OVERVIEW OF ELECTROMAGNETIC THEORY

As the examples will be selected from electromagnetics, we will briefly revise the

governing equations, field and source quantities and notations used in

electromagnetic theory.

STATIC FIELDS

The sources of static electric fields are stationary charges and satisfy the following

differential equations:

0

. v

XE

D

E field, is a conservative field. i.e.

0XE

Static electric field intensity is a vector field in the direction where the

electrostatic potential V decreases with the maximum rate.

E V

For a volume charge of density3/v C m

V can be written as:

3

0 '

( ')1'

4 '

v

v

rV dv

r r

Where 'r is the vector from the origin to a point on the source region (charge

region) and r is the vector from the origin to the point of observation.

The sources of static magnetic fields are steady currents and they satisfy:

. 0

XH J

B

Static magnetic flux density is a solenoidal field. i.e.

. 0B

The steady current satisfies:

. 0J

Electrostatic Boundary Value Problems

We will consider electrostatic problems where only electrostatic conditions (charge and potential) at some boundaries are known. For these kinds of problems, it is desired to find the electric field and the electrostatic potential by making use of the boundary conditions. Such problems are usually solved using Laplace’s and Poisson’s equations which are in general referred as Boundary Value Problems.

Laplace’s and Poisson’s Equation’s

Poisson’s and Laplace’s equations are derived from the Gauss’s Law (in a linear medium):

. . vD E and,

Substituting, E V into this equation we get,

2 vV

which is known as Poisson’s equation. In the source free region:

2 0V

This equation is known as the Laplace’s Equation. The Laplace equation in two dimensions is:

2 2

2 20

V V

x y

Time-Varying Fields

Electromagnetic Field and Source Quantities

Time varying electromagnetic fields are related to each other by the following

differential equations:

Curl Equations:

BXE

t

(Faraday’s Induction Law)

DXH J

t

(Ampere’s Circuital Law)

We can see from these curl equations that time-varying electric fields create

magnetic field and vice versa.

Divergence Equations:

. Gauss Law for Electric FieldsD

. 0 Gauss Law for Magnetic FieldsB

Definitions:

Field Terms

E : Electric Field Intensity ( / )V m

D : Displacement Vector 2( / )C m

H : Magnetic Field Intensity ( / )A m

B : Magnetic Flux density 2( , / )T Wb m

Source Terms

v : Electric charge density 3( / )C m

J : Electric Current Density

2( / )A m

The Maxwell’s equations together with the Continuity Equation;

. 0Jt

and the Lorentz Force Equation:

( )F q E u B Newton

form the foundation of the electromagnetic theory.

Integral Forms of the Maxwell’s Equations

Faraday’s Law of Electromagnetic Induction:

ˆ. .emf

C s

Bv E dl nds

t

Ampere’s Law:

ˆ. ( ).C s

DH dl J nds

t

Gauss Law for Electric Fields:

ˆ. v equ

s v

D nds dv Q

Gauss Law for Magnetic Fields:

ˆ. 0s

B nds

Constitutive Relations

In free space, the flux densities and field intensities are related by the constitutive

relations:

0

0

D E

B H

Where 12

0 8.854 10 ( / )X F m is the permittivity of free space, and 7

0 4 10 ( / )X H m is the permeability of free space.

In a material medium,

D E

B H

with

0

0

r

r

Where r and

r are the relative permittivity and permeability of the medium

respectively. In an inhomogeneous medium permittivity and permeability are

functions of position. So,

( , , )

, ,

D x y z E

B x y z H

In a conductive material, the conduction current density is:

J E

where, is the conductivity of the material.

Impressed and Induced Currents

A fixed current that excites the system is referred to as an impressed current

(source current) and the conduction current is the induced current . i.e. indJ E .

The total current is:

ind impJ J J

Coordinate Systems

A coordinate system is required to give explicit numerical values for the sources

and fields which are computed by using the Maxwell’s equations. In the context

of this course the Cartesian coordinate system will be used.

