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110 Antenna Parameters

3.8. Input Impedance

Rf current usually reaches to the antenna through a transmission line. The antenna input

is at the point where the transmission line is connected to the antenna. At this location

the antenna presents a load impedance to the transmission line, for example, ZL of Fig.

22. This impedance is the input impedance of the antenna, and an impedance mismatch

occurs unless the input impedance equals the characteristic impedance of the lineZ0. A

mismatch leads to loss of power transfer and creates a standing wave on the line. Aside

from the loss of power transfer, standing waves are the source of increased voltage along

the transmission line. Consequently, because input impedance of the antenna in relation

toZ0 controls the standing wave ratio, it is a very important parameter. Measurement of

the antenna input impedance is discussed in sec. 9.11.

In addition to its importance in determining whether or not there will be a standing

wave on the transmission line, the input impedance determines how large a voltage mustbe applied at the antenna input terminals to obtain the desired current flow and hence

the desired amount of radiated power. The impedance is in fact equal to the ratio of the

input voltageEi to the input currentIi, by definition:

ZE

Ii

i

i

= (331)

This impedance is in general complex. If the antenna input is at a current maximum,

and if there is no reactive component to the input impedance,Zi will be equal to the sum

of the radiation resistance and the loss resistance; that is:

Z R R Ri i r= = + 0 (332)

The input impedance will be nonreactive for this feed point if the antenna is resonant (as

in the case of a wire approximately an integral number of half wavelengths long, dis-

cussed in secs. 4.24.4) or if the antenna is terminatedby a resistance of proper value

at its far end so that there is no standing wave of current on it (as described in sec. 4.4).

In other cases, however, the input impedance may consist effectively of the radiation and

loss resistances in series with a very high reactance.If this reactance has a large value, the antenna input voltage must be very large to

produce an appreciable input current. If in addition the radiation resistance is very small,

the input current must be very large to produce appreciable radiated power. Obviously

this combination of circumstances, which occurs with the short dipole antennas that

must be used at very low frequencies, results in a very difficult feed impedance-match-

ing problem. Methods of solving it are described in sec. 4.1 and 4.11.

The concepts of radiation resistance and feed point or input impedance have been

discussed here in terms of antennas that have currents flowing in linear conductors as

the basis of the radiation. As will be discussed in secs. 4.7, 4.8, and 4.10, other types of

antennas exist with radiating properties difficult to analyze in these terms. They are, forexample, fed by waveguides rather than by transmission lines. The equivalent of an input

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impedance can be defined at the point of connection of the waveguide to the antenna,

just as waveguides have a characteristic wave impedance analogous to the characteristic

impedance of a transmission line. It is difficult to define a radiation resistance for

such antennas.Even for some types of antennas consisting of current-carrying conductors this is dif-

ficult, and it may even be difficult to define an input impedance. This is true, for example,

of an array of dipoles, when each dipole is fed separately; sometimes each dipole, or

groups of dipoles, will be connected to separate transmitting amplifiers and receiving

amplifiers. The input impedance of each dipole or group may then be defined, but the

concept becomes meaningless for the antenna as a whole, as does also the radiation

resistance. Both these terms have meaning primarily for simple linear-current radiating

elements; but they comprise a very large class of antennas.

3.9. Bandwidth

All antennas are limited in the range of frequency over which they will operate satisfac-

torily. This frequency range, whatever it may be, is called the bandwidth of the antenna.

If an antenna were capable of operating satisfactorily from a minimum frequency of

195 MHz to a maximum frequency of 205 MHz, its bandwidth would be 10 MHz. It

would also be said to have a 5 percent bandwidth (the actual bandwidth divided by the

center frequency of the band, times 100). This would be considered a moderate band-

width. Some antennas are required to operate only at a fixed frequency with a signal that

is narrow in its bandwidth; consequently there is no bandwidth problem in designingsuch an antenna. But in other applications much greater bandwidths may be required;

in such cases special techniques are needed. Such techniques are known and will be

described in secs. 71 and 72. In fact, some recent developments in broad-band anten-

nas permit bandwidths so great that they are described by giving the numerical ratio of

the highest to the lowest operating frequency, rather than as a percentage of the center

frequency. In these terms, bandwidths of 20 to 1 are readily achieved with these antennas,

and ratios as great as 100 to 1 are possible.

The bandwidth of an antenna is not as precise a concept as some of the other parame-

ters that have been described, because many factors are involved in what is meant by

operating satisfactorily. The principal ones are input impedance, radiation efficiency,gain, beamwidth, beam direction, polarization, and sidelobe level.

