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Microwave and Antennas extended reading list . Look here for useful textbooks on antennas and propagation.
Relativity, electromagnetism and Maxwell's equations
(DJJ's pages offsite) a speculative article
Electromagnetics and antennas
Slot, notch, and rectangular loop antennas
Electromagnetic scattering cross section of a matched antenna
The SMITH chart, with antennas examples
Method of moments remarks
Antennas measurements laboratory, MSc.
Measurements of driving point impedance of dipoles and folded dipoles; network analyser plots.
Array antennas and pictures.
Aperture antenna principles.
Antenna height parameter.
Transmission line notes.
Tidbinbilla 70 metre antenna
Problems on Antennas.
Antennas exam questions spring 1998
Antennas exam questions spring 1999
Antennas exam questions spring 2000
Antennas exam questions spring 2001
Antennas exam question spring 2002
Antennas exam paper spring 2003 (pdf file)
Antennas exam paper spring 2004 (pdf file)
Antennas exam paper spring 2005 (pdf file)
Collected problems on antennas
Antenna gain and beamwidth.
The radiation impedance of rod and wire antennas.
The Yagi-Uda antenna.
Intermediate-sized loop antennas.
Antennas above a ground plane.
Mobile phone handset antennas - field strengths.
The Poynting vector, power flow, and the Crossed Field Antenna (CFA)
Some topics in antenna theory.
The Zagi antenna (pdf file).
Zagi element measurements (pdf file).

Resources elsewhere on the web

For antenna modelling see L B Cebik's pages at which contains much introductory and advanced material on using the Numerical Electromagnetic Code NEC software.

This Java applet can be used to visualise wave motion in 2 dimensions close to antennas and other conducting structures. Highly recommended.

There is a complete set of notes by Prof Georgieva of McMaster University in pdf format at which will take time to download, but contain reasonably complete maths and descriptions and novel insights.

For a propagation-of-microwaves tutorial see by Dr Michael Willis, G0MJW (and the Rutherford-Appleton Lab).

Fundamental properties and definitions

The purpose of a transmitting antenna is to radiate electromagnetic waves into "free space" (usually, but not necessarily, air). The power for this is supplied by a "feeder" which is often a length of transmission line or waveguide having a well-defined characteristic impedance. One can regard an antenna as a kind of "transducer" to turn generated electrical energy into radiating energy. The acoustic equivalent of an antenna is a loudspeaker, (or microphone in the case of a receive antenna).

Antennas are also used in "receive mode" to collect radiation from "free space" and deliver the energy contained in the propagating wave to the feeder and receiver. Usually, antennas are reciprocal devices; their essential properties do not depend on whether they are used as transmit or receive devices. So an efficient transmit antenna can also be used as an efficient receive antenna for the same kind of signal. The directional pattern also does not depend on the transmit or receive mode usage. These properties are collected together and called reciprocity .

Isotropic radiator

An "isotropic radiator" has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the antenna. It is a reference radiator with which other antennas are compared. If the power supplied to the isotropic radiator is P watts, the energy density (watts per square metre) at a distance R metres from the centre of the radiator is P/(4 pi R^2). This is because we are spreading the power P uniformly across the area (4 pi R^2) of a sphere of radius R.

Since propagating electromagnetic waves are transverse, the electric and magnetic field vector directions are at right angles to each other and also at right angles to the direction of travel of the wave.

Clearly then, we cannot realise an isotropic radiator in practice since there will be places on the unit sphere where we cannot specify a unique "polarisation direction" for the direction of the electric field. (For example, the lines of longitude on a sphere all meet at the poles, and the directions N and S are not defined at the poles).

This is sometimes called the "hairy ball" problem. Can you comb a hairy ball so that there is no parting or point of baldness anywhere on the ball? No; there must be a discontinuity in hair direction somewhere.

For this reason, it is impossible to construct, or even envisage, a perfect isotropic radiator. (If one allows more than one polarisation mode, one can approximate an isotropic radiator for some practical purposes. However, this can be argued to be a superposition of two antennas, and the polarisation properties are not isotropic even though the power summed over the two modes may be, approximately. To receive isotropic radiation from such an antenna over the sphere, one would have to construct a receive antenna whose polarisation properties depended on where it is located with respect to the source. An example of this dual-polarisation use is provided by the turnstile antenna, and tailored stacked turnstiles may be made a good approximation to an isotropic source in this sense.) It is however possible to have uniform radiation in all azimuth (see below) directions, or in all elevation directions at a particular azimuth plane.

Such an antenna, having uniform radiation azimuthally, is called "omnidirectional". This term is a misnomer, as the antenna is not isotropic and the radiation strength will decrease if we increase the size of the elevation angle. Thus an "omni" antenna does not radiate equally in all directions.

Radiation direction

Imagine you are standing upright on the ground. If you look straight ahead you are looking "along boresight". The boresight direction of an antenna is usually taken to be the direction along which the radiation is most highly concentrated. There can be, therefore, more than one boresight direction (eg, a vertically orientated half-wave dipole has uniform radiation in the azimuth plane, and any direction in this azimuth plane may be defined as boresight.)

If you look around you horizontally, you are looking in various "azimuth" directions. The azimuth angle varies from 0 to 360 degrees, allowing you to look in every horizontal direction.

If you look up or down with respect to your local horizon, the angle of view up or down is termed the "elevation". The elevation angle varies from -90 degrees (straight down) to +90 degrees (overhead).

