Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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1.2 Boundary Conditions 21

1.2.2 The Short-Circuit Surface

The free charges inside a conductor or on the surface of the conductor are

mobile in that they move when the slightest electric ﬁeld exerts a force on

them until an electrostatic equilibrium state is reached. In such a state, no

charge remains inside the conductor and all charges reside on its surface.

The surface charges must be distributed so that no electric ﬁeld exists inside

the conductor or tangentially to it’s surface, and the electric ﬁeld outside the

conductor is normal to the conducting surface. The state of electrostatic equi-

librium itself is independent of the conductivity, but according to Maxwell’s

equations, the time required for approximate equilibrium, namely the relax-

ation time, is inversely proportional to the conductivity of the medium, as

can be seen in problem 1.2. This is the basis for distinguishing conductors

from insulators. If the conductivity of the medium is suﬃciently large and

the relaxation time is suﬃciently small that the approximate equilibrium is

achieved in a negligibly small time compared with the period of our experi-

ment, the medium is considered to be a conductor. On the contrary, if the

conductivity is suﬃciently small and the relaxation time is suﬃciently large

that the charges remain approximately motionless within the period of our

experiment, it is considered to be an insulator.

For time-varying ﬁelds, as we will see in Section 2.1.3, there are electric

ﬁelds as well as magnetic ﬁelds in the form of damping waves inside conductive

media. The time-varying electric and magnetic ﬁelds inside the conductor

vanish only when the conductivity of the conductor tends to inﬁnity, σ → ∞,

which means the conductor is considered as a perfect conductor.

Consider a surface with the unit vector n directed outward from a perfect

conductor on one side into a nonconducting medium on the other side of

the boundary, see Fig. 1.4(b). The charges inside the perfect conductor are

assumed to be so mobile or the relaxation time is assumed to be so small that

charges move instantly in response to changes in the ﬁelds, no matter how

rapid. Then just as in the static case, there must be neither a time-varying

electric ﬁeld nor electric charge inside the perfect conductor and the electric

ﬁeld outside the perfect conductor must always be normal to the surface.

All the charges concentrate in a vanishingly thin layer on the surface of the

perfect conductor and produce the surface charge density ρ

. The correct

surface charge density is always produced in order to satisfy the boundary

condition (1.117),

n · D = ρ

and gives zero electric ﬁeld inside the perfect conductor.

According to Maxwell’s equations, in isotropic medium it is not possible

to have a time-varying magnetic ﬁeld alone without an electric ﬁeld. So there

must be neither a time-varying magnetic ﬁeld nor an electric current inside

the perfect conductor and the magnetic ﬁeld outside the conductor must be

tangential to the surface. All the currents ﬂow in a vanishingly thin layer and

become the surface current density J

on the surface of the perfect conductor.

22 1. Basic Electromagnetic Theory

The correct surface current density J

is always produced in order to satisfy

the boundary condition (1.116)

n × H = J

and gives zero magnetic ﬁeld inside the perfect conductor.

Whereas only superconductors have inﬁnite conductivity, it is a very good

approximation in many practical problems to treat good conductors as perfect

conductors in considering the ﬁelds outside the conductor.

In conclusion, the ﬁelds in medium 1, a perfect conductor, and medium

2, an insulator, are given by

= H

= D

= B

= 0,

= E, H

= H, D

= D, B

= B.

The boundary equations (1.115)–(1.118) become

n × E = 0, n ×H = J

, n · D = ρ

, n · B = 0.

This means

= 0, H

6= 0. (1.125)

We see that the tangential component of the electric ﬁeld on the surface

of a perfect conductor is zero, which means that the tangential component

of the electric ﬁeld satisﬁes the Dirichlet homogeneous boundary condition,

and the tangential component of the magnetic ﬁeld is not zero. In analogy

with the short-circuit condition in circuit theory, this kind of surface is called

a short-circuit surface or electric wall. Any surface on which the tangential

component of the electric ﬁeld is zero and the tangential component of the

magnetic ﬁeld is not zero is recognized as an equivalent short-circuit surface.

1.2.3 The Open-Circuit Surface

On the contrary, any surface on which the tangential component of the mag-

netic ﬁeld is zero and the tangential component of the electric ﬁeld is not zero

can be considered as an equivalent open-circuit surface or magnetic wall.

