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For Monopole antenna

A typical mast radiator monopole antenna of an AM radio station in Chapel Hill, North Carolina. The metal mast itself is connected to the transmitter and radiates the radio waves. It is mounted on a ceramic insulator to isolate it from the ground. The other terminal of the transmitter is connected to a radial ground system consisting of cables buried under the field.

A monopole antenna is a class of radio antenna consisting of a straight rod-shaped conductor, often mounted perpendicularly over some type of conductive surface, called a ground plane. The current from the transmitter is applied, or for receiving antennas the output signal voltage to the receiver is taken, between the monopole and the ground plane. One side of the feedline to the transmitter or receiver is connected to the lower end of the monopole element, and the other side is connected to the ground plane, which may be the Earth. This contrasts with a dipole antenna which consists of two identical rod conductors, with the current from the transmitter applied between the two halves of the antenna. The vertical monopole is an omnidirectional antenna with a low gain of 2 - 5 dBi, and radiates most of its power in horizontal directions or low elevation angles. Common types of monopole antenna are the whip, rubber ducky, helical, umbrella, inverted-L and T-antenna, inverted-F, folded unipole antenna, mast radiator, and ground plane antennas.

The monopole is usually used as a resonant antenna; the rod functions as a resonator for radio waves, oscillating with standing waves of voltage and current along its length. The length of the antenna is determined by the wavelength of the radio waves it is used with. The most common form is the quarter-wave monopole, in which the antenna is approximately one quarter of the wavelength of the radio waves. It is said to be the most widely-used antenna in the world. . Half-wave monopoles, one-half wavelength long, and monopoles shorter than one-quarter wavelength, called electrically short monopoles, are also common. Monopoles five-eights (5/8 = 0.625) of a wavelength long are also widely used, because at this length a monopole radiates a maximum amount of its power in horizontal directions. A capacitively loaded or top-loaded monopole is a monopole antenna with horizontal conductors such as wires or screens attached to the top of the monopole element, to increase radiated power. Large top-loaded monopoles, the T and inverted L antennas and umbrella antenna are used as transmitting antennas at longer wavelengths, in the LF and VLF bands.

The monopole antenna was invented in 1895 by radio pioneer Guglielmo Marconi; for this reason it is also called the Marconi antenna although Alexander Popov independently invented it at about the same time.

Types and uses

Due to their omnidirectional radiation pattern, vertical monopole antennas are commonly used in terrestrial radio communication systems in which the direction to the transmitter or receiver is unknown or constantly changing, such as broadcasting, mobile two-way radios, and wireless devices like cellphones and Wi-fi networks, because they radiate equal radio power in all horizontal directions but little power up into the sky where it would be wasted. The quarter-wave monopole is the smallest antenna that is resonant, making it an efficient radiator; it is said to be the most widely used antenna in the world.

Lower frequency monopole antennas

200 foot mast radiator of AM radio station, USA

Amateur radio cage T antenna, used to communicate with Europe at frequency of 1.5 MHz, 1922

VLF flattop antenna at Grimeton Radio Station, Sweden

Trideco umbrella antenna of the VLF transmitter at Anthorn military radio station, UK, transmitting at 19.6 kHz

Amateur inverted-L antenna for shortwave reception, showing construction

Large monopoles are the main transmitting antennas used in the lower frequencies below 3 MHz, the MF, LF, and VLF bands, because the radio propagation mode used in these bands, ground waves, requires a vertically polarized antenna with good horizontal radiation characteristics. At these frequencies, the Earth itself is used as the antenna's ground plane. The most common antenna is the mast radiator, a vertical mast mounted on the ground but insulated from it electrically. One side of the feedline from the transmitter is connected to the conductive metal mast which serves as the radiating element, and the other to an Earth ground connection at the base of the antenna. To reduce ground resistance this is usually a radial network of buried wires stretching outward from the base of the antenna. This design is used for AM radio broadcasting antennas in the MF and LF bands. Another variant is the folded unipole antenna. At lower frequencies in the LF and VLF band, construction limitations mean monopoles are electrically short, shorter than one-quarter wavelength. Simple monopoles this short are inefficient due to their very low radiation resistance, so to increase efficiency and radiated power, capacitively toploaded monopoles such as the inverted-L, T antenna and umbrella antenna are used.

At higher frequencies variations such as the folded monopole, J-pole antenna, and normal mode helical are used.

Higher frequency monopole antennas

Retractable quarter-wave whip antenna for FM reception on a portable radio, 88 - 108 MHz

Rubber ducky antenna on walkie-talkie

Cell phone UHF whip antenna on car

3 fiberglas half-wave whip antennas

US Navy broadband conical monopole antenna

VHF ground plane antenna.

Dual band 2.4 and 5 Ghz monopole antenna on a home Wi-fi router

At higher frequencies in the VHF and UHF bands, the size of the ground plane needed is smaller, so artificial metal ground planes of screen or rods are used to allow the antenna to be mounted above the ground. A common type for mounting on masts or stationary structures is the ground plane antenna, consisting of a quarter-wave whip antenna with a ground plane of 3 or 4 wires or rods a quarter-wavelength long radiating horizontally or diagonally from its base, connected to the ground side of the feedline. Another variation is the discone antenna, which is notable for having a very broad bandwidth. At frequencies above 30 MHz an automobile or aircraft body makes an adequate ground plane, so whip antennas for two-way radios and cell phones are mounted on car bumpers or roofs, and aircraft communication antennas frequently consist of a short conductor in an aerodynamic fairing projecting from the fuselage; this is called a blade antenna.

The quarter-wave whip and rubber ducky antennas used with handheld radios such as walkie-talkies and portable FM radios in the VHF and UHF bands are also monopole antennas. In these portable devices the antenna does not have an effective ground plane, the ground side of the transmitter or receiver is just connected to the chassis ground connection on its circuit board. Since these "ground" conductors are no larger than the element itself the antenna usually functions more like an asymmetrical dipole than a monopole antenna.

A monopole type widely used in wireless devices and cell phones operating at microwave frequencies is the inverted F antenna (IFA). The monopole element is bent over in an L shape parallel to the ground area on the circuit board, to make it compact enough to be enclosed in the device case; the antenna may be fabricated of copper foil on the printed circuit board itself. To improve the impedance match with the feed circuit the antenna is shunt fed, the feedline is connected to an intermediate point along the element, and the base of the element is grounded. Many variants of this antenna are used in handheld devices, such as multiband versions and meander antennas.

History

Development of Marconi's monopole antenna from Hertz's dipole antenna

Hertz's dipole antenna transmitter, invented 1886

Marconi first tried enlarging the dipole antenna with 6×6 foot metal sheet "capacity areas" (r), 1895 Metal sheets and spark balls not shown to scale.

Illustration from Marconi's 1896 patent showing his first monopole antennas, consisting of suspended metal plates (u,w) attached to one terminal of the transmitter (left) and receiver (right), with the other terminal grounded (E). Later he found that the plates were unnecessary and a suspended wire was adequate.

Marconi's first monopole transmitter

One of Marconi's early monopole antennas at his Poldhu, Cornwall transmitting station, 1900, consisting of a small metal plate suspended from a wooden arm with a long wire running down to the transmitter in the building.

Early vertical antennas. (A) Marconi found suspending the metal plate "capacity area" high above the ground increased range. (B) He found that a simple elevated wire worked just as well. (C-F) Later researchers found that multiple parallel wires were a better way to increase capacitance. "Cage antennas" (E-F) distributed current more equally between wires, reducing resistance

The monopole antenna was invented in 1895 and patented in 1896 by radio entrepreneur Guglielmo Marconi during his first experiments in radio communication. He began by using dipole antennas invented by Heinrich Hertz consisting of two identical horizontal wires ending in metal plates, and by his mentor Augusto Righi consisting of four metal spark balls, but was unable to transmit further than about a half mile. He found by experiment that if instead of the dipole, one side of the transmitter and receiver terminals was connected to a wire suspended overhead, and the other side was connected to a conductor buried in the Earth, he could transmit for longer distances. For this reason the monopole is also called a Marconi antenna, although Alexander Popov independently invented it at about the same time for his lightning detection receiver.

