electromagnetics review unit 1: introduction to antennas...

Introduction Maxwell's Eqns & Related Formulae Scalar & Vector Potentials Radiation Mechanism Ackgt

Unit 1: Introduction to Antennas +

Electromagnetics Review

Antenna Theory ENGI 9816

Khalid El-Darymli, Ph.D., EIT

Dept. of Electrical and Computer EngineeringFaculty of Engineering and Applied Science

Memorial University of NewfoundlandSt. John's, Newfoundland, Canada

Winter 2017

K. El-Darymli http://www.engr.mun.ca/~eldarymli/ Winter 2017 |1/71|


Outline

1 Introduction

2 Maxwell's Eqns & Related FormulaeMaxwell's EquationsConstitutive RelationshipsMiscellaneous RelationsSome Propagation ParametersTime-Harmonic FieldsBoundary Conditions

3 Scalar & Vector PotentialsReviewNew Material on Potentials

4 Radiation MechanismAntenna, a Circuit Point of ViewTwo-Wire AntennasHelpful Animation and Applets

5 Ackgt


http://www.engr.mun.ca/~eldarymli/



Antennas act as transducers associated with the region of transitionbetween guided wave structures and free space, or vice versa.The guiding structure could be, for example, a two-wire transmission lineor a waveguide (hollow pipe) leading from a transmitter or receiver tothe antenna itself.Generally, the antennas are made of good-conducting material and aredesigned to have dimensions and shape conducive to radiating orreceiving electromagnetic (e-m) energy in an ecient manner.As we shall see, the antenna structure may take many dierent forms:e.g., wires, horns, slots, microstrips, reectors, and combinations ofthese.While we shall be able to examine many important basic characteristicsusing mathematics appearing earlier in the programme, in most practicalsituations antenna design must be carried out using sophisticatedsoftware i.e., ecient numerical techniques and packages must beemployed.The main purpose of this course is to introduce the basics so that futureexposure to the engineering software will be meaningful.For Antennas simulations in this course, you're encouraged to use theStudent Edition of FEKO,

http://www.altairuniversity.com/feko-student-edition/K. El-Darymli http://www.engr.mun.ca/~eldarymli/ Winter 2017 |3/71|


Timeline

Ref.: W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd edition, 2013, John Wiley & Sons, Inc.

Pre-modern civilization (up to 2 million years ago)

Acoustical communications: DrumsOptical communications: Smoke signals, ags

This 14th century BC image of Akhenaton is the rst known image thatdepicts that light travels in a straight line.[Ref.:http://chemistry.berkeley.edu/ahmed-zewail-nobel-winning-science-envoy-to-middle-east]


http://www.altairuniversity.com/feko-student-edition/


http://chemistry.berkeley.edu/ahmed-zewail-nobel-winning-science-envoy-to-middle-east



Timeline

1844 TelegraphThe beginning of electronic communication

Samuel Morse

1864 Maxwell's equationsPrinciples of radio waves and theelectromagnetic spectrum

James Clerk Maxwell

1866 First lasting transatlantic telegraph cable

1876 TelephoneWireline analog communication over long distance

Alexander Bell

1887 First Antenna

Heinrich Hertz

1897 First practical wireless (radio) systems

Guglielmo Marconi

1901 First transatlantic radio

Guglielmo Marconi



Timeline

1920 First broadcast radio station

World War II Development of radar; horn, reector, and array antennas

1950s Broadcast television in wide use

1960s Satellite communications and ber optics

1980s Wireless reinvented with widespread use of cellular telephones





Single-Element Antennas

Wire antennas (examples)

(a) Monopoleantenna.

(b) Dipoleantenna.

(c) Loopantenna.

(d) Helicalantenna.




Examples of monopole type antennas used in cellular and cordlesstelephones, walkie-talkies, and CB radios (taken from Balanis).

The monopoles used in these units are either stationary or

reactable/telescopic.






Aperture antennas (examples)

Corner reector (example from Balanis)




This 500 meter Aperture Spherical Telescope (FAST) in china is the world's

largest single-aperture telescope (courtesy of: Xinhua).






