TELE465 Mobile and Satellite Communication Systems Lecture 3 Antenna Theory A radio antenna, whether transmitting or receiving, is an integral component of any wireless communication system, whether it is a mobile phone, satellite, WLAN, or other system. An antenna acts as a transducer that converts the current or voltage generated by the feeding-based circuit, such as a transmission line, waveguide, or coaxial cable, into electromagnetic field energy propagating through space, and vice versa. In both mobile and satellite systems communication occurs through the propagation of unguided electromagnetic waves through the atmosphere, and antennae are the interface of these radio waves into the electronics of the transmitter and the receiver. The behaviour and structure of the antenna determine wave field strength, polarisation, and direction of propagation. In the case of mobile communications the performance of the system is entirely dependent on the bi-directional radio link between the base station and the handset. While the base station will typically be equipped with a high-gain antenna and a transmitter capable of delivering in the order of tens of watts of RF power, the handset must contain an antenna that is fundamentally constrained both by the dimensions of the device and the available power, and would typically be able to transmit a total effective radiated power of only about 1 Watt. In addition, base station antennae are typically mounted in a clear location 10 m or more above ground level, while the MS antenna can be obstructed by the user s hand or head, and is at most 1.5 m or so above the ground. It is clear that BTS and MS antenna design present two very different technical challenges, and for practical purposes, the shortcomings of the MS antennae are the single most intrinsic loss factor in the system. Antenna design tends to be carried out virtually independently of the other components in the communication system, and is a very active area of research and development. In the case of mobile telephony, the move towards 3G networks, where phones provide services ranging from telephony, high-speed data services, location and navigation services, and entertainment, has put a demand on antenna engineers to build ever smaller antennae with ever increasing bandwidths of operation and diverse range of operating frequencies (ranging from AM radio (540-1600 khz) up to Bluetooth (.4 GHz), all for the same device). Moreover, the advent of these services means the antenna must compete for space in the small devices, and must operate in close proximity to devices like cameras, flash units, loudspeakers, and batteries, without degrading their performance. In this lecture we will firstly make some general comments on the characteristics by which we classify and quantify the performance of an antenna. Then we will begin with the simplest antenna, the elementary dipole, and use classical electromagnetic theory to derive all of the important characteristics to describe its performance. We will then show how we can extend these results to obtain the performance of a practically realisable antenna, linear half-wave antenna. We ll then describe the most
common practical antenna designs, before extending our discussion to antenna arrays and smart antennae an emerging 4G technology. Antenna Characteristics In this section we will describe the basic concepts of an antenna and its fundamental figures of merit, such as radiation patterns, directivity, gain, bandwidth, polarisation, and impedance, which describe the performance of any antenna. The first important point is that the beautiful symmetry of nature means that an antenna must have exactly the same properties when transmitting as when receiving. For instance, if a certain current applied to the antenna produces a certain power of electromagnetic energy transmitted in a particular direction, then the same radiation power incident onto the antenna from this same particular direction produces exactly the same current in the antenna when receiving, subtracting any losses due to internal resistance or other, which in turn must be the same magnitude when transmitting or receiving. This nice fact of life halves the amount of work we need to do and the number of quantities with which we describe an antenna. Our basic principle for understanding the operation of an antenna comes from Maxwell s equations an oscillating current or voltage in the antenna produces oscillating electric and magnetic fields. For this reason nearly all practical antennae are metallic in structure. A component of these generated electric and magnetic fields will be self-propagating the electromagnetic waves that Maxwell predicted and Hertz discovered. If we are sufficiently far from the antenna all other components of the generated fields can be neglected and we need only consider these electromagnetic waves, which are then called the radiation produced by the antenna. As a consequence we divide the radiation field generated by an antenna into three regions, or zones. These are illustrated in the diagram below.
