thumb|UHF half-wave dipole
thumb|upright|Dipole antenna used by the [[radar altimeter in an airplane]]
thumb|upright=1.5|Animated diagram of a [[half-wave dipole antenna receiving a radio wave. The antenna consists of two metal rods connected to a receiver . The electric field (<span style="color:green;">, green arrows</span>) of the incoming wave pushes the electrons in the rods back and forth, charging the ends alternately positive <span style="color:red;">(+)</span> and negative <span style="color:blue;">(−)</span>. Since the length of the antenna is one half the wavelength of the wave, the oscillating field induces standing waves of voltage (<span style="color:red;">, represented by the red band</span>) and current in the rods. The oscillating currents (<nowiki/>black arrows) flow down the transmission line and through the receiver (represented by the resistance ).]]
In radio and telecommunications, a dipole antenna or doublet
is one of the two simplest and most widely used types of antenna; the other is the monopole. The dipole is any one of a class of antennas producing a radiation pattern approximating that of an elementary electric dipole with a radiating structure supporting a line current so energized that the current has only one node at each far end. A dipole antenna commonly consists of two identical conductive elements such as metal wires or rods.
Dipoles are frequently used as resonant antennas. If the feedpoint of such an antenna is shorted, then it will be able to resonate at a particular frequency, just like a guitar string that is plucked. Using the antenna at around that frequency is advantageous in terms of feedpoint impedance (and thus standing wave ratio), so its length is determined by the intended wavelength (or frequency) of operation.
Although they may be used as standalone low-gain antennas, dipoles are also employed as driven elements in more complex antenna designs
For the low frequencies Marconi employed to achieve long-distance communications, this form was more practical; when radio moved to higher frequencies (especially VHF transmissions for FM radio and TV) it was advantageous for these much smaller antennas to be entirely atop a tower thus requiring a dipole antenna or one of its variations.
In the early days of radio, the thus-named Marconi antenna (monopole) and the doublet (dipole) were seen as distinct inventions. Now, however, the monopole antenna is understood as a special case of a dipole which has a virtual element underground.
Dipole variations
Short dipole
A short dipole is a dipole formed by two conductors with a total length substantially less than a half wavelength Short dipoles are sometimes used in applications where a full half-wave dipole would be too large. They can be analyzed easily using the results obtained below for the Hertzian dipole, a fictitious entity. Being shorter than a resonant antenna (half a wavelength long) its feedpoint impedance includes a large capacitive reactance requiring a loading coil or other matching network in order to be practical, especially as a transmitting antenna.
To find the far-field electric and magnetic fields generated by a short dipole we use the result shown below for the Hertzian dipole (an infinitesimal current element) at a distance from the current and at an angle to the direction of the current, as being:
:<math>\begin{align}
H_\mathrm{\phi} \quad &= \quad j\ \frac{\ I_\mathrm{h}\ \ell\ k\ }{4\pi\ r}\ e^{\ j\ \left( \omega t\ -\ k r \right)}\ \sin(\ \theta\ ) \\
E_\mathrm{\theta} \quad &= \quad \zeta_\mathrm{o}\ H_\mathrm{\phi} \quad = \quad j\ \frac{\ \zeta_\mathrm{o}\ I_\mathrm{h}\ \ell\ k\ }{4\pi\ r}\ e^{\ j\ \left( \omega t\ -\ k r \right)}\ \sin(\ \theta\ ) ~.
\end{align}</math>
where the radiator consists of a current of <math>\ I_\mathrm{h}\ e^{\ j \omega t}\ </math> over a short length and <math>\ j^2 \equiv -1\ </math> in electronics replaces the customary mathematical symbol for the square root of . is the angular (radian) frequency and is the wavenumber is the impedance of free space which is the ratio of a free space plane wave's electric to magnetic field strength.
