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Chap. 1 Fundamental Concepts

Important Laws in Electromagnetics• Coulomb’s Law (1785)• Gauss’s Law (1839)• Ampere’s Law (1827)• Ohm’s Law (1827)• Kirchhoff’s Law (1845)• Biot-Savart Law (1820)• Faradays’ Law (1831)•Maxwell Equations(1873)• The governing equations of macroscopic electromagnetic

phenomena.• Predict the existence of electromagnetic waves.• Predict light to be electromagnetic waves.• Verification of electromagnetic waves: Hertz (1887-1891)• Radio communication: Marconi (1901)

: Electric field intensity (V/m): Electric flux density (A/m): Magnetic field intensity (C/m2): Magnetic flux density (W/m2): Electric current density (A/m2): Volume electric charge density (C/m3)

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Continuity Equation

Conservation of charges

Integral Forms of Maxwell Equations

: total magnetic flux.: total current.: total charge.

Note: differential form only works when derivatives exist.

Constituent relationshipIn general,

In free space,

where

(F/m): Permittivity or capacitivity of

vacuum. (H/m): Permeability or inductivity of vacuum.

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Note:

In simple media,

Not true when or are very large, or the time derivatives of or are large.

In general, for linear material,

Category of material:• conductor: large , . Good or perfect conductor: .• dielectric: small , . Good dielectric or insulator: .• diamagnetic: , (of the order of 0.01 percent).• paramagnetic: , (of the order of 0.01 percent).• ferromagnetic: .

Generalized Current Concept

, where t, c and i are total,

conduction and impressed currents. is magnetic current.

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xa

b

(a,b)y

(r, )

r

Vector Analysis

What is a coordinate system?Examples:Rectangular coordinate systemPolar coordinate system

Suppose a coordinate system described by three ordered variable. When we say a point’s coordinate is , we mean the

point is located at the interception of the three surfaces defined by. When the three surfaces are always orthogonal,

the coordinate system is called orthogonal coordinate systems.

What is a vector?A vector is a quantity defined by a direction and a magnitude.

What is a vector field?A vector field is a distribution of vectors whose directions andmagnitudes are a function of location.

Base vectors

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In order to define the direction everywhere, the surface normals of theconstant surfaces at each point are used as base vectors to definedthe direction of the vector. The directions of the surface normals arechosen to point to the direction at which the coordinate increases. In ageneral right-handed, orthogonal, curvilinear coordinate system, thebase vectors are arranged in such a way that the followingrelations are satisfied:

,

Note that in general the base vectors are functions of coordinate.

Then a vector field can be represented as

! Vector Algebra" Addition and subtraction.

" Dot product:

" Cross product :

" Identities

-

-

! Orthogonal coordinate systems" Cartesian" Cylindrical

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" Spherical! Vector calculus

" Integration" Gradient" Divergence" Curl

Metric coefficientSome coordinate variables may not correspond to the actual length,therefore a conversion factor is needed to convert a differentialchange, say to a change in length by a factor , i.e.,

Line Integral

Surface Integral

Volume Integral

! Cartesian coordinates

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"

"

"

"

"

"

! Cylindrical coordinates

"

"

"

"

"

Coordinate transform

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! Spherical coordinates

"

"

"

"

"

" Coordinate transform

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Vector Calculus

Integration

! Volume integration

! Line integration

" Scalar:

" Vector:

! Surface integration:

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Gradient

!

" Cartesian:

" Cylindrical:

" Spherical:

!

" is normal to the surface " represent both the magnitude

and the direction of the maximumspace rate of increase of a scalarfunction.

" Proof:

From the above, the maximum of occurs when . Let be

a tangential vector on the surface, , therefore.

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Divergence

!

" Cartesian:

" Cylindrical:

" Spherical:

! Definition:

! Divergence Theorem

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Curl

!

" Cartesian:

" Cylindrical:

" Spherical:

! Definition:

! Stoke’s theorem:

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Two Null Identities

Ex:

Ex:

Helmhotz’s TheormLet

In other words, if the curl and divergence of a vectorfield are known, the vector field can be uniquelydetermined within a constant.