The first zone is called the reactive near-field region. For distances less than 3 approximately 0.6 D λ, where D is the dimension of the antenna and λ is the wavelength of radiation, the fields and currents within the conductor dominate everything else. When we move further away from the antenna the internal fields become less predominate and we enter the radiation zone. The radiation near-field region or Fresnel zone last up to the order of D λ. In this region the radiation components dominate, though in a complicated and irregular way. Further out the radiation field becomes smooth and follows the familiar 1 r relationship. This is called the far-field region or Fraunhofer zone. We have moved sufficiently far from the antenna that the waves radiated from the different elements have all combined and interfered in a regular way. In the angular field distribution of radiation from the antenna is independent of distance. We will usually only consider the farfield region. The first quantity of interest in describing an antenna is the radiation pattern, which quantifies the amount of electromagnetic energy transmitted in a certain direction. We use spherical polar coordinates, ( r,θ,φ), to describe the radiation field. These are shown in the diagram below. In the far-field region we can express the radiation field θ, φ. in terms of only the elevation and azimuth angles, ( ) There are two ways to express the radiation pattern from an antenna. We can talk of the amplitude field pattern, which represents the magnitude of the electric field amplitude of the radiation being transmitted in by the antenna in the far-field region in E θ,φ. Alternatively, we can talk about the intensity pattern, the direction, ( ) quantifying the electromagnetic power radiated in this particular direction, ( θ,φ) The two are naturally related, since we know for waves that the intensity is proportional to the field amplitude squared, I.
1 I η where η 10π is the impedance for free space. ( θ, φ) E( θ, φ) If we drive an antenna with twice the power then we would expect that twice as much power would be radiated, so it usual to normalise these radiation patterns to make them independent of applied power. There are two ways this can be done. The radiation pattern can be divided by its maximum value, so that its value in the direction of maximum radiation is 1, E( θ, φ ) F( θ, φ ) Emax However, when comparing two different antennae it usually more useful to divide the intensity pattern by the total power radiated by the antenna. Thus we obtain the power radiated in different directions by the antenna if unity power is applied to the antenna. To find the total radiated power we simply integrate the radiation intensity over all directions, P rad ππ 0 0 I ( θ φ) ( θ φ), sinθ dθ dφ The average power per solid angle is thus P rad 4π. Dividing the intensity pattern by this is known as the directivity pattern, 4π I( θ, φ) D, The radiation pattern is naturally a function of two angles, and as such would require a 3D plot to represent. Usually, though, symmetry would dictate that the radiation pattern is uniform in one direction, so a D plot is sufficient to illustrate the variation in radiation intensity with respect to other axis. In other case two D polar plots of perpendicular cuts could be used to represent the radiation field. This is shown in the diagram below for the radiation pattern from an elementary dipole. The radiation P rad
pattern in independent of the azimuth, φ, clear from the symmetry, since the antenna looks exactly the same regardless of which azimuth φ we are coming from. Thus, the polar plot of intensity variation with altitude θ is sufficient to represent this radiation pattern. An example of a radiation pattern of a typical antenna is shown below. A directional antenna will have a preferred direction of radiation a direction in which it transmits the largest proportional of its power, and on receiving is most sensitive to. This preferred direction is responsible for the main lobe. The width of the main is called the beamwidth and much like bandwidth in a frequency spectrum can be measured in numerous ways, be it half-power beamwidth or null-to-null beamwidth. A typical antenna will also have some directions to which secondary maxima are transmitted, producing what are called side lobes or back lobes (or radiation opposite to the principle direction). The polar plot can also be unwrapped to produce plots like intensity in decibels against elevation, as shown too. An omni-directional antenna, also called an isotropic radiator, is one that transmits the same amount of power in all directions. The radiation patterns of an omni-directional antenna are unity, F ( θ, φ) 1, D ( θ, φ) 1 The directivity of an antenna measures its deviation from omni-directionality essentially how much more power is transmitted in the direction of maximum radiation as compared to the average power level. I max ( θ, φ) 4π I max ( θ, φ ) D I av P rad As illustrated in the diagram, the directivity must always be greater than 1, D 1, with equality if and only if the antenna is omni-directional.