left|Diagram of a short dipole antenna
The feedpoint is usually at the center of the dipole as shown in the diagram. The current along dipole arms are approximately described as proportional to <math>\ \sin(\ k\ z\ )\ </math> where is the distance to the nearest end of the arm. In the case of a short dipole, that is essentially a linear drop from <math>\ I_\mathrm{o}\ </math> at the feedpoint to zero at the end. Therefore, this is comparable to a Hertzian dipole with an effective current <sub>h</sub> equal to the average current over the conductor, so <math display="inline">\ I_\mathrm{h} = \tfrac{\ 1\ }{ 2 }\ I_\mathrm{o} ~.</math> With that substitution, the above equations closely approximate the fields generated by a short dipole fed by current <math>\ I_\mathrm{o} ~.</math>
From the fields calculated above, one can find the radiated flux (power per unit area) at any point as the magnitude of the real part of the Poynting vector, , which is given by <math display="inline">\ \tfrac{\ 1\ }{ 2 } \mathbf{E} \times \mathbf{H}^* ~.</math> Because and are at right angles and in phase, there is no imaginary part and the cross product is equal to <math display="inline">\ \tfrac{\ 1\ }{ 2 }\ E_\mathrm{\theta}\ H_\mathrm{\phi}^*\ ;</math> the phase factors (the exponentials) cancel out, leaving:
:<math>\begin{align}
S &= \frac{\ 1\ }{ 2 }\ E_\mathrm{\theta}\ H_\mathrm{\phi}^*
&= \frac{\ 1\ }{ 2 }\ \frac{\ \zeta_\mathrm{o}\ I_\mathrm{h}^2\ \ell^2\ k^2\ }{(4\pi r)^2}\ \sin^2\!(\ \theta\ )
&= \frac{\ \zeta_\mathrm{o}\ }{ 32 }\ I_\mathrm{o}^2\ \left(\frac{\ \ell\ }{\lambda}\right)^2\ \frac{ 1 }{\ r^2 }\ \sin^2\!(\ \theta\ ) ~.
\end{align}</math>
We have now expressed the flux in terms of the feedpoint current and the ratio of the short dipole's length to the wavelength of radiation . The radiation pattern given by <math>\ \sin^2\!(\ \theta\ )\ </math> is seen to be similar to and only slightly less directional than that of the half-wave dipole.
right|thumb|Radiation pattern of the short dipole (dashed line) compared to the half-wave dipole (solid line)
Using the above expression for the radiation in the far field for a given feedpoint current, we can integrate over all solid angle to obtain the total radiated power.
:<math>P_\text{total} = \frac{\ \pi\ }{ 12 }\ \zeta_\mathrm{o}\ I_\mathrm{o}^2\ \left( \frac{\ell}{\lambda} \right)^2 ~.</math>
From that, it is possible to infer the radiation resistance, equal to the resistive (real) part of the feedpoint impedance, neglecting a component due to ohmic losses (presumed smaller). By setting to the power supplied at the feedpoint <math display="inline">\ \tfrac{\ 1\ }{ 2 }\ I_\mathrm{o}^2\ R_\mathrm{radiation}\ </math> we find:
:<math>\ R_\mathrm{radiation} = \frac{\ \pi\ }{6}\ \zeta_\mathrm{o}\ \left(\frac{\ell}{\lambda}\right)^2 \approx \left(\frac{\ell}{\lambda}\right)^2 (197\ \Omega) ~.</math>
Again, these approximations become quite accurate for Setting despite its use not quite being valid for so large a fraction of the wavelength, the formula would predict a radiation resistance of 49 Ω, instead of the actual value of 73 Ω produced by a half-wave dipole, when more correct quarter-wave sinusoidal currents are used.
Dipole antennas of various lengths
The fundamental resonance of a thin linear conductor occurs at a frequency whose free-space wavelength is twice the wire's length; i.e. where the conductor is wavelength long. Dipole antennas are frequently used at around that frequency and thus termed half-wave dipole antennas. This important case is dealt with in the next section.
Thin linear conductors of length <math>\ \ell\ </math> are in fact resonant at any integer multiple of a half-wavelength:
:<math>\ \ell = n \times \frac{\ \lambda\ }{2}\ </math>
where is an integer, <math>\ \lambda = \frac{\ c\ }{ f }\ </math> is the wavelength, and is the reduced speed of radio waves in the radiating conductor ( the speed of light). For a center-fed dipole, however, there is a great dissimilarity between being odd or being even. Dipoles which are an odd number of half-wavelengths in length have reasonably low driving point impedances (which are purely resistive at that resonant frequency). However ones which are an even number of half-wavelengths in length, that is, an integer number of wavelengths in length, have a high driving point impedance (albeit purely resistive at that resonant frequency).