We shall call the elevation angle "theta" and the azimuth angle "phi". The distance from the antenna is the radius "R". These are commonly termed "spherical polar co-ordinates".

The set of angles phi(0 to 360 degrees) and theta(-90 to 90 degrees) allows us to specify any radiation direction uniquely. In the far field region (see below), the electric and magnetic fields fall off proportional to distance R; that is, they go as 1/R. The power therefore falls off as (1/R)^2 and the total radiated power over the entire sphere surrounding the antenna is independent of distance R.


Given a set of spherical polar co-ordinates (R, theta, phi) we can determine the power density in watts/(square metre) for both the antenna being investigated, and the isotropic reference antenna which is radiating the same total power. The ratio of these power densities gives us the "directivity" of the unknown antenna in the direction (theta, phi) at a distance R from the antenna. If the direction (theta, phi) is not specified, the "directivity" is taken to be the maximum directivity of any of the directions of radiation. The quoted definition is

"The directivity of an antenna is defined as the ratio of the radiation intensity in a given direction from the antenna, to the radiation intensity averaged over all directions. This average radiation intensity is equal to the total power of the antenna divided by (4 pi). If the direction is not specified, the directivity refers to the direction of maximum radiation intensity". We should caveat that this definition is for power radiated in the FAR FIELD region.

Here, the distance R from the antenna is taken out of the considerations by defining the radiation intensity as "the power radiated from an antenna per unit solid angle". For a directional antenna the radiation intensity will depend on direction from the antenna. However, in the far field it will not depend on the distance from the antenna.

In both cases the power at large distances falls off with distance as 1/(R^2). This is called the "inverse square law" and is because the area of a sphere increases with distance as (R^2). The "solid angle" subtended at the origin by an area A on a sphere is defined as A/(R^2). The units of solid angle are called "steradians". There are 4 pi steradians in a complete surrounding surface of the antenna.

The power radiated into any solid angle by an antenna is constant in the "far field region". Thus the directivity in the far field is not a function of the distance from the antenna.


The "gain" of an antenna takes account of the antenna efficiency as well as its directivity. The formal definition of gain in any direction (theta, phi) is "power density radiated in direction (theta, phi) divided by the power density which would have been radiated at (theta, phi) by a lossless (perfect) isotropic radiator having the same total accepted input power." If the direction is not specified, the value for gain is taken to mean the maximum value in any direction for that particular antenna, and the direction along which the gain is maximum is called the "antenna boresight".

Allied to the concept of gain is the concept of efficiency. The efficiency of an antenna is the gain divided by the directivity, in any direction.

The efficiency and gain are limited by resistive losses in the antenna structure, and by resistive loss in objects which may lie inside the "near field" region of the antenna. (see below.)

The IEEE standards specifically exclude reductions in total transmitted signal arising from impedance mismatch (reflection loss) or polarisation mismatch. These reduce the transmitted signal in any particular application by further amounts, and have to be considered in any link budget calculation.

As an example of loss produced by objects close to the antenna radiating structure, there is a very substantial reduction in gain of the 10 element Yagi array antennas on the roof of BB building at about 4pm in the afternoon, when all the birds come and roost on the antennas. Birds have a high dielectric loss tangent.

Both Directivity and Gain are quantities relative to a reference antenna. Most often this is taken to be an isotropic antenna, which radiates equally in all (spherical) directions, and which cannot be made in practice. If the gain is expressed as dB it is usual to suffix the "isotropic reference" with the letter i; thus the gain with respect to an isotropic antenna in dB is called


However, antenna designers often want to know how much better a given antenna (possibly a Yagi-Uda) is than a reference dipole antenna. A dipole antenna has directivity 2.2 dBi. So it is frequently the case that we see the term dBd, where the directivity is referenced (by the suffix letter d) to a perfect dipole. For the reference dipole, the gain and directivity are assumed to be the same, that is, the efficiency is assumed to be 100%. Thus the gain with respect to a perfect half-wave dipole antenna in dB is called


Immediately, we see that there is the additive relationship

Gain(dBd) = Gain(dBi) - 2.2

Now since 10^(2.2/10) = 10^(0.22) = 1.6596 we see that the numerical gain of a reference dipole is about 1.66 over isotropic, and that in numerical terms

Gain(numerical over dipole) x 1.66 = Gain (numerical over isotropic)

These figures are good to a percent, which is adequate for practical antenna design and measurement purposes.


The propagating wave has a transverse direction for the electric field called the "polarisation direction". This normally lies along the direction of electric field in the waveguide feed, or along the conducting driven rod element in a linear antenna.

It is of course possible to radiate from a conductor which is not constructed in a straight line. However, there will still be a preferred polarisation direction.

The polarisation direction is necessarily at right angles to the line of sight joining the observer to the transmitting antenna. It is also at right angles to the magnetic field direction, which is also "transverse".

It is possible for the plane of polarisation, or the polarisation direction, to change with time, and to change with distance away from the source antenna. This is called "rotation of the plane of polarisation".

If the antenna consists of a helix, or a crossed array of dipoles fed in quadrature, then the plane of polarisation can rotate one complete cycle every wavelength. The wave is then said to be "circularly polarised". It is possible to have right hand and left hand circular polarisation.

Clearly we can rotate the plane of polarisation in time and distance by spinning the antenna physically about an axis lying along the boresight direction.

