Time-Harmonic Electromagnetic Fields (IEEE Press Series on Electromagnetic Wave Theory)

IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY

The IEEE Press Series on Elcctromagnetic Wave Theory consists of new lilIes as well as reprints and revisions of recognized classics that maintain long-term archival significance in ela:tromagnelic waves and applications. Series Edilor Donald G. Dudley

Universit)' ojArizona Advisory Board

Robef1 Il Collin

Case Western Re;fe/w Uniw!rsity Akira lshimaru Unil'ersio' of Washing/on

D. S. Jones UnillfrsiIYQ!D,llldee

Associatt Edilors ElECTROM ...ONnIC THEORV, ScATIEIUNC. ANI) DWFIlACTION

Ehud Heyman

INTEGRAL EQUATION MF.THODS

Donald R. Wilton University of Hou£tol1

Tel-Aviv Ulliw!r$ily DIFFEREr-'TIAL EQUATIO:-" METHOOS

ANTENNAS. PROl'AGATION. AND MI(:ROWAVIOS

Andreas C. Cangellaris

David R. Jackson

University ofAri:ona

Unil'ersity ofHouslOn

BOOKS IN THE IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY

Christopoulos. C. The Transmission-Une MQdeling Me/hads: TLM Clcmmow. P. c.. The Plane WaveSpeclrnm Represemmion o[£leclromagnetic Field~' Collin. R. B.. FieM Theoryo[Guided Waves. Second Edition Collin. R. E.. Foundations [Qr MicrQwave ElJgilJeering Dudley, D. G.. Mathematical FQundatiollS[or Elec/romagne/ic Thl'oJy Elliot, R. S., Electromagnetics: lIis/Qty. TheQry. and Applicll/iQns Felscn. L. 11.. and Marcuvit"z. N.• Radiation and Scallerillg o[Wal-'t's Harrington. R. F.• Field ComplllariQII by MOIllI'Il/ Metho/-' :JC·d1--+i dt

Similarly, the equation of continuity in mixed field and circuit form is

4.1J.ds = _ 'Jr

dq dt

(1-7)

Finally. the various equations can be written entirely in terms of circuit quantities. For this, we shall use the notation that :z denotes summation over a closed contour for a line-integral quantity, and summation over a closed surface for a surface-integral quantity. cuit forms of Eqs. (1-6) are

1>--~~

1>

=

In this notation, the cir-

(1-8)

ddt' +i

and the circuit form of Eq. (1-7) is (1-9)

Note that the first of Eqs. (1~8) is a generalized form of Kirchhoff's voltage law, and Eq. (1-9) is a generalized form of Kirchhoff's current law. It is apparent from the preceding summary that many mathematical forms can be used to present a single physical concept. An understanding of the concepts is an invaluable aid to remembering the equations. While an extensive exposition of these concepts properly belongs in an introductory textbook, let us here summarize them. Consider the sets 01 Eqs. (1-1), (1-3), (1-6), and (1-8). The first equation in each sct i. essentially Faraday's law of induction. It states that a changing magnetic flux induces a voltage in a path surrounding it. The second equation in each set is essentially Ampere's circuital law, extended to the time-varying case. It is a partial definition of magnetic intensity and magnetomotive force. The third equation of each set states that magnetic flux haa no "flux source," that is, lines of l of a material in the time-harmonic case. 1-4. The Generalized Current Concept. It was Maxwell who first noted that AmpAre's law for statics, V X X ... :1, was incomplete for time-varying fields. He amended the law to include an elutr* di.3plaament currem O'D/at in addition to the conduction current. He visualized this displacement current in free space as a motion of bound charge in an Uether/' an ideal weightless Buid permeating all space. We have since discarded the concept of an et.her for it bas proved undetectable and even somewhat illogical in view of the theory of relativity. In dielectrics, part of the term aD/at is a motion of the bound particles and is thus a current in the true sense of the word. However. it is convenient to consider the entire O'D/at term as a current. In view of the symmetry of Maxwell's equations, it also is convenient to consider the term (}(B/at as a magnetic di8ploament current. Finally, to represent sources, we amend the field equations to include impre88ed currents, electric and magnetic. These are the currents we view as the cause of t.he field. We shall see in the next section that the impressed currents represent energy sources. The symbols .g and ml will be used to denote electric and ma.gnetic currents in genera.l with superscripts indicating the type of current. As discussed above we define total currents j

j

j

j

j

.!l' - a:D at ol.

~

wC) the current and voltage are almost in phase, and the power is principally dissipative. The element in this case could be classified as a resistor. The angle between Ie and I is called the loss angle 8, as shown in Fig. 1-13c. Let US idealize the problem to a capacitor with perfectly conducting plates. Furthermore. we shall approximate the field by

V

E=(f

I

J=A

1 We are using t.he convention P - VI-. Some authors define P .. IV-, in which case the lign of react.ive power is opposite to that. which we get.

31

FUNDAMENTAL CONCEPTS

where A is the area. of the plates and d is their separation. The constitutive relationship for the field between the capacitor plates is

J - fiE

~~

(.

+ wi' + i",,')E

Substituting for E and J from the preceding

where we have taken 4 = u. equations, we have

I =

~

&-0

V = (u

+ we" + j~') ~

V

A eompnrison of this with Eqs. (1-83) shows that

Y -

A

fI d

G - (.

A

A C - l d

+ wi') d

Thus, for our idealized circuit element, the admittance is proportional to the admittivity of the matter between the plates. The equivalency of l
Re (8

+E

X Hel"-I)

Why is 5 not related to S by Eq. (1-41)7 1-11. Consider the unit cube shown in Fig. 1-16 which has sl1 sides except the face z - 0 covered by perfect conductors. If B• .. 100 sin (ry) and 11. - e/r'l sin (ry)

z 11"'-- fll and jj = jja. The intrinsic phase velocity and wavelength in a dielectric arc also less than those of free space. In the high·loss case (see Table 2-1), we have

k'=~ k" =

1,1

~

~w;o. fo;j,

in good conductors (, - Re (P,) - Re

(P.).-~·

The fate of decrease in ~I versus z equals the time-average power dissipated per unit length (1)01, or

<J>" ... - d<J>, = 2a(f>,

d,

Thus, the attenuation constant is given by

.--

IJ>, 21J>,

(2-76)

While this equation is exact if d'>" and ~I are determined exactly, its greatest use lies in approximating a by approximating ".), the mode propagates. At frequencies less than f. (wavelengths greater than A~), the mode is nonpropagating. A knowledge of /~ or A~ is equivalent to a knowledge of k~i so they also arc eigenvalues. In particula.r, from Eqs. (2-79), (2-81), and (2-82), it is eviden t that (2-83)

Using the last equality and k as

CI

27:/ W in Eq. (2-80), we can express 'Y

I> I. (2-84)

1 :: 2b. Thus, wave propagation can take place in a rectangular waveguide only when its widest side is greater than a half-wavelength.! A sketch oC the instantaneous field lines at some instant is called a mode pattern. The mode pattern of the TEo! mode in the propagating state is shown in Fig. 2-17. This figure is obtained by determining E and JC from the E and H of Eqs. (2-87) and specializing the result to some instant of time. AB time progresses, the mode pattern moves in the z direction. It is admittedly confusing to learn that many modes exist on a given guiding system. It is not, however, so bad as it seems at first. If only one mode propagates in a. waveguide, this will be the only mode of appreciable magnitude except near sources or discontinuities. The rectangular waveguide is usually operated so that only the TE ol mode propagates. This is therefore the only wave of significant amplitude along the guide except near sources and discontinuities. Because of the importance of the TED I mode, let us consider it in a little more detail. Table 2-4 specializes our preceding equations to this mode and includes some additional parameters which we shall now consider. The power transmitted along the waveguide can be found by integrating the axial component of the Poynting vector over a guide cross section. This gives

PI'"

foG

fob

E~H: dx dy

=

IEol! 2i:

which, above cutoff, is real and is therefore the time-average power transmitted. Below cutoff, the power is imaginary, indicating no time-average I We are referring to the intrinaic wlLvelengt,h of the dieleet,rie filling the waveguide, which is usually free space.

71

INTRODUCI'ION TO WAVES

TABi..E 2-4. SUMMARY OF WAVEOUIDE PARAMETERS FOR THE DO),[INANT MODE (TE ol ) IN A RECTANGULAR WAVEOUIDE

E~ ... Eo sin TY e-"f'

Complex field

Cutoff frequency Cutoff wavelength

Propagation constant

b Eo . TY 1 H - -8m -e ' v Zo b Eo I. TJI H ---cos-e"'" • ;" I b

I. -

1

V III

2b

).., - 2b

{j~ "1'-

-

jkVI 2...

a - -

'.

Characteristic impedallce

Guide wavelength

Guide phase velocity

Power transmitted

Attenuation due to lossy dielectric

Attenuation due to imperfect conductor

V1

u.;n'

f >1.

(fll.)!

I 1. I
p _ IEoltab

2Z.

a. -

2"VI

WI"

a. - a"

(f
power transmitted. (The preceding equation applies only at z = 0 below cutoff unless the factor cta. is added.) It is also interesting to note that the time-average electric and magnetic energies per unit length of guide are equal above cutoff (see Prob. 2-32). In contrast to the transmission·line mode, there is no unique volta.ge and current associated with a waveguide mode. However, the amplitude of a modal traveling wave (Eo in Table 2-4) enters into waveguide reflection problems in the same manner as V in transmission-line problem.s.

72

TIME-HARMONIC ELECTROMAGNETIC FIELDS

To emphasize this correspondence, it is common to define a mode voltage V and a. mode current] such that

Zo =

v

T

P = VI-

,,

(2-90)

From Table 2-4, it is evident that

V

=

EO~e-T.

(2-91)

satisfy this definition. Remember that we are dealing with only 0. +z traveling wave. In the -z traveling wave, I = - V /Zo. When waves in both directions are present, the ratio V/ I is a. function of z. Other definitions of mode voltage, mode current, and characteristic impedance can be found in the litera.ture. These alternative definitions will always be proportional to our definitions (see Prob. 2-34). Our treatment has 80 far been confined to the ideal loss-free guide. When losses are present in the dielectric but not in the conductor, all our equations still apply, except that most parameters become complex, There is no longer a real cutoff frequency, for 'Y never goes to zero. Also, the characteristic impedance is complex at all frequencies, The behavior of'Y = a + jfJ in the low-loss case is sketched in Fig. 2-18. The behavior of 'Y for the loss-free case is shown dashed, The most important effect of dissipation is the existence of an attenuation constant at all frequencies. In the low-loss case, we can continue to use the relationship

provided f is Dot too close to f~,

Letting k = k' - jk" and referring to

Flo. 2--18. Propagation const.a.nt (or & l066y waveguide (loss-free. case shown dashed).

•

o

f

73

lNTRonVC'I'lON TO WAVES

Table 2-1, we find (2-92)

This is t!le attenuation constant due to a lossy dielectric in the guide. Even more important is the attenuation due to imperfectly conducting guide walls. Our solution is no Jonger exact in this case, because the boundary conditions are cbarrged. The tangential component of E is now not quite zero at the conductor. However, for good conductors. the tangential component of E is very small, and the field is only slightly changed, or "perturbed," from the loss-free solution. The loss-free solution is used to approximate H at the conductor, and Eq. (2-42) is used to approximate the power dissipated in the conductor. Such a procedure is called a perturbational mdJwd (see Chap. 7). The power per unit length dissipated in the wall y = 0 is

lP,

L. - (!l f." IH.I' =

C(;ialized to the uniform plane traveling wave are all intrinsic parameters. This is, by definition, the meaning of the word "intrinsic."

88

TIME-HARMONIC ELECTROr..fAGNETIC FIELDS

PROBLEMS

2-1. Show that E. - Er[/b satisfies Eq. (2-6) but not Eq. (Z-5). Show tha.t it does not Hlllisfy Eq. (2-3). This ill nol II. possible c.1ectromagnctic field. 2-2. Derive the "wave equatioDs" for inhomogeneous media VX(f-1VXE)+tiE-O V X (f)-IV X B) !H .,. 0

+

Are these valid for nonisotropie mcdia1 Do Eqs. (2-5) hold for inhomogeneous mediar 2-3. Show that for any lossless nonmagnetic dielectric

,. ,- -v.;

, ,,-v.; where

t. i$ the dielectric constant and k o, '10, ),,0, and.c are the intrinsic parameters of vacuum. 2·4. Show that the quantities of Eqa. (2-J 8) satisfy Eq. (1.-35). &peat {orEqa. (2-21), (2-27), and (2-20). 2-15. For the field of Eqs. (2.20), show tha.t the velocity of propagation of energy 8.9 defined by Eq. (2-19) is

,

_ ~

II. -

v,"

1

sin 2kt sin 2wt < _,_ cos 2h: cos ~ - V;

2-6. For the field of Eqs. (2-22), show that the phase velocity i.e 1

tI" - . . . ; ; ;

2-7. For the field of Eqs.

(A+C A-C.) A _ C cost kz + A + C kz Sln

(2~28),

1

show that the z-direeted wave impedances are

Would you e;\:pect Z~. + - Z.~ + to be true for al1ll....(; fields? 2-8. Given a uniform plane wave traveling in the +z direction, show that the wave is circularly polarized if

E. -

E.

- ±.,

being right-handed if the ratio is +j and left-handcd if thc ratio is -j. 2-9. Show that the uniform plane traveling wave of Eq. (2-25) can be expressed as the sum of a right-hand circularly polarized wave and a left-hand circularly polarized wave. 2-10. Show that the uniform plane traveling wave of Eq. (2-25) can be expressed as

E - (El

+ jE1)C /h

89

n."Tl\ODOCl'JON TO WA YES

where B l and E t are real vecton lying in the %1J plane. ReJate E 1 and E z to A and B. 2-11. Show that the tip of the arrow representing t for 80 arbit.rary complex E i 1m (E) and usc the rcsulta traccs out an elliptl6 in space. [Hint: let E - Re (E) of Prob. 2.10.\ 2-HJ. For the frequencies 10, 100, and 1000 megacycles, determine k - k' - ik" and., _ lJI. + iOC for (a) polyst.yrene, Fig. 1-10, (b) Plexit!u. Fig. I-II, (c::) Fernunic A, Fig. 1-12, f. - 10, and (d) copper, " - 5.8 X 107• 2-1S. Show that. when all1088ell are of the maplctie type (" - e" - 0),

+

2-101. Show th8t for nonm8gnetic dielectrics

Q»I

where Q is defined by Eq, (1-79). i-Hi. Show that (or nonmagnetic conducton

.' ~ ~(I +~) ." ~ ";" (I -~) lJI. ;(1 +~)

Q

«1

h~(I-~) where Q is defined by Eq. (1-79). 2-16. Show that for metals

"'"'" lJI.(1

+11

I '--(1-" •

"'-1

.1

where Gl is the surface resistance, II is the skin depth, and" is the conductivit.y. 2-11. Derive the following formulu

(Jl

Gl (silver) Gl (copper) Gl (gold) (aluminum) Gt (braaa)

-

2.52 X 10-7 VI 2.61 X 10- 7 v1 3.12 X 10- 7 VI

3.26 X 10-' 5.01 X 10-7

where I is the frequency in eycles per lICCond.

Vi VI

90

TlllE-HAlWONlC ELECI'aOJolAONETIC FIELDS

2-18. Find t.he power per square meter dissipated in II. copper aheet if the rma magnetic intensity at it.e lIurfae6 is 1 ampere per meter at (a) 60 cyclCtl, (b) 1 megacycle, (6) 1000 mcgacyeletl. 2,.19. Make a sketch similar to Fig. 2-6 for Il. circularly polarized standing wave in dissipative media. Give a verbal deacription of 8 and :te. 2-20. Given a uniform plane wave normally incident. upon a plMC air-to-dielcctric interface, abow t.hat the standing-wave ratio is

V; - index of refraction

SWR -

wbere ... is the dielectric constant of the dielectric (ILSSUmed nonmaloetic and 100000free). 2-21. Take the index of refraction of water to be 9, and calculate the percentAge of power reflected and tranamitted wben a plane wave is normally incident on a calm lake. 2-22. Calculate the two polarizing angles (internal and external) and the critical angle for a plane interface between air and (a) water, f< - 81, (b) high-density gla8ll, f< - g, and (e) polystyrene, f. is an integer. 2-46. For an antenna of resonant length (Prob. 2-45), show that the radiation resistance referred to I .. is H. - 4:

Ie

+ log 2n...

- Ci(2nr)J

where 11. - 2L/". C - 0.5772, and Ci is as defined in Prob. 2-44. resistance for a loss-free antenna with feed point at :: - a). is

R(-.

Show that the input

R~

SIn 2r(a

+ n/4}

Specialize t.his result to L - >"/2, a - 0 (the half-wave dipole) and show that R, - 73 ohms.

CHAPTER

3

SOME THEOREMS AND CONCEPTS

3-1. The Source Concept. The complex field equations for linear media arc (3-1) -v X E - £H+M vxH-gE+J where J and M are sources in the most general sense. We have purposely omitted superscripts on J and M because their interpretations vary from problem to problem. In one problem, they might represent actual sources, in which case we would call them impressed currents. In another problem, J might represent a conduction current that we wish to keep separate from the 1]E term. In stln another problem, M might represent a magnetic polarization current that we wish to keep separate from the ~H term, and so all. We can think of J and M as If ma.the· matical sources/' regardless of their physica.l interpretation. For our first illustration, Jet us show how to represent 1/ circuit sources II in terms of the "field sources" J and M. The current source of circuit theory is defined as one whose current is independent of the load. In terms of field concepts it can be pictured as a short filament of impressed electric current in series with a perfectly conducting wire. This is shown in Fig. 3-la. That it has the characteristics of the current source of circuit theory can be demonstrated as follows. We make the usual circuit assumption that the displacement current through the surrounding medium is negligible. It then follows from the conservation of charge that the current in the leads is equal to the impressed current, independent of the load. The field formula. for power, Eq. (1-66), reduces to

---I

FlO. 3-1. Circuit sources in terms nf impressed currents. (a) Current Bouree; Cb) voltage

I

+ V

I'

K'~

D

source. Ca)

9'

(b)

•

+ V

96

TllLE-BAJUlONIC ,£LEcraoMAGNETIC nELDS

the circuit Cormula (artius source. p. - -

We have only electric currents; hence

III E'J"dT -

-I"

I E·d1- VI'

The "internal impedance" of the source is infinite, since 8. removal of the impressed current leaves an open circuit. The voltage &aurce of circuit theory is defined as onc whose voltage is independent of the load. In terms of field concepts it can be pictured as a small loop of impressed magnetic current encircling a perfectly conducting wire. This is illustrated by Fig. 3-lb. To show that it has the characteristics of the voltage source of circuit theory, we neglect displacement current and apply the field equa.tion K "'" -:J'E· d1 to a path

coincident with the wire and closing across the terminals. The E is zero in the wire; 80 the line integral is merely the terminal voltage, that is, K' "" - V. The impressed current, and therefore the terminal voltage, is independent of load. The field formula for power, Eq. (1-66), reduces in this case to p. = -

III H-· MfdT -

-K't H-·dl = VI-

which is tbe usual circuit formula.. The internal impedance of the source is zero, since a removal of the impressed current leaves a short circuit. We can use the circuit sources in field problems when the source and input region are of "circuit dimensions." that is, of dimensions small compared to a wavelength. Given a pair of terminals close together, we can apply the current source of Fig. a-la, that is, 8 short filament of impressed electric currcnt. Given a conductor of 5Inall cross section, we can apply the voltage source of Fig. a-Ib, that is, a small loop of impressed magnetic current. As an example of the use of a circuit source, consider the linear antenna of Fig. 2-23. The geometry of the physical antenna is two sections of wire separated by a small gap at the input. To excite the antenna, we can place a current source (a short filament of electric current) across the gap, which causes a current in the antenna wire. An exact solution to the problem involves a determination of the resulting current in the wire. This is difficult to do. Instead, we approximate the current in the wire, drawing on qualitative and experimental knowledge. We then use this current, plus the current source across the gap, in the potential integral formula to give us an approximation to the field. We shall find much use for the concept of current sheets, considered in Sec. 1-14. As an example, suppose we have a J. over the cross section of a rectangular waveguide, as shown in Fig. 3-2. Furthermore, we postulate that this current should produce only the TEol waveguide mode,

97

SOME THEOREMS AND CONCEI"I"a

x

•

/1

J.

/L __

+ 1,-

/'

/: z

/

I

// /

I ,I ,L __

,-

FlO. 3-2. A sheet of current in a rectangular waveguide.

which propagates outward from the current sheet. Table 2-4, we have the wave

E£+

=

A sin

1r: ci4~

H"+=~sin7rY_i6~ Zo b .

H~+

=

Abstracting from

z>O

f

~ cos 1r: e-i6•

where the constant A specifies the mode amplitude. The -z traveling wave is of the same form with {J replaced by -{J and Zo by -Zoo Thus,

E£-

=

B sin

i: eJt'.

B."1/.. H,- = - Zosmb(7~'

z 0, and Eq. (3-7) cannot be satisfied for any r. In this case, only the a solution vanishes as r --+ 00. It is therefore the dp..sired solution in loss-free media. 3-4. Image Theory. Problems for which the field in a given region of space is determined from a knowledge of the field over the boundary of the region are called boundary~value problema. The rectangular waveguide of Sec. 2-7 is an example of a boundary-value problem. We shall now consider a class of boundary-value problems for which the boundary surface is a perfectly conducting plane. The procedure is known as image theory. The boundary conditions at a perfect electric conductor are vanishing tangential components of E. An element of source plus an "image" element of source, radiating in free space, produce zero tangential components of E over the plane bisecting the line joining the two elements. According to uniqueness concepts, the solution to this problem is also the solution for a current element adjacent to a plane conductor. The necessary orientation and excitation of image elements is summarized by Fig. 3-5. Matter also can be imaged. For example, if a. conducting sphere is adjacent to the plane conductor in the original problem, then two conducting spheres at image points are necessary in the image problem. In other words, we must maintain symmetry in the image problem. The procedure also applies to magnetic conductors in a dual sense. The application of image theory in a-c fields is much more restricted than in d-c fields. It is exact only when the plane conductor is perfect. As an example of image theory, consider a current element normal to the ground (conducting) plane, as shown in Fig. 3-6a. This must produce the same field above the ground plane as do the two elements of Fig. 3-6b. Let us determine the radiation field. The radius vector from each current element is then parallel to that from the origin and given by ro=r-dcosO) r. = r + d cos 0

r»d

_II

where subscripts Q and i refer to original and image elements, respectively. The radiation field of a single clement is given by Eq. (2-114); so the radiation field of the two elements of Fig. 3-6b is the superposition

(e-fl: + -e-il:"). sm 8 r,

r • j II H. = -2A To

"" j Il ~r

rile.

cos(kd cos 8) sin 0

~

FlO. 3-5. A sum-

(3-10)

mary of image theory.

104

TIME-DARMONIC ELECTROMAGNETIC FIELDS

z

Z

r

e

'"

e

n

r;

r

II

n

(a)

FlO. 3..{t A current clement adjacent to a ground plane.

(6) (a) Original problem; (b)

image problem.

and E, = 7JH~. According to image theory, this must also be the solution to Fig. 3-6a above the ground plane. The problem of Fig. 3-6a represents the antenna system of a short dipole antenna adjacent to a ground plane. The total power radiated by the system is

~,

=

JJ B,B: ds

= 2'l1'"7J

fo·

n

11i.lt r t sin () dO

hemi_

.phere

where integration is over the large hemisphere z tuting from Eq. (3-10) and integrating, we have

I

Ill' [1:3 -

is', - 2""i: As kd - t

00.

cos 2kd (2kd)'

> 0,

r --+

2kd] + sin (2kd)'

00

Substi-

(3-l1)

the power radiated is equal to that radiated by an isolated

element [Eq. (2-116)]. As led --+ 0, the power radiated is double that radiated by an isolated clement. The gain of the antenna system over an omnidirectional radiator, according to Eq. (2-130), is g -

4.1rT 211I H40l t

is',

2 J = _'I-_---=C=OS'"2ki-. (maximum gain). Image theory also can be applied in certain problems involving more than one conducting plane. Two such cases are illustrated by Fig. 3-8. In the case of a conducting tube (Fig. 3-Sa), an infinite lattice of images is needed. In the case of a conducting wedge (Fig. 3-8b), a finite set of images results. Image theory can be used for conducting wedges when the wedge angle is 180o/n (n an integer).

3-5. The Equivalence Principle. Many source distributions outside a given region can produce the samo field inside the region. For example, the image current element of Fig. 3·6b produces the same field above the plane z = 0 as do the currents on the conductor of Fig. 3-6a. Two Rources producing the same field within a region of space are said to be equivalent within that region. When we are interested in the field in a given region of space, we do not need to know the actual sources. Equivalent sources will serve as well. A simple application of the equivalence principle is illustrated by Fig. 3-9. Let Fig. 3-9a represent a source (perhaps a transmitter and antenna) internal to S and free space external to S. We can set up a problem equivalent to the original problem external to S as follows. Let the original field exist external to 8, and the null field internal to S, with free space everywhere. This is shown in Fig. 3-9b. To support this field, there must exist surface currents J., M. on S according to Eqs. (1-86). These currents are therefore

J. -

n X H

M.

~

E Xn

(3-13)

where n points outward and E, H are the original fields over S. Since the currents act in unbounded free space, we can determine the field from them by Eqs. (3-4) and (3-5). From the uniqueness theorem, we know that the field so calculated will be the originally postulated field, that is, E, H external to S and zero internal to S. The final result of this procedure is a formula for E and R everywhere external to S in terms of the tangential componentS'" of E and H on S.

(r;)

Flo. 3·9. The equivalent currents original eources.

prod1J~

the 5&ffie field c.xlernal to S

lill

do the

107

SOME TIlEOREJd.S AND CONCEPTS

. ___ I\

E'H'

/--

..... ,

1-... \

s:...

(0)

!n

E',H'

I

--' /

(b)

\,; /~__.Jn (....... " \

\

E"H",

J,J.

-~

S' (0)

M,

Flo. 3-10. A general formulation of t.he cquivalcnce prhu::iple. (a) Original 4 problem; (b) original b problem; (e) equivalent. to a external to S and to b internal to S; (d) equivalent to b exwmal to Saud to a internal to S.

We were overly restrictive in specifying the null field internal to S in the preceding example. Any other field would serve as well, giving us infinitely many equivalent currents as far as the external region is eoncerned. This general formulation of the equivalence prineiple is represented by Fig. 3-10. We have two original problems consisting of currents in linear media, as shown in Fig. 3-1Oa and b. We can set up a problem equivalent to a external to S and equivalent to b internal to S as follows. External to S, we specify that the field, medium, and sources remain the Barne as in the a problem. Internal to S, we specify that the field, medium, and sources remain the same as in the b problem. To suP"' port this field, there must be surface currents J. and M. on S. According to Eqs. (1-86), these are given by

J.

= D X (Ho - H')

M. - (Eo - E') X D

(3-14)

where Ea, Ha is the field of the a problem and E', II' is the field of the b problem. This equivalent problem is shown in Fig. 3-1Oc. We can also set up a problem equivalent to b external to S and to (J internal to S in I\n Analogous manner, as shown in Fig. 3-1Od. In this case the necessary ~urface currents are the negative of Eqs. (3-14). Note that in each case we must keep the original sources and media. in the region for which we keep the field. Note also that we cannot use Eqs. (3-4) and (3-5) to

108

TIME-HARMONIC ELECTROMAONETIC FIEL06

!

E,B /

E,B

....; ; _......

{ I Sources \/

D

Zero field

\

Electric

J

"

S ......- - - (a)

_/

D

s

conductor

E,B

--~

D

.:ftI

M., - EXn (0)

(e)

Flo. 3-11. The field external to S is the Mme in (a), (b), and (e). Ca) Original problem; (b) magnetic current backed by all electric conductor; (e) eledric current backed by a magnetic conductor.

determine the field of the currents unless the equivalent currents radiate into an unbounded homogeneous region. Finally, note that the restricted form of the equivalence principle (Fig. 3-9) is the special case of the general form for which all a sources and matter lie inside S and all b sources are zero. So far, we have used the tangential components of both E and H in setting up our equivalent problems. From uniqueness concepta, we know that the tangential components of only E or H arc needed to de~rmine the field. We shall now show that. equivalent. problems can be found in terms of only magnetie eurrents (tangential E) or only elect-ric currents (tangenlial H). Consider a problem for which all sources lie within S, as shown in Fig. 3-110. We set up the equivalent problem of Fig. 3-11b as foUowa. Over S we place 0. perfect electric conductor, and on top of this we place a sheet of magnetic current MI' External to S we specify the same field and medium as in the original problem. Since the tangential components of E are zero on the conductor (just behind M.), and equal to the original field components just in front of M I , it follows from Eqs. (1-86) that

MI = E X n

(3-15)

We now have the same tangential components of E over S in both Fig. 3-11a and bj so according to our uniqueness theorem the field outside of S

must be the same in both cases. We can derive the alternative equivalent problem of Fig. 3-11c in an analogous manner. For this we need the perfect magnetic conductor, that is, a boundary of zero tangential components of H. We then find that the elcctric current sheet ]. -

D

X H

(3-16)

over a perfect magnetic conductor coveri.ng 8 produces the same field external to S 88 do the original sources. By now, the general philosophy of the equivalence principle should be

109

SOME THEOREMS AND CONCEPTS

appa.rent. It is based upon the one-to-one correspondence bet\"een fields and sources when uniqueness conditions are met. If we specify the field and matter everywhere in space, we can determine all sources. We derived our various equivalences in this manner. Considerable physical interpretation can be given to the equivalence principle. For example, in the problem of Fig. 3-9b, the field interna.l to S is zero. It therefore makes no difference what matter is within S as far as the field external to S is concerned. We have previously assumed that free space existed within S, so that the potential integral solution could be applied. We could just as well introduce a perfect electric conductor to back the current sheets of Fig. 3-9b. It can be shown by reciprocity (Sec. 3-8) that an electric current just in front of an electric current conductor produces no field. (We can think of the conductor as shorting out the current.) Therefore, the field is produced by the magnetic currents alone, in the presence of the electric conductor, which is Fig.3-11b. Alternatively, we could back the equivalent currents of Fig. 3-9b with a. perfect magnetic conductor and obtain the equivalent problem of Fig. 3-He. When matter is placed within S in Fig. 3-9b, the partial fields produced by J. alone and M. alone will change external to 8, but the total field must remain uncha.nged. Perhaps it would help us to understand the equivalence principle if we considered the analogous concept in circuit theory. Consider a source (active network) connected to a passive network, as shown in Fig. 3-12a. We can set up a problem equivalent to this as (ar as the passive network is concerned, as follows. The original source is switched off, leaving the source impeda.nce connected. A current source 1, equal to the terminal current in the original problem, is placed across the terminals. A voltage I

~

Source

tv

Passive network

Source Impedance

(0)

Passive network

Passive network (b)

t"''---_~_"'_tw_O_'_k_.J Passive

I

(d)

FIG. 3-12. A circuit theory analogue to thQ equivalence principle. (0) Original problem; (b) equivlilent sources; (c) source impedance replaced by a short circuit; (d) source impedance replaced by an open circuit.

uo

TIME-HARMON'IC ELECTROMAGNETIC FIELDS

source V, equal to the terminal voltage in the original problem, is placed in series with the interconnection. This is illustrated by Fig. 3-12b. It is evident from the usual circuit concepts that there is no excitation of the source impedance from these equivalent sources, whereas the excitation of the passive network is unchanged. Thus, Fig. 3-12b is the circuit analogue to Fig. 3-9b. Since there is no excitation of the source impedance in Fig. 3-12b, we may replace it by an arbitrary impedance without affecting the excitation of the passive network. This is analogous to the arbitrary placement of matter within S in the field equivalence of Fig. 3-9b. In particular, let the source impedance be replaced by a short circuit. This short-circuits the current source and leaves only the voltage source exciting the network (recall circuit theory superposition). Thus, the voltage source alone, as illustrated by Fig. 3-12c, produces the same excitation of the passive network as does the original source. This is analogous to the field problem of Fig. 3-lIb. Now consider the source impedance of Fig. 3-12b replaced by an open circuit. This leaves only the current source exciting the network, as shown in Fig. 3-12 0, as shown in Fig. 3·13a. An application of the equivalence concepts of Fig. 3-lIb yields the equivalent problem of Fig. 3-13b. This consists of the magnetic currents of Eq. (3-15) adjacent to an infinite

z-o

I I

E,H Sources and

matter

@

E.H

I I I

I I I I

E,H

Zero field

E.H

Ima~e

fie d

~

~

..," C

M. = El

(3-51)

where tP is given by Eq. (3-48). This is evidently the magnetic field of a current clement 11 = be. Hence, G t is a solution to r ,. r'

(3-52)

We now apply Eq. (3-46) with A = E and B = G t to the region enclosed by Sand, in Fig. 3-20. The result. is' 4.e . V' X E

~

1ft (G. X V X E -

E X V X G.) . ds

(3-53)

s This is a formula for Vi X E (bence for H) at r in terms of n X E and n X V X E on S. Equation (3-53) does not require E to be continuous on S, nor do we need to know n . E on S. Thus, Eq. (3-53) is a substantial improvement over Eq. (3-SO). In fact, Eq. (3-53) can be shown to be identical to the formula obtained from the equivalence principle of Fig. 3-9, applied to a homogeneous medium. Another useful Green's function is

G,=vxvxcq,

(3-54)

where ¢ is given by Eq. (3-48). This is proportional to the electric field of an elect.ric current elementi 80 G, also satisfies Eq. (3-52). An application of Eq. (3-46) would yield a formula for E at r' , similar in form to Eq. (3-53). All of t.he G's considered 80 far are tlfrce-space" Green's functions, that is, they.are fields of sources radiating into unbounded space. We can choose other G's such that t.hey satisfy boundary conditions on S. I J. R. Menber, "Scat.tering and Diffract.ion of Radio Waves," p. 14, P@tgamon Press, Kew York, 1955. I The left-hand aide of this equation is a function only of the primed coordinates. Bence, a prime is placed on v' to indicate operation on r' instead of r.

THEORE~IS

SOME

123

AND CONCEPTS

For example, let Gt = G2

+G

(3-55)

t'

such that G 4 satisfies Eq. (3-52) and n X V X G4

"'"

on S

0

(3-56)

The physical interpretation of Gt is that it is the magnetic field of a current element II = hc rad.iating in thc presence of a perfect electric conductor over S. The G 2 is the incident field, and the G t ' is the scattered field. Application of Eq. (3-46) with A = E and B = G 4 results in Eq. (3-53) with the last term zero, because of Eq. (3-56). Thus, be· v· X E ~

1ft (G. X V X E) . ds

(3-57)

s

which is a formula for V' X E in terms of only n X V X E over S. The same formula can be obtained from the equivalence principle of Fig. 3-11, as it applies to a homogeneous region. Similarly, defining a G~ such that on S

n X G6 = 0

(3-58)

we can obtain a formula 41fc • V' X E =

-1}

(E X V X G 6 )

•

ds

(3-59)

and so on. All these various formulas, and many more, can be directly obtained from the equivalence principle. We have discussed the Green's function approach mel'cly bec:luse it has been used extensively in the literature. 3-10. Tensor Green's Functions. We shall henceforth usc the term /lOreen's function" to meau "field of a point source." Suppose we have a current clement II at r' and we wish to evaluate the field E at r. The most general linear relationship between two vector quantities can be represented by a tensor. Hence, the field E is related to the source 11 by E

~

Ir]!l

(3-60)

where IfJ is called a tensor Green's function. and ma.trix notation, Eq. (3-60) becomes Elf [E.] E.

=

[r.. r.. I'lIz

1'1/11'

r ..]

I',l'

r az rill r..

In rectangular components

[II.] Il"

(3-61)

Il.

Thus, r,j is the ith component of E due to n unit j-directed electric current clement. The E might be the free-space field of 11, ill which case

124

TIME-HARMONIC ELECTROMAGNETIC FIELDS

[rJ would be the "free-space Green's function." Alternatively, E might be the field of 11 radiating in the presence of some matter, and [1') would then be called the uGrean's function subject to boundary conditions." Still other Green's functions are those relating H to 11, those relating E to Kl, and so on. Our principal use of tensor Green's functions will be for concise mathematical expression. For example, the equation (3-62)

where [r] is tbe free-space Green's function defined by Eq. (3-60), represents the solution of Eq. (2-111), which is

-jw,.A

E -

+ Jw, .,!.- v(v . A)

-Iff J.-'"'-'''

=--:;rdT , ':t'lrlf - r I

A -

(3-63)

Equation (3-62) also represents the field of currents in the vicinity of a material body if {rl represents the appropriate Green's function, and so on. In other words, Eq. (3-62) is symbolic of the solution, regardless of whether or not we can find [r]. Even though we shall not use tensor Grecn's functions to find explicit solutions, it should prove instructive to flnd an explicit IrJ. Let us take [rJ to be the free-space GreenJs function defined by Eq. (3-60). If II is z-directed, I Le-Jl:lr-r'l

and

A , -- 4;1, "I . A + 1 a'A, E '" = -Jwp.

'"

-.-

JWt

-a x'

E __1_ alA", jwe ayax

tI

E __1_ alA", I

jWf.

az ax

Comparing this with Eq. (3-61) for 1"" = Il. = OJ we sec that

r"""

=

a')

. 1 , f ( -JwJl+-·-JWf. aX

1 a 1/1 r "'" -jWf. ay'ax

r"" where

=

J- dZiJ"Ytax

JWt

e-Jl:1r-r'1

f -

4rI' _ "

(3-80)

This reduces Eqs. (3-78) to V2A V2F

+k A = 0 + k F '- 0 2

2

(3-81)

SolutioDs to these equations arc called wave potentials. Note that the rectangular components of the wave potentials satisfy the scalar wave equa.tion, or Helmholtz cquation, (J.il)

(c>/l')

+~v y

+ 'j1 V X

X

V

X

(c>/l') (3-91)

V

X (C~f)

where the y,'s are solutions to Eq. (3-82). We must therefore study solutions to the scalar Helmholtz equation to Jearn how to pick the y,'s. If the region is not source-Cree but is still homogeneous, our starting equations are -V X E = ZH + M (3-92) vxH=yE+J instead of Eqs. (3-77). General solutions to Eqs. (3-92) call be constructed as the sum oC any possible solution, called a particular solution, plus a solution to the source-free equations, called a complementary solution. We already have a particular solution, namely, the po~ential in~e­ gral solu~ion of Sec. 3-2. ThereCore, solu~ions in a homogeneous region containing sources are given by

E - E~ + E"

H ~ H•• + H"

(3-93)

where the particular solution (pa) is formed according to Eqs. (3-4) and (3-5), and the complementary solution (cs) is constructed according to Eqs. (3-91). We can think of the particular solution as the field due to

132

TIME-BARlIONIC ELECTROMAGNETIC FIELDS

sources inside the region and the complementary solution as t.he field due to sources outside the region. 3-13. The Radiation Field. It is easier to evaluate tbe radiation (distant) field from sources of finite extent than to evaluate the near field. (See, for example, Sees. 2-9 and 2-10.) In this section, we shall formalize the procedure for specializing solutions to the ra.d.iation zone. Consider a distribution of currents in the vicinity of the coordinate origin, immersed in a homogeneous region of infinite extent. The complete 80lution to the problem is represented by Eqs. (3-4) and (3-5). If we specialize to the radiation zone (r r:.....). as suggested by Fig. 3-22, we have Ir - ['1-+ r - r' cos t (3-94)

»

where t is tbe angle between rand r'. Furthermore, the second term of Eq. (3-94) can be neglected in the Umagnitudc factors/' Ir - r/l-I, of Eqs. (3-5). It cannot, however, be neglected in the flphase factors,1I exp (-jkjr - ell), unless r~ «"-. Thus, Eqs. (3-5) reduce to A

~ : ' JJJ J(r')&"-'dT'

F -

~ JJJ M(r')&"'-'dT'

(3-9.)

in the radiation zone. Not-e tha.t we now ha.ve the T dependence shown explicitly. Many or the opera.tions or Eqs. (3-4) can therefore be performed. Rather thnn blindly expanding Eqs. (3-4), let us draw upon some previous conclusions. In Sec. 2-9 it was shown that the distant field or an electric current clement was esscntinUy outward-traveling plane waves. The same is true of a ma.gnetic current element, by duality. Hence, the

z

To distant

POi"1

field r - ,-

Source r'

r

Fro. 3-22. Geometry for evaluating the radiation field.

y

x

133

SOME THEOREMS AND CONCEPTS

z

Flo.

