Module 1 - Electromagnetic Wave Propagation - UBC · PDF file1 EECE 483 - Antennas and...

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EECE 483 - Antennas and Propagation

Module 1 - ElectromagneticWave Propagation

Basic antenna concepts; antennas for low, medium and high frequencies; terrestrial and satellite propagation links;environmental effects on electromagnetic radiation. [3-0-0]

EECE 483 - Antennas and Propagation (Fall 2013) Prof. David G. Michelson

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Introduction

• The electromagnetic force is one of four fundamental forces (gravity andthe strong and weak nuclear forces are the other three); only gravity andelectromagnetism manifest themselves at macroscopic scales

• In past courses, you studied:

– static electric and magnetic fields;

– time-varying electromagnetic fields;

– guided waves (transmission lines and waveguides);

– propagating plane waves.

• If one is to excel in EECE 483, one requires an excellent grasp of thefundamentals of electromagnetics, hence this review.


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Assumptions

• In EECE 483, our focus is on the theory and analysis of electromagneticphenomena that vary sinusoidally with time, i.e., time-harmonic fields,especially those with frequencies between 9 kHz and 275 GHz.

• In this course, we will view such electromagnetic phenomenon from themacroscopic viewpoint, i.e., cases in which:

– linear dimensions are large compared to atomic dimensions,

– charge magnitudes are large compared to atomic charges.

• In general, we will not consider mechanical forces associated withelectromagnetic fields.

• We will assume that relative velocities between sources and observers aresufficiently low that relativistic effects can be ignored.


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Objectives

Upon completion of this module, the EECE 483 student will be able to:

• recount key milestones in the history of electromagnetics,

• describe the key electromagnetic quantities and their physical significanceand express Maxwell’s equations in both point and integral form,

• solve problems involving plane waves propagating in free space andvarious material media,

• solve problems involving spherical waves,

• solve problems involving plane waves incident upon material boundariesand,

• solve problems involving diffraction by a knife edge.


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Outline

1. A Very Brief History of Electromagnetics

2. Electromagnetic Quantities

3. Maxwell’s Equations

4. Wave Propagation in Space

5. Boundary Conditions

6. Wave Propagation at Boundaries


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1. A Very Brief History of Electromagnetics

Electromagnetics has a long and rich history:

• Static electric and magnetic fields have been known since antiquity.

• In 1747, the modern concept of electric charge was introduced byBenjamin Franklin.

• In 1764, Joseph Louis Lagrange discovered the divergence theorem inconnection with the study of gravitation. In 1813, it was rediscovered inconnection with electrostatic fields and became known as Gauss’s law.

• In 1820, the relationship between electric current and magnetism wasfirst noted by Oersted. This discovery ushered in a period of intenseactivity in this area.


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• In 1825, Ampere, referred to by some as “the Newton ofelectricity,” published an amazingly complete and modern descriptionof electromagnetism.

• In 1831, the relationship between changing magnetic flux and voltagewas first noted by Michael Faraday.

• In 1837, Faraday introduced the notion of the dielectric constant. In1846, he speculated that light might be transverse electromagnetic waves.

• In 1849, Fizeau repeated Galileo’s famous hilltop experiment with a 9 kmseparation distance and a rapidly rotating toothed wheel. He estimatedthat c = 3.15× 108 m/s.

• In 1850, William Thomson (Lord Kelvin) introduced the notion ofmagnetic permeability and susceptibility, along with the concepts of Band H.


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• In 1864, after a decade of work, James Clerk Maxwell presented acomplete mathematical description of electromagnetism to the RoyalSociety.

• In 1888, Heinrich Hertz became the first person to artificially generateand detect radiofrequency waves.

• By the early 1900’s, wireless technology (or radio) began to achieveconsiderable economic and practical significance, especially for maritimecommunications.

• By the 1920’s, broadcast radio had become a potent consumer force.

• Initially limited to lower frequencies, radio technology quickly movedupward to shortwave, VHF/UHF, and microwave frequencies.