Polar radiation patterns

The general dependence of directivity and gain on the angles (theta, phi) is called the "radiation pattern".

In the case of a linear polarised antenna having fixed direction of polarisation, one can draw polar sectional plots in the "E-plane" and in the "H-plane". The E-plane contains the direction of propagation and the electric field vector. The H-plane contains the direction of propagation and the magnetic field vector. The E-plane is at right angles to the H-plane.

E-plane and H-plane plots are normally regarded as sufficient to characterise an antenna.

The radiated power density may fall into well-defined regions called "lobes", separated by regions of low intensity called "nulls". Strictly speaking the nulls can only be precisely zero intensity for particular directions (points from a continuous set). There is the "main lobe", which is usually where the wanted power from the antenna is directed, and "side lobes" where the antenna sends radiated energy which is regarded as "wasted" or may even interfere with other transmitting systems.

It is possible for there to be more than one main lobe having a given maximum value of gain. For example, a linear array of dipoles can have main lobes spaced 180 degrees apart, and both having the same gain.

Omnidirectional Antennas

Any radiating structure which has rotational invariance around a vertical axis will radiate equally in all directions in the horizontal plane, because there is nothing to define a preferred direction of (horizontal) radiation.

Examples are a vertical whip antenna, or a vertical dipole, or a monopole over a ground plane. These antennas radiate with the electric field vertical, and the magnetic field horizontal.

In the case of a horizontal loop or coil, the radiation is also omnidirectional but the magnetic field is vertical and the electric field is horizontal.

An alternative method of producing horizontally polarised (nearly) omnidirectional radiation is to use crossed horizontal dipoles fed in phase quadrature. Such an arrangement is called a turnstile antenna. Each dipole produces a characteristic figure-of-eight radiation pattern in the horizontal plane; these are superposed in quadrature so the pattern, looked at from above, rotates about the axis once a cycle of radiation. Turnstile antennas also radiate circular polarisation vertically; the radiation may be concentrated in the horizontal plane by stacking turnstile antennas one above the other and feeding them in phase with each other. Turnstile antennas are commonly used as transmitting antennas when horizontal polarisation is required together with omnidirectional radiation.

In the early days of FM band II broadcasting, transmitters were horizontally polarised and the electric field was in the horizontal plane. This made reception on vertical whip antennas on motor vehicles unsatisfactory, as the polarisations were crossed. An ideal receiving antenna for this configuration would have been a horizontal loop above the top of the vehicle. Occasionally one can see such an arrangement on the streets. However, this has overheads of complexity and so the band II transmitters nowadays are slant polarised or elliptically polarised, so that there is a vertical component of electric field.

The horizontal polarisation was adopted because it was found that the reception on antennas spaced around 10 metres from the ground could be maintained over a slightly greater service area than was the case when using vertical polarisation. Also it was found that the interference from unsuppressed car ignition systems was less for the horizontally polarised case, and it was believed that the multi-path reflections from aircraft flying overhead were less troublesome for this configuration. If one measures the signal strength of a band II transmission somewhere near a major airport, it is constantly fluctuating due to these multi-path effects from the moving reflecting surfaces of the aircraft.

Power density, field strength, and impedance.

The power density in watts per square metre is numerically equal to the rms E field in the wave times the rms H field in the wave. We remember the rms values are 0.707 times the peak values, or 1/sqrt(2) times the peak values.

We recall the SI unit of the electric field E is volts/metre, and the SI unit of the magnetic field H is amps/metre. Thus the product is (volts amps)/metre^2 or watts per square metre, as expected.

The characteristic impedance of free space (that is, vacuum or air) is 377 ohms or 120 pi ohms. This is the ratio of E field to H field. It is called Zo. Thus the power density in watts per square metre is (H^2)*Zo or (E^2)/Zo. The field strengths are therefore proportional to the square root of the power density, and they therefore fall off as 1/R, or linearly with distance.

The field strengths therefore fall by a factor of 2 every time the distance from the antenna is doubled; the radiated power density falls by a factor of 4 (or 6dB) every time the distance from the antenna is doubled.

Exercise for the reader. Calculate the electric field strength on boresight, 100 kilometres away from a transmit antenna which has boresight gain 20dB and accepted input power 10 kilowatts. Compare the received voltage from a 1 metre length of wire antenna (assumed short compared to a wavelength) with the thermal noise voltage produced by a resistance of 75 ohms across a bandwidth of 10 kHz at a temperature of 300K. Hint. Boltzmann's constant k = 1.38 * 10^(-23) watts per degree K. Express the signal to noise ratio in dB.

Radiation mechanism

Apart from quantum electronic devices such as lasers and masers, radiation is always the result of accelerated charge. Since ions are so much heavier than electrons, in practice this means that radiation is almost entirely due to accelerated electrons. These do not have to be "free electrons" or even "conduction electrons"; it is possible for radiation to occur from the acceleration of electrons bound to positive ionic charge in a dielectric. We note especially that radiation does not occur from the displacement current term in Maxwell's equations if this term does not include dielectric effects. That is, normally the relative dielectric constant of a physical medium is greater than unity because of polarisation effects (displacement of physical charge). There can be a contribution to the radiated fields from this bound charge which is represented by the displacement current contribution (epsilon-relative - 1)(epsilon0)dE/dt. However, in a vacuum, the displacement current (epsilon0)dE/dt does not contribute to the radiated far fields. For a full proof of this, the reader is directed to the book "Classical electromagnetism via relativity" by W V G Rosser, Butterworths, 1968.