3~23.

,

Conventional

tlounIiull.W oricutu.tioll.

y

y

x

-------

radiation zone must be characterized by (3-96)

since it is a superposition of the fields from many current elements. We can evaluate the partia.l H field due to J according to H' =- V X A (see Sec. 3-2). Retaining only the dominant terms (r 1 variation), we ha.ve H~ =- (V X A), = jkA. H; ~ (V x A). - -jkA.

with E' given by Eqs. (3-96). Simila.rly, for the partial E field due to M, we have, in the radiation zone, E;' ~ - (V X Flo - -jkF. E~ = -(v X F). =jkF,

with R" given by Eqs. (3-96). The total field is the sum of these partial fields, or E, = -jwj.LA, - jkF. (3-97) E. = -jwj.LA. + jkF, in the radiation zone, with H given by Eqs. (3-96). Thus, no differentiation of the vector potentials is necessary to obtain the radiation field. Also, for future reference, let us determine r' cos ~ as a function of the source coordinates. The three coordinate systems of primary interest are the rectangular, cylindrical, and spherical, as illustrated by Fig. 3-23. For the conventional orientation shown, we have the transformations

x=rsin8cost/J y = rain 8ain t/J z=rcos8 To obtain r' cos

~,

X=PCOSt/J

y=-paint/J z= z

(3-98)

we form

rr' cos ~ = r . r' = xx'

+ yy' + zz'

(3-99)

134

TWE-HARMONlC ELECTROMAGNETIC FiELDS

Substituting for

X,

r' cos

Y. z from the first set of Eqs. (3-98), we obtain ~ =

(x' cos tP

+ v' sin 4»

ein

(J

+ %' C08 8

(3-11lO)

which is the desired form when rectangular coordinates are chosen for the source. Substituting into Eq. (3-100) for x', Vi, z' from the second set of Eqs. (3-98), we obtain

r' cos

~ =

p'ain Seos (41 - q/)

+ z' cos (J

(3-101)

which is the desired form when cylindrical coordin:ltes are chosen for the source. Finally, substituting into Eq. (3-100) for x'. y', z' from the first set of Eqs. (3-98), we have

r

cos E = r'[cos 8 cos 0'

+. sin 8 sin 8' cos (I/>

- ,p')]

(3-102)

wruch is the desired form when spherical coordinates arc chosen for the source. PROBLEMS

S-1. Show that a current sheet

J-

u.J,

over the: - 0 plane produces the out.ward-traveling plane waves

HI< -

- ~"-'-

,> 0

_,

{ -Tel .,.,. 'u

J

roducea a field

,>0 , a and !loro field r < 4. 3-10. If E is well-behaved ill a homogenoous region bounded by S, and if fH - -v X &, show that the currente

J-

1

-~E-!VXVXE

will support this and only t.hi& field among a claas E, H having identieal lange.nti&! components of E on S. Show that the same H, but different H, can be obtained within this (ll&sfl if magnetic sources K are allowed in addition to J. 8-11. Suppose there exiBte within t.he rectangular cavity of Fig. 2-19 a field ,-rY'h E . - E ,Bln"bBln 'YJ:

where eAD be

V

(-rib)' ,tl and 1 is complelt (lOll8Y dielectric). Show that thia field supported by the lJOurce

'Y -

M, - -u.E,sin

'Z sinh

'YC

at the wall z - c. Show tbat for a low~loS6 dielect.ric, M. almost vanishes at the resonant frequency (Eq. {2-95)J, that iB, a small M. produces a large E. 3-U1. Consider a ~rected current element II a distance It in front of a ground plane covering the l' - 0 pla.ne, aa shown in Fig. 3-25. Show that the radiation field ia given by

E, _ -'Ill .-AI' ain 6 lin (kd sin • ain 6)

"

137

SOME THEOREMS AND CONCEPTS

and 1/110 _ H,. referred to / is

Find the power radiated and show that the radiation resist.a.nce R _ '1""11 [~ • A' 3

For d

_

sin 2kd _ cos 2A:d + sin 2kd] 2kd (2kd)l, (2kd)l

.s: k/4, the ffilH'imurn radiation is in the 1/ direction.

Show that

and that the gain is 7.5 for d small, 4.15 for d - A/4, and approximately 6 for d large.

z

,

• FlO. 3·25. Current element parallel to a ground plaoe.

II y

3-13. In Fig. 3-&, suppose we have a small loop of electric current. with ~irccted moment IS, instead of the current element. Show that the radiation field is given by '1 . (kd cos ' ) .' E • - ;'lj2...IS ~e-"8m Sln

and ?lB, _ -E•. referred to I is

Find tbe power radiated and show that the radiation resistance 2kd ('8)' [1"3 + cos(2kd)l -

R. - 2>01); For small d,

sin 2kd] (2kd)'

i'1""ISkd

. E. ~ - - - e-/" sm 28 u ..... O Air

sin 4>0) J 2,..,. k(a/2) (sin. sin fla) cos "'a

Show that the eeho area is the same as obtained in Prob. 3·20. 3-22. Usc rcdprocit.y to evaluat.e t.he radiation field of the dipole antenna of Sec. 2-10. To do this, place n 9-direeted current element at. large r, tIond apply Eq. (3-36), obtaining Eq. (2-125). 3-23. By applying voltage lIOurces t.o the network of Fig. 3-18. show that the admittance matril( luI defined by

_["" "u] [v.] [ I,] It 1111 lin VI satisfies the reciprocity relationship Un - ~u when Eq. (3-38) is valid.

141

SOME THEOREMS AND CONCEPTS

Flo. 3-30. lIClatt.ering.

Differential Obstacle

3·24. Let Fjg. 3-30 represent two antennas in the presence of an obst.acle.

Let

VI be the voltage received at antenna 1 when a unit current source is applied at

antenna 2 and V, be the voltage received at antenna 2 when & unit current source is applied at antenna 1. Let VI' and V,I be the corresponding voltages when the obstacle is absent. Define the scattered voltages as

and show that V,' _ V t·. 3·2li. For the problem of Fig. 3-2, define the input impedance of the sheet of current as

z- -(0,0) p where (0,0) is the self-reaction of the currents and I is the total current of the sheet. Evaluate Z when the field is given by Eqs. (3-3). 3-26. Repeat. Prob. 3-25 lor the current sheet and field of Prob. 3--1. 3-27. In the vector Green's t.heorem (Eq. (3-16) I, let A_E· and B - E~ in a homogeneous isot.ropie region, and show that it reduces to Eq. (3-35). 3-28. Use the vector identity v . (A X I• (4-26)

1 ...).1 (/.).. ("-)... TEll

FOR TIlE

RrCUNOULAlt

TE u TM II

TEll TM l1

TEn TM n

2.236 3.162 4.123

2.828

4

2.236 2.500 2.828

6

3.606

6.083

6.325

•

•

•

•

TE••

TEll

TM II

TE H

TEll

I 1 1 1 1

1 l.5

1.414 1.803 2.236 3.162

2 2 2 2 2

2 3

1

1.5 2 3

•

2 3

•

•

,

W.\VEQUlDE, b

3.606 4.472

>c -

TEll 3 3 3 3 3

b = 2 centimelers, then}... - 4 centimeters for the TEol mode, and we are operating welt above cutoff. The next modes to become propagating are the TE IO and TEn: modes, at. a frequency of 15,000 megacycles. The TEll and TM l l modes become propagating at 16,770 megacycles, and so on. The mode pa.tterns (field lines) a.re also of interest. For this, we determine E and H from Eqs. (3-86) and (4-19) or Eqs. (3-89) and (4-21), and then determine S, :JC from Eq. (1--41). The mode pattern is a plot of lines of t and 3C at some instant. (A more direct procedure for obtaining the mode patterns is considered in Sec. 8-1.) Figure 4-2 shows sketches of cross-sectional mode patterns for some of the lowCI'-'Oroer modes. When a line appears to end in space in these patterns, it actually loops down the guide. A more complete picture is shown for the TEo l mode in Fig. 2-17. In addition, each mode is chnrocterized by a constant (with respect to

(a)

TEo1

(c) TMIl

(b) TEn

~~' , '/

(d) TEo,

...

,,'-

..

",

.

'-' ' ... ~"r:f-

(e) TEI2

e

)Io.!J( -

- - - -;.,..

Flo. 4-2. Rectangular waveguide mode patterns.

152

TIME-HARMONIC ELEC"rROMAONETIC FIELDS

y) z-directcd wa.ve impedance. For the TE•• modes in loss-free media, we h.ve from Eq•. (3-89) .nd (4-21)

2:,

·H '"

]WIJ.

"'"

-1·k8~ -. - = -1·k-.0, P

8•

. H .. = -1·kI 8~ fJy == J·kE ••

JWIJ

The TEa. characteristic wave impedances are t.herefore

f > f. (4-27)

f < f. Similarly, for the TM•• modes, we h:lve from Eqa. (3-86) and (4-19)

j~.

-

-jk.: = jk.N,

jwtElI = -jk.

:t

= -jl.,.ll,.

Thus, the TM... characteristic wave impedances are

(Z) ~ 0 ••

E. H"

-E. 11.

k'-l~-.-

-------Wf

a JW'

f > f. (4-28)

f < f.

It is interesting to Dote t.hat the product (Zo}••T&(Zo}...n1 = 'It at all fr~ quencies. By Eq. (4-26), P < k for propagating: modesj so the TE characteristic wave impedances are always greater than 'I, and the TM characteristic wave impedances are always less than '1. For non propagating

modes, the TE characteristic impedances aro inductive, and the TM characteristic impedances are capacitive. Figure 4-3 illustrates this behavior. Attenuation of the higher-order modes due to dielectric losses is given by the same formula as for the dominant mode (see Table 2-4). Attenuation due to conductor losses is given in Prob. 44. 4-4. Alternative Mode Sets. The classification of waveguide modes into sets TE or TM to z is important because it applies also to guides of nonrectanguln.r cross section. However, for many rectangular waveguide problems, more convenient e1a.ssifications can be made. We now consider these alternative sets of modes. If, instead of Eq. (3-84), we choose (4-29)

153

PLANE WAVE FUNCTIONS

\ Zo

1\

t > t, ~ {Ro "jXo t < t,

\RoTE

"

, _Ko TAt

/

X,TE/

1/

/

"-

':;;;,TM

\

1/

/

o

3

2

I

FlO. 4-3. Chnracteristic impedance of wfl.vcguide modes.

we have an electromagnetic field given by a set of equations differing from Eqs. (3-86) by a cyclic interchange of x, y, z. To be specific, the field is given by H. - 0

II ~

,

a.a,

(4-30)

a;, Hr=.-• ay This field is 'I'M to x.

Similarly, if, instead of Eq. (3-87), we chooso (4-31)

we have an electromagnetic field given by

E __

,

E _

•

a.

ay

a.az

(4-32)

154

TUdE-HAR.YONIC ELECTROMAGNETIC FIELDS

This field is TE to x. According to the concepts of Sec. 3-12, aD arbitrary field can be C!'nstructed as a superpositioo of Eqs. (4-30) and (4-32). The choice of ¥'s to satisfy the boundary conditions for the rectangular waveguide (Fig. 2-16) is relatively simple. For modes TM to z (TMx•• modes) we have (4-33)

where m - 0, 1, 2, . . . ; n = 1, 2, 3, ' .. ; and k. is given by Eq. (4-26). The electromagnetic field is found by substituting Eq. (4-33) into Eqs. (4-30). For modes TE to x (TEx..... modes) we have (4-34) where m - 1,2,3, . . . ; n = 0, 1,2, . . . ; and k. is again given by Eq. (4-26). The field is obtsinOO by substituting Eq. (4-34) into Eqs. (4-32). Note that the TAu.. mOOes are the TEo. mOOes of Sec. 4-3, and the TEx. o modes are the TE.o modes. All other modes of Eqs. (4-33) and (4-34) are linear combinations of the degenerate sets of TE and TM modes. Note that our present set of modes have both an E. and H. (except for the O-order modes). Such modes are called hybrid. The mode patterns of these hybrid modes can be determined in the usual manner. (Determine E, H, then S, :JC, and specialize to some instant of time.) The TEa.. o mode patterns are those of the TE.. modes, and the TMxo.. mode patterns are those of the TEo. modes. Figure 4-4 shows the mode patterns for the TF.a: u and TMxll modes, to illustrate the character of the higher-order mode patterns. The characteristic impedances of the hybrid modes are also of interest. For the TM% modes, we have from Eqs. (4-30) and (4-33) H, - -jk,~

Hence, the z-directed wave impedances are

f > f. (4-35)

f ", this is the mode corresponding to the TJ\uol mode of the empty guide, whicb is also the TE u mode of the empty guide. For a given n, Eq. (4-45) has a. denumerably infinite set of solutions. We shall let m denote the order of these solutions, as follows. The mode with the lowest cutoff frequency is denoted by m = 0, the next modo by m = 1, and 50 on. Tbis numbering system is chosen so that the TMx... partially filled waveguide modes correspond to the TMx... empty-guide modes. The dominant mode of the partially filled guide is then the TMxol mode when b > a. lIenee, the propagation constant of the dominant mode is given by the lowest-ordcr solution to Eq. (4-45) when the k.'s a.re given by Eqs. (4-42) with n = 1. Fign.re 4-7 shows some calculations for the case E ",. 2.451'1). When k l is DOt very different from k l , we should expect k. 1 a.nd k.z to be small (k. is zero in an empty guide). If this is 50, then Eq. (4-45) can be approximated by

k..'d

_-:;k",,-,'(",0_d:=!)

1'1

1'1

--~

(4-48)

With this explicit relationship between kd and kd , we can solve Eqs. (4-42) simultaneously for k.r and k. (given Cal). Note that when kd is real, k.z is imaginary, and vice versa. The cutoff frequency is obtained hy oetting k, - 0 in Eqa. (4-42). Uaing Eq. (4-48), we have for the

161

PLANE WAVE FUNCTIONS

1.6

1.2

•

, ,

T.

1 t-- .b---l :«i

~

~ 0.8

/'

.-

/

/

0.4

o

0.1

0.2

0.3

0.5

0.4

0.6

0/).0

Fla. 4-7. Propagation constant for a rectangular waveguide pll.rtially filled with dielectric, • - 2.45-.. alb - 0.45, dla - O.SO. (Alta Prank.)

dominant mode kd

+ (i)' =

'

WlflJll

fl(~~d) k + (i)' "" zl

l

These we solve for the cutoff frequency r ColC"=-b

fl

valid when Eq. (4-48) applies.

l W ftJll

==

Col

flea d) (a d - ) fUll

When

J11

obtaining

We,

+ ftd

+

==

(4-49)

J

E2UfIJlI

J11

==

J1,

this reduces to (4-50)

Note that thi.s is the equation for resonance of a parallel-plate mission line, shorted at each end, and baving

L ==

J10

~

==

EIES

fl(a - d)

traDS~

+ Etd

per unit width. All cylindrical (cross section independent of z) waveguides at cutoff are tw b. The dominant mode of the empty guide is then the TEz u mode, or TE lo mode. The dominant mode of tbe pa.rtially filled guide will a.lso be a

162

TWE-HARMON1C ELECTROHAONETIC FIELDS

TEz modej SO the eigenvalues are found from Eq. (4-47) with n "'" O. We shall order the modes by m 88 followa. That with the lowest cutoff frequency is denoted by m = 1, that with the next lowest by m = 2, and 50 on. This numbering system corresponds to that (or the empty guide, the dominant mode being the TEx lO mode. When k 1 is not too different. from k" we might expect k Z1 and kd to be close to the empty-guide value k., = ria. An approximate solution to Eq. (4-47) could then be found by perturbing k. 1 and kd about 7/a. For the cutoff frequency of the

t:= d-----j T 1

"I

0

b

Zo -

'II

{j '""' kl

I

(2,1'2

I-d

I

Zo - '12 13 :: k2

I

1

"I'

(oj

I

0-

d----l

(bj

FIo. 4-8. (a) Partially filled waveguide; (b) tran.amission·line resonator. The cutoff frequency of t.he dominant. mode of (0) ia the reIIOnAot frequency of (b).

1.6

12

-

I-d-i

,

~l t--o--j

die

l.-?=' ~ ::..-V V ---

r7k::

fl; / . /V f/ '/1# ~ Iff

... o

~I 0.8

Ie

1

.8 .6 0.5 0.375 0.280 0.167 0

0.4

o

0.2

0.4

0.6 o{>..

0.8

1.0

FIo. 4-9. Propagation constant for a rectangular waveguide partially 6UOO with (After PraM.)

dielectric, • - 2.451..

163

PLANE WAVE FUNCTIONS

x Flo. 4-10. The dielectriclliab waveguide.

dominant mode, Eqs. (4-42) becomo k"'ll k l d

= kl~2 = wc2el~1

= k 2c I

. . We 2e 2/JI

and Eq. (4-47) becomes I

I

- cot k1cd = - - cot (k..(a -

",

",

dlJ

(4-51)

It is interesting to note that this is the equation for resonance of two shortcircuited transmission lines having Z.'s of '11 and 'Ill and P's of k 1c and k'k,

as illustrated by Fig. 4-8. The reason for this is, at eutoll, the TEz lo mode reduces to the parallel-plate transmission-line mode that propagates in the z direction. This viewpoint has been used extensively by Frank. 1 Some calculated propagation constants for the dominant mode are shown in Fig. 4-9 for the case «: - 2.45«:0. Similar results for a centered dielectric slab arc shown in Fig. 7·10, and the characteristic equation for that case is given in Prob. 4-19. 4.-7. The Dielectric-slab Guide. It is not necessary to have conductors for the guidance or localization of waves. Such phenomena also occur in inhomogeneous dielectrics. The simplest illustration of this is the guidance of waves by a dielectric slab. The so-called slab waveguide is illustrated by Fig. 4-10. We shall consider the problem to be two-dimensional, allowing no variation with the y coordinate. It is desired to find z-trllveling waves, that is, rj!~ variation. Modes TE and TM to either x or z can be found, and we shall choose the latter representation. For modes TM to 2, Eqs. (3-86) reduce to E = -k. '"

WE

a"ax

E. _

.,!... (k'

_ k.')"

H, _ - "'"

)we

ax

(4-52)

We shall consider separately the two cases: (1) .p an odd function of x, denoted by ~, a.nd (2) '" an even function of x, denoted by 1/t'. For case IN. H. Frank, Wave Guide Handbook, MIT Rad. L4b. Rept. 9,1942.

164

TIME-HAUMONIC ELECTROMAGNETIC FIELDS

(I), we choose in the dielectric region

[xl

< 2~

(4-53)

and in the air region

a

x>'2 x


2

JWEo

E. = .B v1e""e-ik,s ]WED

Continuity of E. and HI/at x =

± a/2 requires that

The ratio of the first equation to the second gives ~tan~ = ~va

2

2

EO

2

(4-56)

This, coupled with Eqs. (4-55), is the characteristic equation for determining k/s and eutoff frequencies of the odd TM modes,

165

PLANE WAVE FUNCTIONS

For TM modes which are even functions of

%,

we ehoose (4-57)

The separation parameter equations are still Eqs. (4-55). The field components are still given by Eqs. (4-52). In this case, matching E. and H. at :t - ± 0{2 yields ua

ua

E.,va

(4-58)

- -cot- = - 2 2 t6 2

This is the characteristic equation for determining the k.'s and cutoff frcquencies of the even TM modes. There is complete duality between the TM and TE modes of the slab waveguidej so the characteristic equations must be dual. For the TE modes with odd 1/t we have ua ua -tan-

2

2

Il4va

( 4-59 )

~--

PO

2

as the characteristic equation, and for the TE modes with even 'It we have _

~cot

2

ua = 2

1A1l~ lAO

(4-60)

2

as the characteristic equation. The u's and v's still s:l.tisfy Eqs. (4-55). The odd wave functions generating the TE modes are those of Eqs. (4-53) and (4-54), and the even wa.ve functions generating the TE modes are those of Eqs.. (4-57). The fields are, of course, obtained from the +'s by equations dual to Eqs.. (4-52), which are, explicitly, H _ - k, a~ •

WI!

ax

fI, _

.,.!.. (k' JWP

_ k,')~

E-~ • ax

(4-61)

These are specializations of Eqs. (3-89). The concept of cutoff frequency for dielectric waveguides is given a somewha.t different interpretation than for metal guides. Above the cutoff frequency, as we define it, the dielectric guide propagates a mode unattenuated (k. is real). Below the cutoff frequency, there is attenu~ atoo propa.gation (k. "'" fJ - ;0.). Since tbe dielectric is loss free, this ll.ttenuation must be accounted for by radiation of energy as the wave progresses. Dielectric guides operated in a radiating mode (below cutoff) are used as antennas. The phase constant of an unattenuatcd mode lies between tbe intrinsic phase constant of tbe dielectric and that of air; tbat is,

166

TUfE-BARMONIC ELECTROMAGNETIC FIELDS

This can be shown as follows. Equations (4-55) require that u and v be either real or imaginary when k. is real. The characteristic equations have solutions only when v is real. Furthermore, II must be positive, else the field will increase with distance from the slab [see Eqs. (4-54) or (4-57)J. When v is real and positive the characteristic equations have solutions only when 'U is also real. Hence, both u and II are real, and it follows from Eqs. (4-55) that ko < k. < ka. This result is So property of cylindrical dielectric waveguides in general. The lowest frequency for which unattenuated propagation exists is called the cutoff frequency. From the above discussion, it is evident that cutoff occurs a.s k. -. ko, in which case v -+ O. The cutoff frequencies are therefore obtained from the characteristic equations by setting u = ylkd' ko' and v = O. The result is

ko')

tan (; ylkd '

=

0

cot (~ylkd'

which apply to both TE and TM modes. when

kO') =

0

These equations are satisfied

n = 0, 1,2,

This we solve for the cutoff wavelengths

>.~

= 2a

n

IfdJld _ 1

'\I fO~O

n = 0, 1, 2,

(4-62)

and the cutoff frequencies

f~

n

= 2a

V fd~d

fO~O

n "'" 0, I, 2, . . .

(4-63)

The modes are ordered as TM.. and TE" according to the choice of n in Eqs. (4-62) and (4-63). Note thatf~ for the TEo and TM omodes is zero. In other words, the lowest-order 7'E and TM modes propagate unattenuated no maUer how thin the slab. This is a general property of cylindrical dielectric waveguides; the cutoff frequency of the dominant mode (or modes) is zero. However, as the slab becomes very thin, k. --+ ko and t1 --+ 0, so the field extends great distances from the slab. This characteristic is considered further in the next section. Finally, observe from Eq. (4-62) that when fdJld» EQJto, the cutoffs occur when the guide width is approximately an integral number of half-wavelengths in the dielectric, zero half~wavelengtb included. Simple graphical solutions of the characteristic equations exist to determine k. at any frequency above cutoff. Let us demonstrate this

167

PLANE WAVE FUNCTIONS

for the TE modes. ou l

Elimination of k. from Eqs. (4-55) gives

+ Vi =

kill -

k ol =

WI(E,iPd -

~llPll)

Using this relationship, we can write tho TE characteristic equations at'!

ua2 \ I(wa)' _.'ua cot ua - V"2 ('m ~uatan

,u." 2

,u." 2

,,,,,) -

(ua)' "2

2

Values of uaj2 for the various modes are the intersections of the plot of the left-hand terms with the circle specified by the right-hand term. Figure 4-11 shows a plot of the left-hand terms for Jld = ,u.ll. A representative plot of the right-hand term is shown dashed. As w or e" is varied, only the radius of the circle changes. (For the case shown, only three TE modes arc above cutoff.) If Pd F Po, the solid curves must be redrawn. The graphical solution for the TM mode eigenvalues is similar. Sketches of the mode patterns afe also of interest. Figure 4-12 shows the patterns of the TEo and TM 1 modes. These can also be interpreted as the mode patterns of the TM o and TEl modes if g and 3C are intercba.ngcd, fOf there is complete duality between the TE and TM cases.

r ••

~

f--

--

~

(UB) -,- ''" (UB)

"

.

[

I

I

I

I

I

/

I

I

/

I I

cot 2/~

,

I

"2

~2

~

1/

\

I I

/

,

I [

I',\\

/i

\

,I

,

. + + (U~.) (U~.)

Fio. 4-11. Grp.phical solution of the characteristic equation for the Blab waveguide.