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Power Relationship

From vector identity

or for simply medium

or in integral form

That is,

where: supplied energy ( ): flow out energy ( ), Poynting vector.: dissipated energy ( )

: stored electric and magnetic energy

( )

: total stored electric and magnetic energy in aclosed surface .

! Boundary Conditions

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! Time-Harmonic Fields

Time-harmonic:

: a real function in both space and time.: a real function in space.

: a complex function in space. A phaser.

Thus, all derivative of time becomes.

For a partial deferential equation, all derivative of timecan be replace with , and all time dependence of can be removed and becomes a partial deferentialequation of space only.

Representing all field quantities as

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,then the original Maxwell’s equation becomes

! Power Relationship

! Poynting vector:

Complex Constitutive Parameters

Similarly,

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DC approximation

Let,

Note,: loss due to free charge.: loss due to bound charge.

Conductor: Dielectric: Correction to power relationship:

Homogeneous: is independent of position.Isotropic: is independent of direction.Linear: the relationship between and is linear.Simple medium: Homogeneous, isotropic, and linear.

In general,

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Anisotropic:

Nonlinear: Inhomogeneous:

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Chap. 2 Introduction to Waves

The Wave EquationAssume simple media and source free ( ).

Taking curl to the first two equations, we have.

Let (wave number), then

(Complex vector wave equation)

Applying vector identity,

we have,

In Cartesian coordinates, assume only exist,

.

Assume is independent of x and y, then

.The solutions are

( real)

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Consider the minus solution. In time domain,

Constant phase

, intrinsic impedance

In general,

where

The solution is a uniform plane wave.

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On Wave in General

A wave function can be specified in complex domain asbelow:

where: the magnitude, real,: the phase, real.

The corresponding time domain wave function is

Equal phase surface are defined as

Definitions:1. Plane, cylindrical, or spherical waves: equal phase

surfaces are planes, cylinders, or spheres.2. Uniform waves: amplitude is constant over

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the equal phase surface.3. Wave normal: surface normals of the equiphase

surfaces. is the direction and is the curve alongwhich the phase changes most rapidly.

4. Phase constant: the rate at which the phase decreasesin some direction is called the phase constant in thatdirection (note: phase constant is not necessary aconstant). Phase constant can be written in vectorform as . The maximum phase constant istherefore .

5. Phase velocity: the speed the constant phase surfacemoves at in a given direction. The instantaneousequiphase surface of a wave is

For ant increment , the change in is

To keep the phase constant for an incrementalincrease in time, corresponding incremental changein is necessary. We have

.In cartesian coordinates

The phase velocity along a wave normal is

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and is the smallest.

Alternatively, the wave function can be expressed as

where is a complex function. A complexpropagation constant can be defined as

where is the vector phase constant and is thevector attenuation constant.

6. Wave impedance: the ratios of components of to. Follow right-hand cross-product rule of

component rotated into . For example,

, ,

Example:

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Wave in Perfect Dielectrics

Energy relations

Define velocity of propagation of energy as

Then,

In general, or

Standing Waves

Let

and in time domain,,

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Properties:

1. The field oscillates in amplitude in stead of traveling;Hence the name standing wave.

2. E reaches maximum when H reaches minimum. Inother words, E and H are out of phase.

3. Planes of zero E and H are fixed in space. Zeros of Eand H are separated by quarter- wavelength.

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PolarizationIn general,

If and , then

1. Elliptically polarized: in general.2. Linearly polarized: .3. Circularly polarized:

Ex: (RHC)

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No change in energy and power densities with time orspace, steady power flow.

Ex: (circular polarized

standing-wave)

1. and are always parallel to each other.2. and rotate about the axis as time progresses.3. Amplitude is independent of time.4. Energy and power densities are independent of time.

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Oblique incidents

Perpendicular Polarization (TE),

,

,,

,.

The above fields must satisfy Boundary conditions

,

which lead to.

.

