Radiation resistance

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Radiation resistance is that part of an antenna's feedpoint electrical resistance that is caused by the emission of radio waves from the antenna.^[1][2] In radio transmission, a radio transmitter is connected to an antenna. The transmitter generates a radio frequency alternating current which is applied to the antenna, and the antenna radiates the energy in the alternating current as radio waves. Because the antenna is absorbing the energy it is radiating from the transmitter, the antenna's input terminals present a resistance to the current from the transmitter.

Radiation resistance is a virtual resistance due to the power carried away from the antenna as radio waves.^[1][2] Unlike conventional resistance, radiation resistance is not due to the impedance of current (resistivity) due to the imperfect conducting materials the antenna is made of. These radiate power as heat, and their dissipated power is often called "Ohmic" loss and their (conventional) resistance "Ohmic" resistance (${\displaystyle R_{\mathsf {L}}\,}$).^[1] The radiation resistance (${\displaystyle \,R_{\mathsf {R}}\,}$) is typically defined as the value of resistance that would dissipate the same amount of power (as radiated heat) as is dissipated by the radio waves emitted from the antenna, with the same input current passing through it.^[1][3][4]

From Joule's law, it is equal to the total power ${\displaystyle P_{\mathsf {R}}}$ radiated as radio waves by the antenna divided by the square of the RMS current ${\displaystyle I_{\mathsf {RMS}}}$ into the antenna terminals:^[4] ${\displaystyle \,R_{\mathsf {R}}=P_{\mathsf {R}}/I_{\mathsf {RMS}}^{2}~.}$^[5]

The radiation resistance is determined by the geometry of the antenna, the operating frequency, and the antenna location (particularly with respect to the ground).^[a][6][1] The total feedpoint resistance at the antenna's terminals is equal to the radiation resistance plus the loss resistance due to "Ohmic" losses in the antenna and the nearby soil (${\displaystyle \,R_{\mathsf {R}}+R_{\mathsf {L}}\,}$). In a receiving antenna the radiation resistance represents the source resistance of the antenna, and the portion of the received radio power consumed by the radiation resistance represents radio waves reradiated (scattered) by the antenna.^[7][8]

At least two factors can change the feedpoint resistance of an antenna. For one, proximity to the earth or to other metal objects close enough to couple magnetically to the antenna. The other is the location of the feedpoint: The feedpoint resistance is transformed up from the "standard" ${\displaystyle \,R_{\mathsf {R}}+R_{\mathsf {L}}\,}$ in a non-linear way that depends on the distance between the feedpoint and the reactance-free point on the antenna (if any). Conventional formulas are based on the minimum value at that special point.^[1]

Cause[]

Electromagnetic waves are radiated by electric charges when they are accelerated.^[2][9] In a transmitting antenna radio waves are generated by time varying electric currents, consisting of electrons accelerating as they flow back and forth in the metal antenna, driven by the electric field due to the oscillating voltage applied to the antenna by the radio transmitter.^[10][6] An electromagnetic wave carries momentum away from the electron which emitted it. The cause of radiation resistance is the radiation reaction, the recoil force on the electron when it emits a radio wave photon, which reduces its momentum.^[11][12][2] This is called the Abraham–Lorentz force. The recoil force is in a direction opposite to the electric field in the antenna accelerating the electron, reducing the average velocity of the electrons for a given driving voltage, so it acts as a resistance opposing the current.

Radiation resistance and loss resistance[]

The radiation resistance is only part of the feedpoint resistance at the antenna terminals. An antenna has other energy losses which appear as additional resistance at the antenna terminals; ohmic resistance of the metal antenna elements, ground losses from currents induced in the ground, and dielectric losses in insulating materials. The total feedpoint resistance ${\displaystyle R_{\mathsf {in}}}$ is equal to the sum of the radiation resistance ${\displaystyle R_{\mathsf {R}}}$ and loss resistance ${\displaystyle R_{\mathsf {L}}}$

${\displaystyle R_{\mathsf {in}}=R_{\mathsf {R}}+R_{\mathsf {L}}}$

The power ${\displaystyle P_{\mathsf {in}}}$ fed to the antenna is split proportionally between these two resistances.^[1][13]

${\displaystyle P_{\mathsf {in}}=I_{\mathsf {in}}^{2}(R_{\mathsf {R}}+R_{\mathsf {L}})}$

${\displaystyle P_{\mathsf {in}}=P_{\mathsf {R}}+P_{\mathsf {L}}}$

where

${\displaystyle P_{\mathsf {R}}=I_{\mathsf {in}}^{2}R_{\mathsf {R}}\quad }$ and ${\displaystyle \quad P_{\mathsf {L}}=I_{\mathsf {in}}^{2}R_{\mathsf {L}}}$

The power ${\displaystyle P_{\mathsf {R}}}$ consumed by radiation resistance is converted to radio waves, the desired function of the antenna, while the power ${\displaystyle P_{\mathsf {L}}}$ consumed by loss resistance is converted to heat, representing a waste of transmitter power.^[1] 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 (${\displaystyle \eta }$) of the antenna.

