73 The above field equations for the Hertzian dipole of length With that substitution, the above equations closely approximate the fields generated by a short dipole fed by current where the conductor is 1/2 wavelength long. 0 2 The feedpoint impedance of a dipole antenna is sensitive to its electrical length and feedpoint position. Therefore, this is comparable to a Hertzian dipole with an effective current Ih equal to the average current over the conductor, so {\displaystyle \ \mathbf {H} \ } In the next section, we'll consider the radiation pattern of dipole antennas. is an integer multiple of a wavelength. I ) z 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. {\displaystyle (\sin \theta )^{2}} consequently the fields are linearly polarized. {\displaystyle \ \lambda ={\frac {\ c\ }{f}}\ } For receipt, of course, this is in reverse. k direction, centered at the origin where This means the dipole elements have zero cross sectional area, there is zero bandwidth, and the phase difference between voltage and current of the standing wave is precisely 90. terms remain significant:[34]. Up: Dipole Antennas Note that the Mikls Kirly. The length of the total wire, which is being used as a dipole, equals half of the wavelength (i.e., l = /2). y r is applied to the terminals on the right antenna in Figure 1, the current distribution will The impedance of the dipole antenna will depend on the frecuency of operation and the length of the antenna. Typically, the width d of the folded dipole antenna is much smaller than the length L.. Because the folded dipole forms a closed loop, one might expect the input impedance to depend on the input impedance of a . r E [26] Other balun designs are mentioned below. The ends can be attached to existing buildings, structures, or trees, taking advantage of their heights. However coax is a single-ended line whereas a center-fed dipole expects a balanced line (such as twin lead). {\displaystyle \ I_{0}\ ,} j R For simplicity, this discussion assumes a theoretical ideal lossless dipole in free space with infinite Q. [9][10] Dipoles are frequently used as resonant antennas. Dipole antennas are frequently used at around that frequency and thus termed half-wave dipole antennas. 1 The Hertzian dipole is quite difficult to realize in practice, and nearly all practical electrically-short dipoles exhibit a current distribution that is closer to that of the ESD. 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). where the wave number Three dimensional radiation pattern of a vertical half-wave dipole antenna. r is the radian frequency ( In spherical coordinates we find[34] that the magnetic field {\displaystyle \ \phi \ } would then predict the actual fields for a short dipole using that effective current A half-wave dipole antenna consists of two quarter-wavelength conductors placed end to end for a total length of approximately = /2 . . we can find the flux averaged over all directions XC + XL = 0 at resonance. So now, rather than being perpendicular to the array direction as in a broadside array, the directivity is in the array direction (i.e. Although they may be used as standalone low-gain antennas, dipoles are also employed as driven elements in more complex antenna designs[3][5] such as the Yagi antenna and driven arrays. . The short dipole, with a feedpoint current of However these are rarely used. The dipole is any one of the varieties of antenna that produce a radiation pattern approximating that of an elementary electric dipole. 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. Impedance of an antenna is a function of frequency. R {\textstyle \ {\frac {1}{2}}E_{\theta }H_{\phi }^{*}\ } E 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 1 For instance, amateur radio antennas designed as half-wave dipoles at 7MHz can also be used as 3/2-wave dipoles at 21MHz; likewise VHF television antennas resonant at the low VHF television band (centered around 65MHz) are also resonant at the high VHF television band (around 195MHz). sin Impedance = resistance + signed reactance; Z = R + j X . {\displaystyle \ \langle S_{\mathsf {r}}\rangle \ } / I dipole antennas are given in Figure 1. resonant. E 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. 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. A 5/4-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. I During transmission, the antenna is fed at the center terminals of the two rods. 0 fields very close to the source are almost 90 out of phase, thus contributing very little to the Poynting vector by which radiated flux is computed. = 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). {\displaystyle \ \mathbf {A} \ } 2 Doing so with horizontal dipole antennas retains those dipoles' directionality and null in the direction of their elements. Before examining the fields radiated by a dipole antenna, consider the input impedance of a dipole I The impedance seen at the feedpoint of a dipole of various lengths has been plotted above, in terms of the real (resistive) component Rdipole and the imaginary (reactive) component j Xdipole of that impedance. 