For instance, a full-wave dipole antenna can be made with two half-wavelength conductors placed end to end for a total length of approximately <math>\ \ell \approx \lambda\ .</math> This results in an additional gain over a half-wave dipole of about 2 dB. Full wave dipoles can be used in short wave broadcasting only by making the effective diameter very large and feeding from a high impedance balanced line. Cage dipoles are often used to get the large diameter.
A -wave dipole antenna has a much lower but not purely resistive feedpoint impedance, which requires a matching network to the impedance of the transmission line. Its gain is about 3 dB greater than a half-wave dipole, the highest gain of any dipole of any similar length.
:{| class="wikitable" style="text-align:center;"
|+ Gain of dipole antennas
:<math>I(z) = I_0 \cos(\omega t) \cos(k z)\ ,</math>
where and runs from to .
In the far field, this produces a radiation pattern whose electric field is given by
Instead of altering thickness or spacing, one can add a third parallel wire to increase the feedpoint impedance to 9 times that of a single-wire dipole, raising the impedance to 658 Ω, making a good match for open wire feed cable, and further broadening the resonant frequency band of the antenna. More extra parallel wires can be added: Any number of extra parallel wires can be joined onto the antenna, with the feedpoint impedance given by
: <math>\ R_\mathsf{rad} \approx n^2\ \times\ 73\mathsf{\ \Omega\ ,}</math>
where <math>\ n\ </math> is the number of parallel halfwave-long wires laid side-by-side in the antenna, and connected at their ends. It is also possible to modify the so-called flattened-loop design, and get nearly as good performance, by making each of the parallel wires too short by the same amount, but connecting a single capacitive loading wire (going off in nearly any direction, most often dangling) on each of the antenna ends. The loading wire length is equal to the single missing length of one of the parallel wires.
Other variants
There are numerous modifications to the shape of a dipole antenna which are useful in one way or another but result in similar radiation characteristics (low gain). This is not to mention the many directional antennas which include one or more dipole elements in their design as driven elements, many of which are linked to in the information box at the bottom of this page.
- The bow-tie antenna is a dipole with flaring, triangular shaped arms. The shape gives it a much wider bandwidth than an ordinary dipole. It is widely used in UHF television antennas.
thumb|Cage dipole antennas in the Ukrainian [[Ukrainian T-shaped Radio telescope, second modification|UTR-2 radio telescope. The 8 meters (30 feet) long by 1.8 meters (6 feet) diameter galvanized steel wire dipoles have an operating frequency range of 8–33 MHz.]]
- The cage dipole is a similar modification in which the bandwidth is increased by using fat cylindrical dipole elements made of a cage of wires (see photo). These are used in a few broadband array antennas in the medium wave and shortwave bands for applications such as over-the-horizon radar and radio telescopes.
- A halo antenna is a half-wave dipole bent into a circle for a nearly uniform radiation pattern in the plane of the circle. When the halo's circle is horizontal, it produces horizontally polarized radiation in a nearly omnidirectional pattern with only a little power wasted toward the zenith, compared to a straight horizontal dipole. In practice, it is categorized either as a bent dipole or as a loop antenna, depending on author preference.
- A turnstile antenna comprises two dipoles crossed at a right angle and feed system which introduces a quarter-wave phase difference between the currents along the two. With that geometry, the two dipoles do not interact electrically but their fields add in the far-field producing a net radiation pattern that is rather close to isotropic, with horizontal polarization in the plane of the elements and circular or elliptical polarization at other angles. Turnstile antennas can be stacked and fed in phase to realize an omnidirectional broadside array or phased for an end-fire array with circular polarization.
- The batwing antenna is a turnstile antenna with its linear elements widened as in a bow-tie antenna, again for the purpose of widening its resonant frequency and thus usable over a larger bandwidth, without re-tuning. When stacked to form an array the radiation is omnidirectional, horizontally polarized, and with increased gain at low elevations, making it ideal for television broadcasting.
