Antennas · Volume 2
RF Fundamentals
Wavelength and frequency, propagation modes, polarization, near vs far field, the Friis equation, and the working subset of electromagnetics every later volume relies on
Contents
1. About this volume
This is the foundation chapter — the working subset of RF and electromagnetic theory every later volume in the series assumes the reader carries in their head. A working engineer already knows most of it; the volume’s job is not to teach EM from first principles (Balanis and Stutzman & Thiele do that better, and the volume’s resource list points at them) but to fix notation, fix units, and fix the working numbers so every subsequent chapter can be read at speed without the reader stopping to ask “wait, was that dBi or dBd, and were they assuming free space or over real ground?”
Skip this volume if you teach RF for a living; skim it if you’ve been doing antenna work for ten years and just want the conventions; read it if you want the cleanest possible mental model for the rest of the series. Cross-references in later chapters land here for the canonical statement of every quantity (Vol 6 §3 returns here for half-wave length math, Vol 11 §4 returns here for the gain-vs-boom-length curve derivation that depends on Friis, Vol 24 §9 returns here for the near-field caveat on NanoVNA measurements).
2. Wavelength, frequency, and the speed of light
Wavelength λ and frequency f are the same physical quantity expressed in different units, related by the speed of light:
$$ \lambda \cdot f = c $$
In vacuum, c = 299 792 458 m/s exactly — a defined constant since the 1983 SI redefinition fixed the metre to it. For engineering work, the canonical form is:
$$ \lambda \text{ (m)} = \frac{300}{f \text{ (MHz)}} $$
— accurate to 0.07 %, which is below every other tolerance in any real antenna build. Use it everywhere. The full-precision form (λ = 299.7925 / f_MHz) shows up only in microwave length-critical applications like sub-mm-wave waveguide design, well outside the Hack Tools envelope.
2.1 Half-wave length at every band of interest
The half-wave length (λ/2) sets the physical size of the canonical resonant element — dipole, folded dipole, EFHW, Yagi driven element. Here it is for every band the radios in the hub care about, both in metric and Imperial (because vendors still spec antenna kits in feet):
| Band | f (MHz) | λ (m) | λ/2 (m) | λ/4 (m) | λ/2 (ft) |
|---|---|---|---|---|---|
| 160 m | 1.85 | 162.2 | 81.1 | 40.5 | 266 |
| 80 m | 3.75 | 80.0 | 40.0 | 20.0 | 131 |
| 60 m | 5.36 | 56.0 | 28.0 | 14.0 | 91.9 |
| 40 m | 7.15 | 42.0 | 21.0 | 10.5 | 68.9 |
| 30 m | 10.13 | 29.6 | 14.8 | 7.4 | 48.6 |
| 20 m | 14.18 | 21.2 | 10.6 | 5.3 | 34.7 |
| 17 m | 18.12 | 16.6 | 8.3 | 4.1 | 27.2 |
| 15 m | 21.23 | 14.1 | 7.07 | 3.53 | 23.2 |
| 12 m | 24.95 | 12.0 | 6.01 | 3.00 | 19.7 |
| 10 m | 28.85 | 10.4 | 5.20 | 2.60 | 17.0 |
| 6 m | 52.0 | 5.77 | 2.88 | 1.44 | 9.45 |
| 2 m | 146.0 | 2.054 | 1.027 | 0.514 | 3.37 |
| 1.25 m | 223.5 | 1.342 | 0.671 | 0.336 | 2.20 |
| 70 cm | 435.0 | 0.690 | 0.345 | 0.172 | 1.131 |
| 33 cm ISM | 915.0 | 0.328 | 0.164 | 0.082 | 0.538 |
| 23 cm | 1270.0 | 0.236 | 0.118 | 0.059 | 0.387 |
| L1 GPS | 1575.4 | 0.190 | 0.095 | 0.048 | 0.312 |
| 13 cm Wi-Fi | 2442.0 | 0.123 | 0.0614 | 0.0307 | 0.2014 |
| 13 cm ISM | 2450.0 | 0.122 | 0.0612 | 0.0306 | 0.2007 |
| 5 cm Wi-Fi | 5500.0 | 0.0545 | 0.0273 | 0.0136 | 0.0894 |
Two practical observations from the table that drive most antenna decisions in this series:
- Below 30 MHz, half-wave dipoles are large objects. A 40 m dipole is the size of a tennis court’s long axis; an 80 m dipole barely fits in the average suburban lot diagonally. This is why HF antennas always involve a real-estate compromise — the wire-folding tricks of Vols 7 (multi-band dipoles) and 10 (end-fed and random wire) and 14 (magnetic loops) all exist because nobody has room for the “right” antenna.
- Above 1 GHz, half-wave dipoles are jewelry. A 2.4 GHz dipole element is shorter than a credit card. This is why Wi-Fi antennas hide inside the device case, why a Yagi with 12 elements at 5 GHz is only 30 cm long, and why microwave antennas trade per-element gain for array gain — building one big aperture out of many tiny resonators.
