Basic Electromagnetic Theory

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Basic ElectromagneticTheory

Electromagnetic analysis has been an indispensable part of many engineering and sci-entific studies since J. C. Maxwell completed the electromagnetic theory in 1873 [1].This is due primarily to the predictive power of Maxwell's equations as proven over theyears and the pervasiveness of electromagnetic phenomena in modern technologies.Examples of these technologies are radar, remote sensing, geoelectromagnetics, bio-electromagnetics, antennas, wireless communication, optics, high-frequency/high-speed circuits, and so on. Moreover, electromagnetic theory is valid from the staticto optical regimes and from subatomic to intergalactic length scales.

The problem of electromagnetic analysis is actually a problem of solving a set ofMaxwell's equations subject to given boundary conditions. In this chapter we reviewbriefly some basic concepts and equations of electromagnetic theory that are usedfrequently in this book. Our emphasis is on the presentation of various differentialequations and boundary conditions that define boundary-value problems to be solvedby finite element analysis. The solution of Maxwell's equations in free space is alsogiven in the form of an integral expression that relates the field to its source, followedby the description of Huygens's principle for calculating the exterior fields from thefield on a closed surface. For a complete presentation of electromagnetic theory,the reader is referred to textbooks such as [2-7]. This chapter may be skipped ifthe reader is familiar with the theory. Since the entire treatment of electromagnetictheory depends on vector analysis, we first review briefly the basic concepts andtheorems of vector calculus.

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1

2 BASIC ELECTROMAGNETIC THEORY

1.1 BRIEF REVIEW OF VECTOR ANALYSIS

Perhaps the most useful concepts in vector analysis are those of divergence, curl,and gradient. In this section we present definitions and related theorems for thesequantities.

Assume that f is a vector function, a quantity whose magnitude and direction varyas functions of space. The divergence of the vector function f is defined by the limit

V-f = lim -*-\(H)f-ds\ (1.1)A.̂ O At' [Jfa J

where s is the surface enclosing volume At' and ds is normal to s and points outward.From the definition of divergence, we can show that

if the vector f and its first derivative are continuous in volume V as well as on itssurface S. Equation (1.2) is known as the divergence theorem or Gauss's theorem.

The curl of the vector function f is defined as

V x f = lim - ^ - | $ d s x f | (1.3)

whose magnitude in the direction of n is given by

1 [f 1h • (V x f) = lim — * f • dl (1.4)

v ; A*-O As [Jc \

where c is the contour bounding surface As and n denotes the unit vector normal toAs. The directions of h and I are related by the right-hand rule. From the definitionof curl, it can be shown that

(I ( V x f ) - d s = I f -d\ (1.5)

if the vector f and its first derivative are continuous on surface S as well as alongcontour C that bounds S. Equation (1.5) is known as Stokes's theorem.

Now let / be a scalar function of space. The gradient of the scalar function / isdefined as

whose magnitude in the direction of n is given by

,.V/=£. (1.7)

fa'*]s

1

At'limV - f (1.1)

£* fdv fa•ds (1.2)

V X f : lim ——At;-^0 A f

If(itdsxf

(1.3)

h • (V x f) lim ——As—0 As

1 f 1ff -dl

Jc J

(1.4)

Js(V x f) • ds =

>cf -dl (1.5)

V / : lim —Ai;->0 At' Is

fda\ (1.6)

dfdn

h • V / = - (1.7)

whose magnitude in the direction of h is given by

where c is the contour bounding surface As and n denotes the unit vector normal to

1.1 BRIEF REVIEW OF VECTOR ANALYSIS 3

From the definition of gradient, we can show that

fff Vfdv= H fds (1.8)

if / and its first derivative are continuous in volume V as well as on its surface S.Equation (1.8) is known as the gradient theorem.

Two very useful identities involving the three basic vector differential operatorsare

V x (V/) - 0 (1.9)

V - ( V x f ) = 0 (1.10)

where / and f are any functions that are continuous with a continuous first derivative.Both identities can be proven with the aid of the divergence and curl theorems andare easily verified in Cartesian coordinates. Another useful identity is

V x V x f = V V f - V 2 f (1.11)

where V2 is known as the Laplacian. Other useful vector identities are given inAppendix A.