- A V antenna is a dipole with a bend in the middle so its arms are at an angle instead of co-linear.
- A quadrant antenna is a 'V' antenna with an unusual overall length of a full wavelength, with two half-wave horizontal elements meeting at a right angle where it is fed. Quadrant antennas produce mostly horizontal polarization at low to intermediate elevation angles and have nearly omnidirectional radiation patterns.
One implementation uses cage elements (see above); the thickness of the resulting elements lowers the high driving point impedance of a full-wave dipole to a value that accommodates a reasonable match to open wire lines and increases the bandwidth (in terms of SWR) to a full octave. They are used for HF band transmissions.
- The antenna is a dipole antenna fed indirectly, through a carefully chosen length of 300 Ω or 450 Ω twin lead, which acts as an impedance matching network to connect (through a balun) to a standard 50 Ω coaxial transmission line.
- The sloper antenna is a slanted vertical dipole antenna attached to the top of a single tower. The element can be center-fed or can be end-fed as an unbalanced monopole antenna from a transmission line at the top of the tower, in which case the monopole's ground connection can better be viewed as a second element comprising the tower or transmission line shield.
- The inverted 'V' antenna is likewise supported using a single tower but is a balanced antenna with two symmetric elements angled toward the ground. It is thus a half-wave dipole with a bend in the middle. Like the sloper, this has the practical advantage of elevating the antenna but requiring only a single tower.
- The AS-2259 antenna is an inverted-‘V’ dipole antenna used for local communications via Near Vertical Incidence Skywave (NVIS).
Vertical (monopole) antennas
250px|thumb|right|A monopole antenna and its ground image together form a dipole that radiates only in the upper half of space.
The vertical, Marconi, or monopole antenna is a single-element antenna usually fed at the bottom (with the shield side of its unbalanced transmission line connected to ground). It behaves essentially the same as half of a dipole antenna. The ground (or ground plane) is considered to be a conductive surface that works as a reflector (see effect of ground). Vertical currents in the reflected image have the same direction (thus are not reflected about the ground) and phase as the current in the real antenna.
Dipoles whose length is approximately half the wavelength of the signal are called half-wave dipoles and are widely used as such or as the basis for derivative antenna designs. These have a radiation resistance which is much greater, closer to the characteristic impedances of available transmission lines, and normally much larger than the resistance of the conductors, so that their efficiency approaches 100%. In general radio engineering, the term dipole, if not further qualified, is taken to mean a center-fed half-wave dipole.
thumb|300px|Feedpoint impedance of (near-) half-wave dipoles versus electrical length in wavelengths. Black: [[radiation resistance; blue: reactance for four different values of conductor diameter.]]
A true half-wave dipole is one half of the wavelength in length, where in free space. Such a dipole has a feedpoint impedance consisting of 73 Ω resistance and +43 Ω reactance, thus presenting a slightly inductive reactance. To cancel that reactance, and present a pure resistance to the feedline, the element is shortened by the factor for a net length <math>\ \ell\ </math> of:
:<math>\ \ell = \frac{\ 1\ }{ 2 }\ k\ \lambda = \frac{\ 1\ }{ 2 }\ k\ \frac{\ c\ }{ f }\ </math>
where is the free-space wavelength, is the speed of light in free space, and is the frequency. The adjustment factor which causes feedpoint reactance to be eliminated, depends on the diameter of the conductor,
as is plotted in the accompanying graph. The relative scale-size ranges from about 0.98 for thin wires (diameter, 0.00001 wave) to about 0.94 for thick conductors (diameter, 0.008 wave). This is because the effect of antenna length on reactance (upper graph) is much greater for thinner conductors so that a smaller deviation from the exact half wavelength is required in order to cancel the 43 Ω inductive reactance it has when exactly For the same reason, antennas with thicker conductors have a wider operating bandwidth over which they attain a practical standing wave ratio which is degraded by any remaining reactance.
thumb|300px|Length reduction factor for a half-wave dipole to achieve electrical resonance (purely resistive feedpoint impedance). Calculated using the [[#Induced EMF method|induced EMF method, an approximation that breaks down at larger conductor diameters (dashed portion of graph).]]