These may not all be involved in every case, because one or another of them may be

so much more critical than the others, in a particular case, that it alone determines the

bandwidth. The two basic factors involved, between which a distinct separation may be

made, are the antenna pattern and the input impedance. Accordingly, the terms pattern

bandwidth and impedance bandwidth are sometimes used to emphasize this distinction.

The beamwidth, gain, sidelobe level, beam direction, and polarization are parameters

associated with the pattern bandwidth, whereas input impedance, radiation resistance,

and efficiency are associated with the impedance bandwidth.

The definition of the bandwidth of an antenna is less precise than the definitions ofother parameters for still another reason: there is no established criterion of satisfactory

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operation. In some applications, for example, an impedance variation of a factor of 2

over the operating frequency band may be acceptable; in others only a 10 percent varia-

tion may be tolerated. Similar statements apply to variations of beamwidth, gain, and so

forth. When an antenna bandwidth figure is given, therefore, it should always be accom-panied by a statement of satisfactory operation on which it is based, unless these criteria

are fairly well established for the particular application involved.

Two categories of bandwidth requirement exist, and sometimes they allow different

approaches. In some cases (category 1) the antenna is required to handle frequencies over

a wide band simultaneously. This is called instantaneous bandwidth. An example is an

antenna that must function properly when passing short pulses (wide bandwidths). There

are also cases (category 2) in which the range of frequencies is covered over a period of

time, but at any one time the bandwidth requirement is only a fraction of the long-term

requirement. This is sometimes called tunable bandwidth. An example of this is a com-

munication antenna that may operate with its carrier frequency anywhere within a bandof 15 MHz, for instance, in the region of 300 MHz, a bandwidth requirement of 5 percent.

But when it is operating at a particular time, its transmitted signal may cover a bandwidth

of, for example, 50 kHza bandwidth of less than 0.02 percent. It is possible, in such

a situation, for an operator to readjust the antenna impedance, if necessary, when the

frequency of operation is changed.

These examples illustrate fundamental differences between instantaneous and tunable

bandwidths. The first category presents the most difficult problem of course, because the

antenna must meet the requirement without the possibility of any adjustments being made

in going from one frequency to another. However, even when the bandwidth requirementis of the second type, it may be desirable to meet it without requiring adjustments to be

made; the decision is primarily an economic one.

3.10. Polarization

The radiation of an antenna may be linearly, elliptically, or circularly polarized, as defined

in sec. 1.1.4 (also see Appendix B). Polarization in one part of the total pattern may be dif-

ferent from polarization in another. For example, the polarization may be different in the

minor lobes and in the main lobe, or may even vary in different parts of the main lobe.

The simplest antennas radiate (and receive) linearly polarized waves. They are usuallyoriented so that the polarization (direction of the electric vector) is either horizontal or

vertical. Sometimes the choice is dictated by necessity, at other times by preference based

on technical advantages, and sometimes one polarization is as good and as easily achieved

as the other.

For example, at the very low frequencies it is practically impossible to radiate a hori-

zontally polarized wave successfully (because it will be virtually canceled by the radia-

tion from the image of the antenna in the earth); also, vertically polarized waves propagate

much more successfully at these frequencies (e.g., below 1,000 kHz). Therefore vertical

polarization is practically required at these frequencies.

At the frequencies of television broadcasting (between 54 and 698 MHz), horizontalpolarization has been adopted as standard. This choice was made to maximize signal-to-

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noise ratios; it was found that the majority of man-made noise signals are predominantly

vertically polarized. (Interestingly, in Great Britain the opposite choice was made, because

from the pure wave-propagation point of view, vertical polarization provides maximum

signal strength.)At the microwave frequencies (above 1 GHz) there is little basis for a choice of hori-

zontal or vertical polarization, although in specific applications there may be some pos-

sible advantage in one or the other. Of course in communication circuits it is essential

that the transmitting and receiving antennas be polarization matched.

Circular polarization has advantages in some VHF, UHF, and microwave applications.

For example, in transmission of VHF and low-UHF signals through the ionosphere, rota-

tion of the polarization vector occurs, the amount of rotation being generally unpredict-

able. Therefore if a linear polarization is transmitted it is advantageous to have a circularly

polarized receiving antenna (which can receive either polarization), or vice versa.

Maximum power transfer is realized when both antennas are either left- or right-circularlypolarized. (This applies, for example, to transmission and reception between the earth

and a space vehicle.) Circular polarization has also been found to be of advantage in

some microwave radar applications to minimize the clutter echoes received from rain-

drops, in relation to the echoes from larger targets such as aircraft.