43 is the lower feedpoint impedance of the resonant halfwave dipole. The dipole antenna ( Figure 5.19) is one of the simple and widely used antennas in ZigBee applications. over a tiny or infinitesimal length ( k is not reached along the Normalized 3d radiation pattern for the 1.5-wavelength dipole antenna. f.d. is increased[15] by the factor 1/cos2(kx): This equation can also be used for dipole antennas of other lengths, provided that Rradiation has been computed relative to the current maximum, which is not generally the same as the feedpoint current for dipoles longer than half-wave. Much more common is the use of common 300 twin lead in conjunction with a folded dipole. slightly less than 0.5 Radiation pattern of vertical half-wave dipole; vertical section. ) Consequently, it can be seen that the radiation resistance computed for the short dipole is one quarter of that computed above for the Hertzian dipole. In consideration of the known gain of a half-wave dipole, 0 dBd is defined as 2.15 dBi; all gains in dBi are 2.15 higher than gains in dBd. {\displaystyle \ I\!\left(\mathbf {r} \right)\ ,} Although the value of {\displaystyle \ell } They are usually constructed of two lengths of wire joined by a strain insulator in the center, which is the feedpoint. {\textstyle \ I_{h}={\frac {1}{2}}I_{0}\ .} 4 G An alternative realization of a uni-directional antenna is the end-fire array. A Folded dipole antenna is a half-wave dipole antenna with an additional parallel wire or rod connecting its two ends and folded to form a cylindrical closed shape. is very nearly the same as sin applying to the short dipole, resulting in a very similar radiation pattern as noted above.[13]. 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 0.001wavelengths. Jan 15, 2022. 2 [27][28], A so-called current balun uses a transformer wound on a toroid or rod of magnetic material such as ferrite. 6,719. direction), while the magnetic field is at right angles to that, in the Standing Waves on a Dipole Antenna. Then using, we can solve for the magnetic field It it also my understanding that such an antenna is not actually resonant (Meaning the impedance is purely resistive) when L a = 0.5 , but rather something a tad smaller. One common modification has the radials forming the ground plane sloped down, which has the effect of raising the feedpoint impedance to around 50, matching common coaxial cable. For a dipole oriented parallel to {\displaystyle I_{0}} i the direction of the line connecting their feedpoints) but with one of the opposite directions suppressed. + I This results in an additional gain over a half-wave dipole of about 2dB. Normalized 3d radiation pattern for the 1-wavelength dipole antenna. j r i | the retardation of the field is converted just into a phase factor {\displaystyle \zeta _{0}\approx 377{\text{ }}} Dividing Ptotal by 4 R2 supplies the flux at a large distance, averaged over all directions. r The Hertzian dipole is an electrically-short dipole with a current distribution that has uniform magnitude over the length of the dipole. [ As a function of position, taking the elementary current elements If the additional wire has the same diameter and cross-section as the dipole, two nearly identical radiating currents are generated. If a voltage H sin | This makes the Hertzian dipole useful for analysis of more complex antenna configurations, where every infinitesimal section of that real antenna's conductor can be modelled as a Hertzian dipole with the current found to be flowing in that real antenna. a dipole is a function of its length. {\textstyle \ k={\frac {\omega }{\ c\ }}\ } L a: The length of the radiating element of the antenna. ) cos | 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 to the direction of the current, as being:[12]. {\displaystyle \ I_{0}\ .}. Such a dipole has a feedpoint impedance consisting of 73 resistance and +43 reactance, thus presenting a slightly inductive reactance. where a is the radius of the conductors, k is again the wavenumber as defined above, 0 is the impedance of free space: 0 377 , and Figure 1. direction (the direction of the current): where ( . I 1 is dominated by a negative reactance value (and a relatively small real impedance or resistance). 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. 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. {\displaystyle \ {\hat {\mathbf {r} }}\ } It is constructed of two telescoping rods that can each be extended out to about 1 m length (one quarter wavelength at 75MHz). {\displaystyle 4\pi r^{2}} I The dipole is the simplest type of antenna from a theoretical point of view. as dictated by the inverse square law. R {\displaystyle \ \mathbf {A} \!