Dipole antenna

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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. (p 3) The driving current from the transmitter is applied, or for receiving antennas the output signal to the receiver is taken, between the two halves of the antenna. Each side of the feedline to the transmitter or receiver is connected to one of the conductors. This contrasts with a monopole antenna, which consists of a single rod or conductor with one side of the feedline connected to it, and the other side connected to some type of ground. A common example of a dipole is the rabbit ears television antenna found on broadcast television sets. All dipoles are electrically equivalent to two monopoles mounted end-to-end and fed with opposite phases, with the ground plane between them made virtual by the opposing monopole.

The dipole is the simplest type of antenna from a theoretical point of view. Most commonly it consists of two conductors of equal length oriented end-to-end with the feedline connected between them. 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. The most commonly used is the center-fed half-wave dipole which is just under a half-wavelength long. The radiation pattern of the half-wave dipole is maximum perpendicular to the conductor, falling to zero in the axial direction, thus implementing an omnidirectional antenna if installed vertically, or (more commonly) a weakly directional antenna if horizontal.

Although they may be used as standalone low-gain antennas, dipoles are also employed as driven elements in more complex antenna designs such as the Yagi antenna and driven arrays. Dipole antennas (or such designs derived from them, including the monopole) are used to feed more elaborate directional antennas such as a horn antenna, parabolic reflector, or corner reflector. Engineers analyze vertical (or other monopole) antennas on the basis of dipole antennas of which they are one half.

History

German physicist Heinrich Hertz first demonstrated the existence of radio waves in 1887 using what we now know as a dipole antenna (with capacitative end-loading). On the other hand, Guglielmo Marconi empirically found that he could just ground the transmitter (or one side of a transmission line, if used) dispensing with one half of the antenna, thus realizing the vertical or monopole antenna. (p 3) For the low frequencies Marconi employed to achieve long-distance communications, this form was more practical; when radio moved to higher frequencies (especially V.H.F. transmissions for F.M. radio and T.V.) 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 ℓ substantially less than a half wavelength ( ⁠1 2 ⁠ λ ). 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 r from the current and at an angle θ to the direction of the current, as being: (p132)

Math summary: Equations compute far-field electric and magnetic fields from a short dipole antenna. They use current, length, wave number, and distance to determine field strengths. Free space impedance scales the magnetic field to find the electric field.

where the radiator consists of a current of I h e^ jomega t over a short length ℓ and j to the 2equiv -1 in electronics replaces the customary mathematical symbol i for the square root of −1. ω is the angular (radian) frequency ( omega equiv 2pi f\) and k is the wavenumber ( kequiv 2pi /lambda \). ζ 0 is the impedance of free space ( zeta _o approximately 376.7 mathsf Omega \), which is the ratio of a free space plane wave's electric to magnetic field strength.

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 sin ( k z ) where z is the distance to the nearest end of the arm. In the case of a short dipole, that is essentially a linear drop from I o at the feedpoint to zero at the end. Therefore, this is comparable to a Hertzian dipole with an effective current I h equal to the average current over the conductor, so textstyle I h =tfrac 1 2 I o ~. With that substitution, the above equations closely approximate the fields generated by a short dipole fed by current I o ~.

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, S, which is given by textstyle tfrac 1 2.E. times H ^*~. Because E and H are at right angles and in phase, there is no imaginary part and the cross product is equal to textstyle tfrac 1 2 E theta H phi ^* the phase factors (the exponentials) cancel out, leaving:

Math summary: Flux calculation finds radiated power per area. It relates electric and magnetic fields to antenna current, length, distance, and angle.

We have now expressed the flux in terms of the feedpoint current I o and the ratio of the short dipole's length ℓ to the wavelength of radiation λ. The radiation pattern given by sin ^2 ( theta ) is seen to be similar to and only slightly less directional than that of the half-wave dipole.

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 summary: Total radiated power equals a constant multiplied by free space impedance, feedpoint current squared, and the ratio of dipole length to wavelength squared.

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 P total to the power supplied at the feedpoint textstyle tfrac 1 2 I o ^2 R radiation we find:

Math summary: Radiation resistance is computed from the ratio of the antenna length to the wavelength, scaled by a constant of approximately 197 Ohms.

