Appearance
7.6 — Antennas and Propagation
An antenna is a piece of metal that converts a guided wave in a wire into a radiating wave in space, and back again. That is a remarkably strange thing for a piece of metal to do, and understanding roughly how it does it explains most of what antennas are and why they are the size they are.
An antenna launches the electromagnetic wave that Volume IV Chapter 4.7 derives from Maxwell's equations, and it does so by exploiting the resonance of Volume IV Chapter 2.2. This chapter takes both as given and works out what shape and size of metal to use.
The letters this chapter uses, said out loud
\lambda — "lambda", wavelength in metres, the distance the wave advances in one cycle. It is tied to frequency by \lambda = c/f, and it is the ruler every antenna is measured against: a "half-wave dipole" is half a lambda long, and that is why the same design shrinks as the frequency rises.
G — gain, usually in decibels. It does not mean amplification. An antenna is a passive piece of metal and adds no energy. Gain means concentration: how much more power it sends in its best direction compared with a hypothetical antenna radiating equally in all directions. A 12 dBi antenna is not sixteen times more powerful, it is sixteen times more focused, and it is correspondingly worse everywhere else.
dBi — decibels relative to an isotropic radiator, the imaginary antenna that radiates equally in every direction. It does not exist and cannot exist, and it is used as the reference precisely because it is a clean geometric fiction.
\eta — "eta", radiation efficiency: the fraction of the power fed in that leaves as radio waves rather than heating the antenna and the ground beneath it. A short antenna at a low frequency can have an efficiency of a few percent, which is the real reason such antennas perform badly.
\theta and \phi — "theta" and "phi", the two angles that specify a direction in three dimensions, exactly as in the spherical coordinates of Volume II Chapter 5.7. A radiation pattern is a plot of how much power leaves in each (\theta, \phi) direction.
Z_0 — the antenna's impedance in ohms, which must match the feed cable's impedance or power reflects back down the cable instead of radiating — the standing-wave problem of Chapter 7.7. A half-wave dipole in free space is about 73 Ω, which is why 75 Ω cable exists.
P_t, P_r, L — transmitted power, received power and path loss. Section 5's link budget is nothing more than these three added up in decibels.
1. Why a wire radiates
Chapter 1.5 established that a changing current produces a changing magnetic field, and Faraday's law says a changing magnetic field produces an electric field. A changing electric field, in turn, produces a magnetic field — that last piece was Maxwell's addition in 1865, and it completes a loop.
Once the two fields sustain each other, they detach from the wire and travel away at the speed of light. That is electromagnetic radiation, and it is why the same equations describe radio and light.
Not every current radiates appreciably. Two conditions decide it:
The current must be accelerating — that is, changing. A steady current produces a static magnetic field that stays put.
The conductor must be a significant fraction of a wavelength. In an ordinary circuit, the outgoing and returning currents are close together and their fields cancel almost perfectly. Separate them by an appreciable fraction of a wavelength and the cancellation fails, and what escapes is radiation.
\lambda=\frac{c}{f}, \qquad c=3\times10^8\ \text{m/s}
| Frequency | Wavelength | Quarter wave |
|---|---|---|
| 100 kHz | 3000 m | 750 m |
| 1 MHz | 300 m | 75 m |
| 100 MHz | 3 m | 0.75 m |
| 1 GHz | 30 cm | 7.5 cm |
| 2.4 GHz | 12.5 cm | 3.1 cm |
| 28 GHz | 1.07 cm | 2.7 mm |
That table is the whole reason antennas look the way they do, and the reason a long-wave broadcast station needs a mast hundreds of metres tall while a Wi-Fi antenna hides inside a laptop lid.
An antenna is not a mysterious component. It is a piece of wire whose length is chosen so that the current on it swings back and forth in step with the wave it is trying to launch.
Follow one cycle. The source pushes charge into the top arm and pulls it out of the bottom, so the antenna becomes briefly positive at one end and negative at the other, and an electric field arcs between them. Then the source reverses, the charge sloshes back, and the field has to reverse too.