For a typical of about 0.95, the above formula for the corrected antenna length can be written, for a length in meters as , or a length in feet as where is the frequency in megahertz.
Dipole antennas of lengths approximately equal to any odd multiple of are also resonant, presenting little or no reactance (which can be removed by making a small length adjustment). However, these are rarely used. One size that is a much more efficient radiator both in terms of Watts out and in direction radiated is a dipole with a length of Not being close to this antenna's impedance has a large (negative) reactance and can only be used with an inductive impedance matching network (a tapped loading coil or a so-called antenna tuner). It is a desirable length because such an antenna has the highest gain for any dipole which isn't a great deal longer.
Radiation pattern and gain
A dipole is omnidirectional in the plane perpendicular to the wire axis, with the radiation falling to zero on the axis (off the ends of the antenna). In a half-wave dipole, the radiation is maximum perpendicular to the antenna, declining as <math>\ (\ \sin \theta\ )^2\ </math> to zero on the axis. Its radiation pattern in three dimensions (see figure) would be plotted approximately as a toroid (doughnut shape) symmetric about the conductor. When mounted vertically this results in maximum radiation in horizontal directions. When mounted horizontally, the radiation peaks at right angles (90°) to the conductor, with nulls in the direction of the dipole.
Neglecting electrical inefficiency, the antenna gain is equal to the directive gain, which is 1.50 (1.76 dBi or -0.39 dBd) for a short dipole, increasing to 1.64 (2.15 dBi or 0 dBd) for a half-wave dipole. For a dipole the gain further increases to about 5.2 dBi, making this length desirable for that reason even though the antenna is then off-resonance. Longer dipoles than that have radiation patterns that are multi-lobed, with poorer gain (unless they are much longer) even along the strongest lobe. Other enhancements to the dipole (such as including a corner reflector or an array of dipoles) can be considered when more substantial directivity is desired. Such antenna designs, although based on the half-wave dipole, generally acquire their own names.
Feeding a dipole antenna
Ideally, a half-wave dipole should be fed using a balanced transmission line matching its typical 65–70 Ω input impedance. Twin lead with a similar impedance is available but seldom used and does not match the balanced antenna terminals of most radio and television receivers. Much more common is the use of common 300 Ω twin lead in conjunction with a folded dipole. The driving point impedance of a half-wave folded dipole is 4 times that of a simple half-wave dipole, thus closely matching that 300 Ω characteristic impedance. Most FM broadcast band tuners and older analog televisions include balanced 300 Ω antenna input terminals. However twin lead has the drawback that it is electrically disturbed by any other nearby conductor (including earth); when used for transmitting, care must be taken not to place it near other conductors.
Many types of coaxial cable (or coax) have a characteristic impedance of 75 Ω, which would otherwise be a good match for a half-wave dipole. However, coax is a single-ended line whereas a center-fed dipole expects a balanced line (such as twin lead). By symmetry, one can see that the dipole's terminals have an equal but opposite voltage, whereas coax has one conductor grounded. Using coax regardless results in an unbalanced line, in which the currents along the two conductors of the transmission line are no longer equal and opposite. Since you then have a net current along the transmission line, the transmission line becomes an antenna itself, with unpredictable results (since it depends on the path of the transmission line). This will generally alter the antenna's intended radiation pattern, and change the impedance seen at the transmitter or receiver.
A balun is required to use coaxial cable with a dipole antenna. The balun transfers power between the single-ended coax and the balanced antenna, sometimes with an additional change in impedance. A balun can be implemented as a transformer which also allows for an impedance transformation. This is usually wound on a ferrite toroidal core. The toroid core material must be suitable for the frequency of use, and in a transmitting antenna it must be of sufficient size to avoid saturation. Other balun designs are mentioned below.
Current balun
A current balun uses a transformer wound on a toroid or rod of magnetic material such as ferrite. All of the current seen at the input goes into one terminal of the balanced antenna. It forms a balun by choking common-mode current. The material isn't critical for 1:1 because there is no transformer action applied to the desired differential current.
A related design involves two transformers and includes a 1:4 impedance transformation.