Using the Cartesian coordinate components, in a three dimensional space, a

vector can be represented as:

ˆ ˆ ˆx x y y z zA a A a A a A

The magnitude of the vector A is:

2 2 2

x y zA A A A

Gradient, Divergence, Curl

The source and field quantities in Maxwell’s equations depend on position and

are represented by vector fields.

ˆ ˆ ˆ( , , ) ( , , ) ( , , )x x y y z zA A x y z a A x y z a A x y z a

The spatial derivatives of vector fields appear in the Maxell’s equations such as

gradient, divergence and curl. In the rectangular coordinate system, the del

operator is:

ˆ ˆ ˆx y za a a

x y z

The gradient of a scalar field ( , , )f x y z is the vector field:

ˆ ˆ ˆx y z

f f ff a a a

x y z

The divergence of a vector field is a scalar function:

.yx z

AA AA

x y z

The curl of a vector field is another vector evaluated as:

ˆ ˆ ˆx y z

x y z

a a a

XAx y z

A A A

Laplacian

The Laplacian operator is:

2 2 22

2 2 2x y z

When applied to a vector field A :

2 .A X A A

When applied to a scalar field f :

2 .f f

Wave Propagation

For some of the simulation examples we will consider the wave propagation.

Therefore, we will review the basic wave equations and their solutions.

Maxwell’s Equations are first-order coupled equations. But, they can be written as

single second-order partial differential equations. Consider a linear, isotropic and

homogeneous dielectric (simple medium) medium. The curl of both sides of the

Faraday’s Law equation is:

X XE XHt

Substitute the Ampere’s Law equation in the above equation considering source

free ( 0impJ J ), non-conducting region 0J E :

EXH

t

Results,

2

2

2.

EX XE

t

X XE E E

In the source free region . 0E :

2

2

20

EE

t

In the Cartesian coordinate system:

2 2 2 2

2 2 2 20

E E E E

x y z t

Denote the wave velocity, 1

/u m s

,

2 2 2 2

2 2 2 2 2

10

E E E E

x y z u t

In free space 8

0 0

13 10 /c X m s

2 2 2 2

2 2 2 2 2

10

E E E E

x y z c t

This equation is valid for any scalar component of the electric field intensityxE , yE

and zE . i.e. for the

xE component:

2 2 2

2 2 2 2 2

10x x x xE E E E

x y z c t

For a special case where ˆx xE a E z ,

2 2

2 2 2

10x xE E

z c t

The solution of the above wave equation is:

,E z t E z ct E z ct

Where E represents the wave travelling in the +z direction and E represents the

wave travelling in the –z direction.

Boundary Conditions

In order to solve electromagnetic problems including more than one material with

different constitutive parameters, it is necessary to know the boundary conditions

that the field vectors ( , , ,E D H B ) satisfy at an interface. These conditions are

derived by using the integral forms of the Maxwell’s equations across the

interface of two materials.

Boundary condition for the tangential components of E :

2 1ˆ 0n E E or

2 1t tE E

Boundary condition for the tangential components of H :

2 1ˆ

sn H H J or 2 1t t sH H J

Boundary condition for the normal component of D :

2 1ˆ. sn D D

or

2 1n n sD D

Boundary condition for the normal component of B :

2 1ˆ. 0n B B or

2 1n nB B

The subscripts 1 and 2 are used to represent the fields in material 1 and material

2. n̂ is a unit vector perpendicular to the interface pointing into the second

material. sJ is the electric surface current density in /A m and s is the surface

charge density in 3/C m .

Good conductors can be approximated as perfect electric conductors (PEC). Inside

a PEC, all time-varying fields are zero.

The boundary conditions for the PEC’s are reduced to:

ˆ 0n E or 0tE

ˆsn H J or

t sH J

ˆ. sn D

ˆ. 0n B

Where n̂ is normal to the PEC.