2.2 Trim factors and the “wire is shorter than the wavelength predicts” effect
A wire antenna’s electrical half-wave is shorter than the free-space λ/2 by a few percent. The trim factor k depends on wire diameter, end termination, and ground proximity:
$$ L_{\text{wire}} = k \cdot \frac{\lambda}{2} $$
Typical values:
- Thin (#14-#18 AWG) bare wire in free space: k ≈ 0.95-0.96
- Thin insulated wire (the same gauge with PVC jacket): k ≈ 0.93-0.94 (the dielectric increases capacitance to surroundings)
- Fat tubing (1/2” aluminum) for VHF dipoles: k ≈ 0.94
- Wire near ground (
h < λ/4): k ≈ 0.90-0.94 (ground proximity adds loading capacitance)
The classical engineering shortcut: cut the wire 5% long, hang the antenna, sweep with a NanoVNA (Vol 24), trim from each end until resonant. Modeling (Vol 28) gives a tighter initial cut but never matches measurement under realistic ground.
2.3 Velocity factor inside transmission lines
Inside a transmission line, the wave travels slower than c by the velocity factor (VF):
$$ v_{\text{line}} = \text{VF} \cdot c $$
For coax, VF is set by the dielectric constant ε_r of the insulator between center conductor and shield:
$$ \text{VF} = \frac{1}{\sqrt{\varepsilon_r}} $$
Quick-reference table (full coax catalogue with loss numbers is in Vol 5 §4):
| Dielectric | ε_r | VF | Where you see it |
|---|---|---|---|
| Solid polyethylene | 2.25-2.30 | 0.66 | RG-58, RG-213, RG-8 |
| Foam polyethylene | 1.40-1.60 | 0.78-0.85 | LMR-240, LMR-400, LMR-600 |
| Solid PTFE (Teflon) | 2.05 | 0.70 | Mil-spec coax, high-temp |
| Air (with spacers) | 1.05-1.10 | 0.95-0.98 | Hardline, satellite TV |
| FEP (foamed teflon) | 1.65 | 0.78 | Some lab-grade coax |
This matters for any length-critical application: phasing harnesses, quarter-wave matching transformers, beverage-antenna terminators. A λ/4 matching stub at 144 MHz is 34.4 cm in RG-58 (VF 0.66), not 52.0 cm. Get this wrong and the match fails.
3. The electromagnetic spectrum in the bench envelope
The Hack Tools radios collectively cover from a few hundred kilohertz to about 6 GHz. The spectrum is conventionally divided into bands by the ITU; the names below show up in every datasheet, regulation, and band plan you will encounter:
| ITU band | f range | λ range | Examples in the hub |
|---|---|---|---|
| VLF | 3-30 kHz | 100-10 km | Submarine comms (not in scope) |
| LF | 30-300 kHz | 10-1 km | LORAN, WWVB time signal |
| MF | 300-3000 kHz | 1000-100 m | AM broadcast, NDB beacons |
| HF | 3-30 MHz | 100-10 m | Ham bands 160-10 m, shortwave, CB |
| VHF | 30-300 MHz | 10-1 m | FM broadcast, 2 m / 6 m / 1.25 m ham, aircraft AM, NOAA |
| UHF | 300-3000 MHz | 100-10 cm | 70 cm / 33 cm / 23 cm ham, cellular, Wi-Fi 2.4, BLE, ZigBee, GPS |
| SHF | 3-30 GHz | 100-10 mm | Wi-Fi 5/6 GHz, weather radar, satellite downlinks, point-to-point microwave |
| EHF | 30-300 GHz | 10-1 mm | mm-wave / 5G FR2 / automotive radar (out of scope) |
Inside this taxonomy, three sub-divisions matter most for the rest of this series:
- HF (3-30 MHz) — the “long-haul DX” range. Ionospheric reflection dominates; antennas are physically large; the half-wave-dipole-or-equivalent is the canonical antenna; the radials and ground system of Vol 20 become load-bearing for vertical designs.
- VHF/UHF (30 MHz - 3 GHz) — the “line-of-sight” range. Antennas shrink to wieldy sizes; gain antennas (Yagi, Vol 11) become practical; ionospheric propagation drops out above ~50 MHz except for sporadic-E openings; gain and directivity drive the antenna choice.
- Low SHF (3-6 GHz) — the “small-aperture” range. Antennas are tiny; arrays of patches and biquads dominate; cable loss climbs sharply (LMR-400 is 6.6 dB/100 ft at 2.4 GHz vs 0.4 dB at 7 MHz, see Vol 5 §5); a foot of cheap coax becomes a meaningful link-budget hit.