• Today, we couldn’t imagine a world without technology based uponelectromagnetics!


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2. Electromagnetic Quantities

E - electric field strength (V/m)

H - magnetic field strength (A/m)

D - electric flux density (C/m2)

B - magnetic flux density (Wb/m2 = T)

J - electric current density (A/m2)

ρv - electric charge density (C/m3)

• Review questions:

– What is the physical significance of each quantity?

– Which of these quantities can be directly observed or measured?


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Constitutive Relations for Linear Isotropic Media

The constitutive relations give the relationship between field strength andflux density, or between field strength and current density.

D = εrε0E

B = µrµ0H

J = σE

ε0 permittivity of free space (F/m)εr relative permittivity of a material (unitless)µ0 permeability of free space (H/m)µr relative permeability of a material (unitless)σ conductivity (siemens/m)


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Physical Constants and Derived Values

ε0 ≈ 1

36π× 10−9 F/m

µ0 = 4π × 10−7 H/m

c =1√µ0ε0

≈ 3× 108 m/s

η =

√µ0

ε0≈ 377 Ω


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Scalar Electric Potential

• Electric field strength E can be obtained from knowledge of theconfiguration of electric charge.

• In many cases, it is more convenient to employ a scalar electric potentialor voltage defined by

V =

∫

v

ρ dv

4πε0R.

• E is given by the negative of the gradient of V , i.e.,

E = −∇V .

• The scalar electric potential is based upon a zero reference at infinity.


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Vector Magnetic Potential

• In a similar way, magnetic field strength H can be obtained fromknowledge of the configuration of electric curents.

• A vector magnetic potential A can be defined such that

∇×A = B = µH .

• As such, A serves as an intermediate quantity from which B and henceH can be calculated.

• This definition is consistent with the requirement that ∇ ·B = 0.

• Why? Recall that ∇ · (∇×A) = 0.


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• For the case of a current filament, the magnetic vector potential at aparticular point is given by

A =

∮µId`

4πR.

• Here, R is the distance from the current element Id` to the point atwhich the vector magnetic potential is being calculated.

• The vector magnetic potential is also based upon a zero reference atinfinity.


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3. Maxwell’s Equations

• James Clerk Maxwell’s unification of electric and magnetic field theory,first presented to the Royal Society in 1864, is one of the great triumphsof mathematical physics. It allowed him to state that:

“We have strong reason to conclude that light itself - includingradiant heat and other radiation, if any - is an electromagneticdisturbance in the form of waves propagated through the electro-magnetic field according to electro-magnetic laws”

• Maxwell expressed his results using Hamilton’s quaternions, but ourmodern vector notation is due to Josiah Willard Gibbs.

• Review question: What were Maxwell’s main contributions, and whichaspects were anticipated by others?


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Maxwell’s equations - General Set

• If the electromagnetic field is well-behaved (i.e., its derivatives arecontinuous), it is convenient to express Maxwell’s equations in pointform:

∇ ·D = ρ

∇ ·B = 0

∇×E = −∂B/∂t∇×H = Jc + ∂D/∂t

• These can be related to the integral forms of the equations via thedivergence theorem and Stokes’ theorem.


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The Divergence Theorem

• Consider a closed surface S which encloses a volume v.

• The Divergence Theorem states that the integral of the normalcomponent of a vector field F over the closed surface S to a volumeintegral of a scalar field f which is the divergence of F.

• This allows us to turn a volume integral into a surface integral, and viceversa.

∮

S

F · dS =

∫

v

(∇ · F)dv

e.g.,

∮

S

D · dS =

∫

v

(∇ ·D)dv = Qencl


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Stokes’ Theorem

• Consider an open surface S whose boundary is a closed curve C.

• Stoke’s theorem states that the integral of the tangential component of avector field F around C is equal to the integral of the normal componentof ∇× F over S.