Again, these approximations become quite accurate for ℓ ≪ ⁠1 2 ⁠λ. Setting ℓ = ⁠1 2 ⁠ λ 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; that is where the conductor is ⁠12⁠ 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 ell are in fact resonant at any integer multiple of a half-wavelength:

Math summary: Conductor length equals an integer multiplied by half the wavelength. This calculates resonant lengths for dipole antennas.

where n is an integer, lambda =( c over f) is the wavelength, and c is the reduced speed of radio waves in the radiating conductor (c ≈ 97percenttimesc o, the speed of light). For a center-fed dipole, however, there is a great dissimilarity between n 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 ell approximately lambda \. This results in an additional gain over a half-wave dipole of about 2 decibel. 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 54-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 decibel greater than a half-wave dipole, the highest gain of any dipole of any similar length.

Table summary: Directive gain varies by dipole length. The greatest gain occurs at 1.25 wavelengths, while performance is poor for lengths much less than 0.5 wavelengths.

Other reasonable lengths of dipole do not offer advantages and are seldom used. However the overtone resonances of a half-wave dipole antenna at odd multiples of its fundamental frequency are sometimes exploited. For instance, amateur radio antennas designed as half-wave dipoles at 7 megahertz can also be used as 32-wave dipoles at 21 megahertz; likewise V.H.F. television antennas resonant at the low V.H.F. television band (centered around 65 megahertz) are also resonant at the high V.H.F. television band (around 195 megahertz).

Half-wave dipole

Image summary: A half-wave dipole antenna shows voltage V(x) and current I(x) distributions along its length, with a length of lambda/2.

Animation of a transmitting half-wave dipole showing the voltage V(x) ( red, ) and current I(x) blue due to the standing wave on the antenna. Since the standing wave is mainly storing energy, not transporting power, the current is not in phase with the voltage but 90° out of phase. The transmission line applies an oscillating voltage V icos omega t from the transmitter between the two antenna elements, driving the sinusoidal oscillation. The feed voltage step has been increased for visibility; typical dipoles have a high enough Q factor that the feed voltage is much smaller in relation to the standing wave.

Since the antenna is fed at its resonant frequency, the input voltage is in phase with the current (blue bar), so the antenna presents a pure resistance to the feedline. The energy from the driving current provides the energy radiated as radio waves. In a receiving antenna the phase of the voltage at the transmission line would be reversed, since the receiver absorbs energy from the antenna.

A half-wave dipole antenna consists of two quarter-wavelength conductors placed end to end for a total length of approximately ℓ = ⁠12⁠ λ. The current distribution is that of a standing wave, approximately sinusoidal along the length of the dipole, with a node at each end and an antinode (peak current) at the center (feedpoint): (pp98–99)

Math summary: Current along a half-wave dipole antenna is calculated. Initial current is scaled by cosine functions of time and position.

where k = ⁠2 π λ ⁠ and z runs from ⁠−12⁠ℓ to ⁠+12⁠ℓ.

In the far field, this produces a radiation pattern whose electric field is given by (pp98–99)

Math summary: Electric field strength from a half-wave dipole antenna is proportional to the antenna's characteristic impedance and peak current, divided by distance. This is scaled by a directional factor and a time-varying sinusoidal term.

The directional factor textstyle (cos [ tfrac pi over 2) cos theta ]sin theta is very nearly the same as sin θ applying to the short dipole, resulting in a very similar radiation pattern as noted above. (pp98–99)

A numerical integration of the radiated power textstyle frac |E theta |^2 2zeta _0 over all solid angle, as we did for the short dipole, obtains a value for the total power P total radiated by the dipole with a current having a peak value of I 0 as in the form specified above. Dividing P total by textstyle 4pi r to the 2 supplies the flux at a large distance, averaged over all directions. Dividing the flux in the θ = 0 direction (where it is at its peak) at that large distance by the average flux, we find the directive gain to be 1.64. This can also be directly computed using the cosine integral:

G=(4 over Cin) (2pi ) approximately 1.64~ (2.15 dBi)

( The Cin(x) form of the cosine integral is not the same as the Ci(x) form; they differ by a logarithm. Both matlab and Mathematica have inbuilt functions which compute Ci(x), but not Cin(x). See the Wikipedia page on cosine integral for the relationship between these functions. )

We can now also find the radiation resistance as we did for the short dipole by solving:

Math summary: Total power equals one half times the square of the peak current, times the radiation resistance. This calculates total power radiated by a dipole.

to obtain:

Math summary: Radiation resistance is approximately seventy three point one ohms.

Using the induced E.M.F. method, (p224) the real part of the driving point impedance can also be written in terms of the cosine integral, obtaining the same result:

Math summary: Radiation resistance equals a constant times a cosine integral. The integral is approximated as a resistance of about 73.1 ohms.