168

TIME-HARMONIC ELECTRO?,IAGNETIC FIELDS

•

•

0

•

• .1 !t1

~~~

".1.

0 0 0

14

l\~t1 (ltZ; >t Q'\~r2:.i 0)8) jJ~~

~~'4 COCO~.. It)D1.1:\1 Q x

It

0

It

0

•

•

•

•

• I

•

•

•

•

• (b)

FlO. 4-12. Mode pfl.ttern.s for the dieJectric-6lab wll.Vcguidc. dashed); (b) TM 1 mode (t lines IlOlid).

(a) TEo mode (X lines

As the mode number increases, more loops appear within the dielectric,

but not in the air region. 4~8. Surface-guided Waves. We shall show that any "reactive boundary" will tend to produce wave guidance along that boundary. The wa.ve impedances normal to the dielectric-to-air interfaces of the slab guide of Fig. 4-10 can be shown to be reactive. A simple way of obtaining a single reactive surface is to coat a conductor with a dielectric layer. This is shown in Fig. 4-13. The modes of the dielectric-coated conductor arc those of the dielectric 0 plane. These are the TM.., slab having zero tangential E over the x n 0, 2, 4, . . . ,modes (odd if;) and the TE., n = 1,3,5, ...• modea =0

=0

169

PLANE WAVE FUNCTIONS

(even '/I) of the slab. We shall retain the same mode designations for the coated conductor. The characteristic equations for the 'I'M modes of the coated conductor are therefore Eq. (4-56) with 'a/2 replaced by t (coating tl},ickness). The characteristic equation for the TE modes is Eq. (4-60) with a/2 replaced by t. The cutoff frequencies are specified by Eq. (4-63), which, for the coated conductor, becomes

n

f. -

41

V"""

(4-64)

d

where

(4-67)

The wave impedance looking into the corrugated surface is Z

-.

,." E. = ju 1I, W~O

(4-68)

Note that this is inductively reactive; so to support such a field, the interface must be an inductively reactive surface. (The TE fields of Sec. 4-7 require a. capacitively reactive surface.) In the slots of the corrugation, we assume that the parallel-plate transmission-line mode

PLANE WAVE FUNCTIONS

171

exists. These are then shorkircuited transmission lines. of characteristic wave impedance 'l0. Hence, the input wave impedance is

Z_. - iv. tan k,.J For kod < T/2. this is inductively reactive. (4-69), we have • = k. tan k,.J

(4-69)

Equating Eqs. (4-68) and (4-70)

and, from Eq. (4-67), we have k, = k,

Vi + tan' k,.J

(4-71)

It should be pointed out that this solution is approximate, for we have only approximated the wave impedance at x "'" d. In the true solution, the fields must differ from those assumed in the vicinity of x = d. (We should expect E a to terminate on the edges of the ~th.) When the teeth are considered to be of finite width. an approximate solution can be obtained by replacing Eq. (4-69) by the average wave impedance. This is found by assuming Eq. (4-69) to hold over the gaps, and by assuming zero impedance over the region occupied by the teeth. The result is' k,

~ k. ~1 + G~ ,)' tan' k,./

where g = width of gaps and t := width of teeth. While at this time we lack the concepts for estimating the accuracy of the above solution, it has been found to be satisfactory for small kdl. Note that, from Eq. (4-70), the wave is loosely bound for very small kod, becoming more tightly bound as kod becomes larger (but still less than 11/2). Tho mode pattern of the w&ve is similar to that for the TM o coated-eonductor mode (Fig. 4-14), except in tho vicinity of the corrugations. 4-9. Modal Expansions of Fields. The modes existing in a waveguide depend upon the excitation of the guide. The nonpropaga.ting modes are of appreciable magnitude only in the vicinity of sources or discontinuities. Given the tangential components of E (or of H) over a waveguide cross section, we can determine the amplitudes of the various wave-guide modes. This we shall illustrate for the rectangular waveguide. Consider the rectangular waveguide of Fig. 2-16. Let E~ = 0 and E, "" J(x.y) be known over the z = 0 cross section. We wish to determine the field z > 0. assuming that the guide is ma.tched (only outwardtraveling waves exist). The TEx modes of Sec. 4-4 have no Ea; 50 let uS

Ie. C. Culler, Electromagnetic Waves Guided by Corrugated Conducting Surfaces, BeU TdqMru Lab. Rept. MM-44-160-218, October, 1944.

172

TI:M.E-HAR~(ONIC

flLECI'ROKAGNETlC FIELDS

take a superposition of these modes.

. .

'I'

2: 2:

-=

A •• sin

This is

m;z cos n~ rr-'

(4-72)

._1 ...0

where A•• are mode amplitudes and the -r•• are the mode-propagation constants, given by Eq. (4-23). In terms of !fi. the field is given by Eqs. (4-32). In particular, E, at z = 0 is given by

E~

. . '\' '\' ... Lt L..t I 0

=

. m.. a

"r.... A... am

nry cos b

",_I .... 0

Note that this is in the form of a double Fourier series: a sine series in :t and a cosine aeries in y (sec Appendix C). It is thus evident that 'Y.... A ... sre the Fourier coefficients of E" or ll

"roo.A ... - E•• = 2t ab

{"

)0

dx

ior dy E, I._0 sin mn a cos n b

ry

(4-73)

where f .. = I for n = 0 and E. = 2 for n > 0 (Neumann'& number). The .A••• and hence the field, are now evaluated. The solution for E. = /(x,Y) and E, .,. 0 given over the z - 0 cross section can be obtained from the above solut.ion by a rotation of axes. The general case for which both E~ and E, are given over the z .= 0 cross section is a superposition of the two cases E• ... 0 and E, = O. The solution for the ease H ~ and H, givCD over the z = 0 cross section can be obtained in a dual manner. For a large class of waveguides, when many modes exist simultaneously, each mode transmits energy as if it existed alone. We shall show that the rectangular waveguide has this property. Given the wave function of Eq. (4-72), specifying a field according to Eqs. (4-32), the z-direetoo complex power at z ... 0 is

P••

Because of the orthogonality relationships for the sinusoidal {unctions,

173

PLANE WAVE FUNCTIONS

Incident wave

L z ---------FIG. 4-16. A capacitive wa.veguide junction.

this reduces to (4-74)

where (yo) .... are the TEx wave a.dmittances, given by the reciprocal of Eqs. (4-36). The above equation is simply a. summation of the powers for the individual modes. In a lossless guide, the power for a propagating mode is real and that for a non propagating mode is imaginary. To illustrate the above theory, consider the waveguide junction of Fig. 4.~16. The dimensions are such that only the dominant mode (TE IO) propagates in each section. Let there be a wave incident on the junction from the smaller guide, and let the larger guide be matched. For an approximate solution, assume that Ell at the junction is tbat of the incident wave Ell

I { "'"

...0

...

y c

sma 0

(4-75)

From Eq, (4-73), the only nonzero mode amplitudes are E lo

=

E 1.. =

C

'Y10A IO

=b . nll'c

2

'YhAh = -

5ln-

(4-76)

b Thus, only the m = 1 term of the m summation remains in Eq. (4-72). Let us usc this solution to obtain an "aperture admittance" for the junction. From Eqs. (4-74) and (4-76), the complex power at z = 0 is p

~

n~

•

a ),,/2 if it is air-filled. However, if the smaller guide is dielectric-filled, we can have wave propagation in it when c < >./2. Moreover, the aperture susceptance is defined only in terms of E~ in the aperture and has significance independent of the manner in which this E~ is obtained. 4-10. Currents in Waveguides. The problems of the preceding section might be called II aperture excitation >1 of waveguides. We shall now consider" current excitation" of waveguides. This involves the determination of modal expansions in terms of current sheets over a guide cross section. The only difference between aperture excitation and current excitation is that the former assumes a knowledge of the tangential electric field and the l::Ltter assumes a knowledge of the discontinuity in the tangential magnetic field. The equivalence principle plus duality can be used to transform a.n aperture-type problem into a current-type problem, and vice versa. To illustrate the solution, consider a rectangular waveguide with a sheet of z-directed electric currents over the z = 0 cross section. This is illustrated by Fig. 3-2, where J. = uz!(z,y) is now arbitrary. We shall assume that only waves traveling outward from the current are present, thllt is, the guide is matched in both directions. At z = 0 we must have E~, E~, and H ~ continuous. Hz must also be antisymmetric about z = 0; hence it must be identically zero, and it is convenient to use the TMx modes of Sec. 4-4. (Note that J and its images are x-directcdi so it is to be expected that an x-directed A is sufficient for representing the field.) Superpositions of the TMx modes are

. .

L 2:

(4-84)

B.....- cos

7

z k. imaginary parts as

G. =

....!.a1 >t"1

4 B• -),"1(11 -

f'

-J:

sin' (k.a/2) dk. k,.l v'k1 k,.l

(f-' + J.') k,,'sin'v'k,,' (k.a/2) dk k' .. __

J:

The above integrals can be simplified to give

X,p. - 2 ),"1B.

=

2

f, J.o

w'

tD

dtD

v' (ka/2)'

w'

(4-105)

sin' w dw

b/2

For srna.ll ka, these

ainl

b/2

w'

v'w'

(ka/2)l

0.00 1

x,p. = r [I >..."B.

t>::

(';1'] }

a

I: < 0.1

(4-106)

3.135 - 2 log ka

For intermediate ka, the aperture conductance and susceptance are plotted in Fig. 4-22. For large w, we have I The formula for B. ill a qU&!i-et.at.ic result. The direct specialUation of the seeond of Eq•. (4-lo.s) to small ka gives a numerical facwr of 4.232 instead of 3.135.

184

TIME~BAlWONlC

ELECTROllAGNETIC VJELDS

!

~"o. ~ ~

~.B. ~ (:.)' [1 - ~.faC08(~' + DT] j

~ > 1 (4-107)

The aperture is capacitive, since B. is always positive. Another problem of practical interest is that of Fig. 4-21 when the incident wave is in the dominant TE mode (TE to y). In this case, E. will be the only component of E, and we shall take E. as our scalar wave fundion. Analogous to the preceding problem, we construct 1 E. = 2...

f"_. f(k.)eJk.z,p.· dk.

(4-108)

In terms of Fourier transforms, this is

2. -

(4-109)

f(k.) ....'

From the field equations, we find the trans(orm of H to be

R.

:::s

-k,S.

H, _ k. 2.

WJl

WJl

(4-110)

Thef(k,,) is evaluated by specializing Eq. (4-109) to y "'" 0, which gives

2.1'.0 ~

f(k.} -

f_'. E.(x.O)..-"·· dx

(4-11 I}

For an approximate BOlution, we assume the E. in the aperture of Fig. 4-21 to be that of the incident TE mode, tbat is,

E.

I,-0

!

COS

~

TX

a

(4-112)

0

Substituting this into the preceding equation, we find

E.I,.0

= f(k.} _ 2T' cos (k,af2) w- t - (k,..a)t

(4-113)

The choice of the root for k, is the same as in the preceding example, given by Eq. (4-104). This completes the formal solution. Let us again calcula.te the aperture admittance. The power transmitted by the aperture is P -

f. [E.H:)~.

dx -

~

where we have used Parseval's theorem.

f.

[E.n:),•• dk.

From Eqs. (4-110) and (4-113),

185

PLANE WAVE FUNCTIONS

0.8

tX

I

f-

\La

_E

0.6 f- I-

IT

f- I- \

( .{l<JG.

E """ cos ('lIx/a)

0.4

I I

0.2

V

V

(.;lJB~

V

t-

V

o

1.5

1.0

0.5

ai' FIo. 4-23. Apert.ure admittance of an inductive Blot radiator.

this becomes p

~

...=!.

21fwIJ

f"

_..

k*IE.I' dk. ~ -2..' WIJ

1/

f"

k: cos' (k.a/2) dk.

_ .. 1~2 -

(k.a)2J2

We shall refer the aperture admittance to the voltage per unit length or the aperture, which is V = 1. This gives

y .,., p.

. rvr

= _2~a2

Wa

f"-" [r'cos (k.a)'l' 2

kll

(k~aj2) dk

•

The integrand is real for Ik~! < k and imaginary for Ik.,l > k. Aseparation of Y. into real and imaginary parts is therefore accomplished in the same manner as in the preceding example. The result is '1

1

_

(0/2

2 }o

); G. -

vi (ka/2)2

w 2 cost W

w'J'

[(r/2)'

dw

• vlw% (kaj2)tcos2w -B =-If." -dw }.,.. 2 h/2 [(orr /2)2 w 2J2

For small ka, we have

~G ~ ~(~)' }.,..

1r).,

~ B. ~ -0.194

l

);a < 0.1

(4-114)

(4-115)

For intermediate ka, the aperture conductance and susceptance are plotted in Fig. 4-23. For large ka, • a ~ G.. = 2>.

a

); >

1.5

(4-116)

186

TIME-HARMONIC ELECTROMAGNETIC FIELDS

z

FIo. 4-24. A sheet. of z-di~

J.

rccted currents in the

y

y -

0 plane.

x and B.. is negligible. The aperture is inductive since B" is always negative. 4-12. Plane Current Sheets. The field of plane sheets of current can, of course, be determined by the potentia.l integral method of Sec. 2-9. We DOW reconsider the problem from the alternative approach of constructing transforms. The procedure is similar to that used in the preceding section for apertures. In fact, if the equivalence principle plus image theory is applied to the results of the preceding section, we have complete duality between apertures (magnetic current sheets) and electric current sheets. However, rather than taking this short cut, let us follow the more circuitous path of constructing thc solution from basic concepts. Suppose we have a sheet of z-directcd electric currents over a portion of the y = 0 plane, as suggested by Fig. 4-24. The field can be expressed in terms of a wa.ve function representing the z-component of magnetic vector potential. (This we know from the potential integral solution.) The problem is of the radiation. type, requiring continuous distributions of eigenvalues. We anticipate the wave functions to be of the transform type, such as Eq. (4-93). From Eqs. (3-86), we have the transforms of the field components for the TM to z field, given by

n. - jk,~ D,

~ -jk.~

D. - 0

(4-117)

187

PLANE WAVE FUNCTIONS

We construct the transform of y, as

These are dual to Eqs. (4-95).

if!'" = f+(k..k.) ~

,,-

f-(k.,k.)

y>O y 0 we have A jll '" = 8...,

where k"

= k,+

II, _. J ~-~·~e-ii .• dx dz = 4r

_.

=

u.J/! where

f" f" _.

I _. -k" eJA··~.lIeJA·· dk~ dk.

is given by Eq. (4-119).

(4-123)

In this example,!Jt as well as the

188

TIME-HARMONIC ELECTROMAGNETIC nELDS

field is unique. the identity rib

-

r

Hence. equating Eqs. (4-122) and (4-123), we have

f" f-

1 21f-j _..

= -

e-i,..;l. l.' .1:,'

_. yk 2

k. t

ks t

...••...··dk.dk

(4-124)

•

This bolds for all y, since kll changes sign as y changes sign. We have considered explicitly only sheets of z-dirccted current. The solution for x-directed current can be obtained by a rotation of coordinates. When the current sheet bas both x and z components, the solution is a superposition of the x-directed case and the z-direct.ed case. The solution for magnetic current sheets is dual to that for electric current sheets. Finally, if t-he sheet contains y-directed electric currents, we can convert to the equivalent x- and z-directed magnetic cunent sheet (or a solution. and vice versa for ~irecLcd magnetic currents. A ttl.·o-<Jimensional problem to which we shall have occasion to refer in the next chapter is that of a ribbon of axially directed current. uniformly distributed. This is shown in Fig. 4-25. The parameter of interest to us is the If impedance per unit length." defined by (4-125)

where P is the complex power per unit length and I is the total current. Rather than work through the details. let u.s apply duality to the aperture problem of Fig. 4-22. According to the concepts of Sec. 3-6. the field y > 0 is unchanged if the aperture is replaced by a magnetic current ribbon K = 2V. This ribbon radiates into whole space; so the power per unit length is twice that from the aperture. The admittance of the magnetic current ribbon is thus

z

y ......lb

=

jP:pott 2Vlt =

p. TKft -

lL)" 7:::

."'~

where the aperture admittance

"'"' G. + jB. is given by Eq. (4-105). which we can represent by 1 Y.~" = .X J(k4) Y.pe,~

J.

y

x

By duality, we have the radiation impedance of the electric current ribbon given by FIa. 4-25. A ribbon

or current.

Z"M'" ~

1 11

2 ~ J(ka)

-

11'

"2

Y. M "

(4-126)

ltLANlf: WAVE FUNQTIONS

180

(Compare this with Prob. 3-1. The fo.ctor--0

eoe-.. orJI d

01-27. Conslder the junction of two parallel-plate tran&niaeion linel of height c for

: < 0 and beight b lor. > 0.

with the bottom plate continuous. (The C10M fJeCtion Using the formulation or Prob. 4-26. show that t.he aperture sWlCCptance per unit width relerred to the aperture voltage is is that of the eecond drawing of Fig, 4-16.)

•

B •

O¥

4 \' sin l (n ..c!b) II>' "" (O"C/b)1 VOl (26/>')1

" .,

where a constant E. has been assumed in the aperture. Compare this with Eq, (4-78). (-28. The centered capacitive waveguide junction is shown i.n }o'ig. 4-26. Show that tbe aperture llU8Ceptanee referred to the maximum aperture voltage fa given by Eq. (4-78) with>" replaced by 2>... It is UlIumed that E. in the aperture is that of the incident mode.

FlO. 4-26. A centered capacitive waveguide junctioD

194

TWE-HABllONlC ELECI'ROllAGNETIC I'lELDS

tx

fx

I

-

Incident

w.W>

FlO. 4·27. A centered inductive waveguide junction. 4.-29. Consider the centered inductive 1I"&veguide junction of Fir,:. 4-27. Aasumin& t.hat B, in t.be aperture is that of the incident mode, &how that the apertl1l'6 IU8CePt.. anee referred to t.he muimum apertuJ't! voltage is giVeD by

B.

_

"

~ (~)' "..'0

./(!!!)' _(~)' 2 ).

[""" (m..r~)]'

k\ '

a

1

"V

(rne/a)!

3,11.7, ••.

•-so. In Eq. (4-83), note that M c/a - 0 the summation becomes similar to an int.egration. Uso the analogy rM./a ,..,. z and cia "'" th to show that

--8._b, 1 0 I _). ,/0_0 ...'/."(";''')' ZI

%d%

Integrate by pArliI, and use the identity'

toahow that

• sin ~ - - d% /.0%1-1 _ b.,

/.2. 0

sin 11 -r- d1l - Si(Z,..)

B.-.... 8i(2

),

c/a_O

) _ 0.226

2...

4.-31. Let there be ••beet of .,..directed current J. over the z - 0 plane of a parallel· plate waveguide formed by conductors over the 11 - 0 and" - b p1aOell. The guide ill matched in both tbe +z and -z direetiotlll. Show that the 6eld produced by the

c\U"rent sheet is

"

• >0 • ee that the coax to waveguide junction of Prob. 4-33 ia changed to that of Fig. 4-286. Show that the input resiatance Ren by the coax is now R _ ~ (z) I

Co

'11

lain (..a/bUrin k(c + d) kacosk(c+d)

sin b:1}'

f.-36. By expanding (sin VJ/w)l in a Taylor aeries about of Eqll. (4-105) becomea

),'1'/1. -

'If

I [ 1 - 6

(ta)' I (to)' 2" + 60"2

tD -

0, sbow that the first

(ta)' + ... ]

I 1008 2"

1m

4.-86. Consider the second of (4-105) as the contour integral , B

" •-

Ile [/.

Eqs.

(I - ,;N)dw ] (14/2)1

c, w' v' w'

where C1 ia shown in Fig. 4-29. Cooaider the closed contour C1 + C, + C. + C" &lid eJ:press M,B. in terma of a contour integral Over C, aDd C.. Show that as ta/2 becomes l&rge, this last conwur integral rOO.ueee to the eeoond of Eqa,

w plane

c, Co

Re

(4-107).

Flo. 4-29. CootoW1l for Prob. 4--36.

196

TIl&E-HAB140NJC ELECTROMAGNETIC FlELDe

4.-11. By expanding coal tD/[(1I'/2)· - will in a Taylor seric. about that the first of Eq•. (4-114) bccomCll

•

..,

tD _

0, abow

(0)"

• 2 \' );0.-; ~ b. >: bl b, 6, b, b, bl

-

_

+1.0 -0.4.67401 +0.189108 -O.05M13 +0.012182 -0.002083

40-88. Specialir;e the second of Eq•. (4-114) to the and U8C the identity (8ee l"rob. 4-30)

f,-

jt

C!\8e

a - 0, integrate by partll,

r

lIin 2z. dz _ ~ rain 1/ d _ ~ SiC..) (11'"/2)' :r:' 1I'}o II 1/ 11'

- !!' 1. SiC...} - ..I ~ - 0.194 ). B. --I> _02..

to show that

4.-St. Show that the 6n;t of Eq•. (4-ll4) reduce. to the contour integral

'G [f

•

(I

+ ''''')",If! dW]

~G. ka_!o"8 Re le, [(.../2)1 where C, ia 8hown in Fig. 4-30.

Conllider the clOlled contour C.

and CXPre&8 G. in terml of a conlour integral over CJ and C,.

+ C. + C. + C.. Evaluate this Ill.8t

contour integral, and ahow that

• '0 -0._4a).

,b-o_

x

1m

ED pltlne

---c+,--E------' Co C,

_/2

-r

h Cl

Fro. 4·30. Conloul1l for Prob. 4-39.

R.

FlO. 4-31. Two parallel-plate trR~ sion lines radiating into half-spllcc.

,-to. Two parallel-plate trans.misaion lines opening onto & conducting plane are excited in oppoeit.e phase and equal magnitude, lUI sbown in Fig. 4-31. Assume E. in

PLANE WAVE FUNCTIONS

197

the aperture is a con.stant lor each line, and show that the aperture susceptance referred to the aperture voltage of one line Us

G • -

B. _

8

J..b

;\"

0

8in' to dto

WI

! (. ;\" J.to to l

V (bip

ttll

sin w dID 4

vw

l

(ka) I

4:·4:1. Construct the vector potential A - UN {or & sheet of t-diteeted currents over tbe 11 - 0 plane (Fig. 4-24) by (0) tbe potential integral method and (b) tbe transform method. Show by use of Grcen'a second identity [Eq. (3-44)] that the twO.p1 are equal. Specialize the potential integral801ution to , _ 10, and show that

f

e-i~

.,.

--+ -4~.

J.( -k cos q, sin

where J.(k.,k.) is given by Eq. (4-121). 4:-4:2. Supposo that tho current in Fig. 4-25 Bnd of magnitude

~

8, -k COB 8)

z.directed rather than z..directed,

..,

J.-C08a

Show tha.t the impedance per unit length, defined by Eq. (4-125), where I is the current per unit length, is given by Eq. (4~126), where Y.p"r' is now the llperture admittance of Fig. 4-23.

C',

CHAPTER

5

CYLINDRICAL WAVE FUNCTIONS

6-1. The Wave FUDctions. Problems having boundaries which coin· cide with cylindrical coordinate surfaces are usually solved in cylindrical coordinates. 1 We shall usually orient the cylindrical coordinate system as shown in Fig. 5-1. We first consider solutions to the scalar Helmholtz equation. Once we have these scalar wave functions, we can construct electromagnetic fields according to Eqs. (3-91). The scalar Helmholtz equation in cylindrical coordinates is

! ~ ( af) + .l.- a.." + a.."2 + k'f p up

p up

p! oq,t

dz

_ 0

(5-1)

which is Eq. (2-7) with the Laplacian expressed in cylindrical coordinates. Following the method of separation of variables, we seck to find solutions of the form

f -

R(p)~(¢)Z(z)

(5-2)

Substitution of Eq. (5-2) into Eq. (5-1) and division by '" yields 1 d (dR) pR dp p dp

1 d'Z

1 d'4>

+ p'4> d¢' + Z dz' + k

,

- 0

The third term is explicitly independent of p and q,. It must a.lso be independent oC z if the equation is to sum to zero Cor all p, q" z. Hence,

.! d'Zt Z dz

_ -k'

(5-3)

•

where k. is a constant. Substitution of this into the preceding equation and multiplication by pi gives

.e.R dp ~ ( dR) + .! d'~ + (k' p dp ~ d~' Now the second term is independent of

p

_ k ') , _ 0 • p

and z, and the other terms are

I The term "cylindrical" is often used in flo more general sense to include cylinders of arbitrary cross section. We are at present using the term to mean "circularly cylindrical."

198

199

CYLINDRICAL WAVE FUNCI'IONS

independent of ¢.

z

Hence,

Id'4> - = '" d~'

-n'

(5-4)

p

where n is a constant. The preceding equation then becomes -p -d ( p -dR) - n l Rdp dp

+ (k l -

• y

k~l)pl

- 0

x

(5-5)

which is an equation in p only. The wave equation is now separated. k.'

+ k.'

Flo. 6-1. Cylindrical coordioat.ell.

To summarize, define k_ as

- k'

(5-6)

and write the separated equa,ions [Eqs. (5-3), (5-4), sud (5-5)1 as d ( p dR) p dp dp

+ [(k.p)' -

n')R - 0

d'''' d~' + n'4> ~~ + k.IZ

- 0 0=

(5-7)

0

The cia and Z equations are harmonic equations, giving rise to harmonic functions. These we denote, in general, by h(n¢) and h(k.,z). The R equation is Buw', equation of order n, solutions of which we shall denote in general by B.(k_p).1 Commonly used solutions to Bessel's equation ar. B.(k.p) ~ J.(k.p) , N.(k.p), H."'(k.p), H."'(k.p)

(5-8)

where J .. (k,p) is the Bessel function of the first kind, N.(k,p) is the Bessel function of the second kind, H.(I)(k,p) is the Hankel function of the first kind, and H.0

Infinite number alon..

..)' (1:"-"2-, ...) ,h. r.l:"ain

r(.I:,,)·

kp-o :tj-

I: lm.... inary-two flvall_lIt fi""d. t floropln-loc..Hre
.''''''di". w..ve

.I: real--.b"dinC .... ve

" _ 0

[H.(»(.I:...) _ HoU)(.I:,,)]

2'(" - I)!

4

I"finil.o nuroher ..10... 'hfl ru,1 axi!

axit

" >0

.l:P" 0 .I: ima&illllry-two eva....ee"t 5eldt

,~,

tp-o i j .. t comp\eor-loc&lired .u"di... waves

• "When I: .. -ja. the functiolll'.(jh) .. 1. (a") .. j'J.(-j.,,,) and K.(jtp) .. K.(..,,) .. i{-j)··'H.UI(-j..,,) are "lied. t

fi~d

kp ..... ; ..

...."

-~IO"(~) ., TI:P

-

i~na~van~nt

t complex-attenu.. ted travtllnc wa"e

-flOll

20 .. 1

,

.I.:

t rul_.."di.". wave "_ 0

H[II.(OI{kp) + H.I'I(.I:"lJ

-

.l.:p ..... -j..

j"s-i'p

" >0

(.1:,,).

H.{k,,)

1:..... 0

" - 0

l+j;IOCC:... )

When I: .. O. tbe Beuel f"netion. are I alld lOll", .... O. and

p' .. nd p-' • .....

O.

204 which are sufficiently general to express any TE (no E.) field existing in a homogeneous source-free region. An arbitrary field (one having both an E. and an H.) can be expressed 8S a superposition of Eqs. (5-18) and (5-19). 5-2. The Circular Waveguide. The propagation of waves in a. hollow conducting tube of circular cross flection, called tho circular waveguide, provides a good illustration of the use of cylindrical wave functions. Qualitatively, the phenomenon is Bimilar to wave propagation in the rectangular waveguide, considered in See. 4-3. The coordina.tes to be used are shown in Fig. 5-2. For modes TM to %, we may express the field in terms of an A havin« only & z component 'It. The field is finite at. p - 0; 80 the wave (unctions must be of the form of Eqs. (5-13). It is conventional to express the 4J variation by sinusoidal functions; hence '" _ J.(k ) ,p

ISin cos n~) nq, ,-f'"

(5-20)

is the desired form of the mode functions. Either sin nq, or COB nq, may be chosen; 80 we have 8. mode degeneracy except for the cases n "'" O. The TM field is found from Eqs. (5-18) applied to the above y,. In particularJ 1

E. - - (k' -

9

k.·)~

which must vanish at the conduct-jng walls

p -

/I.

Hence, we must have

J .(k.a) - 0

(5-21)

from which eigenvalues for k, may be determined. The functions J M(Z) afC shown in Fig. D-l. Note that for each n there are a denumerably infinite number of zeros. These are ordered a.nd designated by X"JO. the

x z Flo. 6-2. The circulu waveguide.

y

205

CYLrNDRlCAL WAVE nJNCI'JONS

x I

2 3

•

TABLE &-2. ORDERED ZEROS ~. 0'

0

I

2

3

2.40.5 5.520 8.054 J1 .7fi2

3.832 7.016 10.173 13.324

6.136 8.417 11.620 14.796

6.380 9.761 13.015

J .(:)

•

•

7.688

8.771 12.339

11.005 14.372

first subscript referring to the order of the Bessel function and the second to the order of the zero. The lower order %•• are tabulated in Table 5-2. Equation (5-21) is now satisfied if we choose k

.--

(5-22)

'" a

Substituting this into Eq. (5-20), we have the TM •• mode functions

~. ~ ~ I'

J. ( •••

a

p) Jlcosn¢o sin n¢) .-".,

(5-23)

where n "'" 0, 1, 2, . . . , and p = I, 2, 3, . . .. The electromagnetic field is then determined from Eqs. (5-18) with the above y,. The mode phase constant k. is determined according to Eq. (5-6), that is,

(.~.)' + k.' -

k'

(5-24)

Subscripts np on the k. are sometimes used to indicate explicitly that it depends on the mode number. Modes TE to % are e;'(pressed in terms of an F having only a % component J/I. This wave function must be of the form of Eq. (5-20), with the field determined by Eqs. (5-19). The E. component is 81/I/iJp, which must vanish at p - a; hence the condition J:(k.a) ~ 0

(5-25)

must be satisfied. The J .. are oscillatory fUDctions; hence, the J~ also are oscillatory functions. (For example, J~ "'" -J l .). The J~(%) have a dcnumerably infinite number of zeros, which we order as x~Jl' (The prime is used to avoid confusion with the zeros of the Bessel function itself.) The lowcr-order zeros are tabulated in Table 5-3.

x 1 2

3

•

0

1

2

3

3.832 7.016 10.173 13.324

1.841 5.331 8.536 11.706

3.054 6.706

4.201 8.015 11.340

9.969

13.170

•

•

5.317

6.416 10.620 13.987

9.282

12.682

206

TIME-HAn~rONIC

ELECTROMAGNETIC FIELDS

We now satisfy Eq. (5-25) by choosing

,

k• =~

(5-26)

a

Using this in the wa.ve function of Eq. (5-20), we have the TE..p mode functions

~ TO ~

J

11)1

where n

=

(~)

"a

0, 1, 2, ... ,and p

=

I

sin n¢J

cos nq,

"I',.

1, 2, 3, . . ..

(5-27)

The electromagnetic

field is given by Eqs. (5-19) with the above if. The mode propagation constant is dotermined by Eq. (5-6), which with Eq. (5-26) becomes

(~)' + k.'

- k'

(5-28)

This completes our determination of the mode spectrum for the circular waveguide. The int.erpretation of the mode propagation constants is the same as for those of the rectangular guide and, in fact, is the same for all cylindrical guides of arbitrary cross section if the dielectric is homogeneous. (This we show in Sec. 8-1.) The cutoff wave number of a mode is that for which the mode propagation constant vanishes. Hence, from Eqs. (5-24) and (5-28), we have (k)

,

~"p

If k k~ =

TM

x"p

_

a

-

(k.) .. pTE = ~

(5-29)

a

> k., the mode propagates, and if k < k. the mode is cutoff. 'hI. Y;;, we obtain the cutoff frequencies ~

(I) ."p

Alternatively, setting

=2 k~

=

x.,

_I

:Ira v EIJ

TE_ (I) • "p -

Letting

,

x" P

(5-30)

211'"a VEIJ

'br/X., we obtain the cutoff wavelengths ') TE ( A. "p

= 2'1fa

x'

••

(5-31)

Thus, tho cutoff frequencies are proportional to the X"p for TM modes, and to the x~p for the TE modes. Referring to Tables 5-2 and 5--3, we note that the zeros in ascending order of magnitude are X~l, X01, X~h Xu, and X~I' etc. Hence, the modes in order of ascending cutoff frequencies are TEll, TM o1 , TE u , TM I1 , and TEo I (a. degeneracy), etc. Circular waveguides are used in applications where rotational symmetry is needed. The dominant TEn umode" is actually a pair of degenerate modes (sin 4> and cos 4> variation); hence there is no frequency

207

CYLINDRICAL WAVE FUNC'I'rONS

(a) TEll

(d) TMll

(b)

e-~.~

(c) TBn

TMOl

(e) 2"Eo'l.

9£---

(f)

7M21

FIG. 5-3. Circular wtlveguidc mode patterns.

range for single-mode propagation. (Recall that single-mode operation over a 2: 1 frequency range is possible in the rectangular waveguide.) Note that, except for the degeneracies betwcen TE op and TM 1p modes, TE and TM modes have different cutoff frequencies and hence different propagation constants. The modes of the circular waveguide have HJirected wave impedances of the same form as we found in the rectangular waveguide. For example, in a TE mode, (Z,)" _ E. ~ _ E. _ ~ H. Hp k.

(5-32)

which is the same as Eq. (4-27). The behavior of the Zo's is therefore the samc as in the rectangular waveguide, which is plotted in Fig. 4-3. Attenuation of waves in circular waveguides due to conduction losses in the walls is given in Frob. 5-9. Modal expansions in circular waveguides can be obtained by the general treatment of Sec. 8-2. The mode patterns for some of the lower-order modes are shown in Fig. 5-3. These can be determined in the usual manner (find £ and :JC, and specialize to some instant of time). Field lines ending in the crosssectional plane loop down the guide, in the same manner as they did in the rectangular waveguide. Solutions for cylindrical waveguides of other cross sections also can be expressed in terms of elementary cylindrical wave functions. Representative cross scc;tions arc shown in Fig. 5-4. Note that all or these

208

TIME-HARMONIC ELECTROMAGNETIC FIELDS

b

Ca)

(d)

Cb)

(e)

(.)

(I)

FIo. 5-4. Some waveguide cross sections for which the mode functions arc elementary w&.ve functions. (a) Coaxial; (b) coaxial with baffle; (e) circular with bame; (d) semicircular; (e) wedge; (f) sectoral.

are formed by conductors covering complete p = constant and 4> = constant coordinate surfaces. Wave functions for the guides of Fig. 5-4 are given in Probs. 5-5 to 5-7. 5-3. Radial Waveguides. In the circular waveguide we have plane wa.ves, that is, the cquiphase surfaces arc parallel planes. Wave functions of the form '" - B.(k.p)h(k.z).±i··

with B,.(kpp) and h(k~z) real, have equiphase surfaces which arc int.crsecting planes (the q, = constant surfaces). Such waves travel in the circumferential direction, and we shall call them circulating waves. Examples are given in Prob. 5-10. Finally, we might have wave func· tions of the form

.H.Ol(k,p)j

'" ~ h(k.z)h(n~) (ll."'(k.p)

with h(k.z) and h(nq,) real. These waves have cylindrical cquiphase sur· faces (p = consta.nt), and travel in the radial direction. We shall call them radial waves. l In this section some simple waveguides capable of guiding radial waves will be considered. Radial wa.ves can be supported by parallel conducting plates. DependI These arc true cylindrical waves as defined in Sec. 2-11, but we are using the term "cylindrical wavo function" to mean "a wave function in the cylindrical coordinate system," regardll$S of il.l:l cquiphllJle surfaces.

209

CYLINDRICAL WA VI'J FUNCTIONS

z y

(b)

(0)

FIa. 5-.5. Radial waveguides.

(a) PBrBl1cl plate; (b) wedge; (e) hom.

iog upon the excitation, waves between the plates may be either plane or radial. When the waves are of the radial type, we call the guiding plates a parallel-plate radial waveguide. Figure 5-5a shows the coordina.te sy&tern we shall use. The TM wave functions satisfying the boundary conditions E_ """ E. :z 0 at Z = 0 and z ... a arc l{!••

~

_ (mT) a Z cosn41 IH,O'(k,p») IJ. m (k_p)

- cos

where m = 0, 1,2, . . . ,and n - 0, I, 2, .. k, _

k' _

(5-33)

, and, by Eq. (5-6),

("'.r)'

The electromagnetic field i!5 given by Eqs. (5-18) with the above l{!. TE wave functions satisfying the boundary conditions arc

. . TE -51n _ . (mT JH""(k,p») aZ) cosn411H.. cIJ(k,p)

.,.-..

(5-34)

The

(5-35)

where m '""' 1, 2, 3, . . . , a.nd n -=- 0, 1, 2, . . . , and Eq. (5-34) still applies. The electromagnetic field for the TE modes is found from Eqs. (&-19) with the above!/t. In both the TM and TE cases, the 11..(I)(k,p) represent inward-traveling wa.ves (toward the Z axis), and the IJ... u1 (k,p) represent outward-traveling waves. For a complete set of modes, those with sin nq, variation must also be included. Radial waves are characterized by a phase constant which is a function of radial distance. Following the general definition of Sec. 2-11, we have the phase constants for the above ~'s given by

p ~~ -

[tan-'

N.(k,p)] J .(k,p) 2 1 - TpJ.'(k,p) N.'(k,p) ap

+

(5-36)

210

TW}:-HARJ,lONIC ELECI'RO.l.lAGSETIC FIELDS

Using asymptotic formulas (or the Bessel functions, we find that (or real k, fJI'

II,,,....!

(5-37)

k,

This is to be expected, because a.t large radii the waves should be similar to plane waves on the parallel-plate guide. Note that the phase constant of Eq. (5-36) is that of tbe mode function and not that for the field. Components of E and H transverse to p are not generally in phase. They become in phase at large radii. Each mode of the radial waveguide is also characterized by a single radially directed wave impedance. Using Eqs. (~) and (5--18), we find for outv.·ani-traveling TM modes ~

__ E. _

~

H."'(k,p)

(5-38)

k, H ..Ol(k,p} E. Z -, ~ = 11. = - jWf. H .. (I)I(k,.p}

(5-39)

Z

H.;I.>E H.UI'(k,p)

+"

while for inward-traveling TM modes

Note that for real k, we have Z_,TW = Z+,TI.I*. we find

Z

TIl:

+"

=

E. H• ...

E.

jWIl

Similarly, for TE modes

H,,(l)'(k,p)

T; H.l2>(k,p} -;w~

B ..(II'(kPJ) Z_,TE "" - H • .". ~ H.lll(k,.p)

(5-40)

where the first equation applies to outward-traveling waves and the second equation to inward-traveling waves. Note that the TE wave admittances are dual to the TM wave impedances. It is seen from Eq. (5-34) that k,. is imaginary if mr/a > k. In this case, let k,. - -ja, and

where K. is the modified Bessel function (see Appendix D). The mode functions are now everywhere in phase, and there is no wave propagation. The radial wave impedances become imaginary, indicating no power flow. For example, from Eq. (5-38), if k,. "" -ja,

Z +,.

~ ~ -ja H."'(-jap) ~ ja K.(ap)

j<M. H.(2)'(-jap)

Wf

K~(ap)

(5-41)

which are always capaeitivcly reactive, since K. is positive and K~ is negative. Hence, whenever a < >'/2, the modes m > 0 are nonpropagating (evanescent). For small 0, only the TM o.. modes propagate, for

211

CYLINDRICAL WAVE FUNCTIONS

which Eq. (&-33) reduces to ~,.

)!

~ _ /H.Ol(k p - cos n~ lH.u'(kp)

(5-42)

From Eqs. (5-38) and (5-39) we have the wave impedances for these modes given by Z+~TM

=:;

~

Z_;n.t·

=:;

. H.'''(kp) H..U)'(kp)

-JY]

IH.,,,7(kp)i'

l.ip -

jIJ.(kp)J;(kp)

+ N.(kp)N;(kp)]!

(5-43)

A consideration of the behavior of the Bessel functions (Figs. D-l and D-2) reveals that {or arguments kp < n the N" functions and their derivatives become large in magnitude. Hence, when 2-,;p < n)." the wave impedances become predominantly reactive. Figure 5-6 illustrates this behavior by showing XI R. where Z+~Ttd = R + jX, (or the first five TMo" modes. We shall call kp = n the point of gradual cutoff, the wave impedances being predominantly resistive when kp > n and predominantly reactive when kp < n. Note that these gradual cutoffs occur when the circumference of the radial waveguide is an integral number of wavelengths. From the above discussion it is evident that the TM oo mode is dominant, that is, propagates energy effectively at smaller radii than any other mode. For this mode we have (5-44)

representing inward-traveling waves, and

k'

E.+ = -.-HoW(kp)

JW. H.+ = kH 1(2'(kp)

(5-45)

which represent outward-traveling waves. Note that there are no p components of E or H, the mode being TEM to p. It is called the transmission-line mode of the parallel-plate radial guide, because of its similarity with plane transmissionline modes. For example, at a given radius we can calculate a unique voltage between the plates a.nd a net radially directed current on one of

