# Introduction An antenna is defined by Webster’s Dictionary as “a usually metallic device (as a rod or wire) for radiating or receiving radio waves”. The IEEE Standard Definitions of Terms for Antennas (IEEE Std 145–1983) defines the antenna or aerial as “a means for radiating or receiving radio waves”. In other words the antenna is the transitional structure between free-space and a wave guiding device, as shown in the figure below. The guiding device or [[Physical Layer#Transmission Lines and Characteristic Impedance of Interconnects|transmission line]] may take the form of a [[Printed Circuit Boards|PCB]] track, a coaxial line or a hollow pipe (waveguide), and it is used to transport electromagnetic energy from the transmitting source to the antenna, or from the antenna to the receiver. In the former case, we have a transmitting antenna and in the latter a receiving antenna. ![[Pasted image 20241231180650.png]] >[!Figure] >_Antenna as a transition device_ (source: #ref/Balanis ) ## A Quantum Interpretation We know that the ether, assumed by Maxwell and Hertz, does not exist. There is no elastic medium for the waves to propagate in, so it follows that the waves Hertz thought he had discovered are not at all what he supposed either. What he actually generated was a stream of radio quanta, identical with the photons of light except for their energy, and small enough to have wave-like properties. Particles can perfectly well move through empty space so the ether is irrelevant to quantum theory, and it is unnecessary to make any implausible assumptions about forces acting at a distance. Apart from its position, we can characterize the state of a particle if we specify its energy or momentum, while for a wave the corresponding parameters are frequency and wavelength. Quantum mechanics relates these pairs of parameters together, linking the wave and particle properties of quanta, in two monumentally important equations: $\large \varepsilon = hf$ where $h$ is Planck's constant and $\varepsilon$ is energy. $m=h/\lambda $where $m$ is momentum. Planck's constant, relating the wave and particle sides of the quanta, is one of the constants of nature, and has the amazingly small value of $6.626 \times 10^{-34} \, \mathrm{J \cdot s}$. The tiny magnitude of this number explains why the classical theories work so well. ==Quanta have such very small energy (and hence mass) and in any realistic rate of transfer of energy (power flow) they are so very numerous that in almost any situation their individual effects are lost in the crowd, and all we see is a statistically smooth average, well represented by the classical theory.== ![[Pasted image 20250101013054.png]] > [!Figure] > _A plane electromagnetic wave_ (source: #ref/Gosling ) In quantum mechanics the [correspondence principle](https://en.wikipedia.org/wiki/Correspondence_principle) states that valid classical results remain valid under quantum mechanical analysis (but the latter can also reveal things beyond the classical theory). However, it is good to know what is really going on (quite different from what Maxwell imagined) and there are times when thinking about what is happening to the quanta can actually help us to a better understanding. What is the significance of the electromagnetic waves in quantum theory? We know that the power per unit area of the wave front (the power density of the advancing wave) can be shown to be given by the Poynting vector $P$ where: $\large \mathbf{P} = \mathbf{E} \times \mathbf{H}$ Then, the power flow per unit area is proportional to the square of the wave amplitude $\large |\mathbf{P}| = \frac{\mathbf{E^2}}{120\pi}$ Now consider a parallel stream of quanta. In a time $\Delta t$ they travel $c\cdot \Delta t$ so the number emerging through a surface of area $A$ at fight angles to their flow must be: $\large n\Delta t = \rho_qAc\Delta t$ Where $n$ is the number emerging per unit of time and $\rho_q$ is the density of radio quanta. We know that each quantum carries a fixed amount of energy $hf$, so: $\large |\mathbf{P}| = nhf = \rho_q A c h f = \frac{\mathbf{E}^2}{120\pi}$ It wouldn't bee too wrong to say that the probability $p$ of finding a quantum in a small volume must be proportional to the density of quanta $\rho_q$ so: $\large p \propto \mathbf{E}^2$ ==The physical significance of the electromagnetic wave is that **it tells us how likely we are to find a radio quantum, because the square of the wave amplitude (its power level) is proportional to the probability of finding a quantum near the location concerned**, and the Poynting vector just gives us the rate of flow of quanta at the point where it is measured.