In the next few years, using the monopole antenna Marconi steadily increased the range of his radiotelegraphy communication system to hundreds of kilometers, convincing the world that radio was a practical communication method. In 1901 he achieved transatlantic radio transmission using a monopole transmitting antenna consisting of 50 vertical wires suspended in a fan shape from 60 meter poles.

Before Marconi, several inventors experimented with wireless communication between vertical aerials, although without creating a practical system. In October 1866 Mahlon Loomis demonstrated communication between two grounded 183-meter (600-foot) wire aerials supported by kites on mountaintops 22 kilometres (14 miles) apart in the Blue Ridge Mountains. When one aerial wire was touched to a grounded contact, currents of atmospheric electricity in it apparently generated radio waves which induced currents in the other wire, detected by a sensitive galvanometer. Starting in 1882, Amos Dolbear also used grounded kite-supported vertical wire antennas during his development of a ground conduction telephone, but his system seems to have worked by electrostatic induction instead of radio waves, and by 1895 he had only achieved distances of 1/4 mile. A suit claiming Marconi infringed Dolbear's 1882 and 1886 wireless patents was dismissed in 1901. In 1885 Thomas Edison patented a system of ship communication between vertical towers on shore and vertical wires suspended from a ship's mast, but this also worked by electrostatic induction and was never tried.

In the primitive spark transmitters used in Marconi's time, in addition to radiating the radio waves the antenna also served as the resonator which generated the oscillating currents which determine the frequency and thus the wavelength of the waves. Marconi's new antenna functioned as a quarter-wave monopole which radiated with a wavelength of approximately four times its height. This longer antenna greatly increased the wavelength, reducing the frequency of Marconi’s transmitter from the VHF and UHF bands generated by Hertz's antennas which could not transmit beyond the horizon, to the MF band. Also, it emitted vertically polarized radio waves, instead of the horizontally polarized waves produced by the Hertz antenna. Longer radio waves have less attenuation with distance. These longer vertically polarized waves could propagate as ground waves which can follow the curvature of the Earth, and could also reflect off the ionosphere (called the 'skip' or skywave mechanism), and thus travel beyond the visual horizon. This explains the increased range.

Marconi, who was self-educated in physics, didn't understand any of this at the time; he merely discovered an empirical relation between antenna height and transmission distance. He credited Prof. Moisè Ascoli of Rome with first calculating in 1897 that the antenna radiated at a wavelength of four times its height. An integral equation for the current in wire antennas was derived by Henry Pocklington in 1897, who showed the current was approximately a sinusoidal standing wave. In 1906 Prof. John Ambrose Fleming, engineer for the Marconi company, mathematically explained the radiation pattern of the monopole using image theory. A more useful version of the Pocklington equation, the Hallen equation, was derived by Erik Hallén beginning in 1938. These integral equations are the starting point in modern analyses of monopoles, and are solved numerically in modern computer antenna simulation programs.

Multiwire monopoles used during the radiotelegraphy era, 1900 - 1930s

Marconi 210 ft (64 meter) inverted cone monopole transmitting antenna, Poldhu, UK, built 1902

An inverted-L, "triatic" or "flattop" antenna, a monopole used in long distance radiotelegraphy transmitting stations at frequencies below 50 kHz, invented by Marconi in 1904

Umbrella antenna of radiotelegraph station near Newcastle, UK in 1910

T antenna of radio station WBZ, 833 kHz, Springfield, MA, built in 1921.

During the radiotelegraphy era, the first two decades of radio from 1900 until the 1920s, radio communication systems used long wavelengths in the MF, LF and VLF bands. Monopoles were the main antennas used. At the longer wavelengths used for long distance communication the antenna masts that could be practically constructed were electrically short, shorter than the resonant length, one-quarter wavelength. Simple monopole antennas this short are inefficient; due to their low radiation resistance of 5 to 20 ohms, a large fraction of the transmitter power was wasted in the ground system resistance. At this time the main technique known for increasing radiated power was to add conductors to the top of the antenna, to increase the capacitance to ground and thus the antenna current. Marconi and others developed huge multiwire capacitively top-loaded monopole antennas which were more efficient at these frequencies, such as the harp, inverted cone, inverted L, T antenna and umbrella antenna. These were the main antennas during this period, and are still the main transmitting antennas used at these low frequencies. When radio broadcasting began in the MF band in the early 1920s, the typical transmitting antenna was the T-antenna. This required two masts, an extensive land area, and currents in the masts distorted the radiation pattern.

Two papers published in 1924 by Stuart Ballantine led to the adoption of the single mast monopole radiator. One derived the radiation resistance of a vertical monopole antenna over a perfect ground plane. He found that the radiation resistance increased to a maximum at a length of a half wavelength, so a mast around that length had an input impedance that was much higher than the ground resistance, reducing the fraction of transmitter power that was lost in the ground system, eliminating the need for capacitive toploads. In a second paper the same year he showed that the amount of power radiated horizontally in ground waves reached a maximum at a mast height of 5/8 wavelength (.625$\lambda$). Due to these discoveries, by 1930 the disadvantages of the T antenna led broadcasters to adopt the half-wave mast radiator antenna in the medium frequency band. Radial wire ground systems were developed at the same time to reduce ground losses.

The advent of handheld radios in the 1950s-60s, the transistor radio and walkie-talkie, made possible by the invention of the transistor in 1947, motivated the development of compact monopole antennas for them, like the retractable quarter-wave whip and rubber ducky antenna.

Elementary description of operation

A monopole antenna, like the dipole antenna from which it is derived, is a resonant antenna; it not only emits and receives radio waves but acts as an electrical resonator. When the radio frequency alternating current applied to its feedpoint is near one of its resonant frequencies, in addition to radiating the power as radio waves, energy is stored in the antenna as oscillating electric currents called standing waves. The advantage of this is that the stored energy is larger than the energy fed to the antenna each cycle by the transmitter (or in a receiving antenna the energy absorbed from the radio waves), so most of the current in the antenna is due to this stored energy. As a result the antenna current at resonance is larger than the current when the antenna is driven at other frequencies. The radio wave power radiated by an antenna is proportional to the square of the antenna current, so an antenna fed at a resonant frequency radiates much more power than the same antenna fed with the same voltage at some other frequency. An antenna only absorbs all the input power from the feedline when it is in a condition of resonance.

The vertical conductor acts somewhat like a transmission line stub, open-circuited at the top. The oscillation modes are analogous to the mechanical oscillations of an elastic beam anchored at one end. The current and voltage along the element are sinusoidal waves. The current in the antenna element bounces back and forth between the ends, and the two equal but opposite current waves interfere to form a standing wave. The standing wave has a current node at its top and either a node or an antinode at bottom. Due to these end conditions the antenna is resonant (has pure resistive input impedance) at a length of a quarter wavelength or multiples of it.

In the common quarter wave monopole, the top end of the vertical rod and the ground plane act as capacitor plates which have opposite charges, storing energy in an electric field, while the middle of the rod acts as an inductor which stores energy in a magnetic field, so the entire antenna acts like a series-resonant tuned circuit. If the top of the rod is negatively charged and the ground plane positively charged at the beginning of the cycle, the current begins to flow up the rod from the ground plane, creating a circular magnetic field around the rod. The negative charge at the top and positive charge on the ground plane decrease until they reach zero. However the current continues, because the inductance of the rod resists changes in current. The current begins to charge the top of the rod positive and the ground plane negative. From Faraday's law of induction the energy to create this separation of charge comes from the magnetic field, which decreases. Finally when the magnetic field reaches zero the current stops with the charges reversed, the top of the rod is charged positive and the ground plane negative. Then the current begins to flow in the opposite direction, down the rod, generating a magnetic field circling in the opposite direction, until the charges reverse again to their original polarity, with the top of the rod negative and the ground plane positive. This oscillation keeps repeating, with the energy stored alternately in the electric field and the magnetic field each half-cycle of the applied alternating current.

Most of these oscillating electric and magnetic fields are near fields (also called reactive or induction fields) which store energy in the space around the antenna, but some of the fields leave the antenna and travel away as electromagnetic waves, radio waves, carrying energy with them. The radiated power is provided by incoming power from the feedline. Due to this power loss, an antenna acts as if it has a resistance, the radiation resistance, at its feedpoint.