Lens antennas with index n > 1 (examples from Balanis)

Lens antennas with index n < 1 (examples from Balanis)




An example for a microstrip (or printed) monopole antenna

(courtesy of: Amitec)





Antenna Arrays

Log-periodic array (courtesy of: BAZ Spezialantennen, Germany)



Antenna Arrays

Patch antenna array (courtesy of: Amitec)





Antenna Arrays

Triangular array of dipoles used as a sectoral base-station antenna for mobile

communication (taken from Balanis).



Antenna Arrays

Very Large Array, National Radio Astronomy Observatory, Socorro County, USA

(courtesy of: http://www.vla.nrao.edu/)



http://www.vla.nrao.edu/



The Electromagnetic Spectrum

Taken from: W. L. Stutzman et al., Antenna Theory and Design, 3rd edition,

2013, John Wiley & Sons, Inc.



The Electromagnetic Spectrum

Microwave bands:





Maxwell's Equations

We have previously seen that to properly describe any time-varying

electromagnetic phenomenon, the following may be invoked:

In point form

~∇×~E˜ =−∂~B˜∂ t

(1.1)

~∇×~H˜ =~J˜+∂~D˜∂ t

(1.2)

~∇ ~D˜ = ρ˜v (1.3)

~∇ ~B˜ = 0 (1.4)

where the subscript ∼ has been used to represent any time variation. Equations (1.1) -(1.4) are referred to as Maxwell's equations.

Of course,

~E˜ ≡ E (~r ,t)≡ Electric eld intensity in V/m.~H˜ ≡H (~r ,t)≡ Magnetic eld intensity in A/m.~D˜ ≡D (~r ,t)≡ Electric ux density in C/m2.~B˜ ≡ B (~r ,t)≡Magnetic ux density in Wb/m2 or T (tesla).~r ≡ position vector as measured from some origin to a eld or observation point.t ≡ time.~J˜ ≡ J (~r ,t)≡ current density in A/m2.

ρ˜v ≡ charge density in C/m3.

The last two quantities are source terms or supports for the eld quantities ~E˜ , ~H˜ , ~D˜ , and ~B˜ .



Maxwell's Equations

Equation (1.1) is Faraday's law;

Equation (1.2) is a modication of Ampères law with∂~D˜∂ t being the

so-called displacement current density (in A/m2 of course);

Equation (1.3) is Gauss' law (electric); and

Equation (1.4) is Gauss' law (magnetic) and it precludes the

possibility of magnetic monopoles i.e., the ~B˜ eld lines do notterminate on a magnetic charge.





Maxwell's Equations

Develop Maxwell's Equations in Integral Form

Recall (for a vector eld ~A):

(a) Geometry in (I) (b) Geometry in (II)

Using (I) and (II), equations (1.1) to (1.4) may be written in integralform.



Maxwell's Equations

Equation (1.1):





Maxwell's Equations

Equation (1.2):



Maxwell's Equations

Equation (1.3):

Equation (1.4):





Constitutive Relationships

In addition to Maxwell's equations, we have the following constitutiverelationships:

~D˜ = ε~E˜ (1.5)

~B˜ = µ~H˜ (1.6)

where ε, measured in F/m, is referred to as the permittivity of the mediumin which the eld exists, and µ, in H/m, is the permeability of themedium.

For homogeneous, isotropic media, ε and µ are simply scalars. In ourproblems, this will always be true (at least, it will be considered to betrue).

Whenever free space is being considered, ε and µ take on the specialnotation and values given as

ε0 = 8.854×10−12 ≈ 10−9

36πF/m , and µ0 = 4π×10−7 H/m.

In general, ε=ε0(1+χe) and µ=µ0(1+χm) where χe and χm are the

electric and magnetic susceptibility, respectively. The former wasencountered in Term 5 and the latter in Term 6.



Constitutive Relationships

Using the constitutive relationships and the forms of ε and µ

as given, equations (1.5) and (1.6) become

~D˜ = ε0~E˜+ ε0χe

~E˜ = ε0~E˜+~P˜

and

~B˜=µ0~H˜+µ0χm

~H˜=µ0

(~H˜ + ~M˜

)where ~P˜ is called the polarization (due to bound e-m charges)

and ~M˜ is called the magnetization (due to bound currents).