The gain of an antenna is closely associated with its directivity. The gain of the antenna represents the fraction of radiation transmitted in the direction of maximum
sensitivity as a fraction of the input power, or really the ratio of the power transmitted by this antenna as referenced to an ideal (lossless) isotropic radiator with the same input power: 4π I max ( θ, φ) G The antenna efficiency, ε, measures the ratio of the power that is input to the power radiated by the antenna. It can be expressed in terms of the radiation resistance R rad and the loss resistance R loss (representing the resistance of any conductive or dielectric materials used to construct the antenna), Prad Rrad ε Pin Rrad + Rloss We ll define these quantities shortly. Clearly, G εd When gain is expressed as compared to an isotropic antenna, it is usually listed in units dbi. A useful quantity related to the directivity is the antenna effective area, A eff, also called the aperture. It is defined as the ratio of the available power at the terminals of the antenna, P rec, to the power flux density from the plane wave incident on the antenna from that direction, I wave, assuming the polarisations are in alignment. Prec A eff I wave It measures essentially how much of the incident radiation intensity is captured by the antenna on reception. It can be shown that the effective area of an antenna on reception of radiation coming from a particular direction, ( θ, φ ), relates to its directivity by λ A eff ( θ, φ) D( θ, φ) 4π In most situations we will assume the antenna is pointed in its direction of maximum sensitivity, and express its effective area simply as A λ eff D 4π. An important property of electromagnetic waves, and so also antennae, is polarisation. Polarisation encapsulates the direction of vibration of the electric field of the EM wave, which in turn determines the way this wave will interact with matter (the electric force being in general stronger than the magnetic force). The electric field can be defined with respect to two arbitrary axes, E x E0x cos ( ωt ) E E cos( ωt δ ) y 0 y The polarisation state of the wave depends on the relative phase of these two components, δ, and their amplitudes. If δ 0 or π, then the two components are in phase and linear polarisation results. If δ ±π/ the two components are out of phase and we obtain circular polarisation if the amplitudes are the same. Other cases result in elliptical polarisation, described by P in
E E x 0x ExE y cosδ E y + sin δ E E E 0x 0 y 0 y A mismatch in polarisation between the incident radiation and the antenna s polarisation axis will result in a loss of energy. This is called the polarisation loss factor (PLF), and is given by PLF cos γ where γ is the angle between the two polarisation axes. The remaining characteristics of the antenna refer to its properties as an electrical circuit element, and hence how we can construct electrical circuits to drive the antenna and produce the desired radiation. From the perspective of the electric circuit, the antenna is taking energy away from the circuit, where this energy is the power that the antenna is radiating. It clear from Maxwell s equations that the power radiated by an antenna is proportional to the square of the current running through it (the electric field is proportional to the current, as is the magnetic field, and the radiated power is 1 given by the Poynting vector S { E H av Re }). From an electrical point of view, then, this antenna can be thought of a resistor dissipating power from the circuit. This leads to the definition of the radiation resistance of the antenna, as the resistance of a resistor that would dissipate the same power as the antenna radiates when the same average current i runs through it, Prad i R rad The radiation resistance is determined by the physical properties of the antenna, as we shall see in the next section. In general the antenna will appear to have some net impedance to the driving circuit,
Z A RA + jx A where the equivalent antenna impedance is the sum of the radiation and the lossy parts, as before, R A Rrad + Rloss. Driving the antenna then we know that maximum efficiency is obtained when the impedance of the driving circuit is matched to the antenna impedance. Moreover, the high frequencies involved typically mean that the system is best understood in terms of transmission line theory the wavelength of the signal is comparable to the size of the circuit elements. The important parameter to quantify performance is the voltage reflection coefficient, Z A Z 0 Γ Z A + Z 0 where Z 0 is the characteristic impedance of the transmission line or waveguide circuit that is driving the antenna. The fraction of input power reflected at the antenna is Preflect Γ Pin This quantity is often expressed in terms of the Voltage Standing Wave Ratio (VSWR), 1+ Γ VSWR 1 Γ and this figure expressed in decibels is called the return loss, Return Loss (db) 0log Γ 10 Bandwidth is once again a quantity that is a little vague to define for antenna. One would define the bandwidth of an antenna as the frequency range over which the performance of the antenna, with respect to some characteristic, conforms to specified standard. Exactly what that characteristic is and what the required standard can vary and depends on the application. Loosely two bandwidths are defined for an antenna. The pattern bandwidth represents how features within the radiation pattern of the antenna, such as the gain, side-lobe level, beamwidth, polarisation, and beam
direction, vary with operating frequency. The impedance bandwidth, on the other hand, encapsulates the variation of the electrical characteristics with frequency. Electrically we treat the antenna as a resonance circuit, with a characteristic frequency, and the bandwidth is then related to the Q-factor, or quality, of the resonance ( Q X R ). L Below illustrates a typical data sheet as one might find for an antenna. This example is for a particular antenna used for WiFi. It is a directional polarised antenna, used for access to stationary lap-top computers. Also shown on these data sheets are the mechanical properties which we have not discussed in these notes, as this information should be fairly self-explanatory.