∮

C

F · d` =

∫

S

(∇× F) · dS

e.g.,

∮

C

A · d` =

∫

S

B · dS = Φ


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Maxwell’s equations - General Set

• The restriction on continuous derivatives can be relaxed if we use theintegral form of Maxwell’s equations:

∮

S

D · dS =

∫

v

ρ dv

∮

S

B · dS = 0

∮

C

H · d` =

∫

S

(Jc +

∂D

∂t

)· dS

∮

C

E · d` =

∫

S

−∂B∂t· dS


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Maxwell’s Equations - Free Space Set

• In free space, there are no charges and currents. Accordingly, the pointform of Maxwell’s equations reduces to:

∇ ·D = 0

∇ ·B = 0

∇×E = −∂B/∂t∇×H = ∂D/∂t


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Maxwell’s Equations - Free Space Set

• Similarly, in free space, the integral form of Maxwell’s equations (whichsome regard as more physically intuitive) reduce to:

∮

S

D · dS =

∫

v

ρ dv

∮

S

B · dS = 0

∮

C

H · dl =

∫

S

∂D

∂t· dS

∮

C

E · dl =

∫

S

−∂B∂t· dS


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Why Do Maxwell’s Equations Take the Form They Do?

• Even if we knew nothing about electromagnetics, we could still predictthat a complete description of the electromagnetic field is given byequations of the form:

∇ ·D = something

∇ ·B = something

∇×E = something

∇×H = something

• How do we know this?


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• Ans. Helmholtz’s theorem tells us that this set of field equations will becomplete!

• Once the divergence and curl of a vector field are known, that field iscompletely specified.

• The “somethings” on the right hand side are simply experimentalobservations.

• Interestingly, unlike Newton’s description of gravity, Maxwell’s descriptionof electromagnetics is consistent with special relativity.


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Displacement Current

• For static fields,∇×H = Jc .

• For time-varying fields, this would imply that the continuity equationwould be

∇ · Jc = ∇ · (∇×H) = 0

instead of the correct

∇ · Jc = −∂ρ∂t

.

• James Clerk Maxwell postulated that

∇×H = Jc + JD

where JD = ∂D/∂t.


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• With the inclusion of the displacement current density, we can show thatthe continuity equation is satisfied.

• Assuming the time dependence ejωt for E, the total current density isgiven by

J = Jc + JD

= σE +∂

∂t(εE)

= σE + jωεE .

• It can thus be shown thatJcJD

=σ

ωε.


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Interpreting Maxwell’s Equations

• The electric and magnetic divergence equations indicate that free electriccharges exist, but free magnetic charges do not.

• The magnetic curl equation (Ampere’s Law) indicates that a constantcurrent will give rise to a magnetic field that is everywhere perpendicularto it.

• Both the magnetic and electric curl equations indicate that a time-varyingmagnetic field gives rise to a time-varying electric field and vice-versa.

• The curl equations further indicate the electric and magnetic field vectorswill be oriented perpendicular to each other at a given point in space.

• Furthermore, time-varying electric and magnetic fields cannot existindependently of each other, or in isolation.


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4. Wave Propagation in Space

1. Time-Harmonic Electromagnetic Fields

2. The Plane Wave Solution

3. Solutions for Perfect Dielectrics

4. Solutions for Partially Conducting Media

5. Solutions for Good Conductors

6. Spherical Waves


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4.1 Time-Harmonic Electromagnetic Fields

• If one assumes a time-factor ejωt and a source-free region, Maxwell’sequations reduce to:

∇ ·H = 0

∇ ·E = 0

∇×H = Jc + jωD = (σ + jωε)E

∇×E = −jωB = − jωµH

• The coupling between the electric and magnetic fields is obvious!

• Look for a plane wave solution in which the wave propagates in the +zdirection, and for which E and H do not vary in x or y.


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4.2 The Plane Wave Solution

• The Helmholtz equations associated with a plane wave propagating inthe ±z direction:

∂2Ex∂z2

+ k2Ex = 0

∂2Ey∂z2

+ k2Ey = 0

where k = ω√µ0ε0 = ω/c = 2π/λ.