== When there is a flow of quanta all of the same frequency, the radiation is referred to as **monochromatic** (if it were visible light it would all be of one color), and if it all comes from a single source, so that the quanta all start out with their wave functions in phase, the radiation is said to be **coherent**. Radio antennas produce coherent radiation, as do lasers, but hot bodies produce incoherent radiation, experienced in radio systems as noise. By contrast, incoherent radiation is interesting to radio astronomers, for whom hot bodies are primary sources. In the case of coherent radiation, very large numbers of radio quanta are present, but the wave functions associated with each photon (quantum) have the same frequency and are in a fixed phase relationship, so we can treat them as simply a single electromagnetic wave, which is why Maxwell's mathematical theory works so well in practice. ## Antenna Theory ### The Isotropic Radiator The isotropic radiator is the simplest conceivable transmitting antenna, radiating equally in all directions. The concept is useful in developing theory, but could any real antenna have this property? Obviously we shall be looking for a system with the maximum possible spherical symmetry. Let us begin with the idea of a point (or very small sphere), isolated in space, carrying a charge $q$. Fortunately this is one of the cases where the classical analytical solution is easy. The electrostatic potential at range $r$ is: $\large \phi = \frac{q}{4\pi \varepsilon r}$ where $\varepsilon$ is the permittivity, in space $\varepsilon_0$. Now if If $q$ varies sinusoidally with angular frequency $\omega = (2\pi f)$ then: $\large \phi = \frac{q_0 \sin \omega \left( t - r/c \right)}{4\pi \varepsilon r}$ where $r/c$ is the time the field takes to reach $r$ while traveling at $c$ meters each second. This expression is known as the retarded potential, retarded because of the replacement of $t$ by $(t - r/c)$. The field $\mathbf{E}$ at $r$ is obtained from: $\large \mathbf{E} = -\nabla \phi$ Since the system has complete spherical symmetry, $\phi$ can vary only with $r$. So: $\large \mathbf{E} = \frac{\partial \phi}{\partial r} \cdot \frac{\mathbf{r}}{|\mathbf{r}|}$ where $\mathbf{r}/\mathbf{|r|}$ is a unit vector in direction $r$, indicating the direction of $\mathbf{E}$ (radial). Differentiating the above expression for $\phi$ with respect to $r$, using the formula for a product, two terms will be obtained. So we can write: $\large |\mathbf{E}| = \mathbf{E_r} = E_{\text{near}} + E_{\text{far}}$ Where: $\large E_{\text{near}} = \frac{1}{4\pi \varepsilon r^2} q_0 \sin \omega \left( t - \frac{r}{c} \right)$ And: $\large E_{\text{far}} = \frac{-1}{4\pi \varepsilon r} \left( \frac{\omega}{c} \right) q_0 \cos \omega \left( t - \frac{r}{c} \right)$ To sum up, for this simple case of an alternating point charge we conclude: 1. That the field is radial only, and therefore the same in all angular directions from the antenna. 2. That there are two field components added together, the near field $E_{near}$ and the far field $E_{far}$, where the near field varies as $1/r^2$ and the far field as $1/r$. At the same time, the far field is smaller near the sphere because of the $\omega/c$ term. Thus, there will exist a critical value of the range such that for shorter ranges the near field will predominate, whilst at longer ranges the far field will be the larger, hence the names. The critical range will correspond to $c/\omega$ or $\lambda/2\pi$. This dimension, the near-far transition radius, has the greatest possible significance in antenna theory, as we shall see. What is quite clear is that at a few wavelengths from the source, the near field becomes quite negligible compared with the far field. Although we have already found that an antenna which radiated equally in all directions would indeed be a useful thing, there are two major snags in trying to realize it this way. The first is simply that practically it seems impossible to build. It is easy enough to suspend a small sphere in space, but to vary the charge on it would require attaching a wire, and the charge flowing to and fro through the wire constitutes a current which would completely alter the solution of Maxwell's equations. It would certainly not be an isotropic radiator. The second snag is worse: as we have already seen, the power flow $\mathbf{P}$ in an electromagnetic wave is at right angles to both the $\mathbf{E}$ and $\mathbf{H}$ vectors. But in this case the $\mathbf{E}$ vector is radial, so the power flow therefore cannot be. Although the system is perfectly symmetrical and has a field, it does not launch electromagnetic energy in the way we require. In fact it is not hard to show that a truly isotropic radiating element, with radio energy flowing out from it only radially and equally in all directions, is not possible. But if a truly isotropic antenna is physically unfeasible why bother with it at all? Even if it is impossible to build one, we can still use it as a kind of bench mark with which to compare other, more complicated, antennas we will discuss. ### The Isotrope as a Receiving Antenna Antennas can both transmit, emitting radio energy when driven by electrical energy, or they can equally well receive, capturing radio quanta as they approach the antenna and converting their energy into electrical power which is then available to pass, through a feeder, to radio