As a result, a monopole acts electrically like a lossy tuned circuit; in general it has both electrical resistance and reactance at its feedpoint. The input resistance has two components; the radiation resistance (normally the largest part) and the loss resistance due to ohmic losses in the antenna conductor and ground plane. At resonance the input impedance is just this pure resistance; at other frequencies it has reactance in addition to the resistance, and thus a higher impedance.

A transmitting antenna will absorb all the power applied to its feedpoint only if it is conjugate impedance matched to the feedline from the transmitter. This means the resistance of the antenna and line must be equal, and the reactance of antenna and line must be opposite. If it is not impedance matched, some of the transmitter power from the feedline will be reflected back down the line toward the transmitter, causing a high SWR, resulting in inefficiency and possibly overheating the transmitter or line, or causing arcing. Similarly, a receiving antenna will only transfer a maximum amount of radio power to the receiver if it is impedance matched to the line.

The ground plane

Most monopoles have a conducting surface under the vertical rod, a ground plane, connected to the ground side of the feedline. The ground plane is an integral part of the antenna; it has two functions. First, it reflects the downward directed radio waves, increasing the power radiated above the ground. Second, it acts as a capacitor plate, receiving the displacement current (alternating electric field) from the rod, returning it to the ground side of the feedline. Without it there will be induced currents on the outside of the shield conductor of the feedline, which will act as an antenna.

The current in the ground plane is radial, directed alternately toward and away from the ground terminal at the base of the antenna. Therefore far from the antenna the radio waves radiated by the currents in opposite sides of the plane have opposite phase and largely cancel. So the plane itself does not radiate; it acts as a mirror for the radio waves from the rod.

The electric field is vertical where it enters the ground plane, identical to the field of a vertical dipole antenna at its symmetry plane. If the ground plane is large enough, due to the waves reflected from it the antenna acts as if it has an image antenna identical to the monopole underneath the plane. The antenna rod and its image together act like a dipole antenna of twice the length, so a monopole over an infinite, perfectly conducting plane has a radiation pattern identical to the top half of the pattern of a vertical dipole of twice the length. For the common quarter wave monopole element the antenna acts like a half wave dipole. Because the antenna only radiates its power into half the space of a dipole antenna, its gain is twice (or in decibels, 3 dB greater than) the gain of an equivalent dipole.

The actual gain and radiation pattern is dependent on the size and conductivity of the ground plane. To function as a mirror the plane must extend least a half wavelength from the monopole element. Low frequency monopole transmitting antennas use the Earth itself as the ground plane. They require a good low resistance connection to the Earth for efficiency, since the soil has significant resistance which is in series with the antenna and consumes transmitter power. These use a radial ground system consisting of many bare copper wires buried shallowly in the earth, radiating from a ground terminal at the base of the antenna, preferably to a distance of a quarter to a half wavelength.

Because of the unbalanced impedance of the ground plane, monopole antennas are most easily fed from an unbalanced transmission line, usually coaxial cable.

Current distribution on antenna

Calculating the current distribution along a thin linear antenna, which determines the radiation pattern and electrical characteristics, requires solving Maxwell's equations for the coupled current, electric and magnetic fields at the surface of the element, driven by the electric field of the sinusoidal feed voltage from the transmitter applied to the antenna's feedpoint (or in a receiving antenna by the incoming electric field of the radio wave). The Pocklington integral equation (Henry Pocklington, 1897) or Hallen integral equation (Erik Hallén, 1938) give the current on thin cylindrical antennas. In general, accurate calculation of an antenna's electrical properties is mathematically difficult, and antenna simulation computer programs like NEC are usually used.

If the ground plane is a good conductor larger in radius than the height of the element (which will be assumed in this section), it approximates a perfect infinite ground plane, and the current and radiation can be calculated by replacing the monopole and plane with a vertical dipole antenna of twice the height. For smaller planes, for accurate results resonances in the ground plane and refraction around the edges must also be taken into account, so the current distribution in the plane must be calculated, making analysis more difficult.

The current in the monopole element is approximately a sinusoidal standing wave $I(z)$ composed of two superimposed traveling current waves, one $i_{\text{up}}(z,t)$ moving up the antenna and reflecting from the top, the other $i_{\text{down}}(z,t)$ moving down and reflecting from the ground plane.

$i(z,t)=I(z)\cos {\omega t}=i_{\text{up}}(z,t)+i_{\text{down}}(z,t)$

From the Pocklington equation, to a first approximation, the current on a thin antenna is given by the Helmholtz equation

${\frac {{\partial ^{2}}I(z)}{\partial z^{2}}}+k^{2}I(z)=0$

This is the same as the Telegrapher's equation for a lossless transmission line, explaining why linear antennas behave like transmission lines. Solving for the monopole element, and applying the boundary condition $I(h)=0$ the peak radio frequency current $I(z)$ at a height of $z$ on the monopole is approximately a sinusoidal standing wave

   $I(z)=I_{\text{max}}\sin {k(h-z)}=I_{0}{\sin {k(h-z)} \over \sin {kh}}$ 

where

$k={2\pi \over \lambda }={2\pi f \over c}$ is the wavenumber in radians per meter

$I_{\text{max}}$ is the loop current, the current at the antinode of the standing wave. In a monopole of $\lambda /4$ or longer it is the maximum current on the antenna.

$I_{0}$ is the input current at the base of the element, for base-fed monopoles

$h$ is the length of the monopole element.

$z$ is the height on the element measured from the ground plane

This approximation assumes the Q_factor of the antenna is much greater than one; in other words the stored energy is much larger than the feed energy per cycle which is equal to the radiated energy. This is a good approximation for a thin antenna driven at resonance. It is numerically accurate for a diameter-to-wavelength ratio less than .05, but applies qualitatively even to thick monopoles. For finite width monopoles the current does not quite go to zero at the nodes, and the 180 degree phase change there is not abrupt but occurs continuously over a short distance centered on the node.

Since this approximation assumes the energy applied by the feedline, and the energy lost to radiation, are negligible, the voltage across the feedline and the radiation resistance are implicitly assumed to be zero. A second approximation takes these factors into account. The current is the sum of two terms: the original sinusoidal wave which is 90° out of phase with the feed voltage, and a second smaller wave in phase with the feed voltage which supplies the radiated power. This decreases with height along the element

$I(z)=I_{0}\sin {k(h-z)}+jk_{1}I_{0}[\cos kz-\cos kl]$

(The input resistance is due to the radiation from the antenna which is neglected in this first "asymptotic" approximation to the current.

Radiation resistance and reactance

Since the input impedance varies depending on where on the monopole rod the feedline is connected, radiation resistance and reactance is customarily defined at the current antinode on the antenna, or else at the current maximum. For a monopole $\lambda /4$ or shorter this is at the bottom, while for a longer monopole the antinode occurs $\lambda /4$ below the top. Over a perfectly conducting infinite ground plane, the input impedance of a monopole is half that of a center-fed dipole twice the length.

For fast calculation the following approximate formulas are convenient
$R_{\text{R}}={\begin{cases}10(kh)^{2}\qquad \quad 0<h<\lambda /8\\12.35(kh)^{2.5}\quad \lambda /8<h<\lambda /4\\5.57(kh)^{4.17}\;\quad \lambda /4<h<0.3183\lambda \end{cases}}$
where $k={2\pi \over \lambda }$

Resonant frequencies and lengths

A monopole antenna is resonant (has pure resistive input impedance, no reactance) at a series of frequencies, which depend on its length $h$. For precise results antenna simulation computer programs must be used. However, for most monopole and dipole antennas in which the element is not excessively thick, the resonant frequencies are often calculated approximately by regarding the conductor as an open-ended single wire transmission line (resonant stub). As in a resonant stub, the standing wave is mainly storing energy, not transporting it, so the phase difference between the current and voltage standing waves is close to 90°. This means the voltage standing wave has an antinode (maximum) at each current node (minimum), and a node (minimum) at each current antinode (maximum).

Series resonances

The condition for resonance in a monopole, analogous to a vibrating string, is that when the sinusoidal current wave travels a round trip from one end of the monopole element to the other and back, the reflected wave must arrive at its starting point in phase with the original wave, so the two waves reinforce.

