Electromagnetic radiation
A car's one-meter antenna has, in the AM band, a radiation resistance of thousandths of an ohm. A half-wave dipole has 73 Ω and connects straight to a coaxial cable.
01Why a conductor radiates
A charge at rest creates an electric field; a charge in uniform motion also creates a magnetic one. But only an accelerated charge sheds fields that move away on their own and carry energy off to infinity: that is radiation. In a conductor carrying alternating current the electrons accelerate and brake all the time, so any conductor with a varying current radiates something.
A transmission line hardly radiates because its two conductors carry opposite currents very close to each other, and their far fields cancel. If the ends of a two-wire line are opened out into a T shape, the two currents end up in the same direction, no longer cancel and the structure radiates: it is a dipole, the most elementary antenna.
02The elementary dipole
The starting point of the calculation is a segment of current \( I \) of length \( \ell \) much smaller than the wavelength. Solving Maxwell's equations with retarded potentials, far from the dipole we get:
The fact that the field falls as \( 1/r \) is what makes communication possible: the power density falls as \( 1/r^2 \), exactly what is needed for the total power that crosses a sphere of any radius to be the same. The factor \( \sin\theta \) says that the dipole does not radiate along its axis and that it radiates more in the perpendicular plane.
03Radiation resistance
The radiated power leaves the source just as if it were dissipated in a resistor. That equivalent resistance, the radiation resistance, is defined by \( P = \tfrac12 I^2 R_r \):
A short dipole has a tiny radiation resistance: one of \( \lambda/10 \) has 2 Ω, and the one-meter antenna of a car in the AM band, a few thousandths of an ohm. Against that, the ohmic resistance of the conductor is no longer negligible, and the efficiency \( R_r / (R_r + R_p) \) drops. The half-wave dipole has a radiation resistance of 73 Ω, and that is why it is the reference for practical antennas: it matches effortlessly to a 50 or 75 Ω cable. Its impedance also has a reactive part of about \( +j42 \) Ω, which is canceled by shortening it a little.
04Radiation pattern and directivity
The radiation pattern is the far field as a function of direction. The directivity compares the intensity in the direction of maximum radiation with what an isotropic source radiating the same power in all directions would have:
A longer dipole concentrates the radiation more in the perpendicular plane, up to 1.25 λ, where the directivity reaches 5.2 dBi. Longer still, the current reverses direction in parts of the conductor and the pattern opens up into several lobes.
A thin center-fed dipole, with a sinusoidal current distribution. The graph shows the relative far field in a plane that contains the dipole (vertical, at the center). The values are calculated by integrating the pattern; the efficiency assumes copper wire 2 mm in diameter.
05Near field and far field
Close to the antenna, besides the field that moves away, there are fields that store energy and return it on each cycle, as in a capacitor or an inductor. They decrease as \( 1/r^2 \) and \( 1/r^3 \), and dominate up to a distance of about \( \lambda / 2\pi \): 48 m for a 1 MHz AM station, 5 cm at 1 GHz. In that region neither the relation \( E/H = 377\ \Omega \) nor the radiation pattern holds.
For antennas that are large compared with the wavelength, the pattern only forms at the Fraunhofer distance, \( 2D^2/\lambda \), with \( D \) the largest dimension of the antenna: a 1 m parabolic dish at 10 GHz needs 67 m. That is why antennas are measured on long test ranges or in anechoic chambers with near-field techniques.
Some systems work in the near field on purpose: contactless cards and NFC, at 13.56 MHz (wavelength of 22 m), communicate through inductive coupling over a few centimeters, radiating almost nothing.
06Where it appears in electronics
| Case | What decides the radiation |
|---|---|
| Antenna of a transmitter | Radiation resistance that is high compared with the losses, and a pattern toward where the receivers are |
| Electromagnetic compatibility of a board | Every current loop and every connected cable is an unintentional antenna; a λ/4 cable radiates very well |
| AM antenna of a receiver | Electrically short: compensated with a tuned ferrite coil |
| Contactless card or NFC | Works in the near field, through magnetic coupling |
How dipoles are combined to give more gain, and the budget of a complete radio link, are covered in the antennas unit of this course and in antennas, in the technician program.
07In the lab
With a 433 MHz generator and a half-wave dipole as the transmitter, rotate the receiving dipole, connected to a field strength meter, in 15° steps and plot the polar pattern.
Rotate the receiving dipole around the axis of the link and measure the drop in the signal when it ends up perpendicular to the transmitter.
With a near-field probe and a spectrum analyzer, scan a board with a microcontroller and locate the traces and cables that radiate the most at the harmonics of the clock.
08Common mistakes
- Measuring the pattern in the near field. Within a few wavelengths, the field does not have the shape of the pattern.
- Confusing radiation resistance with losses. The power in \( R_r \) is the power that is radiated: it is the useful one.
- Ignoring the efficiency of a short antenna. With 0.02 Ω of radiation resistance, a few hundredths of an ohm of loss in the conductor or in the loading coil take most of the power.
- Believing that a short cable does not radiate. At the harmonics of a clock of tens of MHz, a cable one meter long is an efficient antenna.
09Self-assessment
What radiation resistance does a short dipole of \( \lambda/10 \) have?
\( 20\pi^2 \cdot 0.01 \approx 2 \) Ω.
In what direction does a dipole not radiate?
Along its axis, where \( \sin\theta = 0 \).
Up to what distance does the near field of a 100 MHz antenna dominate?
Up to about \( \lambda/2\pi = 3 / 2\pi \approx 0.48 \) m.
How much more directive is a half-wave dipole than a short one?
1.64 versus 1.5: barely 0.39 dB.
10Further reading
- Constantine A. Balanis. Antenna Theory: Analysis and Design. 4th ed., Wiley, 2016. The reference on antenna theory: the elementary dipole, the finite-length one and the field regions, fully derived.
- John D. Kraus and Ronald J. Marhefka. Antennas for All Applications. 3rd ed., McGraw-Hill, 2002. A classic with a lot of physical intuition about how an antenna radiates.
- Henry W. Ott. Electromagnetic Compatibility Engineering. Wiley, 2009. Radiation seen from the other side: how to keep a board or a cable from radiating unintentionally.