basic antenna theory

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Basic Antenna Theory Ronald E Goans Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA (Dated: April 8, 2008) The theory of antennas is as rich a field as it is complex. The theory was first investigated in the late 19th century, and later expanded upon during World War II. In this paper, we will examine a few of the basics of antenna theory. Additionally, we will look at some of the fundamental parameters surrounding one of the most widely used antennas, the half-wavelength dipole antenna. Finally, we will examine the physics behind loop antennas. INTRODUCTION An antenna is defined as an object, often a metallic wire or rod, designed to radiate or receive electromag- netic radiation. Additionally, antennas are required to optimize the radiation in some directions, while suppress- ing it in others. [1] Antennas may take several different forms, depending on the particular need. Examples in- clude wire, loop, and horn antennas. Their utility has spurred decades of research into the theory and applica- bility of antennas. Of particular interest historically is the research and development of antennas during World War II; particularly with respect to radar, and the intro- duction of microwave antennas. Arguably, the most important aspect of antennas is the radiation pattern. The radiation pattern describes the spatial variations in the electric and magnetic fields. The problem of analyzing an antenna is essentially re- duced to analyzing its radiation pattern. Now, radiation is a byproduct of the current being driven by an AC volt- age through input terminals connected to the antenna. These accelerated electrons radiate energy in the form of electromagnetic waves, creating the radiation pattern. This pattern is usually measured (determined) in what is known as the far-field regime. The far-field regime is defined as 2πr λ, where r is the distance from the antenna and λ is the wavelength of the EM wave. Typi- cally, most measurements are made at distances greater than the wavelength, so that we can examine the radi- ation patterns in the far-field regime. In this limit, the fields drop off as r -1 ; this is typical behavior of radiation. [2] As per usual, the instantaneous total power P is de- fined through the Poynting vector S as P = II S · ˆ nda. (1) where da = r 2 sin(θ)dθdφ is the infinitesimal area of the closed surface. Now, the average radiated power can be found from (1) by P rad = 1 2 II S rad · ˆ nda. (2) where S rad is the time-averaged Poynting vector. Hence, the radiation pattern for an antenna depends on its av- erage power density. To analyze the radiation pattern, we need to know the intensity and distribution of the radiation. The intensity, U, of the radiation is simply U = r 2 S ave . Calculating the average energy density is accomplished through the con- venient form of the Poynting vector, S ave = 1 2 Re(E×B * ). The problem of knowing the radiation problem reduces further to simply determining the electric and magnetic fields produced by the antenna of interest. To determine the electric and magnetic fields, we rely on the scalar and vector potentials, φ and A respectively. The magnetic field B can be defined through B=∇×A. The electric field is determined through E=- φ - A ∂t , where A and φ are determined in the usual way (see, for example, Jackson, Classical Electrodynamics). For transmitting low-frequency waves over short dis- tances, i.e. for minimal loss, transmission lines are more suited for the task. However, for higher-frequency EM waves over larger distances in a lossy medium, the use of antennas is a better choice. The reason for this can be illustrated through the concept of radiation resistance. The antenna loses the real part of its power (since it depends on the real part of the Poynting vector) as a con- sequence of radiation resistance. Radiation resistance, R r , is the effective opposition to the current, causing it to attenuate energy. This energy is radiated in the form of electromagnetic waves, and appears to be caused by a resistance - in this case, the resistance is due to the radiation field. Analogous to power dissipated in DC cir- cuits as a result of resistors, the radiated power from an antenna is related to the radiation resistance. This rela- tionship, through the average power P ave and the current through the input terminals I in , can be defined as, R r = P ave I * in I in (3) DIPOLE ANTENNAS The simplest, and most widely used, of the antennas is the dipole antenna. A dipole antenna consists of two

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Page 1: Basic Antenna Theory

Basic Antenna Theory

Ronald E GoansDepartment of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA

(Dated: April 8, 2008)

The theory of antennas is as rich a field as it is complex. The theory was first investigated in thelate 19th century, and later expanded upon during World War II. In this paper, we will examine a fewof the basics of antenna theory. Additionally, we will look at some of the fundamental parameterssurrounding one of the most widely used antennas, the half-wavelength dipole antenna. Finally, wewill examine the physics behind loop antennas.