The contribution to the E field at large distances is proportional to the [amount of charge being accelerated] times [its acceleration]. This has dimensions Coulombs-metres/sec/sec, which is the same dimensions as the rate of change of the quantity (IL) for a current I in a little length L of conductor. Thus we see, in order to increase the radiation from a short length of antenna, we either

Thus we see that antennas become (on the whole) physically smaller as the frequency increases (wavelength decreases), and it becomes progressively harder to get them to radiate large total amounts of power.

Effective area

The effective area multiplied by the wave incident power density in watts per square metre gives the total power delivered to the antenna's feeder. This is for a receive antenna.

The effective area A of an antenna is related to the boresight gain G and the free space wavelength lambda of the radiation by the formula G = (4 pi A)/(lambda^2). This is a most important formula.

A half-wave dipole has effective area of 0.13 lambda^2, which is roughly an area lambda/2 by lambda/4. The directivity of a half wave dipole, in the azimuth direction or H-plane, is about 1.67 or about 2.23 dB. Within elevation angles of size about 32.6 degrees the dipole has higher directivity than an isotropic source; outside this range it has lower directivity.

Considering an antenna as a transmitter, if it is fed with power P (accepted power) then the power density on boresight is G P/(4 pi R^2) at distance R. Here, G is a straight number calculated from the directivity and the efficiency. It is also possible to give the gain G in decibels; remember G is a power gain so in dB a gain G of 10 is 10dB, a gain of 100 is 20dB, a gain of 1000 is 30dB and so on.

If we transmit between two antennas each of gain G, spaced by a distance R, the field strength at the second due to the first is G P/(4 pi R^2) watts per square metre, and the effective area of the second is A=G*(lambda^2)/(4 pi) so the total power transferred from transmitter to receiver is the product of these factors. The received power is therefore P*(G lambda)^2/(4 pi R)^2. This can be factorised into three parts as follows; the gain of the transmitting antenna times the gain of the receiving antenna times a "divergence factor" because not all of the power transmitted is picked up by the receiver. This latter factor is [lambda/(4 pi R)]^2 and the reciprocal of this, namely [(4 pi R)/lambda]^2 is often referred to as the "free space loss". We note that it is not really a "loss" as free space itself is a lossless propagating medium.

These antenna transmission formulae only apply in the far field region, so we need to know when we are in the far field.

The near field, the far field, and the Rayleigh distance

In the near field region, the polar radiation pattern depends on distance from the antenna and there is reactive power flow in and out of the region. One can imagine that the energy, instead of propagating uniformly and steadily away from the antenna, has an oscillatory longitudinal component. Energy is transferred to and from the near field region which represents the reactive part of the antenna driving point impedance. As one moves further away, this oscillatory energy flow reduces leaving just the regular power flow in the resistive characteristic impedance (377 ohms or 120 pi ohms) of free space.

In the far field the polar radiation pattern is completely independent of distance from the radiating source.

The transition from near to far field happens at the "Rayleigh distance", sometimes called the "far field distance". An estimate for this distance may be made from the formula (2 d^2)/(lambda) where d is the maximum dimension of the radiating structure. In the case of a circular dish this is just the diameter; but in the case of a rectangular horn it is the diagonal distance across the mouth. This is only an estimate, and nothing suddenly happens at the far field distance thus estimated.

As an example, for a 10GHz antenna having dish diameter 30 cm, the wavelength is 3 cm and 2d^2/lambda = 2*30*30/3 cm or 6 metres. This is a sizeable distance compared to the dish dimensions.

If we consider the Tidbinbilla dish at 6GHz, shown elsewhere in this collection of notes, the wavelength is .05 metres and the diameter 75 metres so the Rayleigh distance is 2*75*75/0.05 = 225 kilometres. Thus when the dish is pointing upwards we need to be above the atmosphere before we arrive at the far field region.

For this reason, it is impossible to measure the far field radiation pattern of a deep space antenna on a terrestrial antenna range. One has to resort to complete measurements of the near field response, and computer calculation to turn them into a far field pattern. Alternatively one can measure the beamwidth by scanning across a small radio star. However it is often difficult to obtain reliable measurements of the sidelobe responses.

Radiation resistance and antenna impedance

Part of the function of an antenna is to match the impedance of the feeder, or driving transmission line, to the impedance of free space.

A half wave dipole presents a resistive impedance of 73 ohms to a transmission line. It also has a small inductive reactance, of about 11 ohms. (The size of the reactive part depends on the length/diameter ratio of the rods of the antenna). The impedance close to resonance varies in a similar manner to a series tuned circuit. If the dipole is shortened from lambda/2 there is an additional series capacitative impedance and if it is cut too long there is an additional series inductive impedance. Thus to make a dipole which has entirely resistive impedance it must be cut a few percent shorter than lambda/2. The precise amount of shortening needed depends on the diameter of the rod elements. In general, the amounts of reactive impedance depend on the ratio of diameter to length of the antenna rods. A good discussion may be found in R S Elliott "Antenna Theory and Design", Prentice-Hall 1981, pp275ff, ISBN 0-13-038356-2, in section III.

Now we see why coaxial cable is often designed to have 75 ohms characteristic impedance.