4 n

3

4

3

1\

\

2

K-

1 0

o

1

2

3

4

5

kp

FIG. 5-6. Ratios of wave reactance to wave resistance for the TM Oto radial modes on the parallel-plate waveguide.

212

TIME-HARMONIC ELEcrnoMAoNETIC FIELDS

the plates. Also, the radial transmission line can be a.nalyzed by the classical transmissioll-line equations with Land C a function of p (Prob. 5-13). Radial waves also can be supported by inclined conducting planes, called a wedge radial waveguide, as shown in Fig. 5-5b. We shall assume no z variation of the field, considering the problem as two-dimensional. TM wave functions satisfying the boundary condition E. "'" 0 at t/> = 0

and t/>

=

cPo are Vtt)'rM "'"

sin (pr ,po

•.(kp»)

~) IHr~/

H,..'4>.(kp)

(5-46)

where p = 1, 2, 3, __ . ,and the electromagnetic field is given by Eqs. (5-18). TE wave functions satisfying the boundary condition E~ = 0 at = tPo are l/!pT";

= cos

(1'cPo< ~) IHf~/··(kp) I Hp.,•• (kp)

(5-47)

where p = 0, 1, 2, . . . , and the elccLromagnetic field is given by Eqs. The interpretation of the modes is essentially the same as that for the TM o.. parallel-plate modes, except that nonintegral orders of Hankel functions appear. This introduces no conceptual difficulties, but if numerical results are desired we would be hampered by a lack of tables for functions of arbitrary fractional order. The radial wave impedances for the wedge-guide modes are of the same form as for the parallel-pla.te guide [Eqs. (5-38) to (5-40)]. We nccd only replace n by 1J7r/q,o and k, by k. These wave impedances exhibit the same characteristic of gradual cutoff for fractional-order Hankel func~ tions as they do for integral-order Hankel functions. Again the transitional point is that for which the argument and order are equal, that is, '[J7f/q,g = kp. The radii so determined correspond to those for which the arc subtending the wedge is an integral number of half-wavelengths long. This is as we should expect from our knowledge of plane waves between p$l.r~lIel plates (the limiting ease rJ'O'- 0). The dominant mode is evidently the TE g mode, in which case, from Eqs. (5-47) and (5-19), we have (5-19).

(5-48)

for inward-traveling waves, and (5-49)

for outward-traveling waves.

This is a transmission-line mode, chamc·

213

CYLINDRICAL Wit. VE Pt1NCTIONB

terised by no E, or H, and possessing a unique voltage and cuceent at aD1 given radii. This mode also can be analyzed by the classical teall&mission-line equatdons for nonunifonn lines (L and C a function of pl. Not< tbat tbe field is dual to tbat of the pamllel-plate line (Eqs. (5-44) Illd (5-45»). Finally, simple radial waves can be supported by the hom-shaped ~e of Fig. 5-5c. called a sectoral horn w<MJ6fluiM. The TM modes are specified by the wave functions 1/I...TM

_

where m - 0, 1, 2,

cos

(~z) sin ("", ~) (H~2,··(k,p)1 a q,o H •." •• (k,p)

, and p ". 1, 2, 3, . ..

(5-50)

The field is given by

Eq•. (5-18), and

k, =

~k' _

(m:)'

(5-51)

The TE modes are specified by the mode functions

fo,T< _

..

sin(~%)C IHI' d.

rf>, - H ,.(!l(kpp)

(5-59)

1 E. Jahnke and F. Erode, "Tables of FunctioD.ll," p. 146. Dover Publications, New York, 1945 (reprint).

217

CYLINDRICAL WAVE FUNcrlONS

z

z

Conductor

z

(c)

FlO. 6-9. Some radial waveguides. (a) Partially filled; (6) dielectric slab; (c) coated conductor; (d) corrugated conductor.

where n = 0, 1, 2, . . .. The subscripts 1 and 2 refer to the regions z < d and z > d, respectively. We have anticipated that the p and 4J variations must be the same in both regions to satisfy boundary conditions at z ... d. Equations (5-59) represent outward-traveling waves. Inward-traveling waves would be of the same form but with H .(!) replaced by 8.(1). The k's in each region must, of course, satisfy the separation relationships k,! k,!

+ +

kIll kl!!

DO

kl! = k!! =

W!~1PI

(5-00)

W 1t!}l1

The field vectors themselves arc obtained from Eqs. (5-18), using the y,'s or Eqs. (5-59). To evaluate the G's and k" we must satisfy the conditions that E" E., H" and H. be continuous at z = d. For E, we ha.ve

1[ a' (1- tJll - -1)] tJI!

[B,I - E"J..., = -:-

JW

~

up uZ

«=l

fl

.-1

=

0

which reduces to

(5-61)

, kit CI sin kold = -k., C sin k. , (a - Ii) ft

For E. we have

fl

1[ (1

1)]

(E. l - E. , ]-.:I = -.- - a' -tJlt - -1/11 Jwi/. iJ4J 81. fl ~1

.-1

=

0

218

TIME-HARMONIC ELECTROMAGNETIC FIELDS

which also reduces to Eq. (5-61).

IH,. -

For H, we have

H,,]..... - ;

[aa~ ("'. - "',)]... . ~ 0

which reduces to (5-62)

0 1 cos kdd = C S COB kd(o - d)

Finally, for H. we have

- [!-ap ("'. -",,)] which a.gain reduces to Eq. (5-62). yields k.. tan k"d tl

~

.-4

~0

Division of Eq. (5-61) by Eq. (5-62)

_ k.. tan Ik.,(a - d») ts

(5-63)

The kd and kd are fun~tions of k, according to Eq. (5-60); so Eq. (5-63) is a. transcendental equation for determining possible k,'s. Once k, is evaluated, the ratio CI/Ct may be obtained from either Eq. (5-61) or Eq. (5-62). For fields TE to z we can satisfy the condition E, = E. = 0 at z = a by choosing 1/11 = C1 sink. 1zcosnq,H..U)(k,p) (5-
=

0

(5-76)

for TE modes (n = 0). We must now pick the proper F functions for the various cases, For the partially filled circular waveguide (Fig. 5-10a), the field must be finite at p = 0; hence

FI To satisfy E. = 0 at

p =

= P, = JII(k,la)

(5-77)

b, we choose

F, - J.(k,..)N.(k,ob) - N.(k,..)J.(k"b)

Furthermore, to satisfy E. "'" 0 at

p "'"

(5-78)

b, we choose

F. ~ J.(k,..)N;(k,ob) - N.(k"a)J:(k,ob)

(5-79)

The dominant mode is the lowest-'

FIo. 5-12. Phase constant for the circular dielectric rod.

(After M. C. GrG1I.)

223

CYLINDRICAL WAVE FUNc-rIONS

satisfy the condition E. = 0 at

p =

b, we should choose

F, = J.(k.,a)N.(k.,b) - N.(k.,a)J.(k..b)

and, to satisfy E• .", 0 at

F1

-

P .",

(5-81)

b,

J.(k,la)N~(k,lb)

- N.(k,la)J~(k ..lb)

(5-82)

For this guide the dominant mode is the lowest n r: 0 TM mode, which bas no cutoff frequency. (Compare it with the dominant mode of the plane coated conductor of Seo. 4-8.) Copper wire with an enamel coating can be used &8 an efficient waveguide for some applications. I Finally, the corrugated wire of Fig. 5-1Od can be analyzed in a mAnner similar to t.hat used for the corrugated plane (Fig. 4-15). The field extel'nal to the corrugated wire will be essentially the dominant TM (n ... 0) mode of t.he coated wire. The field in the corrugations will be essentially that of tho shorted parallel-plate radial transmission line. The characteristic equation is obtained by matching wave impedances at the corrugated surface. As the radius of tbe corrugated cylinder becomes large, the solution approaches tha.t for the corrugated plane. fi·6. Sources of Cylindrical Waves. In this section we shall consider two-dimensional sources of cylindrical waves, that is, sources independent of the z coordinate. The extension to three dimensions can be effected by a Fourier transformation with respect to z (see Sec. 5-11). Suppose we h.&ve an infinitely long filament of constant &.--Q current along the z axis, 0.8 shown in Fig. 5-13a. From the theory of Sec. 2--9, we should expect the field to be TM to z, expressible in terms of an A having only a z component "'. From symmetry, '" should be independent 1 G. Goubau, Surface-wave Tra.nlfmission Linee, Proc. IRE, vol. 39, no. 6, pp. 619624, June, 1951.

z

y

I

'-I" Y

,

P'

X

p

X Ca)

(b)

FIo. 5--13. An infinite filament of collltaDt a-e current (0) along the I" axis aDd (b) placed parallel to the I" axis.

m..

224

of ¢ and z.

TIM.E-BARMONIC ELECTROMAGNETIC FIELDS

To represent outward-traveling waves, we choose A. - ~ - CH,"'(kp)

where C is a constant to be determined according to lim '+'H.pd¢ = I ~o'f

Evaluating H

:=

V X A, we find

H. _ - Of _ -C 2. [H,"'(k,)] ~ j2C 8p

dp

t_O -rp

The preceding equation then yields I

C=4j I A. - ~ - 4j H,"'(k,)

Hence,

(5-83)

is the desired solution. The line current is the elemental two-d.imensional source, just as the current element (Sec. 2-9) is the elemental threedimensional source. The electromagnetic field is obtained from Eqs. (5-18), using the 1ft of Eq. (5-83). The result is (5-84)

Thus, lines of electric intensity run parallel to the current, and lines of magnetic intensity encircle it. Equiphase surfaces are cylinders, but E and H are not in general in phase. However, at large distances we have E. -

-.kl ~s;rp rT

H. = kl

'-;"1

(5-85)

~8.,;kp j g-li,

which is essentially an outward-traveling plane wave. The amplitude of the wave decreases as p-~t, in contrast to the r 1 variation in the threedimensional case. The outward-directcd complex power crossing a cylinder of unit length and radius p is

P,

1PE H··ds 102'r E.H:pd4J _";! Ikl!'H,"'(kp)[H,""(kp)]'

=

X

= -

The real part of this is the time-average power flow

(SJ"

(5-86)

which, by virtue

225

OYLINDRICA.L WAVE FUNcrIONB

of the Wronskian [Eq. (0-17)1, reducee to /PI -

Re (PI)

_ .k Ill'

(5-87)

4

Hence, the time-avemge power is independent of the distance from the source, as we should expect. It could be more simply obtained from Eqs. (&-85). If the current filament is not along the z axis but parallel to it, we can extend Eq. (5-83) by replacing p by the distance (rom the current to the field point. In radius vector notation, we specify the field point by p-uzX+UI/Y

and the source point (current filament) by '1' - Uz:z;'

as shown in Fig. 5-13b. point is then

+ u,y'

The distance (rom the source point to the field

I. - .'1 - V(x = .,;pI

x')'

+ p'l

+ (y

if)'

2pp'

cos (I/>

1/>')

We emphasize that A. is evalua.ted at '1 by writing A.(p) and that J is located at p' by writing I(ri). We can now generalize Eq. (5-83) to read

A.(.) ~ l~./ H."'(kl. - .'1)

(5-88)

This is our (re&space Green's (unction (or two-dimensional fields. The solution for two or more filaments o( z...directed current can be represented by a summation of the A/a from each current clement. Suppose we have two filaments of equal magnitude but opposite phase, as represented by Fig. 5-14a. As the separation 8--+ 0 and the magnitude 1_ 00 such that 18 remains constant, we have a two-dimensional dipole y

y

-I

j.-."'-lO""'+:-I;-'---X"" (aj

Flo.

~I4.

Sources of higher-order wave-.

(0) Dipole aource; (6) quadrupole aoUl'Ce.

226

TIME-HARMONIC ELECTROMAGNETIC FIELDS

source. Note that A. at a point (xtY) due to a current filament at (x',O) is the same as A. at (x - X',Y) due to a current filament at (0,0). Hence, for Fig. 5-14a, the vector potential is

A. - A+ -~.y) - A.'(Z +~.y) where A,l is that due to a single current filament at the origin [Eq.

(5-83)].

In the limit 8 --+ 0 the above equation becomes A,_ _0

(JA.l

-8-

ax

18

a

= - -. - (H o(2)(kp)] 4J ax

The differentiation yields kl. A.... 4j H 1 (2 J(kp) cos ¢

(5-89)

Thus, the vector potential of a dipole line source is a cylindrical wave function of order n """ 1. For the quadrupole BOUTee of Fig. 5-14b we have, by reasoning similar

to that above,

where (5-89).

is the vector potential of the dipole 8Ouroe, given by Eq. Hence, \ -kIsts! a A. 4; oy [H,"'(kp) cos ~J

A.(2)

which reduces to

A. =-

k2~?8t

1:l 2(2l(kp) sin 2q,

(5-90)

Thus, the vector potential of a quadrupole line SOurce is a. wave function of order 11. = 2. This procedure can be continued to obtain sources for the higher-order wa.ve functions. It can be shown (Prob. 5-29) that, when A. is a wave function of order 11, a possible souree consists of 211. current filaments equispa.coo on an infinitesim.al cylinder. We shall call such a SOurce a multipole source of order n. The dual analysis applies to the case of magnetic current filaments. It is merely necessary to replace I by K and A by F in the various vector-potential formulas of this section. For example, from Eq. (5-88), the electric vector potential at p due to a magnetic current filament at p' is F.(p} =

K~r) H ,'''(kip

- p'[)

(5-91)

Using both electric and magnetic multipoles, we can generate an arbitrary source-free field in homogeneous space (P > 0).

CYLINDRICAL WAVE PUNCI'lON8

z

Fro. 5-15. A cylinder of uniform cmrcnt.

y

The field due to a cylinder of Currents can be obtained quite simply by treating the problem as a boundary-value problem. We shall consider here only a cylinder of uniform z-directed surface current. (The general case is considered in Prob. 5-30.) The geometry of the problem is illuSotrated by Fig. 5-15. Because of the rotational symmetry, we choose

of

I A.- l A.+

~ C,J.(kp)

~ CI H.(t'(kp)

pa

The boundary conditions to be satisfied are

where J. is the density of the z-dirccted current sheet. Using Eqs. (5--18) with the above ¥to and satisfying the boundary conditions, we obtain

- ; .kaJ.H,'''(ka)J.(kp) E. -" 1- -2 ,kaJ,J.(ka)fl,"'(kp) T

p

a

as the only component of E. Let us calculate an impedance per unit length for this source, as we did for the ribbon of current in Sec. 4-12. By definition, p Z - jl[i where P is the complex power per unit length r

p = - Jo'J E.J:ad4l =

-2raJ:E.I..._

228

TIME-HARMONIC ELECTROMAGNETIC FIELDS

a.nd I is the total z-directed current

f

J =}o

2.

J.a

d~ ~ 2~aJ.

Hence, the impedance per unit length is

z ~ ,k4 J.(ka)H,"'(ka)

(5-93)

Using small-argument formulas for J a and H 0(2) I we obtain

. r ka ) z __O2"-, ( "--J21og2

(5-94)

la.....

where"Y = 1.781. Compare this with the Z of a ribbon of current [Eq. (4-127)]. Tho resistances (real parts) are identical. The reactance of a cylinder of current of small diameter d is approximately equal to the reactance of a ribbon of current of width w = 2d. More generally, it CaD be shown l by a quasi-static approximation that the impedance per unit length of a small elliptic cylinder of minor axis a and major axis b is the same as that of a circula.r cylinder of diameter d ~ ).f(a

+ b)

)

A ribbon is the special case a = 0 and b = w. 6-7. Two-dimensional Radiation. We can construct the solution {or an arbitrary two-dimensional distribution of currcnts by dividing the source into elemental filaments of current and summing the fields {rom all elements. For example, if' we have a J~, independent of z, each element J ds' produces &. vector potential 0

dA.

~J418' Ho"'(kle

-e'l)

where ds' is an element of area perpendicular to z. entire sourcc, we have

A.

~

;j II

Summing over the

J.(e')H."'(kle - e'l) dB'

where the integration extends over a. cross section of thc source. Since the equations for A z due to J~ and for Al' due to Jl' a.re of the same form as those for A~ due to J o, the above equation also applies for z replaced by x or y. Combining components, we have the vector equation

A(e) -

;j II

J(e')H,"'(kl. - e'l) d8'

(5-95)

I R. W. P. King, "The Theory of Linear Antennas," pp. 16-20, Harvard University PreM, Ca.mbridge, M88S" 1956,

CYLINDRICAL WAVE PUNcrlONB

229

representing the solution for an arbitrary two-dimensional distribution of electric currents. The cases of surfa.ce currents and current filaments are included by implication. The electromagnetic field is obtained, as usual, from H - V X A. The electric vector potential due to two-dimensional magnetic currents M is given by the formula dual to Eq. (5-95), or F(p) -

i ff j

M(p')H."'(klp - p'l) M

(5-96)

'the electromagnetic field in this case is given by E = - V X F. When the field point is distant from the source, our formulas simplify to a. form similar to those for three-dimensional radiation (Sec. 3-13). For klo - fl'l large, the Hankel function can be represented by the asymptotic formula

Furthennore, when p» p', as shown in Fig. 5-16, we have

If' - f"I--+

P -

p'

cos

(~

-

~')

(5-97)

The second term must be retained in the phase factor, exp (-jkle - £1'1), but not in the magnitude factor, 10 - (l'l-~. Hence, the vector potential. of Eq•. (5-95) and (5-96) reduce to

(5-98)

provided p »p~w

These arc the radiation-zone formulas corresponding

to Eqs. (3-95) in the three-dimensional ca.sc.

y Fio. ,&-16. Geometry (or determining the radia.-

So",,,,

tion field.

x

230

TIME-HARMONIC ELECTROMAGNETIC FIELDS

We now have the p variation explicitly shown in Eqs. (5-98), and simplified formulas for the radiation field can be obtained. A13 evidenced by Eq. (5-85), the distant field of a single current filament is essentially an outward-traveling plane wave; so the superposition of fields from all current elements should also be of this type. Hence, in the radiation zone, E, - .H.

E. - -.H,

(5-99)

which can be verified by direct expansion of Eqs. (3-4), using Eqs. (5-98). To obtain the field components, let us again divide the field into that due to I. given by H' = V X A, and that due to M, given by E" = - V X F. Retaining only the dominant terms (p-~ variation), we obtain

H.

= jkA.

H~ =

-jkA.

E': = -jkF. E~' =

jkF.

in the radiation zone. The corresponding E~, E~, H~', and H:' can be determined from Eqs. (5-99). The total field is simply the sum of the primed and doublo--primcd components, or )

E. = -jwp.A. - jkF. E. - -jwp.A. + jkF.

(5-100)

in the radiation zone, with H given by Eqs. (5-99). These formulas correspond to Eqs. (3-97) in the three-dimensional case. Note that, except for the contrasting p-J1 and r- 1 dependences, the radiation fields are of similar mathematical forms in two and three dimensions. 6-8. Wave Transformations. It is often convenient to express the elementary wave functions of one coordinate system in terms of those of another coordinate system.' We refer to expressions of this type as wave transformations. Some representative wave transformations are derived in this section. Others will be derived as they arc needed. Suppose we have the plane wave e-iz , which we wish to express in terms of cylindrical waves. (The conventional coordinate orientation of Fig. 5-1 is assumed.) This wave is finite at the origin and periodic in 2'11' on 1/>. Hence, it muet be expreesible 80S rf:r< = e-i,.-.:o::

• L ,,-- ..

a..J .. (p)e"'·

where the a.. are constants. To evaluate the a.., multiply each side by r~ and integrate from 0 to 2'11" on p'

It _ _ •

wbere the b.. are constants. To cvalua.te thc b.. , let p' -+ 00 and 1/>' - 0, and use the asymptotic formulas for the Hankel functions. Our original I R. V. Churcbill, "Fourier Series and &undary Value Problems," p. 141, MeGrawHill Book Company, Inc., New York, 1941.

232

TIM.&RAR.MONIC ELECTROMAGNETIC FIELDS

and our constructed expression for !f becomes

These are now representations of a plane wave, and, from Eq. (5-101),

it follows that b.. ... 1. Thus, •

2:

H ."'(p')J.(p)""f-'"

F


O. Figure 5-23 shows some radiation patterns (or tbe special case a = O. When q,' - a we have the solution (or a radiating slit in a conducting wedge. Finally, for plane-wave incidence we can specialize the first oC Eqs. (5-130) to the case p' --+ co. The procedure is analogous to that used to establish Eq. (5-128), and the result is H. = TH, '\' ••j'J.(kp) cosvW - a) 7f -

a

1::1

cosv(~ -

a)

(5-133)

•

This is the field due to a plane wave polarized orthogonally to z incident at an angle tIJ' on a wedge of angle 2~ The case a = 0 gives (5-134)

which is the solution for a plane wave incident on a conducting half plane. 6-11. Three-dimensional Radiation. A thrro-dimensionnl problem having cylindrical boundaries can be reduced to a two-dimensional problem by applying a Fourier transformation with respect to z (the cylinder

Flo. 5-23. Radiation patterM for a magnetic current filament adjacent. to a conducting half plane, p' - II, . ' _ ../4. (A.fUr J. R. Wait)

243

CYLINDRICAL WAVE FUNCTIONS

z

axis). 1 For example, if t/;(x,y,z) is a solution to the three-dimensional wave equation

a' a' a' ) ( ax' + ay' + az' + k' '" then J,(x,y,w) =

f

-- -

r'

l(z)

- 0

_*".. t/;(x,y,z)e-i¥· dz

y p

will be a solution to the two-dimensional wave equation

(:;, + :;, + «,) J, =

x

0

where «' = k' - w'. Once the twodimensional problem for J, is solved, the three-dimensional solution is obtained from the inversion 1 ~(x,y,,) - 2.

f"_"

FlO. 5-24. A filamCDt of curreDt aloDg

tho z a.xis.

~(x,y,w)"" dw

This is usually a diffioult operation. Fortunately, in the radiation zone the inversion becomes quite simple. We shall now obtain this far-zone inversion formula. Consider the problem of a filament of z-directed current along the z axis, as illustrated by Fig. 5-24. The only restriction placed on the current l(z) is that it be Fourier-transformable. In the usual way, we construct a solution H-VXA (5-135) A = u.t/; where t/; is a wave function independent of ¢ and representing outwardtraveling waves at large p. Anticipating the need for Fourier transforms, we construct

~ - i. J-"" f(w)H,"'(p yk'

w')...• dw

which is of thegenersJ form of Eq. 105·)1). Tne Fourier transform 0/ jP is evidently ~ - f(w)H ,'''(p yk' w') The I(w) is determined by the nature of the source, according to

10'1.. R. p d¢ -;::t lew) I

Thill applies to cylinders of arbitrary cross sectioD as well as to circular cylinders.

TIME-HARMONIC ELECTROWAGNETIC FIELDS

where D. and I are tDe transforms of HI and I. ment formula for a,en, we have

!If

From the small-argu-

2·

B. - - -iJp --+ ..1.f(w) ,......0 rp and the preceding equation yields few) -

It')

Hence, the "transform solution" to the problem of Fig. 5-24 is

f"

-1.., 8rJ

f

_ ..

l(w)H,"'I...v'k'

lew) -

where

w')...• dw

f-"" I(z')clw'dz'

(5-136) (5-137)

The field is obtained from !/I according to Eqs. (5-135). Compare the equations of t.his paragraph to those of the second paragraph of Sec. 5-6. The transformed equations in t.he three-dimensional problem are of the same form as the equations in the twCHiimensional problem. Another solution to the problem of Fig. 5-24 is the "potential integral solution II oC Sec. ~9. This is J.

•

~

f"_.

I(z')

.-J"/~+-lished by contour integration, using the method of steepest descent. I Finally, we shall need a formula similar to Eq. (5-142) valid for Hankel functions of arbitrary order. The desired generalization can be effected by considering the asymptotic expression

f2i j"~ --'Vn

H ..U}(:x:) ----+ from which it is evident that

--

8 ..(2)(%) ---+ j"B ,(I) (:x:) AAlong as" ¢ 0 or 7', we have p -+ 00 as r --+ co I since P = r sin 8. Also, if k is complex (some dissipation assumed), then .y k' - 10' is never zero on the path of integration. We are then justified in using the asymptotic formula for Hankel functions and can replace the HoC') of Eq. (5-142) by ;....8 ..(1). The result is

f-·.

rI'O'

l(w)H.U'(P y'k'

w·)...·dw--+ 2-j'+'1(-k cos B) ~.

r

(5-143)

We shall have use for this formula in the radiation problems that follow. 6-12. Apertures in Cylinders.' Consider a conducting cylinder of infinite length in which one or more apertures exist. The geometry is I

A. Erde1yi, "Aaymprotic Expanaiona," pp. 26-27, Dover Publieationa, New York,

1956. t Silver &nd Saunders, The External Field Produced by a Blot in an Infinite Cir· cular Cylinder, J. Appl. Ph"., vol. 21, 00. 5, pp. 153-158, February, 1950.

246

TDUi-HAlWONIC ELECTROMAGNETIC FIELDS

z r

FlO. &-25. An aperture in a condu.ctiog cylinder.

y

p

x

------ -

shown in Fig. 5-25. We seck a solution for the field external to the

cylinder in terms of the tangential components of E over the apertures. Anticipating that we shall use transforms of the fields, let us define the "cylindrical transCorms l l of the tangential components of E on the cylinder as

J." d~ f" E.. (n-,tD) = -2 1 J." dljl f" B.(n,1D) = -2 1 r

0

T

-.

0

_.

dz E.(a.~.z)c~e-/W· (5-144)

dz E.(a,~,z)r~rjw·

The inverse transformation is

2: ".. f.", 2~ 2: ,;"' !-", .. --. 1

"

= 2T

E.(n,.,),·" dw (5-145)

,

E.(.,¢,z) -

E.(n,")&-· d.,

Note that these nre Fourier series on 41 and Fourier integrals on %. The field external to the cylinder can be expressed as the sum of a TE component and TM component. According to the concepts of Sec. 3-12, the field is given by E - -'0 X F -

;w"A +.,!.. '0'0' A

H ."" V X A - jwiF where

A "" u.A.

JW'

+ JW. J- vv· F F - uJ'.

(5-146)

(5-147)

247

CYLINDRICAL WAVE FUNCTIONS

We now construct the wave functions A. and F. as

(5-148)

"

F.

=1. \'

~ ,, __ 4w

eM

f"-. g.(w)H."'(p Vk' - w')e'" dw )

which are of the form of Eq. (5-11). We choose the Bessel functions as to represent outward-traveling waves. We choose the rp and z functions such that the field will be of the same form as Eqs. (5-145). To determine the f,.(w) and g.. (w) in Eqs. (5-148), let us calculate E. and E. according to Eqs. (5-146). The result is H,.(I)

" 1..(

E.(p,~,,)

= -2 1 \'

E.(P,tP,z)

"".!. \' 2r L.

rJWf

tJ·. f"-_ (k' -

"

.... --

ei ••

f"__ [- ~w f .. JWf

(w)H .. (ll(p Vlc 1

+ g.. (w) v'lcl -

Since these equations specialized to have

f.(w) _

w')!.(w)H."'(p Vk'

p =

-

w 1 1I,,(I)'(p Vk 1

w')t!-· dw

Wi)

-

Wi)] eiw' dw

a must equal Eqs. (5-145), we

jw.E.(n,w) (k 1

_

w 1)H"U)(a Vk2

g.. (t.r) = vk! _

Wi

fI..

Wi)

(~l'(a Vlcl _

Wi) [

2.(n,w)

(5-149)

+ a(k,nw w') E.(n,W)] This completes the solution. The inversions of Eqs. (5-148) are difficult except for the tar zone, in which case we ean use Eq. (5-143). Hence, we have

(5-150)

248

TWE-HARMONIC ELECI'ROKAGNETIC FIELDS

z

Z

-a3

f-a':::

I

I

-J.~I--'

I

II 'f. ---

Flo. 5-26. A conducting cylinder and (a) an a,;ia.I slot, (b) a circumferential slot.

11--·a--1

v--

I

I

-~,

-~,

(b)

(a)

Finally. in the radiation zone Eqs. (3 97) apply j hence R

E, --+ jwp ,.......

rib ain

rr

8

-

\ ' eJrl.j"+!j.. ( -k cos 6)

Lt

n--"

(&-151)

1t _ _ OO

Thus, the radiation pattern of apertures in cylinders is relatively easy to calculate. The only difficulty is that the number of significant terms in the summation becomes veri)' large for cylinders of large diameter. To illustrate the theory, let us consider the thin rectangular slot in the two orientations shown in Fig. 5-26. For the axial Blot we shall assume in the aperture

E.

=

a. V

rz

-cosL

!

-~/-

~ T ~. \ ' •• j'J,(kp' Bin B) coo ,(~' ex Lt ,

where

- a) COB ,(~ - a)

(5-162)

....'"

';'0 = K l - eft.. _· hr

v = 2(...

ex)

m =

(5-163)

.9,

I, 2,

The electromagnet.ic field is found from'" according to Eqs. (5-19). To relate this solution to the field from an aperture in a conducting wedge, we again apply reciprocity [Eq. (3-35)}. This reduces to

ff

.,..,

(E,'H: - E,'lI.·) d.

~ KIH,'

(5-164)

where superscripts a and b refer to the fields of Fig. 5-2& with Il replaced by Kl, and of Fig. 5-28b, respectively. From Eqs. (5-19) and (5-162) we 11. N. Sneddon, "Fourier Tranaforma," p. 6, McGraw-Hili Book Company, Inc., New York, 1951.

253

CYLINDBJCAL WAVE FUNctIONS

calculate

H: ...

1fkl sin /1 cos /1 ( ) wp. 1f a

l H.' = .1fk ( sin I 8) 1f1J JWIJ. 1f a

L:. L: ¥t,

,

.

f.J·J~(kP' Stu 8)

cos v(t/>' - a) cos vet/> - a)

f.j·J.(kP' sin 8) cos v(t/>' - a) cos vet/> - a)

•

Finally, we evaluate Eq. (5-164) and use the radiation-zone relationship

E. _ -"fB,

= ~H. Sin

8

The result is

"'-~. E. - 4r(1f a)

where

'\' e.j· cos vet/> L, ,

Q'.(w,u) =

- a)(cos 8 g.(k cos 8, k sin 8)

(&-165)

+ j sin 8 h.(k cos 8, k sin 8)]

J--.. e

it..

dz fo - J~(up) dp E.(p,a,z)

h.(w,u) ~ /_"" .... dz / ; J.(up) dp E.(p,a,z)

(5-166)

We now have a complete solution for the radiation field from apertures in conducting wedges. As an example, let us calculate the radiation from a narrow axial slot of length L, as shown in Fig. 5-29. We shall assume that in the slot

E.

~

..

VI(p - 0) cas L

(5-167)

is the only tangential component of E. The I, Q', and h functions [EQs. (5-161) and (5-166)] are then found to be I.~O

8,-0 cas (wL/2) J ( ) h• ~ 2.VL 7'1 (Lw)! • ua From Eq. (5-160) we see that E. - 0, and from Eq. (5-165) we have E•

""

I() T

· 8 caslk(L/2) cas 8) ... 1 (kL cos 8)1

8m

L: .,j·cos,(~

- a) J .(kosin 8)

(5-168)

•

where v =~, 1,

%, . . ..

Plots of

FlO. 6-29. A narrow axial slot in a conductine halr plane.

254

TIME-HARMONIC ELEctROMAGNETIC Fl.ELDS

Flo. &-.30. Radiation patterns for axial slot.ll in a conducting half plaoe (the slot in fLn infinite ground plane is shown dashed).

the radiation pattern in the plane 8 - 90 0 arc shown in Fig. 5-30 for the case ex - 0 (half plane). The cases a ... 0.16A and a = 0.96>. are sbown, with the infinite ground-plane pattern shown dashed for comparison. PROBLEMS

6-1. Show that Eq. (5-12) is a IlOlution to the ecalar Bclmholtl equation. 6-2. Show that", - Oog p)e-'k ia a 1I01ution to the sealar llelmholu equation. Determine the TM field generated by this" according to Eqa. (5-18). Sketch the t and :Je Jines in a ~ - constant plane. What pbysicalllystem IIUpporla this wave? Repcat for the TE case. 6-S. For two-dimensional fields (no :e variation) ahow that an arbitrary field in a llOurce-frce homogeneous region can be eJl:prCMC'd in ten.Ila of two lICalar wave fuoctioDa, ,p, and lJ-1, according to Eqs. (3-79) whcro A - u"py.,

Note that. ihis corresponds to choosing

*

j;P(~~)

~

_ _j~(F:)

illlltead of Eqs. (3-80). 6-4. A circular waveguide has & dominant mode cutoff frequency of 9000 megacycles. What ia ita inside diameter if it. is air-filled! Determine the cutoll frequencies for the next ten loweJ~rder modeJ. Repeat for the case f. _ 4. 6-6. All the waveguides whose CtOSll eeetiona are shown in Fig. f).4 are characl.erized by wave functions of the form 'I- - B.(.c,p)h(n.)eslA'••

where TM modes are detenninoo by Eqs. (&-18) and TE IDOdea by Eqs. (&-19). phue constant ia given by

The

CTLTh"DRICAL WAVE FUNCTIONS

255

Let. a denote the inner ndius and b the outer radius of the eoaxiaJ. waveguide of Fig. 5-411. Show that for TM modes B..(i",) _ N ..(i,o)J.(i"p) - J ..(i,a)N..(i"p)

h(n.) _ &in n.

or

COlI

n•

..here n - 0, 1.2, . . . , and i, is a root of

Show that for TE modes B.(k",) - N~(kptJ)J..(k,p) - J:(k,a)N..(k,p) h(n.) - sin n. or cos n •

..here n _ 0, 1.2, . . . ,and k, is a root of

G·8. Show that the modea of the coaxial waveguide with a bame (Fig. 6-4b) are ehal'$Cterizcd by the lAme B.(k",) fWlction. u the coaxial guide (prob. &-5), but for Tl\f modes n - ~. 1.

H.2, .

&lid for TE modes n -

..here the baffle iI at • - O.

0, }i, 1, M•.•.

The dominant mode is !.he lowen TE mode with

• -)i.

5·7. Show that the ,.,.edge waveguide of Fig. 6-k supportA TlI,l modes specified by ~na

....... &lid k,IJ is a

_ J .(l-".) ain "'. cua,T 2... :w

n--.-,-.·· . • a .... KJ'O

of J.(lllS).

Show that it aupporta TE modes specified by

....

I/-TZ _ J.(i".) COIn. e*ia..

• 2• n-O-.-,··· '

• h""

and kpa is a tero of J~(k,a). The guides of Fip. 5-4c and d are the special cases ... - 2.. and T, respectively. G-8. Show that the cutoff wavelengtb for tbe dominant mode of the circular waveguide witb bame (Fig. 5-4c) iI

1-'. Using the perturbational method of Sec. 2-7, ahow tbat tbe attenuation eonIl&nta due to conductor loaaes in a circular waveguide arc given by

.. '" "'1

~ - :;;-:;7i~:7:ff,iii for all TM modes, and by

.. - '" VI"

V.tn'

V.tn'

[(>:,>:'- .' + GY]

256

TU.I];-RARMONlC

ELEC'I'ROMAGN'ETlC Fl,ELDS

(or all TE modes. Note that for the "circular electric" modes (n - 0) the attenua,.tion decreases without limit as f -. ... 6-10. Consider the two-dimensional "cireulatinc waveguide" formed of concentric conduct.iDg cylinders, - CJ a.nd p - b. ShOw that the wave function

specifies circulating modes TM to

I:

according to Eqa. (5-18) if n is a root. of

Show that tbe above wave function specifics modes TE to z according to Eqa. (6-19) if n ill a root of J~(ko)

B .A

J:(kb) N .(kb)

------= N~(ko)

6-11. For the TM radial wave specified by Eq. (5-33), show that the radial phaae constant of E. is given by Eq. (b-36), while tbe radial phase constant of H. i.

IJ'-.![l-(~)']. 1, k r, IJ.(k,,)!' + IN.(},,»)' Tp

Show that Eq. (6-37) is also valid for this phase constant.

6-12. Consider the TM radial wave impedances of Eqs. (5-38) and (5-39). that for luge radii Z+,.nI _ Z_..TN _ 11

Show

.......

and that for small radii

Z+.~ - Z_.~·~ .......

{

~k"(d;I"+) k [(2• )'(k..»~ + J'J ",p ft.! "'2 n

.-0 •

>0

where y _ 1.781. 6-18. Conaider the radial parallel-plate w&veguide of Fig. 5-50. For the transmission-lino mode IEq8. (5-45)J. one can define a voltage and current. lUI V(P) _ -oB.

Show that V &nd I sati8fy the transmiMion-line equatiool!l dV dp

-

, LJ

-J(jI

dl -jwCV dp -

where Land C are the ".static" parametera L

_.e 2.,

c _ 2rtp

•

Why.should we expect circuit conecpta to apply for this mode? 6-1'. Cooaidenhe wedge guide of Fig. 5-Sb. For t.he dominant mode (Eq. (5-49»). one can define a voltage and eurre.nt. as 1(P) - H~

257

CYLINDRICAL WAVE FUNCTIONS

Show that V and 1 aatiafy tbe tranamission-liDe equation (prob. 6-13) with

c-~

•••

6-16. Show that the re8Qnant frequenciel of the two-dimensional cylindrical cavity (no I: variation, conductor over p - 0) arc equal to the cutoff frequencies of tbe circular waveguide. 6-18. Following the perturbational metbod used to derive Eq. (5-58), sbow tbat the Q due to conductor lOIlllel for tbe various modes in the circular cavity of Fig. 6-1 are

1-17. The circular cavity of Fig. 6-7 baa dimensions II - d - 3 centimetera. Detennine the first ten resonant. frequeDciei and tbe Q of the dominant mode if the ...alls are copper. 6-18. Consider tbe dominant mode of the partially filled radial waveguide of Fig. &-90. Show tbat for small a and large ,. the phase constant is

Compare tbill to the uniform transmiasion-line formula IEq. approximations

L _ lAid

+ "1(0

(~)I,

using the .tatic

- d)

2r. 6-19. Conlider tbe dieleetric-elab radial guide of Fig. S-9b. Let II - 41. and ,.. and 0 - )... Which model can propagate unattcnuated in the a1ab? Rcpeat the problem for the coatcd~onduetorguide of Fig. 5-9c witb t - 0/2. 6-20. For the partially IDled circular waveguide (Fig. 5-10a), show tbat tbe characteristic equation (Eq. (5-74)] for the n - 1 modell reduces to

PI -

IANI(A:_.l!) where

+ BJI(k_.l!)IlAN;(k,ib) + BJI(k,tb)l

- 0

A - k,IJ;(k,10)J1(k_lO) - k,J'I(k_IO)J1(k_10) B - k,aN';(k,tO)J 1 (k,la) - k_IJ;(k,lo)N.(k,,a)

6-21. Consider the dominant. (n - 1) mode of the dielectric-rod waveguide of Fig.5-1Ob. Show tbat for small a tbe cbaracteriatic equation beeomes (,.1

+ ,.f)(11 + If) 2,... ,K.(PXl)

Note tha.t t.bere ia no cutoff frequency.

258

TIME-HARMONIC ELECTROMAGNETIC FIELDS

6-22. The field external to a dielectric-rod waveguide varies as K1{vp). Using the results of Prob. 5-21, show that for a small (4 « A,), nonmagnetic &01 - #1) rod I 2 1 ~l + t l og:;va ... (k1o)l h - h

where,. - L 781. Take I I - 9'1 and a - 0.1).1, and calculate the distance from the lUis for which the field is 10 per cent of its value at the surface of the rod. 6-23. Consider the circular cavity with concentric dielectric rod, as shown in Fig. &-310. Show that the dominant resonant frequency is the smallest root of

~ J;(kc) "" ~ [No(ktIJ)J;(kl$:) - Jo(kotl)N;(ktcJ] "Jo(ke)

1'/0

N.(koa)Jo(koe)

Jo{koa)N.(koc)

For small c/o., show that resonant frequency"" is related to theempty-eavity resonll.nce %"

2.405

:1:01 -

according to

where

I,. -

./to.

1

~a

I

I

--a

l ~ ------ 'J -----....

r

b L

d

• (b)

(a) FlO.

6-31.

Plirti~ly

filled cavities.

6-24. Consider the circular cavity with a dielectric slab, as shown in Fig. 5-31b. Show that the characteristic equation for tbe resonant frequency of the dominant mode is

-~tank,b

-

•

k.t _ kt _

where

(X;IY

Show that when both d and b are small Iolr ...

lola

•

lr.l;:::;J(~l::::JIZ/.~, )~bId l)bjd

-V~l + ~

where lola is the cmpty-cavity reaooant frequency, given in Prob. 5-23, and and ~ - /JI/J •.

f. -

.1"

259

CYLINDRICAL WAVE FUNClIONS

FlO. 5-32. Wedge jn A circular cavity.

1-26. Consider the circular cavity "dth a conductiDg wedge, as showD in Fig. 5-32 Show that, for daman, the resoDant frequency of the dominant mode is given by

whcre 1D is the firet root of J.(w) - 0 and values of ware

•

11 -

.. /(2..

- ..,. Somc representative

0.5

0.6

0.7

0.8

0.9

1.0

3.14

3.28

3.4.2

3.56

3.70

3.83

1-26. Figure 5-330. shows a linear den.e.ity of z-directcd current elementa alODg the r axil. Show that the field is given by H - V X A where

Show that t.be field is idcntical to that. produced by t.he magnetic dipole formed of a-direct.ed magnet.ic currents +K at 11 - -'/2 and -K at 11 - ,/2 in tbo limit, _ O. &-27. Show that the field of the magnetic-dipole source of Fig. s-33b in the limit ._ 0 is given by E _ -v X u.'" where

6-28. Consider the quadrupole 80uree of Fig. s-33c in the limit '1 - 0 and '1 _ O. Show that the field is given by H - V X where

u."

a-2ft Figure &-33d represents a 50urce of 2n current filaments, equal in amplitude but alternating in sigo, on a cylinder of radius p _ a. Show t.hat, in the limit a --. 0,

260

TIME-HARMONIC ELECl'nOMAGNETIC FIELDS

y

y

•

J.l

, +

T

X

••_K ...1.

CG)

(b)

Y

+1

Y

A

,A ,

X

- I -I

f.--,,-.:.j