receiving equipment. If we consider a beam of radio energy falls on an isotropic antenna, how many quanta will be captured and how many will pass fight by? Each antenna is characterized by an aperture, or capture area, centered on the antenna structure. If the quanta pass within this aperture they are captured, outside it they pass by. In the case of the isotrope this boundary corresponds to the edge of the area where the near field is predominant; outside this quanta can move away freely. Near field is an induction effect, and, from a quantum physics perspective, results from the emission and almost immediate recapture of radio photons. Any radio quanta that stray within this area are very likely to be captured, whereas outside it the probability of capture falls off sharply. The radius at which the near field falls below the far field has already been shown above to be $c/\omega$, or $\lambda/(2\pi)$. Photons that stray within this radius will be quickly reabsorbed, whilst those outside it have much less chance of being captured. It is not a sharp boundary; a few photons will be captured from further out while a few from nearer in will escape. These two effects cancel, however, and on average it is as if all the quanta are captured within $\lambda/2\pi$ and all those beyond escape. This, therefore, is the radius of a circle corresponding to the aperture $A_i$ of the isotropic antenna when receiving. Hence: $\large A_i = \pi \left( \frac{\lambda}{2\pi} \right)^2 = \frac{\lambda^2}{4\pi}$ is a very important result, because the aperture of any 'real' antenna can be compared with this basic value for the theoretical isotrope in order to obtain a measure of its performance. Combining it with the expression already derived for received power gives the power received by an isotropic antenna as $P_r$ where: $\large P_r = \frac{P_T}{4\pi r^2} \cdot \frac{\lambda^2}{4\pi}$ Real antennas do better (and often very much better) than an isotrope, as we shall see. Nevertheless it is a useful standard for comparison. ### Radiation Mechanism One of the first questions that may be asked concerning antennas would be “how is radiation accomplished?” In other words, how are the electromagnetic fields generated by the source, contained and guided within the [[Physical Layer#Transmission Lines and Characteristic Impedance of Interconnects|transmission line]] and antenna, and finally “detached” from the antenna to form a free-space wavefront? The best explanation may be given by an illustration. However, let us first examine some basic sources of radiation. #### Single Wire Conducting wires are material whose prominent characteristic is the motion of electric charges and the creation of current. Let us assume that an electric volume charge density, represented by $q_v$ (in Coulomb/m3), is distributed uniformly in a circular wire of cross-sectional area $A$ and volume $V$, as shown in the figure below. The total charge $Q$ within volume $V$ is moving in the $z$ direction with a uniform velocity $v_z$ (in meters/sec). It can be shown that the current density $J_z$ (in amperes/m2) over the cross section of the wire is given by: $\LARGE J_z = q_vv_z$ ![[Pasted image 20241231172517.png]] >[!Figure] > _Charge uniformly distributed in a circular cross section cylinder wire._ (source: #ref/Balanis ) If the wire is made of an ideal electric conductor, the current density $J_s$ (amperes/m) resides on the surface of the wire and it is given by: $\LARGE J_s = q_sv_z$ where $q_s$ (coulombs/m2) is the surface charge density. If the wire is very thin (ideally zero radius), then the current in the wire can be represented by: $\LARGE I_z = q_lv_z$ where $q_l$ (coulombs/m) is the charge per unit length. Instead of examining all three current densities, we will primarily concentrate on the very thin wire. The conclusions apply to all three. If the current is time varying, then the derivative of the current of the equation above can be written as: $\large \frac{dI_z}{dt} = q_l\frac{dv_z}{dt} = q_la_z$ Where $dv_z/dt= a_z$ (meters/sec2) is the acceleration. If the wire is of length $l$, then the previous equation can be written as: $\large l\frac{dI_z}{dt} = lq_l\frac{dv_z}{dt} = lq_la_z$ The equation above is the basic relation between current and charge, and it also serves as the fundamental relation of electromagnetic radiation. It simply states that to create radiation, there must be a time-varying current or an acceleration (or deceleration) of charge. We usually refer to currents in time-harmonic applications while charge is most often mentioned in transients. To create charge acceleration (or deceleration) the wire must be curved, bent, discontinuous, or terminated. Periodic charge acceleration (or deceleration) or time-varying current is also created when charge is oscillating in a time-harmonic motion. Therefore: 1. If a charge is not moving, current is not created and there is no radiation. 2. If charge is moving with a uniform velocity: a. There is no radiation if the wire is straight, and infinite in extent. b. There is radiation if the wire is curved, bent, discontinuous, terminated, or truncated, as shown in the figure below. 