Elementary Dipole Antenna Our aim here is to determine the radiation emitted by our small elementary dipole antenna, of length l << λ, where λ is the wavelength of the radiation emitted. We ll see that the purpose of this assumption is that we can assume that the current in the element is uniform, and does not vary along its length. The theory of radiation here comes from Maxwell s equations for electromagnetism. In differential form, they are dh E μ D ρ dt de H ε + J B 0 dt where B μh, D εe, and J σe for a linear medium. The beautiful symmetry of these equations should be readily apparent. The sources, charge density ρ and current density J, produce the electric (E and D) and magnetic (B and H) fields, dependent on the properties of the material, permittivity ε, permeability µ, and conductivity σ. As we are dealing with electromagnetic waves, we make each of these fields and j t currents sinusoidal ( e ω ), which allows us to remove the time-dependence from the equations to obtain the harmonic form of Maxwell s equations, E jωμh D ρ
H j ωε E + σe B 0 In addition, our waves are generated by currents and charges moving inside a conductor. The ability of charges to move freely inside a conductor gives us two sets of boundary conditions. The E-field must always be perpendicular to the surface, n E 0 while the H-field must be tangential to the surface, n H 0 where n is the normal vector to the surface. In a medium with zero conductivity and no sources, termed free space, we can obtain wave equations for the fields, E H E εμ 0 H εμ 0 t t from which we identify the wave speed as c 1 με. The solutions in free space are harmonic travelling waves, j( ωt kz ) j( ωt kz ) E E 0 e H H 0 e where we have taken the z axes as the direction of propagation. The symmetry between the two fields is readily apparent. If we define u as the direction of propagation, we can relate between the fields as μ E H u ε so the fields and the direction of propagation are all mutually orthogonal. The power transported by this electromagnetic wave is given by the Poynting vector, 1 S E H To solve Maxwell s equations for sources it is generally easier to work with potentials. The vector potential A is defined so that B A and for an appropriate choice of gauge, the scalar potential V gives A E V t For these potentials Maxwell s equations become, for σ 0, V ρ V με t ε A A με μj t The solution space to this set of equations is spanned by the familiar Green s functions,
V 1 4πε A V V jk r e ρ() r dv r jk r 1 e J() r dv 4πε r Let s now turn our attention to the simple elementary diploe antenna, as shown in the diagram below. It is fairly apparent, from the symmetry of the situation, that only one component of the vector potential is non-zero, jβr μ e Az il 4 π r with β π λ. This follows from the current distribution, J ( 0,0,ilδ ( z) ). As we have defined the axes, the current i only runs down the z-axis over this infinitesimal length of wire. Transferring to spherical polar coordinates and taking the curl of the vector potential, we obtain the following components for the fields,
and E θ E H r φ 1 il sinθ e 4π H r H θ 1 il cosθ e π 1 il sinθ e 4π jβ jβr jβr jβ 1 + r r 0 η + r 1 jωεr r jωμ η + + r r 3 1 jωεr 3 Now, the imaginary terms at 1 r represent the radiation field in the far-field (oscillating, wave-like terms, hence imaginary). The 1 r terms represent the induced 3 field, while the1 r terms are the electrostatic components. Thus, in the far-field, we have j j( ωt βr ) H φ ( r, t) il sinθ e λr j μ j( ωt βr ) Eθ ( r, t) il sinθ e λr ε Note that Eθ ( r, t) μ H ( ) ε φ r, t which is η 10π in free space, as we expect for an electromagnetic wave. Note that the fields are totally independent of φ, the azimuth, as we expect from symmetry. We know that in the far-field we have waves propagating in free space. The Poynting θ, φ, vector gives the field carried away in a direction ( ) ( il) μ S 8λ r sin θ rˆ ε Having obtained expression for the fields in the far-field, we can easily normalise to get the radiation patterns. The amplitude pattern is, normalising so that the maximum value is unity, F ( θ, φ) sinθ As we expect from symmetry, there is no dependence with the azimuth, φ. The maximum amount of radiation is transmitted in a direction perpendicular to the orientation of the element ( broadside ), and no radiation is emitted along the axis, in the direction of current flow.