It is apparent that the polarization properties of an antenna are an important part of

its technical descriptiona parameter of its performance. Sometimes it may be desirable

to provide a polarization pattern of the antenna, that is, a description of the polarization

radiated as a function of angles within a spherical coordinate system, although such a

complete picture of the polarization is not ordinarily required.

3.11. Interdependencies of Gain, Beamwidths, andAperture Dimensions

Aperture area controls directivity and gain, linear dimensions control beamwidths and

area, and thus directivity and gain depend on beamwidths. These interdependencies of

directivity and gain, beamwidths, linear dimensions, and area on one another are impor-

tant to the antenna design process, and are outlined below.

Equation (325) relates effective receive areaAe to gain G and wavelength l. Because

of reciprocity, it is applicable to both effective transmit and receive areas. The generalrelationship from antenna theory (Silver 1949, sec. 6.4) for directivity D is related to

maximum effective areaAem and wavelength las follows:

D Aem= (4 )/ 2 (333)

Since gain G=kD from (323), we can express G in terms of an effective area Ae and

las

G Ae= (4 )/ 2 (334)

where k=Ae/Aem.

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The definitions ofAem andAe are defined by (333) and (334). Physical aperture Ap

is another frequently used aperture term, but it is less clearly defined. Obviously,Ap is a

measure of the physical size of an antenna and it is an area over which radiation is trans-

mitted or received. Physical area for a horn or reflector antenna can be readily visualized.PresumablyAp for a stub antenna is its cross-sectional area, but what about one attached

to an automobile which affects its radiation pattern? Another definition sometimes used

is aperture efficiency eap, where

ap e pA A= / (335)

Depending on how Ap is defined, eap can exceed unity. For a uniformly continuouslyilluminated aperture, that is, one for which the amplitude and phase are both constant

over the physical area,Aem=Ap. This can be shown to be true from results of integrating

the total radiation pattern for a uniformly illuminated aperture and calculating D from(333), as done by Silver (1949, p. 183). Thus, for a lossless, uniformly continuously-

illuminated aperture eap=Ae/Ap= 1 and

D Ap= (4 )/ (lossless, uniform illumination)2 (336)

The actual gain of an antenna is of course less thanD of (336), because of losses that

may occur from large and extraneous sidelobes, impedance mismatches, and dissipative

(I2R) losses.

Theoretically, the half-power beamwidth qHP for a uniformly illuminated linesource is

HP HPd d= ( )50 8. in degs., dimensionless

or (337)

HP HPd d= ( )0 887. in rads., dimensionless

For the equations of (337), land dare in the same units. The equations of (337) can

be derived from the pattern for a uniformly illuminated aperture, which is discussed insec. 6.10.

Now consider a rectangular aperture. Its gain can be determined with (336) by

expressing physical aperture Ap in terms of principal plane beamwidths, qHP and fHP,using (337). Accordingly, let dq and df denote the linear dimensions that control the

beamwidths in qand fdirections, respectively. Then, theoretically, the gain G for a loss-less, uniformly continuously-illuminated aperture is

G d d HP HP HP= 4 ( )/ = 4 [(50.8 ) /( )]/ = 32,429/(2 2 2 HHP ) (338)

with G dimensionless and qHP and fHP in degrees.

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Stutzman (1998) investigated equations for a number of theoretical patterns, including

those for this case of a rectangular aperture. He notes that, ordinarily, patterns of real

antennas do not have perfectly deep nulls. Also, because of the effects of installation

environments, they do not have sidelobes that decrease uniformly at wide-angles, as dotheoretical patterns. Further, the gain of an antenna is also reduced by lossy materials

(I2R losses), and the radiated power may be reduced by impedance mismatches. Stutzman

suggests, without detailed measurement data being available, that a reasonable approxi-

mation, as a general rule, for an antenna having a beam normal to the aperture, is

G GHP HP HP HP= 26,000/( ) ( dimensionless, and in ddegs.) (339)

The approximate, typical gain of (339) is nearly 1.0 dB less than that of (338), which

is for a lossless, uniformly illuminated rectangular aperture. Furthermore, the use of

(339) is recommended only if the known antenna efficiency (k in G=kD) is nearly

1. The point being that directivity and thus gain are reduced by spurious radiation not

readily detected and thus not ordinarily accounted for when determining directivity.

For example, if only the usual principal-plane measurements are made, there may be

unsuspected out-of-plane spurious radiation.

References

Friis, H. T., A Note on a Simple Transmission Formula, Proceedings of the IRE, vol.34, May, 1946, pp. 25456.