INTRODUCTION

An antenna is defined as an object, often a metallicwire or rod, designed to radiate or receive electromag-netic radiation. Additionally, antennas are required tooptimize the radiation in some directions, while suppress-ing it in others. [1] Antennas may take several differentforms, depending on the particular need. Examples in-clude wire, loop, and horn antennas. Their utility hasspurred decades of research into the theory and applica-bility of antennas. Of particular interest historically isthe research and development of antennas during WorldWar II; particularly with respect to radar, and the intro-duction of microwave antennas.

Arguably, the most important aspect of antennas isthe radiation pattern. The radiation pattern describesthe spatial variations in the electric and magnetic fields.The problem of analyzing an antenna is essentially re-duced to analyzing its radiation pattern. Now, radiationis a byproduct of the current being driven by an AC volt-age through input terminals connected to the antenna.These accelerated electrons radiate energy in the formof electromagnetic waves, creating the radiation pattern.This pattern is usually measured (determined) in whatis known as the far-field regime. The far-field regime isdefined as 2πr λ, where r is the distance from theantenna and λ is the wavelength of the EM wave. Typi-cally, most measurements are made at distances greaterthan the wavelength, so that we can examine the radi-ation patterns in the far-field regime. In this limit, thefields drop off as r−1; this is typical behavior of radiation.[2]

As per usual, the instantaneous total power P is de-fined through the Poynting vector S as

P =∮ ∮

S · nda. (1)

where da = r2sin(θ)dθdφ is the infinitesimal area of theclosed surface. Now, the average radiated power can befound from (1) by

Prad =12

∮ ∮Srad · nda. (2)

where Srad is the time-averaged Poynting vector. Hence,the radiation pattern for an antenna depends on its av-erage power density.

To analyze the radiation pattern, we need to know theintensity and distribution of the radiation. The intensity,U, of the radiation is simply U = r2Save. Calculating theaverage energy density is accomplished through the con-venient form of the Poynting vector, Save = 1

2Re(E×B∗).The problem of knowing the radiation problem reducesfurther to simply determining the electric and magneticfields produced by the antenna of interest.

To determine the electric and magnetic fields, we relyon the scalar and vector potentials, φ and A respectively.The magnetic field B can be defined through B=∇×A.The electric field is determined through E= - ∇φ - ∂A

∂t ,where A and φ are determined in the usual way (see, forexample, Jackson, Classical Electrodynamics).

For transmitting low-frequency waves over short dis-tances, i.e. for minimal loss, transmission lines are moresuited for the task. However, for higher-frequency EMwaves over larger distances in a lossy medium, the use ofantennas is a better choice. The reason for this can beillustrated through the concept of radiation resistance.

The antenna loses the real part of its power (since itdepends on the real part of the Poynting vector) as a con-sequence of radiation resistance. Radiation resistance,Rr, is the effective opposition to the current, causing itto attenuate energy. This energy is radiated in the formof electromagnetic waves, and appears to be caused bya resistance - in this case, the resistance is due to theradiation field. Analogous to power dissipated in DC cir-cuits as a result of resistors, the radiated power from anantenna is related to the radiation resistance. This rela-tionship, through the average power Pave and the currentthrough the input terminals Iin, can be defined as,

Rr =PaveI∗inIin

(3)

DIPOLE ANTENNAS

The simplest, and most widely used, of the antennasis the dipole antenna. A dipole antenna consists of two

Page 2: Basic Antenna Theory

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conductors separated by a small gap. The terminals ofthe gap are connected to the appropriate electronics - asignal generator for radiation, and to a load for receiv-ing. For the dipole antenna, the radiation pattern canbe maximized by having the length of the dipole d becomparable to the wavelength λ.

The most common of the dipole antennas is the linear,half-wavelength dipole antenna. This antenna is illus-trated in Figure 1. For this antenna, the length of thedipole d is half of a wavelength, d = λ

2 .