The crucial moment is the pinch-off. The old field loop cannot reverse instantly, because change propagates outward at a finite speed. So the inner part of the loop reverses while the outer part has not heard yet, the loop is squeezed in the middle, and it detaches. Once detached it is a self-sustaining ripple of electric and magnetic field — the electromagnetic wave of Volume IV Chapter 4.7 — and it carries on without the antenna.
That is why antenna length matters. Make the wire half a wavelength long and the charge's natural sloshing period matches the driving frequency, so the antenna is at resonance in exactly the sense of Volume IV Chapter 2.2, and the current for a given drive is large. Make it much shorter and it still radiates, badly, which is what every stubby antenna in your house is doing.
2. The basic antenna parameters
Radiation pattern — how much is radiated in each direction, plotted in three dimensions or as two cuts through it.
Gain — how much more power is sent in the best direction compared with an ideal isotropic radiator that spreads power equally in all directions.
G_{dBi}=10\log_{10}\left(\frac{\text{power density in the best direction}}{\text{power density from an isotropic radiator}}\right)
An antenna does not amplify. It cannot produce more power than it is fed. Gain is concentration — taking power from directions you do not need and putting it where you do.
Beamwidth — the angular width where the power is within 3 dB of the peak. Gain and beamwidth are inverse, and a useful approximation is
G\approx\frac{41{,}000}{\theta_E\theta_H}
for beamwidths in degrees. A 10° by 10° beam gives about 26 dBi.
Bandwidth — the range of frequencies over which the antenna works acceptably, usually defined by its impedance match.
Polarisation — the orientation of the electric field. Vertical, horizontal, or circular.
Polarisation mismatch is a real and often overlooked loss. A vertically polarised transmitter and a horizontally polarised receiver are theoretically infinitely mismatched and practically 20 to 30 dB down. That is why a portable radio's reception changes so much when you tilt it, and why broadcast standards specify polarisation.
Circular polarisation — where the field rotates as it travels — is used for satellite links because the satellite's orientation and the ionosphere's rotation of the polarisation both make linear polarisation unreliable. The cost is a fixed 3 dB loss when receiving with a linear antenna, which is accepted because it is predictable.
3. The half-wave dipole
The reference antenna against which everything is compared: two quarter-wave conductors fed at the centre, total length half a wavelength.
Why half a wavelength. At that length the antenna is resonant, meaning its reactance is zero and its impedance is purely resistive. This is exactly the resonance of Chapter 1.6, with the antenna's distributed inductance and capacitance in place of discrete components. At resonance the current distribution is a half sine, maximum at the centre and zero at the ends.
Its properties:
- Gain: 2.15 dBi. Modest, because it radiates in a doughnut around itself.
- Impedance: 73 Ω, which is why 75 Ω coaxial cable exists.
- Pattern: figure-of-eight in the plane containing the wire, with nulls off the ends — a dipole radiates nothing along its own axis.
- Bandwidth: roughly 10% of the centre frequency for a thin wire, more for a fat one.
The quarter-wave monopole is half a dipole standing on a conducting ground plane, which supplies the missing half by reflection. Half the size, half the impedance (36 Ω), and 2 dB more gain because it radiates into a hemisphere rather than a sphere.
Every car radio whip, every mobile phone antenna and every Wi-Fi dongle is a monopole, and the ground plane is the car body, the phone's circuit board, or whatever is available. A monopole with an inadequate ground plane performs badly and unpredictably, which is why phone antenna design is so difficult — the "ground plane" includes the user's hand.
4. Directional antennas
Yagi-Uda
A driven dipole with one reflector slightly longer behind it and several directors slightly shorter in front. The parasitic elements are not connected to anything; they are excited by the driven element's field and re-radiate with phases that reinforce forwards and cancel backwards.
Gain: 7 to 20 dBi depending on the number of elements, with roughly 1 dB more for each doubling of the boom length. The classic rooftop television aerial, invented by Shintaro Uda in 1926 and published in English by Hidetsugu Yagi, whose name stuck to it.