Recall also, the notion of: (1) relative permittivity:

εR = ε/ε0 and (2) relative permeability: µR = µ/µ0 .





Miscellaneous Relations

Besides equations (1.1) to (1.6), we have the following usefulresults:

Continuity of Current (or Conservation of Charge)

~∇ ~J˜=−∂ρ˜v∂ t

(1.7)

Convection Current Density

~J˜cnv = ρ˜v~v (1.8)

where ~v is the charge velocity.



Miscellaneous Relations

Ohm's Law

~J˜= σ~E˜ (1.9)

where σ is conductivity in mhos per metre (0/m) or siemens per metre(S/m).

Lorentz Force Equation

~F = Q(~E +~v ×~B

)(1.10)

where Q is charge in coulombs and ~F is force in newtons.





Some Propagation Parameters

Phase Velocity, in general,

vp =1√

µε(1.11)

or for free space,

c =1

√µ0ε0

(1.12)

where c=3×108 m/s .

Also,vp = f λ (1.13)

where f is the frequency in hertz (Hz) and λ is wavelength in metres.




Wave Number and Wave VectorWe dene a wave number k for lossless media as

k =2π

λ= ω√

µε (1.14)

where k is in radians per metre and the radian frequency ω = 2πf . In the case of the

lossless medium, k is equivalent to the β of the Term 6 course.

If the medium is lossy, k may be complex and we dene

jk = α + jβ (1.15)

In this case, k is referred to as the complex propagation constant and jk is the sameas γ of the Term 6 course. The quantity α is the attenuation coecient in nepers per

metre (Np/m). In the lossy case, β = 2π

λ.

For plane wave propagation in lossless isotropic media, we may dene a wave vector ,

~k, such that k = |~k|= 2π

λ,

and the direction of ~k is the direction of wave energy ow.






In Term 6 we also arrived at the Helmholtz equation (point form,time-harmonic elds with no source),




Intrinsic Impedance

Finally, we dene the intrinsic impedance, η , (in ohms) of amedium in which a waveeld exists as

η =

√µ

ε(1.16)

which for free space becomes

η0 =

√µ0

ε0(1.17)

It may be noted that η0 ≈ 120π Ω.





Time-Harmonic Fields

We discovered previously that one solution to Maxwell's equations wasindicative of plane waves travelling through a medium whose properties arestipulated by the values of ε, µ, σ , and so on. In arriving at this conclusion, westarted with time-harmonic (sinusoidal) elds.

Recall, for a time harmonic eld, ~A˜, of the form ~A˜ = ~A0 cos(ωt + φ), that

~A˜ (r ,t)≡~A˜ = Re~Ase

jωt

(1.18)

where ~As is the phasor form of the eld.

Also, recall that the time derivative ∂/∂ t transforms to jω in the phasor domain.

From now on, since it is generally to be understood almost everywhere IN THISCOURSE that the elds are time-harmonic, we shall drop the s subscript on thephasor and use simply the form ~A.



Time-Harmonic Fields

Equations (1.1)(1.4) in phasor form become

~∇×~E =−jω~B (1.19)

~∇×~H =~J + jω~D (1.20)

~∇ ~D = ρv (1.21)

~∇ ~B = 0 (1.22)

In these equations, for free space, ~J = 0 and ρv = 0.

ASIDE: Many texts use E , etc. to denote phasors.





Boundary Conditions

There are many instances in which e-m energy impinges a boundary betweentwo electromagnetically distinct media. For example,

In general, for two media as shown, where n is the unit normal to the boundaryor interface, the following important relationships hold:

(1) The tangential ~E -eld ( ~ET ) is continuous across the boundary; i.e.,

n×~E1 = n×~E2 ⇒ ~ET1 = ~ET2.

What does this imply if medium 2 is a perfect conductor?

(2) If no surface current density (~K) exists on the boundary, then the tangential~H-eld ( ~HT ) is continuous across the boundary i.e.,

n×~H1 = n×~H2 ⇒ ~HT1 = ~HT2;

else ~HT is discontinuous by an amount equal to ~K ; i.e.,

n×[~H1−~H2

]= ~K .