${\displaystyle \eta ={P_{\mathsf {R}} \over P_{\mathsf {in}}}={R_{\mathsf {R}} \over R_{\mathsf {R}}+R_{\mathsf {L}}}}$

To transfer maximum power to the antenna, the transmitter and feedline must be impedance matched to the antenna. This means the feedline must present to the antenna a resistance equal to the input resistance ${\displaystyle R_{\mathsf {in}}}$ and a reactance (capacitance or inductance) equal but opposite to the antenna's reactance. If these impedances are not matched, the antenna will reflect some of the power back toward the transmitter, so not all the power will be radiated. For "large" antennas, the radiation resistance is usually the main part of their input resistance, so it determines what impedance matching is necessary and what types of transmission line would match well to the antenna.

Effect of the feedpoint[]

In a resonant antenna, the current and voltage form standing waves along the length of the antenna element, so the magnitude of the current in the antenna varies sinusoidally along its length. The feedpoint, the place where the feed line from the transmitter is attached, can be located at different points along the antenna element. Since radiation resistance depends on the input current, it varies with the feedpoint.^[14] It is lowest for feedpoints located at a point of maximum current (an antinode), and highest for feedpoints located at a point of minimum current, a node, such as at the end of the element (theoretically, in an infinitesimally thin antenna element, radiation resistance is infinite at a node, but the finite thickness of actual antenna elements gives it a high but finite value, on the order of thousands of ohms).^[15] The choice of feedpoint is sometimes used as a convenient way to impedance match an antenna to its feed line, by attaching the feedline to the antenna at a point at which its input resistance is equal to the characteristic impedance of the feed line.

In order to give a meaningful value for the antenna efficiency, the radiation resistance and loss resistance must be referred to the same point on the antenna, usually the input terminals.^[16][17] Radiation resistance is usually calculated with respect to the maximum current ${\displaystyle I_{\mathsf {0}}}$ in the antenna.^[14] If the antenna is fed at a point of maximum current, as in the common center-fed half-wave dipole or base-fed quarter-wave monopole, that value ${\displaystyle R_{\mathsf {R0}}}$ is the radiation resistance. However, if the antenna is fed at another point, the equivalent radiation resistance at that point ${\displaystyle R_{\mathsf {R1}}}$ can easily be calculated from the ratio of antenna currents^[15][17]

${\displaystyle P_{\mathsf {R}}=I_{\mathsf {0}}^{2}R_{\mathsf {R0}}=I_{\mathsf {1}}^{2}R_{\mathsf {R1}}}$

${\displaystyle R_{\mathsf {R1}}=\left({I_{\mathsf {0}} \over I_{\mathsf {1}}}\right)^{2}R_{\mathsf {R0}}}$

Receiving antennas[]

In a receiving antenna, the radiation resistance represents the source resistance of the antenna as a (Thevenin equivalent) source of power. Due to electromagnetic reciprocity, an antenna has the same radiation resistance when receiving radio waves as when transmitting. If the antenna is connected to an electrical load such as a radio receiver, the power received from radio waves striking the antenna is divided proportionally between the radiation resistance and loss resistance of the antenna and the load resistance.^[7][8] The power dissipated in the radiation resistance is due to radio waves reradiated (scattered) by the antenna.^[7][8] Maximum power is delivered to the receiver when it is impedance matched to the antenna. If the antenna is lossless, half the power absorbed by the antenna is delivered to the receiver, the other half is reradiated.^[7][8]

Radiation resistance of common antennas[]

In all of the formulas listed below, the radiation resistance is the so-called "free space" resistance, which the antenna would have if it were mounted several wavelengths distant from the ground (not including the distance to an elevated counterpoise, if any). Installed antennas will have higher or lower radiation resistances if they are mounted near the ground (less than 1 wavelength) in addition to the loss resistance from the antenna's near electrical field that penetrate the soil.^[a][1]