Parabolic dish
A parabola reflects all rays from its focus into a parallel beam. The gain is set by the aperture area:
G=\eta\left(\frac{\pi D}{\lambda}\right)^2
with \eta the efficiency, typically 0.5 to 0.7.
Worked example. A 1 m dish at 12 GHz (\lambda = 2.5 cm), \eta=0.6:
G=0.6\left(\frac{\pi\times1}{0.025}\right)^2=0.6\times(125.7)^2=9480 = 39.8\ \text{dBi}
Beamwidth:
\theta\approx\frac{70\lambda}{D}=\frac{70\times0.025}{1}=1.75°
Under two degrees. That is why a satellite dish must be aimed carefully and why wind loading and mount rigidity matter — a few degrees of movement loses the signal entirely.
Note the frequency dependence. Gain rises as the square of frequency for a fixed dish size, which is exactly the effect that Chapter 7.1's path loss formula reflected from the other side. A fixed-size dish at higher frequency more than compensates for the higher path loss, which is why satellite links use high frequencies.
Phased array
Many small elements, each fed with a controlled phase. Adjusting the phases steers the beam electronically, with no moving parts.
The steering angle:
\sin\theta=\frac{\Delta\phi\,\lambda}{2\pi d}
for element spacing d and phase increment \Delta\phi.
Element spacing must be at most \lambda/2, or grating lobes appear — additional unwanted beams in other directions, which is the spatial equivalent of aliasing (Chapter 4.7) with position in place of time. The mathematics is identical, and the \lambda/2 requirement is exactly the Nyquist criterion applied to space.
Where they are used: military radar since the 1960s, weather radar, 5G base stations at millimetre wave, and the Starlink user terminal, which is a flat phased array that tracks satellites electronically as they cross the sky.
5. Impedance matching
An antenna presents a complex impedance, and the transmitter expects a specific resistance, usually 50 Ω. Mismatch reflects power back down the cable.
The reflection coefficient:
\Gamma=\frac{Z_L-Z_0}{Z_L+Z_0}
The standing wave ratio:
\text{SWR}=\frac{1+|\Gamma|}{1-|\Gamma|}
| SWR | |\Gamma| | Power reflected | |---|---|---| | 1.0 | 0 | 0% | | 1.5 | 0.20 | 4% | | 2.0 | 0.33 | 11% | | 3.0 | 0.50 | 25% | | 5.0 | 0.67 | 44% |
An SWR of 2 loses 11% of the power, which is only 0.5 dB. That is a genuinely small loss, and it is worth saying because SWR is often treated with more anxiety than it deserves.
The real danger of high SWR is not lost power but reflected power. A solid-state transmitter's output devices see the reflected wave adding to their own output, and the resulting voltage can exceed their rating. Every modern transmitter therefore has a protection circuit that reduces power when SWR rises, and the classic symptom of a broken antenna cable is a transmitter that will not produce full output.
Matching techniques: an L-network of two reactive components (which is the resonance idea of Chapter 1.6 used as a transformer), a quarter-wave transformer of impedance \sqrt{Z_0Z_L}, a stub of transmission line, or a broadband transformer wound on a ferrite core. Chapter 7.7 covers the transmission-line versions.
6. Propagation
How the signal actually gets from one antenna to the other, and it depends dramatically on frequency.
Ground wave
Below about 3 MHz, the wave follows the earth's curvature by diffraction, with the ground acting as part of the waveguide.
Range: hundreds to thousands of kilometres, better over seawater than dry land because seawater conducts.
This is how AM broadcast covers a region during the day, and how very low frequency signals reach submarines — at 20 kHz the wave penetrates tens of metres of seawater, which nothing higher does.
Sky wave
Between about 3 and 30 MHz, the wave refracts off the ionosphere and returns to earth far away.