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Note that the condition holds.If at (Brewster (polarization) angle), then

Therefore,

. No

solution for .Snell’s Law

Definition of refraction index

Total reflection occurs at critical angle .

When , is real. At angle larger than criticalangle, surface wave exists in dielectric. The wavedecays inside the dielectric. For , the propagationconstant in media 2 becomes,

Note that the minus sign is chosen such that theresulting field quantity in media 2 won’t grow toinfinity. In view of the phase term of the propagation

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wave in media 2,

the wave decay in the media in direction.

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Parallel Polarization (TM)

,

,,

,,

.

.

.

Note that the condition holds.

If at Brewster angle , then .

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Therefore,

.

For nonmagnetic media, ,

,

Applications: Polarization separation, anti-flare glasses

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TEM WavesAssume and dependence of the form .Substitute to Maxwell’s equations, we have

These lead to

1. The propagation constant of any TEM wave is theintrinsic propagation constant of the media.

Also,

2. . The z-directed wave impedance of any TEMwave is the intrinsic wave impedance of the medium.

Let , then from wave equation we have

.

Similarly,

The boundary conditions at perfect conductors are

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Also,

Therefore, there exists unique such that.

3. The boundary-value problem for and is the sameas the 2-dimensional electrostatic and magnetostaticproblem. Thus, static capacitances and inductancescan be used for transmission lines even though thefield is time-harmonic.

4. The conductor must be perfect, otherwise willexist.

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Multiplying both , we have

.

Equating both , we have

Note: in fact, for TEM wave

where

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Radiation

(vector potential)(scala

r potential)

Let , then (Helmholtz equation, or complex wave

equation)Assume the source is an z-directed infinite small currentelement or electric dipole of moment Il located at theorigin in free space. Then, outside the source,

.Due to spherical symmetry, . We have

.

The solutions are . Choose the minus solution due

to out-going wave assumption. Then,. Since

(Spherical wave)

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1. Near field approximation. Quasi-static.( )

No power flow.

2. Far field approximation.( )

TEM wave.

(HW#1)

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Chapter 3 Some Theorems and Concepts

DualityDue to symmetry of Maxwell’s Equations, systematicexchange of variables leads to the same form ofequations. Thus the solutions will be of the same form.

Suppose we have as our source producing and in medium and a source producing and inmedium . If is made to be equal to in terms offunctional form, and , , then

Example:

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Example:The magnetic field of an electric dipole is

The electric field due to a magnetic dipole is

While for a electric current loop ,

.

Therefore, an electric current loop can be consideredequivalent to a magnetic dipole according to thefollowing equation.

(Homework #2)

Uniqueness Theorem

Let be two sets of solutions of the sameexcitation , then

Similar to the way Poynting Theorem is proved,

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The surface integral vanishes if1. , tangential electric field specified, or2. , tangential magnetic field specified.then

.

Assuming that real medium is always lossy, then for thesecond term to be zero, it is necessary that

everywhere in . Lossless case can be consider as thelimit of the lossy case.

Therefore, in a enclosed volume, if the source in thevolume and the tangential fields on the boundary arethe same, the fields are the same everywhere insidethe volume. If the given sources is in an unboundedlossy region, the surface integral term vanish too. Thus,the uniqueness theorem still hold.

Image TheoryAn application of Uniqueness Theorem.

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Equivalence Principle— Another application of Uniqueness Theorem

In a volume, two sets of fields produced by two sets ofsources are equivalent(the same) if the tangential fieldson the boundary of the volume and the sources in thevolume are the same.

Case 1. Zero field replacement

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Case 2. Exchanging two fields

Case 3. Perfect electric or magnetic conductorreplacement

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Field in Half-space

Approximate:

The Induction Theorem

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Define:• Total field : the field generated by the source

with the obstacle present.• Incident field : the field generated by the

sources with the obstacle absent.• Scattered field : the difference of the total

and incident field, that is,

Let exist inside the obstacle and existoutside the obstacle. By equivalence principle, must exist on the surface of the obstacle and satisfy

Usually, are known. However, the fieldgenerated by with the obstacle present is notknown. Approximation is needed.