Boundary Conditions

(3) The normal component of the ~D-eld is discontinuous by an amount equal tothe surface charge density, ρs , on the boundary; i.e.,

n ·[~D1−~D2

]= ρs .

(What's the implication if medium 2 is a perfect conductor? The answercan also be arrived at by applying Gauss' law.)

(4) The normal component of the ~B-eld is continuous across the boundary; i.e.,

n ·~B1 = n ·~B2 .





Review

In earlier electromagnetics courses, it was observed that implementingconstructs referred to as potentials facilitated the calculation of the ~E - and~B-eld quantities.

In general, it was found that the potential integrals were easier to calculatethan the eld expressions appearing, for example, in Coulomb's law (staticelectric eld) or in the Biot-Savart law (steady magnetic eld).

Furthermore, on determining the potentials, the ~E and ~B elds were readilyfound using derivatives rather than integrals.

Recall the following geometry for the specication of a eld point P(x ,y ,z):

~r ≡ position vector for observation or eld point, P.~r ′ ≡ position vector for points in the source region.~r −~r ′ is as shown.



Review

Electrostatic Field

The source consists of electric charges. It was discovered that if a scalarpotential, say Φ, existed at P, then the electric eld was simply given by

~E (~r) =−~∇Φ(~r) (1.23)

Note that Φ is the V of Term 6 and

Φ(~r) = V (~r) =∫v ′

ρv (~r ′)dv ′

4πε|~r −~r ′|(1.24)

where (in Cartesian coordinates),∫v ′. . .dv ′ ≡

∫z ′

∫y ′

∫x ′. . .dx ′dy ′dz ′ and ρv is the

charge density. Equation (1.24) is the solution to Poisson's equation

~∇2Φ =−ρv

ε(1.25)

Note that equation (1.24) could be also written explicitly as a 3-dimensionalspatial convolution:

Φ(x ,y ,z) =∫z ′

∫y ′

∫x ′

ρv (x ′,y ′,z ′)dx ′dy ′dz ′

4πε√

(x−x ′)2 + (y −y ′)2 + (z−z ′)2=

ρv (x ,y ,z)

ε

∗3d

1

4π|~r |(1.26)

with |~r |=√x2 +y2 + z2 while

∗3d represents a 3-dimensional convolution.

Recall:K. El-Darymli http://www.engr.mun.ca/~eldarymli/ Winter 2017 |45/71|




Review

Magnetostatic Field

The source consists of a steady current. We have seen that steadycurrents (i.e., dc) produce steady magnetic elds. In our analysis weintroduced a vector potential, ~A, dened as

~B = ~∇×~A (1.27)

Making the substitution into Maxwell's equations and invoking theCoulomb gauge (~∇ ·~A=0) which we proved had to be true for steadyelds as a result of there being no (∂/∂ t) terms it was shown that

~∇2~A =−µ0~J (1.28)

On comparing (1.28) with (1.25), while keeping an eye on (1.24) and(1.26), we may immediately write that

~A(~r) =∫v ′

µ0~J(~r ′)dv ′

4π|~r −~r ′|= µ0

~J(x ,y ,z)∗3d

1

4π|~r |(1.29)

Generally, equation (1.27) and (1.29) together give a simpler way ofcalculating the magnetic ux density, ~B, (or, equivalently, the magneticeld intensity, ~H) than is available via the Biot-Savart law which containsa cross product.



New Material on Potentials

So far, we have considered only the non-time-varying eld. However, inantenna theory, the elds are time varying.

In fact, as we shall elaborate, the production of radiated energy requirescharges to be accelerating.

Therefore, the steady eld forms for the vector and scalar potentials mustbe revisited for this new case.

We shall assume time-harmonic sources and elds.

From equation (1.19),

Suppose

Then,

(Since the curl of the gradient is always zero).

However, if ~∇×~E = 0, there cannot be a time-varying ~B-eld (seeequation (1.1) and remember that the curl is a spatial operator).

Since the discussion has now turned to time-varying elds it is clear that

our old (i.e., steady eld) scalar potential cannot be used.