+I

X

- I, + I, - I' + I' - I' , +1

,+1 ,- I G

'+1 '-I •-'+1 I

(e)

X

(d)

FIo. S.33. Some two. a~'

~

(6-6)

0

Note that there is now no interrelationship between separation constants. The ~ equation is the familiar harmonic equation, giving rise to solutions h(m41). The R equation is closely related to Bessel's equation. Its solutions are called spherical Bessel functions, denoted b.(k1'), which are related to ordinary Bessel functions by (&-7) (see AppendiX" D). The 8 equation is related to Legendre's equation, and ita solutions are called a3sociaied Legerulre functiona. We shall denote solutioDS in general by L."(C08 8). Commonly used solutions are

L..... (C08 0) "..." P .... (C08 8), Q.... (cos 8)

(6-8)

where P ."(cos 8) are the associated Legendre functions of the first kind and Q.-(C09 8) are the associated Legendre functions oC the second kind. These are considered in some detail in Appendix E. We caD DOW form

266

TIME-JiARMONIC ELECTROMAGNETIC FIELDS

product solutions to the Helmholtz equation as

f ... - b.(kr)L.-(cos

9)h(m~)

(&-9)

These are the elementary wave functions (or the spherical coordinate system.

Again we can ;construct more general solutions to the Helmholtz equation by forming linear combinations of the elementary wave functions. The most general form that we shall have occasion to use is a summation over possible values of m and n

~

• •

II C_.•b.(kr)L.-(cos 9)h(m~)

-.

(6-10)

where the C.... are constants. Integro.tions over m and n are also solu· tions to the Helmholtz equation, but such forms are not needed (or OUf purposes. The ho.rmonic functions h(mlj) have already been considered in Sec. 4-1. If a singlc-valued y, in the range 0 to 2.. Oil 4J is desired, we must choose h(m;.) to be a linear combination of sin (mQ) and cos (m.p), or of ~ and r~, with m an integer. A study of solutions to the associated Legendre equation shows that aU solutions have singula.rities a.t 6 - 0 or 6 = 'I' except the P.-(cos 6) with n an integer. Thus, if oJ- is to be finite in the range 0 to 'I' on 6, then n must also be an integer and L.-(C08 6) must be P.-(cos 9). The spherical Bessel functions behave qualitatively in the same manner 88 do the corresponding cylindrical Bessel functions. Thus, for k real, i.(1cr) and n .. (kr) represent standing waves, h.(II(kr) represents au inward-traveling wave, and h..(l)(kr) represents an outwnrdtraveling wave. IncidentaUy, it turns out that the spherical Bessel functions are simpler in form than thc cylindrical Bessel functions. For examplc, the zero-order functions are . (k r ) .... sin kr Jo kr

(&-11)

no(kr) .... _ cos kr kr

The higher-order functions are polynomials in l/kr times sin (kr) and C06 (kr), which can be readily obtained from the recurrence formula. The only spherical Bessel functions finite at r = 0 are the i.(Jrr). Thus, to represent a finite field inside a sphere, the elementary wave functions are (&-12) r 0 included "'_.• - i.(kr)P.-(cos 9).... :IZ

SPHERICAL WAVE FUNCTIONS

267

with m and n integers. To represent a finite field outside of a sphere, we must choose outward-traveling waves (proper behavior at infinity). Hence, '!'",.... = h"U)(kr)P,,"'(cos 8)ei"'. r --+ !Xl included (6-13) with m and n integers, arc the desired elementary wave functions. To represent electromagnetic fields in terms of the wave functions ,!" we can use the method of Sec. 3-12. This involve.s letting y, be a rectangular component of A or F. The z component is most simply related to spherical components; hence the logical choice is A = u.y, = Ury, cos fJ - uey,sin fJ

(6~14)

which generates a field TM to z. Explicit expressions for the field components in terms of,!, are given in Prob. 6-1. The dual choice is F = u.'" = ur'!' cos 8 -

U6'"

sin 8

(6-15)

which generates a field TE to z. Explicit expressions for the field components are given in Prob. 6-1. An arbitrary electromagnetic field in terms of spherical wave functions can be constructed as a superposition of its 'I'M and TE parts. An alternative, and somewhat simpler, representation of an arbitrary electromagnetic field is also possible in spherical coordinates. Suppose we attempt to construct the field as a superposition of two parts, one TM to r and the other TE to r. For this we choose A = urA, and F = urF'" with the field being given by Eq. (3-79). The A, a.nd F, are not solutions to the scalar Helmholtz equation, because 'V 1 A. ¢ ('V 2A),. To determine the equations that A. and F r must satisfy, we return to the general equations for vector potentials [Eqs. (3-78)]. For the magnetic vector potential we let A = u,A, and expand the first of Eqs. (3-78). The 0 and q, components of the resulting equation are, respectively,

where )

The power dissipated in the conducting walls is approximately

tl', -

Illlff> 11l1'd. = ill 8; 1,'(2.744)

(1)-36)

Hence, the Q of the cavity is _ Q _ ~W _ ~

~"(1.14)

kIllJ ,'(2.744)

= 1.01

l! ill

(1)-37)

I E. Jahnke and F. Emde, "Tablea of Functiona." p. 146, Dover Publications, New York, 1945 (reprint).

273

SPHERICAL WAVE FUNCTIONS

Comparing this with Eqs. (5-58) and (2-102), we see that the spherical cavity has a. Q th3.t is 25 per cent higher than the Q of a circular cavity of height equal to its diameter and 35 per cent higher than the Q of a cubic cavity. The Q's of higher-order modes are given in Prob. 6-4. 6-3. Orthogonality Relationships. In many ways the Legendre polynomials are qualitatively similar to sinusoidal functions. For example, the P.. (cos 8), sometimes called zonal harmonics, form a complete orthogonal set in the interval 0 to 11" on 8. An arbitrary function can therefore be expanded in a series of Legendre polynomials in this interval, similar to the Fourier series in sinusoidal functions. The functions P.... (cos 8) cos mq, and P.."'(cos 0) sin 1»4', sometimes called te88eral harmonics, form a complete orthogonal set on the surface of a sphere. Hence, an arbitrary function defined over the surface of a sphere can be expanded in a series of tessel'al harmonics. We shall, in this section, derive the necessary orthogonality relationships. For our proof it is convenient to use Green's theorem [Eq. (3-44)], which is

1ft (~, ~~ - ~, ~:,) ds -

Iff (~,V"", - ~,V"",)

dT

(6-38)

The right-hand side vanishes if 1ft and 1f, are well behaved solutions to the same Helmholtz equation. Assuming this to be the case and applying Eq. (6-38) to a sphere of radius r, we have rl

10

2 .,

d¢

fa" dO sin 0 ( 1fl 0;1 -

1f,

at)

=

0

(6-39)

In particular, choose '" ~ j.(kT)P .(cos e)

~, ~

j.(kr)P.(cos e)

which are solutions to the Helmholtz equation. becomes

27rkT'(j,.j~ - jqj~) This must be valid for all Hence,

Tj 80,

10" p,.pq sin 0 dO

Equation (6-39) then = 0

if n ,e q, the integral itself must vanish.

fo" P ..(cos O)Pq(cos 0) sin 0 dO =

0

(6-40)

When n "'" '1, we have

for [P.. (cos O)p sin 0 dO = 2n ~ 1

(6-41)

which can be obtained by using Eq. (E-lO) and integrating by parts.

274

TIM:E-BAltMONIC

ELECTROMAGNETIC FIELDS

To obtain a Legendre polynomial representation of a function /(8) in

o to 'It: on 6, we assume

fee) -

l•

.-.

(6-42)

a.P.(COB e)

Multiply each side by P ,(cos 8) sin 8 and integrate from 0 to

'If

on B.

•

J: f(e)p,(c,," e) sin ede - l a. J: P.(c,," e)p,(c,," e) sin ede

.-.

Each integral on the right vanishes by Eq. (6-40), except the one n - P, which is given by Eq. (6-41). The result is a. -

2.+11.' 2

(6-43)

• f(B)P .(c,," B) sin B de

Equation (6-42) with the coefficients determined by Eq. (6-43) is called a

Fourier-Legendre &erie.. It converges in the same sense as the usual Fourier series. For a more general result, define tbe tesseml harmonics as

T••·(e,~) - P.-(cos e) COB m~ T...~(81t/J) = P..... (cos 8) sin mt/J

(6-44)

and assume two solutions to the HelmholLz equation as

These are well behaved within a sphere of radius r; hence Eq. (&-39) applies and reduces to

kr2(j,J~ - ivj~)

102• d¢ 10" dB T....·1',/ sin Bde "" 0

The term outside the integral vanishes for arbitrary hence

T

only when n - q; n"q

For the t/I integration, we have the known orthogonality relationships

102" sin mI

:;I!

P, q, i

(6-51)

p, q, j, we have

d~ j,' dB [Sin B(a~;.;)' + do e~;';)'] "",,(n + 1) 2n + 1 ~ 2",,(n + I) (n + m)! 2n + 1 (n m)!

l

m = 0, i"'" e

(6-52)

which can be obtained by integrating once by parts and using Eq. (6-47). 6-4. Space as a Waveguide. We have seen that in a complete spherical-shell region (0 ::; 8 ::; 71,0 ::; q, ::; 211") only spherical wave functions of integral m and n give a finite field. The fields specified by these wa.ve functions can be thought of as the Hmodes of free space." When viewed in this manner, the space is oft-en called a 8pherical waveguide, even though there is no material guiding the waves. The spherical coordinate system is defined in Fig. 6-1. There exists a set of modes TM to T, generated by

n.'O)(h)!

, - 7' '(B) J ( A) r "''' .... ,q, \ 11,,(21(kr)

(6-53)

where n = I, 2, 3, . . . ; m = 0, 1, 2, ' , . , ni and i = e or o. T functions are defined by Eqs. (6-44), and the field is given by

The

(6-54)

a..

Inward-traveling waves are represented by the (!) and outwardtraveling waves by the 11.. (2), In the dual sense there exists a set of

277

SPHERICAL WAVE FUNCTIONS

modes TE to r, genera.ted by

,- , In.W(Ier»)

(F,)•• - T.. (e,~)

(6-55)

n.m(kr)

where n = 1, 2, 3, . . . ; m = 0, 1, 2, . . . , n; and i = e or o. field is given by HT£;

••

1 = --.-VXE~~i

!w.

The

(6-56)

Thc set of TM plus TE modes is complete, that is, a summation of them cRn be used to represent an arbitrary field in a source-free region. Mode patterns for the TM ol and TE ot modes are sketched in Fig. 6-4. The 'fM and TE modes are dual to each other; so an interchange of E by H and H by -E in Fig. 6-4 gives the TEo I and TM ot mode patterns. The spherical modes are qualitatively similar to the radial modes of Sec. 5-3. There is no well-defined cutoff wavelength but rather a Ir cutoff radius." To illustrate, consider the radially directed wave impeda.nces for the TM modes E,+

E.+. B .. C21 / (kr)

Z+,'" = H.+ = - H,+ "'" 1'1 B,,(2)(kr)

E,E.. B,,{l)/(kr) = - H.- = H,- = -JY] O,,(I)(kr)

(6-57)

where the superscripts + and - denote outward- and inward-traveling waves, respectively. Note that, for real Y] a.nd k, Z_/'M = (Z+r™)*. For

9(---_

Fro. 6-4. Mode patterns lor the

(a) TM ol

and

(b) TE ol

(b)

modes ollrce space.

278

TDlE-HARMONIC ELEC'rROYAGNETlC FIELDS

the TE modes the radially directed wa.ve impedances are

Z

n:

0::

+r ZTE

-

E,+ __ E.+ '"'" _j." B.(t)CJ.;r) lJ.+ H.+ O.Ull(kr) o. E.- E.- J":;11~("ik::rl H.H."' 11 ,,(l)'(kr)

(6-58)

The behavior of these wave impedances is qualitatively similar to the behavior of the twtrdimensional wave impedances, illustrated by }~ig. 5-6. In other words, the wave impedances of Eqs. (6-57) and (6-58) are predominantly reactive when kr < n, aod predominantly resistive when Icr > n. The value kr = n is the point of gradual cutoff. Nate that this cutoff is independent of the mode number m. The frequency derivative of the various wave impedances is of interest for determining the bandwidth of various devices (see Sec. 6-13). A Dovel way of representing this frequency derivative, which also illustrates the above cutoff phenomenon l was devised by ProCessor Chu. 1 He took the wave impedances and, using the recurrence formulus for spherical Bessel functions, obtained a partial fraction expansion. For example, for the TM impedance of outward-traveling waves Z+rTV. "'" 1'1

{j~ + 2n _ jkr

1 +l

1

2.3+ jkr (6-.\9)

1 +-31 -+~­ jkr j~ + 1

This can be interpreted as a ladder network of series capacitances and shunt inductances, as shown in Fig. ~5a. The equivalent circuit (or the TE.. modes is shown in Fig. 6-5b. Those of us familiar with filter theory will recognize thc equivalent circuits as high-pass filters. The dissipation in the resistive element at the end of the network represents the transmitted power in the field problem. It is therefore apparent that, for fixed r, the higher the mode number n the less easily power is transmitted by a spherical waveguide mode. I L. J. Chu, Physical Limitations of Omnidirectional Antennas, J. Appl. Phl/., vol. 19, pp. 1163-1175, December, 194.8.

279

8PHERICAL WAVE F'ONcrION8

"

2n-3

- - - - f--...,.--i"f---,-- - - --ZTII _ _

••

"'

L 2n-I '--

la,

"

"

C· 2n-1

2n-5

~------,,-""'H"--..,...--i'f-- -

-.-

z'£ __

••

'"

FIG. 6--5. Equivalent circuits for the (0) TM.. And (b) TE... model of free apaee.

A quality factor Q. for modes of order

n

can now be defined

88

(6-60)

'W.. > 'W. where W. and W.. are the average electric and magnetic energies stored in the C's and L's, and (J' is the power dissipated in the resistance. In TM waves 'W. > OW.., while in TE wa.ves 'W. > OW.. However, the two eases are dual to each other; so the Q's of TM waves are equal to the Q's of the corresponding TE waves. An approximate calculation of the Q's for Q > 1 is shown in Fig. 6-6. Note that for kr > n the wave impedances are low Q and for kr < n they arc high Q_ This again illustrates the cutoff phenomenon that occurs at kr - n. 6-6. Other Radial Waveguides. A number of structures capable of supporting radially traveling waves can be obtained by covering 8 = con.. stant and ~ - constant surfacee with conductors. Such II radial waveguides" are e.ho·.rn in Fig. &-7. We can have waves outside or inside eo single conducting cone, 88 shown in Fig. &-70 and b. These two cases are actually a single problem with two different values of 81• The fields must be periodic in 2.. on 1/1 and

280

-

TWE-RARMONIC ELECTROWAGNE'rIC I'l.ELDS

kr 10'10.6-6. Quality factor'll Q. for the TM•• and TE... modes of free llP:t.ee.

finite at 6 = O.

Hence, we choose the TM to

r

mode functions

I I '"

(A,)" - P,'(eos 9) cos . mq, • n,"'(kr)

am m",

(6-61)

where m = 0, 1,2, . . .. To satisfy tho boundary condition E r = E. - 0 at () = 01, the parnmet.er v must be a solution to P ,,(cos 9,) - 0

Also, we choose the TE to

T

(6-62)

mode functions

'" cosm,pl B,'~(lT) (F,),. - P,'(eos 8 ) (.

smmep

(6-63)

where m = 0, 1, 2, .. _ _ To satisfy the boundary condition E. = 0 at 8 = 81, the parameter" must be a solution to (6-64)

281

SPHERICAL WAVE FUNCTIONS

zl

~

"

(0)

(0)

(e)

(d)

(.)

(f)

Flo. 6-7. Borne spherically radial waveguides. (a) Conical (wavC8 external); (b) conical (waves internal); (c) biconical; (d) couial; (e) wedge; (f) born.

Because of a scarcity of tables for the eigenvalues v, it is difficult to obtain numerical values. The field components are, of course, obtained from the A, and F, by Eqa. (1)-26). The biconical and coaxial guides of Fig. 6-7c and d are again a single mathematical problem. Now both 8 = 0 and 8 = 'If are excluded kom the region of 6eldj so two Legendre solutions, P.-(cos 8) and Q.-(cos 8), or P.-(cos 8) and P.-( - coe 8), are needed. Choosing the latter two sohfiions, we find modes TM to T defined by (A,),. = [P.'(eoa 8) p."( -eoa 8,) - p."( -eoa 8) P."(eoa 8,)]

lc?S m4» 18lD m4J where m = 0, 1,2, . . .

I

fJ.l::(kf')

and the v are determined by the rootB of

P.-(eos e,)p,,(- cos e,) - P.-(- cos e,)P.-(eos e,) = 0 I

(6-65)

(6-66)

282

TW&-IIARMONIC ELECTROMAGNETIC :rtELDS

Similarly, for the modes TE to r we have

(F,)_, _ [p,-(COS 8) dP,-( ~8'cOS 8,)

P,_( _ cos 8)

dP,-~~~s

lc?Sm4>\ sm mq,

where m

=

0, 1,2, . . . ,and the

1:1

8,)]

n.al(kr)

are determined by the

TOOts

dP,-(cos 8,) dP,-( - cos 8,) _ dP,-( - cos 8,) dP,-(cos 8,) _ 0

de!

d(h

dB,

(~7) of (6-65)

dB I

Again the field components are found from 'the A r and F r of Eqs. (6-65)

and (6-67) according to Eqa. (6-26). The dominant mode of the biconical and coaxial guides is a. TEM, or

transmission-line, mode. The eigenvalues m = 0, v - 0 satisfy both Eqs. (6-66) and (6-65), but the A, and F, of Eqs. (6-65) and (6-67) vanish. We could redefine Eq. (6-65) such that the limit v - 0 exists, but instead let us separately define the TEM mode 88 a TM oo mode defined by

B

(II

(A,) .. - Q,(cos 8)B,"'(kr) - log cot 2 ('l'J)~tt

(6-69)

The field components of this mode, determined from Eqs. (6-26), are (

Eif' _

'k J. e±iAr uxr 810 8

H

+ 1'Sl08 ---i- e±i

•

'f

=

(6-10)

kr

where the upper signs refer to inward-traveling waves and the lower signs to outward-traveling waves. The wave impedance in the direction of travel is

(6-11)

which is the same as for TEM waves on ordinary transmission lines. The characteristic impedance defined in terms of voltage and current. is of great-er interest.. At a given r, t.he volLage is defined as

V

1:3

J.

'-h

I,

E d ' 1 cot (8./2) .", ' r 8 -= 311 og cot (8J2) e

(6-72)

and the current as

1

=

f02~ H.,. sin 8 d,p - +21r;je±/i'r

(6-13)

SPHJ!:RICAL WAVE FUNCTIONS

283

At small r these are the usual circuit quantities. The characteristic impedance is v+ V-, cot (8,/2) (6-74) Z, - [+ = - [_ = 2r log cot (8.12) Note that the various equations are the same as for the usual uniform transmission lines. For this reason the biconical and coaxial radial lines are called uniform radial transmission lines. Spherical waves on the wedge waveguide of Fig. 6-7e exist for all fJ but only for restricted fjI. Hence, the wave functions will contain only the PIl"(cos 0) with n an integer and to determined by the boundary conditions. We then find TM modes defined by

CA,)., = P.'CC08 8) 'in w~ n.lUCkr)

(6-75)

where n = 1, 2, 3, . . . ,and

pr

w=-

C6-76)

~,

with p = 1, 2, 3,

The TE modes are defined by

'"

(F,)_ = P .'(C08 8) coo w~ l1.'''(kr)

(6-77)

where n - I, 2, 3, . . . , and to is given by Eq. (6-76) with p "'" 0, 1, 2, . . .. There is no TEM spherical mode, the TEM mode being a cylindrical wave defined by Eq,. (5-48) and (5-49). Finally, the spherical-horn waveguide of Fig. 6-7/ will require Legendre functionsL."(cos fJ) of nonint.egral v and w. The TM modcs can be defined by Eqs. (6-65) and (6-66) with m changed to wand only the sin wq, functions allowed. The values of to are those of Eq. (6-76). Similarly, the TE mode, can be defined by Eq,. (6-67) and (6-68) witb m changed to w and only the cos wq, functions allowed. Again, to is given by Eq. (6-76). There will, of course, be no TEM mode. 6·6. Other Resonators. Resonators having modes expressible in terms of single spherical wave functions can be obtained by closing each of the radial waveguides of Fig. 6-7 by one or two conducting spheres. Some examples are shown in Fig. 6-8. The fields in each case can be expressed in terms of mode functioll8 which are the same as for the radial wave-guides of the preceding section, except that the traveling-wave functions 11.(l)(kr) and 11..(I)(kr) are replaced by standing-wave functions J.(kT) and IV.. (kr). Numerical calculations are hampered by a scarcity of tables of eigenvalues. Let us calculate the Q's for the dominant modes of the first three cavitics of Fig. 6-8. For the hemispherical cavity of Fig. 6-80, the dominant mode is the dominant TM to T mode of the complete spherical cavity,

284

TWE-BARMONIC ELECTROMAGNETIC FIELDS

z

"

zrt

zl ~a---ol

NJ f---a---l

(a)

(b)

'r

~!J

(0)

Z!

....~ ;, (.)

(d)

f (f)

FlO. &-8. Some cavities having modes expressiblo in terms of singlc spherical wft.ve functions. (a) Hemispherical; (6) hemisphere with cone; (e) biconical; Cd} conical; (e) wedge; (f) !legmen".

considered in Sec. 6-2. The magnetic field is ll. -

~ JI (2.744~) sin 8

and the stored energy is one-half that for the complete spherical cavity [Eq. (6-35»); hence

w

4"

where the c,. are constants. If we let the source recede to infinity, the field in the vicinity of the origin is a plane wave. Using the asymptotic formula

we have for the leCto-hand side of the preceding equation holU (1r - rl')

and for the right-hand side - je~ir' r'-o.. r .'_0

je-ir'

>-.

-I' -,-

r'_oo

L-

.-.

r

eir-'

c,.j"(r)P ,,(cos 9)

292

TutE-HAlUlONlC ELECTROMAGNETIC FIELDS

z r

y

I

Flo. &.11. A plane wave incident on a conducting sphere.

1 ,I

x

-J

t Incident plane wave

A comparison of these two expressions with Eq. (6-90) shows that c" = 211. + 1j hence

-

h,"'(lr - r'J) =

L: (2. + l)h."'(r')j.(r)P.(cos!l ._0 - (2. + l)j.(r')h."'(r)P.(cos n L: ._0

r

< r' (6-94)

r> r' I

This is the addition theorem for spherical Hankel functions. Since ha(l) = ham., Eq. (6-94) is also valid for superscripts (2) replaced by (1). The real part of Eq. (6-94) is nn addition theorem for io(lr - rD, and the imaginary part is an addition t.heorem for nD(lr - t'l). Finally, one can express the zonal harmonics P .(cos t) in terms of the tesseraJ harmonics P.-(cos 8)h(m4». In other words, a wave function referred to the t = 0 axis of Fig. 6-10 cnn be expressed in terms of wave functions referred to the 8 = 0 axis. The identity is

.

\'

.

(n - m)' P .(co. !) ~ __ '-'I '_ (n m)! P .-(co. 8)p.-(co. 8') co. m(~ - ~')

+

(6-95)

where too is Neumann's number (1 for m = 0 and 2 for m > 0). The proof of Eq. (6-95), plus some ot.her wave transformations that we have not treated explicitly, can be found in Stratton's book. 1 Equation (6-95) is an addition theorem for Legendre polynomials. 6-9. Scattering by Spheres. Figure 6-11 represents a conducting sphere illuminated by an incident plane wave. Take the incident wave I J. A. Slrat.ton, "Elec:l.romagnetic Theory," pp. 406-414, MeGra.w-HiIl Book Company. Inc., New York, 1941.

293

SPHERICAL WAVE FUNCTIONS

to be x-polarized and z-traveling, t.hat. is, E~;

=

Ecl.... _

H;

=

Eo eft. = Eo c11ir_'

EI1~~-flir_'

'. .

(6-96)

For convenience in applying boundary conditions, we express t.his incident field as the sum of components TM and TE to r, that is, in terms of an Fr and an Ar _ From Eqs. (6-26) we see that AT can be obtnined from Er, nnd Fr from Hr. The r component of E' is

Er; - cos q, sin 9 E~' _ Eo c~:/ :9 (e-i l'r_') Using Eq. (6-90), we can write this as

•

Ei = E,

ej~r~ 2>-'(2n + l)j.(kr) :0 P.(eDs 8)

._0

Finally, using Eq. (6-23) and the relationship

E,' - - jEt:;:

~

2>-'(20

.-,

aP./ao

=

P.I, we obtain l

+ I)J.(kr)P.'(eDs 8)

Noting the form of E r \ we construct. the magnetic vector potential as

•

w.

Ai = E, CDS

~ L." a.J.(kr)P.'(cos 8) ,

..

and evaluate E,' by Eqs. (6-26). we obtain •

E,'

~

-

jEtk:~: ~

(6-97)

Simplifying the result by Eq. (6-24),

L .-,

a.n(n

+ I)J.(kr)P.'(cDs 8)

Compa.ring this expression with the preceding formula for Er', we see that

a. =

j-'(20 + I) n(n + I)

(6-98)

A similar procedure using H,' and F r ' gives

Pi = where the a. I

Me

Note that. t.he

~. sin ~

•

L .-,

(6-99)

a.J.(kr)P.'(CDS 8)

again given by Eq. (6-98).

II. _

0 term of the lIummation drops out because

ptl -

O.

294

TIME-HARUmnC ELECTROUAGNETIC FIELDS

Now that the incident field is expressed in terms of radially TE and TM modcs. the rest of the solution parallels the cylinder problem (Sec. 5-9). The scattered field will be generated by an A. and P. of the same form as the incident field with J. replaced by 0 ...l is shown in Fig. 6-12. For small ka, the n = 1 term of Eq. (6-105) becomes dominant and

~

A • --+ 9X' (1m)' .to--oo 4....

+ 1)

f1

--.

V V Lr

0.1

(6-106)

which is a good approximation when a/~ < 0.1. Equation (6-106) is known as the Rayleigh scattering law. It states that the echo area. of small spheres varies as >.-4 and was

0.0 1

o

02 0.4

0.6 0.8

1.0 1.2

a/' FlO. 6-12. Echo Ar'e& of a ooDdue;ting apbere of radiua a (optical approrima. tien shown dashed).

296

TIME-HARMONIC ELECTROPoL\ONETIC FlELDS

first used to expla.in the blueness of the sky. A.

kG......

J

For large spheres

lI'a 2

(6-107)

which is the physical optics solution. The region between the Rayleigh and optical approximations is called the resonance region and is charac· terized by oscillations of the echo area. Let us now look at the field scattered by the small conducting sphere. Using small-argument formulas for the spherical Bessel fUllctions, we find from Eq. (6-102) aod (6-98) that b.. _ _ la_O

n

+1

---c.~ n la-O

(2n)! 1)']' (ko)"+' . ),,+1

[2'(n -

(6-108)

so the n = I terms of Eqs. (6-104) become dominant for small /ro. Hence, at large distances from small spheres,

e- jkr B,' - + Eo - k (ka)l cos 4> (cos 8 k..-O

r

(6-109)

rrlJ..

E.'

-10

la-O

~)

Eo - k (ka)! sin 4> r

(~

cos 0 - 1)

A comparison of this result with the radiation field of dipoles shows that the scattered field is the field of an x-directcd electric dipole Il = Eo

~~{ (ka)'

(6-110)

plus the field of a y-directed magnetic dipole

2.

Kl - E. jk' (ka)'

(6-111)

The ratio of the magnetic to electric dipole moments is IKif Ill. = '1'//2. Figure 6-13 illustrates the origin of these two dipole moments. A surface

z

z J.

x (aJ

x

(bJ

FlO. 6-13. Components of surface current giving rise to the dipole moments of a conducting sphere. (a) Electric moment; (b) magnetic moroent.

297

SPHERICAL WAVE FUNCTIOr.."S

current in the same direction on each side of the sphere gives rise to the electric moment, while a circulating current gives rise to the magnetic moment. In general, the scattered field of any small body can be expressed in terms of an electric dipole and a magnetic dipole. For a conducting body, the magnetic moment may vanish, but the electric moment must always exist. Now consider the case of a dielectric sphere, that is, let the region r < a of Fig. 6-11 be characterized by t.I, lSd, and the region r > a by to, 1010. In addition to the field externnl to the sphere, specified by potentials of the form of Eqs. (6-101), there will be a field internal to the sphere, specified by • A,- ~ E, cos ~ \ ' d.J.(k.,.)P.'(cos 0)

..L.t,

WJlO

F,- -

~: sin ~

(6-112)

•

2: ..

.-,

J.(k.,.lP.'(cos 0)

The superscripts - denote the region r < 0, and superscripts + denote the region r > a. Boundary conditions to be met at r :.: a are H,+ = lJ,lJ.+ = H.-

E,+ = E,E.+ = E.-

that is, tangential components of E and H must be continuous. Determining the field components by Eqs. (6-26), using Eqs. (6-101) for r > a and Eqs. (&-112) for T < 0, and imposing the above boundary conditions, we find b.. "'"

- v;;,;; J~(koa)J.(k,a) + ...r.;;;; J.(koa)J~(k,al v;;,;; fl.''''(koa)J.(k,a) - vi"., fl."'(koa)J~(k,a)

a

•

- vi;;;;; J.(koa)J~(k,a) + ...r.;;;; J~(koa)J.(k,a)

c. d.. =

e _

vi,,,,, fl."'(koa)J~(k,a)

- j

_

vi,,,,, fl.,n'(koa)J.(k,a) a.

v';;;;;

vi"", fl.''''(koa)J.(k,a) - vi,,,,, n."'(koalJ~(k,al

(6-113)

0"

i~ 0 ''''' n.''''(koalJ .(k,al •

• v;;,;; fl."'(koalJ~(k,a)

where a" is given by Eq. (6-98). The conducting sphere can be obtained as the specialization J.ld _ 0, Ei --. co, such that k" remains finite. ote that, in contrast to static-field problems, t.I- co is not sufficient to specialize to a conductor. In the special case of a small dielectric sphere, the n = 1 coefficients

298

TIM"L-HARYOSIC ELECrROaLAGNETIC FIELDS

are dominant nod reduce to

(IH14)

tlt

~ 2jl,(2

+ p,)

where f r cc fd!fG and /J. "'" IldllJo. A calculation of the scattered field reveals that it is the field of the two dipoles _

47rj

1(,-1 2

II - u.Eo .k' (ka) ... The pattern for the very small sphere is the usual dipole pattern. For a very large sphere it approaches the pattern of a dipole on a ground plane but always with some diffraction around the sphere. The radiation field for dipoles of other orientations, and also for magnetic dipoles, can be obtained in a similar manner. The ficld in t-he entire region r > b can be determined from the radiation

300

TIME-HARMONIC ELECTROMAGNETIC FIELDS

I FlO. 6-Hi. Radiation patterna for the radially directed dipole on a conducting Bphere of radius a.

field as follows. From symmetry considerations (Fig. 6-14a) we conclude that H = u.H+, and therefore the field can be expressed in terms of an A = urA r. Also, A, must be independent of q, and represent outward traveling waves; hence A. ~

• L... .., fl.'''(kr)P. (cos 0)

r>b

(6-119)

From this we can calculate E, by Eqs. (6-26), obtaining

(6-120)

The a.. are then evaluated by equating this expression to the radiation

301

SPHERICAL WAVE FUNCTIONS

field previously determined. For example, in the special case b = a we equate Eq. (6-120) to Eq. (6-118) and obtain a" _ 1l(2n 1) (6-121) 41fkB ,,(2)'(ka)

z

+

•

Tbe field everywhere can now be obtained from Eq,. (6-26), (6-119), nud (6-121). x 6-11. Apertures in Spheres. In Sec. 4-9 we saw how to express the field in a matched rectangular waveguide in terms of the field over a FlO. 6-16. Slotted conducting sphere. cross section of the guide. In Sec. 6-4 we saw that space could be viewed as a spherical waveguide. A given sphere r = a is a cross section of the spherical guide. If r > a contains only free space, then the guide is matched, that is, there are no incomir.g waves. By writing the general expansion for outward-traveling waves and specializing to r - G, we obtain the field r > G. When apertures exists in a conducting sphere of radius r = a, tbe tangential components of E are zero except in the apertures. Our f9rmulas for the field r > a then reduce to ones involvinl?, only the tangential components of E over the apertures. A general treatment of the problem is messy; so let us restrict consideration to the rotationally symmetric TM case, that is, one having only an H.. The slotted. conducting sphere of Fig. 6-16 gives rise to such a field if there exists only an E, independent of ~ in the slot. The field is expressible in terms of nn A r of the form

•

•

A, -

..l ,a.n.'''(h)P.(co, 0)

(6-122)

From Eqs. (6-26) we calculate

E.

=

• k \' a jWf.T' L, a"J1.(t)I(lcr) a8 P.(cos 8)

.. ,

(6-123)

Noting iJP./a8 = p.l, we multiply each side of the above equation by P.,I(COS 8) sin 8 and integrate from 0 to 'II' on 8. By the orthogonality relntionship [Eqs. (6-46) nnd (6-47»), we obtain

+ 1) 4,.n,,('I'(kr) 2T1l(n ",::;2:'';:'-.=-' fao• E.P"l(COS 8) sm. 8 d8 -- )r..,...."" 2n+l

302

TIME-HARMONIC ELECTROMAGNETIC FIELDS

I

FIG, 6-17. Radiation patterns for the slotted sphere, 8D .. 11:/2.

Specializing this to r = a, we have the coefficients a.. determined as

+

r'

ja(2n 1) aft = 172'1m(n + l)fl,,(z)'(ka»)o E,

I ' r_a

.

P .. (cos 8) sm 8 de

The field simplifies to some extent in the radiation zone. asymptotic forms for fl.. m in Eq. (6-123), we obtain

(IH24)

Using the

• E, - . ! l e-II:< '\' a,.j"P"I(cos 0) kr-o .. r

i..J

(6-125)

"-,

This result could also be obtained from the plane-wave scatter result of Sec. 6-9, using reciprocity. For tho slotted sphere of Fig. 6-16, let us assume a small slot width, so that E, is essentially an impulse fundiOIl at r = a. Hence. we assume (6-126)

303

SPHERICAL WAVE PUNcrlONS

where V is the voltage across the slot.

+ l)P.. J(C08 8 sin 8 .2rn(n + I)O.""(ka)

jV(2n G. -

Then Eq. (6-124) reduces to 0)

0

and the radiation field [Eq. (6-125)] becomes • jV....." . 9 \ ' ;'(20 + 1)P.'(cos 9.) P '( 9) E, 2S1" SID 0 Lt n(n + l)fl..uJI (ka) .. cos

(6-127)

.-,

Figure 6-17 shows radiation patterns for the case 80 = -r/2, that is, when the conductor is divid~d into hemispheres. Patterns for sphcres of radii >../4 and 2" are shown. Very small spheres produce a dipole pattern, while very large spheres produce an almost omnidirectional pattern with severe interference phenomena in the 8 "'" 0 and 8 = -r directions. In the limit 80 """, 0 we obtain tho patterns of Fig. 6-15, which is to be expected in view of the equivalence of a small magnetic current loop and an electric current element. The general problem of finding the field in terms of arbitrary tangential components of E over a sphere is treated in the literature. l 6-12. Fields External to Cones. The general treatment of the probz lem of sources external to a. conduct-. ing cone is also messy but can be found in the literature. l We shall here l"C3trict consideration to the e, rotationally symmetric case of "ringsource" excitation of a conducting y cone. The geometry of the problem Current filament is shown in Fig. 6-18. The special case of a magnetic current ring on e, the conical surface gives the field of X a slotted cone. The limit as the magnetic current ring approaches the cone tip gives the field of an axially directed electric current element on FlO. 6018. Ring excitation of a conductiog oooe. the tip. Consider first the case of an electric current ring. From symmetry considerations, it is evident that E will have only a 4> componentj so the field is TE to T. The modes of the "conical waveguide" are considered in Sec. 6-5, Eqs. (6-61) to (6-64). In the region r < a we have standing waves, while in the region r > a we have outward-traveling waves. I L. Bailin and B. Silver, Exterior Electromagnetic Boundary Value Problems for Spheres and Cones, IRE TroBl., vol. AP-4, no. I, ~p. 5-15, January, 1956.

304

TIME-HARMONIC ELECrROllAGNETIC FIELDS

Hence, we construct

F~

1a.P.(eas o)B.ln(kT) "'" • I 1 b,P.(eo, O)J.(kT)

r>a (6-128)

r a Eq. (6-137) becomes A r = : sin! Ih

LA}u p~

(cos 81)P.. (cos 8)J..(ka)Il,,(l)(kr)

• Using the asymptotic form for 8 m and evaluating E, by Eq. (6-26), we find for the radiation field 10

+ l)(.P.(cos O)/.oJ J.(ka) + 1)(.P.(eo, O,)/QuJ

E. _ V ,_;'" \ ' j·(2u Jr u(u

L< •

(6-146)

Some radiation patterns for slotted cones with cone angle 30° are shown in Fig. 6-19. A discussion of the problem of plane-wave scattering by a cone is given by Mentzer. I I J. R. Mentzer, "Scattering and Diffraction of Radio Waves," pp. 81-93, Pergamon Press, Inc., New York, 1955.

307

SPHERICAL WAVE PUNCTIONS

6-13. Maximum Antenna Gain. The general form of the field in a spherical spnce external to aU sources is Eqs. (6-26) with A, -

2: a••fl.,o(kr)P.·(cos 8) cas (m~ + a••) 2: b••fl.'O(kr)P.'(cos 8) cos (m~ + P••)

~.

F, -

(&-147)

-.'

Given an arbitrary field at T = Ti, the field can be projected backward toward tbe origin as far as desired. At some sphere T - a we can determine sources by the equivalence principle (Sec. 3-5), which will support this field. Hence, it appea.rs that sources on an arbitrarily small sphere can support any desired radiation field. The gain of an antenna. is defined by Eq. (2-130) in general. We shall here consider the largest gain g -

4n"(S,)._ ~I

(&-148)

where (8.)... is the maximum power density in the radiation zone and ~I is the power radiated. By the discussion of the preceding paragraph, it appears that arbitrarily high gain can be obtained, regardless of antenna size. In practice, however, the gain of a directive antenna is found to be related to its size. A uniformly illuminated aperture l type of antenna is found to give the highest practical gain. This apparent discrepancy betwoon theory and practice can be resolved if the concepts of cutoff and Q of spherical waves are properly applied. Let us orient our spherical coorclinatc system so that maximum radiation is in the tJ = 0 direction. The radially directed power flux in this direction is then (&-149) (8.)... = E.H: - E.H: From Eqs. (&-147) and (&-26) we find

e-ik'

E~ = 2jr

\' L..t n(n + l)j"(l1 alii cos

"'Ill -

h. sin (hI!)

• (&-150)

I The term "uniformly illuminated aperture" ill u.sed to describe antcnnss for which the &Duree (primary or aeeondary) is COl1Iltant in 3mplitudc and phase over a given area on .. plane, and zero e1aewhere.

308

TU.£E-.HARMONIC ELECTROBLAONETIC FIELDS

in tbe 8 = 0 direction of the radiation zone. The total radiated power is found by inwgrating tbe Poynting vector over a. large sphere. The result is

~ _ ,- '\' n(n+ I)(n+m)! (I ~/- ~ '-' ..(2n + 1)(n m)!

.a...1'+!lb .'.I')

~

(&-151)

where f . - 1 for m - 0 and t. - 2 for m > O. We used the ortbogonality relationships of Eqs. (6-51) in the derivation of Eq. (6-151). Equations (6-148) to (6-151) give a. general formula for gain in terms of spherical wa.ves. We shall now consider under what conditions g is a. maximum. Note that the Ilumerator of Eq. (6-148) involves only the aIR and bhl coefficients. Hence, the denominator caD be decreased without changing the numerator, by setting

a.... = b... =0 Also, both numerator and denomina.tor of 9 arc independent of

(&-152) 0'1"

and

PI.. ; so they may be chosen for convenience without loss of generality. In particular1 let al. = T and Pl. = T/2 1 and the gain formula reduces to

12: (A. +E.) [' g - ---.=.----'-;'- - - - 22: 2n ~ 1 (IA.I' + IB.I')

(6-153)

•

(&-154)

where

The denominator of Eq. (6-153) is independent of the phases of A. and B.; so we ean maximize the numerator by choosing A. and B. real. Furthermore1 g is symmetrie in A. and B.. ; hence the maximum exists when A .. """ B .. = real

(&-155)

The maximum gain thorefore will be found among those specified by

(&-156)

• where A .. is real. As long as n is unrestricted, this g is unbounded, as we anticipated earlier. If the field, specified by Eqs. (6-147), contains only wave functions of order n ~ N, then an upper limit to g exists. Setting iJgjaA i = 0 for

SPHERICAL WAVE FUNCTIONS

309

all A i, we find N

U_ -

and also

.l-.

(2n

A .. =