Another narrow-band design is to use a length of metal pipe. The coaxial cable is placed inside the pipe; at one end the braid is wired to the pipe while at the other end no connection is made to the pipe. The balanced end of this balun is at the end where no connection is made to the pipe. The conductor acts as a transformer, converting the zero impedance at the short to the braid into an infinite impedance at the open end. This infinite impedance at the open end of the pipe prevents current flowing into the outer coax formed by the outside of the inner coax shield and the pipe, forcing the current to remain in the inside coax. This balun design is impractical for low frequencies because of the long length of pipe that will be needed.
Common applications
"Rabbit ears" TV antenna
thumb|200px|"Rabbit-ears" VHF [[television antenna (the small loop is a separate UHF antenna).]]
One of the most common applications of the dipole antenna is the rabbit ears or bunny ears television antenna, found atop broadcast television receivers. It is used to receive the VHF terrestrial television bands, consisting in the US of 54–88 MHz (band I) and 174–216 MHz (band III), with wavelengths of 5.5–1.4 meters (18 feet to 4 feet 8 inches). Since this frequency range is much wider than a single fixed dipole antenna can cover, it is made with several degrees of adjustment. It is constructed of two telescoping rods that can each be extended out to about 1 meters (3 feet) length (one-quarter wavelength at 75 MHz). With control over the segments' length, angle with respect to vertical, and compass angle, one has much more flexibility in optimizing reception than available with a rooftop antenna even if equipped with an antenna rotor.
FM-broadcast-receiving antennas
In contrast to the wide television frequency bands, the FM broadcast band (88-108 MHz) is narrow enough that a dipole antenna can cover it. For fixed use in homes, hi-fi tuners are typically supplied with simple folded dipoles resonant near the center of that band. The feedpoint impedance of a folded dipole, which is quadruple the impedance of a simple dipole, is a good match for 300 Ω twin lead, so that is usually used for the transmission line to the tuner. A common construction is to make the arms of the folded dipole out of twin lead also, shorted at their ends. This flexible antenna can be conveniently taped or nailed to walls, following the contours of moldings.
Shortwave antenna
Horizontal wire dipole antennas are popular for use on the HF shortwave bands, both for transmitting and shortwave listening. They are usually constructed of two lengths of wire joined by a strain insulator in the center, which is the feedpoint. The ends can be attached to existing buildings, structures, or trees, taking advantage of their heights. If used for transmitting, it is essential that the ends of the antenna be attached to supports through strain insulators with a sufficiently high flashover voltage, since the antenna's high-voltage antinodes occur there. Being a balanced antenna, they are best fed with a balun between the (coax) transmission line and the feedpoint.
These are simple to put up for temporary or field use. But they are also widely used by radio amateurs and short wave listeners in fixed locations due to their simple (and inexpensive) construction, while still realizing a resonant antenna at frequencies where resonant antenna elements need to be of quite some size. They are an attractive solution for these frequencies, when significant directionality is not desired, and the cost of several such resonant antennas for different frequency bands, built at home, may still be much less than a single commercially produced antenna.
Dipole towers
Antennas for MF and LF radio stations are usually constructed as mast radiators, in which the vertical mast itself forms the antenna. Although mast radiators are most commonly monopoles, some are dipoles. The metal structure of the mast is divided at its midpoint into two insulated sections to make a vertical dipole, which is driven at the midpoint.
Dipole arrays
thumb|upright=0.5|Collinear folded dipole array
Many types of array antennas are constructed using multiple dipoles, usually half-wave dipoles. The purpose of using multiple dipoles is to increase the directional gain of the antenna over the gain of a single dipole; the radiation of the separate dipoles interferes to enhance power radiated in desired directions. In arrays with multiple dipole driven elements, the feedline is split using an electrical network in order to provide power to the elements, with careful attention paid to the relative phase delays due to transmission between the common point and each element.