\left(\mathbf {r} \right)\ } {\displaystyle \ P_{\text{avg}}\ } H 2 ^ Determination of each matrix element requires at least one double integration involving the weighting functions, which may become computationally intensive. A common construction is to make the arms of the folded dipole out of twin lead also, shorted at their ends. {\displaystyle \ \phi \ } at a specified position. {\displaystyle \ r\equiv \left|\ \mathbf {r} \ \right|~.} 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.[16]. {\displaystyle \ P_{\text{avg}}\ } A 1 An antenna is an instrument that converts alternating voltage electrical signals into electromagnetic fields. [20](p164) The conductor and its image together act as a dipole in the upper half of space. In radio and telecommunications a dipole antenna or doublet[1] is the simplest and most widely used class of antenna. Note that the input impedance is specified as Z=R + jX, where R is the resistance and X is the reactance. using the formula for the retarded potential. r 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. Figure 4. {\displaystyle \ \mathbf {E} \ } {\displaystyle \ \sin(k\ z)\ } {\displaystyle \ I\ e^{i\omega t}\ } 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. 0 Coax and antenna both acting as radiators instead of only the antenna, A folded dipole (300 ) to coax (75 ) 4:1 halfwave balun, Detailed calculation of dipole feedpoint impedance, This computation using the induced EMF method is identical to the computation of the, Practical Wire Antennas 2 (I. Poole, G3YWX), Baluns: What They Do And How They Do It (W7EL), Learn how and when to remove this template message, "An exact impedance step-up impedance-ratio chart of a folded antenna", "Single wire quadrant antenna: Horizontal omnidirectional shortwave aerial comparisons", "The design of antennas for use in radio telemetry", ycars.org - Reflections and standing wave ratio, http://www.eznec.com/Amateur/Articles/Baluns.pdf, "A Cost Effective Current-mode 1:1 Balun", "A Cost Effective Current-mode 1:4 Balun", "Chaotic behavior in receiver front-end limiters", https://web.archive.org/web/20070926195106/http://www.nt.hs-bremen.de/peik/asc/asc_antenna_slides.pdf, Reflections on Hertz and the Hertzian Dipole, https://en.wikipedia.org/w/index.php?title=Dipole_antenna&oldid=1158218040, This page was last edited on 2 June 2023, at 17:36. {\displaystyle \ j^{2}=-1\ } Dividing the flux in the = 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. 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. Current distributions on finite-length dipole antennas. f the z-axis, the E-field is in the z-direction, and the dipole antenna is referred to as vertically polarized (note that if a dipole is mounted horizontally, it will be horizontally polarized). H {\displaystyle \ \omega ~.} The near field solution for an antenna element (from the integral using this formula over the length of that element) is the field that can be used to compute the mutual impedance between it and another nearby element. 0 It forms a balun by choking common-mode current. avg h {\displaystyle \ \sin ^{2}(\theta )\ } has only a component in the H Antennas List , Thin linear conductors of length {\displaystyle \ \omega \ } 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). Dipole Impedance (G9B08) - There is no conductive path at the antenna ends for the electrons and positive charges to continue upon, so they tend to accumulate with greater density toward the antenna ends. is the wavelength, and c is the reduced speed of radio waves in the radiating conductor (c 97%co, the speed of light). In the next section, we'll look at the most common dipole antenna, the half-wave dipole antenna. S 2 R Such antenna designs, although based on the half-wave dipole, generally acquire their own names. {\displaystyle \ \mathbf {E} \ } 1 It is a desirable length because such an antenna has the highest gain for any dipole which isn't a great deal longer. A true half-wave dipole is one half of the wavelength in length, where = c/f in free space. 377 A short dipole is a dipole formed by two conductors with a total length substantially less than a half wavelength (.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2). They are also widely used as driven elements for rooftop Yagi television antennas. 21 Figure 1. In the case of a short dipole, that is essentially a linear drop from {\displaystyle \ \mathbf {A} \ } = The radius of the dipole is assumed to be very small compared to the length. the direct signal. of: where is the free-space wavelength, c is the speed of light in free space, and f is the frequency. 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. 2 . z ( = For a dipole antenna of length L is an example of an omnidirectional antenna. 