For the directivity pattern we need to integrate the intensity distribution over all solid angles to find the total radiation power emitted, i l sin θ η 8π sinθ θ φ 8λη ππ i l 8 3 0 0 λ P rad r d d r The directivity pattern is 4π I ( ) ( θ, φ) 3sin θ D θ, φ P rad The direction of maximum radiation is θ π, giving a directivity of D 1. 5. Assuming the antenna is loss-less, this is also the antenna gain. The effective aperture of this elementary dipole antenna, when receiving radiation from its direction of maximum sensitivity, is 3λ A eff 8π The polarisation of the radiation emitted by this antenna is clear from the components of the electric field. The fact that only the E θ of the electric field is non-zero means that the direction of polarisation is always given by the unit vector, θˆ. The other quantity that is easily obtained is the radiation resistance, l R rad 0π λ Determination of the other important electrical characteristics, in particular the impedance, would require assumptions and discussion of its internal structure and driving circuit. We ll instead turn our attention to how we can use these results of the elementary dipole antenna to find the characteristics of a practical, realisable antenna. The one we will choose to focus on will be the linear antenna, whose length is taken to be one-half of the wavelength of the emitted radiation. This requirement on the length of the antenna is purely to simplify the fairly complicated mathematical expressions that we will obtain.
Half-wave Antenna Our basic strategy here is to exploit the property of linearity of electromagnetic fields. We could imagine a real, finite sized antenna to be built from a large (well, really infinite) number of elemental dipole antenna, each of which is generating its own radiation field. The total radiation field created by this antenna is the sum of the radiation fields of each of the component elementary dipole antennae. So, in theory at least, the hard work has already been done. Let s consider a linear dipole antenna of length l excited in the centre with a current of amplitude i. The motivation for choosing this arrangement, with the excitation in the centre, is that it corresponds to the typical practical geometry of situating the antenna over a large, grounded sheet, as shown in the diagram. The properties of the grounded sheet mean that the charge distribution within the sheet will be such that we will have an identical image of the antenna within the spatial region encompassed by the grounded plate. This is naturally important in the design of mobile handsets, where we are inherently pushed for space. In implementing the antenna in the handset as an antenna above a grounded plate, being the handset case, we can essentially double the length of the antenna, increase its radiation resistance, and increase its operational bandwidth. We will break our antenna up into elements of length dz and find the contribution of the field from this element at the point Q ( r,θ,φ). The total field at Q will then be the vector sum of the fields from all elements along the length of the dipole. The first complication from our calculation for the elementary dipole is that we can no-longer assume that the current is uniformly distributed along the length of the antenna, since for practical purposes we can assume the wavelength of radiation is of the same scale as the antenna length. A reasonable assumption for the current is to take it as sinusoidal and time varying along the conductor. Our current distribution will be assumed to be i sin( k( l z) ) for 0 < z < l I ( z) i sin( k( l + z) ) for l < z < 0
Denoting the distance of our current element from Q as r and its relative elevation as θ, we can obtain the fields in the far-field as, identical to our results from the previous section, jωμ I( z) dz jkr de dh θ η φ sinθ e 4πr The far-field approximation (r large, r >> l ) lend to some simplifying assumptions, 1 1, θ θ, and r r + z zr cosθ r z cosθ r r The total field at Q is then found by integrating all the contributions from all elements along the length of the conductor, E θ With the aid of the following standard integral, ax ax e e sin( bx + c) dx [ asin( bx + c) b cos( bx + c) ] a + b the field is easily calculated as jηi jkr cos( kl cosθ ) cos( kl) Eθ e πr sinθ Having obtained the field all other quantities regarding the radiation pattern easily follow. For the practical case of the half-wave antenna, l λ, and the field reduces to The amplitude radiation pattern is E θ l l de π cos cosθ 60i r sinθ θ
F ( θ, φ) π cos cosθ sinθ The directivity power pattern is D ( θ, φ) C in 4 ( π ) π cos cosθ sinθ 1 τ cos d τ where C ( x ) τ in x 0. The directivity is then found to be D 1. 64 (.15 dbi), implying that the half-wave dipole achieves a slightly more directed beam than the elementary dipole antenna. The radiation resistance of the half-wavelength antenna is found to be R rad 73. 09Ω. However, it should be noted that the radiation resistance of the quarter-wave dipole sitting above a conducting sheet is only one half of this value, since radiation can only be emitted in directions with θ < π (no radiation can propagate within the conductor). The impedance of the half-wave antenna can similarly be shown to be Z A 73 + 4 j Ω Practical Antennae Dipole antennae are fairly popular for radio reception and cordless phones. These antennas are omni-directional in the azimuth plane, and are sensitive to polarised beams, which makes them suitable for reception of ground-based waves in fixed, vertical orientations. And, of course, linear dipole antennae are very easy to use and cheap to build. Another popular antenna design for the reception of TV signals is the familiar Yagi-Uda, seen on the tops of many suburban houses.