3.1 Where every radio in the hub sits
A spectrum chart showing the operating range of every radio in the Hack Tools hub, on a log axis from 100 kHz to 10 GHz:
100k 1M 10M 100M 1G 10G Hz
──────────┬────────┬────────┬────────┬────────┬─────
│ . . │ . . │ . . │ . . │ . . │ λ ↑
│ RTL-SDR V4 :══════════════════════════════: 2.5cm
│ HackRF One :══════════════════════════════════: 5cm
│ PortaRF :══════════════════════════════════:
│ :
│ Flipper sub-GHz :════════: 33cm
│ UV-K5 :═══: 2m
│ Marauder / Pineapple 2.4 :══: 12cm
│ Pineapple AC / Banshee 5 :══: 5cm
│ PWNagotchi / DSTIKE Wi-Fi :══: 12cm
│ Rayhunter (cellular) :════════: 15-50cm
│ Nyan Box NRF24 :══: 12cm
──────────┴────────┴────────┴────────┴────────┴─────The chart shows: (1) the HF-capable receive-only radios (RTL-SDR V4, HackRF One in RX mode) cover the full sweep; (2) the dedicated-band radios sit in narrow windows in UHF; (3) the 2.4 GHz Wi-Fi/BLE cluster is densely shared territory; (4) the 5 GHz cluster (Pineapple AC, AWOK C5, Banshee) is the gap-filler the 2026 acquisitions targeted.
3.2 Regulatory bands inside the engineering spectrum
The engineering bands above don’t map cleanly onto regulatory bands. Within HF, for example, the US has ham (Part 97), broadcast, commercial fixed-service, military, and amateur bands all coexisting. The regulatory geography is the subject of Vol 31; for now, the only piece that matters in this volume is that transmit on any band requires authority: a Part 15 device certification, a Part 97 amateur license at the appropriate class, or a Part 22/24/27/90 specific authorization. Receive has essentially no regulatory exposure, modulo ECPA carve-outs (US cellular traffic, paging) and some trunked-radio decoding limits. ../_shared/legal_ethics.md is the cross-tool baseline.
4. Propagation modes — ground wave, sky wave, line-of-sight, NVIS
A radio signal between transmitter and receiver gets there by one (or several, simultaneously) of four physical mechanisms. Which mechanism dominates depends on frequency, time of day, season, geometry, and luck. The four modes:
THE FOUR PROPAGATION MODES
Sky wave (HF, 3-30 MHz):
╔═══════════════════════════╗
║ F-layer (ionosphere) ║ ~250-400km
╚════╗ ╔════════════════════╝
║ ║
▼ ║
TX ─→ ▼
─→ RX (500-2000+ km hop)
Ground wave (LF/MF/lower HF, <5 MHz):
TX ──→─→─→─→─→─→─→─→─→ RX
(along Earth surface, ~100-3000 km)
Line-of-sight (VHF/UHF, >50 MHz):
TX ──→──→──→──→ RX
(visible horizon, <50 km typical)
NVIS (HF, 3-10 MHz):
F-layer
════╦═══════════
╱║╲
╱ ║ ╲
TX ──→╱ ║ ╲─→ RX (≤500 km regional)
╱ ║ ╲
─────╨───── 4.1 Ground wave
Below approximately 5 MHz, radio waves travel along the surface of the Earth as a ground wave — the field induces currents in the conductive ground (or seawater), which guides the wave along the curvature. This is how AM broadcast (MF, 530-1700 kHz) reaches receivers 100+ km away even with the F-layer absent during daylight, and how LORAN-C navigation worked at 100 kHz across thousands of kilometres.
Ground-wave range scales roughly inversely with frequency: a 1 kW transmitter at 500 kHz reaches ~1500 km of useful ground-wave coverage over good soil; the same power at 5 MHz reaches ~200 km. Above 10 MHz, ground wave is effectively dead for any useful distance.
Practical implication for this series: ground wave is not the propagation mechanism for any radio in the Hack Tools hub. None of the in-scope radios transmit below 1 MHz, and HF receive of MF broadcast (via random wire + 9:1 UNUN on an RTL-SDR or HackRF, see Vol 10 and Vol 16) is the only ground-wave-adjacent use case.
4.2 Sky wave — the HF DX mechanism
Sky wave is the ionospheric reflection mechanism that makes HF (3-30 MHz) into the long-haul propagation band. The Earth’s ionosphere — a region from ~60 km to ~400 km altitude where ultraviolet radiation from the Sun ionizes atmospheric gases — develops several distinct layers that behave as frequency-selective mirrors for radio waves:
- D layer (~60-90 km, daytime only): absorbs HF energy, particularly below 7 MHz. The reason 80 m and 40 m operation is mostly an after-dark activity.
- E layer (~90-120 km, day and night, stronger by day): reflects HF moderately; supports “sporadic-E” openings at VHF up to ~150 MHz during summer months.
- F1 / F2 layer (~150-400 km, F1 only by day, F2 day and night with diurnal variation): the workhorse for HF DX. F2 is the high-flying layer responsible for trans-oceanic propagation on 20 m, 17 m, 15 m, and (during solar maxima) 10 m and 6 m.
Two derived quantities dominate practical HF planning:
- MUF (Maximum Usable Frequency): the highest frequency that reflects off the ionosphere for a given path geometry, time, and solar conditions. Above the MUF, the wave punches through into space rather than reflecting. MUF rises with solar activity (peaks during solar maxima ~every 11 years) and varies sharply with time of day, season, and latitude.