In order to increase antenna gain in horizontal directions (at the expense of radiation towards the sky or towards the ground) one can stack antennas in the vertical direction in a broadside array where the antennas are fed in phase. Doing so with horizontal dipole antennas retains those dipoles' directionality and null in the direction of their elements. However, if each dipole is vertically oriented, in a so-called collinear antenna array (see graphic), that null direction becomes vertical and the array acquires an omnidirectional radiation pattern (in the horizontal plane) as is typically desired. Vertical collinear arrays are used in the VHF and UHF frequency bands at which wavelengths the size of the elements are small enough to practically stack several on a mast. They are a higher-gain alternative to quarter-wave ground plane antennas used in fixed base stations for mobile two-way radios, such as police, fire, and taxi dispatchers.
thumb|A [[reflective array antenna for radar consisting of numerous dipoles fed in-phase (thus realizing a broadside array) in front of a large reflector (horizontal wires) to make it uni-directional]]
On the other hand, for a rotating antenna (or one used only towards a particular direction) one may desire increased gain and directivity in a particular horizontal direction. If the broadside array discussed above (whether collinear or not) is turned horizontal, then the one obtains a greater gain in the horizontal direction perpendicular to the antennas, at the expense of most other directions. Unfortunately, that also means that the direction opposite the desired direction also has a high gain, whereas high gain is usually desired in one single direction. The power that is wasted in the reverse direction, however, can be redirected, for instance by using a large planar reflector, as is accomplished in the reflective array antenna, increasing the gain in the desired direction by another 3 dB
An alternative realization of a uni-directional antenna is the end-fire array. In this case the dipoles are again side by side (but not collinear), but fed in progressing phases, arranged so that their waves add coherently in one direction but cancel in the opposite direction. So now, rather than being perpendicular to the array direction as in a broadside array, the directivity is in the array direction (i.e. the direction of the line connecting their feedpoints) but with one of the opposite directions suppressed.
Yagi antennas
The above-described antennas with multiple driven elements require a complex feed system of signal splitting, phasing, distribution to the elements, and impedance matching. A different sort of end-fire array which is much more often used is based on the use of so-called parasitic elements. In the popular high-gain Yagi antenna, only one of the dipoles is actually connected electrically, but the others receive and reradiate power supplied by the driven element. This time, the phasing is accomplished by careful choice of the lengths as well as positions of the parasitic elements, in order to concentrate gain in one direction and largely cancel radiation in the opposite direction (as well as all other directions). Although the realized gain is less than a driven array with the same number of elements, the simplicity of the electrical connections makes the Yagi more practical for consumer applications.
Dipole as a reference standard
Antenna gain is frequently measured as decibels relative to a half-wave dipole. One reason is that practical antenna measurements need a reference strength to compare the field strength of an antenna under test at a particular distance to. While there is no such thing as an isotropic radiator, the half-wave dipole is well understood and behaved, and can be constructed to be nearly 100% efficient. It is also a fairer comparison, since the gain obtained by the dipole itself is essentially "free," given that almost no antenna design has a smaller directive gain.
For a gain measured relative to a dipole, one says the antenna has a gain of (see Decibel). More often, gains are expressed relative to an isotropic radiator, making the gain seem higher. In consideration of the known gain of a half-wave dipole, 0 dBd is defined as 2.15 dBi; all gains in "dBi" are shifted 2.15 higher than gains in "dBd".
Hertzian dipole<span class="anchor" id=Hertzian></span>
right|thumb|Hertzian dipole of tiny length <math>\ \delta\ell\ ,</math> with current <math>\ I\ ,</math> and field sensed at a distance <math>\ r\ </math> in the <math>\theta\ </math> direction
The Hertzian dipole or elementary doublet refers to a theoretical construction, rather than a physical antenna design: It is an idealized tiny segment of conductor carrying a RF current with constant amplitude and direction along its entire (short) length; a real antenna can be modeled as the combination of many Hertzian dipoles laid end-to-end.
The Hertzian dipole may be defined as a finite oscillating current (in a specified direction) of <math>\ I\ e^{i\omega t}\ </math> over a tiny or infinitesimal length <math>\ \delta\ell\ </math> at a specified position. The solution of the fields from a Hertzian dipole can be used as the basis for analytical or numerical calculation of the radiation from more complex antenna geometries (such as practical dipoles) by forming the superposition of fields from a large number of Hertzian dipoles comprising the current pattern of the actual antenna. As a function of position, taking the elementary current elements <math>\ I\!\left(\mathbf{r}\right)\ ,</math> multiplied by infinitesimal lengths <math>\ \delta\ell\ ,</math> the resulting field pattern then reduces to an integral over the path of an antenna conductor (modeled as a thin wire).