• The two cases refer to horizontally and vertically polarized waves,respectively.

• Use superposition of the results to obtain the general case - an arbitrarilypolarized wave.


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• The solution to the first equation is

Ex = Ae−jkz +B e+jkz V/m

Hy =1

Z0

(Ae−jkz +B e+jkz

)A/m.

• The e−jkz solution is a plane wave travelling in the +z direction.

• The e+jkz solution is a plane wave travelling in the −z direction.

• For each wave separately, |Ex/Hy| = Z0.


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• The power flux density is given by the time-averaged Poynting vector,

S =1

2ReE×H∗ (W/m

2).

• For the e−jkz solution,

S =A2

2Z0z .

• For the e+jkz solution,

S = − B2

2Z0z .


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4.3 Solutions for Perfect Dielectrics

• A material is classified as a perfect dielectric if σ = 0.

• In this case,

α = 0 β = ω√µε η =

√µ

ε.

• Implications:

– waves do not suffer attenuation.

– E and H are in phase.

– v =ω

β=

1√µε

.

– E and H are everywhere perpendicular to each other.


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4.4 Solutions for Partially Conducting Media

• For a region which is slightly conductive, one can assume an electric fieldof the form:

E = E0 e−γz x .

• Because∇×E = −jωµH

it can be shown that

H =

√σ + jωε

jωµE0 e

−γz y .

• Here, the ratio Ex/Hy gives the intrinsic impedance of the medium.


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• In this case, the intrinsic impedance is complex and can be expressed inthe form |η|6 θ.

• We can express the electric and magnetic fields in the form

E(z, t) = E0 e−αzej(ωt−βz)x

H(z, t) =E0

|η| e−αzej(ωt−βz−θ)y

• Exercise: What is the physical significance of the parameters α and θ?

• Exercise: What are the velocity of propagation and wavelength in termsof µ, ω, ε, and σ?


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4.5 Solutions for Good Conductors; Skin Depth

• A material is classified as a good conductor if σ >> ωε.

• Exercise: What is the significance of the ratio σ/ωε?

• In this case,

γ = α+ jβ

α = β =

√ωµσ

2=

√πfµσ

η =

√ωµ

σ6 45


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• We can express the fields in the form

E(z, t) = E0 e−αzej(ωt−βz)x

H(z, t) =E0

|η| e−αzej(ωt−βz−π/4)y

• Exercise: What are the velocity of propagation and wavelength in termsof µ, ω, ε, and σ?

• What is the significance of the skin depth, δ =1√πfµσ

?

• Ans. Skin depth is the the distance over which the fields decay by 1/e.At 5δ, the magnitude is 0.67% of its initial value, or essentially nil.

• Exercise: How else is skin depth significant?


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4.6 Spherical Waves

• Plane waves are idealizations that cannot exist in practice. Why?

• Practical antennas radiate spherical waves.

• In spherical coordinates, the Helmholtz equation is given by:

1

r2∂

∂r

(r2∂ψ

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂ψ

∂θ

)+

1

r2 sin2 θ

∂

∂θ

(sin θ

∂2ψ

∂φ2

)+k2ψ = 0.

• Solve using the method of separation of variables. We begin by assumingthat the solutions takes the form

ψ = R(r)Θ(θ)Φ(φ) .


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• We can show that:

– The R equation gives rise to spherical Bessel functions.

– The Θ equation gives rise to associated Legendre functions.

– The Φ equation is the familiar harmonic equation.

• In the simplest case, we assume spherical symmetry and ignore sphericalharmonics.

• The result is satisfied by solutions of the form

Ψ(r) = Ce−jβr

rand Ψ(r) = D

ejβr

r.

• The first solution corresponds to an outward travelling wave while thesecond corresponds to an inward travelling wave (of no practical interest).


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5.Wave Propagation and Boundary Conditions

• What happens to the field strength and flux density when fields cross theinterface between two dielectrics?