The wave travels along the element at a velocity close to the speed of light $c$. The distance the wave travels in one period $T=1/f$ is the wavelength $\lambda$ where

$\lambda ={c \over f}$

Therefore the phase change in radians of the wave from one end of the element to the other is $2\pi {h \over \lambda }$. For a round trip the phase change is twice this, $4\pi {h \over \lambda }$. At the ends of the element there can be an additional phase change, which depends on the end conditions. For a monopole at the so-called series resonances:

• The current reflects from the top with a 180° ($\pi$ radians) phase change: At the top of the element, the total current must be zero because there is no place for it to go, making this point a current node (zero) of the standing wave. So the upward and downward traveling waves must have equal but opposite amplitude there, $i_{\text{up}}(h,t)=-i_{\text{down}}(h,t)$, The upward current wave is said to "reflect" from the top end of the element with opposite phase.
• The current reflects from the ground plane with no phase change: The downward wave travels down the element, through the feedline to the transmitter and back, and at the bottom reflects from the ground plane to become the upward wave. The ground plane, which can be modeled as a large capacitor plate connected to the ground conductor of the feedline, acts as a current source or sink, its voltage is approximately zero regardless of the current into it. Therefore the element has a voltage node (zero) and a current antinode (maximum) there. Since the voltage waves must be equal and opposite at the ground plane, $v_{\text{up}}(t)=-v_{\text{down}}(t)$, and there is a sign change due to the opposite directions of the currents, the upward and downward current waves are always equal in amplitude there, they are in phase

$i_{\text{up}}(0,t)=-C{dv_{\text{up}}(0,t) \over dt}=-C{d[-v_{\text{down}}(0,t)] \over dt}=i_{\text{down}}(0,t)$

The sine wave repeats every $2\pi$ radians (360°). So for resonance the total phase change $\phi$ during a round trip along the antenna element, including the $\pi$ (180°) phase shift at the top, must be $2\pi$ or an integral multiple of it

$\phi =4\pi {h \over \lambda }+\pi =2\pi m\qquad m\in 1,2,3,...$

Solving for $h$ and substituting $m=n+1$ the monopole antenna is resonant at a length of a quarter wavelength or an odd multiple of it

    $h={\lambda \over 4}+n{\lambda \over 2}={\big (}{1 \over 4}+{n \over 2}{\big )}{c \over f}\qquad n\in 0,1,2,3,...$ 

$h={\lambda \over 4},{3\lambda \over 4},{5\lambda \over 4},...$

(The resonant lengths are actually slightly shorter than this, see End effects section below.) For a given length $h$ the corresponding resonant frequencies $f_{\text{n}}$ are

    $f_{\text{n}}={\big (}{1 \over 4}+{n \over 2}{\big )}{c \over h}\qquad n\in 0,1,2,3,...$

The lowest resonant frequency, $f_{\text{0}}$, at which the antenna is a quarter-wavelength ($\lambda /4$) long, is called the fundamental resonance, while the higher resonances, which are multiples of the fundamental, are called harmonics.

These are sometimes called the series resonant frequencies because the antenna acts similar electrically to a series resonant tuned circuit. Because the feedpoint at the bottom of the antenna is a voltage node (minimum) and current antinode (maximum), at these frequencies the input impedance, equal to the ratio of voltage to current, is a minimum. For a $\lambda /4$ monopole it is 36.5 ohms (as shown below).

Parallel resonances

The monopole can also resonate at a second series of lengths, at which the bottom end of the element is a current node (minimum) instead of a current antinode (maximum). So the element has a node of the current standing wave at both top and bottom, it is equivalent to an end-fed vertical dipole antenna (many sources classify this antenna as a dipole instead of a monopole). The resonant frequencies can be calculated by a derivation similar to that in the previous section, but it is easier to note that a standing wave has a node at intervals of one-half wavelength. Therefore the antenna is resonant at lengths at which it is a half-wavelength long or a multiple of it

    $h=n{\lambda \over 2}={nc \over 2f}\qquad n\in 1,2,3,...$ 

$h={\lambda \over 2},\lambda ,{3\lambda \over 2},2\lambda ,...$

For a given length $h$ the corresponding resonant frequencies are

    $f_{\text{n}}={nc \over 2h}\qquad n\in 1,2,3,...$ 

These are sometimes called parallel resonances or antiresonances because the antenna acts similar to a parallel resonant (antiresonant) tuned circuit. When fed at the bottom, due to the current node and voltage antinode there the antenna has a very high input resistance, which is difficult to calculate. It would be infinite for an infinitely thin element, but for a typical finite thickness monopole element it is around 600 - 4000 ohms depending on thickness. It also has a very high rate of change of reactance with frequency about the resonance point, which gives the antenna a narrower bandwidth than at the series resonances.

To reduce the impedance enough to match a transmission line, either an impedance matching circuit, shunt feed, or a very thick monopole element must be used. An advantage is that since it acts as a dipole the current in the ground system is low, so ground losses are minimized; for thin antennas a ground plane is not needed at all.

In practice monopoles are mainly used at the two lowest resonant frequencies; where the element is one quarter of the wavelength long ($\lambda /4$), the quarter-wave monopole, or one half of the wavelength long ($\lambda /2$), the half-wave monopole, because their radiation patterns consist of a single lobe in horizontal directions, perpendicular to the antenna axis. Higher harmonics are little used since they have more complicated radiation patterns consisting of multiple lobes directed at angles into the sky with nulls (directions of minimum radiation) between them.

End effects

Reduction of resonant length of monopole as a function of element diameter. The resonant length decreases with increasing diameter of the element. If $\delta$ is the reduction factor above, the resonant length of a quarter wave monopole is $h=\delta (\lambda /4)=\delta c/4f$. As the ratio of rod diameter to wavelength increases, the node of the standing wave moves further above the end of the element, so at a given frequency the antenna is resonant at a shorter length.

The lengths at which physical monopoles are resonant are a little shorter than the theoretical resonant lengths $h$ calculated in the previous section, and depend on the element's diameter.

This is due to the shape of the electric field (fringing field) at the top end of the element which spreads out in a fan. This adds capacitance and reduces the inductance at the end. Due to this ability to store more charge, near the top the standing wave current profile differs from a sine wave, decreasing faster with distance. When approximated as a sine wave, this is equivalent to the node of the standing wave occurring not at the top of the antenna but some distance above it. So the resonant length of the element is shorter than a multiple of one quarter of the free space wavelength $\lambda =c/f$ from the previous section.

Another common way of saying this is that the resonant frequencies depend on the electrical length of the element (length in wavelengths of the current in the conductor), not the physical length (length measured in wavelengths of the radio wave in free space). The electrical length of a linear antenna is longer than its physical length, so the resonant frequencies are lower than would be calculated from its physical length.

A thin monopole exactly one-quarter free space wavelength long, .25$\lambda$, fed at bottom, has an inductive input impedance of 36.5 + j21.25 ohms. To make it resonant it can be shortened to .237$\lambda$ at which length it has an input impedance of 34 + j0 ohms.

In addition, anything which adds capacitance to the antenna element, such as the presence of grounded objects or high permittivity dielectric materials like insulating coatings or supporting electrical insulators near the end of the element will further decrease the resonant length.

These are collectively called "end effects". Due to these effects the resonant length $h$ of a typical quarter wave monopole antenna is about 5% shorter than its theoretical length from the previous section, or for $h$ in meters

   $h\approx 0.95{c \over 4f}={71.25 \over f{\text{(in MHz)}}}$  

In the rest of the article, when the resonant length of a monopole is mentioned, it is assumed to include this correction.

(The approximate standing wave is $I(z)=I_{\text{max}}\sin {k(h+\delta -z)}$ where $\delta =\delta _{\text{fringe}}+\delta _{\text{cap}}={\frac {\lambda {\hat {X}}(h)}{2\pi {\hat {K}}}}+{\frac {a{\hat {K}}}{60\pi }}$)

Radiation pattern

Showing the monopole antenna has the same radiation pattern over perfect ground as a dipole in free space with the same current but twice the voltage

Vertical radiation patterns of monopole mast radiator antennas over perfect ground. The distance of the line from the origin at a given elevation angle is proportional to the power density radiated at that angle. The horizontal gain increases from a quarter-wave (0.25$\lambda$) through a half-wave (0.5$\lambda$) to a maximum at five-eights wavelength (.625$\lambda$)

Like a vertical dipole antenna, a monopole radiates radio waves in an omnidirectional radiation pattern: it radiates equal power in all azimuthal directions perpendicular to the antenna axis. The radiated power varies with elevation angle, with the radiation dropping off to zero at the zenith on the antenna axis. The radio waves are vertically polarized.