Time and Frequency Domain Representations of Maxwell’s Equations

The computational examples will include both time-domain and frequency-

domain problems. For the frequency-domain problems, sources and fields are

sinusoidal or time- harmonic. Time harmonic quantities are represented by

phasors.

The phasor expressions of the Maxwell’s equations are obtained from the time-

varying equations by replacing:

jt

. 0

. v

E j H

H J j E

B

D

All field and source quantities are complex-valued vectors or scalars.

Plane Waves

Plane waves are the most fundamental solutions to the Maxwell’s equations in

the source free region and are also solutions of the wave equation.

The frequency-domain representation of the Wave equation is:

2 2 0E E

This partial differential equation (PDE) is referred as the Helmholtz equation.

Denote, k , the wavenumber in (rad/m), then 2 2 0E k E

The solution to this PDE can be obtained by using the method of separation of

variables as:

, , x y zjk x jk y jk z

oE x y z E e

The constants ,x yk k and zk determine the direction of the propagation of the wave

and

2 2 2 2

x y zk k k k

related to the dispersion relation:

2 2k

oE , is a constant vector with complex coefficients, determining the polarization of

the wave.

Also note that the wavelength is:

2 u

k f

In free space:

c

f

Example: Assume that, x-polarized uniform plane wave in free space has an

amplitude of 2 / mmV and propagates in the z-direction. Write the expression of

the electric field intensity of this wave at 300MHz.

Solution:

Time-domain:

8

ˆ, 2cos ( / )

ˆ( , ) 2cos 2 3 10 2 ( / )

x

x

E z t a t kz mV m

E z t a X X t z mV m

Frequency Domain (Phasor form):

2ˆ 2 /j z

xE z a e mV m

The magnetic field intensity:

Consider the electric field intensity0

ˆjk z

xE E e a in a simple medium characterized

by 0r , 0r , 0 and has an amplitude 0E . We use the phasor form of the

Faraday’s Induction Law to find the magnetic field intensity as:

X E j H

0

ˆ ˆ ˆ

0 0

0 0

x y z

jk z

a a a

XEz

E e

0ˆ jk z

y

kH a E e

0 0 0

0 0

r r r

r r

k

Denote:

00

0

120 377

the intrinsic impedance of free space. The intrinsic impedance of a lossless

( 0 ) dielectric medium is:

00

0

r r

r r

The real physical fields:

0ˆ cosxE a E t kz

0ˆ cosy

EH a t kz

We can write the following useful results related to the uniform plane waves:

1) E is to H .

2) E

H

3) E and H are in phase.

4) Both E and H are to the direction of the propagation.

Using the following equation the magnetic field intensity can be calculated from

the electric field intensity:

n̂ X EH

In free space,

0

0ˆjk z

xE E e a

0 0 0k

0

0

0

ˆ ˆ jk z

z xa X a E eH

ˆ ( / )120

ojkzoy

EH a e A m

Obtaining Real-Physical Field from the Phasor Field

To obtain the real physical field we multiply the phasor field by j te

and take the

real part. i.e.

( , ) Re[ ( ) ]j tf x t F x e

The time-domain expression of the above magnetic field expression is:

ˆ Re120

ojkzj toy

EH a e e

0ˆ cos

( / )120

y oa E t k zH A m

Conducting Materials

Conducting materials are characterized by permittivity , permeability and

conductivity . These materials are called lossy dielectric materials.

The time-domain expressions of the Maxwell’s equations in a source- free lossy

medium are:

. 0

. 0

HXE E

t

EXH E H

t

The curl equations above will be useful for the simulation of the wave in a lossy

medium.

The equations in the phasor forms are:

. 0

. 0

XE j H E

XH E j E H

We can write the Ampere’s Law equation as:

( ) cXH E j E j E j Ej

Where,

cj

is the complex permittivity.