3. If charge is oscillating in a time-motion, it radiates even if the wire is straight ![[Pasted image 20241231175744.png]] > [!Figure] > _Wire configuration for radiation_ (source: #ref/Balanis ) A qualitative understanding of the radiation mechanism may be obtained by considering a pulse source attached to an open-ended conducting wire, which may be connected to the ground through a discrete load at its open end, as shown in the figure above (d). When the wire is initially energized, the charges (free electrons) in the wire are set in motion by the electrical lines of force created by the source. When charges are accelerated in the source-end of the wire and decelerated (negative acceleration with respect to original motion) during [[Physical Layer#Reflections|reflection]] from its end, it is suggested that radiated fields are produced at each end and along the remaining part of the wire. Stronger radiation with a more broad frequency spectrum occurs if the pulses are of shorter or more compact duration while continuous time-harmonic oscillating charge produces, ideally, radiation of single frequency determined by the frequency of oscillation. The acceleration of the charges is accomplished by the external source in which forces set the charges in motion and produce the associated field radiated. The deceleration of the charges at the end of the wire is accomplished by the internal (self) forces associated with the induced field due to the buildup of charge concentration at the ends of the wire. The internal forces receive energy from the charge buildup as its velocity is reduced to zero at the ends of the wire. ==Therefore, charge acceleration due to an exciting electric field and deceleration due to impedance discontinuities or smooth curves of the wire are mechanisms responsible for electromagnetic radiation.== While both current density ($J_c$) and charge density ($q_v$) appear as source terms Maxwell’s equation, charge is viewed as a more fundamental quantity, especially for transient fields. Even though this interpretation of radiation is primarily used for transients, it can be used to explain steady state radiation. #### Two Wires Let's consider a voltage source connected to a two-conductor in transmission line which is connected to an antenna. This is shown in the figure below, part (a). Applying a voltage across the two-conductor transmission line creates an electric field between the conductors. The electric field has associated with it electric lines of force which are tangent to the electric field at each point and their strength is proportional to the electric field intensity. The electric lines of force have a tendency to act on the free electrons (easily detachable from the atoms) associated with each conductor and force them to be displaced. The movement of the charges creates a current that in turn creates a magnetic field intensity. Associated with the magnetic field intensity are magnetic lines of force which are tangent to the magnetic field. ![[Pasted image 20241231181300.png]] > [!Figure] > _Source, transmission line, antenna, and detachment of electric field lines._ (source: #ref/Balanis ) We have accepted that electric field lines start on positive charges and end on negative charges. They also can start on a positive charge and end at infinity, start at infinity and end on a negative charge, or form closed loops neither starting or ending on any charge. Magnetic field lines always form closed loops encircling current-carrying conductors because, as per [[Physical Layer#Interconnects#Maxwell's Equations|Maxwell's equations]], physically there are no magnetic charges. The electric field lines drawn between the two conductors help to exhibit the distribution of charge. If we assume that the voltage source is sinusoidal, we expect the electric field between the conductors to also be sinusoidal with a period equal to that of the applied source. The relative magnitude of the electric field intensity is indicated by the density (bunching) of the lines of force with the arrows showing the relative direction (positive or negative). As we saw when we discussed [[Physical Layer#Interconnects|interconnects]], the creation of time-varying electric and magnetic fields between the conductors forms electromagnetic waves which travel along the transmission line, as shown in the figure above, part (a). The electromagnetic waves enter the antenna and have associated with them electric charges and corresponding currents. If we remove part of the antenna structure, as shown in the art (b) of the figure above, free-space waves can be formed by “connecting” the open ends of the electric lines (shown dashed). The free-space waves are also periodic but a constant phase point $P_0$ moves outwardly with the speed of light and travels a distance of $λ∕2$ (to $P_1$) in the time of one-half of a period. It has been shown that close to the antenna the constant