The details are a little more complicated this time, but the starting pointis our observation that a time-varying ~B-eld is still solenoidal ordivergenceless, i.e.,

Therefore, a vector potential, ~A, is still generally dened by (1.27)

~B = ~∇×~A.

However, we cannot invoke the Coulomb gauge (~∇ ·~A = 0) and expect ~Ato be useful in determining time-varying elds.

What to do?!! Stay tuned (if you are still tuned)!

Using equation (1.27) in~∇×~E =−jω~B

we get~∇×~E =

which implies~∇× (1.30)




Since the curl of a gradient is zero, equation (1.30) implies that

(1.31)

where Φ is a scalar potential. (Note that the − allows us to write ~E =−~∇Φ

when ~A is not time-varying: recall jω ↔ ∂

∂ t and∂~A˜∂ t =0 when ~A has no time

dependence.)For time-harmonic elds, equation (1.31) gives

~E =−jω~A−~∇Φ (1.32)

and it is seen that the ~E eld depends on both the scalar and vector potential.Now, (1.27) and (1.32) satisfy

~∇×~E =−jω~B , and

~∇.~B = 0

automatically, since that's how we started.What about the remaining Maxwell equations?

(1.33)

(1.34)

Since (1.33) and (1.34) must also be satised by (1.27) and (1.32), it looks like

picking the proper forms of ~A and Φ could be quite messy!






Substituting from equations (1.27) and (1.32) into (1.33) gives

. (1.35)

Invoking the vector identity ~∇×~∇×~A = ~∇(~∇ ·~A)−~∇2~A, equation (1.35)

becomes ~∇2~A+ω2µε~A=~∇((

~∇.~A)

+ jωµεΦ)-µ~J .

Recalling k = ω√

µε,

~∇2~A+k2~A = ~∇((

~∇.~A)

+ jωµεΦ)−µ~J . (1.36)

We have argued in Term 6 that to completely specify a vector, both the curland divergence are required.

(It may be seen that the curl alone is not enough to uniquely dene ~A

since ~∇× (~A+~∇λ) = ~∇×~A for any scalar function λ).

Of course, the curl is specied here by ~∇×~A = ~B..... but what are we todo about the divergence (~∇ ·~A) in (1.36)?In electro/magnetostatics we invoked the Coulomb gauge, which we havesaid is not a good plan for time-varying elds.

With a view to eliminating the gradient on the R.H.S. of (1.36), we DEFINEAND USE the Lorenz gauge:

~∇.~A =−jωµεΦ (1.37)

This is a good choice as, in retrospect, it is seen to lead to all of the properresults for the elds.




On using the Lorenz gauge, equation (1.36) becomes

~∇2~A+k2~A =−µ~J (1.38)

and from (1.32), (1.34) and (1.37)

~∇2Φ +k2Φ =−ρv

ε(1.39)

Equations (1.38) and (1.39) are the inhomogeneous Helmholtz equationsfor vector and scalar potentials.

We note that for the static case, in which k = 0 since ω = 0, (1.38) and(1.39) reduce to the proper forms given by (1.25) and (1.28).

It is possible to develop solutions to (1.38) and (1.39) by analogy to thestatic case (of course, they may be solved rigorously also).

We will not give a rigorous solution in this course. Rather, let's writedown the answers and observe some of the properties which seem tomake intuitive sense.






The answers are very important because they eventually lead to thetime-varying elds produced by time-harmonic sources. We get

~A(~r) =∫v ′

µ0~J(~r ′)e−jk|~r−~r

′|dv ′

4π|~r −~r ′|(1.40)

and

Φ(~r) =∫v ′

ρv (~r ′)e−jk|~r−~r′|dv ′

4πε|~r −~r ′|(1.41)

Note that

Source to the Observation (eld) point.

Since ~E and ~B can be determined from ~A via equations (1.27), (1.33),and (1.38), let's consider (1.40) in detail:




Observations of Similarities Between (1.29) and (1.40)

1. Mathematically, at points removed from the source (i.e.,~r 6=~r ′), thestatic and time-varying forms dier by the presence of a phase term,e−jk|~r−~r

′| in the latter.