If a half-wave dipole is driven at a point other the center, then the feed point resistance will be higher. The radiation resistance is usually expressed relative to the maximum current present along an antenna element, which for the half-wave dipole (and most other antennas) is also the current at the feedpoint. However, if the dipole is fed at a different point at a distance x from a current maximum (the center in the case of a half-wave dipole), then the current there is not I 0 but only I 0 cos( k x ).

In order to supply the same power, the voltage at the feedpoint has to be similarly increased by the factor sec( k x ). Consequently, the resistive part of the feedpoint impedance mathcal R e [tfrac V I ] is increased (p227) by the factor sec 2 ( k x ):

Math summary: Feedpoint resistance is calculated by dividing radiation resistance, approximately 73.1 Ohms, by the square of a cosine function involving a distance value.

This equation can also be used for dipole antennas of any length, provided that R radiation has been computed relative to the current maximum, which is not generally the same as the feedpoint current for dipoles longer than half-wave. Note that this equation breaks down when feeding an antenna near a current node, where cos( k x ) approaches zero. The driving point impedance does indeed rise greatly, but is nevertheless limited due to higher order components of the elements' not-quite-exactly-sinusoidal current, which have been ignored above in the model for the current distribution. (p228)

Folded dipole

A folded dipole is a half-wave dipole with an additional parallel wire connecting its two ends. If the additional wire has the same diameter and cross-section as the dipole, two nearly identical radiating currents are generated. The resulting far-field emission pattern is nearly identical to the one for the single-wire dipole described above, but at resonance its feedpoint impedance R mathsf f.d. is four times the feedpoint impedance of a single-wire dipole.

A folded dipole is, technically, a folded full-wave loop antenna, where the loop has been bent at opposing ends and squashed into two parallel wires in a flat line. Although the broad bandwidth, high feedpoint impedance, and high efficiency are characteristics more similar to a full loop antenna, the folded dipole's radiation pattern is more like an ordinary dipole. Since the operation of a single halfwave dipole is easier to understand, both full loops and folded dipoles are often described as two halfwave dipoles in parallel, connected at the ends.

The high feedpoint impedance R mathsf f.d. at resonance is because for a fixed amount of power, the total radiating current I 0 is equal to twice the current in each wire separately and thus equal to twice the current at the feed point. We equate the average radiated power to the average power delivered at the feedpoint, we may write

Math summary: Radiated power from a half-wave dipole antenna equals radiated power from a folded dipole antenna. The folded dipole uses half the total radiating current. This shows the relationship between the feedpoint impedances of the two antennas.

where R mathsf h.w. is the lower feedpoint impedance of the resonant halfwave dipole. It follows that

Math summary: Folded dipole resistance equals four times half-wave dipole resistance, approximately two hundred ninety-two ohms.

Half-wave folded dipoles are often used for F.M. radio antennas; versions made with twin lead which can be hung on an inside wall often come with F.M. tuners. They are also widely used as driven elements for rooftop Yagi television antennas. The T.².F.D. antenna is a folded dipole with a resistor added on the second wire, opposite the feedpoint.

The folded dipole is therefore well matched to 300 Ω") balanced transmission lines, such as twin-feed ribbon cable. The folded dipole has a wider bandwidth than a single dipole. They can be used for transforming the value of input impedance of the dipole over a broad range of step-up ratios by changing the thicknesses of the wire conductors for the fed- and folded-sides.

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 summary: Radiation resistance approximates a value. The number of parallel wires squared, times a constant, determines the radiation resistance.

where n 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 U.H.F. television antennas.

Image summary: The image shows cage dipole antennas at the Ukrainian UTR-2 radio telescope. The antennas are made of galvanized steel wire and have a cylindrical shape.

Cage dipole antennas in the Ukrainian U.T.R.-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 to 33 megahertz.

- 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 S.W.R.) to a full octave. They are used for H.F. band transmissions").

- The G.5.R.V. 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 A.S.-2259 antenna is an inverted-'V' dipole antenna used for local communications via Near Vertical Incidence Skywave (N.V.I.S.).

Image summary: A quarter-wave monopole antenna above a ground plane is depicted, with a dashed line showing the ground image. The antenna height is labeled as λ/4.

A ⁠ 1 4 λ monopole antenna and its ground image together form a 2 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. (p164) The conductor and its image together act as a dipole in the upper half of space. Like a dipole, in order to achieve resonance (resistive feedpoint impedance) the conductor must be close to a quarter wavelength in height (like each conductor in a half-wave dipole).