phase point in $P_0$ moves faster than the speed of light but approaches the speed of light at points far away from the antenna (analogous to phase velocity inside a rectangular waveguide). The question still unanswered is how the guided waves are detached from the antenna to create the free-space waves that are indicated in the figures as closed loops. Before we attempt to explain that, let us draw again a parallel between the guided and free-space waves, and water waves created by the dropping of a stone in a calm body of water or initiated in some other manner. Once the disturbance in the water has been initiated, water waves are created which begin to travel outwardly. If the disturbance has been removed, the waves do not stop or extinguish themselves but continue their course of travel. If the disturbance persists, new waves are continuously created which lag in their travel behind the others. The same is true with the electromagnetic waves created by an electric disturbance. If the initial electric disturbance by the source is of a short duration, the created electromagnetic waves travel inside the transmission line, then into the antenna, and finally are radiated as free-space waves, even if the electric source has ceased to exist (as was with the water waves and their generating disturbance). If the electric disturbance is of a continuous nature, electromagnetic waves exist continuously and follow in their travel behind the others. When the electromagnetic waves are within the transmission line and antenna, their existence is associated with the presence of the charges inside the conductors. However, when the waves are radiated, they form closed loops and there are no charges to sustain their existence. This leads us to conclude that electric charges are required to excite the fields but are not needed to sustain them and may exist in their absence. This is in direct analogy with water waves. #### The Dipole Now let us attempt to explain the mechanism by which the electric lines of force are detached from the antenna to form the free-space waves. This will be illustrated by an example of a small dipole antenna where the time of travel is negligible. This is only necessary to give a better physical interpretation of the detachment of the lines of force. Although a somewhat simplified mechanism, it does allow one to visualize the creation of the free-space waves. Figure below (a) displays the lines of force created between the arms of a small center-fed dipole in the first quarter of the period during which time the charge has reached its maximum value (assuming a sinusoidal time variation) and the lines have traveled outwardly a radial distance $λ∕4$. For this example, let us assume that the number of lines formed are three. During the next quarter of the period, the original three lines travel an additional $λ∕4$ (a total of $λ∕2$ from the initial point) and the charge density on the conductors begins to diminish. This can be thought of as being accomplished by introducing opposite charges which at the end of the first half of the period have neutralized the charges on the conductors. The lines of force created by the opposite charges are three and travel a distance $λ∕4$ during the second quarter of the first half, and they are shown dashed in the figure legend (b). The end result is that there are three lines of force pointed upward in the first $λ∕4$ distance and the same number of lines directed downward in the second λ∕4. Since there is no net charge on the antenna, then the lines of force must have been forced to detach themselves from the conductors and to unite together to form closed loops. This is shown in part (c) of the figure. In the remaining second half of the period, the same procedure is followed but in the opposite direction. After that, the process is repeated and continues indefinitely and electric field patterns are formed. ![[Pasted image 20241231232442.png]] > [!Figure] > _Formation and detachment of electric field lines for short dipole._ (source: #ref/Balanis ) In order to illustrate the creation of the current distribution on a linear dipole, and its subsequent radiation, let us first begin with the geometry of a lossless two-wire transmission line, as shown in the figure below, legend (a). The movement of the charges creates a traveling wave current, of magnitude $I_0∕2$, along each of the wires. When the current arrives at the end of each of the wires, it undergoes a complete reflection (equal magnitude and 180 degrees phase reversal). The reflected traveling wave, when combined with the incident traveling wave, forms in each wire a pure standing wave pattern of sinusoidal form as shown in figure below, legend (a). The current in each wire undergoes a 180-degree phase reversal between adjoining half-cycles. This is indicated in the figure by the reversal of the arrow direction. Radiation from each wire individually occurs because of the time-varying nature of the current and the termination of the wire. ![[Pasted image 