• In a source-free region, we can show that:

– the normal component of flux density is continuous across the interface;

– the tangential component of field strength is continuous across theinterface.

• This can be proven fairly easily by assuming static conditions. Theresults apply equally well to time varying fields.


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5.1 Boundary Conditions for Electric Fields

• In a charge-free region

Dn1 = Dn2 .

• If there is surface charge,

(D1 −D2) · n12 = −ρs .• In all cases,

Et1 = Et2 .

• In a charge-free region,tan θ1tan θ2

=εr2εr1

.

• Can you prove this?


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5.2 Boundary Conditions for Magnetic Fields

• In all cases

Bn1 = Bn2 .

• If the surface is current-free,

Ht1 = Ht2 .

• If there is a current sheet,

(H1 −H2) · n12 = K .

• If there is no current,tan θ1tan θ2

=µr2µr1

.

• Can you prove this?


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5.3 Plane Waves at Boundaries

1. Normal Incidence

2. Oblique Incidence and Snell’s Laws

3. Perpendicular or TE Polarization

4. Parallel or TM Polarization


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5.3.1 Normal Incidence

• When a propagating wave reaches an interface between two differentregions, it is partly reflected and partly transmitted.

• Application of the appropriate boundary conditions allows us to determinethe transmission and reflection coefficients.

• At the boundary,

Ei0 + Er0 = Et0. Hi0 +Hr

0 = Ht0 .

• Furthermore,

Ei0Hi

0

= η1,Er0Hr

0

= −η1.Et0Ht

0

= η2 .


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• These equations can be combined to yield:

Er0Ei0

=η2 − η1η1 + η2

,Hr

0

Hi0

=η1 − η2η1 + η2

,Et0Ei0

=2η2

η1 + η2,

Hr0

Hi0

=2η1

η1 + η2.

• As shown in §4, the intrinsic impedances for various types of media are:

free space η0 =√

µ0ε0

= 120πΩ

perfect dielectric η =√

µε

partially conducting medium η =√

jωµσ+jωε

conducting medium η =√

ωµσ6 45


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5.3.2 Oblique Incidence and Snell’s Laws

• A wave incident upon a plane interface between two media will lead toa reflected wave in the first medium and a transmitted wave in the second.


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• The plane of incidence contains the normal to the interface and thenormal to the directions of incident, reflected, and transmitted wavepropagation.

• Snell’s law of reflection:θi = θr .

• Snell’s law of refraction:

sin θisin θt

=

√µ2ε2µ1ε1

.

• What is the critical angle of incidence observed when a wave propagatesfrom the denser medium into the less dense medium?


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5.3.3 Perpendicular or TE Polarization

• For perpendicular or TE polarization, E is perpendicular to the plane ofincidence.


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• At the interface between two regions,

Er0Ei0

=η2 cos θi − η1 cos θtη2 cos θi + η1 cos θt

.

andEt0Ei0

=2η2 cos θi

η2 cos θi + η1 cos θt. .

• For normal incidence, θi = θt = 0 and the expressions reduce to thoseof the previous section.

• Does the reflected wave ever drop to zero for TE polarization?

• Ans. No. Prove this.


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5.3.4 Parallel or TM Polarization

• For parallel or TM polarization, E is entirely within the plane of incidence.


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• At the interface,Er0Ei0

=η2 cos θt − η1 cos θiη2 cos θt + η1 cos θi

andEt0Ei0

=2η2 cos θi

η2 cos θt + η1 cos θi. .

• If µ1 = µ2, will there be a particular angle of incidence for which thereis no reflected wave?

• Ans. Yes, this angle is given by the Brewster angle,

θB = tan−1√ε2ε2

.


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6. Diffraction

• When a plane wave is partially blocked by an obstruction, geometricoptics predicts that there is an infinitely sharp transition between theshadow region and the illuimated region.

• In practice, some energy diffracts into the shadow region.

• The effect becomes more significant as wavelength increases relative tothe dimensions of the obstruction.