From the divergence theorem, one can derive some integral theorems that are usedfrequently in the formulation of the finite element method. If we substitute f = aVbinto (1.2), we obtain the first scalar Green's theorem

[[[ {aV2b + Va-Vb)dV= H a~ dS (1.12)

where V 2 / = V • V / . Exchanging the positions of a and b and subtracting theresulting equation from (1.12), we obtain the second scalar Green's theorem

Hi (aV2b - bV2a) dV=H (a^- - b^ ) dS. (1.13)v jfs\ dn dn;

If we substitute f = a x V x b into (1.2), we obtain the first vector Green's theorem

fff [(V x a) • (V x b) - a • (V x V x b)] dV

- (B{8LX V xb)-hdS. (1.14)

Switching the positions of a and b and subtracting the resulting equation from (1.14),we obtain the second vector Green's theorem

£ (b-(VxVxa)-a-(VxVxbpy

(axVxb-bxVxa)-y5. (1.15)s

on on /

db 7 da

Is \

= H (a x V x b) • n dS.

IILwid"§sid'


These are the standard Green's theorems. They can be generalized slightly to containanother function or parameter. These generalized theorems are given in Appendix Aand are actually more useful in the finite element formulation.

Exercise 1.1 Derive Gauss's theorem in (1.2) from the definition of the divergencein (1.1).

Exercise 1.2 Derive the alternative definition of the curl in (1.4) from the originaldefinition given in (1.3). Derive Stokes's theorem in (1.5) from (1.4).

Exercise 1.3 Derive the alternative definition of the gradient in (1.7) from the ori-ginal definition given in (1.6). Derive the gradient theorem in (1.8) from (1.6).

1.2 MAXWELL'S EQUATIONS

Maxwell's equations are a set of fundamental equations that govern all macroscopicelectromagnetic phenomena. The equations can be written in both differential andintegral forms, and here we present both to illustrate applications of some of theintegral theorems discussed in the preceding section.

1.2.1 The General Integral Form

For general time-vary ing fields, Maxwell's equations in integral form are given by

0 E • dl = - — / / B • ds (Faraday's law) (1.16)Jc dt JJs

/ H • dl = -f / / D • ds + [[ J • ds (Maxwell-Ampere law) (1.17)Jc dt JJs JJS

O D - ( i s = : / / / pdv (Gauss's law) (1.18)

B • ds = 0 (Gauss's law—magnetic) (1.19)

E = electric field intensity (volts/meter)

D = electric flux density (coulombs/meter2)

H = magnetic field intensity (amperes/meter)

B = magnetic flux density (webers/meter2)

J = electric current density (amperes/meter2)

p = electric charge density (coulombs/meter3).

Js

where

Js r5

'£>

/ E - d l = -— [[ BdsJc dt JJS

(Faraday's law)

(Maxwell-Ampere law)

(Gauss's law)

(Gauss's law—magnetic)

Jv§B.ds= Pdv

B • ds = 0

1.2 MAXWELLS EQUATIONS 5

In (1.16) and (1.17), S is an arbitrary open surface bounded by contour C, whereasin (1.18) and (1.19), 5 is a closed surface enclosing volume V.

Another fundamental equation, known as the equation of continuity, is given by

#'•*-"I#/-'*• a20)

This equation, which can be derived from (1.17) and (1.18), is the mathematical formof the law of the conservation of charge.

Equations (1.16)—(1.20) are valid in all circumstances regardless of the mediumand the shape of the integration volume, surface, and contour. They can be consideredas the fundamental equations governing the behavior of electromagnetic fields.

1.2.2 The General Differential Form

Maxwell's equations in differential form can be derived from (1.16)—(1.20) by usingGauss's and Stokes's theorems. Consider a point in space where all the field quantitiesand their derivatives are continuous. Application of Gauss's and Stokes's theoremsto (1.16)-( 1.20) yields

Among the five equations above, only three are independent for the case oftime-varying fields and thus are called independent equations. Either the first threeequations, (1.21)-(1.23), or the first two equations, (1.21) and (1.22), with (1.25)can be chosen as such independent equations. The other two equations, (1.24) and(1.25) or (1.24) and (1.23), can be derived from the independent equations and thusare called auxiliary or dependent equations.

1.2.3 Electro- and Magnetostatic Fields

When the field quantities do not vary with time, the field is called static. In this case,(1.21), (1.22), and (1.25) can be written as

V x E = 0 (1.26)

V x H = J (1.27)

V - J = 0 (1.28)

v—fdt

8B

V x H = f + Jatdt

8T>

V D = p

V - B - 0

V . J - 4atdtdo

(Faraday's law)

(Maxwell-Ampere law)

(Gauss's law)

(Gauss's law—magnetic)

equation of continuity).