2 Now, however, the "monopole" antenna is understood as a special case of a dipole which has a virtual element "underground". H {\displaystyle \ \operatorname {\mathsf {G}} \left(\theta \right)\ :}. and from that (dependent on us having chosen the Lorenz gauge) the electric field The graph confuses feed point impedance with wave impedance and gives the reader the impression that the wave impedance is the feed point impedance by plotting standing wave voltage and current . {\textstyle {\frac {1}{r}}} I Abstract: A low-profile dual-polarized planar ultrawideband modular antenna (PUMA) array is proposed in this paper. k I know that an antenna that has half a wavelength does not need the ground plane (example a dipole antenna) and when using a wavelength of 1/4 it is usually necessary to have the ground plane (planar antennas [PCB]). Often, however, the signals contain only components with frequencies within a narrow band. is the linear distance between the point being considered to the origin (where we assumed the current source to be), so This can also be directly computed using the cosine integral: We can now also find the radiation resistance as we did for the short dipole by solving: Using the induced EMF method,[14] the real part of the driving point impedance can also be written in terms of the cosine integral, obtaining the same result: If a half-wave dipole is driven at a point other the center, then the feed point resistance will be higher. With E and H being at right angles and in phase, there is no imaginary part and is simply equal to After setting an appropriate weighting function the cost may be minimized through the inversion of a NN matrix. This solution includes near field terms which are very strong near the source but which are not radiated. The better the antenna is, the less of the wave that reaches it will be reflected back into the cable, and the more will propagate through free space. Many types of coaxial cable (or "coax") have a characteristic impedance of 75, which would otherwise be a good match for a half-wave dipole. : To cancel that reactance, and present a pure resistance to the feedline, the element is shortened by the factor k for a net length The Hertzian dipole is an electrically-short dipole with a current distribution that has uniform magnitude over the length of the dipole. This is not an actual performance advantage per se, 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!) , as we assumed the source to be at the origin). is the net power radiated due to that current, so equating the above to 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. The combiner contains 7 regular Tee-s. As we know, a Tee's one arm has to shift . A numerical integration of the radiated power By making the dipole antenna feed impedance equal to the source or load impedance, antenna efficiency is maximized. In order to supply the same power, the voltage at the feedpoint has to be similarly increased by the factor 1/cos(k x) . However a short conductor fed with a RF voltage will not have a uniform current even along that short range. Next: Half-Wave Dipole Antennas 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). This is usually wound on a ferrite toroidal core. 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. with amplitude which Its gain is about 3dB greater than a half-wave dipole, the highest gain of any dipole of any similar length. . = 1 The Hertzian dipole is quite difficult to realize in practice, and nearly all practical electrically-short dipoles exhibit a current distribution that is closer to that of the ESD. = [8] 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. [9][10] 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). By placing a capacitive hat, such as a metallic ball, at the end of the conductor, it is possible for its self capacitance to absorb the current from the conductor and better approximate the constant current assumed for the Hertzian dipole. is specified as Z=R + jX, where R is the resistance and X is the reactance. Figure \(\PageIndex{1}\): Radiation pattern plots for a half-wave dipole antenna. Many types of array antennas are constructed using multiple dipoles, usually half-wave dipoles. 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. 2 The solution depends on an assumption for the form of the current distribution along the antenna conductors. The theoretical impedance of a half wave dipole antenna is 73 + j 42.5. This is the most widely used antenna because of its advantages. + {\displaystyle \ \langle S_{\mathsf {\phi }}\rangle \ } By definition, the radiation resistance 2 2 {\displaystyle \ I_{h}\ e^{\ j\ \omega \ t}\ } / For VHF and UHF bands, the radiating and ground plane elements can be constructed from rigid rods or tubes. A folded dipole is a half-wave dipole with an additional parallel wire connecting its two ends. The HPBW is 78 degrees. It follows that. Common mode noise filters are HERE. A with the phase factors (the exponentials) cancelling out leaving: We have now expressed the flux in terms of the feedpoint current I0 and the ratio of the short dipole's length to the wavelength of radiation . Figure 5. component. The dipole antenna is symmetric when viewed azimuthally (around the long axis of the dipole); as a result the radiation pattern is not a function of . rad As seen in the accompanying animation, the ( {\displaystyle \ r~.