A monopole can be visualized (see diagram) as being formed by replacing the bottom half of a vertical dipole antenna (c) with a conducting plane (ground plane) at right-angles to the remaining half. If the ground plane is large enough, the radio waves from the remaining upper half of the dipole (a) reflected from the ground plane will seem to come from an image antenna (b) forming the missing half of the dipole, which adds to the direct radiation to form a dipole radiation pattern. So the pattern of a monopole with a perfectly conducting, infinite ground plane is identical to the top half of a dipole pattern.

Up to a length of a half-wavelength (${\lambda \over 2}$) the radiation pattern has a single donut-shaped lobe with maximum radiated power in horizontal directions, perpendicular to the antenna axis. As the length is increased above ${\lambda \over 4}$ the lobe flattens, radiating less power at high angles and more in horizontal directions.

Above a half-wavelength the pattern splits into a horizontal main lobe and a small second conical lobe at an angle of 60° elevation into the sky. However the horizontal radiated power and gain keeps increasing and reaches a maximum at a length of five-eighths wavelength: ${5 \over 8}\lambda =.625\lambda$ (this is an approximation valid for a typical thickness antenna, for an infinitely thin monopole the maximum occurs at ${2 \over \pi }\lambda =.637\lambda$) The maximum occurs at this length because the opposite phase radiation from the two lobes interferes destructively and cancels at high angles, leaving more power to be radiated in horizontal directions.

Above $.625\lambda$ the high angle lobe gets larger, becoming the main lobe, and the horizontal lobe rapidly gets smaller, reducing power radiated in horizontal directions, so very few antennas use lengths above this. At the 4th harmonic, $h=\lambda$, the horizontal lobe disappears and all the power is radiated in the high angle lobe. As the antenna is made longer, the pattern divides into more lobes, with nulls (directions of zero radiated power) between them.

The general effect of electrically small ground planes, as well as imperfectly conducting earth grounds, is to tilt the direction of maximum radiation up to higher elevation angles and reduce the gain.. When mounted on the Earth, due to the finite resistance of the soil the portion of the ground wave propagating horizontally in contact with the ground is attenuated exponentially and vanishes at long distances, so in the (far field) radiation pattern the radiated power declines smoothly to zero at the horizon (zero elevation angle).

With an asymmetrical ground plane, such as a whip antenna mounted on a car’s bumper,the pattern will no longer be omnidirectional, but will have stronger horizontal radiation on the side with the larger ground plane area.

Gain and input impedance

Characteristics of a thin monopole over perfect ground

Length $h$ wavelengths	Gain   dBi	3dB Beamwidth degrees	Radiation resistance ohms
≪0.25	4.76	45°	$40\pi ^{2}(h/\lambda )^{2}$	electrically short
0.25	5.19	39°	36.5	fundamental
0.375	5.75	32°	92.8
0.5	6.82	24°	>600	2nd harmonic
0.625	8.16	16°	53.2	maximum gain
0.75			52.7	3rd harmonic

Radiation pattern of 3/2 wavelength monopole. Monopole antennas up to 1/2 wavelength long have a single "lobe", with field strength declining monotonically from a maximum in the horizontal direction, but longer monopoles have more complicated patterns with several conical "lobes" (radiation maxima) directed at angles into the sky.

Because it radiates only into the space above the ground plane, or half the space of a dipole antenna, a monopole antenna over a perfectly conducting infinite ground plane will have a gain of twice (3 dB greater than) the gain of a similar dipole antenna, and a radiation resistance half that of a dipole. Since a half-wave dipole has a gain of 2.19 dBi and a radiation resistance of 73.1 ohms, a quarter-wave ($\lambda /4$) monopole will have a gain of 2.19 + 3 = 5.19 dBi and a radiation resistance of about 36.5 ohms. The antenna is resonant at this length, so it's input impedance is purely resistive. The input impedance has capacitive reactance below $\lambda /4$, inductive reactance from $\lambda /4$ to $\lambda /2$, and capacitive reactance from $\lambda /2$ to $3\lambda /4$.

The gain figures given in the table above are never approached in practice; they would only be achieved if the antenna was mounted over an infinite perfectly conducting ground plane. The ground plane is part of the antenna, and the gain is highly dependent on its size and conductivity. An artificial ground plane a wavelength or more in radius is equivalent to a infinite plane, but for smaller planes, which are often used at high frequencies, the gain will be 2 to 5 dBi lower, because some of the horizontal radiated power will diffract around the plane edge into the lower half space. Similarly over a resistive Earth ground the gain will be lower due to power absorbed in the Earth.

For electrically short monopoles below $\lambda /4$ the gain decreases slowly; it is 4.76 dBi at $\lambda /20$.

As the length is increased to a half-wavelength ($\lambda /2$), the gain increases to about 1.7 dB over the $\lambda /4$ gain. Since at this length the antenna has a current node at its feedpoint, the input impedance is very high.

The gain continues to increase up to a maximum of about 3 dB over a quarter-wave monopole at a length of five-eighths wavelength ($.625\lambda$) so this is a popular length for ground wave antennas and terrestrial communication antennas. The radiation resistance drops to about 53 ohms at that length. Above $.625\lambda$ the horizontal gain drops rapidly because more power is radiated at high elevation angles in the second lobe.

Directivity equation

The power density $s$ of radio waves radiated by a monopole mounted over a perfectly conducting infinite ground plane, as a function of elevation angle $\theta$ relative to horizontal [$s(0^{\circ })=1$] is

    $s(\theta )={\cos({h \over \lambda }\sin \theta )-\cos {h \over \lambda } \over (1-\cos {h \over \lambda })\cos \theta }$ 

where

$G={h \over \lambda }$ radians, or $360^{\circ }{h \over \lambda }$ degrees, is the electrical length of the monopole element

Impedance matching

A transmitting antenna will absorb all the power supplied by the transmitter feedline, and in a receiving antenna the receiver will absorb maximum power from the antenna, only if the antenna is impedance matched to the transmitter or receiver. This means the impedance presented by the feedline must be the complex conjugate of the input impedance of the antenna; they must have equal resistance and opposite reactance at the operating frequency. For efficiency most transmitting monopoles are impedance matched to the feed circuit. For receiving antennas below 30 MHz matching is not very important since losses in the feed circuit can be compensated by amplification in the receiver.

If the monopole's length is resonant at the operating frequency, the antenna impedance is a pure resistance, so for matching it is only necessary to transform the characteristic resistance of the feedline, if needed, to match the antenna resistance. Monopoles, as unbalanced loads, are usually fed with coaxial cable. The impedance of standard 50 or 75 ohm coaxial cable is a poor match to the 36.5 ohm radiation resistance of a quarter-wave monopole. A small monopole's impedance can be raised to match 50 ohm coax by an impedance matching network, tilting the ground plane rods diagonally down, or adding a cylindrical sheet metal 'sleeve' around the bottom, this is called a sleeved monopole.

In many circumstances it is not convenient or not possible to use an antenna of resonant length. If a transmitting monopole antenna is operated at a frequency different from its resonant frequencies, the antenna will have reactance and will reflect some of the input power back down the feedline toward the transmitter, causing a high VSWR on the line, wasting energy and possibly overheating the line or transmitter or causing arcing.

In this case the antenna can be made resonant at the desired frequency by canceling the reactance of the antenna by adding an equal but opposite reactance, an inductor or capacitor, in series with the feedline at the base of the antenna. The antenna and reactance together act as a tuned circuit resonant at the feed frequency, with an input impedance that is purely resistive.

• A monopole shorter than a quarter wavelength will have capacitive reactance, so it can be made series resonant by adding an inductor called a loading coil at its base. This is called electrically lengthening the antenna. This technique is widely used to match a transmitter to an electrically short antenna that is a more convenient length.
• A monopole between a quarter and a half-wavelength will have inductive reactance, so it can be made series resonant by adding a capacitor at its base. This is called electrically shortening the antenna.