In this upper side of space, the emitted field has the same amplitude of the field radiated by a similar dipole fed with the same current. Therefore, the total emitted power is half the emitted power of a dipole fed with the same current. As the current is the same, the radiation resistance (real part of series impedance) will be half of the series impedance of the comparable dipole. A quarter-wave monopole, then, has an impedance (p173) of textstyle ( 73 + j 43 over 2)=36 + j 21 mathsf Omega ~. Another way of seeing this is that a true dipole receiving a current I has voltages on its terminals of +V and −V, for an impedance across the terminals of ⁠2 V I ⁠, whereas the comparable vertical antenna has the current I but an applied voltage of only V.

Since the fields above ground are the same as for the dipole, but only half the power is applied, the gain is doubled to 5.14 dBi. This is not an actual performance advantage per say, since in practice a dipole also reflects half of its power off the ground which (depending on the antenna height and sky angle) can augment (or cancel!) the direct signal. The vertical polarization of the monopole (as for a vertically oriented dipole) is advantageous at low elevation angles where the ground reflection combines with the direct wave approximately in phase.

The earth acts as a ground plane, but it can be a poor conductor leading to losses. Its conductivity can be improved (at cost) by laying a copper mesh. When an actual ground is not available (such as in a vehicle) other metallic surfaces can serve as a ground plane (typically the vehicle's roof). Alternatively, radial wires placed at the base of the antenna can form a ground plane.

For V.H.F. and U.H.F. bands, the radiating and ground plane elements can be constructed from rigid rods or tubes. Using such an artificial ground plane allows for the entire antenna and ground to be mounted at an arbitrary height. One common modification has the radials forming the ground plane sloped down, which has the effect of raising the feedpoint impedance to around 50 Ω, matching common coaxial cable. No longer being a true ground, a balun (such as a simple choke balun) is then recommended.

Image summary: A line graph shows the impedance of a dipole antenna, where conductor diameter is 0.001 wavelengths. Resistive impedance (black line) and reactive impedance (blue line) are plotted against dipole length in wavelengths. Resistive impedance is generally higher than reactive impedance, except near dipole lengths of 0.75 and 1.25 wavelengths, where reactive impedance sharply decreases.

The feedpoint impedance of a dipole antenna is sensitive to its electrical length and feedpoint position. Therefore, a dipole will generally only perform optimally over a rather narrow bandwidth, beyond which its impedance will become a poor match for the transmitter or receiver (and transmission line). The real (resistive) and imaginary (reactive) components of that impedance, as a function of electrical length, are shown in the accompanying graph. The detailed calculation of these numbers are described below. Note that the value of the reactance is highly dependent on the diameter of the conductors; this plot is for conductors with a diameter of 0.001 wavelengths.

Dipoles that are much smaller than one half the wavelength of the signal are called short dipoles. These have a very low radiation resistance (and a high capacitive reactance")) making them inefficient antennas. More of a transmitter's current is dissipated as heat due to the finite resistance of the conductors which is greater than the radiation resistance. However they can nevertheless be practical receiving antennas for longer wavelengths.

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.

Image summary: This graph shows impedance in ohms versus dipole length in wavelengths. The black line represents reactance, while the blue lines represent resistance for varying conductor diameters. Reactance increases with dipole length. Resistance is 50 ohms at a dipole length of 0.5 wavelengths.

A true half-wave dipole is one half of the wavelength λ in length, where λ = ⁠ c f ⁠ 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 k for a net length ell of:

Math summary: Antenna length equals one half of an adjustment factor times wavelength. Wavelength is the speed of light divided by frequency. The equation calculates the length of a shortened dipole antenna.

where λ is the free-space wavelength, c is the speed of light in free space, and f is the frequency. The adjustment factor k 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 k 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 ⁠1 2 ⁠ λ. 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.

Image summary: A graph shows the length reduction factor for a half-wave dipole relative to the conductor diameter in wavelengths. The length reduction factor decreases from approximately 0.99 to 0.93 as the conductor diameter increases.

For a typical k of about 0.95, the above formula for the corrected antenna length can be written, for a length in meters as ⁠143 f ⁠, or a length in feet as ⁠468 f ⁠ where f is the frequency in megahertz.

Dipole antennas of lengths approximately equal to any odd multiple of ⁠1 2 ⁠λ 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 ⁠5 4 ⁠ wave. Not being close to ⁠3 2 ⁠ wave, 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.

Image summary: A toroid represents the radiation pattern of a dipole antenna in three dimensions.

Image summary: Radiation pattern of a 5/4 wave dipole antenna, showing a multi-lobed pattern with the strongest lobes at approximately +/- 60 degrees from the dipole axis.

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 ( sin theta )^2 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 ⁠5 4 ⁠ wave 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.