- LUF (Lowest Usable Frequency): the lowest frequency that gets through the absorbing D layer with usable strength. Below the LUF, absorption eats the signal. LUF rises during the day (when D-layer absorption is strongest) and falls at night.
The right band for any HF QSO at any given moment is bracketed: LUF < band < MUF. VOACAP and propagation-prediction software (PropQuest, VOACAP-Online) computes both numbers as functions of time-of-day, season, and the smoothed sunspot number SSN; they are the reference tool for any HF planning.
For antenna implications: an HF antenna at moderate height (h < λ/2 above ground) radiates significantly upward, which is the right radiation pattern for sky-wave operation. An HF dipole at 0.5λ is a respectable DX antenna; the same dipole at 0.15λ is a poor DX antenna but a respectable NVIS antenna (§4.4 below).
4.3 Line-of-sight
Above ~50 MHz, ionospheric reflection drops out for most practical paths (sporadic-E openings excepted), and propagation reverts to the basic geometric mechanism: the signal travels in a straight line from transmitter to receiver, attenuated by free-space path loss (§7 below) plus any obstacles in the path.
The “line-of-sight” range is gated by:
- Earth curvature — for an antenna at height h metres on a flat-water surface, the geometric horizon is roughly d (km) ≈ 3.57·√h. A 10 m mast sees ~11 km to the geometric horizon.
- Refraction in the troposphere — the lower atmosphere has a slowly-decreasing refractive index with altitude, which bends radio waves slightly downward. The “radio horizon” is roughly 15% farther than the geometric horizon: d (km) ≈ 4.12·√h. A 10 m mast has a radio horizon of ~13 km.
- Fresnel zone clearance — the radio path is not a ray but a 3-D ellipsoid; obstacles intruding into the first Fresnel zone (60% of the bare-line geometry between TX and RX) cause attenuation even though the geometric line is clear.
- Tropospheric ducting — anomalously sharp refractive-index gradients (often above ocean, in summer high-pressure conditions) can trap a wave between Earth and an atmospheric layer, extending propagation to hundreds of kilometres at frequencies that “shouldn’t” go that far. The 2 m and 70 cm ham bands see ducted openings several times a year.
For Hack Tools radios: every VHF/UHF use case (Flipper sub-GHz, Marauder Wi-Fi, Pineapple, UV-K5, PWNagotchi) is line-of-sight propagation. The antenna’s elevation pattern (§5.1 of Vol 4) needs to put energy at the right angle for the geometry; a vertical with a peak elevation of 25° is wasting most of its energy upward if you’re trying to talk to a station at the geometric horizon.
4.4 NVIS — Near Vertical Incidence Skywave
NVIS is a deliberate variant of sky wave: rather than firing energy at a low elevation angle for long-distance DX, the operator fires energy up (60-90° elevation) so the F-layer reflects it back down with a steep return angle. The result is a circular coverage footprint of roughly 50-500 km, with no skip zone in the centre — exactly the geometry for regional / emergency / inter-city communication on HF.
NVIS is used heavily by military HF (HF/SSB tactical comms over rough terrain), by emergency-services HF networks (ARES/RACES regional traffic nets), and by hams operating from low-mast deployments where the antenna height can’t be optimized for low-angle DX.
The antenna implication: an NVIS antenna is a horizontal dipole at low height (h ≈ λ/8 to λ/4). The low height forces a high-angle radiation pattern — what would be a flaw for DX is the feature for NVIS. The “Inverted-V at 10 ft” deployment that field-day operators use is a textbook NVIS antenna by accident. See Vol 6 §4 for the dipole pattern as a function of height.
5. Polarization — linear, circular, slant, and why it matters
A radio wave’s electric-field vector points in some direction, perpendicular to the direction of propagation. Polarization is the description of how that vector orients itself as the wave propagates. The four practical cases:
Polarization vectors (looking
along direction of propagation)
Vertical Horizontal Slant 45°
│ ━━━━━━ ╱
│ ╱
▼ ╱
Right circ. Left circ.
↻ ↺
(clockwise as (counterclockwise as
wave leaves wave leaves source)
source — IEEE
convention) 5.1 Linear polarization (vertical, horizontal)
The E-field oscillates along one fixed line — either vertical (perpendicular to Earth) or horizontal (parallel to Earth). The defining axis of the antenna sets the polarization: a vertical dipole, a vertical monopole, a J-pole hanging vertically all radiate vertical polarization. A horizontal dipole, a Yagi with horizontal elements, an inverted-V all radiate horizontal polarization.
Which to choose:
- HF DX (sky wave) — usually horizontal. The F-layer reflection scrambles polarization moderately on its way through, but starting horizontal gives a tighter low-angle pattern over real ground.
- HF mobile / portable — usually vertical. A vertical sticks up; a horizontal needs a long mast and supports.