} 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. S The current distributions for the quarter-wavelength (left) and full-wavelength (right) H 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. {\displaystyle R_{\text{rad}}} Share Improve this question Follow asked Oct 8, 2019 at 18:35 hjf 1,189 9 20 Add a comment 2 Answers Sorted by: 2 Read it off this chart: Interferometrist [ CC BY-SA 4.0 ], via Wikimedia Commons A folded dipole with equal-sized conductors has 4x the impedance of an ordinary dipole. P Rather, a short dipole in real life has a current equal to the feedpoint current at the feedpoint but falling linearly to zero over the length of that short conductor. z {\textstyle \ {\frac {1}{2}}\mathbf {E} \times \mathbf {H} ^{*}\ .} The fundamental resonance of a thin linear conductor occurs at a frequency whose free-space wavelength is twice the wire's length; i.e. I It is used to receive the VHF terrestrial television bands, consisting in the US of 54 to 88MHz (band I) and 174 to 216MHz (band III), with wavelengths of 5.5 to 1.4m. Since this frequency range is much wider than a single fixed dipole antenna can cover, it is made with several degrees of adjustment. 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. {\displaystyle \omega \equiv 2\pi f\ } More often, gains are expressed relative to an isotropic radiator, often for advertising reasons as this makes the gain appear higher. This flexible antenna can be conveniently taped or nailed to walls, following the contours of mouldings. Dipoles that are much smaller than one half the wavelength of the signal are called short dipoles. t Note that the peak value of the current It behaves essentially as a dipole antenna. ) and k is the wavenumber ( The feedpoint is usually at the center of the dipole as shown in the diagram. ] V ( are not always obvious. {\textstyle \ {\tfrac {\ 1\ }{2}}I_{0}~.} 0 . The ground (or ground plane) is considered to be a conductive surface which works as a reflector (see effect of ground). 0 is the impedance of free space ( Question: Design a horizontally polarized dipole antenna that resonates at 950MHz. The current along dipole arms are approximately described as proportional to using the simpler far field expressions for are in fact resonant at any integer multiple of a half-wavelength: where n is an integer, S {\displaystyle \ \gamma _{e}=0.57721566\ } Cage dipoles are often used to get the large diameter. ), which is the ratio of a free space plane wave's electric to magnetic field strength. Ideal antenna? . = 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. For a dipole of total length L, the resistive and reactive components of the driving point impedance can be expressed as:[35][b]. A quarter-wave monopole, then, has an impedance[20](p173) of One easy way to make a balun is to use a length of coaxial cable equal to half a wavelength. {\displaystyle R_{\text{f.d.}}} 0 r {\displaystyle \ I_{0}\ } These are simplified if the weighting functions are simply delta functions, which corresponds to fitting the boundary conditions for the current along the conductor at only N discrete points. Note that this 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). at resonance is because for a fixed amount of power, the total radiating current The calculation of Xdipole is more difficult. {\displaystyle \ r\ } has an average current over each conductor of only The Hertzian dipole is similar to but differs from the short dipole, discussed above. 2 e One might be tempted to say "infinite," since the current is zero independently of the voltage. R Consequently, infinite impedance occurs whenever the dipole antenna and , Figure 6. E {\displaystyle \ I_{0}~.} 1 Using the above expression for the radiated flux given by the Poynting vector, it is also possible to compute the directive gain of the Hertzian dipole. Its conductivity can be improved (at cost) by laying a copper mesh. 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 . A Folded Dipole Antenna of length L.. This important case is dealt with in the next section. A folded dipole is a dipole antenna with the ends folded back around and connected to each other, forming a loop as shown in Figure 1.. E pattern for the quarter- and half-wave dipole antenna. is the impedance of free space. Figure 2. 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. At VHF frequencies, a sleeve balun can also be built to remove feeder radiation.[32]. Longer dipoles than that have radiation patterns that are multi-lobed, with poorer gain (unless they are much longer) even along the strongest lobe. cos k ranges from about 0.98 for thin wires (diameter, 0.00001 wavelengths) to about 0.94 for thick conductors (diameter, 0.008 wavelengths). As the frequency is swept across the resonant frequency of a tuned circuit or antenna, the reactance passes through zero, and at that frequency the antenna is purely resistive. | , where k = 2/ and z runs from /2 to +/2.
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