1 (Note that this is similarly the case for the Φ's of equations (1.26)and (1.41)).

2 That is, physically, as compared to the static case, the ~A viewed at

the eld position~r is phase-delayed by an amount determined by

the distance R = |~r −~r ′| as shown:

3 That is, the phase delay is determined by the distance from thesource to the observer.






2. If we take (1.40) to the time domain, we would see that ~A˜ at~r isdetermined by the state of the source at time (R/c) seconds earlier.

Of course, (R/c) is simply the time necessary for a phenomenontravelling at the speed of light to cover the distance R. For thisreason the potentials in (1.40) and (1.41) are referred to asretarded potentials.

3. The amplitude of ~A decreases as 1/R.

We have, then, the following procedure for nding the ~E and ~B elds due

to a source:




It may be noted that ~Jdv ′ could be replaced by ~KdS ′ or Id`′ ˆ′ forsurface and line currents, respectively, and the triple integral in ~Awould accordingly reduce to a double or single integral.

The important point is that ~A is derived from the current source, nomatter what that source might be.

We are nally in a position to begin our discussion of radiatedenergy due to time-varying sources.

In what follows, the sources are antenna currents.





Antenna, a Circuit Point of View

From a circuit point of view, a transmitting antenna behaves like anequivalent impedance that dissipates the power transmitted.

The transmitter is equivalent to a generator.

[Taken from: Ulaby, Fawwaz T., Eric Michielssen, and Umberto Ravaioli. "Fundamentals of

Applied Electromagnetics, 6th edition, 2010, Boston, Massachussetts: Prentice Hall.]



Antenna, a Circuit Point of View

A receiving antenna behaves like a generator with an internalimpedance corresponding to the antenna equivalent impedance.

The receiver represents the load impedance that dissipates the time

average power generated by the receiving antenna (taken from

Ulaby).





Two-Wire Antennas

Antennas are in general reciprocal devices, which can be used bothas transmitting and as receiving elements.

The basic principle of operation of an antenna is easily understood

starting from a two=wire transmission line, terminated by an open

circuit (taken from Ulaby).



Two-Wire Antennas

Imagine to bend the end of the transmission line, forming a dipoleantenna.Because of the change in geometry, there is now an abrupt changein the characteristic impedance at the transition point, where thecurrent is still continuous.

The dipole leaks electromagnetic energy into the surrounding space,

therefore it reects less power than the original open circuit ⇒ the

standing wave pattern on the transmission line is modied (taken

from Ulaby).





Two-Wire Antennas

Current distribution on linear dipoles (taken from Balanis)



Two-Wire Antennas

Current variation as a function of time for λ/2 dipole (taken from

Balanis)





Two-Wire Antennas

Source, transmission lines, antenna and detachment of electric eldlines:

Antenna and electric eld lines (taken from Balanis)



Two-Wire Antennas

Antenna and free space wave (taken from Balanis)





Two-Wire Antennas

In the space surrounding the dipole we have an electric eld. Atzero frequency (dc bias), xed electrostatic eld lines connect themetal elements of the antenna with circular symmetry.

Antenna and electric eld lines (taken from Ulaby).



Two-Wire Antennas

At higher frequency, the current oscillates in the wires and the eldemanating from the dipole changes periodically. The eld linespropagate away from the dipole and form closed loops (taken fromUlaby).





Helpful Animation and Applets

Electric eld lines of a radiating vertical half-wave dipole antenna,

goo.gl/0Af2an

What is the relationship between the dipole length, radiated current,and radiation pattern?

http://www.amanogawa.com/archive/DipoleAnt/DipoleAnt.html

3-D radiation pattern vs. dipole length,

http://demonstrations.wolfram.com/DipoleAntennaRadiationPattern/



These slides were developed (with permission) from the notesof Prof. Dr. Eric W. Gill. Some modications/additions wereincorporated.


http://goo.gl/0Af2an

http://www.amanogawa.com/archive/DipoleAnt/DipoleAnt.html

http://demonstrations.wolfram.com/DipoleAntennaRadiationPattern/



electromagnetics review unit 1: introduction to antennas...

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