Example: Scattering by a conducting plane

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Approximate the finite plate to infinite plate, then imagetheory can be applied to get at far field

where A is the area of the plate.Effective area or radar cross section

Reciprocity Theorem

Two sets of solutions with two sets of excitation in thesame space:

satisfying Maxwell’s Equations

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Then by vector identity , we have

Note that for the above equation to be true, it is required

that ,

that is, and .

1. No sources

2. Bound by a perfect conductor

3. Unbounded

Example: proof of circuit reciprocity. For instance, in atwo port network,

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Since.

Therefore,

Example: the receiving pattern and transmission patternof an antenna are the same.

Let be the antenna ininterest. When is excitedby current causingvoltage at antenna , is proportional to the transmission pattern of antenna .When is excited by current causing voltage atantenna , is proportional to the receiving patternof antenna . By reciprocity,

.Thus, the receiving pattern and transmission pattern arethe same.

Example: an electric current impressed along the

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surface of a perfect conductor radiated no field.Proof:

Let be on the conductor and anywhere outsidethe conductor. Then

Green’s Functions

Reciprocity relationships are formulas symmetrical intwo field-source pairs. Mathematical statements ofreciprocity are called Green’s theorems.

From identity

apply divergence theory, we have Green’s first identity.

Interchange and , then subtract each other, we haveGreen’s second identity

Similarly, based on vector identity

we have Green’s first vector identity

and second vector identity

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Ex: Let . If , then is the

vector potential generated by an infinite small current at . Then, any in a source free region

enclosed by can be express as (Use Green’s

second vector identity)

Homework #3: prove Eq. (3-53)

Here the field of a point source is called the Green’sfunction.

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Ex: Suppose in a volume bounded by surface wherethe tangential electric field or magnetic field vanishes,the vector potential and the source satisfy the followingequation

and the required boundary condition.

If is a unit delta source located at pointing in direction, that is

.Let the solution be . Then,

From linear system point of view, for any arbitrary, since

.

That is, is superposition of , thus,the solution is also a superposition of . Therefore, thesolution is

.

Tensor Green’s Functions

In general, caused by point source at can beexpressed in tensor form as

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where is called tensor Green’s functions. Inrectangular coordinate, the equation can be expressed inmatrix form as

or

The electric field generated by arbitrary source can beexpressed as

Ex: derive free space Green’s function from followingequations.

Alternatively, consider be a linear operator.Define inner product . Let and are

the solutions of the wave equations in satisfying therequired boundary conditions. From Green’s identity

.Then,

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The surface integral vanishes due to the boundarycondition since

,

the stronger condition is or . Or, if the volumeis unbounded,

the surface integral will vanish too.

Too sum up, .

That is is a self-adjoint operator.

Then,

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That is,

Therefore, we can write

or in dyadic form.

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Integral Equations

An integral equation is one for which the unknownquantity appears in an integrand.

Ex: the induction theorem of Fig. 3-16.Let be the tenser Green’s function of generated by . Then, the total scattered field for theproblem of is

.

Since and on the conductorsurface, then

for on S.

Ex: inhomogeneous material. Assume , .In the obstacle

Let , then theobstacle can be removed

and the whole space can be treated as free space. Thus,free space Green’s function can be used. Insidethe obstacle, we have

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Solutions in a Homogeneous Source-free Region

Because sourceless and homogeneous, we have

and

Also, and satisfy

How to divide the field between and ?

Let us choose an arbitrary direction, say , andconstruct the vector potential as

.Then, we have

Since , it is called transverse magnetic to z (TM).

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Like wise, if choose , we have the dual case,

Since , it is called transverse electric to z (TE).Note that and satisfy

Now suppose we have a field neither TE nor TM, wecan determine a and according to

and .

Thus, an arbitrary field in a homogeneous source-free region can always be expressed as the sum of aTM field and a TE field.

In general, let be a constant vector, then

If not sourceless in the region, then

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The Radiated Field

Consider far field:

At far field,

HW#3 Prove the above two equations.HW#4 Problem 3-7, Babinet’s principle.