In this case, there are equivalent surface magnetic charge %

and equiva-

lent surface magnetic current J

on the surface. The boundary equations

(1.119)–(1.122) become:

n × E = −J

, n × H = 0, n · D = 0, n · B = ρ

The boundary conditions for the tangential components of the electric and

magnetic ﬁelds on the open-circuit surface are

= 0, E

6= 0. (1.126)

1.2 Boundary Conditions 23

Figure 1.5: Short-circuit surface, open-circuit surface and impedance surface

on the shorted transmission line or waveguide.

The tangential component of the magnetic ﬁeld on the surface of an open-

circuit surface is zero, but the tangential component of the electric ﬁeld is

not zero. The open-circuit surface and the short-circuit surface are dual

boundary conditions.

There is no such material that can form a real short-circuit or open-circuit

surface. The good-conductor surface can be considered as an approximate

short-circuit surface. The boundary between vacuum or air and dielectrics

with suﬃciently high permittivity can be considered as an approximate short-

circuit surface looking from the vacuum or air, and can be considered as an

approximate open-circuit surface looking from the high permittivity dielec-

tric. See section 6.8.

1.2.4 The Impedance Surface

In the general case, there is both a tangential component of the electric ﬁeld

and a tangential component of the magnetic ﬁeld on the surface. The ratio

of these two complex components is deﬁned as the surface impedance Z

and

this kind of surface is called an impedance surface. The reciprocal of Z

the surface admittance Y

, Y

. (1.127)

As an example, we consider a shorted transmission line shown in Fig. 1.5.

There is a real short-circuit surfaces at the shorted end of the line and there

are equivalent short-circuit surfaces at multiples of λ/2 or even multiples of

λ/4 apart from the shorted end, which are electric ﬁeld zeros and magnetic

ﬁeld maximums. The equivalent open-circuit surfaces appear at odd multiples

of λ/4 apart from the shorted end, which are electric ﬁeld maximums and

magnetic ﬁeld zeros. Any cross section in between the short-circuit surface

and the open-circuit surface is an impedance surface.

24 1. Basic Electromagnetic Theory

1.3 Wave Equations

Maxwell’s equations are simultaneous vector diﬀerential equations of ﬁrst

order, which describe the interactions between electric ﬁelds and magnetic

ﬁelds. The wave equations are derived from Maxwell’s equations, which give

the space and time dependence of each ﬁeld vector and explain the wave

nature of the time-varying electromagnetic ﬁelds. The wave equations are

three-dimensional vector partial diﬀerential equations of second order.

1.3.1 Time-Domain Wave Equations

In homogeneous, non-dispersive, isotropic and linear media, i.e., simple me-

dia, ² and µ are constants. Taking the curl of (1.35), and substituting ∇×H

from (1.36), we obtain

∇ × ∇ × E = −µ²

∂

∂t

− µσ

∂E

∂t

− µ

∂J

∂t

The left-hand side may be expanded by using the following vector identity

(B.45)

∇ × ∇ × A = ∇(∇ · A) − ∇

and substituting ∇ · E from (1.37), to give

∇

E − µσ

∂E

∂t

− µ²

∂

∂t

∇% + µ

∂J

∂t

. (1.128)

Similarly, by taking the curl of (1.36), using identity (B.45) and substi-

tuting ∇ × E and ∇ · H from (1.35) and (1.38), respectively, we have

∇

H − µσ

∂H

∂t

− µ²

∂

∂t

= −∇ × J . (1.129)

Equations (1.128) and (1.129) are the inhomogeneous generalized wave

equations in uniform simple medium. They are time-domain equations. On

the right-hand side, % and J are the true charge density and the true current

density, respectively, which are the sources of the ﬁelds. On the left-hand

sides, the second-order time-derivative terms are oscillating terms or wave

terms, and the ﬁrst-order time-derivative terms are damping terms.