(1.21)

(1.22)

(1.23)

(1.24)

(1.25)

V x E = 0

V x H = J

V - J = 0

(1.26)

(1.27)

(1.28)

dt ,

d

JV


whereas (1.23) and (1.24) remain the same. It is evident that in this case there is nointeraction between electric and magnetic fields; therefore, we can have separatelyeither an electrostatic case described by (1.23) and (L26) or a magnetostatic casedescribed by (1.24) and (1.27), with (1.28) being a natural consequence of (1.27).

1.2.4 Time-Harmonic Fields

When field quantities in Maxwell's equations are harmonically oscillating functionswith a single frequency, the field is referred to as time-harmonic. Using the complexphasor notation [3], (1.21), (1.22), and (1.25) can be written in a simplified form as

VxE=-jo;B (1.29)

V x H = ju;D + J (1.30)

V-J = -jup (1.31)

where the time convention e3ujt is used and suppressed and UJ is angular frequency. Itis evident that in this case, the electric and magnetic fields must exist simultaneouslyand they interact with each other, and it is also evident that the static case is thelimiting case of the harmonic fields as the frequency u approaches zero.

The use of time-harmonic fields is not as restrictive as it first appears. UsingFourier analysis, any time-varying field can be expressed in terms of time-harmoniccomponents via the the Fourier transforms

/»OO

E(t) = / E(u)ejuJt du (1.32)J — oo

1 f°°E(CJ) = — / E{t)e~3UJt dt. (1.33)

^ J-oo

Therefore, if a time-harmonic field is known for any LU, its counterpart in the timedomain can be obtained by evaluating (1.32).

Exercise 1.4 Show that the Fourier transforms defined in (1.32) and (1.33) do notviolate causality. In other words, show that E(ti) as evaluated from (1.32) usingE(UJ) given by (1.33) does not depend on Efo) for £2 > î-

1.2.5 Constitutive Relations

The three independent equations among the five Maxwell's equations describedabove are in indefinite form since the number of equations is less than the number ofunknowns. Maxwell's equations become definite when constitutive relations betweenthe field quantities are specified. The constitutive relations describe the macroscopic

V x E = -JUJB

V x H = juuB + J

V J = -jup

(1.29)

(1.30)

(1-31)

2W-OO

1 f°°— / EMe-^dt.

— oc

/»OO

E(t) =

EM =

E(u,'Wwf duj (1.32)

(1.33)

1.3 SCALAR AND VECTOR POTENTIALS 7

properties of the medium being considered. For a simple medium, they are

D = eE (1.34)

B = fjH (1.35)

J = crE (1.36)

where the constitutive parameters e, /i, and a denote, respectively, the permittivity(farads/meter), permeability (henrys/meter), and conductivity (siemens/meter) of themedium. These parameters are tensors for anisotropic media and scalars for isotropicmedia. For inhomogeneous media, they are functions of position, whereas forhomogeneous media they are not.

1.3 SCALAR AND VECTOR POTENTIALS

To solve Maxwell's equations, one may first convert the first-order differential equa-tions involving two field quantities into second-order differential equations involvingonly one field quantity. This is demonstrated here by considering the electro- andmagnetostatic cases.

1.3.1 Scalar Potential for Electrostatic Field

As we mentioned above, the electrostatic field is governed by (1.23) and (1.26). Thelatter can be satisfied by representing the electric field E as

E = - V 0 (1.37)

where <fi is called the electric scalar potential. Substituting (1.37) into (1.23) withthe aid of (1.34), one obtains

-V-(eV<£) =p (1.38)

which is the second-order differential equation governing (p. Equation (1.38) is thewell-known Poisson equation.

1.3.2 Vector Potential for Magnetostatic Field

The magnetostatic field is governed by (1.24) and (1.27). Equation (1.24) can besatisfied by representing the magnetic flux density B as

B = V x A (1.39)

where A is called the magnetic vector potential. Substituting (1.39) into (1.27) withthe aid of (1.35) yields the second-order differential equation

V x (-V X A ) = J . (1.40)

D = eE

B - fill

J - a E

(1.34)

(1.35)

(1.36)


This equation, however, does not determine A uniquely because, if A is a solution to(1.40), any function that can be written as A' = A + V / is also a solution regardlessof the form of / . Thus, to determine A uniquely, one must impose a condition onits divergence. Such a condition is called a gauge condition, and a natural choice forthis condition is

V - A = 0. (1.41)

It is important to understand that B is always unique even if A is not. Therefore, ifour objective is to calculate B, it is not necessary to impose the gauge condition.