20241231233128.png]] > [!Figure] > _Current distribution on a lossless two-wire transmission line, flared transmission line, and linear dipole._ (source: #ref/Balanis ) For the two-wire balanced (symmetrical) transmission line, the current in a half-cycle of one wire is of the same magnitude but 180-degree out-of-phase from that in the corresponding half-cycle of the other wire. If in addition the spacing between the two wires is very small ($s ≪ λ$), the fields radiated by the current of each wire are essentially cancelled by those of the other. The net result is an almost ideal (and desired) non-radiating transmission line. This is what we discussed when we discussed [[Physical Layer#Differential Interconnects|differential interconnects]]. > [!tip] Note how during the discussion of [[Physical Layer#Interconnects|interconnects]], radiation was an effect to avoid at all cost, whereas when discussing antennas, radiation is what's desired. This is the nature of engineering: an an effect can be undesired under certain circumstances or very much desired, depending on the design requirements! > [!attention] The dipole, in addition to being an antenna, is also a resonator. When fed at its resonant frequency, in addition to radiating, it stores energy in oscillating near-field electric and magnetic fields around the antenna, created by a wave of voltage and current bouncing back and forth between the ends of the antenna. The oppositely directed waves interfere to form a standing wave of voltage and current on the antenna. As the section of the transmission line between $0 ≤ z ≤ l∕2$ begins to flare, as shown in part (b) of the figure above, it can be assumed that the current distribution is essentially unaltered in form in each of the wires. However, because the two wires of the flared section are not necessarily close to each other, the fields radiated by one do not necessarily cancel those of the other. Therefore, ideally, there is a net radiation by the transmission-line system. Ultimately the flared section of the transmission line can take the form shown in part (c) of the figure. This is the geometry of the widely used dipole antenna. Because of the standing wave current pattern, it is also classified as a standing wave antenna (as contrasted to traveling wave antennas). If $l < λ$, the phase of the current standing wave pattern in each arm is the same throughout its length. In addition, spatially it is oriented in the same direction as that of the other arm as shown in part (c) of the figure. Thus the fields radiated by the two arms of the dipole (vertical parts of a flared transmission line) will primarily reinforce each other toward most directions of observation (the phase due to the relative position of each small part of each arm must also be included for a complete description of the radiation pattern formation). If the diameter of each wire is very small ($d ≪ λ$), the ideal standing wave pattern of the current along the arms of the dipole is sinusoidal with a null at the end. However, its overall form depends on the length of each arm. For center-fed dipoles with $l ≪ λ$, $l = λ∕2$, $λ∕2 < l < λ$ and $λ < l < 3λ∕2$, the current patterns are illustrated in figure below (a–d). The current pattern of a very small dipole (usually one $λ∕50 < l ≤ λ∕10$) can be approximated by a triangular distribution since $sin(kl∕2) ≃ kl∕2$ when $kl/2$ is very small. This is illustrated in figure below, part (a). ![[Pasted image 20241231233700.png]] > [!Figure] > _Current distribution on linear dipoles_ (source: #ref/Balanis ) To derive the radiation pattern of a dipole antenna, we begin with the fundamental concepts of electromagnetic radiation from an oscillating current and apply solutions of Maxwell's equations in free space. ##### Current Distribution in a Dipole Consider a thin, linear dipole antenna of length $L$ with an oscillating current: $\large I(z,t)= I_0 \cos\left(\omega t - k z\right)$ where $I_0$ is the maximum current amplitude, $\omega = 2 \pi f$ is the angular frequency, $k = \frac{2 \pi}{\lambda}$ is the wavenumber, and $z$ is the position along the dipole. For simplicity, we consider the case where the current is sinusoidal and symmetric about the center of the dipole. ##### Vector Potential from a Dipole To find the field is necessary to solve Maxwell's equations, which is by no means easy. A short cut is to use the retarded vector potential $\mathbf{A}$ which is related to the current in the element by deriving $\mathbf{A}$ from the oscillating current: $\large \mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}}{r} e^{-j \omega (t - r/c)} dV$ where $\mathbf{J}$ is the current density, $r$ is the distance from the source to the observation point, and $c$ is the speed of light. For a linear dipole along the $z$-axis, the vector potential has only a $\hat{z}$-component: $\large A_z(\mathbf{r}) = \frac{\mu_0}{4 \pi} \int_{-L/2}^{L/2} \frac{I(z') e^{-j k r}}{r} dz'$ where $z'$ is the source coordinate along the dipole. In the far field ($r≫L$), we approximate $r$ as constant over