• According to Huygens’ principle, each element of a wavefront may beregarded as the centre of a secondary disturbance that gives rise tospherical wavelets.

• This approach can be used to model diffraction of a plane wave by aperfectly absorbing half screen.


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6.1 Diffraction by a Knife Edge


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• Diffraction by a perfectly absorbing half screen is an important canonicalproblem in wave propagation.

• The contributions from an infinite number of secondary sources above theedge are summed in a manner that accounts for their relative amplitudesand phases.

• We’ll leave the detailed formulation and solution of the integral for alater date.

• The final result can be expressed as excess path loss Lke in dB in termsof a diffraction parameter ν that depends on the problem geometry,

Lke(ν) = −20 log

(∣∣∣∣EdEi

∣∣∣∣)

= −20 log |F (ν)|.


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• F (ν) is given by

F (ν) =1 + j

2

∫ ∞

ν

exp

(−jπt

2

2

)dt

which can be reduced to

|F (ν)| = 1

2

(1

2+ C2(ν)− C(ν) + S2(ν)− S(ν)

)

where C(ν) and S(ν) are the well-known Fresnel sine and cosine integrals.

• For ν > 1, Lke can be well-approximated by

Lke(ν) ≈ −20 log1

πν√

2≈ −20log

0.225

ν.


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Knife Edge Diffraction Attenuation as a Function of ν


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Knife Edge Diffraction Geometry

• The geometrical parameters that define the Fresnel diffraction parameterν are given by:


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• The exact value of ν is given by

ν = h′

√2(d′1 + d′2)λd′1d

′2

= α

√2d′1d

′2

λ(d′1 + d′2).

• For d1, d2 >> h, ν is well approximated by the more convenientexpression

ν ≈ h√

2(d1 + d2λd1d2

= α

√2d1d2

λ(d1 + d2).

• Why is the latter expression more convenient?


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Fresnel Zones

• The nth Fresnel zone is the region inside an ellipsoid defined by thelocus of points where the distance (a+ b) is larger than the direct pathbetween transmitter and receiver (d1 + d2) by n half-wavelengths.


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• The radius of the nth zone rn is given by applying the condition

a+ b = d1 + d2 +nλ

2.

• If we assume rn << d1, d2, then

rn ≈√nλd1d2d1 + d2

.

• The diffraction parameter ν can be determined from the Fresnel zoneclearance h/rn through

ν ≈ h√

2(d1 + d2)

λd1d2=

h

rn

√2n .


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Fresnel Zone Clearance

• If the inner 60% of the first Fresnel zone is penetrated by an obstacle,the link will experience excess attenuation.

• If a half-plane penetrates to the line-of-sight, the signal will be attenuatedby 6 dB.


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Summary

We have reviewed (or, in some cases, introduced):

• The history of electromagnetics, including key milestomes.

• Electromagnetic quantities and their physical significance.

• The scalar electric and vector magnetic potentials.

• Maxwell’s equations in various forms.

• Plane wave propagation in free space, partially conducting media, andgood conductors.

• Spherical wave propagation in free space.

• Boundary conditions at the interface between two different materials.


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• Wave propagation across the interface between two different media.

• Huygens’ principle and diffraction by a half screen.

• Fresnel zones and Fresnel zone clearance.


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References

[1] R.F. Harrington, Introduction to Electromagnetic Engineering.McGraw-Hill, 1958.

[2] R.F. Harrington, Time Harmonic Electromagnetic Fields. McGraw-Hill, 1961.

[3] J.A. Edminister, Electromagnetics, 2nd. ed., Schaum’s Outline series,New York: McGraw-Hill, 1993.

[4] J. Kraus and R. Marhefka, Antennas: For All Applications, 3rd Ed.,McGraw-Hill, 2002.

[5] W. Hayt and J. Buck, Engineering Electromagnetics, McGraw-Hill,2006.


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[6] S. R. Saunders and A. Aragon-Zavala, Antennas and Propagation forWireless Communication Systems, 2nd ed., Wiley, Chap. 3, 2007.


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