+ I) - N' + 2N 2n + 1 A 3 I

(6-157) (6-158)

Equation (6-151) represents the highest possible gain using spherical waveguide modes of order n 5" N. A similar limitation to the nearzone gain also exists. l To relate gain to antenna si7.e, we define the radius a of an antenna M the radius of the smallest sphere that can contain the antenna. We saw in Sec. 6-4 that spherical modes of order n were rapidly cut 01T when ka < n. Hence, it is reasonable to assume that modes of order n > ka are not normally ·present to any significant extent in the field oC an antenna of radius a. We define the normal gain of an antenna oC radius aas (6-159) u_••• = (ka)' + 2ka which is obtained by sett-ing N "'"' ka in Eq. (6-151). Hence, the normal gain is maximum gain obtainable when only uncutoff modes are present. It is interesting to note that, for large 1M, a circular, uniformly illuminated aperture of radius a h3S the same gain as the above-defincd normal gain." It is thereCore not surprising that the uniformly illuminated aperture gives the highest antenna gn.in in practice. The normal gain is not an absolute upper limit to the gain of an antenna. Antennas having higher gain are a distinct possibility and will be called supergain antennQ.$. We shall use the Q concept of Sec. 64 to show that (1) supcrgain a.ntennas must necessarily be narrow-band devices, and (2) supergain techniques yield only a smaU increase in gain over normal gain for large antennas. Other characteristics which we shall not demonstrate here are (3) supcrgain antennll8 have high field intensities at the antenna. structure and (4) they tend to have excessive power loss in the antenna structure. The Q of a loss-free antenna is defined as

'11.>'11.

w. > W.

(6-160)

I R. F. Harrington, Effect of A.ntenna Siu on Gain, Bandwidth, and Efficiency, J. ReuGrcA NBS, vol. 640, no. I, pp. 1-12, January, 1960. t S. Ramo and J. R. Whinnery, "Fields and Waves in M.odern Radio," 2d ed., p. 533, iohn Wiley &: Sons, Inc., New York, 1953.

310

TlllE-HARMOXIC ELECTROI.LAGNETIC FIELDS

Ill'

e-

10'

0

Ill'

30

0'

25 \ Ill'

20

10

10

I

i~5

5

\

\ \ 1\

\.

\';'-\ o

\

FlO. 6-20. Quality factofft for ideal loss-free anten· nas adjusted for mlUimum gain using modes of order n ::; N.

10

15

20

25

ka

where 'W. and OW. are the time-average electric and magnetic energies Bnd {j>, is the power radiated. 'Vc shall define an ideal loss-free antenna of radius a as ODC having no energy storage r < a. The Q of this ideal antenna. must be less than or equal to the Q of any other loss-free antenna of radius a having the same field r > a, since fields r < a can only add to energy storage. If the Q of an antenna is large, it can be interPreted as the reciprocal of the fractional bandwidth of the input impedance. If the Q is small, the antenna has broadband potentialities. Antennas adjusted for maximum gain according to Eq. (5-158) have equal excitation of 'I'M and TE modes. The Q.. of spherical modes, defined by Eq. (6-60) and plotted in Fig. 6-6, involve OW. for 'I'M modes slid "XI", for 1'E modes. We need Q's defined in terms of the same energy for aU modes, and it is convenient to deal with Q's for equal TM and TE modes. The Q for equal TM,. and TE" modes is

ka < N

(6-161)

because the 'W, is essentially that of the TM.. mode alone a.nd the rJJ is twice that of the TM II mode alone. When QII < I, we take it as unity. Because of the orthogonality of energy and power in the spherical modes, the tolal encrgy and power in any field is the sum of the modal energies and powers. Hence, the Q of our ideal loss-free antenna is

Q~

2: p.Q.",m - ='=o---'='i:--=--:::t.....,,.2: A.' (2n ~ I) Q. 2: p. 22:A"(2n~l)

311

SPHERICAL WAVE FUNCTIONS

where P. is the transmitted power in the TM. and TE. modes. Eq. (6-158), thie becomes

Using

N

l

(2n

+ 1) Q.(ka)

.-1 ::-:---:2"'N""·;-+-;--;4"N,.------

Q-

(6-162)

where the Q. are given in Fig. 6-6. Curves of antenna Q for several N nre shown in Fig. 6-20. Note that the Q rises sharply for ka < N, showing that supergain antennas must necessarily be high Q, or frequency sensitive.·, The Q of Fig. 6-20 is a lower bound to the Q of any loss-free antenna. of radius a. By picking a Q, we can calculate an upper bound to the gain of an antenna of radius a. Figure 6-21 shows the ratio of this upper bound to the normal gain. Note that for large ka the increase in gain over normal gain possihle by supergain techniques is small. For small ka supergain can give considerable improvement over normal gain. In fact, ns ka --+ 0 the supergain condition is unavoidable. All very small antennas are supergain antennas by our definition. The problems of narrow bandwidth and high losses associated with small antennas are well-known in practical antenna. work.

10

9 ~

II .~

"

8 7

Ii

6

~

5

3

..

4

.!!

3

0

2

g ~

~

.'\

\"-

'"

--

~l~ '- ........ lO'" '-..

1

0

Q = 10·

10

....

20

"-

30

40

50

60

70

80

Fro. 6-21. Ma.ximum poasible iocrea.ee in gain over normal gain tor a given Q.

90

312

TIME-HARMONIC ELECTROMAGNETIC FIELDS

PROBLEMS

6-1. Use Eqs. (3·85) p.nd t.he wave potential of Eq. (6-14) to show that a general expression for fields TM to t: is . E. - -1fsJJJy, cos

8 ali} . + 1r

where ~ is the aogle between rand r'. 6·24. CoIlBider the scattering of a plane-polariled wave by a smrill conducting sphere (Fig. 6-11). Show that the distant. llCatt.ered field is plane pol/lri1ed in t.he direction 8 _ 60'. 6-26. Considcr an z.polariled, z traveling plane wave incident on a conducting sphere encased in llo concentric dielectric coating, IUl shown in Fig. 6-23. Show that the 6eld ill given by Eqll. (6-26), o where for r > b the A. and F. are given by Eqs. (G-lOl), and for (1 < .,. < b

,

z

• A. _ E,

COli

0.680

(a) -2.02

(b) 0.680

Table 7-1 gives the value of C in Eqs. (7-6) for cavities of several geometries for a., located at (a) maximum E and (b) maximum H. These values have been obtained using the crude approximations of replacing E, H by Ee, H o in Eq. (7-3). They are therefore valid only for smooth, shallow deformations. In general, the frequency shift depends on the shape of the deformation as well as on the shape of the cavity. The formulas for deformations of the form of small spheres or small cylinders caD be obtained from the results of the next section by letting E _ 00 and 11- O. 7-3. Cavity-material Perturbations. Let us now investigate the change in the resonant frequency of a cavity due to a perturbation of the material within the cavity. Figure 7-2a represents the original cavity containing matter E, IJ, Figure 7-2b represents the same cavity but with the matter changed to E + AE , IJ + All.

322

TDrE-RAIUIONIC ELECTROMAGNETIC FIELDS

n

'.

,

n

Flo. 7-2. Perturbatwn of mat.ter in a cavit.y. (4) Original cavity; (b) per· turbed cavit.y.

s

(a)

(6)

Let E Ot HOI Wo represent the field and resonant frequency of the original cavity. and let E, H, (oj represent the corresponding quantities of the perturbed cavity. Within S the field equations apply, that is, - V X E = jw(p V X H = jW(E

- V X Eo = jwopHo V X H o "'" jwotE o

+ .1p)H

+ ..6.E)E

(7-9)

AB in the preceding section, we 8calarly multiply tbe last equation by E: and the conjugate of the first equation by H, and add the resulting two equations. This gives

v . (H

+ AE)E • E:

X E:) = jW(E

- j(J},pB: . H

Analogous operation on the second and third of Eqs. (7-9) gives V.

CD:

X E) = jwCp

+ dp)R· H:

- jW(ltE: . E

The sum of the preceding two equations is integrated throughout the cavity, and the divergence theorem is applied to the left-hand terms. The left-hand terms then vanish, because both n X E = 0 on Sand n X Eo = 0 on S. The result is 0=

III IIw(. + 6.) -

w"jE· E:

+ [wu. + 6,)

- w.,IH· H:I dT

Finally, this can be rearranged as w - Wo w

- IIIIII

(llEE· Et (tE. E:

+ ~pH· Hn dT

+ pH . Ht) d,.

(7-10)

This is an exact formula for the change in resonant frequency, due to a change in t and/or p within a cavity. Once again our development bas assumed the loss-free case, that is, E and p are real. The general formulation when losses are present is given in Prob. 7-5. In the limit, as ~t -+ 0 and IIp -+ 0, we can approximal-e E, H, w by Eo, H o, Wo and obtain

III (6·IE,I' + 6,IH.,') dT III (·IE,,' + ,IH.l') dT

(7-11)

PERTURBATIONAL AND VARL-\TIONAL TECHNIQUES

323

This slates that any small increau in E and/or ~ can only decrease the resonant frequency. Any large change in E ,and/or ~ can be considered as n succession of mnny small changes. Henco, any imrease in f and/or ~ within a cavity can only decroou the reMnant frequ.emy. We can recognize t.he various terms of Eq. (7-11) as energy expressions and rewrite it as w -

w,

w, ::::s -

-

1

'"

fl! (d' - + d" - )d -10. E

-"'_

~

T

(7-12)

where W is the total energy contained in the original cavity. Now if the change in E and ~ occupies only a small region AT, we can further approximato Eq. (7-12) by

(7-13) where tb is the space average of W. The parameters C1 and C! depend only on the cavity geometry and the position of aT. Note that 0. small change in E at a point of zero E or a small change in p. at a point of zero IJ does not. change the resonant frequency. If we compare Eq. (7-13) wit.h Eq. (7-6), it is evident that. C - C! - CJ. For the cases considered.in Table 7-1, aT' is either at a point of zero H, in which case C! = 0, or at a point of .tero E, in which case CI = O. To be explicit, for a material perturbation at (a) of Table 7-1 we have C1 "'" -C and C! = 0, while for a material perturbation at (b) of Table 7-1 we have C 1 = 0 and C! = C. The preceding approximations require that AE, Ap., and AT all be small. We shall now cODsider a procedure for removing these restrictions on AE and Ap.. This introduces the further complication that the change in frequency depends on the shape of tl.r, as well as on its location. The modification is accomplished by using a. quasi-static approximation to the field internal to tl.T. This assumes that the ficld internal to AT is related to the field external to AT in the same manner as for static fields. The procedure is justifiable, because, in a region small compa.red to wavelengt.h, the Helmholtz equation can be approximated by Laplace's equation. There are Cour types oC samples for which this quasi~static modification to the pert.urbat.ional solution is very simply accomplished. These are shown in Fig. 7-3 for the dielectric case. For the magnetic case, it is merely necessary to replace E by Hand E by p.. For the thin slab wit.h E normal to it (Fig. 7-3a), we must have continuity of t.he normal com-

324

TIME-HARr.l0NIC ELECTROMAGNETIC FIELDS

O'----_.:.JD (cj

(d)

FlO. 7-3. Some small dielectric objects {or which the quasi-.static solutions are simple.

ponent of D. so that (7-14)

This approximation is valid regardless of the cross-sectional shape of the cylinder. For the long thin cylinder with E tangential to it (Fig. 7-311), we must have continuity of the tangential component of E, so that (7-15)

Again this approximation is independent of the cross-sectional shape of the cylinder. For E normal to a long thin circular cyliuder (Fig. 7-3c), we can use the static solution, I which is (7-16)

Finally. for E normal to a small sphere (Fig. 7-3d), we can use the static solution,' which is (7-17)

The static solution for a dielectric ellipsoid In a uniform field is also known but is not very simple in form.' To use the above quasi-static approximations, we approximate E (and H in the magnetic case) in the numerator of Eq. (7-10) by E 1D , of the preceding equations. In the denominator we can stilJ use the approximations E = Eo and H = H o, because the contribution from AT is small compared to that from the rest of T. Hence, our quasi-static correction to the perturbational formula is (oj

-

WD

~

JII AtE

IDI •

Eci dT

2jjj.IE.I'dT

(7-18)

I W. R. Smyth, "Static and Dynamic Electricity," pp. 67-68, McGraw-Hill Book Company, Inc., New York, 1950. I J. A. Stratton, "Electromagnetic Theory," pp. 205-213, McGraw-Hili Book Com.

pany, Inc., New York, 1941.

325

PERTURBATIOYAL AND VARIATIOXAL TECHNIQUES

,

A , - -71

1 d--W=-a-----..j

(a)

(e)

(b)

FIo. 7-4. CAvities used t.o illustrate t.be pcrturbatKlnal fonnulas.

for the case t1p. "'" O. (The denominator bas been simplified by equating W.. to 'W•. ) The corresponding formula for the frequency shift due to a magnetic material would be of same form, but with E replaced by H and E by p. throughout. Equation (7-18) is, of course. most valuable for problems for which the exact solution is not known. However, so that we may gain confidence in the results as well as pr:lCuce in the procedure, let us apply Eq. (7-18) to problems for which we have the exact solution. These are illustrated in Fig. 7-4. For a dielectric slab on the base of a rectangular cavity (Fig. 7-44), we have E l ., given by Eq. (7-14). The field and energy expressions for the unperturbed cavity are given in Sec. 2-8. Application of Eq. (7-18) then yields lE,-ld

(7-19)

- 2-,-,-.

where d is the slab thickness and a is the cavity height. A comparison of this with the result of Prob. 4-17 for P.l = P.I "'" p.o and EI = EO shows that our answer is identical to the fU'St t-erm of the expansion for cal in powers of dla. ~n fact, if tip. is also nonzero a.nd we treat it to the same degree of approxima.tion (match tangential H), we again get the correct first term of the expansion. To illustrate the improvcment obtained by using the quasi-static field, we can compare Eq. (7-19) to the result obtained from Eq. (7-11), which is I)

~ o

It is apparent that the above formula is accurate only {or when tu is small.

Er F:S

I, that is,

A nonmagnetic dielectric slab at a. side wall of the rectangular cavity (Fig. 74b) has but little effect on the resonant frequency, because E is zero at the wall. In this case E is tangential to the air-dielectric interrace; so Eq. (7-15) should apply. Note that Eqs. (7-18) and (7-Jl) give

326

TIME-HARMONIC ELECTROMAGXE'I'lC FIELDS

identical approximations in this case. ",-wo

In particular, we obtain

".. - (t~-l)f,J.t'll'"Xd sm :t:

a

Wo

0

~ _ T' (. _ 3'

1)

a

(~)'

(7-20)

•

A comparison of this with the answer to Prob. 4-18 shows that we again ha.ve the correct first term of the expansion when tip. "'" O. As a final example, consider the spherical cavity with a concentric dielectric sphere (Fig. 7-40). The field of the unperturbed cavity is defined by H.

=~JI(2.744&)Sin9

and tbe stored energy is given by Eq. (6-35). using the quasi-static Eq. (7-17), we obtain w "'0

Wo ===

-0.291

Applying Eq. (7-18),

fOr 1 (2.744 ~)' t.+2 b

where a is the radius oC the small dielectric sphere and b is the radius of the conductor. This we can compare to the exact solution (Peob. 6·8), which is the same. The perturbational method used in conjunction with the quasi-static approximation gives excellent a.ecurncy when properly used. This sbift in resonant frequency caused by the introduction of & dielectric sample into a resonant cavity can be used to measure the constitutive parameters of matter. 7-4. Waveguide Perturbations. We shall now consider waveguides cylindrical in the general sense, that is, all z: = constant cross sections are identical. Figure 7-5a represents a cross section of the unperturbed wavegUide, Fig. 7-5b represents a wall perturbation, and Fig. 7-5c repre.sents a material perturbation. All perturbations must, of course, be independent of z. The guide boundary is taken as perfectly conducting in aU cases.

'.

,

S

.c

S-

C (0)

E. H

E.H

Eo.Bo

•

n

n

C' (b)

Flo. 7-5. Perturbat.ions of eylindrieaJ waveguides. (b) wall per~urbalion; (c) material perturbation.

(,j

c

O. However, calculations have not been made for this choice. 7·10. Stationary Formulas for Scattering. Let us first treat the ba.ckscattering, or radar echo, type of problem by the variational method. The problem is represented by Fig. 7-14. It consists of a source and one or more obstacles, and we wish to determine the field scattered back to the source. For simplicity, the obstacle will be considered a perfect conductor and the source a current element n. The more general case of dielectric obstacles is considered in Sec. 7-11. Let the incident field, that is, the free-space field of the source alone, be denoted by Ei. The total field E with the obstacle present is then the sum of the incident field Eo plus the scattered field E·. The reaction of the sca.ttered field on the current element is (8,i)

~

liE,'

~

-IV'

(7-108)

where V· is the scattered voltage appearing across l. Let the echo be defined as the ratio of E,- to n. Then, using reciprocity, we ha.ve

EI' (S,1) (i,s) Echo - If - (II)' ~ (II)' -

(I~)' 1P E; . J. d,

(7-109)

where J. is the current induced on the perfectly conducting obstacle. The boundary condition at the obstacle is n X E = 0, or on S

n X E; - -n X E'

(7-110)

Hence, Eq. (7-109) ca.n be written as (c,c) -1 Af.. Echo - (II)' 'Jr E' . J. d, - - (ll)'

where (c,e) stands for the self-reaction of the on the obsta.cle hy the source.

U

(7-111)

correct" currents induced

356

TI~RARMONIC

ELEC1'ROMAONETIC FIELDS

For a stationary formula, we assume a current J- on S and approximate (c,c) by (a,a), subject to the constraint (7-112)

(a,a) - (e,a) - - (i,a)

The last equality results from Eq. (7-110). To express tbis constraint in a form for which (ala) is insensitive to the amplitude of J., we take

(i,a)' (a,a) - - ( a,a ) and, replacing (c,c) by {a,a} in Eq. (7-111), we have -(i a)' Ecbo = (Tl) '(a,a) =

(1fE"

d.)'

Jo

(TI)' 1f Eo.

JO d.

(7-113)

where E- is the field produced by the assumed currents Ja, This is the variational formulation of the problem. Note the close similarity of the echo problem to the impedance problem of the preceding section. The impedance problem is essentially an echo problem for which the source is at the obstacle. A more general formulation of the echo problem can be made by replacing I l with an arbitrary source. The tensor Green's functions alSee. 3-10 can be used to put Eq. (7-113) into a more descriptive form. Define [r(r,r')] as the tensor of proper· tionality between a current element dJe at r' and the field dEe that it produces at r. that is, dEo(r) _ (r(r,r») dJo(r) Then Eq. (7-113) can be written as

- [i11f E'(r) . Jo(r) ds

Echo """

r

-,,---'7;:-:=-------""--

1f d.1f d.' J"(r) . (r(r,r')] Jo(r') This equation is in a form characteristic of variational solutions in general. A commonly calculated parameter is the echo area, defined by Eq. (3-30). For linearly polarized fields, the echo area is given by (7-114) If, in Fig. 7-14, we let TZ be z4rected and located on the.:z: axis, and then

let r = ::c: -+ co, we have, in the vicinity of the obstacle,

E' = u. i"ll tJh 2~T

=

u.BoeJb

357

PERTURBATIONAL AND VAlUATIONAL TECHNIQUES

2.0

••

1:, m

1.5

-1

..

( \ Llo -

150

"

1'"'20

1.0

r'\

~\Lla - 1600

0.5

o

~ .L: \\: 2

........

\

L~o =

4

8

6

2f.OOO

10

12

1L Flo. 7-]5. Broad,ide echo

.rea A. of .. wire.

(Aflu Y. Y. Bu.)

Also, by definition, we have echo = E,'fll; hence from Eq. (7-113)

E J•

~E.

(1f> u, . J'e d8)' -'1'21\r 1f> E' . J'd.

= _--"-il-,,

i"

Therefore, by Eq. (7-114), our stationary formula for echo area is

." (1f> J.·oi" d.)' ,

A. = .. - X

1P E'· J'd.

(7-115)

when the incident plane wave is .-polarized and -x traveling. As an example, consider the scattering of a plane wave by a thin conducting wire, as represented by the insert of Fig. 7-15. The integral in the denominator of Eq. (7-115) is just the self-reaction of the assumed current on the wire. This is the same type of reaction that we encountered in the linear-antenna problem, approximated by Eq. (7-105). Defining A as the self-reaction, we have A

= A E- . J- dB 'it'

=

-~- fL/2

•

tHi ITI-t'"

E'

7 /

o

---

0.2

../

0.4

0.6

0.8

1.0

aI'

FIG. 7-16. Echo width L, of a conduct,.. ing ribbon of width 4.

(7-123)

if the incident field is z-polarized and

-:J:

traveling.

(f J.,"h dl)'

SimilarlYj

, (7-124)

fE'. J' dl

•

if the incident field is y-polarized and -x traveling. From symmetry, J" in Eq. (7-124) should have no z component. In both Eqs. (7-123) and (7-124), it is assumed that thc scatterers are cylinders generated by elements parallel to the z axis and the line integrals are in a transverse (z = constant) plane. For an example of a two-dimensional problemj consider a z-polarized plane wave normally incident on a conducting ribbon of width a. This is illustrated by the insert of Fig. 7-16. Assume that the current induced on the ribbon is uniform j that is,

J.' = 1

(7-125)

Because the current is real, the integral in the denominator of Eq. (7-123) is

J'/2

- ../2

E.oJ.fJ dy =

J'/2

- ../2

E."J.fJ$ dy = -P

where P is the complex power per unjt length supplied by J .-. But we have already analyzed the ribbon oC uniform current in Sec. 4-12, the result being

P =- 1[21Z

=

where Y.~rt is plotted in Fig. 4-22.

at

I• Y_~rt

The echo width j according to Eq.

360

TIME-HARMONIC ELECTROMAGNETIC FIELDS

Receiver

./ FIG. 7-17. scattering.

Differential

Transmitter

(7-126)

A plot of this is shown in Fig. 7-16. and obtain

For large a we can use Eq. (4-107) (7-127)

which is also the physical optics approximation (see Fig. 3-21). The more general case of differential scattering, or transmission,' is represented by Fig. 7-17. The problem consists of a transmitter, which illuminates the obstacle, and a receiver at which we wish to evaluate the scattered signal. For simplicity, let us consider both the source and receiver to be unit electric currents. Then, according to Eq. (3-39), the voltage across the receiving current due to the transmitting current is (7-128)

where t and r refer to the source or field of the transmitter and receiver, respectively. The total signal received is the superposition of the inci· dent field, due to the transmitter alonc. plus the scattered field. due to the currents c on the obstaclc. Hencc, (7-129)

where (t,r) is calculated with the obstacle absent and (c,r) involves the free-space field of the currents on the obstacle. The transmitter and receiver currents are assumed to be known (they are current clements in our simplified case); so V r' can, in principle, be ealculatcd exactly. Our problem is to obtain the variational formula for V r -. We shaH here consider only the simple case of a pcrfectly conducting obstacle, the general case being considered in Sec. 7-11. Applying reciI A traDsmission problem involves the evaluation of the total field at the receiver, while a. scattering problem involves the evaluation of only the scattered field.

PERTURBATIONAL AND VARIATIONAL TECHNIQUES

361

procity. we have, for the scattered voltage at the receiver, - V.' ~ (c,r) - (r,c)

-1ft (E')-· a,')' tU

(7-130)

where 0/)' is the surface current induced on the obstacle by the transmitter and (Ei)r is the field of the receiver current calculated with the obstacle absent (the incident field). Our boundary conditions on the various true fields are n X E = 0 at the obstacle boundaryj hence n X (E~- - -n X (E')' n X (E')' - -n X (E') ,

(7-131)

where superscripts i and 8 refer to incident and scattered components, and t and r refer to transmitter and receiver sources. Hence, by Eqs. (7-130) and (7-131), we have

V.'

-1ft (E')-· (J,')' d8 -

(c.,c,)

(7-132)

where (er,c,) stands for the reaction between the field of the II correct " currents induced on the obstacle by the receiver and the" correct" currents induced by the transmitter. For our stationary formula, we approximate (c"c,) by (a,.,4,), where the a's denote assumed currents on the obstacle, and constrain the latter according to Eq. (7-65), which is

(a.,..) = (c.,a,) - (..,c,)

(7-133)

In the language of t·he reaction concept, Eq. (7-133) says that the assumed currents look the same to each other as to their respective true currents. By Eqs. (7-131) and reciprocity, Eqs. (7-133) become (a"a,) = (Cr,a,) = - (r ,4,) (a"a,) = (a"c,) = (CI,a,) = -(t,a,.)

(7-134)

Substituting from Eqs. (7-134) into Eq. (7-132), we have for our variational formula a) -= (r,4,}(t,4,) V ,, = (a r,' Gr,a,

(>

=

[1ft (E~' . (J,o), d8] [1ft (E')' . a:>- dB] 1ft (EO)' . a:)' d8

(~lM)

where (Ea)r is the field due to the assumed currents (J~.)", which approximate the currents induced by the receiver. Note that Eq. (7-135) involves the assumption of currents on tho obstacle due to sources at both the transmitter and receiver. Note also that Eq. (7-135) reduces to the formula for back-scattering [Eq. (7-113)J when the transmitter and receiver coincide.

362

TIME-HARMONIC ELECl'RQAlAGNETIC FIELDS

7-11. Scattering by Dielectric Obstacles. I The problem of differential scattering by a. dielectric obstacle is represented by Fig. 7-17 if the obstacle is now considered a.s a dielectric body. We shall assume it to be nonmagnetic (}ol = 1£0), but it may be lossy if E is complex. The extension to magnetic obstacles is given in Prob. 7-42. When the obstacle is excited by a source, there will be induced in it polarization currents given by

J' - jw(. - "0/2, as shown in the insert of Fig. 7-1.8. For our first approximation, let us take (7-147)

where k - (oJ V;;; is the wave number of the dielectric. This very crude assumption yields curve (b) of Fig. 7-18. For a better approximation, which yields curve (c) of Fig. 7-18, take (7-148)

where A is a variational parameter to be determined either by the Ritz procedure or by the reaction concept. While Eq. (7-148) is a better approximation than Eq. (7-147), it is still crude. The integrations occurring in the various reactions were accomplished by expressing the exponentials and Hankel functions as Bessel function series, according to Sec. 5-8. The resulting series converged fairly rapidly. An alternative procedure for treating dielectric obstacles can be given

~

0.00012

I

I

,

hi

/

,, i

--+l",1+-

~ 0.00008

---~

/.

-, r-, ,

~--

V

'"

/

0.00004

/

o1.00

1/ 1.04

1.08

1.12

..

(a)

r--.. .... f-j

1',

1.16

Ie,) ,(b)

~

1.20

"-

1.24

1.28

./

Flo. 7-18. Scat.tering by a dielectric cylinder (0) exact. aolut.ion, (b) Jirsw,rder varia· ~ionalllOlution, and (c) aec:ond..<Jrder variat.iollAl solution. (AfUr' Cokm.)

365

PERTURBATIONAL AND VARIATIONAL TECRNlQUES

in terms of cQuivalent currents over the surface of the obstacle. I This method leads t.o more t.han one formula for the desired parameter, and Rumsey discusses how to choose the best approximation according to the react.ion concept. 7-12. Transmission through Apertures. The problem of transmission through apertures in an infinitely thin, perfectly conducting plane is closely related to the problem of scattering by plane obstacles. The precise interrelationship is shown by the following extension of Babinet's principle for optics. Consider the three cases of a given source (a) radiating in free space, (b) radiating in the presence of an electrically conducting screen, and (e) radiating in the presence of a magnetically conducting screen, as shown in Fig. 7-19. The electric and magnetic screens are said to be eomplemen/.aTY if the two screens superimposed cover the entire V = 0 plane with no overlapping. (The apert.ure of one is identical to the obstacle of the other.) Let the fields V > 0 be designated (EI,H'), (Eo,Ho), and (E"',H"') for the cases (a), (b), and (e), respectively. Then Babinet'a principle for complementary screens states that H'

+ H" =

H'

(7-149)

proved as follows. Let S. be the screen surface of Fig. 7-19b, and S. be the aperture surface of Fig. 7-19b. The total field in each case is the incident field E' plus the scattered field E' produced by the currents on the screen. An element of electric current produces no components of H tangential to any plane containing the element (see Sec. 2-9). The currents induced on the screen thus produce no tangential H over the V - 0 plane; hence n X H- ... n X HI over S.

On the screen itself we havc the boundary condition n X E- "" 0

over S.

For the complementary magnetic screen, following similar reasoning, we find nXE"'=nXEi over S, n X H- = 0 over S.

By the above four equations, the sum E' + E-, H- + H- satisfies nx(E'+E-)=nxE' n X (H' + H-) "" n X Hi

overS. over S.

IV. B. Rumsey, The Reaction Concept in Electromagnetic Theory, PAil" Rtl1., vol. 94, no. 6, pp. 148&-1491, June 15, 1954.

2 aer.,

366

Tl!ld.E-HARldO~'lC

ELECTROMAGNE'l'IC FIELDS

+

Hence, the e m. field has the same n X E as the incident field over part of the y - 0 plane and the same n X H over the rest of the y .... 0 plane. These conditions are sufficient to determine E, H in the region y > 0 according to the uniqueness theorem (Sec. 3-3); so Babinet's principle [Eq. (7-149») follows. An alternative statement of Babinet's principle can be given in terms of the dual problem to Fig. 7-19c, shown in Fig. 7-19d. If the original repla.ced by K), the magnetic screen source is replaced by its dual replaced by an electric screen, and the medium replaced by its "reciprocal" (11 by 1/,,), then E will be numerically equal to -H- and H numerically equal to E'" (see Table 3-2). If the field of this dual problem is

a

I I I

Electric conductor S.

I I

EJ. HI 1JO

E-, He,

I

I

t

Is. I

I I I

Source

I

~n

I I

r-+- n

1

,-0

,~O

(a)

(6)

Is.

Is.

II

I

I

t

Source

'10

I

EM, Hili. "10

I S.

Magnetic conductor

I

*

S. Electric conductor

Dual source

I

I

IS.

~n

Is. ~n I

I

,-0

,-0

(0)

(d)

Flo. 7-19. Illustration of Babinet'a principle.

367

PERTURBATIONAL AND VARIATIONAL TECHNIQUES

Electric conductor

E' M,g"t;c cood"to,

"I.

,,-1,

",I,. Transmitter

I

E'+&

1,/ Receiver

Transmitter

Receiver

(6)

(0)

Flo. 7-20. The trnll8mitted field E' of (a) ill equal to the scattered field E' of (b).

denoted by E", R", Babinet'e principle [Eq. (7-149)} becomes H'" - E" = Hi

(7·150)

The problem of Fig. 7-19d is more easily approximated physically than is the problem of Fig. 7-19c. The direct application of Babinct's principle to the problems of Fig. 7-20a and b shows that the field transmitted by an aperture in a planeconducting screen is equal to thc negative of the field scattered by the complementary obstacle. Hence, stationary formulas for..the signal at a receiver on the shadow side of a screen are of the same form as the stationary formulas for the scattered signal at a receiver in the complementary problem. In Fig. 7-20b, let the sources at the transmitter and receiver be magnetic currents across the "terminals" I, and 1.. Then, dual to Eq. (7-135), we have at the receiver

H- ·1.

= -

[fJ

(H')'· (Moo)' dS]

[fJ (H')'· (MtV dS]

ff (Ho),. (Mt)' ds

(7-151)

where Moll denotes the assumed magnetic current on the obstacle. It approximates the true magnetic current M. = (E+ - E-) X n = 2E' X n

(7-152)

where E+ and E- denote E in the regions y > 0 and y < 0, respectively, and n -= U lt• The interrelationships between Fig. 7-20a and b can be expressed as

368

TDlE-H.A.IU10NIC ELEcraOMAGr.'"ETIC FlELDS

Hence, from Eq. (7-151), we obtain for the aperture problem

[JJ

o.

PERTURBATIONAL AND VARiATIONAL TECHNIQUES

369

wave be specified by

H' =

Ue-ih

E' = '7H' X

UI/

(7-155)

"1/'

In the proof of Babinet's

n X H' - n X H'

(7-156)

where u is any unit vector orthogonal to principle, we noted that in the aperture

because the currents on the conducting screen produce no tangential components of H in the y = 0 plane. Equation (7-155) chooses Hi to be real in the y "'" 0 plane; so by Eq. (7-156) n X Hi is real in the aperture. Hence,

ff E' X H'· . ds

, = Re

= Re

.. per~

ff EI X H' • ds

(7-157)

..pen

Now consider the problem of Fig. 7-21b, which for

M.=E'xn is equivalent to Fig. 7-21a in the region y > 0, problem,

~ - Re


u·n H4. n X Eo dB

(7-162)

370

TI1lE-HARMONIC ELECTROMAGNETIC FIELDS

2.0 1.5 h 1.0

\.

1

E,t 1

I

"'act Variational

0.5

o

0.2

0.4

0.6

0.8

1.0

where E· is the assumed tangential electric field in the aperture and H· is the magnetic field calculated from E· by the methods of Sec. 3~. As an example, let us consider tbe two-dimensional problem of transmission through a slot, u.s shown in the insert of Fig. 7-22. If we assume E· in the slot to be real, then E" X H"* = (E- X H·)*

01" FIG. 7·22. TransmissiOll coefficient. for a slot.ted conductor, incident. wave polarized traosvcnro to slot. ≪U!. (Alter

and the denominator of Eq. (7-162) is

II HO • n X Eo dB =

llfiJu.)

(ff Eo X Ho' • ds)' 10 Sec. 4-11 we defined the admittance of an aperture as

y.~.. -I~I'

ff

EX H' ·ds

and calculated it for a slot for particular assumed E's. Hence, applying Eq. (7-162) to a unit length of our two-dimensional slot, we have

[U

, I u·Eo X 7 - - Re ---r;;r.;;V 'y' I'l-~_ ..

"a

dJ)']

(7-163)

where a is the width of the slot. When the incident wa.ve is polarized transverse to the slot, we have the case of Fig. 4-22; hence we take

Eo _ I in the slot.

(7-164)

Now Eq. (7-163) reduces to T =

where Y.""•• = G. have for smaH a

~ He (+) "a y.""••

+ jB" is shown in

Fig. 4-22.

r'

To;:::, kolog ko

(7-165)

From Eqs. (4-106) we (7-166)

and from Eqs. (4-107) we have for large a

.....

T_I

(7-167)

371

PERTURBATIONAL AND VARIATIONAL TECHNIQUES

This last result is the geometrical optics approximation. The variational solution is compared to the exact solution, which can be obtained by solving the wave equation in elliptic coordinates l (Fig. 7-22). The case of a. plane wave at an arbitrary angle of incidence is considered by Miles. t If the incident wave is polarized parallel to the axis of the slot, we have the case of Fig. 4-23; so to make use of the analysis of Sec. 4-11 we would assume

..

Ed = cos-

(7-168)

a

in the slot.

Equa.tion (7-163) then reduces to T = 4a 1r;"

where Y.... ,t = G have for small a

d

Re(_1) Y:...

+ jB.. is shown in Fig. T

(7-169)

fI

4-23.

From Eqs. (4-115), we

~ 6.85 (~y

(7-170)

For large a we should expect the field in the aperture to be uniform. Hence, we should not expect the trial field of Eq. (7-168) to give good results for large a, say a > X. Equation (7-169) actually approaches 0.81 for large a, instead of the expected value 1. PROBLEMS

7-1. Suppose the cavities of Fig. 7-1 cOlltaiu lossy material characterized by /l. Show thlLt thc pertllJ'batiolllll formula. corresponding to &t. (7-3) is

IT, t,

and

jdbH X Eo·ds

~CT~lf~.,-

"'-"'0·-;

_

///I.E' E, -.H· Hold.

Note that both wand .

"d'

"'e

4D.b

-~-

w.

where "'. - 2r/b Hence, the mode separat.ion is increased. 7-13. Consider t.he rectangular waveguide with t.he bottom covered by fI, t.hin dieleetric slab (Fig. 4-6 wit.h d Sa). U8C 8 pert.urba.t.ional method and quasi-stat.ic approximatiou to show that the phase constant is

v'l

where (10 - k, the same 88 the first given in Prob. 4-14. 1-14. Consider the shown in Fig. 7-240. show that

(J.In' is the empty-guide phase constant. Note that this is term of an expansion of the exact characteristio equation, sa rectangulnr waveguide with 8. centered dielectric cylinder, all Use a perturbational method and quasi-6tatic approximation to

(1-fJ.

rdle.-l

1

-,-,- ~ 2-00- ,-,-+-, :v'~"l=~(.=.'il.=)'"

where

til.

can be taken

88

the cutoff frequency of the perturbed guide, given in Frob

375

PERTURBATIONAL AND VAIUATIONAL TECHNIQUES

7-11, if u il cloac to u 4 •

Sbow thA\ at the unpert.urbed TE,u cutoff frequency

/J .. k. 7-16. Suppose t.hat a waveguide ill filled wit.h 108lIY material, and comrider .. pert.\U'batKJD of illl pcrfect.ly conducting walls. ReprC8eot the unperturbed 6eldl (lUbecript OJ and tbe perturbed fiddll (no lubscripto) by toe....,.~

E. -

E - £e,,,.

H _

Alii....,...

H. -

Note t.he oppoeit.e directions of propagation. to Eq. (7-29) ia

'Y -.,.. -

II

Ae'"

Show t.hat the formula corTCllponding

,,(. 1:o Xft'ndl 'f .c ct. X 11- 2 xli.) 'u.do

s'

Show that thia reduces to Eq. (7-29) in the lose-free case. 7-16. Consider t.he pert.urbation of material in a lossy waveguide from I, 1/-, , to I + 111, I/- + 6p, , + two Represent t.he fielda lUI in Prob. 7-15, and show t.hat. t.he formula corresponding to Eq. (7-30) ia

.fJ(••• -;..

)2,

.,. -.,.. -

-1

IJ (t.

X

to - ....11 ,11.1 do

11- t xli.) 'u.do

Show that. t.his reduees to Eq. (7..:30) in the lOllS-free ease. 7-11. Use the reeults of Prob. 7-16, and let. t.he unperturbed guide be loa-free. Denote t.he propagation constant. of the pert.urbed guide by .,. - a + j/J, and let E ... E: and H .. -~. Show that. t.he resultant. approximation for /J is Eq. (7-33) aod a "

_~IJT"_III...,..I'_d'_

2Refft.x:a:.u.d4

Note that this is an Approximate form of Eq. (2-76). 7-18. Consider the perturbation of the walla of a waveguide from ductor to an impedance sheeL Z. 8uch t.hat.

D.

perlcet con-

nXE-Z.H

Represent the unperturbed and perturbed fields a.s in Prob. 7-15, and show that