For the following derivation, we shall take the current to be in the <math>\ z\ </math> direction, centered at the origin where <math>\ x = y = z = 0\ ,</math> with the sinusoidal time dependence <math>\ e^{i \omega t}\ </math> for all quantities being understood. The simplest approach is to use the calculation of the vector potential <math>\ \mathbf{A}\!\left(\mathbf{r}\right)\ </math> using the formula for the retarded potential. Although the value of <math>\ \mathbf{A}\ </math> is not unique, we shall constrain it by adopting the Lorenz gauge, and assuming sinusoidal current at radian frequency <math>\ \omega\ </math> the retardation of the field is converted just into a phase factor <math>\ e^{-ikr}\ ,</math> where the wave number <math display="inline">\ k = \frac{\omega}{\ c\ }\ </math> in free space and <math>\ r\ </math> is the linear distance between the point being considered to the origin (where we assumed the current source to be), so <math>\ r \equiv \left|\ \mathbf{r}\ \right| ~.</math> This results
in a vector potential <math>\ \mathbf{A}\ </math> at position <math>\ \mathbf{r}\ </math> due to that current element only, which we find is purely in the <math>\ z\ </math> direction (the direction of the current):
:<math>\mathbf{A}\!\left(\mathbf{r}\right) = I\ \delta\ell\ \frac{\mu_0}{\ 4\pi r\ }\ e^{-ikr}\ \hat\mathbf{z}\ </math>
where <math>\ \mu_0\ </math> is the permeability of free space. Then using
:<math>\ \mu\mathbf{H} = \mathbf{B} = \nabla\times\mathbf{A}\ </math>
we can solve for the magnetic field <math>\ \mathbf{H}\ ,</math> and from that (dependent on us having chosen the Lorenz gauge) the electric field <math>\ \mathbf{E}\ </math> using
:<math>\mathbf{E} = \frac{\ \nabla\times\mathbf{H}\ }{i\omega\epsilon}</math>
In spherical coordinates we find
:<math>\begin{align}
R_\mathsf{dipole} = \frac{ \zeta_0 }{\ 2 \pi \sin^2\left(\tfrac{1}{2} k L \right)\ } \Biggl\{\
\gamma_e + \ln( k L ) - \operatorname{Ci}( k L ) + {} &\tfrac{1}{2} \sin( k L )\,\Bigl[ +\operatorname{Si}( 2 k L ) - 2\operatorname{Si}( k L )\ \Bigr] \\
{} + {} &\tfrac{1}{2} \cos( k L )\,\Big[ +\operatorname{Ci}( 2 k L ) - 2\operatorname{Ci}( k L ) + \gamma_e + \ln\left( \tfrac{1}{2}kL \right)\ \Bigr]
\ \Bigg\}\ , \\
X_\mathsf{dipole} = \frac{ \zeta_0 }{\ 2 \pi \sin^2\left( \tfrac{1}{2} k L \right)\ } \Biggl\{ {}
+ \operatorname{Si}( k L ) + {} &\tfrac{1}{2}\cos( k L )\,\Bigl[ - \operatorname{Si}( 2 k L ) + 2\operatorname{Si}( k L )\ \Bigr] \\
{} + {} &\tfrac{1}{2}\sin( k L )\,\Bigl[ +\operatorname{Ci}( 2 k L ) - 2\operatorname{Ci}( k L ) + \operatorname{Ci} \left(\tfrac{\; 2 k a^2\ }{ L }\right)\ \Bigr]
\ \Biggr\}\ ,
\end{align}</math>
where is the radius of the conductors, is again the wavenumber as defined above, is the impedance of empty space, which very nearly the same as impedance of air: and <math>\ \gamma_e = 0.57721566\ \ldots \ </math> is Euler's constant. There is an equivalent alternate form favored by some authors that uses a different function, .
Integral methods
The induced EMF method is dependent on the assumption of a sinusoidal current distribution, delivering an accuracy better than about 10% as long as the wavelength-to-element diameter ratio is greater than about 60.