Loss Tangent

tan c

is called the loss tangent of the medium and it is a measure of the power loss in

the medium. In practice, it is found that the loss tangent increases, for lower end

of the microwave frequencies.

The following approximations are very useful for conducting materials:

The medium is said to be;

1) a good dielectric, if 1

2) a good conductor, if 1

.

Complex Intrinsic Impedance

Since the lossy materials have complex permittivity, the intrinsic impedance is

also complex:

c

c

For good conductors:

12

c j

The x-polarized electric field intensity propagating in the + z direction will

attenuate by a factor of ze and will take the following form:

ˆ cos( )z

x oE a E e t z

Where is the attenuation constant (Nepers/m) and is the phase constant in

(rad/m). (The attenuation of 1 / 8.69 /Np m db m ).

For good conductors:

f

The phase velocity is:

( / )pu m s

The wavelength is: 2

The Skin Depth

Skin depth is another important characteristic of a lossy dielectric. It is defined as the distance measured from the surface of the lossy medium over which the

magnitude of the field is reduced to 1/ e or approximately 37%, of the field at the surface of the medium. For good conductors:

1 2

The corresponding H -field is:

ˆ

c

n X EH

We see that, for the lossy dielectrics E and H fields are out of phase.

Propagating, Standing and Evanescent Waves

There are three basic types of waves: propagating, standing and evanescent. The

plane wave discussed above represents a single wave propagating in one

direction ( z ). In a more general case, there are waves propagating in two

directions. i.e. Plane waves propagating in the positive and negative ( z )

directions. Thus the general solution for the wave equation:

2 2 0x xE z k E z

Is,

0 0

jkz jkz

xE z E e E e

A standing wave is the superposition of two plane waves of equal amplitudes

propagating in the opposite directions. A standing wave is formed, when the

coefficients0E ,

0E and k are real and the wave is reflected from a conducting

plate.

In certain cases, it is possible to have complex wave numbers (i.e k j ). Then

the term jkze becomes

ze . If is positive, the wave groves as z increases. Such

waves are known as evanescent waves. Evanescent waves occur when a plane

wave strikes on a dielectric interface at an angle larger then the incident wave or

when a mode is excited in a waveguide at a frequency that is lower than the

cutoff frequency.

A waveguide is a guiding structure which consists of four conducting walls. So,

waveguide wave modes have the form of a standing wave in the coordinate’s

transverse to the axis of the waveguide.

Energy and Power

We would like to derive equations governing electromagnetic (EM) energy and power. Starting with Maxwell’s equations:

(1)

(2)imp ind

BXE

t

DXH J J

t

Apply .H to the first equation and .E to the second:

.( )

.( )imp ind

BH XE

t

DE XH J J

t

Subtracting:

.( ) .( ) .( ) .( )imp ind

B DH XE E XH H E J J

t t

Since:

.( ) .( ) .( )

.( ) .( ) .( )imp ind

EXH H XE E XH

B DEXH H E J J

t t

Integration over the volume of interest:

.( ) [ .( ) .( ) ]imp ind

v v

B DEXH dv H E J J dv

t t

Applying the divergence theorem we obtain the Poyntings Theorem:

ˆ. . .( )imp ind

s v v

B DEXH nds H dv E J J dv

t t

Explanation of different terms:

Poynting Vector in 2( / )W m :

P EXH

The power flowing out of the surface S in (W ):

0ˆ.

s

P P nds

Dissipated Power:

2

( . ) .d ind

v v v

P E J dv E Edv E dv

Supplied Power (W ):

( . )s impv

P E J dv

Magnetic Power (W ):

2

. .

1

2

m

v v

m

v

B HP H dv H dv

t t

H dv Wt t

mW : Magnetic energy

Electric Power (W ):

2

. .

1

2

e

v v

ev

D EP E E dv

t t

E dv Wt t

eW : Electric energy

Conservation of EM energy:

0 ( )s d e mP P P W Wt

The time-averaged power:

*1Re ( ) ( )

2avP E r X H r