

k m

a

n

bc mnΤΕ =

+

π π * *

This is in terms of which the wa5e number and the wa5e im6edance for Tmn are:

( ) ( )k k mn c mnΤΕ ΤΕ = −ω µ ε * * -!&0

( )

( ) ( )η

ω µ

ω µ ε ΤΕ

ΤΕ mn

c mnk =

−* * -!!0

L%ote: m, n are integers that define the number of half wa5elengths that will fit in the -a0

and -0 dimensions, res6ecti5el9 a, b are the wa5eguide dimensions. -see igure #.*0

Example 8./:This e=am6le will show for the T1mn modes of a rectangular wa5eguide and it will show

that k cT1mn k cTmn. The subscri6ts T and T1 can be dro66ed from all modal 6arameters

of rectangular guides sa5e the wa5e im6edance.

/btain the analogues of -"0 D -!*0 for T1mn.

Analogous to -#0,

( ) ( )Ε z x x x x y y y y x y C k x D k x C k y D k y- , 0 cos sin cos sin= + +

%otes b9: ;ebbie Prestridge

b

a

Figure 8.$ Rectangular wa5eguide -end 5iew0

"

8/20/2019 Chapter8 (2)


( ) ( )where k k k k x y c

* * * * *+ = ≡ −ΤΜ ΤΜω µ ε

But now the boundar9 conditions are:

Ε Ε z z y a y and - , 0 - , 0& &= = Ε Ε z z y x b- , 0 - , 0& &= =This will re7uire that:

C k

m

a C k

n

b x x y y= = = =& &

π π

where m n, , , , ....= ! * ( %ote that neither m nor n is 8ero in a T1 mode.The re7uired formulas are:

Ε Ε zmn mn x ym x

a

n y

b- , 0 sin sin=

π π -!0

k m x

a

n y

bk c cmn mnΤΜ ΤΕ =

+

=

π π * *

-*0

k k mn mnΤΜ ΤΕ = -(0

( )η

ω ε ΤΜΤΕ

mn

k = ->0

elo"it1 and Wavelength in Waveguides:

igure #.( illustrates the geometr9 for two wa5e com6onents sim6lified for sake of

illustration. There are three different wa5e 5elocities to consider with res6ect to

wa5eguides: free space velocity -c0, group velocity -Qg0, and phase velocity -Q 60.The s6ace 5elocit9 of 6ro6agation in unbounded free's6ace, i.e., the s6eed of light

-c ( L !&# mEs0.

The group velocity is the straight line 5elocit9 of 6ro6agation of the wa5e down the

center'line -8'a=is0 of the wa5eguides. The 5alue of Qg is alwa9s less than c, because theactual 6ath length taken as the wa5e bounces back and forth is longer than the straight

line 6ath -i.e., 6ath ABC is longer than 6ath AC0. The relationshi6 between c and Qg is:

Qg c sin a

L%ote: g is the grou6 5elocit9 in -mEs0, " is the free s6ace 5elocit9 -( L !&# mEs0, and a

is the angle of incidence in the wa5eguide.The 6hase 5elocit9 is the 5elocit9 of 6ro6agation of the s6ot on the wa5eguide wall where

the wa5e im6inges -e.g., 6oint B in igure #.>0. This 5elocit9 is actuall9 faster than

both the grou6 5elocit9 and the s6eed of light. The relationshi6 between the 6hase andgrou6 5elocities can be seen in the Beach analog9. $f we consider an ocean beach that

wa5es will arri5e from offshore at an angle other than "&, meaning the arri5ing wa5e

fronts will not be 6arallel to the shore. The arri5ing wa5es at Qg as it hits the shore willstrike a 6oint down the beach first, and the 6oint of strike races u6 the beach at a faster

%otes b9: ;ebbie Prestridge !&

8/20/2019 Chapter8 (2)


Antenna

a

b

c

F

This end is o6enS

6hase 5elocit9, Q 6, that is faster than Qg. $n a microwa5e wa5eguide the 6hase 5elocit9

can be greater than ".

&ode Cuto Fre2uen"ies

The 6ro6agation of signals in a wa5eguide de6ends in 6art u6on the o6erating fre7uenc9of the a66lied signal. The angle of incidence made b9 the 6lane wa5e to the wa5eguide

wall is a function of fre7uenc9. As the fre7uenc9 dro6s, the angle of incidence increases

towards "&.$n 6ractice one ma9 deal with fre7uencies and not wa5e numbers. $t is desirable to

re6lace the conce6t of the critical wa5e number -k c0 b9 one of the cutoff frequency - f c0.

This was accom6lished in the e=am6le for -k c0 -refer to 6age ( of these notes0:

( ) f

uk k c

o

c c= =*

!

*π π µ ε

-!(0

$n terms of the cutoff fre7uenc9 f c and the o6erating fre7uenc9 f f c= >

ω

π *

-!&0, -!!0, and -!*0 will become:


Qg

C

B

A aaλ

>

λ g

>

&

!!