The Yagi-Uda antenna is a dipole pair to act as the feeding circuit, with a set of linear dipole elements to act as directors of the radiation beam. As an antenna by itself, the dipole pair has identical radiation pattern characteristics as the linear dipole, however it boast a lower input impedance and a wider bandwidth of operation. The analysis of a Yagi-Uda antenna is fairly complicated and requires computational calculations. They are quite common in practise as they are simple to build, lightweight, low-cost, and have desirable characteristics for many applications, including a uni-directional beam, and acceptable bandwidth and impedance characteristics. For satellite communications a highly directed antenna is required, and the most common solution in these systems is the parabolic reflector antenna. Reflector antennae use a large, most commonly paraboloid-shaped dish to focus the incident beam into a receptor antenna, usually a horn or dipole element antenna, depending on the polarisation of the signal. The purpose of these antennae is to achieve a very high gain and directivity, to be able to pick up very weak satellite signals that have travelled a very long distance. The largest parabolic antennae ever built, used as radio telescopes, have maximum gains in the order of 80 db! We already made the point in the preamble that antenna design for mobile handsets provides a unique challenge. The physical size constraints and the need for power efficient operation are competing, contradictory elements in the design. The desirable characteristics of a mobile station antenna are to be omni-directional, have a widebandwidth, high efficiency, and good shielding of other equipment. There are two common solutions for handset antennae. The first solution was the external helical or spiral antenna. These antennae are considerably smaller than the wavelength of the radiation would imply, however their performance is aided in having the base of the handset act as a ground sheet. Increasingly these days microstrip antennae are becoming the chosen solution. The attractions of microstrip antennae are that they are small and planar in shape, so easy to enclose in a handset, and they are cheap and easy to manufacture using modern printed circuit board technology. Moreover they are very versatile in terms of operating frequency, pattern, polarisation, and impedance, all of which can easily be tuned by appropriate selection of the geometry of structure. Additionally, by adding loads between the patch and the ground plane, such as with pins or varactor diodes, adaptive elements with variable
frequency, polarisation, patterns, and impedance can be obtained. The major operational disadvantages of microstrip antenna are their low power, low efficiency, high Q, narrow bandwidth, and poor polarisation purity, though much research is currently being conducted to improve their suitability for portable devices. RF circuits Before turning our attention to antenna array, one should just make a brief summary of some of the important issues that go with RF circuits. In mobile and satellite communications, our transmission frequency is very high, in the order of GHz or above. Electronic devices operating at these high frequencies present two major differences as compared to conventional, low frequency circuits. The first is called the skin effect. At high frequency the current in a conductor tends to gravitate to the surface, or skin of the conductor. This focusing of the current to the surface region naturally increases the effective resistance of the conductor, and hence the power loss in the circuit. At an operating frequency of f, the current will be limited to an approximate depth of d from the surface, where 1 d πfσμ
The second effect is that, at such high frequency and hence small wavelengths, the current and voltage will not be uniformly distributed within the conductor. This makes these high frequency circuits analogous to transmission lines, and so transmission line theory must be applied to obtain the effective impedance and load matching the familiar smith chart theory. Students will get experience with this in the laboratory. These two factors mean that we attempt to minimise the amount of circuitry that is needed at the high, carrier frequency. Mixers are used to down-convert the signal to a lower, intermediate frequency (IF) for processing (demodulation, digital sampling, etc), and on the transmitter side the final stage will always be the up-converter to mix the signal to the final, RF frequency. High frequency circuits are need to fed the signal from the up-converter to the antenna, and these usually are microwave waveguides, transmission lines, or coaxial guides. Laboratory experiments will explore the issues inherent in these high-frequency waveguide feeds. Antenna Arrays A very important development in the theory of antennae was the antenna array a group of antenna elements geometrically aligned in some way acting jointly to generate a beam on transmission or recover a signal in reception. The positions, amplitudes, and phases can be set to obtain radiation