The discussions above are pertinent to the static case. In the time-harmonic case,the electric and magnetic fields can also be represented by introducing a scalar anda vector potential, in a manner similar to the above [2]. However, this will notbe discussed here because in this book we will work directly with the electric andmagnetic fields for time-harmonic problems.

Exercise 1.5 Formulate time-harmonic fields in terms of a vector and a scalarpotential and discuss possible choices for a gauge condition.

1.4 WAVE EQUATIONS

As just mentioned, we will deal with the time-harmonic case directly in terms ofthe electric and magnetic fields. For this, it is necessary to derive from Maxwell'sequations, which involve both electric and magnetic fields, the governing differentialequations involving only either field.

1.4.1 Vector Wave Equations

The differential equation for E can be obtained by eliminating H from (1.29) and(1.30) with the aid of the constitutive relations (1.34)—(1.36). Doing this, one obtains

V x ( - V x E J - uo2eE = -juJ. (1.42)

Similarly, one can eliminate E to find the equation for H as

V x ^ V x H ) - u2i2H = V x f~j"\ . (1.43)

These equations are called inhomogeneous vector wave equations. It is evident thatthe solution of (1.42) also satisfies (1.23), and the solution of (1.43) also satisfies(1.24).

1.4.2 Scalar Wave Equations

In electromagnetic analysis, whenever possible we simplify problems by using atwo-dimensional model to approximate a three-dimensional problem. Assume that

v / i

/ I~cu2eE = -juJ.x EVx -V

UJ2/JH. = V x [ - J/I -A

7)V x f -V x HJ -

V ef 1X7 v T ^ (1.43)

(1.42)

1.5 BOUNDARY CONDITIONS 9

the fields and the associated medium have no variation with respect to one Cartesiancoordinate, say the z-coordinate. It can then be shown that the z-components of(1.42) and (1.43) become

T" ( — 4~) + T" ( — 4-)+ko€r] Ez=jk0ZQJz (1.44)dx \(j.rdxj dy \firdyj J

and

dx \erdxj dy \er dy J u J

= ~~TT~ I A "I" 7T~ I Jx I (1.45)dx \er J dy \er J

respectively, where er (= e/e0) and jir (= /x//io) denote the relative permittivity andrelative permeability, respectively, which are assumed here to be complex scalar func-tions of position; k0 (= cj^/eo/io ) *s t n e wavenumber in free space; Zo (— y/fio/eo )is the intrinsic impedance of free space; and e0 (= 8.854 x 1CT12 farad/meter) and/io (= 4TT x 10~7 henry/meter) are the permittivity and permeability of free space.Equations of the type of (1.44) and (1.45) are called inhomogeneous scalar waveequations.

1.5 BOUNDARY CONDITIONS

While there are many functions that satisfy the differential equations given above in adomain of interest, only one of them is the real solution to the problem. To determinethis solution, one must know the boundary conditions associated with the domain.In other words, a complete description of an electromagnetic problem should includeinformation about both differential equations and boundary conditions. In this sectionwe present some boundary conditions that apply to many practical problems. Thesecan be derived from Maxwell's equations in integral form (1.16)—(1.19).

1.5.1 At the Interface between Two Media

At a source-free interface between two media, say medium 1 and medium 2, thefields must satisfy four conditions, given by

h x (Ei - E 2 ) = 0 (1.46)

h- (Di - D 2 ) = 0 (1.47)

h x (Hi - H 2 ) = 0 (1.48)

f i . ( B i - B 2 ) = 0 (1-49)

where h is the unit vector normal to the interface pointing from medium 2 into medium1 (Fig. 1.1). These four equations are also known as field continuity conditions.

Ker dx) dy \er dy,+ kliir\Hz

1 d \ d ( \ d\ d ([dx '

d fi \ 0 / i

dx \er J dy \er•Jx) (1.45)

h x (Ei - E 2 ) = 0

h- (Di - D 2 ) = 0

h x (Hi - H2) = 0

h • (Bi - B2) = 0

(1.46)

(1.47)

(1.48)

(1.49)


Medium 1

Medium 2

Fig. 1.1 Interface between two media.

Among these four conditions, only two are independent: one from (1.46) and (1.49),and the other from (1.47) and (1.48).