7-19. Uee lobe reeulta of Prob. 7-18 and let the unperturbed guide be lOllS-free, 110 t.hAt. ,.. - ;/J.. In the pert.urbed guide, let Z - (J( + j~, .,. "'" a + jfJ, E - &:,

376

H - -

TIME-HARMONIC ELEGraOMAGNETIC PIELDS

a:, and abow that

f "'19.1' JJ t, f 1ll19.1· •• ---A,------

fJ, ...

(J -

dl

--17---,---

2Re

Xa:'u.da dl

2Re / /

t.

X

a: ·u,'"

If Z - 11, the intrinsic impedance of metal walls, the above formula for a is the approximation that we have been using to calculate attenuation in metal wAveguides.

7-110. Show that

/ / / .-'Iv

.... -

X

EI' d,

~//"/-.IB-I·d-,-

is a. stationary formula for the resonant frequency of a lOIl-lru cavity, provided n X E _ 0 on S, but is not stationary if 108!le8 a~ pre&ent. 7-21. Show tluLt. Eq. (7-46) is a stationary formula for ..... 1, with no boundary conditions required on H. 7-22. CoOllider the rectangular cavity (Fig. 2-19) and the stationAry fonnula IEq. (7-44)). Use a trial field E - u.. ~z(~ - b)(z - c)

and abow that.

F~.

(7-4.01) gives w,. ..

Vfij ~Ol + cl be ..

In the exact 1lO1ution (Eq. (2-95)1, the numerical factor is 1T inatead of V'iO, 7-23. Consider a arnaU ddormation of the walls of a cavity, such all represented by Fig. 7-1. Tako tho variational formula IEq. (7-45)), which requires no boundary conditions on E, and take the unperturbed cavit.y field E, as a trial field. Show that. Eq. (7-45) reduces to

.

Wi - "': --

•:

/// (,.lll.l· - .IB.I') d, -'~''-''77----/ / / .IB.I'd,

Show that thia formula is essentially the same as Eq. (7-4). '1-111, Figure &-31b shows a partially lilled circular cavity. trial field

H- u.,J

1

(2.405~)

to show that the dominant. mode re8On&nce ia ........ 2.405 a

V;;;

Compare with the results of Prob. 6-24.

1 _ ~(, __ .')

UIMl Eq. (7-46) and ,.

PERTURBATIONAL AND VARIATIONAL TECHNIQUES

377

7·2l5. Consider a waveguide whose cross section is an equilateral triangle of aide length 11. Use variational formulas to approximate the lowest. cutoff frequency. The exact solution ia

.. ---."

3o.y';;

7-26. Considcr the rectangular cavity (Fig. 2-19) and the mhed-field variat.ional Connula lEq. (7-72)1. Choose a trial ficld E

. ... y ....11 -u~smbsmc

H

- U r A ISln1)COS

·..-y

".r,+A

c

ry.".Z u'lcosb'Bln

c

where AI and AI are variational parameters. Determine Al and AI by the Ritz method, and show that the resultant formula for "', is the exact formula IEq. (2-95)1. Why do we get an exact solution in thia case? 7-27. In Fig. 7-25, the surface S represents a perfect electric conductor enclosing a cavity. A variational solution is desired in terms of a trial field aatisfying n X E _ 0 n

I

I n "'-I

t.

(1)

(2)

s

FlO. 7-25. Trial fields are disconLiJJuous over a. on Sand n X (p-lV X E) continuous at a, but with n X E discontinuous at a. that the stationary E-ficld formula is

Show

where subscripts 1 and 2 refer to regions] and 2 (Fig. 7-25). Show also that a varia-tional solution in terms of trial fields satisfying n X E - 0 on Sand n X E continuous at a, but with n X (,l.I-IV X E) discontinuous at 8, is given by Eq. (7-44). 7-28. Show that the variational H-field formula. for Prob. 7-27 is of the same form 9.8 the above E-Beld formula, given by replacing E by H, • by ,l.I, and ,l.I by t. Show tbat no boundary conditions e.t S are required in the H-ficld formula. 7-29. Consider a perturbation of material in a cavity, auch as represented by Fig. 7-2. Take the mixed-Bcld variational formula fEq. (7-72)], a.nd take t.he unperturbed cavity field Eo, H o as a trial field. Sbow that Eq. (7-72) then reduces to Eq. (7-11). 7-30. Repeat Prob. 7-26, using the reaction concept of Sec. 7-7. 7-31. Consider the partially filled rectangular waveguide of Fig. 4-&. Use the E-ficld variational formula [Eq, (7-8)1, and the trial field E

. '" -u.SID'B

378

TWE-BARMONlC ELECTROMAGNETIC FIELDS

and show that Col. _

!

a

[1 + ~ (!! - ..!.. Bin 2lt'd)J-~ flo2ra

Compare BOrne calculated points with the exact eolution (Fig. 4-9). 7-82. Use the reaction concept to derive the mixed-field variational formula ror waveguide phase constants

which corresponds to Eq. (7-85) if n X E _ 0 on C. No boundary conditions are required in the above formula. 7-53. Consider the variational formula of Prob. 7-3'2 and II perturbation of waveguide walls, &8 illustrated by Fig. 7-5« and b. Use the unperturbed field E.. H. as a trial field, and ahow that the formula of Prob. 7-32 reducetl to Eq. (7--32). 7-34. Consider the variational formula of Eq. (7-85) and a perturbation of matter in a waveguide. represented by Fig. 7-00 and c. Use the unperturbed field E" H. lUI a trial field, and show that Eq. (1-85) reduces to Eq. (7-33). 7-3li. Figure 7-26 shows a coaxial stub to parnllcl·plat.e waveguide feed 8yst.em. Aasume II « "- eo that fL rensonablo trial current is a uniform current. Show by the variational method that. the impedance !een by t.he coax is

• Z _ -ka 4

where

l' -

(1

.2 -,-log -7,",) ... 4

1.781.

____---=j=.I o-d matched

~!

~Ioad---'1111

matched

;

load

p

FlO. 7-26. Coax to parallel-plate feed.

7-36. In Prob. 7-35. remove the restriction on .tub

-

II

and assume a trial current on the

I - COli 1:(11 - z) Obtain the input impedance secn by the coax by the variational method. 7-37. Repeat Prob. 7-36 for the sccond-order variationalllOlution, assuming trial currents

/- - cos 1(4 - :I)

/- - 1

Note that only one new reaction is needed in addition to those obtained in Probl. 7-35 and 7-36. Speeialhie the result to II _ "-/4. T-38. Consider tho two-dimensional problem of planc.wave scattering by a conducting ribbon, shown in the insert of Fig. 7-16, but with the opposite polariZl\tion.

PERTURBATIONAL AND VARIATIONAL TEClL.'ilQ1JE8

'In other words, Hi ia pa.rallel to the aria of tho ribbon.

] ... u~ cos

379

Ulle the trial current

'Z

and .how that the variational solution ia L

32"1 "Y...", I' _

•

1

r~

where" Y.pu~ ia given in Fig. 4-23. Show that lLS la - ... this answer reduces to 0.66 times t.he physical optics solution. Why should we expect the above formula to be inaccurate for large ka1 1·59. Consider plane-wave scattering by a wire, represented by Fig. 7-15. At the first resonnnce (L ., )./2), the current ilJ

1- ..

cO.!

b

and we know that. (lJOO Fig. 2-24) (0,0) ., 73

The imaginary put of (0,0) is BelO becaWle the length is adjUllt.cd for naonanee. Using Eq. (1-115), show that. at resonance the echo area is A• .. 0.86).1

This is relatively inaensitive to the diameter of the win. 7-tO, Figure 7-27 represents a re8Onaot length of wire illuminated by a uniform plnne wave at the angle 8, polarized in the r-z plane. Using the approximations of Prob. 7-39, show that the back-ecattcring area is

[ ~,,)']' ooa !co. ~

A • ., 0.86).'

Apin this is relatively inaeollitive to the diameter of the wire.

zl ~'" L.,

T L 1

r (to

receiver) r"(to transmitter)

I

Flo. 7-27. Scattering by a resonant wire (L .. )./2). 1-4.1, Repeat Prob. 7-40 for the ease of differential ecattering, showing that the differential echo area ia

A• .. 0.86).1

[

'o, (~ao,,) ao, (~ao, ")]' "

SID

whl!re .d. ill defined by Eq, (7-114) with

'

lIln

"

E' evaluated in the I' direction.

380

TIME-HARMONIC ELECTROMAGNETIC FIELDS

7-i2. Consider differential scattering by a magnetic obsf.aele (Fig. 7-17) and define

Show tbat, instead of Eq. (7.143), we have

Echo _ where

(',a) FCa,a) (a,a) ....

«i,a}jl)1 F(a,a) - (a,a)

JJJ (E;· J' - H' . M') d. JJJ (,.-'(J')' - '. -'(M')') d.

JfJ

(E·'

1'> - KG.

M") dr

In the above formulas, EI, H' is the incident. field, Jft and M· are the assumed electric and magnetic polarization currents on the obstacle, and E", H" is the field from J", MG, 7-43. Figure 7-2& represents a metlLl. antenna cut from a plane conductor and fed across the slot abo Figure 7-28b represents tho aperture formed by the remainder of the metal plane left lLftcr the metal antenna was cut. The aperture antenna, fed

, , (b)

(a)

FlO. 7-28. (a) A sheet-metal antenna. and (b) its complementary aperture antenna.

across cd, is said to be complementary to the metal a.ntenna. Let Z", be the input impedance or the mctalllntenna and Y. be the input admittance to the slot antenna, and show that

Hint: Consider line integmls of E BOd H from a to band c to a, and use duality. 7-'U" Consider a narrow resonant slot of approximate length "'/2 in a conducting

8creen.

Show that the transmission coefficicnt ill

,

T "" 0.52w where to is the width of the slot. similar to those or Prob. 7-39.

Hinl.: Use the result of Prob. 7-43 and IUIsumptioD8

CHAPTER

8

MICROWAVE NETWORKS

y 8 -1. Cylindrical Waveguides. Several special cases of the cylindrical waveguide, n such as the rectangular and circular guides, already have been considered. We now wish to give a general treatment of cylindrical x s (cross section independent of z) waveguides '---------'c consisting of a homogeneous isotropic dielectric bounded by a perfect electric conductor. FIG. 8-1. Cross section of a cylindrical waveguide. Figure 8-1 represents the cross section of one such waveguide. Our formula.tion of the problem will be similar to that given by Marcuvitz. 1 A1J shown in Sec. 3-12, general solutions for the field in a homogeneous region can be constructed from solutions to the Helmholtz equation

(8-1)

In cylindrical coordinates, this equation can be partially separated by taking '" - i'(x,y)Z(z)

(8-2)

The resultant pair of equations are

+ k. ' '!' ... 0 dtZ + k tZ = 0 dz t •

'v'.t'!'

(8-3) (8-4)

where the separation constants k. and k. arc related by ke t

+ k.'

= kt

(8-5)

and 'Vr is the two-dimensional (transverse to z) del operator

a ,az

l",-l"-u-

(8-6)

IN. Marcuvitz, "Waveguide Handbook," MIT Radiation Laboratory Series, vol. 10, sec. 1-2, McGraw-Hill Book Company, Ino., New York, 1951. 381

382

TJi\lE-HARMONIC ELECTROMAGNETIC FIELDS

Solutions to Eq. (8-4) are of the general form Z(z) - Ae-f',-

+ BeI',-

(8-7)

which, for k. real, is a superposition of +z and -z traveling waves. The k. are determined from Eq. (8-5) aIter the k e (cutoff wave numbers) are found by solving the boundary-value problem. For TE modes, we take F = u•.p (superscript 6 denotes TE) and determine (8-8)

The component of E tangential to the waveguide boundary C is E 1" = 1· (u. X VI'!") = (n . V,'1")Z'

where 1is the unit tangent to C and n is the unit normal to C (sec Fig. 8-1). The boundary is perfectly conducting; hence E l = 0 on C and a",' _ 0

an

on C

(8-9)

The associated magnetic field is given by

H" = - -.-1 v x E' = -.-1

JW~

JWp.

(a'fa'"," u. iJxiJz -- + u - + u.k.~1f·) ayaz W

For morc concise notation, we define a tran8verse field vector as H, = H - u.II. and rewrite the above as I dZH,' = -.- (V,i") -d JWIJ

(8-10) (8-11)

Z

It is evident from Eqs. (8-8) and (8-11) that lines of 8 and :Je, are everywhere perpendicular to each other. For TM modes, we take A = u,'Jt'" (superscript m denotes TM) and, dual to Eq. (8-8), we determine (8-12)

Defining the transverse electric field vector E, by Eq. (8-10) with H replaced by E, we have, dual to Eq. (8-11), 1 dZ' E,'" = -.- (V1'I'''') -d JWE

(8-13)

Z

From the second of these equations, it is evident that for E. to vanish on C we must meet the boundary condition on

C

(8-14)

383

MJCROWAVE NETWORKS

provided Icc ':F O. Note that EQ. (8-14) also satisfies the condition 1 . E. = 0 on C. When the waveguide cross section is multiply connected, such as in coaxial lines, it is possible to have k. = O. In this case, the necessary boundary condition is '1'- - constant on each conductor. The corresponding ficld is TEM to z and is a transmission-line mode. It should be kept in mind that Eq. (8-3) subject to boundary conditions is an eigenvalue problem, giving rise to a discrete set of mades. These modes can be suitably ordered, and the various equations of this section then apply to each mode. It is convenient to introduce mode !urn;tions e(x,y) and h(x,y), mode voltages V(z), and mode currents I(z) according to E' = eoVo H," =- hOI"

(8-15)

Comparing Eqs. (8-15) with Eqs. (8-8) and (8-11), we see that we may choose Vo ... z. eo "" u. X V.i" = h' X u. ]. I_dZ· (8-16) JWIt dz

foc TE modes. and, comparing Eqs. (8-15) with Eqs. (8-12) and (8-13), e- -

1 dZv-= -jwt dz

-V,i'- = b- X u.

h-- -u.XV,i'--u.Xe-

for TM modes. to

z-

1-. =

(8-17)

:Furthermore, we normalize the mode veetors according

II (o')' d. - II (h')' d. II (,-)'d, - II (h-)'d. -

I

(8-18) I

where the integration extends over the guide cross section. Hence, all amplitude factors are included in the V's and l's. We shall now show that all eigenvalues aTe real. Consider the tW(r dimensional divergence theorem

II V,·Ad. and let A - i'·v,\{I.

~A.ndl

Then,

v,· A "'"' V,'I'·· V.V

+ i'·Vli'

"'"

Iv,'!'11 - k/j"irl l

and the divergence theorem becomes

ff (IV,'I'I' - k.'I'I'I')

d. -

~ '1"

: : dl

384

TIllE-HARMONIC ELECTROi\(AGNETIC FIELDS

But the boundary conditions on the eigenfunction "if are either Ilf =- 0 or (Jllfjan = 0 on C. Hence, the right-hand term vanishes a.nd

!! IM'I'd. !!I j, (8-22)

j

< j,

These are, of course, just the relationships that we previously established for the rectangular and circular waveguides. Figure 2-18 illustrates the behavior of a and {:J versus f. When the mode is propagating (f > fo), the concepts of guide wavelength, 2.

~ - -

~

• - P -- ---r,=iTjj' f.

-

JWE

(8-28)

f < f.

Note that these are just the characteristic wave impedances that we previously defined for rectangular and circular waveguides. Figure 4-3 illustrates the behavior of the Za'S versus frequency. Finally, from Eqs. (8-4), (8-16), and (8-17), we can show that V and 1 also satisfy the tram-

missWlIrlim eqootiom (8-29)

where Yo=- 1/Zo is the charaderi'lic admittance.

Hence, the analogy

386

TIKF;-UAJUlONIC E.LECTROYAGNETIC FIELDS

I joI'

I I

VI;'

i-

I I

I-

-I

dz (0)

1----

~/jOH

---~

I

jOJI£

I

i-I

I

I I I I

I

I I I

I

I I

I

I'

·1

dz (b)

FlO. 8-2. Equivalent. transmission lines Cor waveguide modes (IICries element. labeled in ohms, 'bunt elements in mhos). (0) TE modes, (6) TM modes.

with transmission lines is complete, and all of the techniques for analyzing transmission lines caD be applied to each waveguide mode. l We may define an equivaknt trammiuion lim for each waveguide mode as one (or which.., and Zo are t.he same as those of the waveguide mode. Such an equivalent circuit may help us to visualize waveguide behavior by presenting it in terms of the more familiar transmission-line behavior. For a dissipationless transmission line, we have

Zo ~

=

/Z IX VY = VB

- v'ZY

-jyXB Equating the above Zo and 'Y to those of a TE waveguide

(see Sec. 2-6). mode, we obtain

jX "" jWIJ

) 'B

.

=)~

+ -.k.'

JW~

(8-30)

Thus, the transmission line equivalent to a TE mode is as shown in Fig. Similarly, for a TM mode we obtain

&-2a.

.

.

k~'

]X=]Wp+-.-

1""

jB = jWt

(8-31)

I For u&mple, see Wilbur LePage and Samuel Seely, "General Network Analysis," Chape.9 and 10, McGraw_Hill Book Company, Inc., New York, 1952.

387

YICROWAVE NETWORKS

The transmission line equivalent to a. TM mode is therefore 88 shown in Fig. 8-2b. H the dielectric is lossy, the equivalent transmission will also have resistances, obtained by replo.cing jWE by (1 + jWf. in Eqs. (8-30) and (8-31). In the light of filter theory, we can recognize the equivalent t.ransmission lines as high-pass filters. The power transmitted along tlie wavcguide is, of course, obtained by integrating the Poynting vector over the guide cross section. Hence, for the +z direction, p, =

JJ E X H··uld, -

= Vl·

IJ e'da

=

VI·

JJ e X h··ulda (8-32)

VI-

and the time-average power transmitted is

/P. - Re (V[')

(8-33)

Hence, in terDl5 of the mode voltage and current, power is calculated by the usual circuit-theory formulas. It is also worthwhile to note that the modc patterns, that is, pictures of lines of Sand :JC at some instant, can be obtained directly from the -v's. For TE modes, H, is proportional to v.'I", and E is perpendicular to H,. Hence. lines 0/ constant '1" are auo linea of instantaneous S. Lines of instantaneous :re, are everywhere perpendicular to lines of instantaneous 8. Similarly, for TM modcs. lines of constant 'It. aTe aUo linea of instanlaneoua :re, and lines of instantaneous 8. are everywhere perpendicular to lines of instantaneous:JC. It is therefore quite easy to sketch the mode patterns directly [rom the eigenfunctions 'It. Recognizing that the gcneral exposition of cylindrical waveguides has been quite lengthy, let us summarize the results. Table 8-1 lists the more important relationships that we have derived. Those equations common to both TE and TM modes are written centered in the table. Keep in mind that all of the equations apply to each mode and tha.t many modes may exist simultaneously in any given waveguide. Finally, for future reference, let us tabulate the normalized eigenfunctions for the special cases already treated. For the rectangular waveguide of Fig. 2-16. we can pick the w's from Eqs. (4-19) and (4-21) and normalize them according to Eq. (8-18). The result is 'l'••C = _1 T

ab,_,. + (na)' cos

(mb)l

(rnT) a x cos (nT) bY

_,.2/ ab _(rnT)_(nT) = ; V(mb)' + (nap SID a:Z: SID bY

(8-34)

't" .......

where m, n = 0, 1, 2, . . . , (m = n = 0 excepted).

SimilarLy. for the

388

TIME-HARMONIC ELECTROMAGNETIC FIELDS TABLE

8-1. Smoo.RY OF EQUATION15 "OR TUE CTLINDBlCAL WAVEGtllDZ (TEM MODES NOT INCLUDED)

I

TE modes Transverse Helmholtz equation Boundary relations

+ k c i1t -0

Vt 1 '1'

a,,'

-an -0

TM modes 1

on C

qi''' -

0

00

C

." -

-'Vt'l''' -u. X'V,'I:'h" -

e' - u. X'Vt'lt· h' - -'Vj'l" Mode vectors

e-hXu. h - u. X e

JJ

Normalization

Propagation constant

Characteristic Z and Y

,.-jk.-

,'do -

I

"do-l

j~ -jkVl - U,If)' a - k, Vi UII,)' 1

jfoli'

Z," - -,. - Y,' dV

Z."

I> I, I E-X H'·ds =

N

Hence, the

N

._1~ V.-I.'cj>e"Xh"'ds= .-1I V,,"I.'

and the Lorentz reciprocity theorem reduces to N

N

L: V.·I.' - L: V.'I.·

Il_l

(8-45)

._1

To show that Eq. (8-45) is equivalent to Eqs. (8-44), it is merely necessary to consider the special CMes (1) all f." = 0 except It and (2) all f,,' = 0 except 11. Then Vt = zJ;Jt and Vf ... %tifl, and Eq. (8-45) reduces to 'Z(j = zJi. Similarly, taking all V,," = 0 except VI", and all V,,, - 0 except V tin Eq. (8-45) establisbes y" ~ Yii. 8-4. One-port Networks. A one-port network is characterized by a single impedance or admittance element. Visualize a surface enclosing the network such that the field is zero on the surface exccpt where it crosses the input guide, as shown in Fig. 8-4. We then have

P l"

=

-effiE X H··ds = -VI*1Pe X h·ds - Vf*

where Vand f are the mode voltage and current !lot the II reference plane," that is, at the cross section cut by the surface enclosing the network. Because of the conservation of complex power [Eq. (1-62)], we have

V 1° - p •• - i!',

+ j2",(W. -

(8-46)

'N.)

where ~~ is the power dissipated, OW. is the magnetic encrgy stored, and w. is the electric energy stored in the network. The input impedance to the network is therefore

z-

Gf. = dr [i!', + j2w(W. -

(8-47)

W.)J

which is well known for lumped-element network theory.

Similarly, the

FIG. 8-4. A one-port net.-

work and a. surface en· closing it.

~s

394

TtM.&HARMOl\'1C ELECTROMAGNETIC FIELDS

input admittance is

Y -

!Vi' - rvr1 ( ~.J we have

[S] ~ [z -

'.1I' + ',J-'

(8-66)

Similarly, the transmission matrix is related to the scattering matrix by [T) =

su _ SuE" su] Sa Sa Sl1 [ - Sa S121

(8-67)

The derivation of Eqs. (8-66) and (8-67), along with other relationships among the various matrices, can be found in vol. 8 of the Radiation

400

TDlE-IlA.R),[ONIC ELEcrRO»AGNETIC FIELDS

Laboratory Series. I For networks constructed of linear isotropic matter, the reciprocity relationships (Eqs. (8-44») apply. From Eq. (8-00), it is evident that reciprocity requires (lHl8)

in the scattering matrix.

From Eq. (8-67), it follows that reciprocity

fp.quires

TuTu - TuTu =

zZ"

(S-ll9)

"

in the transmission matrix. Equations (8-66) and (8-68) also apply to multipart networks. There are realizability conditions imposed on the matrices by the con~ servation of energy theorem. These conditions can be obtained from the corresponding one-port conditions by terminating the two-port network in various ways to form a one-port. For example, if port 2 is opencircuited (II - 0), then til is the input impedance. Similarly, whon port 1 is opco-i:ircuit-ed, Zu is the input impedance looking from port 2. Hence, by Eq. (8-47) we know

Re (zu)

~

0

Re (z,,)

~

0

(8-70)

Similarly, using the y matrix and short circuits on the ports, we obtain from Eq. (8-48) that

Re (Yu)

~ 0

Re (y,,)

~ 0

(8-71)

More generally, since Eqs. (8-47) and (8-48) must be valid for any passive termination, we can show that

Re (Zl1) Re (zu) - Re (za) Re (Ztl) Re (VII) Re (Yn) - Re (Ya) Re (Y21)

~

0

~ 0

(8-72)

Finally, when the network is loss-frec, the clements of the impedance and admittance matrices become imaginary, and restrictions on them can be obtained from the corresponding restrictions in the one-port case. Such considerations are particularly useful in the theory of filters. l Our principal concern for thc remainder of this chapter win be to obtain equivalent circuits for microwave networks. For any particular network, an infinite number of equivalent circuits will exist. One of oW' tasks will be to choose a .. natural" equivalent circuit, that is, one which suggests the physical nature of the network. For example, a section of I C. D. Montgomery, R. H. Dieke, and E. M. PuN:cll (eda.), "Principles or Microwave Circuita," Chap." MIT Radiat.ion Laboratory Series, vol. 8, McGraw-BiD Book Company, IDe., New York, 1945. • M. Van Valkenburg, "Network ADalYliJI," Chap. ]3, Prentice-Ball, lne., Englewood CIiBI, N.J., 19M.

401

WCROWAVE NETWORKS

FIo_ 8-9. A typical equivalent cireuit tOt a 1000·tree t.wo-port mierowave network.

waveguide would not be represented by an equivalent tee or pi circuit, since t.his would hide the transmission-line character of the guide. For loss-free networks, we shall use the symbolism of Table 8-2 in equivalent circuits. It should be emphasized that it is only the 8ign of a reactance or susceptance that dictates whet.her an inductor or capaoitor is ohosen. The reactance or susceptance does not, in general, have the simple frequency dependence of a lumped-element inductor or capacitor. Figure 8-9 illustrates a t.ypical equivalent circuit for a loss-free two-port network. TABU; 8-2. SUlllOLl&W: Element.

C1RCutTS

USED IN EQutVAL£ST

Lo&&-FREE

NETWOU8

Represents

Symbol

.n

or

jX

P08iLivc reactance

Inductor ---,

.. , jB

Negative S1.l8Ccptance

---4~

Negative reactance

---4ri!!-

Positive lJusceptal)ce

Capacitor

n:l

Ideal transformer

Transmission line

~C Z,

,

~l-----l

Change in impedance level

Waveguidc section

402

TIME-HARMONIC ELECTROMAGNETIC FIELDS

T

T

T

Z.

Z.

~

Zo

Z.

(0)

(b)

FIG. 8-10. (a) A 8ymmetrical obstacle in a cylindrical waveguide, and (b) an equivalent circuit.

In the case of dissipative networks, resistors in series with X or in parallel with B can be used to represent the losses. Similarly, the characteristic impedances and propagation constants of the equivnlent trans· mission lines can be assumed complex to account for losses. Most of the networks used in microwave practice are DIlly slightly lossy, and the small losses introduce only second-order corrections to the reactances calculated on a loss-free basis. 8-6. Obstacles in Waveguides. An object in a cylindrical waveguide can be represented as a two-port network. Figure 8-lOa shows an obstacle, symmetric about the cross section T, in a waveguide. Figure 8-lOb shows a possible equivalent circuit. In the more general case of an unsymmetrical object, the two Z6'S would probably be different from each other, and it might even be desirable to choose two reference planes T. In the loss-free case, the Z's will all be jX's. Before considering the obstacle problem, let us consider "dominantmode sources" in cylindrical waveguides. Figure 8-11 shows the electric source ]. in a waveguide terminated at z = 0 by a magnetic conductor and matched as z -+ - 00. The method of treating this problem is that used in Sec. 3-1 for rectangular guides, as, for example, Fig. 3-2. Let superscripts (I) denote the region -I < z < 0, and superscripts (2) denote the region z < -l. Then in region 1 there will be an incident wave plus a reflected wave such that HI = 0 at z. = O. Hencc, Ej(l)

-= A (e- il"

A

+ 61")e

Hll) ... - (e- I,. Z,

-

= 2A cos (fJz)

e

2A ei,s')b =- -,- sin (ftz) b

l ')

(8-73)

JZ,

where e and h are the mode vectors, fJ is the phase constant, and Zo is the characteristic impedance, all of the dominant mode (see Table 8-1). In region (2) there will be only a wave in the -z. direction; hence Elm = Be;'~e HI(I) _

-8 el" h

--

Z.

Continuity of E l at z ".. -I requires that 2A cos fJl =- Be- uJi

403

I.flCROWAVE NETWORKS

which determines B in terms of A. The boundary condition on H at -l is uJ X tRw - H(ll] = J.

t: "'"

J. - - ~~ ej~le

which leads to

A quantity of interest to us is the self-reaction of the current sheet

If

(8,.) ~

E . J. d. - -

2A' Z, (1

+ &~')

(8-74)

We shall use dominant-mode current sheets as mathematical II waveguide probes" to determine the equivalent circuit im"pedances. Now return to the original problem, Fig. 8-10a. We define even excitation of the waveguide as the case of equal incident waves from both z < 0 and z > 0, phased so that E , is maximum and H, is zero at z = O. By symmetry arguments, the H, scattered by the obstacle will also be zero in the z = 0 cross section i so a magnetic conductor can be placed over the z = 0 plane without changing the field. This divides the problem into two isolated parts, one of which is shown in Fig. 8-12a. The excitation is provided by the dominant-mode source J., which we have just analyzed. The equivalent circuit of Fig. 8-12a is shown in Fig. 8-12b. (The magnetic conductor is equivalent to an open circuit, and the J. is equivalent to a shunt current source I.) We now further restrict the problem to the loss-free case. Then the dominant mode will be a pure standing wave in the region - l < z < 0 of Fig.8-12a. If J. is located where E , = 0, then by the usual tro.nsmissionline formulas

Z Zo ~'or

=

Z.

+Zo2Z. =

the source of arbitrary l, the total reaction on

Reaction -

(8-75)

-jtanpl

J.

is

ff E . J. do = ff (E' + E') . J. d.

- (".) + (c,.) where E' is the field of

~

J.

alone, and E' is the field of the current on the

Matched guide

J·r

Magnetic conductor

-----IIi:.==-;:,==l.17 Fto. S.11. A domiuant-mode source in a waveguide terminated by a magnetio conductor.

404

TIME-IIARMONIC ELECTROMAGNETIC FIELDS

T

Matched ~

guide

T

T

t1-1.-,--.1 J

Mag. condo

T

-----.;-------'..z

1----,-_.1 (b)

(a)

T Matched ~

guide

Za

I

•

V T

T

1I.

Elect. condo

M•

Za

- - - - - . ; - - - - - - - ' +-

,-_.1

T

1+1,--1--'
since the I." is independent of ¢.

The "image " term is a source-iree field in the vicinity of the post and

can therefore be expressed as ElIlm....

=

. l

A ..J.. (kp)e f ...

"--00

(see Sec. 5.-8).

r

Thus,

E.'·..·d¢ - 2< A

oJ'(k;) - 2rJ. (k;) E,'·..··I,.o

and Eq. (8-86) reduces to

(a,a) - at [

Er L~ + J. (k;) E,'..... Lo]

(8-87)

The field of a. single cylinder of constant current was calculated in Sec. 5-6. Abstra.cting Crom Eq. (5-92), we have E II I1OH"",

-

~kIJo(k~)Ho(2)(kp)

p

~~

The field from each image is also of the above form, with p replaced by the distance to the image. Hence, Eq. (8-87) becomes

•

(a,a) - K [ H 0'" (k ;) where

K=

-

+ J. (k;) 2

§ka[!Jo(k~)

L:..

,

(-I)"H,"'(nkb)]

(8-88)

is an unimportant constant. Equation (8-88) is an exact evaluation of (a,a) for the assumed current of Eq. (8--85). Unfortunately, the Hankel function summation in Eq. (8--88) converges slowly and is not. convenient for computation. However, we shall now show that it can be transformed to

..

\ ' (-I)'H,'''(nkb) ~

,-.

~~[

1

~ ~(2b/X)'

1

4

+iG10g2r

b

-I

+8)]

(8-89)

where"Y "'" 1.781 and S is the rapidly convergent summation

1(2b/X)' I] n

(8-90)

409

MICROWAVE NETWORKS

The free-spacc field of a fllament of current is given by Eq. (5-84). Hence, the left-hand side of Eq. (8-89) is the E" from all images of the filament

-2

1--

,k

across the center of the original waveguide. ~This problem is Fig. 8-14 with J." replaced by the above f.) Then, by the method of Sec. 4-10, we can find the total field in the z = 0 cross section due to the above f. It is • E ow ~ ~ [ (n/b) (n./2) (nn/b)] (8-91) , • V(2b/X)' IlL. Vn' (2b/X)'

sin

+. \' sin

sin

•• 2

where only the first term is real bccause it is assumed that 1 < (2b/X) < 2. For large 11., the above summation has terms equal to those of

•

•

2:~sin(';)sinH;+ I)] - 2:

... )

C05no

n

.. _1.3.5....

_ Re

• . \'

4 ..... 1.3.5, ...

(e")" _ Re

(!2 log 1I + eei,') _ Re (!2 log I j sincosoI ) 1 j) _ I I - Re ( 2 10g tan (1/2) - - 2 log tan 2 1

11.

Hence, letting x = (b/2) + p in Eq. (8-91) and 8 = 7rp/b in the above identity, we can add and subtract the latter from the former and obtain

E

~[

00' _

"

_0 r

1 V(2b/X)'

1

+j

(!2 log 2b _ 1 + S)] rp

The free-space Ell from the same filament I is

When this is subtracted from the total Ell, and p set equal to zero, we have the right-hand side of Eq. (8-89). Returning now to the self-reaction, wc substitute Eq. (8-89) into Eq. (8-88) and obtain

Re(aa}-C

,

2

V (2b/X)'

X

-C-' 1 b

.N,(kd/2) 1m (a,a) = C [ - 2J ,(kd/2)

2,b

+ log T - 2 + 2S

]

(8-92)

410

TIME-HARMONIC ELECI'ROMAGNETIC FiELDS

where C is the unimportant constant,

C__ .ka l'J"(k~) 4r

2

Equation (8-92) is still exact for the current assumed in Eq. (8-85). However, because of the crudeness of OUf initial trial current, we caD expect our result to be valid onll:. for small d/>.. Hence, we use smallargument formulas for the Bessel functions and obtain

1m (a,a)

~ C (lOg ;~ -

2

+ 28)

(8-93)

Now, substituting from Eqs. (8-92) and (8-93) into Eq. (8-81), we ha.ve X.

+Zo2X. ~ ~).,....d [lOg 46 _ 2 + 28 (~)] ).

(8-94)

where S is given by Eq. (8-90). For odd excitation (Fig. 8-12c), we llSSume a current

J.- "'" induced on the post. 12

u"sin ~

The appropriate variational formula is Eq. (8-84),

T

-

I

1.0

X';',/Zob 0.8

7>

~

0.6

0.4

~

"lb=2.0

IB.ld ,

.l

End view

Top view

jXII

Zo

l'-..

'f-l

jX. 'X Zo J •

R::: ~I.O ~ ~ I'-....

Equivalent circuit

......

"'" '" "R

1.2-

- J4>,/Zoh

I 0.05

-lot-

L.j

1.4 -

O. 2

o

(8-95)

0.10

--

L--

g;; ~ 0.15

.-

~ :::::

11

0.20

dth FlO. 8-15. The centered circular inductive poet. in a rectAngular waveguide. Mamaih.)

(Afkr

411

!lucnoWAVE NETWORKS

the exact evaluation of which follows steps similar to those used to derive Eq. (8-94). The result is (8-96)

---f--6

I

FlO. S-16. A small obstacle

Figure 8-15 shows X .. and X& as calculated in a waveguide. from a second-order variational solution. 1 Our solution [Eqs. (8-94) and (8-96)] is accurate for small d/b, the error being of the order of 10 per cent for d/b = 0.15. Formulas and calculations for off-centered posts are also available. I A solution for the circular ca.pacitive post (Fig. 8-13b) is given in Prob. 8-12. 8-8. Small Obstacles in Waveguides. Figure 8-16 represents a small obstacle in a waveguide of arbitrary cross section. If the obstacle is symmetrical about a transverse plane, the equivalent circuit is as shown in Fig. 8-lOb. If the obstacle is loss-free, the Z's arejX's. The formulation of the problem for a conducting obstacle is that of Sec. 8-6. An approximate evaluation of tho reactions, made possible because the obstacles are small and not too near the guide walls, will now be discussed. Consider even excitation of the guide (Fig. 8-12a). The effect of a small obstacle is small; hence Z& is small and Z.. is large. Equation (8-81) is then X.. 1 1m (a,a) Zo = 2 Re (a,a)

(8-97)

where (a,a) is the self-reaction of the assumed currents in the waveguide. Let us first make some qualitative observations. In a rectangular waveguide, the reaction (a,a) is the free-space self-reaction of the obstacle plus the mutual reaction with all its images. For real current, the imaginary part of the free-space self-reaction becomes extremely large as the obstacle becomes smaU. Hence, for sufficiently small obstacles, we can let 1m (a,a) "'" 1m (a,a)/reo ..._

(8-98)

In contrast to this, the real part oC the Cree-space reaction approaches a constant, independent of the size of the obstacle, as the obstacle becomes small. The mutual reaction between the obstacle and its images therefore cannot be neglected. However, because the real part of the reaction is independent of the size and shape of the obstacle, we can calculate the dipole moment Il of the free-space obstacle and let

Re (a,a) = Re (Il,Il)

(8-99)

IN. Marcuvitz, "Waveguide Handbook," MIT Radiation Laboratory Series, vol. 10, pp. 257-263, McGraw-Bill Book Company. Inc., New York, 1951.

412

TIME-HARMONIC ELECTROMAGNETIC FIELDS

xt J... to

4-- b----j

T~

-

L~~ End view

z

----'----Side view

FIG. 8-17. A small conducting sphere centered in!L rectangular waveguide.

The right-hand term represents the self-reaction of a current element Il in the waveguide. As a.n example, consider the small sphere of radius c in the ccnter of a rectangular waveguide, as shown in Fig. 8-17. As our trial current, assume J." is that which produces the dipole field external to the sphere. This current, even though we shall not need it explicitly, is approximately

J."

II

.

(8-100)

= -U'-228m 8 ~c

where (J is measured from the x direction. Because the above current produces the same field as an x-directed element of moment Il, the ima.ginary part of the free-space self-reaction is the imaginary part of Eq. (2-115) evaluated at r = c. Hence, 1m (a,a)f'" ...-

= -." 23~

(~l)' (L)' 1\

/Ii(;

Equation (8-98) is therefore .~(Il)'

1m (a,a) "'" 12rJ c

(8-101)

'

For the real part of (a,a), we can use the analysis of Sec. 4-10 for a current sheet

J.-Il+-~),(y-n Because the current is real, we can set Re (!l,ll) ... - Re (P) of Eq, (4-87) and obtain

Re

(Il,If) -

where, from Eq. (4-86), 2

Jo 1 =-Il ab Hence, Eq. (8-99) becomes

Re (a a) ~ - Z. ,

ab

(If)' = -

.~. (II)'

ab).

(8-102)

MICROWAVE NETWOR.KS

413

Sub,tituting from Eq,. (8-101) aod (8-102) into Eq. (8-97), we bave

x.

Atab

Z. ~ - 24~'l.,c'

(8-103)

8ma1l~bst.acle approximation Cor a centered sphere in 8. rectangular waveguide. Our free-space reaction is the Rayleigh approxi· mation (Eq. (6-106)], which is valid for c/). < 0.1. Hence, we should expact Eq. (8-103) to be accurate wheo c/~ < 0.1 and c« a/2.

This is the

Now consider odd excitation of the guide (Fig. 8-12c). The evaluation of X. can then be made according to Eq. (8-84). Taking the current as real, we evaluate the imaginary part of (a,a) according to the free-space approximation [Eq. (8-98)]. However, because of the symmetry of the obstacle and of the excitation, there can be no net electric dipole moment, and Eq. (8-99) does Dot apply. There will be a magnetic moment KI (unless the obstacle has zero axial thickness), which can be calculated from the MSumed current. Then, analogous to Eq. (8-99), we use the approximation Re (a,a) ~ Re (KI,Kf) (8-104) where the right-hand term represents the self reaction of a magnetic current element Kl in the waveguide. Return now to the specific problem of a conducting sphere in a rectangular guide (Fig. 8-17). It is evident from symmetry that, for odd exeitation, the resultant magnetic dipole will be y--directcd. For the trial current. assume that which produces the magnetic dipole field external to the sphere. The free-spaee sclf~rcaction of this current is then just the dual of that for the electric dipole. given by Eq. (8-101). Hence. 1m {a,a)I ... 01'_ =

~(Kl)'

12 2 I (8-105) "re For the real part of (a,a), we evaluate the right-hand side of Eq. (8-104) by methods dual to those used to establish Eq. (8-102). For the centered y-directed magnetic current element iu tbe rectangular guide, we obtain 1m (a,a)

1'::

Y, Re (a,a) ~ Re (KI,KI) - ab (Kf)' -

~

ab.'. (Kl)'

Substituting from this and from Eq. (8-105) into Eq. (8-84), we have

Zo ahA, X, ",. - 12..-2e'

(S.106)

The accuracy of this formula is at least as good as that of Eq. (8-103). The evaluation of ot-her small-obstacle equivalent circuits can be found in the literature. I I A. A. Oliner, Equivalent Circuits for Small Symmetrical Longitudinal ApertW1!l aDd Obstacles, IRE TraM., vol. MTT-8, no. 1, January, 1960.

414

TlME-HAlUIONIC ELECTROMAGNETIC FIELDS

----['------

T

---------..!.. (a)

T

~ (b)

FIG. 8-18. (0) A diaphragm in a waveguide, and (6) an equivalent circuit..