As the dipole is shortened, the radiation resistance falls sharply and it becomes a very inefficient radiator. For example, Elliott (op. cit. p304) has a calculation which indicates that for a rod-radius/wavelength ratio of 0.2% a dipole has to be cut about 5% shorter than a half-wavelength long to have a vanishing radiation reactance (that is, to present an entirely resistive impedance to the feed) and that then, its radiation resistance has fallen to about 63 ohms. To put this into perspective, at a band II frequency of 100 MHz, the wavelength is 3 metres and a rod-radius of 0.2% of a wavelength is 0.6 cm so the rod diameter is about 1.2 cm, which is typical for such an antenna.

Incidentally, those of you who try to model dipoles using NEC (Numerical Electromagnetic Code) software may find that the returned real part of the driving point impedance stays close to 72 ohms, whatever the value of the diameter/length ratio of the rods. There is an argument that this is due to the stray capacitance between the closely-spaced ends of the thick rods. The combination of shunt capacitance with the radiation resistance in series with the residual inductance provides an impedance transformer, as is found in RF power amplifiers for example. This transformer steps up the actual radiation resistance to a higher driving point resistance; at the same time the shunt capacitance resonates with the residual inductance. There is an argument that the radiation resistance which "matters" is the driving point resistance; however, we then find that this is critically dependent on the gap capacitance and varies with the spacing of the rods, and whether they are made from solid metal or tubes.

Thus we see that these notions about the impedance of a half-wave dipole are only a guide to what we would measure in a practical installation. Indeed, a balanced feeder of characteristic impedance about 70 ohms is impracticable; so we have to incorporate some kind of balance-unbalance ("balun") transition between feed and antenna. The separation of the antenna rods also affects the total antenna length and the feed characteristics, and the physical feed structure and balun affect the near-field distribution of the dipole. It is thus possible to prefer cut-and-try methods for matching practical dipole antennas over the carefully calculated nostrums of antenna theorists.

By the time the dipole length has reduced to lambda/10 the radiation resistance has decreased to about 2 ohms and the reactance has increased to between 1 and 5 kilohms depending on the diameter of the rods.

An infinitesimally short dipole is called a "Hertzian" dipole and is important theoretically since in practice all its properties may be calculated analytically. However, it is never used in practice because of its vanishingly low radiation resistance. For many purposes, calculations on a Hertzian dipole give a useful guide to the behaviour of longer dipoles.

For practical reasons, particularly in mobile applications, it is necessary to cut dipoles short or to use monopoles loaded with inductance over a ground plane. The radiation resistance of a short dipole is given by the formula Rrad = 20*(pi*L/lambda)^2 and for a lambda/8 dipole is only 3 ohms. The series capacitative impedance for this length antenna may be as much as 1000 ohms; most of the transmission line voltage is lost across this capacitative reactance unless it is tuned out. One often sees short monopoles with a coil at the foot, to provide inductive tuning for this capacitative reactance. However, this is poor policy as it puts up the Q factor and reduces the bandwidth of the antenna. The tuning can be quite critical, especially in the presence of variable near-field obstacles.


ALL the above properties of a linear passive antenna are identical whether it is used in transmit or receive mode. There is only one exception to this rule called "reciprocity", and that is when the antenna contains magnetically biased magnetic materials such as ferrites with resonantly rotating electron spin systems.

The physical reason for reciprocity is that the only difference between outgoing and incoming waves lies in the arrow of time. Since the electromagnetic equations are invariant except for the signs of magnetic fields and currents, under time reversal, there can be no difference between transmit and receive mode in the physical current and field distributions. However, if we have a magnet providing a steady bias field, under time reversed conditions we would have to reverse the direction of this bias field. But for incoming and outgoing waves, the bias field direction remains the same. Thus it is possible for the system to be non-reciprocal.

Of course, antennas containing amplifiers, or diodes, or spark gaps, may well not be reciprocal for obvious reasons. Also, practical antenna installations having metal-oxide-metal contacts, "rusty bolts", dry soldered joints and other electrical contact imperfections are also likely to behave differently under transmit and receive modes of operation.

Bandwidth and broad-banding

If we recall the definition of the Quality factor or Q factor as being the ratio of the stored energy to the energy dissipated per radian of oscillation, it is clear that in an antenna the part of dissipation is taken chiefly by the radiated energy. The stored energy is held in the near field region of the antenna structure. Since the fractional bandwidth (delta f)/f is just the reciprocal of the Q factor, for a given radiated energy the Q will be smaller and the bandwidth larger if we minimise the amount of energy stored in the near field region of the antenna structure.

One way of doing this is to make the antenna elements fatter in relation to their length. For a very fine wire antenna, the magnetic field for a given current rises as we approach the axis of the conductor, as 1/r, where r is the radial distance out from the conductor. Thus making dipole antennas out of thick rods rather than thin wires is a good method of broad-banding, up to a point.

The biconical antenna, and its derivatives, the broad-banded Yagi, the bow-tie antenna and the phantom conical antenna (which doesn't have a complete conical surface, but just conically disposed rod elements), is a good method of broad-banding a dipole type of antenna. There is a degenerate form of biconical antenna where the rods are arranged as an X with the upper v and lower ^ fed as opposing arms of the dipole. This was very common in the early days of TV broadcasting, and was also relatively broadband compared to a simple dipole. A variant of this kind of X antenna had the upper and lower arms of the > as a dipole, and the < as a reflector. The radiation pattern had a maximum in the direction away from the reflector, but again the antenna structure was more broadband than a simple H antenna. It was also easier to construct.