Note that in (1.47) and (1.48) it is assumed that neither surface currents nor surfacecharges exist at the interface. If indeed there exists a surface electric current densityJ s and a surface charge density ps, these two equations must be modified to read

n - ( D i - D 2 ) =p8 (1-50)

hx ( H i - H 2 ) = Js (1.51)

Exercise 1.6 Derive the boundary conditions (1.50) and (1.51) from (1.18) and(1.17).

1.5.2 At a Perfectly Conducting Surface

The boundary conditions can be simplified when one of the media, say medium 2,becomes a perfect conductor. Since a perfect conductor cannot sustain internal fields,(1.46) becomes

n x E = 0 (1.52)

and (1.49) reduces to

n - B = 0 (1.53)

where E and B are the fields exterior to the conductor and h is the normal pointingaway from the conductor. Note that in this case the boundary can always support asurface current (Js = n x H) and a surface charge (ps = h • D).

1.5.3 At an Imperfectly Conducting Surface

When medium 2 is an imperfect conductor, it can be shown [8] that the electric andmagnetic fields at the surface of the conductor are approximately related by

E - (n • E)n = 7]Zofi x H (1.54)

n /ê,

th>*2

Medium 2

h- (Di ~D 2 ) = p3

nx ( H ! - H 2 ) =JS.

Fig. 1.1 Interface between two media.

(1.50)

(1.51)

n x E = 0 (1.52)

(1.53)n • B = 0

E - (h • E)n = r/Zon x H (1.54)

1.5 BOUNDARY CONDITIONS 11

or alternatively,

ft x E = nZ0 [(n - H)n - H] (1.55)

where q = y/JJhvJ&rt is the normalized intrinsic impedance of medium 2. Equation(1.54) or (1.55) is called an impedance boundary condition. It can be written in amore standard form as

—h x (V x E) - 3— ft x (n x E) = 0 (1.56)

or

— n x (V x H) -jkonn x (n x H ) = 0 (1.57)erl

which is known as a mixed boundary condition or a boundary condition of the thirdkind.

In the two-dimensional case, the impedance boundary condition can be written as

±dE±_JkoEjJLrl On 7]

for the case of E = zEz (usually referred to as the £^-polarization case), and

— ^T-=jkoriHz (1.59)er\ on

for the case of H = zHz (often referred to as the Hz-polarization case).

1.5.4 Across a Resistive and Conductive Sheet

An electrically resistive sheet is a thin sheet of electric current with density propor-tional to the tangential electric field at its surface [9]. Based on (1.46) and (1.51), theboundary conditions across its surface are

n x ( E i - E 2 ) - 0 (1.60)

ftx(Hi-H2) = J s (1.61)

with

hx (fixE) = -RJa (1.62)

where R is the resistivity in ohms per square. Substituting (1.62) into (1.61), weobtain

h x (h x E) - -Rn x (Hi - H2). (1.63)

h x E = nZ0 \(n • H)n - H (1.55)

jh

V

1

Mrln x ( V x E ) - -h x (n x E) = 0 (1.56)

(1.57)—n x (V x H) - jkonn x (n xH)=0erl

1

e r i

1 dEz jkor/jiri dn n

}-E (1.58)

1 dHz

eri dn"- = jkor]Hz (1.59)

n x (Ei - E2) = 0

n x (Hi - H2) = J s

(1.60)

(1.61)

(1.62)n x (h x E) = -RJS

n x (h x E) = -Rn x (Hi - H2). (1.63)


Resistive sheets are useful in numerical simulations because they can be used toapproximate thin dielectric layers with

R=^r-A-rr (L64)

jko{er - l ) rwhere er is the relative permittivity and r is the thickness of the dielectric layer.

The electromagnetic dual of a resistive sheet is a magnetically conductive sheetsupporting only a magnetic current. The boundary conditions across its surface are

h x (Hi - H 2 ) = 0 (1.65)

h x (ft x H) = -Gh x (Ei - E2) (1.66)

where G denotes the conductivity in Siemens per square. A conductive sheet is usefulbecause it can be used to approximate a thin magnetic layer with

G = — — ^ — (1.67)3ko(nr - i ) r

where Yo = 1/ZO and fir is the relative permeability of the layer.In general, a resistive and a conductive sheet can be combined to model a thin

material layer whose permittivity and permeability both differ from those of thesurrounding medium [9].

7.n

jko{er - l ) r

1.6 RADIATION CONDITIONS

When the outer boundary of a domain recedes to infinity, the domain is calledunbounded or open. A condition must be specified at this outer boundary to obtaina unique solution for the problem. Such a condition is referred to as a radiationcondition.