Figure 8.( Antenna radiator in a ca66ed wa5eguide.

Figure 8.) ?a5e 6ro6agation in a wa5eguide

8/20/2019 Chapter8 (2)


f u m

a

n

bcmn

o=

+

*

* *

- Rectangular waveguide0 -!& bis0

k u

f f or f

f

cmn

o

c mn nn

o

c mn

= − =

−

*

!

* *

*

π λ

λ

-!! bis0

η η

ΤΕ mn

o

c mn f

f

=

−

!

*

-!* bis0

whereu

f o

oλ = is the wa5elength of an imaginar9 uniform 6lane wa5e at the o6erating

fre7uenc9 and where η µ

ε o = is the 6lane wa5e im6edance of the lossless

dielectric. The second form of -!! bis0 e=hibits the relation between the operating

wavelength o and the actual guide wa5elength mn. or T1mn wa5es, -!* bis0 is re6laced

b9 Usee -)0V

η η ΤΜmno

cmn f

f = −

!

*

-!>0

The 6hase 5elocit9 of a Tmn or T1mn wa5e is gi5en b9:

u f u

f

f

mn mn

cmn

=

−

λ &

*

!-!40

The meaning of cutoff is made 6articularl9 clear in -!40. As the o6erating fre7uenc9

dro6s to the cutoff fre7uenc9, the 5elocit9 becomes infinite. This is a characteristic, not

of wa5e 6ro6agation, but of diffusion -instantaneous s6read of e=6onentiall9 small

disturbances0.

Example 8.3

+eine the notion of cutoff wavelength.

The cutoff wa5elength c is the wa5elength of an unguided 6lane wa5e whose

fre7uenc9 is the cutoff fre7uenc9 i.e., cL f c uo

$s the cutoff wa5elength an u66er limit on the guide wa5elength, Iust as the cutoff

fre7uenc9 is a lower limit on the guide fre7uenc9@

%otes b9: ;ebbie Prestridge !*

8/20/2019 Chapter8 (2)


%o in fact, the formula λ mn

o

cmn

u

f f =

−* * shows that an -m, n0 mode can 6ro6agate

with an9 guide wa5elength greater than .

+ominant &ode

The dominant mode of an9 wa5eguide is that of the lowest cutoff fre7uenc9. %ow, for a

rectangular guide, the coordinate s9stem ma9 alwa9s be oriented to make a W b.

ince f m

a

n

bcmn

=

+

* *

for either T or T1, but neither m nor n can 5anish in T1, the

dominant mode of a rectangular guide is in5ariabl9 T!&, with

( ) f

u

a a k

u f c

o o

oo

o!& !& *!&

!& !& !&

!&

* ! *

*= =

−≡ = =λ

λ

λ

π λ η

λ

λ

η X

rom -"0, 8!& &, and the e7uations -Trans5erse Com6onents from A=ial

Com6onents ection0:

Η Η Ε z x

x

a!& !& !& &= =cosπ

Η Η Ε Η Η x y x oo

ja x

a j

a x

a!&

!&

!& !& !& !& !&

* *=

= − = −

λ

π η η

λ

π sin sin -!)0

Η y!& &=

or H!& real, the three non8ero field com6onents ha5e the time'domain e=6ressions:

( )Η Η z

x

a t k z !& !& !&=

−cos cos

π ω

( )Η Η x

a x

at k z !&

!&

!& !&

*= −

−

λ

π ω sin sin -!+0

( )Η Η y o

a x

a

t k z !&

!&

!& !&

*=

−η

λ

π ω sin sin

Plots of the dominant'mode fields -!+0 at t & are gi5en in igs. #.4 and #.). BothΕ Η y xand 5ar9 as ( )sin π x a . This is indicated in igure #.4 b9 drawing the lines of

E close together near = aE* and far a6art near = & and = a. The lines of H are showne5enl9 s6aced because there is no 5ariation with 9. This same line'densit9 con5ention is

used to indicated the local 5alue of E = Ε y in igure #.)-a0 and of

%otes b9: ;ebbie Prestridge !(

8/20/2019 Chapter8 (2)


H = +Η Η x z

* *

in igure #.)-b0. %otice that the lines of H are closed cur5es -di5 H &0 the H field ma9

be considered as circulating about the 6er6endicular dis6lacement current densit9 4;.


&

b

9

=

Figure 8./ Trans5erse cross section

a

9

b

&a

H

9

b

& λ !&

>

λ !&

*

(

>

!&λ '8

H

'8

=

a

&

(

>

!&λ λ !&

*

λ !&

>

igure #.) <ongitudinal cross sections

-a0 Y = aE*-b0 Y 9 const

!>

=

8/20/2019 Chapter8 (2)


%otes b9: ;ebbie Prestridge !4

chapter8 (2)

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