Bandwidth

The bandwidth of a monopole antenna, the range of frequencies over which it is resonant and works efficiently, increases with the thickness of the element. The radiated power and standing wave ratio (VSWR) on the feedline as a function of frequency is given by a smooth resonance curve similar to a tuned circuit, which has a peak at its resonant frequency. Surrounding the resonant frequency is a defined band of frequencies called the bandwidth $B$ within which the antenna is considered adequately impedance matched to the feedline and radiates close to maximum power. The radio frequency driving signal from the transmitter and its modulation sidebands, or the received signal in receiving antennas, should remain within its bandwidth for the antenna to operate with rated gain. Outside it the antenna loses impedance match with the feedline, the VSWR and reflected power increases, and the gain and radiated power drops rapidly.

The antenna's bandwidth $B$ is defined as the difference between the frequencies on either side of resonance at which the antenna output power has dropped to half of (3 dB less than) its maximum value. A measure of the bandwidth of an antenna is its Q factor, equal to the ratio of the resonant frequency $f$ to the bandwidth $Q=f/B$. The $Q$ is also equal to 2π = 6.28 times the ratio of energy stored in the antenna to the energy input per cycle from the feedline, which is approximately equal to the radio wave energy radiated per cycle.

However the usable bandwidth of an antenna is more often specified by a maximum voltage standing wave ratio (VSWR) which determines the maximum allowable transmitter power reflected by the antenna back down the feedline. Typical values are 1.5 or 2.

The bandwidth of a monopole increases with the width (diameter) of the conductive element. As the length-to-diameter ratio of the rod decreases, the standing wave current in the antenna departs from a sine wave and the resonance curve broadens. Bandwidth of a "typical" monopole antenna is around 10% of the resonant frequency, but ranges from 2% for a thin wire element to 30% for a thick rod. Where a really broadband monopole is needed a cage monopole is sometimes used, consisting of a fat cylindrical element made of an open latticework of metal rods.

Types of feed

Because in a resonant antenna the energy fed to the antenna by the transmitter each cycle is small compared to the energy stored in the standing wave on the antenna, the feed current can be applied at different points on the antenna without altering the current standing wave pattern much; leaving the radiation pattern the same. The advantage of this is that at different points on the antenna the input impedance has different values, allowing the possibility of impedance matching the antenna to the feedline characteristic impedance without a matching network, by choosing the correct feedpoint.

• Series or base feed - This is the most common type, the type discussed above, in which the feedline is connected between the base of the monopole and the ground plane. For the $\lambda /4$ monopole and odd harmonics the input impedance is a minimum, 36.8 ohms for a quarter-wave monopole. For the $\lambda /2$ monopole, the impedance is very high, requiring a matching transformer or thick monopole.
• Shunt feed - One side of the feedline is connected to ground, and the other to a point along the antenna element, and the base of the element is grounded. The part of the element between the feedpoint and ground acts as a shorted stub. Since the impedance is zero at the base and increases continuously to a very high value, 800 - 4000 ohms at a height of $\lambda /4$, any input impedance between these values can be realized by choosing the correct feed height on the element.

• Gamma match - a shunt feed with a capacitor in the feedline connecting to the element.

• Folded monopole - A monopole can also be fed at the top, by grounding the base of the element, mounting a parallel conductor next to it, attached at the top, and feeding this conductor at the bottom. Because of their proximity the two elements are coupled so the current and voltage are the same in each. The folded monopole has a radiation resistance of 4 times the base fed monopole.

Electrically short monopoles

A monopole shorter than the fundamental resonance length of a quarter-wavelength at its operating frequency is called electrically short. Even a very short rod a small fraction of a wavelength long can be impedance matched to a transmitter so it absorbs all the power from the feedline. Electrically short monopoles are widely used since they are more compact, and for long wavelengths construction limitations make it impractical to build an antenna mast a quarter wavelength high. However as the length is decreased the antenna eventually becomes inefficient due to its low radiation resistance.

Below a quarter wavelength the radiation resistance of a monopole decreases with the square of the ratio of length to wavelength

$R_{\text{R}}=40\pi ^{2}{\Big (}{h \over \lambda }{\Big )}^{2}\qquad h<<{\lambda \over 4}$

The radiation resistance is only part of the feedpoint resistance at the antenna terminals. A monopole has other energy losses which appear as additional resistance at the antenna terminals; ohmic resistance of the metal antenna elements, dielectric losses in insulating materials, feedline losses, and particularly resistive losses in the Earth ground system, often the largest loss factor in low frequency monopoles. The total feedpoint resistance $R_{\text{IN}}$ is equal to the sum of the radiation resistance $R_{\text{R}}$ and loss resistance $R_{\text{L}}$

$R_{\text{IN}}=R_{\text{R}}+R_{\text{L}}$

The power $P_{\text{IN}}$ fed to the antenna is split proportionally between these two resistances. From Joule's law

$P_{\text{IN}}=I_{\text{IN}}^{2}(R_{\text{R}}+R_{\text{L}})$

$P_{\text{IN}}=P_{\text{R}}+P_{\text{L}}$

where

$P_{\text{R}}=I_{\text{IN}}^{2}R_{\text{R}}=40\pi ^{2}{\Big (}{LI_{\text{IN}} \over \lambda }{\Big )}^{2}\quad$ and $\quad P_{\text{L}}=I_{\text{IN}}^{2}R_{\text{L}}$

The power $P_{\text{R}}$ consumed by radiation resistance is converted to radio waves, the desired function of the antenna, while the power $P_{\text{L}}$ consumed by loss resistance is converted to heat, representing a waste of transmitter power. So for minimum power loss it is desirable that the radiation resistance be much greater than the loss resistance. The ratio of the radiation resistance to the total feedpoint resistance is equal to the efficiency $\eta$ of the antenna as a transducer

$\eta ={P_{\text{R}} \over P_{\text{IN}}}={R_{\text{R}} \over R_{\text{R}}+R_{\text{L}}}$

As the monopole is made shorter it's radiation resistance decreases and a greater proportion of the transmitter power is dissipated in the loss resistance. At a length of 0.06$\lambda$, the radiation resistance drops to about 2 ohms, the typical resistance of a buried radial ground system, so in an Earth-grounded antenna about 50% of the transmitter power would be wasted in the ground resistance. In the VLF band the huge top-loaded wire monopoles used by megawatt military transmitters are often less than ___ high and have radiation resistance of only ____ . Even with extremely low resistance ground systems they are often only 15% to 30% efficient.

Another disadvantage of electrically short monopoles is that the low resistance combined with the high capacitance and inductance of the monopole element gives the antenna a large Q factor, it has a narrow bandwidth, which reduces the data rate that can be transmitted or received.

Capacitively top-loaded monopoles

Capacitive "top hat" on mast of AM radio tower in Hamersley, Australia

To increase the radiated power of an electrically short monopole, capacitance to ground can be added to the top by attaching horizontal metal conductors to the top of the element. This is called a top loaded monopole. This results in increased current in the vertical monopole element, to charge and discharge the capacitance each cycle. Since the power radiated by a monopole is proportional to the square of the current in the radiating element, this increases the radiated power. The buried radial wire ground system under the antenna serves as the bottom plate of the 'capacitor'.

Mast radiators sometimes include a circular structure of radial rods at the top of the mast; this is called a 'top hat'. At lower frequencies in the LF and VLF bands larger top loads are used. The T antenna consists of a vertical wire driven at the bottom, rising to attach to the center of a horizontal top load wire insulated at both ends, supported by masts. Multiple parallel top load wires can be used to increase capacitance. The largest top loaded antenna is the umbrella antenna, consisting of a monopole mast radiator with many diagonal top load wires radiating symmetrically from the top, anchored to the ground through insulators. To tune out the high capacitive reactance and make the antenna resonant a large loading coil is required in series with the feedline at the base of the antenna.

At low frequencies, due to the high capacitance and low radiation resistance, the top loaded monopole has a very narrow bandwidth. This may limit the width of sidebands and thus the data rate that can be transmitted. High power transmitting antennas in the VLF band typically have $Q$ of several hundred and bandwidths less than 100 Hz. The energy stored in the antenna, stored alternately as an electrostatic field in the topload and a magnetic field in the loading coil, is hundreds of times the energy input from the transmitter each cycle. The voltage at the ends of the topload wires is very high, $Q$ times the feed voltage, and may be hundreds of kilovolts, requiring very good insulation. The antenna must be tuned to resonance with the transmitter using a variometer coil.

Effective height

The radiation resistance and radiated power output of any monopole antenna can be calculated from its effective height $h_{e}$. The effective height is the moment of its vertical current distribution $I(y)$ divided by the input current, that is the integral of the vertical component of the current along the length of the antenna from ground to top.