FIG. 1: Dipole antenna shown with current distribution.Taken from Jackson [2]

The current through the antenna is taken to be alongthe z-axis, and the distance r is measured from the gap.In the far-field approximation kr 1 - or equivalently,with k = 2π

λ , 2πr λ - the radiation is linearly polar-ized. This can be seen from the fact that, in this ap-proximation, the radial component of the electric fieldEr is much less than Eθ. In other words, the electricfield is always directed along θ. Additionally, E and Bare perpendicular.

Now, we are really interested in the radiation pattern.Looking at equations (1) and (2), the radiated power Pradis found for a half-wave dipole by integrating the averagepower radiated over the volume of a sphere of radius r.Thus, the electric and magnetic field components are,with d = λ

2 , [1]

Eθ ≈ i(√µ0

ε0)I0e

−ikr

2πr(cos (π2 cosθ)

sinθ) (4)

Bφ ≈ iI0e

−ikr

2πr(cos (π2 cosθ)

sinθ) (5)

Now, Save = 12~E × ~B∗ enables us to determine Save as

Save = (õ0

ε0)(

(I0)2

8π2r2)(

cos2 (π2 cosθ)sin2θ

) (6)

According to equation (2), the radiated power can befound by integrating the time-averaged energy densitySave over the infinitesimal area of the enclosed surface.That implies

Prad =õ0

ε0

(I0)2

∫ π

0

(cos2 (π2 cosθ)

sin2θ)dθ (7)

This integral is of the form

2∫ π

0

1− cos(y)y

dy ≈ 2.435 (8)

which evaluates to

Pave ≈√µ0

ε0

(I0)2

8π× 2.435. (9)

Additionally, the radiation intensity U is seen to be

U = r2Save = (õ0

ε0)(

(I0)2

8π2)(

cos2 (π2 cosθ)sin2θ

) (10)

As we would expect, the radiation intensity (10) showsno radial dependence for this antenna. This is seen read-ily by Figure 2, which shows the radiation pattern fora half-wave dipole antenna. However, this is not alwaysthe case. Some antennas, notably horn antennas, havecharacteristic radiation patterns that depend on both rand θ. [5]

We can calculate the radiation resistance for a half-wavelength antenna, using equation (3). Pave was foundin equation (9) to be approximately

√µ0ε0

(I0)2

8π × 2.435.Now,I∗inIin eliminates the complex time dependence ofthe current, leaving only (I0)2. Thus, the radiation re-sistance for a dipole antenna (d = λ

2 ) reduces to

Rr = 80(π)2(dλ

)2 (11)

The radiation resistance for the half-wavelength dipoleantenna is calculated to be about 73 Ω. The resistance ofmany transmission lines is 75Ω. So, the half-wave dipoleantenna radiation resistance is able to match up prettywell with these transmission lines - particularly at reso-nance. If, on the other hand, the resistance of the trans-mission line does not match the radiation resistance ofthe antenna, then there will be additional energy loss.

Page 3: Basic Antenna Theory

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FIG. 2: Elevation plane amplitude patterns illustrating thedipole radiation pattern for dipole antennas of various lengths.Image taken from Balanis [1]

Hence, this is not an ideal situation. Now, antennas arebetter suited for long-range transmissions and receptions.This helps to illustrate the usefulness of the half-wavedipole antenna.

The above derivations, specifically for the radiation re-sistance, were for d = λ

2 . However, these equations canbe recast to account for a dipole of any length. The fol-lowing equations hold for any value of d: [3]

Prad =õ0

ε0(π

3)(I0lλ

)2 (12)

Prad =12

(I0)2Rr (13)

Rr = 80(π)2(dλ

)2 (14)

Using the forms of Prad and Rr given in equations(11) and (12), we can calculate the radiated power andthe radiation resistance for the whole-wavelength dipoleantenna. In this example, l ≈ λ. That implies that theradiation resistance and radiated power are, respectively,80(π)2 and 40(π)2(I0)2.

For further, more in depth discussions on dipole an-tennas, the reader should consult references [1],[2]-[7].