the length of the dipole, except in the phase term where $r$ depends on the angle $\theta$ (the angle between the dipole axis and the observation direction). Thus, the phase factor becomes $r \approx r - z' \cos\theta$. ##### Far-Field Approximation Substituting the current $I(z')$ and integrating, the $A_z$-component of the vector potential in the far field becomes: $\large A_z(r, \theta) = \frac{\mu_0 I_0}{4 \pi r} \int_{-L/2}^{L/2} e^{j k z' \cos\theta} dz'$ Performing the integration yields: $\large A_z(r, \theta) = \frac{\mu_0 I_0}{4 \pi r} \frac{\sin\left(k \frac{L}{2} \cos\theta\right)}{\cos\theta}$ ##### Electric and Magnetic Fields The electric field $\mathbf{E}$ in the far field is related to the vector potential $\mathbf{A}$. For a dipole, the dominant component of $\mathbf{E}$ is in the $\hat{\theta}$-direction: $\large E_\theta = -j \omega A_z(r, \theta)$ Substituting $A_z$, we get: $\large E_\theta(r, \theta) = -j \frac{\omega \mu_0 I_0}{4 \pi r} \frac{\sin\left(k \frac{L}{2} \cos\theta\right)}{\cos\theta}$ The magnetic field $\mathbf{H}$ is orthogonal to $\mathbf{E}$ and propagates in the $\hat{\phi}$-direction. ##### Radiation Intensity and Pattern The radiation intensity $U(\theta)$, which represents the power radiated per unit solid angle, is proportional to the square of the electric field magnitude: $\large U(\theta) \propto |E_\theta(r, \theta)|^2 \propto \left(\frac{\sin\left(k \frac{L}{2} \cos\theta\right)}{\cos\theta}\right)^2$ For a short dipole ($L \ll \lambda$), $kL \ll 1$, and the approximation $\sin(x) \approx x$ simplifies the radiation pattern to: $\large U(\theta) \propto \sin^2\theta$ This shows the radiation is strongest perpendicular to the dipole ($\theta = \pi/2$) and zero along its axis ($\theta = 0, \pi$). The normalized radiation pattern for a short dipole is: $\large F(\theta) = \sin\theta$ where $F(\theta)$ is the relative amplitude of the radiation as a function of angle. This corresponds to a toroidal pattern, with no radiation along the dipole axis and maximum radiation in the plane perpendicular to it. ![[Pasted image 20250101232415.png]] > [!Figure] > Dipole toroidal radiation pattern (source: www.antennatheory.com) Since, for the same power input to the antenna, the total number of radio quanta emitted would be the same for a dipole and an isotropic radiator, clearly the density of photons emitted at low angles by the dipole is going to be greater than that for the isotrope; a remote receiving antenna in this direction will therefore catch more of them, and will therefore receive more signal power. This leads to the idea of an antenna having *gain*. The name is a little misleading; no extra power magically appears from somewhere, and the antenna transmits exactly the same total power in both cases, it is just that the dipole concentrates the flow of radio energy in directions at right angles to its axis, so the power density there is enhanced, and by definition the isotrope does not. The gain of an antenna is normally specified relative to a hypothetical isotropic antenna. Because Maxwell's equations work in the same way whether the antenna is transmitting or receiving, the same gain is also realized in the receiving mode, and more quanta approaching the antenna get converted into electrical energy in the feeder. Physically what is happening is that the dipole can capture radio quanta over a larger area (provided that they are coming from the $\theta = 0$ direction). This leads to an alternative way of looking at antenna gain: it can be regarded as equivalent to an increase in capture area (or aperture). ##### Polarization Electromagnetic radiation is always polarized, depending on the direction of the electric and magnetic field component of their associated wave functions. Once this polarization is established, at the time they are emitted by an antenna, they carry it unchanged until they are absorbed. A vertical dipole will give rise to an electric field vector which is vertical, while the magnetic vectors (which curl round the conductor) constitute magnetic horizontal field vectors. As a matter of convention, the emitted waves are said to be vertically polarized when the electric vector is vertical. Similarly a horizontal dipole will produce a horizontal electric vector, corresponding to horizontal polarization. Obviously the radiation may be polarized at any intermediate angle. Since a vertical dipole can respond only to a vertical electric field component when receiving, in principle it will capture only vertically polarized radio quanta, and similarly a horizontal dipole will receive only horizontally polarized quanta. In practice the separation between these two directions of polarization is not as absolute as this makes it seem; due to imperfections in their construction few antennas really have no response to the orthogonal polarization, while many quanta are absorbed by metal objects in the environment, which may then re-radiate with quite different polarization, throwing energy into the unwanted response. ## Antenna