8-9. Diaphragms in Waveguides. Figure 8-184 represents a cylindrical waveguide of arbitrary cross section with an infinitely thin electric conductor covering part of the z - 0 plane. This conductor is csJled a diaphragm, and the opening in it is called a. window. The diaphragm pill! the window cover the entire: = 0 cross section. The exact equivalent circuit is just a shunt element, as shown in Fig. 8-1&. Depending upon the shape of the diaphragm or window, the susceptance may be p.,

• \'

1 [ « f1. =-V7n:C'C=;'("'2b"'/"'~.T.), J. /(y) cos nTl/]' b dy

=---7[l'.><J-(Y-)-dY~]"'----

(8-115)

Equation (8-112) represents the special case fey) = 1. Better approximations to B/Yo can be obtained by using a better choice for fey), or by applying the Ritz procedure. The stationary formula. in terms of obstacle current [Eq. (8-107)] is specialized to the capacitive diaphragm as follows. The field is TE to x, given by Eqa. (4-32) witb

• !J' = sin ~ \ ' A cos nry e''''~ aLi" b

.-0

where The current on a diaphragm backed by a magnctic conductor (Fig. 8-194) is then

•

nry J.""'HlIl.-0 _abjrtwp cos~\' nA"sin b a '-'

J" - -H.

I

_.0

=

(T/a)' . JWP

•• 0

2: •

k' sin "" a

.-0

A" cos n-' _"_V b

Hence, the current has both x and y componenta, but the A" can be determined from the y component alone. The x component then adjusts itself to make the field TE to x. If we assume a current

J,. ""

g(y) sin ~ a

(8-116)

and define Fourier coefficienta (8-117)

then

418

TIME-BAnMONIC ELECTROMAGNETIC FIELDS

5

,.

(b~/

,,

4

-- -

~~

---

../

-- ---

~ CO) ' Cd)-

. rx \ '

Ell = - sm

0.1

0.2

0.3

0.4

b/'A, Flo. 8-21. The capacitive dia.phragm with c - 2b. (a) Exact solution, (b) crude aperture-field variational solution, (e) crude obstAcle-currcnt variational solution, and (d) crude qU8llistatic solution.

a ..~, ""A" cos nry -b-

Hcnce, in the same manner as Eq. (4-74) was derived, we find the sclfreaction of J." as • (a,a) ~

1

o

Also, at z = O. the tangential electric intensity is given by E" = 0, and •

ab \ ' 1

..,

"2 L.; '.

(Z,).J.'

where the characteristic impedances (Zo)" are the reciprocals of Eqs. (8-114). Because only the dominant mode propagates, only the n = 0 term of the summation is real, and Eq. (8-107) reduces to • (Z,).J.' 2Y, ,.~.~1=~~_ B"" 2ZJo"

L:

Substituting for In from Eq. (8-117) and for (Zo) .. = l/(Ya)l.. from Eq. (8-114), we finally have

(8-118)

This is the stationary formula in terms of obstacle current for the capacitive diaphragm of Fig. 8-20. Figure 8-21 compares various solutions to the capacitive diaphragm problem for the case of a diaphragm covering half the guide cross section. Curve (0) is called the exact solution because the estimated error is less than the accuracy of the graph. This solution is obtained by finding" quasi-static field and then using it in the variational formula, Eq. (8-115).1 Curve (b) is the crude aperture-field variational solution, Eq. (8-112), which is also Eq. (8-115) with f(y) ~ 1. Cu'v. (e) is • crud. I N. Marcuvit.r:, "Waveguide Handbook," MlT Radiation Laboratory Scries, vol. 10, secs. 3-5 and 5-1, McGraw-Hill Book Company, [nc., New York, 1951.

MICROWAVE NETWORKS

419

obstacle-current variational solution, Eq. (8-118), with . 'K(y - c) g(y) - ,m 2(b _ c)

(8-119)

(If the case g = 1 is tried, the solution diverges, because the boundary condition that the current vanishes at 11 = c is violated.) Curve (d) is

a. first-order quasi-static solution to the problem 1 B 8b 'KC - """ -logcscYo X 2b

(8-120)

Q

In practice, waveguides are usually operated with b/XQ < 0.25; so this last solution is a good approximation for most purposes. Note that the aperture-field variational solution, curve (b), is above the true solution, and the obstacle-current variational solution, curve (c), is below the true solution. That this is so for any trial functions /(y) and g(y) follows from the fact that Eqs. (8-115) and (8-118) are positive definite and hence are an absolute minimum for the true fields. Since Eq. (8-115) gives B/Yo and Eq. (8-118) gives Yo/B, the former yields upper bounds and the latter yields lower bounds to the true B/Yo. The existence of variational formulas for both upper and lower bounds is not very common and is a consequence of the self-duality of the problem plus the positive-definite nature of the resulting variational formulaa. Our crude variational solutions give an error of the order of 20 per cent, but it is remarkable that they are as close n.s that. A quasi-sLatic solution to the problem is f( ) _ co, (Ty(2b) y ""in' (Tc(2b) 'in' ("Y(2b)

(8-121)

which actually has a singularity at y = c. Hence, our approximation [(V) = 1 was an exceedingly crude choice, yet it led to usable results. Our approximation to g(y) [Eq. (8-119)] is equally crude. If we were to use Eq. (8-121) in Eq. (8-115), the result would be very close to the true solution. It is interesting to note that the three diaphragms shown in Fig. 8-22 all have the same equivalent circuits. This is evident, because the image systems for all three cases are identical. The treatment of the inductive diaphragm (Fig. 8-23) is similar to that of the capacitive diaphragm. The general variational formulas for upper and lower bounds are given in Probs. 8-14 and 8-15. For a crude aperture-field solution, we ~an assume Eq. (4-75) for E,t in tbe aperture. W. R. Smythe, "Static and Dynamic Electricity," 2d ed., Sec. 15-10, McGraw· Hill Book Company, lne., New York, 1950.

I

420

TDlE-HARAlONJC ELECTROMAGNETIC FIELDS

Ir--iI

,

L.-_ _--l..!. (6)

(aJ

(c)

Flo. 8-22. These three diaphragms give rise to the same shunt. capacitance.

xt-- 6 --f

_ _ _T'---_ _

I

T

1T

Jl

Stde view (a)

T

y,

-

jB

Y,

Y

End view

(6)

FIo. &-23. (0) Induct.ive diaphragm, a.nd Cb) an equivalent circuit..

This procedure gives

~ ~ Yo

_X.a [~ 1(c/a)']' (,6 B) 5m (7fe/a) X C

(8-122)

1I

where B. is the aperture susceptance plotted in Fig. 4-19. The values of -BIY. calculated from Eq. (8-122) will be higher tban the true values (of the order of 20 per cent higher). The problem can also be treated by quasi-static methods) a first-order solution being! B Yo

t::>< _

Xlai' (1 + esc' 2a ~) cot' 'Ire 2a

(8-123)

A combination of the quasi-static and variational methods can be used to obtain solutions of high accurney.' 8-10. Waveguide Junctions. We shall now consider waveguide junctions formed by butting two cylindrical guides together, possibly with a diaphragm covering part of the:;:: = 0 cross section. Figure 8-24 represents the general problem. No longer is there symmetry about the:;:: = 0 cross section; 60 the methods of Sec. &-6 do not apply directly. We there1 W. R. Smyt.he, "Statie and Dynamic Electricity," 2d cd., p. 555, McGraw-HiU Book Company, Inc., New York, 1950. IN. Mareuvitz, "Waveguide Handbook," M1T Radiation Laboratory Series, vol. 10. !leC. &-2, McGraw-Bill Book Company, Inc., New York, 1951.

421

MIcaOWAVE NE'TWORXB

(ore take the more fundamental a.pproach of const.ructing complete solutions in each region and enforcing

.If

E+ X H+· ds

-.£1

(8-124)

E- X H- . ds

where superscripts + and - refer to regions z > 0 and z < 0, respectively. In terms of the reaction concept, we caD think of Eq. (8-124) as stating that the reaction is conserved at the junction. An equivalent network for the junction is shown in Fig. 8-24b. It is evident that only a shunt element is required to represent the junction, because an electric conductor placed across the entire z = 0 cross section presents a short circuit to both waveguides. The characteristic admittances of the equivalent transmission lines are taken to be the characteristic wave admittances of the guides, and the ideal transformer represents the change in admittance level. If the characteristic admittance of the right-band transmission line were chosen as n t times the characteristic wave admittance of the guide, then the transformer would not be needed. We shall usc Eq. (8-124) to obtain stationary formulas for Band n t • It is assumed that the excitation is at z = - gQ ; hence in the region , 0, are now differentj so the superscript - has been retained on Y o- in Eq. (8-137). Finally, the value of -B/Y o- obtained from Eq. (8~137) will be larger than the true solution, because of the positive definiteness of the variational formula. The alternative equivalent circuit of Fig. 8-25 illustrates a very useful way of viewing the waveguide junction T 1 :n T of Fig. 8-24a. We bave separated the shunt susceptanee into two parts, which, by Eq. (8-129), can be identified as

J!...-

:IDCl: FIG. 8-25. Alternative equivalent circuit lor Fig. 8-24a.

LY.V.'

jn-

Yo-

=

-,,~~~

Yo-Vo t

jB+ Y o+ =

425

MICROWAVE NETWORKS

T

T

n : 1

Yo,

r

II Side view

T

Yo,

r

End \liN (0)

(b)

Fla. 8-26. (a) A thin coax.to-waveguide feed, and (b) an equivalent circuit.

where the Vi and V"i are given by Eq. (8-131). Note that B- depends only on guide z < 0, and in particular is one-half tbe shunt susceptance of a diaphragm, assuming E t in the aperture is unchanged. This assumption is, of course, incorrect, but. our formulas are stationary; so B- in the junction problem is approximately B/2 in the corresponding diaphragm problem. Similarly, B+ is approximately B/2 for the diaphragm problem corresponding to the guide z > O. Hence, by defining a.perture susceptances according to Eqs. (8-138), we effectively divide the problem into two parts, each part relatively insensitive to the other. An aperture susceptance calculated for the aperture and ODe guide, such as Figs. 4-17 and 4-19, thereby becomes useful for a wide variety of problems. 8-11. Waveguide Feeds. We shall DOW consider thin coax-to-waveguide feeds, as illustrated by Fig. 8-260. By thin, we meaD that the dimension in the axjal (z) direction is small. The analysis will be exact only for zero-thickness junctions. An equivalent circuit when only one mode propagates is shown in Fig. 8-26b. When morc than one mode propagates, Bay N modes, there will be N ideal transformers in series, each coupling to one mode. The justification for this equivalent circuit will be found in the analysis. Let the feed be viewed as l\ sheet of current J. in the z = 0 cross section. (This neglects the effeet of the gap. which is usually small.) Then, in the region z > 0, we have

(8-139)

where

r,+ is the +z reflection coefficient of the ith mode referred to

426 Z -

TIME-HARMONIC ELECTROMAGNETIC FIELDS

O.

Similarly, for z

< 0,

(8-140)

where rr is the -z reflection coefficient of the ith mode referred to z = O. We have ensured continuity of E, at z = 0 by choosing coeflia cients Vi the same in both Eqs. (8-139) and (8-140). The boundary con~ dition on H at z = 0 is

J.

= u. X (H,+ - H ,-)

-2:, v..

Y..

1.... 0

(~ ~ ~:= + ~ ~ ~::) u. X h ..

(8-141)

Multiplying each side bye, and integrating over the guide cross section, we have

v,y,

G~ ~:= + : ~ ~::) - - II J. -

e, d8

(8-142)

The field is then completely determined if the r's and J. are known. We now use the stationary formula of Eq. (7..s9) to determine the impedance seen by the coax. This formula is

ZOo

~

I~.'

-

II

E-J.d.

where the integration extends over the z = 0 guide cross section and II.. is the current at the reference plane T'. Using the first of Eqs. (8-139) for E, and Eq. (8-141) for ]., we obtain 1 '\'

•

(1 -

ZI.. = II"s ~ V, Y i 1

•

r,-

1 -

r.+)

+ r,- + 1 + r..+

Finally, substituting for Vi from Eq. (8-142), we have

z. (ff J•. e, d.)'

1 '\'

Z,. - I,.'

L, (1

-

r,

)(1

+ r,-)-' + (1

r,+)(l + r,+)-'

(8-143)

where Z, is the characteristic impedance of the ith mode. This is a sta.tionary formula for the input impedance of a zero-thickness COQx-to-waveguide feed. We can put it into a slightly different form by noting that

427

?tfiCROWAVE NETWORKS

the wave impedance of an ith mode referred to z = 0 is ~. ~ Zl

b,

•

+ r,r,

(8-144)

1_

Hence, Eq. (8-143) can also be written as (8-145)

This shows that the guides z > 0 and z < 0 appear in parallel for each mode. Nonpropagating modes decay exponentially from the junction and their r. may be taken as zero unless some obstacle is close to the feed. If we assume that only one mode propagates, then all Z. are imaginary except i = 0, and all r. = except i = 0, provided the terminations are not too close to the feed. Equation (8-143) or (8-145) can then be written as

°

(8-146)

where

(8-147) (8-148)

Equation (8-146) is, of course, just that for the equivalent circuit of Fig.8-26b. As an example, consider a probe in a rectangular guide (Fig. 8-27). Assume

J. -

1~. sin ked -

x) '(y - c)

where k = 2r/X is the wave number of free space. vector is eo

=

xd

(8-149)

The dominant-mode

/2 . "11 b

u., Vab sm

Equation (8-147) is therefore

n "'"

~ foci dx 10

6

dy sin ked - x) a(y - c) sin

(d)

?

TC (8-150) n' - - 2 .SlOt - tan t kk'ab b 2 The summation for X [Eq. (8-148)] divergcs, because the current was

giving

TIME-HARMONIC ELECTROMAGNETIC FIELDS

taken as filamentary. If the probe is taken as circular in cross section, the reactance can be evaluated by methods similar to those used in Sec. 8-7. How~ ever, if the probe is very thick, we shall have to modify the equivalent circuit of Fig. 8-2Gb. The reactance of a short probe can be estimated by the smallFlO. 6-27. Probe in fl, rectangular obstacle approximation of Sec. 8-8. It wAveguide. is evident from the sn;u:t.ll-obstacleanalysis that X is capacitive (negative) for a short probe and is of the order of magnitude of X for a probe over a conducting ground plane. Note that our present solution [Eqs. (8-146) to (8-148)], specialized to a rectangular waveguide matched in both directions, is the same problem treated in Sec. 4-10. From our equivalent circuit (Fig. 8-26), it. is evident that the coax sees

R ,. = under matched conditions.

n! ~o 2

Hence, !

2R I •

(8-151)

n -Z.

where R l • is the quantity calculated in Sec. 4-10. For example, when the probe is connected to the opposite wall of the waveguide, as in Fig. 4-20, we have from Eq. (4-91) ,

2. (tan

lea)! . ~

n=b~

'll"C

SInb"

(8-152)

Other possible feeds are shown in Fig. 4-28. 8-12. Excitation of Apertures. We now wish to consider conducting bodies containing apertures excited by waveguides. The general problem is represented by Fig. 8-28a. As far as the waveguide is concerned, the aperture appcars simply as a load across the reference plane T. A variational solution to the problem can be obtained by assuming tangential E in the aperture, calculating the resultant fields on each side of the aperture, and then conserving the flux of reaction by

II (E X H . ds)... ~ II (E X H . ds),•• ..pen

(8-153)

.. pe~t

This is the same approach that we took in Sec. 8-10 for the waveguide junction. Indeed, we can think of our present problem as a junction between the waveguide and external space.

)(ICROWAVE NETWORKS

Once the tangential E in the aperture is assumed, the problem separates into two parts, external and internal. We have anticipated this separation by taking the equivalent circuit as shown in Fig. 8-28b, where jB represents the internal susceptance of the diaphragm and Yap••, the external admittance of the aperture. The ideal transformer accounts for possible differences of impedance reference in the internal and external problems. The internal problem is identical to one-half of the waveguide-junction problem. Let us therefore abstract from Eq. (8-138)

LY,V,' jB

,

(8-154)

Y ..... Y.V.I

where

-

JJ Eo' . e, d&

V, -

(8-155)

These formulas give the internal shunt susceptance B in terms of an 8.S8umed E,& in the aperture. For tbe external problem, we dofine an aperture admittance as

Y.~.. ~ ~,

ff .,."

E,' X H'· d.

(8-156)

where V is some reference voltage and H- is the external magnetic field calculated from the assumed E 1-. Examples of some aperture-admittance calculatioll8 are given in Sec. 4-11. (These calculations were made on a conservation of power basis, but, beeause E- was assumed real, they are the same as variational solutions.) To determine we note that the dominant-mode voltage coupled to the aperture is V.. but we have referred the aperture admittance to V j hence

"I

n'- -VV',,

(8-157)

where V, is given by Eq. (8-155) applied to the dominant mode.

T Conductor

,-

, ,--.,

----

Side view

,:

,

l' n

I

I

jS

y.. Aperture

I

I

End view Ca)

(b)

FlO. S-28. (a) An aperture excited by a waveguide, and (6) an equivalent circuit..

430

TIME-HARMONIC ELEC'l'ROMAGNETIC FIELD8

0.004

I alb

0.002

1-.,.

/ 'V

o

,-

0.2

/i'--

C

/

-0.002

~G

---- -

-;:;'Ib - k.25

1/ 0;6

alb - I'

-];:JL 0;8

.J 1.0 a/~~

, ,

I I

CEJI

,~I

-0.004

I-a-ol

"

, ,

"

, ,

-0.006 FIG. g..29. Aperture admittance for rectangular apertures in ground planes, referred to the dominant-mode voltage of a rectangula.r waveguide of the same dimcDsiona. (Ailer Cohen, Crowley, and Levi,.)

An aperture of practical importance is the recta.ngula.r aperture in a conducting ground plane, as shown in the insert of Fig. 8-29. The aperture admittance has been calculated for the assumed field E •• =

7'

. U 1I 810-

a

(8-158)

in the aperture, referred to the voltage

v=jJ

(8-159)

which is the dominant-mode voltage for a. waveguide of the same dimensions as the aperture. Hence, when the aperture is simply the flanged open end of a rectangular waveguide, then n - 1. The field due to E,· in the aperture can be found by the methods of Sec. 3-6, and the aperture admittance calculated by Eq. (8-156). The mathematical details are tedious but can be found in the literature.! Figure 8-29 shows the aperture admittances for a square aperture and for a rectangular aperture with Bides in the ratio 1 to 1 and 2.25 to 1. 1 1 Cohen, Crowley, and Levis, The Aperture Admittance of a Rectangular Waveguide Radiating into Half-6paee, Qhw Stal~ UnilJ. Anten1UJ Lab. Rept. ac 21114 SR no. 22, H153. , Additional calculations have been made hy R. J. Tector, The Cavity-backed Slot Antenna, Univ. IUiMi4 Antenna Lab. &pt. 26, 1957.

431

!,(JCROWAVE NETWORKS

As an example, suppose we have a square waveguide of height and width 0 feeding 8. rectangular aperture with sides in the ratio alb - 2.25, as shown in Fig. 8-30. The waveguide is excited in the dominant y-polarized mode, for which

V2 . "" eo = u, sm -

•

•

Hence, by Eqs. (8-155) and (8-158), we have

V o ...

01" a

dz

0

1~ dy sin' ~ 0

=

0

_b_

V2

and eo, hy Eqa. (8-157) and (8-159), n'

IC1

•

b - 2.25

The shunt susceptance B is one-half that for the diaphragm of Fig. 8-22b. An approximation to B is therefore given by Eq. (8-120) with B replaced by B/2, b by 012, and c by b/2, giving B Sa rb a R:: -Iogcsc- = 3.54Y,).. 2a X.

-

Hence, the terminating admittance seen by the waveguide is y

~ j3.54 :. + 2.25Y...,.

where Y.,.•• is given hy the alb""" 2.25 curves of Fig. 8-29. 8-13. Modal Expansions in Cavities. Consider a cavity formed by a perfect conductor enclosing a dielectric medium. Each mode must y

Side view

End view

Fla. 8-30. A square waveguide £eediDg a reeuDgula.r aperture in

&

ground plane.

432

TIME-HARMONIC ELEC'1'ROMAGNETIC FIELDS

satisfy the field equations (8-160)

V X Eo' "'" - jWiJlH..

where i is a mode index. Either Eo or Hi may be eliminated {rom the above pair of equations, giving the wave equations V X (1l- 1V X E,:> - ",-'"E. = 0

(8-161)

V X (elv X Hi) - w,IJlH; - 0

valid even if E and p are functions of position. tions, coupled with the boundary condition n X E, - n X (c'v X H,)

~

Each of these wave equa-on S

0

,.. < 0.2.

,

T

f ,

Side view

End view

Equivalent circuit

Flo. 8-36. Centered capacitive post in a rectangular waveguide.

8-13. Conaidu the inductive diaphragm of Fi&. 8-23. aperture by

Approrimating B, in the

ehow that Eq. (8-122) is a crude variationallOlution (or the shunt IU8Oeptance. 8-If. The inductive diaphragm (Fig. 8-23) baa boundaries cylindrical to r. The incident. mode ill TM to 1/; hence, the entire field must be TM to Jj'. Expreee t.he 6eId aa H - V X uri- where

• tj;-

'\' L

.-1

a '''··

A .IID . .""

In the aperture, tangential E must be of the form

Show that

is a variational (onnu1& (Of the ahunt IU8Ceptance. Note that it gives upper bouads to -B/Y.. Problem 8-J3 is the special cu.ej(%) - sin (77./e). 8-16. Consider the indlJt:t.ive diaphragm (Fig. 8-23) and the variational fonnulaill terms of obstacle current {Eq. (8-107»). On the diaphragm, the t:urrent is of tbeform

].

-...

(.)

443

ltIICROW AVE NETWORKS

Show that

• Y.

a

[!.c [!.,. a d:l:

/:12 v(n/2)'1 'I;'

- 11 - 2>.

0

(4/X)'

c g(x) Sin

TZ

•

,,(x) sm

n.z

4 en

J'

]'

is the variationallormulo.lor lower bounds to -B/Y•. 8·16. Show that the shunt susceptance of the capacitive diaphragm of Fig. 8-37 is given by the same formula 88 applies to Fig. 8-228.

FIG. 8-37. A capacitive diaphragm (metaJ shown dashed).

8-17. Consider the capacitive junction of Fig. 8-38. Show that the parameters of the equivalent circuit are B+ 4b+ ...c - - -log CIlCYo ~ 2b+ B4b...c --I,.,~­ Yo ""' 2b-

-

n

b-

l

- b' -

Use the approximation of Eq. (8-120).

-1-- {~;;I---- --j Side view

End view

Equivalent circuit

FlO. 8-38. A capacitive junction. 8·18. Considc{ the waveguide junction of Fig. 8-24a and the equivalcnt circuit of Fig. 8-25. Show that, Analogous t-o Eqs. (8-138),

Y,jBand n l

-

101/1,t. The mode current.e arc given by

1, -

ff H," !i,do

where H,+ and H,- denote tangential H on the +z and -z sides of the junction, retlpcct.ively. Variational formul!l8 are obtained by M5uming H.+ IUld H,- subject to the restriction H, + - H,- in the aperture.

444

TIME-HARMONIC ELECTROMAGNETIC FIELDS

8-19. Let 0#(Z,1I) - !(P,.) be a 8OIution to the two-dimensional source-free Helmholtz equation p < a. Prove that

where

,i..e is

an operator defined by . D

Sin

,.

1 •

COllD-..,....--

-..,.....-

}k

ax

}k

and ,i.D.J-(O) means e'·D.p(z,y) evaluated at value theorem.

:t -

ax

0, II - O.

This is a. kind of mean-

8-20. Consider the eoa'll: to waveguide feed of Fig. 4-ZO. Let d denote the diameter of the coa.xialstub, and let a «>.. Show that, for the equivalent cireuit of Fig. 8-26b, 2a

1 "r + U ..1' dr de

ds - u.dydz

-

(A-4)

t

and differential elements of vector length are d1 = u.dz = u.dp = u.,.. dr

+ u,dy + u.dz + ..pd~ + u.dz + u,r de + u,.r sin e d,p

(A-5)

The elementary algebraic operations are the same in all right-handed orthogonal coordinate systems. Letting (Ut,Ut,u,) denote the unit vectors and (AI,At,At) the corresponding vector components, we have addition defined by A

+B

= u.{A.

+ B.) + u,{A, + B,) + u,(A. + B.)

(A-6)

scalar multiplication defined by A· B = A IB I

+ AtB t + AlB,

(A-7)

and vector multiplication defined by (A-8)

The above formula is a determinant, to be expanded in the usual manner. The differential operators that we have occasion to use are the gradient (vw), divergence (V· A), curl (V X A), and Laplacian (Vito). In rectangular coordinates we can think of del (v) as the vector operator (A-g)

449

¥ECI'OR ANALYSIS

and the various operations are

U.

Ur

u.

VXA=!..

d

d

(A-lO)

ay iiz A. A. A.

()x

V'w ~ d'w ax l

+ d'w + d'w alii

az l

In cylindrical coordinates we have

In spherical coordinates we have law I aw + U.-+ ur.SID - .0-a¢ r ao la (r S A) 1 a (A ,smO . ) +-.-1 dA. • +-.-v, A =--

vw

=

iiw

u.iir T10T

T SID 0 00

1 [d ( . ) dA.]

r 510 0 o¢

VXA=u.-.- A.sm8 - r 8m e ae of/>

I. dA. + U,-T1 [ - - -a (rA.) ] 8m8 o¢ ar

Vlw ==

.!:. ~ (TS ()w)

,1

ar

(A-l2)

+u.-r1 [a-iJr( r A . )aA.] -a8

1 ~ (Sin 0 iJw) + l 1 l alvJ ar + ,1 sin 0 as ao r sin 0 a¢1

A number of useful vector identities, which are independent of the choice of coordinate system, are as follows. For addition and multiplica-

450

TIM&-HA1UtONlC ELECTROMAGNETIC FlELDS

tiOD we have

A' - A· A

IAI'

= A·A'

A+B=B+A A·B - B·A A

x

B

~

-B

x

A

(A-13)

(A + B) . C = A· C + B . C (A + B) X C ~ A x C + B X C A·BxC=B·CXA-C·AxB A X (B X C) = (A· C)B - (A· B)C For differentiation we have

V(. + w) ~ V. + Vw V . (A + B) = V • A + V . B V X (A + B) = V X A + V X B v(vw) - • Vw + w v. V· (wA) = wV . A + A· Vw V X (wA) - wV X A - A X Vw V . (A X B) - B . V X A - A . V X B V'A ~ V(V· A) - V X V X A v X (v Vto) "'" Vv X Vw v X Vw = 0 V·VxA-O

(A-14)

For integration we have

JJJ V·Ad< - effi A . ds JJ V X A·ds - ¢A.dl JJJ V X A dr - - effi A X ds JJJ Vwdr = effiwds JJ n X vwd. = ¢wdl

(A-IS)

Finally, we have the Helmholtz identity 4rA -

-V

'fJ.(y Irv'·Arl

dr'

+V

X

f'(yv' X A

J. 1r

r[dr'

(A-16)

valid if A is well-behaved in all space and vanishes at least as rapidly as ,.-1 at infinity.

APPENDIX B

COMPLEX PERMITTlVITIES

The following is a. table of relative a-c capacitivities dielectric loss factors E~' where ~, "

"~

= -

EO

= -EOE' -

. E"

J-

Eo

= E,,

E~

and relative

. 11

- JE,

is the relative complex permittivity. The measurements, along with many others, were reported in II Tables of Dielectric Materials" (vol. IV, Mass. Inst. Technol., Research Lab. Insulation, Tech. Rept.). They also appear in Part V of U Dielectric Materials and Applications," Technology Press, M.LT., Cambridge, Mass., 1954.

Frequency, cycles per second

Material

Amber (fOSflil resin) ............

Bakelite (no filler) .............

Beeswax (white) ......... _.....

TOO

,

25

" 10";' ,

24

" lO'e:.' ,

23

Clay soil (dry) ................ Ethyl alcohol (absolute) ...... ..

Fibergla.a BK 174 (laminated) . .

lead~barium .............

25

Glass,

10'

a x 10'

3 X 10'

2.7 34

2.7 49

2.7

2.7 116

2.65

84

148

2.65 180

.... ....

2.0 223

2.6 234

8.2 1100

7.15

6.5 410

5.9 330

5.4 320

4.9 360

4.4 340

.... ....

3.64 190

3.52

585

2.65

....

130

2.43

2.41

2.39

2.35

165

145

....

2.35

205

120

113

,

2.17

2.17

2.17

2.17

2.17

2.1?'

0.9

1

2.17 6

2.17

17

2.17 1

2.17

130

0

3

8

35

4.73 570

3.94

3.27

2.79

2.57

280

170

.... ....

2.38 48

2.16

390

2.44 98

2.27

470

34

28

. ... ....

... . ....

.... . ...

24.5

24.1

23.7

22.3

220

80

150

600

6.5 165

1.7 10

9.8 255

7.2 115

5.9 52

5.3 24

5.0 17

4.8

4.M

4.40

4.37

365

12.5

10

13

16

,,

14.2

lOtf~'

-,,,

tOtO

470

, ... . " lO'f~' ... .

24 25

10'

2.48

" lO'e:.'

25

Glass, pho8phate .. ............ (2 per cent iron oxide)

10'

2.56 680

,

25

10'

2.63 310

10";'

::;

10'

360

,

~

10'

10'f;'

"

Carbon tetrachloride . .......... 25

10'

5.25

5.25

5.25

5.25

5.25

115

95

85

80

75

5.25 85

5.24 105

6.23 130

6.17

10j~'

240

5.00 210

,,

6.78

6.77

6.76

6.75

6.73

6.72

6.70

6.69

160

120

100

65

85

95

115

130

.... ....

6.64 470

lO'f~'

,

Gut.ta-.percha . ................

Loamy soil (dry) ... ...........

Lucite RM-1l9 ................

Myca.lex 400 (micA, gl!U1S) . .....

~

25 25

- 23 25

Neoprene compound .......... (38 per cent ON)

24

Nylon 66 .....................

25

Paper {Royalgrcy) .... ........

Paraffin 132 0 ASTM ...........

25 25

81 Plexiglas . ....................

Polyethylene (pure) ........ ....

27 24

,, 10·':" ,, ,,

2.61

2.60

2.47

2.45

2.40

2.38

s
rder functions of the first kind, and Fig. D-2 shows those for the second kind. For small arguments, we have from the series Jo(x) __ 1 _0

(D-9)

2 'Y" No{:z:) - -log-2 _0 r and, (or v

> 0,

J.(x) -;::t

~ (~y

N.(x) --+ _ (v - 1)1 (~)' _0

'Ir

(D-lO)

:z:

provided He (tI) > O. For large arguments, asymptotic series exist, the leading terms of which are

cos (x _!4 _ or) 2 N.(x) _ . [2 ,in (x _! _ or) _.. v;Z 4 2 /2x _. '\j...

J.(x) --+

(D-ll)

provided Iphase (x)1 < r. For the expression of wave phenomena, it is convenient to define linear combinations of the Bessel functions

+

H.Ol(X) ~ J.(x) jN.(x) H.Ol(X) - J.(x) - jN.(x)

called Hankel functions of the first and second kinds.

(D-12)

Small-argument

BESSEL FUNCI10NS

463

and large-argument fonnulas are obtained from those for J. and N.. particular, the large-argument formulas become

In

(D-Ia)

whicb place into evidence the wave character of the Hankel functions. Derivative formulas and recurrence formulas can be obtained by differentiation of Eqs. (0-2). Letting B.(x) denote an arbitrary solution to Bessel's equation, we have B;(x) = B_ 1

! B•

-

•

8;(x) .... -8-+ 1

(D-14)

+!

•

B•

which, in the special case v .., 0, become (D-IS)

B;(.) - -B,(.)

The difference of Eqs. (0-14) yields the recurrence formula B.(x) "'"

2(, - I)

•

8._ 1

-

B._ 2

(D-16)

which is useful for calculating 8 ..(x), n > I, from a knowledge of Bo(x) and 8 1(x). The Wronskian of Bessel's equation is often encountered in problem solving. This is (D-17)

from which Wronskians for other pairs of solutions cnn be easily obtained. When x = ju is imaginary, modi.fied Bessel functions of the first and second kind can be defined M I.(u)

~

j'J.(-ju)

K.(u) - ; (-J)o+'H."'( -ju)

(D-IS)

These are real functions for real u. General formulas for I. and K. can be obtained from the' corresponding formulas for J. and H.(2). Figure 0-3 shows curves of the zero- and 6rst-order modified Bessel functions. The large-argument formulas, obtained from Eqa. (D-11) and (D-12),

e"

I.(u)~ _,_ ........ V 2'11"'U

K.(u)

~ /72 . --V2U T

(D-19)

464

TUlE-BARMONIC ELECTROMAGNETIC PlELDS

, • • 2

/

'I

,;/ "

.\-K.

\''''

V

/

>L;(u)

(E-25)

which is a useful special case. Finally, some specializations of the argument will be of interest to us.

470

TIME-HARMONIC ELECTROMAGNETIC FIELDS

At 8 = 0, that is, at u = 1, the Qnm functions are infinite and m = 0

(E-26)

m>O At 8 = 1r/2, that is, at u = 0, P nm(O)

Qnm(O) =

=

{

(_I)(n+m>l2 1 . 3 . 5 0 2 .4 .6

{~_1)(n+m+ll/£2'14·6 .3.5

(n

+m

1)

-

(n - m)

(n

+m

-

(n - m)

1)

+ meven n + m odd n + m even n + m odd n

(E-27)

Some specializations involving derivatives are

(E-28)

BWLlOGRAPHY A. Clauical Books 1. Abraha.m, A'I and R. Becker: "The Classical Theory of Electricity," Blackie &; Son, Ltd., Glasgow, 1932. 2. Heaviside, 0.: "Electromagnetic Theory," Dover Publications, New York, 1950 (reprint).

3. Jeans, J.: "Electric and Magnetic Fields," Cambridge University Press, London, 1933. 4. Maxwell, J. C.:" A Treatise on Electricity and Magnetism," Dover Publications, New York, 1954 (reprint). B. lnlrodudory Book8 1. Attwood, S.: "Electric and Magnetic Fields," 3d ed., John Wiley & Sons,

Inc., New York, 1949. 2. Booker, H. G.: uAn Approach to Electrical Science," McGraw-Hill Book Company, Inc., New York, 1959. 3. Harrington, R. F.: "Introduction to Electromagnetic Engineering," MeGraw.Hill Book Company, Inc., New York, 1958. 4. Hayt, W. H.: "Engineering Electromagnetics," McGraw-Hill Book Company, Inc., New York, 1958. 5. Kraus, J. D.: "Electromagnetics," McGraw-Hili Book Company, Inc., New York, 1953. 6. Neal, J. P.: "Electrical Engineering Fundamentals," McGraw-Hill Dook Company, Inc., New York, 1960. 7. Page, L., and N. Adams: "Principles of Electricity," D. Van Nostrand Company, Inc., Princeton, N.J., 1931. 8. Peck, E. R.: "Electricity and Magnetism," MeGraw~HilI Dook Company, Inc., New York, 1953. 9. Rogers, W. E.: "Introduction to Electric Fields," McGraw-Hill Book Company, Inc., New York, 1954. 10. Sears, F. W.: "Electricity and Magnetism," Addison-Wesley Publishing Company, Reading, Mass., 1946. 11. Seely, S.: "Introduction to Electromagnetic Fields," McGraw-Hill Book Company, Inc., New York, 1958. 12. Shedd, P. C.: "Fundamentals of Electromagnetic Waves," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1955. 13. Skilling, H. H.: "Fundamentals of Electric Waves," 2d ed., John Wiley & Sons, Inc., New York, 1948. 14. Spence, D., and R. Galbraith: "Fundamentals of Electrical Engineering," The Ronald Press Company, New York, 1955. 15. Ware, L. A.: "Elements of Electromagnetic Waves," Pitman Publishing Corporation, New York, 1949. 16. Weber, E.: "Electromagnetic Fields," John Wiley & Sons, Inc., New York, 1950.

4"

472

TIME-HARMONIC ELECTROMAGNETIC FIELDS

C. lnUr'mtdiate and Advanced Boo~

1. Jordan, E.: "Electromagnetic Waves and Radia.ting Systems," PrenticeHall, Inc., Englewood CliiTs, N.J., 1950. 2. King, R. W. P.: "Electromagnetic Engineering," McGraw-Hill Book Company, Inc., New York, 1953.

3. Mason, M., and W. Weaver: "The Electromagnetic Field," Univenity of Chicago Press, Chicago, 1929. 4. Ramo, S., and J. R. Whinnery: "Fields and Waves in Modern Radio," 2d ed., John Wiley & Sons, Inc., New York, 1953. 5. Scbelkunoff, S. A.: "Electromagnetic Waves," D. Van Nostrand Company, Inc., Princeton, N.J., 1943.

6. Smythe, W. R.: "Static and Dynamic Electricity," 2d ed., McGraw-Hill Book Company, Inc., New York, 1950. 7. Strat.ton, J. A.: "Electromagnetic Theory," McGraw-Hill Book Company, Inc., New York, 1941.

D. Books on Special Topics 1. Aharoni, J.: HAntennae," Clarendon Press, Oxford, 1946. 2. Bronwell, A., and R. E. Beam: /fTheory and Application of Microwaves," McGraw-Hill Book Company, Inc., New York, 1947. 3. Kraus, J. D.: "Antennas," McGraw-HiU Book Company, Inc., New York,

1950. 4. Lewin, L.: IIAdvanced Theory of Waveguides," Illiffc and Sons, London, 1951. 5. Marcuvitz, N.: "Waveguide Handbook'" MIT Radiation Laboratory Series, vol. 10, McGraw-Hill Book Company, Inc., New York, 1951. 6. Mentzer, J. ft.: "Scattering and Diffraction of Radio Waves," Pergamon Press, New York, 1955. 7. Montgomery, C. G., R. H. Dicke, and E. M. Purcell (eds.): "Principles of Microwave Circuits," MIT Radiation Laboratory Series, vol. 8, McGrawHill Book Company, Inc., 1948. 8. Moreno, T.: HMicrowave Transmission Design Data," Dover Publications, New York, 1958 (reprint). 9. Reich, H. J. (ed.): "Very High Frequency Techniques," Radio Research Laboratory, McGraw-Hill Book Company, Inc., New York, 1947. 10. Schclkunoff and Friis: lly, S., 386 Segmental cavity, 284 Seidel, H., 222 SeIr·reaction, liS Separation of variables, 143, 198, 264,

381 Silver, S., 245, 303, 306 Simple matter, 6, 18 Singular 6eld, 32 Skin depth, 53 Slot in ground plane, 138(17, 18), 181186,261(32),370,430,444(21,22), 445(23) Slotted cone, 306 Slotted cylinder, 238 Slotted sphere, 302 Smythe, W. R., 324, 419, 420, 467 Sneddon, 1. N., 252 Snell's l3W, 58 Source coordinatcs, 80 Source-free regions, 37 Sources, 7, 12,19,95,96 Spherical Bessel functions, 265, 268, 464 Spherical cavity, 269-273 partiaUy-611ed, 313(7, 8), 326 Spherical coordinates, 265, 447 Spherical waves, 79, 85, 276, 286-289 Standing wave, 42-47, 69 Standing-wave pattcrn, 44 Standing·wave ratio, 45, 55 Static mode, 338

479

Stationary formulaa, 317, 34.1 for aperture admitlance, 428-431 for cavities, 331-345 for cavity feeds, 434-440 for impedance, 348--355 for obstacles in waveguides, 402-406 for scattering, 355-365 for transmission, 365--371 for waveguide feeds, 425-428 for waveguide junctions, 420-425 for waveguides, 345-348 Storer, J. E., 354 Stratton, J. A., 121, 324 Supcrgnin antennas, 309 Surface of constant phase, 85 Surface currents, 33 Surface guided waves, 168-171,219 Surface impedance, 53,371(2),375(18)

Tai, C. T., 358 TE, TM, TEM, 63, 67, 130, 202, 267, 382 Tector, R. J., 430 Teichmann, T., 434 Tensor Green's functions, 123-125,356 Tesseral harmonics, 273 Tightly bound wave, 170 Total reflection, 59 Transmission, 360 Transmission area, 368 Transmission coefficient, 55, 368 Transmission lincs, 61-66 biconical, 284-286, 313(13) equivalent, 386 modC8,63 parallel·plate, 90(28), 01 (31), 189(6, 7), 440(1) radial, 211 twin-slot, 135(7) wedge, 212 Transmission matrix, 399 Transverse 6eld vector, 382 Transvcrse fields, 63, 67. 130, 202 Traveling waves, 39 Trial field, 332 Twin-51ot line, 135(,>

480

TIME-HARMONIC ELECTROMAGNETIC FIELDS

Uniform plane wave, 39, 147 Uniform waves, 85

Waveguide junctions, 172-177, 193 (27-29), 420-425, 443(17, 18)

Uniqueness, 100-103 Units, 1

Waveguides, 66 biconical, 284-286, 313(13)

circular (see Circular waveguides) Van Valkenburg, M. E., 397, 400, 435

Variation, 332 Variational methods, 317, 331-380 Vector analysis, 447-450 Vector Green's theorems, 121, 141(28) Veclor potential, 77, 99

Velocity, of energy, 42 of light, 5 of phase, 39, 40, 88, 86, 385 Voltage, 3, 15

Voltage source, 96, 118 Von Hipple, A" 23

Wait, J. R., 240, 242 Wall impedance, 371(2, 3), 375(18) Wave equation, 37 for inhomogeneous matter, 88(2)

Wave functions, 85 cylindrical, 199-204 plane, 143-145 spherical, 264-269

Wave impeda.nce, 39, 55, 86 characteristic, 69, 152 Wave number, 37 Wave potentials, 77, 129 Wave transformations, 230-232, 289292 Waveguide feeds, 179, 195(33,34),425428, 444(20)

corrugsted conduclor, 170, 193(25)

corrugated wire, 223 dielectric slab, 163, 192(22) in general, 381-391 psrallel-plate (see Parallel-plate waveguide) posts in, 406-411, 442(12) probes in, 178, 425-428, 446(26) radial, 208, 279 partially filled, 216 rectangular (see Rectangular waveguide) Wavelength, 40 cutoff, 88, 150,206,384

guide, 68, 384 intrinsic, 40 Waves, in dielectrics, 41-48 in general, 85-87 in lossy matter, 51-54

Wedge cavity, 284 waveguide, 208, 255(7), 256(14) Whinnery, J. R, 309

Wigner, E, P" 434 Windows, 414

Zeros, or Bessel functions, 205 or spherical Bessel functions, 270 Zonal harmonics, 273