- VHF/UHF FM voice — by convention, vertical. The whole 2 m / 70 cm FM repeater ecosystem standardized on vertical decades ago for ease of mobile + handheld + base-station compatibility.
- VHF/UHF SSB and weak-signal — by convention, horizontal. The other half of the VHF community split off in the 1960s when long Yagis became practical; horizontal Yagis are mechanically tighter than vertical Yagis at the same gain.
- Wi-Fi (2.4 / 5 GHz) — usually vertical (router whips, mobile-device antennas). Sometimes circular, sometimes dual-pol via two slanted elements.
5.2 Circular polarization (RHCP, LHCP)
The E-field rotates as the wave propagates. Right-hand circular (RHCP, IEEE convention: the vector rotates clockwise when viewed from behind the wave, i.e. from the transmitter looking forward) is used for:
- GPS L1 (1575.42 MHz): RHCP. GPS receivers reject the LHCP component to suppress ground-reflection multipath.
- Most amateur and commercial satellite uplinks/downlinks: RHCP for satellites whose Faraday rotation isn’t large enough to scramble polarization; some use LHCP for cross-polarization isolation.
Left-hand circular (LHCP) is used in:
- Some GLONASS / BeiDou bands.
- Some amateur satellite cross-band links to differentiate uplink from downlink polarization.
Circular polarization is generated by feeding two crossed (perpendicular) linear elements 90° out of phase — a crossed Yagi or a quadrifilar helical antenna. Or by a single helical antenna with a specific winding direction.
5.3 Slant polarization
A linear polarization oriented at 45° to vertical/horizontal. Two slant antennas at +45° and −45° form a dual-polarization pair — the geometry used in every modern cellular sector antenna (look at any cell tower and the panel antennas are slant-pair pairs). Slant pair feeds two independent receive chains that combine via MIMO; the cross-polarization between the two slants gives ~10 dB of polarization diversity, which is roughly equivalent to space diversity at half the geometric footprint.
Hack Tools radios don’t usually generate slant directly, but a Rayhunter or HackRF tuning a cellular band is receiving signal that was transmitted slant-45. Whether to receive with a vertical, a horizontal, or a slant-pair is a polarization-match question (§5.4).
5.4 Polarization mismatch — the loss math
When the receive-antenna polarization doesn’t match the transmit-antenna polarization, signal is lost. The loss depends on the angle between the polarization vectors:
| TX pol | RX pol | Theoretical loss | Practical loss |
|---|---|---|---|
| V | V | 0 dB | 0-0.5 dB |
| V | H | ∞ (complete null) | 15-30 dB |
| V | Slant-45 | 3 dB | 3-4 dB |
| RHCP | RHCP | 0 dB | 0.5-1 dB |
| RHCP | LHCP | ∞ (complete null) | 15-30 dB |
| RHCP | Linear | 3 dB | 3-5 dB |
The “complete null” cases in theory become 15-30 dB in practice because no real antenna is perfectly polarized — real antennas always have a few percent of energy in the opposite polarization, set by feed asymmetry, ground reflections, and construction tolerance.
Practical consequence: a 30 dB polarization mismatch is the difference between a copyable signal and silence. The classic mistake is to put up a horizontal Yagi to receive an FM repeater (which transmits vertical) — 20+ dB worse than rotating the Yagi 90°.
For the receive-only use cases that dominate this series — RTL-SDR scanning, HackRF wideband receive, signal-intelligence work — polarization-agnostic antennas are the right answer: omnidirectional verticals like the discone (Vol 12) accept vertically-polarized signal natively and capture ~3 dB of any other polarization; magnetic loops (Vol 14, Vol 15) capture H-field rather than E-field and are roughly polarization-agnostic.
5.5 Faraday rotation — the satellite caveat
Radio waves passing through the ionosphere experience Faraday rotation: the polarization vector rotates by an amount that scales with the integrated electron content along the path and with 1/f². Below 100 MHz the rotation can be hundreds of degrees, randomizing linear polarization completely; at 1 GHz it’s a few tens of degrees; at 10 GHz it’s negligible.
This is why HF/VHF satellite operation typically uses circular polarization (rotation has no effect on circular: rotating a circular vector keeps it circular, only changing the phase reference); and why 1 GHz+ GPS works fine with linear-RX-of-circular-TX with only the predictable 3 dB loss.
6. Near field, far field, and the Fraunhofer distance
An antenna’s behaviour changes as a function of distance from it. Close to the antenna, the electric and magnetic fields are dominated by reactive (energy-storing) components and the radiation pattern hasn’t fully formed. Far from the antenna, the pattern is fixed and the field decays as 1/r (power as 1/r²). The transition distance is called the Fraunhofer distance:
$$ d_F = \frac{2 D^2}{\lambda} $$
where D is the largest dimension of the antenna.