Antenna	Radiation resistance ohms	Source
Center-fed half-wave dipole	73.1[b]	Kraus 1988:227 Balanis 2005:216
Short dipole of length ${\displaystyle \lambda /50<\ell <\lambda /10}$	${\displaystyle 20\pi ^{2}\left({\ell \over \lambda }\right)^{2}}$	Kraus 1988:216 Balanis 2005:165,215
Base-fed quarter-wave monopole over perfectly conducting ground	36.5	Balanis 2005:217 Stutzman & Thiele 2012:80
Short monopole of length ${\displaystyle \ell \ll \lambda /4}$ over perfectly conducting ground	${\displaystyle 40\pi ^{2}\left({\ell \over \lambda }\right)^{2}}$	Stutzman & Thiele 2012:78–80
Resonant loop antenna, ${\displaystyle 1\lambda }$ circumference	~100	Weston 2017:15 Schmitt 2002:236
Small loop of area ${\displaystyle A}$ with ${\displaystyle N}$ turns (circumference ${\displaystyle \ll \lambda /3}$)	${\displaystyle 320\pi ^{4}\left({N\,A \over \lambda ^{2}}\right)^{2}}$	Kraus 1988:251 Balanis 2005:238
Small loop of area ${\displaystyle A}$ with ${\displaystyle N}$ turns on a ferrite core of effective relative permeability ${\displaystyle \mu _{\mathsf {eff}}}$	${\displaystyle 320\pi ^{4}\left({\mu _{\mathsf {eff}}\,N\,A \over \lambda ^{2}}\right)^{2}}$	Kraus 1988:259 Milligan 2005:260

The above figures assume the antenna is made of thin conductors and that the dipole antennas are sufficiently far away from the ground or grounded structures.

The half-wave dipole's radiation resistance of 73 Ohms is near enough to the characteristic impedance of common 50 Ohm and 75 Ohm coaxial cable that it can usually be fed directly without need of an impedance matching network. This is one reason for the wide use of the half wave dipole as a driven element in antennas.^[19]

Relationship of monopoles and dipoles[]

The radiation resistance of a monopole antenna created by replacing one side of a dipole antenna by a perpendicular ground plane is one-half of the resistance of the original dipole antenna. This is because the monopole radiates only into half the space, the space above the plane, so the radiation pattern is identical to half of the dipole pattern and therefore with the same input current it radiates only half the power.^[20] This is not obvious from the formulas in the table because the derived monopole antenna is only half the length of the original dipole antenna. This can be shown by calculating the radiation resistance of a short monopole of half the length of a dipole

${\displaystyle R_{\mathsf {R}}=40\pi ^{2}\left({\ell /2 \over \lambda }\right)^{2}=10\pi ^{2}\left({\ell \over \lambda }\right)^{2}\qquad }$ (monopole of length ${\displaystyle \ell \over 2}$)

Comparing this to the formula for the short dipole shows the monopole has half the radiation resistance

${\displaystyle R_{\mathsf {R}}=20\pi ^{2}\left({\ell \over \lambda }\right)^{2}\qquad \qquad \qquad }$ (dipole of length ${\displaystyle \ell }$)

Calculation[]

Calculating the radiation resistance of an antenna directly from the reaction force on the electrons is very complicated, and presents conceptual difficulties in accounting for the self-force of the electron.^[2] Radiation resistance is instead calculated by computing the far-field radiation pattern of the antenna, the power flux (Poynting vector) at each angle, for a given antenna current.^[21] This is integrated over a sphere enclosing the antenna to give the total power ${\displaystyle P_{\mathsf {R}}}$ radiated by the antenna. Then the radiation resistance is calculated from the power by conservation of energy, as the resistance the antenna must present to the input current to absorb the radiated power from the transmitter, using Joule's law ${\displaystyle R_{\mathsf {R}}=P_{\mathsf {R}}/I_{\mathsf {RMS}}^{2}}$^[5]

Small antennas[]

Electrically short antennas, antennas with a length much less than a wavelength, make poor transmitting antennas, as they cannot be fed efficiently due to their low radiation resistance.

At frequencies below 1 MHz the size of ordinary electrical circuits and the lengths of wire used in them is so much smaller than the wavelength, that when considered as antennas they radiate an insignificant fraction of the power in them as radio waves. This explains why electrical circuits can be used with alternating current without losing energy as radio waves.^[c]

Low radiation resistance[]

As can be seen in the above table, for antennas shorter than their fundamental resonant length (${\displaystyle \lambda /2}$ for a dipole antenna, ${\displaystyle \lambda /4}$ for a monopole, circumference of ${\displaystyle \lambda }$ for a loop) the radiation resistance decreases with the square of their length.^[22] As the length is decreased the loss resistance, which is in series with the radiation resistance, makes up a larger fraction of the feedpoint resistance, so it consumes a larger fraction of the transmitter power, causing the efficiency of the antenna to decrease.