A rule of thumb is that a typical half wave dipole with sensible diameter rods has a fractional bandwidth of about 15%.

Antenna calculations

Given a geometrical current distribution on the antenna structure, it is relatively straightforward to calculate the radiation integrals (see any good em theory textbook) to determine the radiation patterns. Often this has to be done numerically.

As with all such activities nowadays, there is commercial software available for the PC platform to do this. One such site is at which also has lots of useful links elsewhere around the web to these topics. I have not tried out this particular software so I cannot vouch for its ease of use or accuracy.

The difficulty with most antenna theory lies in determining the current distributions on the antenna conductors, given an arrangement of feeds, and the terminal voltages at the antenna ends of the feeds.

The Hertzian dipole is artificial in that it assumes there is a uniform current density along the arms of the dipole. There is thus an unphysical current discontinuity at the ends of the arms, which cannot be realised in practice. However, given a uniform current distribution, the properties of the antenna are reasonably straightforward to calculate. That is why this unphysical situation is so often presented in textbooks.

An equivalent problem pertains in simple loop antennas. Here it is often assumed that the current is constant around the loop. That is only a reasonable assumption if the loop perimeter is short compared to a wavelength. There is capacitance between the opposite sides of the loop which can carry displacement current, which results from a build up of charge due to the voltage drop around the loop which has inductive impedance. One can easily see that although there is only a single continuous conductor, the current does not have to be the same everywhere around the loop, as some of it goes to charge the stray capacitance.

Normally, self-consistent calculations are used to calculate together the current distributions and the radiated fields. The "method of moments" is popular. In this method the antenna structure is spilt up into a number of regions, on each of which the current distribution is assumed uniform. The integral equations for the antenna then reduce to solving (what may be a quite large) matrix equation. This is well adapted to computer solution. There are issues of accuracy, and sensitivity to the model framework assumed. This method is also used to work out radar cross sections of complicated objects such as helicopters and aircraft.

In a reflector-aperture antenna fed from the front by a sub-reflector and/or a feed, the far field radiation pattern can be calculated from the Fourier Transform of the field distribution across the aperture, accounting as well for phase variations across the illuminated area. The side lobe behaviour of a reflector antenna is particularly well-suited to this calculation method. A process known as "Apodisation" (after "apod" = "without foot") tapers the amplitude ("amplitude taper") illumination across a dish reflector antenna so that there are no sudden changes of excitation amplitude, especially at the edges of the reflector. We recall from Fourier Transform theory that sudden changes in a function give rise to the presence of high frequencies in the Fourier Transform. In this particular case the sudden change in spatial illumination gives rise to high spatial frequencies in the transform, which directs the energy well away from boresight as the "spatial frequency" translates into the deviation of the propagation direction from boresight.

There is a good discussion of the sidelobe suppression process by using apodisation techniques in volume 1 of the book "The Handbook of Antenna Design", publishers Peter Peregrinus, 1982, on behalf of the British IEE, editors A W Rudge, K Milne, A D Olver and P Knight, ISBN 0-906048-82-6, table 1-3 page 43. As always there is a trade-off, in this case the beamwidth of the main beam is increased by some tens of percent depending on the illumination profile, for the benefit of reducing the sidelobe levels. For reference purposes, this table 1-3 is reproduced here with minor modifications.

       For a circular reflector antenna of diameter D, fed
       with radiation of wavelength lambda, r = fractional distance
       from the centre of the dish towards the rim.

Distribution     -3dB beamwidth   level of 1st sidelobe   Radian angle of 1st 
                   radians                                null (from boresight) 
                  (Full width at
		  half maximum

Uniform             1.02 lambda/D    -17.6dB                 1.22 lambda/D

Tapered to
zero at edge        1.27 lambda/D    -24.6dB                 1.63 lambda/D
as (1-r^2)

Tapered to
zero at edge as     1.47 lambda/D    -30.6dB                 2.03 lambda/D

Tapered to 1/2
at edge as          1.16 lambda/D    -26.5 dB                1.51 lambda/D

distribution"       1.31 lambda/D    <-40.0 dB                no null?

In array antennas, consisting of a number of identical or similar elements driven in synchronism, the problems are very similar to those encountered in filter design situations. Again, one needs to find the individual currents on each element. Often it is assumed that if the elements are identical, the currents must be also. However, consider the situation where there are four dipoles arranged in a straight line. Two of these will be end elements, and two will be interior elements. The coupling impedances between the adjacent elements will thus be different, and the currents necessarily different also.

Antenna source current synthesis

Since the Fourier Transform method mentioned in the previous sections has an inverse, it is in principle possible to start from a knowledge of the required far-field radiation pattern, and also knowledge of the desired geometry of the source structure supporting the generating currents, and use inverse transform methods to derive the geometrical distribution of current amplitudes and phases on the source structure which would give rise to the desired far-field pattern.

Of course, whilst this is in principle no more difficult than is calculating the forward Fourier Transform to derive the far field pattern from the source currents, the difficulty, as always, from a practical point of view, comes in setting up the calculated amplitudes and phases of current elements on the specified source structure from an appropriate feed structure.

Types of antenna

Here we list some of the common types of antenna. Apart from exotic applications, such as the banana tree, most antennas consist of a juxtaposition of conductor and insulator, which may be dielectric or it may be air or free space. This is not necessary; any structure which will support a current on its surface, or guide or modify the direction of propagation of an electromagnetic wave, may be pressed into service as a kind of antenna.