Near-field region Far-field region
┌──────────────────────────┐ ┌──────────────────────────┐
│ Reactive │ Radiating │ │ Pattern fully formed, │
│ near-fld │ near-field │ │ E,H in phase, │
│ (E²,H² │ (radiating │ │ Power ∝ 1/r², │
│ decay │ but pattern │ │ Field ∝ 1/r │
│ fast) │ not formed) │ │ │
│ │ │ │ │
└──────────────────────────┘ └──────────────────────────┘
r=0 r=λ/2π r=2D²/λ (Fraunhofer)
Antenna sits here6.1 Three zones inside the near field
The full picture has three zones:
- Reactive near field, r <
λ / (2π): dominated by stored E and H energy that doesn’t radiate. Fields decay as1/r³. Touching an antenna with a hand at HF transmit power can give a burn here. NanoVNA calibration short / open / load are inside this zone, which is why they work as calibration references. - Radiating near field (Fresnel region),
λ/(2π)< r <2D²/λ: fields are radiating but the pattern hasn’t fully formed; different parts of the antenna contribute to the field with different phase, and the resulting pattern depends on observation distance. - Far field (Fraunhofer region), r >
2D²/λ: pattern fully formed,1/rfield decay, the standard “antenna pattern” applies.
6.2 Worked examples
For the antennas this series covers:
| Antenna | Frequency | D (max dim) | λ | d_F |
|---|---|---|---|---|
| Half-wave dipole | 14 MHz | 10.6 m | 21.4 m | 10.5 m |
| Half-wave dipole | 144 MHz | 1.0 m | 2.1 m | 0.95 m |
| 5-element 2 m Yagi | 144 MHz | 3 m boom | 2.1 m | 8.6 m |
| 5-element 2.4 GHz Yagi | 2.4 GHz | 0.3 m boom | 0.125 m | 1.44 m |
| 2.4 GHz parabolic 60 cm | 2.4 GHz | 0.6 m | 0.125 m | 5.76 m |
| 5 GHz panel 30 cm × 30 cm | 5.8 GHz | 0.42 m (diag) | 0.052 m | 6.78 m |
Two practical implications:
- Bench characterization of a Yagi or panel requires distance. A 5-element 2 m Yagi must be measured at ≥ 9 m for the pattern to be correct; a 2.4 GHz dish at ≥ 6 m. Indoor antenna ranges are physically large for VHF/UHF measurement work; this is why commercial antenna characterization is done outdoors on a range or in an anechoic chamber.
- The NanoVNA’s calibration standards are deliberately in the reactive near field (a few mm from the test port) — the whole point is that the SOL standards interact only with reactive fields. Calibrating a NanoVNA with an antenna 1 m away on the bench is wrong; the antenna under test must be in the far field of its measurement reference plane, which (per Vol 24 §5) is at the test port, not at the antenna.
6.3 Near-field probes for diagnostic work
The reactive near field has practical use for EMC diagnostics and antenna debugging. A near-field probe is a small E-field stub or H-field loop that picks up only one field component at very short range; sweeping the probe around an antenna or a circuit board maps where energy is being radiated (often where you don’t want it). Inexpensive near-field probe sets (Aaronia PBS series, Beehive 100-series) feed straight into a spectrum analyzer (Vol 27) for EMI hunting. A HackRF + a homebrew H-field loop costs effectively nothing and works to 1 GHz; see Vol 27 §12 for the homebrew approach.
7. The Friis transmission equation and free-space path loss
The Friis transmission equation is the canonical link budget for a radio path between two antennas in free space:
$$ \frac{P_{rx}}{P_{tx}} = G_{tx} \cdot G_{rx} \cdot \left( \frac{\lambda}{4 \pi d} \right)^2 $$
where the G values are absolute (linear) gain referenced to isotropic.
In decibels — the form anyone actually uses:
$$ P_{rx}\text{(dBm)} = P_{tx}\text{(dBm)} + G_{tx}\text{(dBi)} + G_{rx}\text{(dBi)} - L_{fs}\text{(dB)} $$
where free-space path loss in dB is:
$$ L_{fs}\text{(dB)} = 20 \log_{10}\left( \frac{4 \pi d}{\lambda} \right) = 20 \log_{10}(d_{km}) + 20 \log_{10}(f_{MHz}) + 32.44 $$
The +32.44 dB constant absorbs 20 log₁₀(4π / (300·1000)) for d in km and f in MHz. The equivalent for d in miles and f in MHz is +36.58 dB.
7.1 FSPL at canonical distances
A quick-reference table of FSPL at distances and frequencies in the Hack Tools envelope:
| Frequency | 100 m | 1 km | 10 km | 100 km |
|---|---|---|---|---|
| 7 MHz | 29.4 | 49.4 | 69.4 | 89.4 |
| 14 MHz | 35.4 | 55.4 | 75.4 | 95.4 |
| 28 MHz | 41.4 | 61.4 | 81.4 | 101.4 |
| 50 MHz | 46.4 | 66.4 | 86.4 | 106.4 |
| 146 MHz | 55.8 | 75.8 | 95.8 | 115.8 |
| 435 MHz | 65.2 | 85.2 | 105.2 | 125.2 |
| 915 MHz | 71.7 | 91.7 | 111.7 | 131.7 |
| 2440 MHz | 80.2 | 100.2 | 120.2 | 140.2 |
| 5800 MHz | 87.7 | 107.7 | 127.7 | 147.7 |
All values in dB. The +20 dB per decade of distance and per decade of frequency is the canonical FSPL slope.