In the source-free region, in a medium with low conductivity, and for fast-

time-varying ﬁelds, % = 0, J = 0, and σ ≈ 0, equations (1.128) and (1.129)

become homogeneous wave equations

∇

E − µ²

∂

∂t

= 0, (1.130)

∇

H − µ²

∂

∂t

= 0. (1.131)

1.3 Wave Equations 25

On the contrary, in a medium with large conductivity, and for slow-time-

varying ﬁelds, the second-order derivative terms can be neglected, and (1.128)

and (1.129) become diﬀusion equations or heat transfer equations,

∇

E − µσ

∂E

∂t

= 0, (1.132)

∇

H − µσ

∂H

∂t

= 0. (1.133)

The solutions of diﬀusion equations are damping or decaying ﬁelds rather

than waves. They are known as slow-time-varying ﬁelds or quasi-stationary

ﬁelds. Circuit theory is well developed and applied for a quasi-stationary

state.

We therefore have two independent equations for the ﬁeld E and for the

ﬁeld H; however, E and H are inextricably related through Maxwell’s equa-

tions.

For a stationary state, the ﬁelds are independent of time. The wave

equations become Poisson’s equations, and the electric ﬁeld and the magnetic

ﬁeld are completely independent, without interaction. They can be discussed

separately.

1.3.2 Solution to the Homogeneous Wave Equations

In the following chapters of this book, we are going to explain a great variety

of waves step by step. At the beginning, we deal with the simplest example,

a one-dimensional solution of the time-domain homogeneous wave equation

in a nonconducting simple medium, which reveals the wave nature of the

electromagnetic ﬁelds and shows that, in space, the electromagnetic wave is

a transverse wave propagating with the velocity of light in free space, c, fully

consistent with Maxwell’s prophecy made more then one hundred years ago.

We begin with the homogeneous time-domain wave equations (1.130) and

(1.131) and the corresponding source-free Maxwell equations. Assume that

E and H are functions of z, one of the space coordinates, and of the time t

only, being independent of x and y, or constant on the plane of constant z.

This is the condition for a uniform plane wave. So we have that

∂

∂x

= 0,

∂

∂y

= 0,

∂

∂z

6= 0,

∂

∂t

6= 0.

The Maxwell equations (1.35) and (1.36) with σ = 0, % = 0, and J = 0

under the above condition are

∇ × E = −µ

∂H

∂t

−

∂E

∂z

∂E

∂z

= −µ

∂

∂t

(

∇ × H = ²

∂E

∂t

−

∂H

∂z

∂H

∂z

= ²

∂

∂t

(

26 1. Basic Electromagnetic Theory

These two vector diﬀerential equations may be decomposed into the following

six scalar diﬀerential equations

∂E

∂z

= µ

∂H

∂t

, (1.134)

∂E

∂z

= −µ

∂H

∂t

, (1.135)

0 =

∂H

∂t

, (1.136)

∂H

∂z

= −²

∂E

∂t

, (1.137)

∂H

∂z

= ²

∂E

∂t

, (1.138)

0 =

∂E

∂t

. (1.139)

We see from (1.136) and (1.139) that E

and H

must be zero except possibly

for static parts, which are not of interest to us in the wave solution, so that

= 0, H

= 0. (1.140)

This means that the uniform plane wave must be a transverse wave or so-

called TEM wave, both the electric ﬁeld and the magnetic ﬁeld have only

transverse components perpendicular to the direction of propagation z. Note

that the electromagnetic wave is not necessarily a transverse wave, they may

have longitudinal ﬁeld components in a variety cases.

We see that there are only E

and H

in (1.135) and (1.137), and that there

are only E

and H

in (1.134) and (1.138). They become two independent

sets of equations. The electric ﬁeld and the magnetic ﬁeld are perpendicular

to each other in each set of equations. We can deal with one set, for example,

the equations containing E

and H

, (1.135) and (1.137).

Take the derivative of (1.135) with respect to z and use (1.137) to elimi-

nate H

on the right-hand side and thus obtain

∂

∂z

− µ²

∂

∂t

= 0 or

∂

∂z

= µ²

∂

∂t

. (1.141)

Similarly, we can have the equation for H

∂

∂z

− µ²

∂

∂t

= 0 or

∂

∂z

= µ²

∂

∂t

. (1.142)

These equations are one-dimensional scalar homogeneous wave equations.