Aperture antennas

This class of antenna contains important technology for satcoms applications.

The simplest kind of aperture antenna consists of a tapered waveguide transition in the form of a "pyramidal horn".

18 dBi X-band pyramidal horn antenna on swivel mount

More pictures of dish antennas here

The TE10 mode in a rectangular waveguide has longitudinal components of magnetic field. As the waveguide is flared to form the horn pyramid, the longitudinal magnetic field components become less and, as the notes on waveguides explain the characteristic impedance of the TE mode approaches that of free space, 377 ohms.

This kind of pyramidal horn aperture antenna is very important in the laboratory, as it is one of the few types of antenna whose boresight gain may be very accurately calculated (to within 0.1 dB). Consequently, it is used for producing reference field strengths and for calibrating the gains of other antennas.

A variant on the rectangular pyramidal horn is the circular horn feed. Such an aperture antenna is commonly used with a circularly symmetric waveguide mode (not the lowest mode in circular guide, NB) to produce uniform illumination of a Cassegrain antenna, which has a circular reflector dish of much larger diameter than the feed. The large reflector dish produces higher gain. The circular waveguide feed can also be used to produce circular polarisation.

Most ground based small broadcast satellite receiver dishes have a small horn feed of low gain placed at the focus of a dish between 0.5 and 1 metre diameter. Often the feed is offset from the boresight direction of the reflector dish; this "offset feed" arrangement directs the main beam away from the feed, and this results in less blockage and improved sidelobe performance.

If the main beam in a Cassegrain antenna hits the feed, or the sub-reflector, it will be diffracted around the obstacles and radiation will be scattered or diffracted into the sidelobe directions. The effective area of the dish is reduced, and the interference with other satellite systems from the sidelobes will be increased. This is not so important in deep space antennas. If we look at the Tidbinbilla deep space tracking antenna, we see there are two reflectors between the main feeds and the main beam. The sub-reflector at the focus of the large 75 metre dish is convex. The feeds are pointing along boresight, and are arranged to have a bean divergence angle which is just sufficient completely to illuminate the sub-reflector. The sub-reflector returns the energy to the main reflector, and again the reflections are arranged so that there is minimal spillover at the edges of the dish, although maintaining uniform illumination as far as is possible.

The feeds are also conveniently located at the centre of the main dish, which moves little as the dish is steered. This has mechanical advantages, and makes the final HPA and LNA electronics more accessible for servicing.

Aperture antennas such as this are used in "very long baseline interferometry" methods. Here, two or more high gain large antennas, having large collecting areas, are separated by many hundreds of kilometres, and used to synthesise an aperture array having the diameter of the baseline separation of the dishes. Radio astronomers use these systems to pinpoint the location of radio sources to great accuracy in elevation and azimuth.

Arrays of antenna elements

If we want to increase the gain of a dipole antenna we can add another dipole antenna alongside it. This is the simplest form of array antenna.

Why is the gain increased, and what is the boresight gain of this "two element array"?

First, we assume the antennas are fed in phase with each other and spaced lambda/2 apart. Considering the radiation in a direction which is normal to the plane containing the dipoles, the contribution from each element arrives in phase with the other. The field strength in this direction is double that for one element, so the radiated power density, which is the square of the field strength, is four times that for one element. However, the two elements together are fed with twice the power of a single element. The increase in gain is therefore a factor 4/2 = 2.

This calculation scales with the number of elements. If we use a 10 by 10 array, the boresight power gain is increased by a factor of 100, which is the number of elements. The field strength is 100 times more along boresight than for a single element, so the power density is 10,000 times greater. But 100 times the power is being fed to the array compared with a single element, so the gain increase is a factor of 100 as stated.

This gain increase is over and above any boresight gain of the individual elements. If we start off with an array of 100 horn feeds at 10GHz, of size 14 cm by 14 cm each, their intrinsic gain is about 20dB and the array factor gives an additional power gain of 100 which is 20dB so the combined structure has a boresight gain of 40dB or so.

Now consider, are we "getting something for nothing" or does this increased gain along the boresight come at the expense of gain elsewhere in the radiation pattern? The answer is clearly that the array concentrates the total radiated power along certain directions at the expense of others.

If we go back to our 2 element dipole array, spaced lambda/2, there can be no radiation along a line joining the centres of the two dipoles as their contributions are in anti-phase in this direction, there being a lambda/2 path difference to get from one to the other.

In general then, the element pattern times the array pattern equals the total radiation pattern of the arrangement. What is the array pattern? It is the pattern you would observe for a set of isotropic radiators spaced as the array elements are actually spaced, and fed with the same amplitudes and phases of signals that the actual array elements receive.

If you want to read more about the fascinating subject of array antenna design, consult R S Elliott [op cit].

More on the directivity-enhancement in array antennas

The argument presented in the section above just happens to be correct when the elements are spaced by lambda/2.

However, for other spacings of the elements, we run into difficulties. Really what we should do is to integrate the array pattern over the sphere surrounding the antenna, to find the averaged isotropic radiated intensity, and then compare that with the intensity on boresight.

This is the approach taken by Constantine Balanis in his book on Antenna Theory and Design. He finds that the gain enhancement depends not only on the number of isotropes, but also on their spacing.

