Radiation resistance

Electromagnetic waves are radiated by electric charges when they are accelerated.[1][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][1] 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.

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 ${\displaystyle R_{\text{IN}}}$ is equal to the sum of the radiation resistance ${\displaystyle R_{\text{R}}}$ and loss resistance ${\displaystyle R_{\text{L}}}$

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

The power ${\displaystyle P_{\text{IN}}}$ fed to the antenna is split proportionally between these two resistances.[2][13]

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

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

where

${\displaystyle P_{\text{R}}=I_{\text{IN}}^{2}R_{\text{R}}\quad }$ and ${\displaystyle \quad P_{\text{L}}=I_{\text{IN}}^{2}R_{\text{L}}}$

The power ${\displaystyle P_{\text{R}}}$ consumed by radiation resistance is converted to radio waves, the desired function of the antenna, while the power ${\displaystyle P_{\text{L}}}$ consumed by loss resistance is converted to heat, representing a waste of transmitter power.[2] 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.

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

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 ${\displaystyle R_{\text{IN}}}$ and a reactance (capacitance or inductance) equal to the opposite of 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. The antenna's radiation resistance is usually the main part of its input resistance, so it determines what impedance matching is necessary and what types of transmission line would be well matched to the antenna.

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 ${\displaystyle I_{\text{0}}}$ 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 ${\displaystyle R_{\text{R0}}}$ is the radiation resistance. However if the antenna is fed at another point, the equivalent radiation resistance at that point ${\displaystyle R_{\text{R1}}}$ can easily be calculated from the ratio of antenna currents[15][17]

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

${\displaystyle R_{\text{R1}}={\Big (}{I_{\text{0}} \over I_{\text{1}}}{\Big )}^{2}R_{\text{R0}}}$

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]

Antenna	Radiation resistance ohms	Source
Center-fed half-wave dipole	73.1[18]	Kraus 1988:227,Balanis 2005:216
Short dipole of length ${\displaystyle \lambda /50<L<\lambda /10}$	${\displaystyle 20\pi ^{2}{\Big (}{L \over \lambda }{\Big )}^{2}}$	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 ${\displaystyle L\ll \lambda /4}$ over perfectly conducting ground	${\displaystyle 40\pi ^{2}{\Big (}{L \over \lambda }{\Big )}^{2}}$	Stutzman & Thiele 2012:78–80
Resonant loop antenna, 1 ${\displaystyle \lambda }$ circumference	~100	Weston 2017:15, Schmitt 2002:236
Small loop of area ${\displaystyle A}$ with ${\displaystyle N}$ turns (circumference ${\displaystyle \ll \lambda /3}$)	${\displaystyle 320\pi ^{4}{\Big (}{NA \over \lambda ^{2}}{\Big )}^{2}}$	Kraus 1988:251, Balanis 2005:238
Small loop of area ${\displaystyle A}$ with ${\displaystyle N}$ turns on a ferrite core of effective relative permeability ${\displaystyle \mu _{\text{eff}}}$	${\displaystyle 320\pi ^{4}{\Big (}{\mu _{\text{eff}}NA \over \lambda ^{2}}{\Big )}^{2}}$	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 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]

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.[1] 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 ${\displaystyle P_{\text{R}}}$ 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 ${\displaystyle R_{\text{R}}=P_{\text{R}}/I_{\text{rms}}^{2}}$[5]

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. 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 ordinary standards, at 15 kHz this is still only about .015 wavelength high, so VLF antennas are electrically short. From the table a .015${\displaystyle \lambda }$ monopole antenna has a radiation resistance of about 0.09 ohm. It is extremely difficult to reduce the loss resistances of the 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 30 MHz this is not such a problem, since the weak signal from the antenna can simply be amplified in the receiver.

At frequencies below 1 MHz the size of ordinary electrical circuits 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.

• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963). The Feynman Lectures on Physics, Vol. I. Addison-Wesley. p. 32.1. 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.

• Antenna efficiency
• Impedance of free space