Equation (1.141) is a partial diﬀerential equation of second order. A gen-

eral solution to it must contain two independent solutions. A direct treatment

of the equation to yield a general solution is not easy. We try to give the

solution by investigating the special property of the equation. We ﬁnd that

1.3 Wave Equations 27

the second-order derivatives with respect to z of the function satisfying this

equation must equal the second-order derivatives with respect to t of the

function multiplied by the constant µ². Suppose that the function E

with

respect to t is,

(t) = Ef (t). (1.143)

Then, the following function will b e the solution of the wave equation for

electric ﬁeld (1.141):

(z, t) = Ef(t ∓

√

µ²z).

This may be veriﬁed by diﬀerentiating it with respect to z and to t:

∂

∂z

(z, t) = E

∂

∂z

[∓

√

µ²f

(t ∓

√

µ²z)] = µ²Ef

(t ∓

√

µ²z),

∂

∂t

(z, t) = Ef

(t ∓

√

µ²z).

The primes in the formula denote the total derivatives with respect to the

variable t ∓

√

µ²z. It shows that (1.141) is satisﬁed. Rewrite the solutions in

the form of two independent terms, we have

(z, t) = E

f(t −

√

µ²z) + E

−

(t +

√

µ²z). (1.144)

The meaning of the ﬁrst term, E

f(t −

√

µ²z), is that the value of the

function at position z and time t equals the value at position z + ∆z and time

t +

√

µ²∆z. In other words, at the time t, the function has a particular value

at z, and after a time interval ∆t =

√

µ²∆z, the above-mentioned value of the

function appears at z + ∆z. This is just a wave-propagation phenomenon or

so-called traveling wave, i.e., a disturbance propagates along z, see Fig. 1.6.

The direction of propagation is +z and the velocity of propagation is deﬁned

∆z

∆t

∆z

√

µ²∆z

√

µ²

. (1.145)

This velocity is the velocity of propagation of a certain phase of the distur-

bance, and is called the phase velocity. In vacuum,

√

= c. (1.146)

The second term of the solution, E

−

f(t +

√

µ²z) represents a wave prop-

agating along −z with the same phase velocity. Constants E

and E

−

rep-

resent the amplitudes of the two opposite traveling waves.

Substituting the solution of E

, (1.144), into (1.135) we have the solution

of H

−µ

∂H

∂t

∂E

∂z

= −

√

µ²E

(t −

√

µ²z) +

√

µ²E

−

(t +

√

µ²z).

28 1. Basic Electromagnetic Theory

Figure 1.6: Traveling waves propagating along +z and −z.

Integrating this expression with respect to t, gives

(z, t) =

µ/²

f(t −

√

µ²z) − E

−

f(t +

√

µ²z)]. (1.147)

We are not interested in static solutions, so the constant of integration is

ignored.

The ratio of E

to H

η =

= ±

, (1.148)

where η denotes the wave impedance of plane wave, the unit of which is ohm

(Ω). In the right-hand coordinate system, the ratio of E

to H

of a wave in

the +z direction is +η, whereas the ratio of E

to H

of a wave in the −z

direction is −η. It depends upon the relations among the choice of coordinate

system, the positive directions of the ﬁelds, and the propagation.

In conclusion, the ﬁelds E

and H

of a uniform plane wave are

(z, t) = E

t −

+ E

−

t +

, (1.149)

(z, t)=H

t−

−

t−

−

. (1.150)

This is a linear polarized uniform plane wave with the electric ﬁeld in the

x direction, called a x-polarized wave. Dealing with (1.134) and (1.138), we

can have another linear polarized uniform plane wave with E

and H

, called

the y-polarized wave. All of them are TEM waves, with ﬁeld components

perpendicular to each other. The phase velocity of a TEM wave is 1/

√

µ²,

and the wave impedance is

µ/².

In this section, the uniform plane wave of an arbitrary time dependence

is described. The sinusoidal time-dependent plane wave will be introduced

in Chapter 2 by solving frequency-Domain Wave Equations or so called

Helmholtz’s equations given in the next section.

1.3 Wave Equations 29

1.3.3 Frequency-Domain Wave Equations

For steady-state sinusoidal time-dependent ﬁelds, the wave equations in com-

plex form are derived from the complex Maxwell equations (1.66)–(1.69).