$h_{e}={1 \over I_{\text{IN}}}\int \limits _{0}^{h}I(y)dy$

where $I(y)$ is the sum of the vertical components of the current in all the antenna elements at height $y$. The radiation resistance $R_{\text{R}}$ of the antenna is

$R_{\text{R}}=160\pi ^{2}{\Big (}{h_{\text{e}} \over \lambda }{\Big )}^{2}$

and the total radiated power $P_{\text{R}}$ is

$P_{\text{R}}=I_{\text{IN}}^{2}R_{\text{R}}=160\pi ^{2}{\Big (}{h_{\text{e}}I_{\text{IN}} \over \lambda }{\Big )}^{2}$

The effective height multiplied by the electric field of the radio wave is equal to the open circuit output voltage of the antenna. The effective height is also equal to the height of the charge center if the antenna were electrostatically charged. The effective height may be difficult to calculate in geometrically complex antennas, but for simple shapes

• Hypothetical constant current in a vertical monopole of height $h$:

$h_{e}={1 \over I_{\text{IN}}}\int \limits _{0}^{h}I_{\text{IN}}dy=h$

• Trapezoidal current distribution:

$h_{e}={1 \over I_{\text{IN}}}\int \limits _{0}^{h}I_{\text{IN}}(1-{y \over h})dy={h \over 2}$

• Quarter wave monopole antenna:

$h_{e}={1 \over I_{\text{IN}}}\int \limits _{0}^{\pi /4}I_{\text{IN}}\sin(\pi /4-\theta )d\theta =-\cos(\pi /4-\theta ){\Big |}_{0}^{\pi /4}$

Definition of variables

Symbol	Unit	Definition
$\phi$	radian	Phase shift of antenna current during round trip along element
$\lambda$	meter (m)	Free space wavelength of radio waves
$\pi$	[none]	Math constant ≈ 3.1416
$\theta$	radian	Elevation angle above horizon
$c$	meters per second (ms−1)	Velocity of light
$f$	hertz (Hz)	Frequency of radio waves
$f_{\mathsf {n}}$	hertz (Hz)	Resonant frequencies of monopole
$G$	[none]	Electrical length, length in wavelengths, of the monopole element
$h$	meter (m)	length of monopole element
$i_{\mathsf {up}}(z,t)$	ampere (A)	Upward current wave on antenna element at elevation z and time t
$i_{\mathsf {down}}(z,t)$	ampere (A)	Downward current wave on antenna element at elevation z and time t
$I(z)$	ampere (A)	Standing wave current on antenna element at elevation z
$I_{\mathsf {in}}$	ampere (A)	RMS current into antenna terminals
$I_{\mathsf {0}}$	ampere (A)	Maximum RMS current in antenna element
$I_{\mathsf {1}}$	ampere (A)	RMS current at an arbitrary point in antenna element
$k$	meter−1	Angular wavenumber $=2\pi /\lambda$
$P_{\mathsf {in}}$	watt (W)	Electric power delivered to antenna terminals
$P_{\mathsf {R}}$	watt (W)	Power radiated as radio waves by antenna
$P_{\mathsf {L}}$	watt (W)	Power consumed in loss resistances of antenna
$Q$	[none]	Q_factor of antenna, the resonant frequency divided by bandwidth
$R_{\mathsf {R}}$	ohm (Ω)	Radiation resistance of antenna
$R_{\mathsf {L}}$	ohm (Ω)	Equivalent loss resistance of antenna at input terminals
$R_{\mathsf {in}}$	ohm (Ω)	Input resistance of antenna
$R_{\mathsf {R0}}$	ohm (Ω)	Radiation resistance at point of maximum current in antenna
$R_{\mathsf {R1}}$	ohm (Ω)	Radiation resistance at arbitrary point in antenna
$t$	second (s)	Time
$z$	meter (m)	Height on monopole element above lower end

References

• Aitken, Hugh G. J (1976). Syntony and spark : the origins of radio. John Wiley and Sons. ISBN 0471018163.
• ARRL (1949). The A.R.R.L. Antenna Book, 5th Ed. American Radio Relay League, Rumford Press, Concord, NH, USA.
• Balanis, Constantine (2005). Antenna Theory: Analysis and Design, 3rd Ed. John Wiley and Sons. ISBN 9781118585733.
• Bevelacqua, Peter J. (2016). "The Monopole Antenna". Antenna Types. Antenna-Theory.com website. Retrieved 20 August 2020.
• Carr, Joseph; Hippisley, George (2012). Practical Antenna Handbook, 5th Ed (PDF). McGraw-Hill. ISBN 9780071639590.
• Chen, Zhi Ning; Chia, Michael Yan Wah (2006). Broadband Planar Antennas: Design and Applications. John Wiley and Sons. ISBN 9780470871751.
• Das, Sisir K. (2016). Antenna and Wave Propagation. Tata McGraw-Hill Education. ISBN 978-1259006326.
• Ellingson, Steven W. (2016). Radio Systems Engineering. Cambridge University Press. ISBN 9781316785164.
• Fleming, John Ambrose (1906). The Principles of Electric Wave Telegraphy. London: Longmans Green and Co.
• Fujimoto, Kyohei (2008). Mobile Antenna Systems Handbook. Artech House. ISBN 9781596931275.
• Griffith, Benjamin Whitfield (2000). Radio-electronic Transmission Fundamentals. SciTech Publishing. ISBN 9781884932137.
• Gosling, William (1998). Radio Antennas and Propagation (PDF). Newnes. ISBN 0750637412.
• Hong, Sungook (2010). Wireless: From Marconi's Black-Box to the Audion. MIT Press. ISBN 9780262514194.
• Jansky, Cyril Methodius (1919). Principles of Radiotelegraphy. McGraw-Hill Book Co.

• Johnson, Richard C., Ed. (1993). Antenna Engineering Handbook, 3rd Ed (PDF). McGraw-Hill. ISBN 007032381X.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Kissick, W. A. (April 2001). NIJ Guide 202-00: Antenna System Guide. by US National Institute of Science and Technology (NIST) for National Institute of Justice, US Dept. of Justice.
• Kumar, Sanjay; Shukla, Saurab (2015). Wave Propagation and Antenna Engineering. PHI Learning. ISBN 9788120351042.
• Laport, Edmund A. (1952). Radio Antenna Engineering. McGraw-Hill Book Co.
• Lewis, Geoff (2013). Newnes Communications Technology Handbook. Elsevier. ISBN 9781483101026.
• Linx (2012). "Understanding Antenna Specifications and Operation" (pdf). Application Note AN-00501. Linx Technologies. Retrieved 24 June 2021.
• MacNamara, Thereza (2010). Introduction to Antenna Placement and Installation. USA: John Wiley and Sons. ISBN 978-0-470-01981-8.
• Orfanidis, Sophocles J. (2016). Electromagnetic Waves and Antennas. Rutgers Univ. website.
• Poisel, Richard (2012). Antenna Systems and Electronic Warfare Applications. Artech House. ISBN 9781608074846.
• Raines, Jeremy Keith (2007). Folded Unipole Antennas: Theory and Applications. McGraw-Hill. ISBN 0071510206.
• Rudge, Alan W.; Milne, K. (1982). The Handbook of Antenna Design, Vol. 2. IET. ISBN 9780906048870.
• Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A. (2006). History of Wireless. John Wiley and Sons. ISBN 978-0471783015.
• Schelkunoff, Sergei A.; Friis, Harold T. (1952). Antennas: Theory and Practice. John Wiley and Sons.
• Schmitt, Ron (2002). Electromagnetics Explained: A Handbook for Wireless RF, EMC, and High-Speed Electronics. Newnes. ISBN 9780750674034.
• Schweber, Bill (2017). "Understanding Antenna Specifications and Operation, Part 2". Article library. Digi-key Electronics website. Retrieved 28 June 2021.
• Sielaff, Gerald W. (1963). Study Guide for Fundamentals of Radio. US Armed Forces Institute.
• Straw, R. Dean, Ed. (2000). The ARRL Antenna Book, 19th Ed. Newington, CT, USA: American Radio Relay League. ISBN 9780872598041.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Stutzman, Warren L.; Thiele, Gary A. (2012). Antenna Theory and Design. John Wiley and Sons. ISBN 978-0470576649.
• US Army (1953). Training Manual TM11-666: Antennas and Radio Propagation. Department of the Army.
• Visser, Hubregt J. (2006). Array and Phased Array Antenna Basics. John Wiley and Sons. ISBN 0470871180.
• Wallace, Richard; Andreasson, Krister (2015). Introduction to RF and Microwave Passive Components. Artech House. ISBN 9781630810092.
• War Department (1943). Technical Manual TM 11-314: Antennas and Antenna Systems (PDF). US Dept. of Defense.
• Williams, Edmund, Ed. (2007). National Association of Broadcasters Engineering Handbook, 10th Ed. Taylor and Francis. ISBN 9780240807515.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Wong, K. Daniel (2011). Fundamentals of Wireless Communication Engineering Technologies. John Wiley and Sons. ISBN 978-1118121092.