The ionosphere is layered, and the layers change dramatically between day and night. The D layer, present only in daylight, absorbs frequencies below about 10 MHz. The F layer refracts them back.
The consequences you can observe. During the day, medium-wave AM stations are limited to their ground wave. At night the D layer disappears and the same stations reflect off the F layer, reaching a thousand kilometres or more — which is why distant stations appear on AM radio after dark and vanish at sunrise.
Shortwave broadcasting works entirely by this mechanism, and the choice of frequency depends on the time of day, the season and the eleven-year solar cycle. Multiple hops can circle the earth, which is why amateur operators can work the other side of the world with 100 watts.
Line of sight
Above about 30 MHz the wave passes through the ionosphere and does not come back. Communication is limited to line of sight, plus a little from diffraction and tropospheric refraction.
d_{horizon} \approx 4.12\left(\sqrt{h_1}+\sqrt{h_2}\right)\ \text{km}
for antenna heights in metres. The 4.12 rather than the geometric 3.57 accounts for atmospheric refraction, which bends the wave slightly downward and extends the horizon by about 15%.
Worked example. A 30 m mast and a 2 m handheld:
d=4.12(\sqrt{30}+\sqrt2)=4.12(5.48+1.41)=28.4\ \text{km}
And that is why antenna height matters so much more than power. Doubling the power adds 3 dB and perhaps 40% to the range in free space; doubling the mast height from 30 to 60 m extends the horizon to 34.9 km with no extra power at all.
Fresnel zones
Line of sight is not enough. The signal travels through a volume, not a line, and obstructions near the path cause loss even without blocking it.
The first Fresnel zone radius at the midpoint:
r=\frac{1}{2}\sqrt{\frac{d\lambda}{1}}\ \ldots \ \text{more usefully} \ \ r_1=17.3\sqrt{\frac{d}{4f}}
with d in km, f in GHz, r in metres.
Worked example. A 10 km link at 5 GHz:
r_1=17.3\sqrt{\frac{10}{20}}=17.3\times0.707=12.2\ \text{m}
So a tree that comes within 12 m of the line of sight causes significant loss, even though nothing blocks the direct path. The rule is to keep 60% of the first Fresnel zone clear, which here means 7.3 m of clearance.
This is the most common mistake in planning a point-to-point link: the two ends can see each other, and the link performs 15 dB below prediction because a ridge halfway along is intruding into the Fresnel zone.
Multipath
Reflections from buildings, ground and water arrive at different times and add with different phases.
Constructive addition gives up to 6 dB of gain. Destructive addition gives deep nulls — 20 to 30 dB is common.
In an urban environment with many reflectors, the received amplitude follows a Rayleigh distribution, which has a long tail towards zero. Deep fades occur regularly, spaced roughly half a wavelength apart in position — 6 cm at 2.4 GHz.
This is why moving a Wi-Fi device a few centimetres can transform the signal, and why a car radio fades in and out when stopped at traffic lights but not when moving.
Where a line-of-sight component exists alongside the reflections, the distribution is Rician and the fading is much less severe.
The countermeasures, each addressing a different dimension:
- Space diversity — two antennas half a wavelength apart, because they fade independently.
- Frequency diversity — spread across a wide band or hop, because a null at one frequency is not a null at another.
- Time diversity — interleaving and coding, so a burst of errors during a fade is spread thinly across many codewords.
- Polarisation diversity — two orthogonal polarisations, which also fade independently.
And MIMO, from Chapter 7.4, which turns multipath from a problem into extra capacity.
7. Path loss models
Free space (Chapter 7.1) is optimistic for anything on the ground. A signal reflecting off the ground produces partial cancellation, and the practical models reflect it.
The two-ray model, valid beyond a breakpoint distance:
L=40\log_{10}d-20\log_{10}h_1-20\log_{10}h_2
Note the 40 rather than 20. Loss grows as the fourth power of distance rather than the square, so doubling the distance costs 12 dB rather than 6.