Clearly, if we have N isotropes collected together in a very small region much less than a wavelength across, then the radiation pattern (given uniform and equal excitations of each element) will be a good approximation to a sphere, and the maximum directivity must be just 1, as it is for a single isotropic element, and not N as predicted by the arguments above.

What Balanis says is that a good approximation to the maximum directivity Dmax is given by

Dmax = 2N(d/lambda)

for N elements spaced d apart. We see that this reduces to N, quoted above, when d = lambda/2 and the spacing is a half-wavelength.

Very Long Baseline Interferometry (VLBI)

If we use two aperture antennas, spaced by a great many wavelengths, as an interferometer, the fringe spacing will be of the order of the angle subtended by an object of diameter one wavelength at a distance equal to the separation of the aperture antennas. For example, at 10GHz the free space wavelength is 3cm or 0.03m, so if we separate the antennas by 3000km or 1E8 wavelengths, we can resolve radio sources about 1E-8 radians across, or about 2 milliseconds of arc. By comparison, the beam width of one of the aperture antennas will be of the order of the angle subtended by a wavelength of radiation at a distance equal to the diameter of the reflector. Thus, if we considered a system where there were two 30 metre diameter antennas separated by 3000km, there would be (3E6)/30 = 100,000 interference fringes within the main beam of one of the apertures. Of course, the sensitivity of the interferometer is still governed by the total capture area of the two dishes; but the resolution is now comparable with that of a dish of diameter 3000km.

The interference fringes from these two circular dishes will form parallel straight lines across the circular beam, as can be seen in the pictures below:-

Two circular apertures spaced a distance apart

Two apertures

The interference bands from the apertures above

In order to get resolution at right angles to these bands it is necessary to add a third aperture at the vertex of a triangle, whose base is delimited by the first two apertures.

Measurements and simulations

Measurements on antennas are difficult. The behaviour of an antenna is best seen by monitoring the reflection coefficient with a network analyser over a band of frequencies, and for convenience a frequency about 1 GHz is appropriate. At 1 GHz a wavelength is 30 cms; the antenna is a reasonable size and it is possible to investigate the effects of adjacent objects, and different feed lengths, without too much difficult physical manipulation. The results may safely be transferred to other frequency bands by thought and analogy.

When this is done, one rapidly appreciates that an antenna can not be considered as a closed, isolated component having well-defined properties. Nearly every electronic measurement on an antenna is grossly affected by its environment and physical mounting. One might well ask the question, "What is an antenna?", or equivalently, "Where does the antenna stop and the outside world begin?". A sensible answer to this question is to consider all objects inside the near field as contributing to the radiation.

A helpful example is a Yagi-Uda antenna. We might regard this as a simple dipole with lots of resonant rods placed in the near field. But if we just consider the properties of the driven "antenna", namely the driven dipole, we know we will be grossly in error in assessing the performance of the installation. So why should we stop considering the effects of metallic structures at the end of the boom? We should add in the scattering from the mast, guys, feed (outer coaxial shields can carry induced current) and even dielectric objects (like the chimney stack or adjacent building) in the near field.

Many people now have access to software which accurately simulates antenna behaviour. To do this it is necessary to construct a model. The process of "modelling" is critical to this enterprise as the simulation has limitations of accuracy depending on the kind of model chosen. In itself, the software is essentially accurate and useful. However, the results it returns, for simulation of real antennas, depends critically on what is built into the model. It is not usually possible, in the NEC2 and miniNEC and NEC4 software, to add in all the local effects which will affect the results. This is not just because it is too difficult; there are difficulties in principle, knowing the correct dielectric and conductivity parameters to put in for a real-world installation. Details of the feed arrangement are also difficult to get right. So it is often difficult to know if the results from the simulation of the model represent the real behaviour of the antenna it was intended to investigate. The process of running the software always returns a result, and the internal checks on validity, while possible, are subtle. Belief in the results often dissolves into a matter of opinion or faith. This can be the subject of strongly-held views, which can only be resolved by recourse to measurements.

Thus, simulation should be regarded (taking the most cautious view) as merely a rough guide to an antenna's behaviour in a real installation. Any modelling process needs careful validation by measurements. One is then presented with the choice of which to believe, if there is disagreement.

I have seen people worry about 1/10 dB in gain in a simulation. This is probably unsound, and one wonders how many hundreds of hours people spend (no doubt happily) in this kind of activity.


An anonymous committed modelling expert tempers the remarks above with the following comment.

Depending upon the conditions of use and application, NEC can be quite accurate. Under some conditions of application, accuracy deviation of dimensions has run to well under 1% in some design projects. In others, accuracy of construction guidance can be considerably off. The real question is this: what are the conditions for each kind of case? However, that question requires an application-by-application analysis, not a wholesale posture toward modeling.

Yes, but application-by-application analysis usually requires experimental measurements to be made to validate the models.

Considering this italic comment, what can be said with confidence is that an unvalidated antenna model is not necessarily a good guide to how a practical antenna may behave.

One of the thrusts of the complex-systems research reported elsewhere on these pages (see complexsim.html, for example) is that simulations are only of use if one knows they are going to be accurate before one validates them against measurements. Most people using antenna models are making extrapolations from validated simulations to applications that they believe to be similar. It is the thesis presented here that this method of proceeding is unsafe and unwise in many cases.

Copyright © D.Jefferies 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005.
D.Jefferies email
8th August, 2005.