Further reading

• Weiner, Melvin M. (2003). Monopole antennas. USA: CRC Press. ISBN 0-8247-4844-1. In depth analysis, covering the equations and computational electromagnetics models used to analyze real-world monopoles, and focusing on the effects of imperfect ground planes
• King, Ronold W. P. (1956). The Theory of Linear Antennas. Harvard University Press. The most comprehensive reference manual before the advent of computer methods

---

For Spark-gap transmitter

Damping and bandwidth

(unfinished)

Unlike modern transmitters which generate a continuous wave oscillating signal, spark-gap transmitters generate a series of damped waves, each consisting of an oscillating sine wave that declines exponentially to zero. The degree of damping, the number of cycles before the wave decays to zero, has an inverse relationship with the bandwidth, the frequency range over which the signal is spread. The damping is determined by Q_factor of the transmitting circuit; the ratio of shunt conductance in the transmitter circuit to the inductance and capacitance.

• Very damped waves, where the wave decays to zero within a few cycles, have very wide bandwidth. Their energy is spread over a wide range of frequencies on either side of the center frequency, so it may overlap the frequencies of other transmitters, interfering with their signals. The early non-syntonic spark transmitters were characterized by high damping.
• Lightly damped "ringing" waves, which decline gradually over many cycles, have a narrower bandwidth and create a "purer" signal, with most of its energy concentrated in a narrow band of frequencies around a single frequency. Later syntonic spark transmitters had lighter damping.
• Continuous waves, consisting of an undamped sine wave, have all their energy concentrated at one frequency. Spark gap transmitters could not produce continuous waves, but the vacuum tube transmitters that replaced them could.

The degree of damping is determined by the amount of resistance in the resonant circuit, compared to the size of the capacitor. The resistance mostly consisted of the radiation resistance of the antenna, plus the dissipation in the spark. In inductively-coupled transmitters it also depended on the degree of coupling of the two coils. Much of the development of later transmitters was directed toward reducing the damping, trying to approach closer to the ideal "pure" continuous wave.

During the wireless era the damping was measured by the "decrement", the fractional decline of the amplitude during one cycle of the wave. The decrement was defined as

$\delta ={\frac {\log A_{1}/A_{2}}{\log e}}\,$

(Stanley) For an LC circuit (Morecroft, p. 62), it is

$\delta ={R \over 2fL}$

Today the same thing is measured by a parameter called the Q. The relationship between decrement and Q factor is

$\delta ={\pi \over Q}\,$

(Terman p. 139)

Power output

The energy in each damped wave is limited to the energy stored in the capacitor, or in transmitters without a capacitor, in the antenna. The energy in joules is

$W={1 \over 2}CV^{2}\,$

where C is the capacitance of the capacitor in farads and V is the peak voltage on the capacitor when the spark gap fires. However only some of this power is radiated as radio waves. Some is dissipated as heat in the transmitter; mostly in the spark, but also in ohmic heating and dielectric losses in the circuit. If η is the efficiency of the transmitter, the fraction of input power radiated as radio waves, the peak output power of the transmitter is

$W={1 \over 2}CV^{2}\eta \,$

The power input is mainly dissipated in two resistances; the radiation resistance of the antenna and the resistance of the spark gap. In a transmitter with a spark rate of f_S sparks per second the average power output in watts is

$P_{\text{avg}}={1 \over 2}CV^{2}f_{\text{S}}\eta \,$

It can be seen that the power output increases with the capacitance and the peak voltage on the capacitor before the spark gap fires. However, this is only the average power output; the communication distance depends more on the "peak" power in an individual damped wave. This also doesn't apply to the quenched gap transmitter, since the spark is extinguished for part of the radiating cycle

For Talk:Negative resistance

• "It is well known that the effect of regeneration... is equivalent to introducing a negative resistance reaction in the circuit." Armstrong (1922) "Some recent developments of regenerative circuits", p.244
• "...if the loop gain [of a feedback circuit] is sufficiently negative (i.e., the feedback is sufficiently positive) a negative resistance is achieved." Rezavi (2001) Design of Analog CMOS Integrated Circuits, p.506
• "One of the simplest ways to produce a negative resistance is to interconnect the input and output terminals of a one-way amplifier. This gives a regenerative arrangement because part of the output energy of the amplifier is fed back into the input." Crisson (1931) "Negative resistance and the Twin-21 type repeater", p.486
• "Negative resistance by means of feedback" in Pippard (2007) The Physics of Vibration, p.314-322
• "Many circuit topologies are capable of producing absolute negative resistance... The simplest type requires an amplifier with voltage gain greater than unity. [...] The input resistance is negative for values of A_v > 1" Aluf (2012) Optoisolation Circuits, p.11
• "As is well recognised, [regeneration] consists of supplying energy by some process akin to feedback... thereby having the same effect as would the introduction of 'negative resistance'." Warner (1922) Superregeneration, p.25
• "Regeneration introduces a negative resistance into a circuit, such that its net positive resistance is reduced." Kitchin (1998) "High performance regenerative receiver design", p.25

For Inductance

In electrical engineering, inductance is a property of an electrical conductor by which a changing electric current through the conductor induces a voltage either in the conductor itself, called self-inductance, or in another nearby conductor, called mutual inductance. Inductance most commonly means self-inductance, which is a property of a conductor which causes it to oppose changes in current through it. A current through a conductor creates a magnetic field around the conductor. A changing current creates a changing magnetic field, causing the magnetic flux, the total magnetic field passing through the circuit, to change. From Faraday's law of induction, any change in the magnetic flux passing through a circuit induces a voltage (electromotive force) in the circuit; this is called by electromagnetic induction. Inductance is defined as the ratio $L$ between this induced voltage $v$ and the rate of change of the current $i(t)$ in the circuit

$v=L{di(t) \over dt}.$

From Lenz's law, this self-induced voltage, or "back EMF", in a conductor will be in a direction so as to oppose the change in current which created it, so inductance causes a conductor to oppose changes in current through it. The inductance of a circuit depends on the geometric shape of the conductors, and the magnetic permeability of nearby materials. An inductor is an electrical component which adds inductance to a circuit. It is usually made of a conductor in a helix or spiral shape.

The term inductance was coined by Oliver Heaviside in 1886. It is customary to use the symbol $L$ for inductance, in honour of the physicist Heinrich Lenz. In the SI system, the unit of inductance is the henry (H), which is the amount of inductance which causes a voltage of 1 volt when the current is changing at a rate of one ampere per second. It is named for Joseph Henry, who discovered inductance independently of Faraday.

Mutual inductance is a property of two nearby electrical circuits by which a changing current in one circuit induces a voltage in the other circuit, by electromagnetic induction. If two circuits are close enough together, the magnetic field created by a current in one circuit will pass through the other circuit. Thus a changing current in one circuit will cause a changing magnetic flux through the other circuit, inducing an electromotive force in the other circuit due to Faraday's law. The mutual inductance $M_{\text{1,2}}$ between two circuits 1 and 2 is defined as the ratio of voltage $v_{\text{2}}$ induced in circuit 2 to the rate of change of current $i_{\text{1}}(t)$ in circuit 1

$v_{\text{2}}=M_{\text{1,2}}{i_{\text{1}}(t) \over dt}$

Like inductance, mutual inductance is measured in henrys. Two circuits which are close enough together to have appreciable mutual inductance are said to be inductively coupled. A common electrical component with operates by mutual inductance is the transformer.