They can also be obtained by the following substitutions in (1.128)–(1.133),

∂

∂t

→ j ω,

∂

∂t

→ −ω

Then we obtain complex equations or frequency-domain equations.

The inhomogeneous generalized complex wave equations in conductive

medium are

∇

E − j ωµσE + ω

µ²E =

∇ρ + j ωµJ, (1.151)

∇

H − j ωµσH + ω

µ²H = −∇ × J. (1.152)

Let

= ω

µ² − j ωµσ, (1.153)

then equations (1.151) and (1.152) become

∇

E + k

E =

∇ρ + j ωµJ, (1.154)

∇

H + k

H = −∇ × J. (1.155)

In the source-free region, ρ = 0 and J = 0, they become homogeneous

generalized complex wave equations

∇

E + k

E = 0, (1.156)

∇

H + k

H = 0. (1.157)

In low-conductivity media and in the high-frequency range, σ ¿ ω², we

have

= ω

µ². (1.158)

Equations (1.154)–(1.157) become complex wave equations and equations

(1.156), (1.157) are well known as Helmholtz’s equations.

In high-conductivity media and in the low-frequency range, σ À ω², we

have

= −j ωµσ. (1.159)

Equations (1.154)–(1.157) become complex diﬀusion equations.

It should be noted that the time-domain wave equations are suitable only

for vacuum or non-dispersive media, but the frequency-domain complex wave

equations are suitable for dispersive media, while the p ermittivity and the

permeability are functions of frequency. For the media with polarization or

magnetization loss, the permittivity or the permeability are complex, ˙² or ˙µ,

respectively.

The time-domain wave equations and the frequency-domain wave equa-

tions given in this section are only applicable for homogeneous, linear and

isotropic media. The wave equations for anisotropic media will be discussed

in Chapter 8.

30 1. Basic Electromagnetic Theory

1.4 Poynting’s Theorem

The energy conservation relation for electromagnetic ﬁelds is derived directly

from Maxwell’s equations and is known as Poynting’s theorem.

1.4.1 Time-Domain Poynting Theorem

The vector identity (B.38) shows that

∇ · (E × H) = H · (∇ × E ) − E · (∇ × H). (1.160)

Substituting the two curl equations (1.107) and (1.108) into the right-hand

side of the above identity and using A · A = A

, yields

−∇ · (E × H) = E ·

∂D

∂t

+ H ·

∂B

∂t

+ σE

+ E · J + H · J

. (1.161)

This may be integrated over a volume V bounded by a closed surface S.

Applying the divergence theorem, we get

−

(E × H)· dS =

E ·

∂D

∂t

+H·

∂B

∂t

+ σE

+ E ·J +H·J

dV. (1.162)

Equations (1.161) and (1.162) are the instantaneous or time-domain Poynting

theorem in derivative form and in integral form, respectively.

For non-dispersive isotropic media, D = ²E, B = µH, the permittivity

and the permeability are constant scalars, independent of time t, we have

∂

∂t

E · D

E ·

∂D

∂t

∂E

∂t

·D =

E ·

∂²E

∂t

∂E

∂t

·²E = E ·

∂D

∂t

= D ·

∂E

∂t

∂

∂t

H·B

H·

∂B

∂t

∂H

∂t

·B =

H·

∂µH

∂t

∂H

∂t

·µH = H·

∂B

∂t

= B·

∂H

∂t

For non-dispersive anisotropic media, D = ² · E, B = µ · H, the permit-

tivity and the permeability are constant tensors, also independent of time t,

yields

∂

∂t

E · D

E ·

∂D

∂t

∂E

∂t

· D =

E · ² ·

∂E

∂t

∂E

∂t

· ² · E, (1.163)

∂

∂t

H · B

H ·

∂B

∂t

∂H

∂t

· B =

H · µ ·

∂H

∂t

∂H

∂t

· µ · H. (1.164)

Applying vector and tensor identity A · a · B = B · a

· A (E.44) to

the above formulae, and for reciprocal anisotropic media, The constitutional

tensors are symmetric tensors, a = a

, we have,

∂

∂t

E · D

= E ·

∂D

∂t

= D ·

∂E

∂t

∂

∂t

H · B

= H ·

∂B

∂t

= B ·

∂H

∂t