For example, navies use radio waves of about 15–30 kHz in the very low frequency (VLF) band to communicate with submerged submarines. A 15 kHz radio wave has a wavelength of 20 km. The powerful naval shore VLF transmitters which transmit to submarines use large monopole mast antennas which are limited by construction costs to heights of about 300 metres (980 ft). Although these are tall antennas by human standards, at 15 kHz this is still only about 0.015 wavelength high, so VLF antennas are electrically short. From the table a ${\displaystyle 0.015\lambda }$ monopole antenna has a radiation resistance of about 0.09 ohm.

Essentially insurmountable loss resistance[]

It is extremely difficult to reduce the loss resistance of an antenna to this level. Since the ohmic resistance of the huge ground system and loading coil cannot be made lower than about 0.5 Ohm, the efficiency of a simple vertical antenna is below 20%, so more than 80% of the transmitter power is lost in the ground resistance. To increase the radiation resistance, VLF transmitters use huge capacitively top-loaded antennas such as umbrella antennas and flattop antennas, in which an aerial network of horizontal wires is attached to the top of the vertical radiator to make a 'capacitor plate' to ground, to increase the current in the vertical radiator. However this can only increase the efficiency to 50–70% at most.

Small receiving antennas, such as the ferrite loopstick antennas used in AM radios, also have low radiation resistance, and thus produce very low output. However at frequencies below about 20 MHz this is not such a problem, since a weak signal from the antenna can simply be amplified in the receiver.

Definition of variables[]

Symbol	Unit	Definition
${\displaystyle \lambda }$	meter (m)	Wavelength of radio waves
${\displaystyle \pi }$	[none]	math constant ≈ 3.142
${\displaystyle \mu _{\mathsf {eff}}}$	[none]	Effective relative permeability of ferrite rod in antenna
${\displaystyle A}$	square meter (m2)	Cross sectional area of loop antenna
${\displaystyle f}$	Hertz (Hz)	Frequency of radio waves
${\displaystyle I_{\mathsf {in}}}$	Ampere (A)	RMS current into antenna terminals
${\displaystyle I_{\mathsf {0}}}$	Ampere (A)	Maximum RMS current in antenna element
${\displaystyle I_{\mathsf {1}}}$	Ampere (A)	RMS current at an arbitrary point in antenna element
${\displaystyle \ell }$	meter (m)	Length of antenna
${\displaystyle N}$	[none]	Number of wire turns in loop antenna
${\displaystyle P_{\mathsf {in}}}$	Watt (W)	Electric power delivered to antenna terminals
${\displaystyle P_{\mathsf {R}}}$	Watt (W)	Power radiated as radio waves by antenna
${\displaystyle P_{\mathsf {L}}}$	Watt (W)	Power consumed in loss resistances of antenna
${\displaystyle R_{\mathsf {R}}}$	Ohm (Ω)	Radiation resistance of antenna
${\displaystyle R_{\mathsf {L}}}$	Ohm (Ω)	Equivalent loss resistance of antenna at input terminals
${\displaystyle R_{\mathsf {in}}}$	Ohm (Ω)	Input resistance of antenna
${\displaystyle R_{\mathsf {R0}}}$	Ohm (Ω)	Radiation resistance at point of maximum current in antenna
${\displaystyle R_{\mathsf {R1}}}$	Ohm (Ω)	Radiation resistance at arbitrary point in antenna

See also[]

• Antenna efficiency
• Impedance of free space

Footnotes[]

References[]

Sources[]

• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963). The Feynman Lectures on Physics. I. Addison-Wesley. ISBN 9780465040858.
• Balanis, Constantine A. (2005). Antenna Theory: Analysis and Design, 3rd Ed. John Wiley and Sons. ISBN 047166782X.
• Kraus, John D. (1988). Antennas (2nd ed.). Tata McGraw-Hill. ISBN 0-07-463219-1.
• Milligan, Thomas A. (2005). Modern Antenna Design, 2nd Ed. John Wiley and Sons. ISBN 9780471457763.
• Schmitt, Ron (2002). Electromagnetics Explained: A Handbook for Wireless RF, EMC, and High-Speed Electronics. Newnes. ISBN 9780750674034.
• Stutzman, Warren L.; Thiele, Gary A. (2012). Antenna Theory and Design. John Wiley. ISBN 9780470576649.
• Weston, David (2017). Electromagnetic Compatibility: Principles and Applications, 2nd Ed. CRC Press. ISBN 9781351830492.