patterns with desired characteristics and features, the latter two factors can be varied electronically in realtime a process called beamforming. With beamforming we can control the direction and width of the main beam, reduce the side-lobe level, and increase the antenna directivity as desired. By altering the relative phases and firing times we can even achieve effects such as beam sweeping. The simplest case is to consider two dipole antennae that are separated by a distance d, and are excited by a common signal but have a phase difference of α. We can find the field at Q by forming the vector sum of the fields produced by the two individual elements of the array. In the far-field we can assume that the travel distance of the two waves is negligible in the difference it will introduce to their relative amplitudes, and it will only affect the relative phase of the two interfering waves. Thus, both waves at Q will have the same amplitude, E 0, which is the amplitude produced by a single dipole antenna at this direction. The phase difference of the two waves at Q is due to the different distance the two waves must travel, ra rb d cosθ, as well as the relative firing phase of the two elements, α. The total wave at Q is thus, dropping the time dependence for simplicity, jkra jα jkrb E E e + E e e Q E e jkra jk ( ra rb ) + jα [ 1 + e ] jkd cosθ + jα [ 1+ e ] jkra E0e The radiation pattern from this two element array is then, jψ F ( θ, φ) E0 1+ e where ψ kd cosθ + α. 0 0 0
This is an important result, as we see we can express the radiation pattern from the antenna array as the product of the radiation pattern of the individual constituent elements, E 0, and an array factor, AF, due only to the positions and firing phases of the elements of the array. Here, AF cos ψ The radiation pattern from this two element antenna is π cos cosθ ( ) ψ F θ, φ cos sinθ It is very easy to extend this to a linear array of M equally spaced antenna elements, each fired with the same amplitude and a constant phase difference of α. The radiation pattern is Mψ sin ( ) F θ, φ E 0 ψ M sin where ψ kd cosθ + α as before. The diagrams below show two radiation pattern for M 6, omni-directional antenna array elements, and a spacing of d λ. The first has α 0, so the six elements are fired in phase. We see the effect of the array is to produce a much more directed beam. The second has α π, and this moves the direction of the main beam to θ π 3. This is the idea of electronic scanning, where we can progressively change the relative firing phase to sweep the main beam direction.
The important characteristics to describe the beam produced by a linear array are: α Main beam direction: cos θ m kd λ Beamwidth: Δ θ Md Side - lobe Amplitude Side-lobe level Main lobe Ampliude 3π The key points are that by increasing the number of elements in the array we can reduce the beamwidth. However, the proportional of energy sent in undesired directions, encapsulated as the side-lobe level, is independent of the number of elements. The only way side-lobe level can be reduced is to alter the relative amplitudes of excitation of the constituent elements. In general the amplitudes of the elements do not need to be the same, and the phase shift between elements does not need to be constant. This is then known as an adaptive antenna. The weights of the signal given to each antenna feed can be expressed as a complex number, jαi wi wi e where w i represents the amplitude of that antenna feed, and α i represents the relative firing phase. The radiation pattern from this array would be jψ jψ Mj F θ, φ E w + w e + w e + K + w e ( ) ψ 0 0 1 M
The above has similar form to a discrete time filter, and so is often referred to as a spatial filter. The process of an adaptive filter is as follows. First, on transmission, a desired radiation pattern must be specified. This must come from some knowledge of the location of the intended recipient and the RF environment. Then an algorithm must be used to find the set of antenna weights that produce a radiation pattern as close as possible to that desired. Common algorithms here are the MMSE, LMS, and RLS, which we will discuss in detail when we talk about digital equalisation. Then these weights are applied to the signals fed to the respective antennae in the array. The process is similar on reception, however the weights represent the relative gain and time shift given to the signal from each antenna when these signals are summed to produce the net, received signal. Adaptive antennae have opened up the possibility to perform SDMA (Space Division Multiple Access). This is where a base station with an adaptive antenna can direct the signal to the user of interest, even to extent where the same channel could be re-used in different directions. SDMA and adaptive antenna promise a increase in the data rates that can be achieved, since the focussing of signal power will lead to a better RF link, and an increase in network capacity through increased channel re-use. Such system are still in the development stage, however, though are much talked about in proposals for 4G networks.