7.2 Worked link budget — Flipper to discone
A Flipper Zero TX at 433 MHz, +10 dBm output, talking to a HackRF One on a discone at 100 m. The Flipper’s PCB whip is ~0 dBi (charitably); the discone is ~0 dBi omnidirectional (Vol 12 §7). Coax to the HackRF: 3 m of LMR-240 at 433 MHz = ~0.3 dB loss (Vol 5 §5).
$$ P_{rx} = +10 + 0 + 0 - 65.2 - 0.3 = -55.5\text{ dBm} $$
The HackRF receive sensitivity at the bandwidth of a Flipper sub-GHz signal (~50 kHz) is roughly −80 to −90 dBm. The link has 25-35 dB of margin — strong, well above the noise floor, copies cleanly.
7.3 Worked link budget — HackRF + Yagi to distant SDR
A HackRF One driving an 8-element 2 m Yagi (G = 11 dBi, Vol 11 §4) at +15 dBm output, talking to a distant RTL-SDR V4 on a discone (G = 0 dBi) at 10 km. Coax at the HackRF: 5 m of LMR-400 at 146 MHz = 0.11 dB. Coax at the RTL-SDR: 3 m of LMR-400 = 0.07 dB.
$$ P_{rx} = +15 + 11 - 0.11 + 0 - 0.07 - 95.8 = -69.98\text{ dBm} $$
About −70 dBm — well within the RTL-SDR’s noise floor (~−115 dBm at narrow bandwidth). 45 dB of margin. The Yagi’s directivity is what makes this work; without it, the link is −81 dBm, still very copyable but less margin.
7.4 Worked link budget — Pineapple to client at the edge of plausibility
A WiFi Pineapple Mark VII + AC at +20 dBm output (the FCC limit, sort of), with a 15 dBi panel antenna, talking to a stock Wi-Fi laptop at 5 GHz, 1 km. Pineapple panel: 15 dBi (Vol 13 §4). Laptop internal antenna: ~0 dBi. Coax at the Pineapple: 1 m of LMR-240 at 5.8 GHz = 0.55 dB. Polarization match between vertical panel and laptop internal: assume 3 dB cross-pol penalty.
$$ P_{rx} = +20 + 15 - 0.55 + 0 - 3 - 107.7 = -76.25\text{ dBm} $$
About −76 dBm — close to the typical laptop Wi-Fi receive sensitivity of −80 dBm at 1 Mbps and well below the −65 dBm needed for 802.11n MCS 7 (full rate). The link works at low data rates but degrades quickly. Useful as a capture surface for handshakes (which need only beacon decoding, working at −90 dBm), less useful as a deauth target at 1 km. Cross-link to Vol 29 §7 for the Pineapple-specific antenna matrix.
8. Atmospheric absorption and rain fade
Above ~5 GHz, the atmosphere itself begins to absorb radio energy. Two molecules dominate the absorption in the Hack Tools envelope and just above it:
- Oxygen (O₂): resonant absorption peak at 60 GHz, broader minor peaks near 118 GHz. At 60 GHz the absorption is ~15 dB/km in a standard atmosphere — the reason 60 GHz is reserved internationally for short-range applications (the atmosphere kills long-haul links naturally).
- Water vapour (H₂O): resonant peaks at 22 GHz, 183 GHz, 325 GHz. The 22 GHz peak is the reason most satellite downlinks above Ku band (11 GHz) avoid 22 GHz specifically.
Below 10 GHz, atmospheric absorption is negligible for terrestrial paths (well under 0.1 dB/km). Above 10 GHz it climbs to several dB/km at the resonance peaks.
8.1 Rain fade
Rain attenuates microwave signals. The attenuation depends on rain rate and frequency:
| Rain rate | 2.4 GHz | 5 GHz | 10 GHz | 30 GHz |
|---|---|---|---|---|
| 1 mm/h light | 0.001 | 0.005 | 0.05 | 1.5 |
| 25 mm/h moderate | 0.02 | 0.1 | 1.0 | 15 |
| 100 mm/h heavy | 0.1 | 0.5 | 5.0 | 50 |
(values in dB/km, approximate, from ITU-R P.838)
At 2.4 and 5 GHz, rain is operationally irrelevant for terrestrial work — even monsoon-rate rain adds tenths of a dB to a 1 km link. Above 10 GHz it becomes the dominant link-budget concern; above 20 GHz rain fade can momentarily kill a microwave link entirely.
8.2 Other atmospheric effects worth knowing about
- Tropospheric scintillation — small fluctuations in refractive index cause minor amplitude variations at GHz frequencies; usually <0.5 dB and operationally invisible.
- Sporadic-E — irregular patches of intense E-layer ionization, mostly summer evenings, can support reflection up to ~150 MHz over ranges of 500-2000 km. The reason 6 m and 2 m occasionally has trans-continental openings.