LOOP ANTENNA

Another simple, versatile antenna geometry is the loopantenna. The loop antenna to be examined is a circularloop, with radius a, oriented in the x-y plane (see figure3). Then, the current is given as I0 = Iφ.

FIG. 3: Loop Antenna. Image taken from Balanis [1]

The electric field and magnetic field components can bedetermined, and the only nonzero components are shownto be, [1]

Eφ =√µ0

ε0

(ka)2I0sinθ4r

(1 +1ikr

)e−ikr (15)

Br = ika2I0cosθ

2r2(1 +

1ikr

)e−ikr (16)

Bθ = −√µ0

ε0

(ka)2I0sinθ4r

(1 +1ikr− 1

(kr)2)e−ikr (17)

With the components of the electric and magneticfields known, the radiated power and radiation resistancecan be calculated. Using the same procedure as before,the radiated density Srad is used to calculate the radi-ated power. Upon integrating equation (2), we find theradiated power (see Figure 4) to be

Prad =õ0

ε0(π

12)(ka)4(I0)2 (18)

Additionally, once the radiated power Prad is known,the radiation resistance can be calculated from equation(3). That implies,

Rr =õ0

ε0(π

12)(ka)4 = 20(π)2(

2πaλ

)4 (19)

Whereas for a coil with N turns, we have

Page 4: Basic Antenna Theory

4

FIG. 4: Elevation amplitude patterns showing the radiationpattern for loop antennas of various diameters. Image takenfrom Balanis [1]

Rr = 20(π)2(2πaλ

)4N2 (20)

It should also be noted that in the far-field region2πr λ, the fields reduce to the following, with A asthe geometrical area of the loop:

Br ≈ Bθ = Er = Eθ = 0 (21)

Bθ = −πAI0e−ikr

(λ2)rsinθ (22)

Eφ =√µ0

ε0

πAI0e−ikr

(λ2)rsinθ (23)

However, the previously determined equations for Pradand Rr (17)-(19)are still valid in the far-field limit.

It is important to note that small loop antennas arenot as useful and practical as dipole antennas. They havesmall radiation resistances, usually lower than their lossresistances. Thus, they do not make good radiators. Assuch, they tend to be used only in the receiving mode ofan antenna in which the signal-to-noise ratio is unimpor-tant. The efficiency of the loop antenna can be improvedby increasing the number of loops, or increasing the cir-cumference. [1]-[7]

The previous discussion on the dipole antenna and loopantenna are only two examples of the many antennasnow available. The type of antenna used depends onthe needs of the project. Single antennas do not alwaysprovide the requirements necessary for a project. Ar-rays are often used to reinforce the radiation patterns insome desired directions, and suppress it in other undesir-able directions. This is a particularly useful method forcontemporary wireless communications, radar, satellitebroadcasting, etc. [9],[10] Again, the reader is directedto the references below for further, detailed discussionsof antenna theory and practical applications.

[1] C.A. Balanis, Antenna Theory: Analysis and De-sign. 1st ed. (Harper and Row, New York, 1982).

[2] J.D. Jackson, Classical Electrodynamics. 3rd ed.(Wiley, New York, 1999).

[3] K.F. Lee, Principles of Antenna Theory. (Wiley,New York, 1984).

[4] D. Corson, P. Lorrain Introduction to Electromag-netic Fields and Waves. (Freeman, San Francisco, 1962).

[5] T. Macnamara, Handbook of Antennas for EMC.(Artech, Boston, 1995).

[6] R.E. Collin, F.C. Zucker Antenna Theory.(McGraw-Hill, St. Louis, 1969).

[7] Y.T. Lo, S.W. Lee Antenna Handbook: AntennaFundamentals and Mathematical Techniques. (Van Nos-trand Reinhold, New York, 1993).

[8] W.L. Stutzman, G.A. Thiele Antenna Theory andDesign. (Wiley, New York, 1981).

[9] H.J. Visser, Array and Phased Array Antenna Ba-sics. (Wiley, San Francisco, 2005).

[10] S. Barbarossa, Multiantenna Wireless Communi-cation Systems. (Artech, Boston, 2005).