The general form used in planning:
L=L_0+10n\log_{10}\left(\frac{d}{d_0}\right)
with n the path loss exponent:
| Environment | n |
|---|---|
| Free space | 2 |
| Urban macrocell | 2.7–3.5 |
| Indoor line of sight | 1.6–1.8 |
| Indoor obstructed | 4–6 |
| Dense urban with obstruction | 4–6 |
The indoor line-of-sight value below 2 is not an error. A corridor acts as a crude waveguide, channelling energy rather than letting it spread spherically, so loss grows more slowly than free space. It is the reason a corridor gives better Wi-Fi range than an open hall.
Building penetration typically costs 10 to 20 dB, and modern low-emissivity window coatings — which are metallic — can add 25 dB on their own. This is a genuine and growing problem for mobile coverage, and it is why indoor small cells and repeaters have become standard in new buildings.
Chapter 7.7 covers the last physical link in the chain: the cable between the transmitter and the antenna, and why it behaves nothing like a wire once it is longer than a fraction of a wavelength.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Link budget
P_{rx}=P_{tx}+G_{tx}-L_{path}-L_{misc}+G_{rx}
L_{fs}=20\log_{10}\frac{4\pi d}{\lambda}=32.4+20\log_{10}d_{km}+20\log_{10}f_{MHz}
Doubling distance costs 6 dB (spherical spreading). Doubling frequency also costs 6 dB — not because space absorbs, but because a fixed-gain antenna's effective area is proportional to \lambda^2.
\text{margin}=\text{SNR}_{available}-\text{SNR}_{required}
Antennas
\lambda=\frac cf
G_{dBi}=10\log_{10}\frac{\text{power density in the best direction}}{\text{isotropic power density}}
G\approx\frac{41{,}000}{\theta_E\theta_H} \qquad\text{(beamwidths in degrees)}
Half-wave dipole: gain 2.15 dBi, impedance 73 Ω, nulls off the ends.
Quarter-wave monopole: impedance 36 Ω, needs a ground plane, 2 dB more gain than a dipole.
Parabolic dish:
G=\eta\left(\frac{\pi D}{\lambda}\right)^2, \qquad \theta\approx\frac{70\lambda}{D}
Effective aperture:
A_e=\frac{G\lambda^2}{4\pi}
This is the formula behind the frequency term in path loss — a fixed-gain antenna's capture area shrinks as \lambda^2.
Phased array steering:
\sin\theta=\frac{\Delta\phi\,\lambda}{2\pi d}
Element spacing must not exceed \lambda/2, or grating lobes appear — the spatial equivalent of aliasing.
Reflection and SWR:
\Gamma=\frac{Z_L-Z_0}{Z_L+Z_0}, \qquad \text{SWR}=\frac{1+|\Gamma|}{1-|\Gamma|}
\text{return loss}=-20\log_{10}|\Gamma|, \qquad \text{power reflected}=|\Gamma|^2
Propagation
d_{horizon}\approx4.12\left(\sqrt{h_1}+\sqrt{h_2}\right)\ \text{km}
The 4.12 rather than 3.57 accounts for atmospheric refraction bending the ray downward.
First Fresnel zone radius at midpath:
r_1=17.3\sqrt{\frac{d_{km}}{4f_{GHz}}}\ \text{m}
Keep 60% of it clear. Obstructions inside it cause loss even with clear line of sight.
Two-ray ground reflection model:
L=40\log_{10}d-20\log_{10}h_1-20\log_{10}h_2
Fourth-power distance dependence — 12 dB per doubling, not 6.
General path loss:
L=L_0+10n\log_{10}\frac{d}{d_0}
n=2 free space, 2.7–3.5 urban, 4–6 obstructed indoor, 1.6–1.8 indoor corridor (which acts as a waveguide).
What the next chapter fixes
An antenna is fed through a cable, and above a certain frequency a cable stops behaving like a wire and starts behaving like a medium with its own impedance and its own echoes. Chapter 7.7 explains when that transition happens, what reflects, and why a mismatched cable can destroy a transmitter.