Hollis, J. S., T. J. Lyon, and L. Clayton,Microwave Antenna Measurements,

Scientific-Atlanta, Inc., 1970. This publication may be available through MITechnologies, Inc. (www. mi-technologies.com).

Jurgenson, R., and R. G. Brown, Geometry, Houghton Mifflin Co., 2000.

IEEE Standard 100-1992, The New IEEE Standard Dictionary of Electrical andElectronicTerms, 1993.

Silver, S.,Microwave Antenna Theory and Design, McGraw-Hill, 1949.

Stutzman, W. L., Estimating Directivity and Gain of Antennas, IEEE Antennas andPropagation Magazine, August, 1998, pp. 711.

Terman, F. E.,Radio Engineers Handbook, McGraw-Hill, 1943, p. 785.

Problems and Exercises

1. Indicate, by placing a letter T or F in each of the boxes, whether you think that

the following statements are true (T) or false (F):

(a) The largest antennas are those used to obtain high gain at high

frequencies.

Problems and Exercises 115

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(b) A steel tower can be used to support an antenna, or sometimes it can itself

be the antenna.

(c) The place at which the feed line connects to an antenna is called its input

terminals or input port.

(d) Guy wires of antenna towers should be interrupted by insulators spaced a

half-wavelength apart.

(e) Solid copper is always the preferred form of conductor for an antenna.

(f ) A radome is a dome-shaped radar antenna.

2. A directional antenna has a maximum electrical dimension of 50 meters, and its

operating frequency is 100 MHz (108 Hz). A field strength measurement is madeat a distance of 1 km from this antenna, in the main beam. (a) Is this a near-fieldor a far-field measurement? (b) What is the approximate distance at which the

near field ends and the far field begins?

3. The solid angle subtended by the sun as viewed from the earth is W= 6 105steradian. A microwave antenna, designed to be used for studying the microwaveradiation from the sun, has a very narrow beam whose equivalent solid angle,in

the sense of the denominators of equations (319) and (321), is approximatelyequal to that subtended by the sun. Assume that this antenna has no minor lobes.What is its approximate directivity D? Express this gain as both a power ratio

and a decibel value.

4. An antenna radiates a total power of 100 watts. In the direction of maximum

radiation, the field strength at a distance of 10 km (104

meters) was measuredand found to beE= 12 millivolts (0.012 volt) per meter. (a) What is the directiv-ityD of this antenna, assuming free-space propagation to the measuring point?

(b) If its efficiency factor is kR= 0.92, what is its gain G? (c) If the wavelengthis l= 3 meters, what is the effective areaAein square meters?

5. If the operating wavelength of an antenna, whose gain G= 30,000, is l= 0.1meter, (a) what is its effective area,Ae, and (b) If the actual cross-sectional area

of the antenna is the same as its effective area, and if the outline of this area iscircular, what is the antenna diameter? (Recall from elementary geometry thatthe area of a circle of diameter dis p(d/2)2.)

6. Derive the two forms of the Friis Transmission Equation given in equation(327). In doing so, start with equation (311) and the definition of gain. (These

two forms of the Friis equation are in fact of greater practical value thanequations that include power density.)

7. A dipole slightly shorter than a half wavelength, fed at the center, has an inputimpedance that is purely resistive and of value 75 ohms. The conductor has someohmic resistance, and the resulting ohmic loss of power is equivalent to what

would result if the dipole were a perfect conductor but had a lumped resistanceof valueR0= 8 ohms connected in series with it at the feed point. (a) What is

the radiation resistance, Rr, of the dipole? (b) What is the radiation efficiency

factor, kr? (c) The directivity of a dipole approximately a half-wavelength longisD= 1.64. What is the gain, G, of this particular dipole?

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8. The two VHF television bands are 54 to 88 MHz (low band) and 174 to 216 MHz

(high band). (a) What are the percentage bandwidths of each of these two bands?(b) If a single receiving antenna is designed to perform satisfactorily at all fre-

quencies between the bottom end of the low band and the top end of the highband, what is its bandwidth, expressed in the conventional way?

9. AssumeNisotropes are placed along a line, and are energized equally in ampli-tude and phase by the same power source. By using equations (19) and (311),

what is the theoretical gain perpendicular to the line of isotropes?

10. List three quantities pertaining to the electric field that should be plotted, asfunctions of the direction angles q, f, to present all possible information aboutthe antenna pattern in the far field. (Note: Do not include the magnetic intensity

or the power density, as these are both derivable from a knowledge of the electricfield.)

Problems and Exercises 117

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