- Aurora propagation — auroral activity in the high-latitude ionosphere can scatter VHF signals during geomagnetic storms; the resulting copy has characteristic “raspy” audio.
None of these dominate Hack Tools work, but they show up in scanner and SDR receive sessions enough to be worth recognising. VOACAP-style prediction tools and DXMaps.com track them in near-real-time.
9. Common gotchas and quick-reference math
A list of the recurring mistakes and the cheats that make them go away. Each one is cross-linked to the volume where it’s treated in depth.
9.1 The unit gotchas
- dBm vs dBW: differ by 30 dB. Confuse them at your peril. (Vol 3 §4)
- dBi vs dBd: differ by 2.15 dB. dBi is referenced to isotropic; dBd to a half-wave dipole. Vendor specs are typically dBi to look bigger. (Vol 3 §6)
- 10 log vs 20 log: 10 log for power ratios, 20 log for voltage / field-strength ratios. Off by 3 dB per element if confused. (Vol 3 §3)
- EIRP vs ERP: EIRP is referenced to isotropic; ERP to a half-wave dipole. EIRP = ERP + 2.15 dB. FCC Part 15 uses EIRP; Part 22/24 uses ERP. (Vol 31 §7)
9.2 The geometric gotchas
- λ/4 vs λ/2: a quarter-wave element is monopole-like (45 Ω free-space-feed, requires ground/counterpoise); a half-wave element is dipole-like (73 Ω feed, free-standing). Not interchangeable.
- Half-wave at the operating frequency, not the design band centre: a “40 m EFHW” cut for 7.15 MHz is the wrong length for 7.3 MHz. Trim for the centre of the band of intended use.
- VF correction: matching stubs and phasing harnesses are electrical
λ/4, which is physicalλ/4 · VF. Forget the VF correction and the match fails. (Vol 5 §6)
9.3 The propagation gotchas
- Polarization match matters more than you’d think: a 20-30 dB penalty for the wrong polarization is realistic, especially at VHF/UHF. Match to the source if known; choose omnidirectional / polarization-agnostic for unknown sources.
- The far-field requirement: pattern measurement requires d >
2D²/λ. Indoor bench measurement of a Yagi gives wrong answers. - Ground proximity changes everything: the free-space pattern of a dipole is irrelevant; the over-real-ground pattern depends on height in wavelengths. NEC models with realistic ground are mandatory for HF work (Vol 28 §10).
9.4 The quick FSPL cheat
Memorize one number: at 1 km, 1 GHz, free-space path loss is 92.4 dB. Every doubling of distance adds 6 dB; every doubling of frequency adds 6 dB. Every halving subtracts 6 dB.
This lets you do FSPL in your head:
- 1 km at 433 MHz ≈ 92.4 − 2·6 ≈ 80 dB (actually 85.2 dB; the approximation is +5 from the 0.5× ratio rounding — close enough).
- 100 m at 2.4 GHz ≈ 92.4 − 6 − 6 + 2·6 ≈ 80 dB (actually 80.2 dB — bang on).
- 10 km at 146 MHz ≈ 92.4 + 6·4 − 2·6 ≈ 105 dB (actually 95.8; the approximation is +10 high — use the table).
The exact formula is better, but the 6-dB-per-octave intuition pays for itself when you’re sanity-checking a link budget in the field.
9.5 The “everything you forgot” reminder
Two final notes that bite even experienced engineers:
- Feedline loss enters the link budget twice on a one-way path (TX feedline, RX feedline). On a transceiver round-trip it enters four times. A 3 dB feedline run is a 6 dB hit on a TX→RX path and 12 dB on a full round-trip.
- Antennas have return-path gain symmetry but receivers don’t. A 10 dBi TX antenna and a 10 dBi RX antenna add 20 dB to the link. A 10 dBi TX antenna only adds 10 dB to a one-way TX, and a noise-floor-limited receiver gains 10 dB of signal but no noise reduction from the receive antenna (unless the receive antenna’s pattern rejects local noise sources — the Vol 15 story).
10. Resources
- ARRL Antenna Book Ch. 1-2 (propagation, fundamentals) — the canonical amateur-radio reference, 25th ed.+
- Balanis, Antenna Theory: Analysis and Design (4th ed., Wiley), Ch. 2 (fundamental parameters)
- Stutzman & Thiele, Antenna Theory and Design (3rd ed.), Ch. 1-3
- Kraus, Antennas (3rd ed., McGraw-Hill) — historical and foundational; the source of “dBi” as a unit
- ITU-R Recommendation P.525 (calculation of free-space attenuation)
- ITU-R Recommendation P.838 (rain attenuation model)
- VOACAP (HF propagation prediction): https://www.voacap.com
- PropQuest (modern propagation prediction with mobile interface): https://propquest.co.uk
- DXMaps (real-time HF/VHF propagation map): https://www.dxmaps.com
- QST magazine “Antenna Topics” series (decades of dipole-and-vertical case studies)