Assuming that all sources and objects are immersed in free space and locatedwithin a finite distance from the origin of a coordinate system, the electric andmagnetic fields are required to satisfy

™r[VX{H)+ikofX(l)]=° (L68)

where r = y/x2 + y2 + z2. Equation (1.68) is usually referred to as the Sommerfeldradiation condition for general three-dimensional fields. For two-dimensional fields,the Sommerfeld radiation condition becomes

^ ^ [ | ( S ) + ^ ( S ) ] = ° (i-69)where p = y/x2 + y2.

'Ez

\HZ,(1.6= 0lim ^fp

p—oc-

ddp'

'Ez

HZJ+ jko

where p = \Jx2 + y2.

E vH y

= 0 (1.68)V H ,

/ E 1

) + jkor x (lim rr—>-oo

V x

>ojko{lJ.r - l ) r

G = (1.67)

n x (Ht - H2) = 0

h x (n x H) = -Gh x (Ei - E2)

(1.65)

(1.66)

1.7 FIELDS IN AN INFINITE HOMOGENEOUS MEDIUM 13

Equations (1.68) and (1.69) are exactly valid at infinity. In numerical analysis, it isoften desirable to reduce the size of a computational domain by using a finite boundaryto truncate the infinite domain. When applied at such a finite boundary, (1.68) and(1.69) can be regarded as the lowest-order radiation conditions with limited accuracy.Better accuracy can be achieved by developing higher-order conditions (Chapter 9).

1.7 FIELDS IN AN INFINITE HOMOGENEOUS MEDIUM

In a general inhomogeneous medium, (1.42) and (1.43) do not have a closed-formsolution, and a numerical method is often required to seek their approximate solution.If, however, the problem at hand involves an infinite homogeneous medium, where eand /i are constant, closed-form solutions may be obtained. In such a medium, (1.42)and (1.43) become

V 2E + /c2E = jujfiJ + - V p (1.70)

V 2H-h& 2H = - V x J (1.71)

with the use of (1.23) and (1.24), where k = ujy/ejl. It is important to recognizethat (1.70) is equivalent to (1.42) only when it is solved in conjunction with (1.23);similarly, (1.71) is equivalent to (1.43) only when it is solved together with (1.24).Using the principle of linear superposition, one can find the solution to (1.70) and(1.71) as

E(r) = JJJ [-j^J(r') - Vp(r')] G0(r,r')dV" (1.72)

H(r) = jjj [V x J(r')]G0(r,r')dV' (1.73)

where Go(r, r') denotes the point-source response, known as the scalar Green'sfunction (see Appendix C), given by

e-jk\r-r'\

Go(r^) = —-—. (1-74)4 T T :

Equations (1.72) and (1.73) can also be written as

E(r) = JJJ [-jc^J(r')Go(r,r') + %(r')V'G0(r, r')] dV

= (-JULL + ~^VV^J jjj J(r')G0(r,r')dV" (1.75)

H(r) - JJJ/^) x V'G0(r,r')dV"

= V x jjj J(r')Go(r,r')d^' (1.76)


with the use of vector identities and integral theorems.Therefore, if a source is known in an infinite homogeneous medium, the fields

can readily be calculated by evaluating appropriate integrals. The results aboveare also applicable to surface or line sources provided that the volume integrals arereplaced by the corresponding surface or line integrals. They can also be reduced fortwo-dimensional problems by using the relation

i:G0{r,r')dz' = G0(p,p') (1.77)J — oc

where

G0(p,p') = ±:H™(k\p-p'\) (1.78)

in which p = xx + yy and HQ \k\p — p'\) is the zeroth-order Hankel function ofthe second kind.

Exercise 1.7 In a homogeneous, source-free region, (1.43) is reduced to V x V xH - jfe2H = 0 and (1.71) is reduced to V 2 H + k2H = 0. Show that H = xHoe-jkx

does not satisfy the former equation, but satisfies the latter. Show that this field doesnot satisfy (1.24) either. This exercise demonstrates that the solution to (1.70) and(1.71) does not necessarily satisfy Maxwell's equations.

Exercise 1.8 Show that (1.75) and (1.76) are equivalent to (1.72) and (1.73).

1.8 HUYGENS'S PRINCIPLE

The preceding section provided expressions to calculate the fields radiated by a givencurrent source. However, in the finite element method we usually solve for the fieldssurrounding an object instead of the currents induced on the object. Therefore, itwould be useful to have expressions to calculate the fields everywhere based onknowledge of the fields surrounding an object. Huygens's principle provides suchrelations.