Arrays Dipoles, short or long, are very useful antennas, but they are limited in the range of polar diagrams that they can have, amounting to nothing beyond a more-or-less flattened 'ring doughnut' shape as we saw before. Because antennas achieve gain by concentrating their emission of radio energy towards particular directions, this limited range of possible polar diagrams means that with simple dipoles, short or long, nothing very remarkable can be achieved in the way of antenna gain either. To escape these limitations it is necessary to break out from the constraints imposed by a single dipole, and the most common way of doing this is by using antenna arrays. The simplest antenna array consists of two half-wave dipoles, shown in the figure below as vertical and side by side. This would be called a two element array. We calculate the polar diagram in the horizontal plane assuming that we are concerned only with far-field radiation and that the distance $d$ between the dipoles is small compared with the distance $r$ of points like $P$ where the resulting signal is of interest. ![[Pasted image 20250106233617.png]] > [!Figure] > Two dipoles as an array. (source: #ref/Gosling ) We can assume that both dipoles are driven from the same transmitter but that the sinusoidal voltage applied to dipole B is phase shifted by an angle $\Psi$. The field at $P$ is: $\large \begin{align*} E_P &= E_A + E_B \\ &= k e_T \left\{ \cos \omega t + \cos \left[ \omega \left( t + \frac{d \cos \phi}{c} \right) + \Psi \right] \right\} \\ &= k e_T \left\{ \cos \omega t + \cos \left[ \omega t + \left( \frac{2\pi d \cos \phi}{\lambda} + \Psi \right) \right] \right\} \end{align*}$ where $k$ is a constant (depending on $r$), and $e_T$ is the voltage applied to each antenna. Using the identity: $\cos \alpha + \cos \beta \equiv 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cdot \cos \left( \frac{\alpha - \beta}{2} \right)$ Then, $ E_P = 2k e_T \cos \left( \frac{\pi d \cos \phi}{\lambda} + \frac{\Psi}{2} \right) \cdot \cos \left( \omega t + \frac{\pi d \cos \phi}{\lambda} + \frac{\Psi}{2} \right)$ The second cosine term is just a phase-shifted sinusoidal wave function, while the first is an angle-dependent amplitude variation, which determines the polar diagram. If the power density at the point $P$ is $p$, this consequently varies with the angle $\phi$ as: $\large p \propto \cos^2 \left( \frac{\pi d \cos \phi}{\lambda} + \frac{\Psi}{2} \right)$ By choosing suitable values of $d$ and $\Psi$ a variety of different polar diagrams can be synthesized. Two are of particular interest. Case 1: $\Psi=0$, $d=\lambda/2$, in which case: $\large p \propto \cos^2 (2\pi \cdot \cos \phi)$ The right-hand side is 0 at $\phi=0$ and $180^\circ$ ($\pi$ radians), and right hand is 1 at $90^\circ$ ($\pi/2$ radians) and $270^\circ$ ($3\pi/2$ radians). This is therefore a broadside antenna array. An antenna like this can be useful to cover a long narrow area (for example, mounted on a bridge over a highway it will give good coverage of the road in both directions). ![[Pasted image 20250107134038.png]] > [!Figure] > _Polar diagram of a broadside array._ (source: #ref/Gosling ) Case 2: $\Psi=90^\circ$ ($-\pi/2$), and $d=\lambda/4$ The required phase shift could be produced by an additional quarter wavelength of feeder cable, or by a suitable LC network. Now in this case: $\large p \propto \cos^2 \left[ \frac{\pi}{4} (\cos \phi - 1) \right]$ In this case the right-hand side is 1 for $\phi = 0$ and falls steadily to zero at $180^\circ$ ($\pi$ radians). The polar diagram is a cardioid, a heart shape, and the array has end-fire characteristics, as depicted in the figure below. ![[Pasted image 20250107134821.png]] > [!Figure] > Polar diagram of an end-fire array. (source: #ref/Gosling ) Since the field vectors from the two dipoles simply add in the direction of maximum propagation, the received power in that direction (which is proportional to the square of the field) is increased by a factor of four (+6 dB), but since the power is shared equally between the two dipoles each receives half-power (-3 dB). Thus the overall power gain is +3 dB relative to a half-wave dipole, and thus (+3 + 2.2) or 5.2 dB relative to isotropic. Its aperture at = 0 is evidently twice that of a half-wave dipole, or $0.260\lambda^2$. The obvious application for this antenna array is to give a good transmission (or reception) path in the direction of its maximum, and it is therefore suited to point-point transmission or broadcast reception. Since in both these cases the dipoles are vertical and side by side, in the $\theta$-plane the polar diagram is not affected for $\phi=\pm90^\circ$, remaining the usual 'flattened ring doughnut', although it is flattened a little more for other $\phi$ directions. In three dimensions, the polar diagram is evidently quite a complicated shape. If the two dipoles had been horizontal, nothing would have altered except that the $\phi$ and $\theta$ polar diagrams are interchanged. > [!warning] > This section is under #development