Consider a surface S enclosing the source of radiation and any other objects sothat the infinite space exterior to the surface is homogeneous (Fig. 1.2). If the fieldson the surface are known, the fields outside the surface are given by

E(r) = H {-jw [n! x H(r')] G0(r, r') + [rY • E(r')] V'G0(r, r')

+ [nf x E(r')] x V'G0(r, r ')} <iS' (1.79)

H(r) = (JJ) {jooe [hf x E(r')] G0(r, r') + [hf • H(r')] V7G0(r, r7)

+ [rV x H(r')] x V'G0(r, r ')} dS' (1.80)

1.8 HUYGENS'S PRINCIPLE 15

Exterior medium

Fig. 1.2 Huygens's principle.

where h' denotes the unit vector normal to S at r' and pointing toward the exteriorregion. These results can be derived rigorously using Green's theorems (see Chapters10 and 14).

If we let n' x H(r') = J s(r ' ) , E(r') x n' = M a(r ' ) , v! • cE(r') = pe(r'), andn' • //H(r') = pm(r'), (1.79) and (1.80) can be written as

E(r) = fj> [-jaâ(r')Go(r,r') + %e(r')V'G0(r,r')

-Ms(r')xV'G0(r,rf)\dS' (1.81)

H(r) = jj^ [js(r') x V'Go^r') - jwcM^rOGoCr,!/)

+ ipm(r /)V /G0(r,r /)ldS /. (1.82)

M JA comparison of these two equations with (1.75) and (1.76) indicates that J s and M s

are equivalent to surface electric and magnetic currents, and pe and pm are equivalentto surface electric and magnetic charges. Because of this interpretation, (1.81) and(1.82) are also known as the surface equivalence principle.

Again, all of the results above can be reduced to two dimensions using the relationin (1.77). In particular, for both Ez- and Hz-polarization cases, we have

*<"> = i {^-^tr1 ~ G^p'^\dT' (1-83)where <j> = Ez for £2-polarization, <fi = Hz for Hz -polarization, and F is a contourthat encloses the source of the field as well as all possible objects.Exercise 1.9 Derive (1.83) from (1.79) and (1.80) using (1.77). Rewrite theright-hand side of (1.83) in terms of equivalent surface currents for both Ez- and#2-polarizations.

Objects

Sources

S

h &e

Fig. 1.2 Huygens's principle.

E ( r ) :Js

- jWMJs(r')Go(r,r') + ±Pe(r>)VG0(r,r')

-MB(r')xV'Go(r,r ') dS' (1.81)

(1.82)

JJs IH(r): Js(r') x V'G0(r,r') - j^Ms(r')G0(r,r')

1

M+ -pm(r')VGo(r,r')\dS'.

*)^r-«w^d4>(p')dG0(p,p>)

dn> dri-Go(p,p') dT' (1.83)HP)

Jr I

A comparison of these two equations with (1.75) and (1.76) indicates that J s and M s

indimensions using the relation

where <j> = Ez for £2-polarization, and F is a contour


1.9 RADAR CROSS SECTIONS

One of emphases of this book is the treatment of open-region scattering problems.An important parameter that characterizes scattering by an object is called radarcross section [10]. This parameter is defined for plane-wave incidence.

In the three-dimensional case, the radar cross section is defined by

ff(^,r,r)=iimwJE^4V ' ^ ' ' ^ ' r->oo | E m c ( ( 9 m c , ( p i n c ) | 2

= lim W ^2'6ML (1-84)Hinc(#incînc)|2

where (ESC,HSC) denote the scattered field observed in the direction (#,</?) and(E i nc , H i nc) denote the incident field from the direction (<9inc, ipinc). When the inci-dent and observation directions are the same, a is called the monostatic or backscatterradar cross section; otherwise, it is referred to as the bistatic radar cross section.

The radar cross section is often normalized to either A2 or m2. The unit for a/\2

is dBsw and for <j/m2 is dBsm. To incorporate information about polarization, theradar cross section can also be defined by

|J5^(r,g,y)|2

\Ei™{6iac,<pinc)\1*n(0,V,P",<S°*) = . l i m ^ , ^ : 1 . . ^ d-85)

where p and q represent either 8 or {p.To evaluate the radar cross section, we have to compute the scattered field in the

far zone with r —» oo using expressions such as (1.81) and (1.82). These expressionscan be simplified greatly for r —> oo. For example, under this condition, (1.81) and(1.82) become

E(r) « e-^— H {jiOfif x [f x J s(r ')] + jkr x M8(r')} ejk*'r'dS' (1.86)

H(r) w € ^ r S {juer x [f x Ms(rf))-jkf x J a ( r ; )} e?k*'r'dS'. (1.87)

In the two-dimensional case, the radar cross section is defined by

V ; p—oo P | E l n c ( ( ^ m c ) | 2

= lim 2.pTgq^Jl (1.88)P-+00 P | H m c ( ( ^ m c ) | 2 V }

which is also referred to as echo width. It is often normalized to A with a unit of dBwor m with a unit of dBm. The scattered field in the far zone can be evaluated usingexpressions such as (1.83). Under the condition p —> oo, (1.83) can be simplified to

|E s c(r ,0,p) |2

E inc(0 inc,<p inc)|2lim 4nr

r—>ooCT(6»,v5,6»inc,^nc)

= lim 47rr2

r—>oor—>oo

Wc(r,0,<p)\2

Uinc(0inc)(/?inc)|(1.84)

IE-(p ) (p)|2

|Einc(<înc)|2= lim 2TI

p—>oooW10)

|Hsc(p,^)|2r |H i n c(^ i n c) |2= l i m Z7T/C (1.88)

e-ifcrE(r)»

H(r)»e-jkr

Aivr Js

Us[jufir x [f x Ja(r ')] +JAX x M s ( r ' )} ejkf"r'dSf (1.86)

{jo;ef x [f x Ma(r ')] - jfcr x J a ( r ; )} e?kf'r'dS'. (1.87)

In the two-dimensional case, the radar cross section is defined by

REFERENCES 17

by using the asymptotic expression for the Hankel function.

Exercise 1.10 Derive (1.86) and (1.87) from (1.81) and (1.82) when r -> oo.

Exercise 1.11 Derive (1.89) from (1.83) when p -> oo.

1.10 SUMMARY

In this chapter we first reviewed the basic concepts and integral theorems in vectoranalysis. We then presented Maxwell's equations in integral form as the fundamen-tal equations, from which Maxwell's equations in differential form and boundaryconditions were derived. The electrostatic problem was formulated in terms of ascalar potential, and the magnetostatic problem was formulated in terms of a vectorpotential. However, the time-harmonic problem was formulated directly in terms ofthe electric or magnetic field, although it was also possible to use scalar and vectorpotentials for formulation. The corresponding partial differential equations werederived in the form of the Poisson, Helmholtz, and vector wave equations. Thiswas followed by the presentation of a variety of boundary conditions and radiationconditions. These will all be used in the finite element analysis of electromagneticsproblems.

This chapter also reviewed the integral expressions that relate the field to itssource and Huygens's principle for calculating exterior fields from the field on aclosed surface. These will be used frequently in the chapters to follow. Finally, wepresented the definition of radar cross section and its calculation because it is usedoften to present the results of application of the finite element method to scatteringproblems.

Although this chapter has not reviewed some other important concepts, suchas the duality principle, uniqueness theorem, reciprocity theorem, and equivalenceprinciple, a good comprehension of these concepts can help greatly in computationalelectromagnetics research. The reader is encouraged to consult [2-7] for this purpose.

REFERENCES

1. J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford: OxfordUniversity Press, 1873. Reprinted by Dover Publications, New York, 1954.

2. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941.

«*)'•{£?-"'I8np</>(p)

read

jk an'

i d<t>(p'y\p-n'Hp') eikp-P' dr, (1.89)


3. R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961.

4. J. Van Bladel, Electromagnetic Fields. New York: McGraw-Hill, 1964. Revisedprinting by Hemisphere Publishing, New York, 1985.

5. J. D. Kraus, Electromagnetics (4th edition). New York: McGraw-Hill, 1973.

6. J. D. Jackson, Classical Electrodynamics (2nd edition). New York: Wiley, 1975.

7. C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley,1989.

8. T. B. A. Senior, "Impedance boundary conditions for imperfectly conductingsurface," AppL Sci. Res. 5, vol. 8, pp. 418-436, 1960.

9. T. B. A. Senior, "Combined resistive and conductive sheets," IEEE Trans. An-tennas Propagat., vol. AP-33, pp. 577-579, May 1985.

10. J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Eds., Electromagneticand Acoustic Scattering by Simple Shapes. Amsterdam: North-Holland, 1969.Revised printing by Hemisphere Publishing, New York, 1987.

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Basic Electromagnetic Theory