Antennas · Volume 4
Antenna Theory
Impedance, resonance, SWR, gain, directivity, radiation patterns, beamwidth, front-to-back, efficiency, Q, bandwidth — the working theory every per-antenna chapter assumes
Contents
1. About this volume
This volume is the third leg of the foundation tripod (Vol 2 propagation, Vol 3 dB units, this volume antenna theory, Vol 5 feedlines). Where Vol 2 dealt with the space the wave moves through and Vol 3 dealt with how we quantify it in dB, Vol 4 deals with the antenna itself as a circuit element: what its feedpoint impedance is, how the radiation pattern is shaped, what gain means, where efficiency goes when it goes, why a narrow-band antenna and a wide-band antenna come from the same physics with different geometric choices.
Every per-antenna chapter (Vols 6–15) opens with “Geometry & theory of operation” and “Feedpoint impedance and matching” sections that quote results from this volume without re-deriving them. A reader who knows antenna theory well can skim Vol 4 once and never come back; a reader newer to the subject should expect to use it as the reference for every “wait, what’s HPBW again?” or “what’s the difference between gain and directivity?” question that comes up in the per-antenna chapters.
The volume assumes the reader has the Vol 2 and Vol 3 material in hand — particularly λ/f relations, dBi vs dBd, and the SWR / return-loss / mismatch-loss triple from Vol 3 §8.
2. Antenna as a transformer between guided and free-space waves
An antenna’s fundamental job is impedance transformation. The transmitter delivers RF energy down a transmission line as a guided wave at some characteristic impedance — typically 50 Ω, the universal modern convention. Free space has its own characteristic impedance, the impedance of free space:
$$ \eta_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} \approx 377,\Omega $$
The antenna’s task is to take guided-wave power at 50 Ω from the coax and convert it into a radiating free-space wave matched to 377 Ω, without introducing significant loss in the process. The transformation isn’t a simple 50:377 ratio change (that wouldn’t be sufficient — radiation is a spatial phenomenon, not a circuit one); it’s a geometric arrangement of conductors that radiates part of the energy out as a propagating wave.
On the receive side, reciprocity (§9 below) says the same antenna does the inverse job: an incident free-space wave induces currents in the antenna structure that present a 50 Ω equivalent source to the receiver, with the antenna’s pattern determining which directions the receiver is sensitive to.
Two consequences of this transformer view that recur throughout the series:
- The antenna is part of the impedance-matching network, not separate from it. The geometry of the antenna (its length, its driven-element spacing, its feed location) sets the feedpoint impedance. BALUNs (Vol 16) and tuners (Vol 17) are auxiliary devices that adjust an antenna’s natural impedance to match 50 Ω coax; they don’t replace the antenna’s geometry.
- Antenna losses are real losses. An antenna that radiates 70% of its input power and dissipates 30% as heat in conductors and ground losses has 70% radiation efficiency. This shows up in the link budget as 1.5 dB less effective TX power than a 100%-efficient antenna at the same nominal directivity. The §5 gain/directivity/efficiency relationship makes this explicit.
3. Feedpoint impedance and resonance
The feedpoint impedance of an antenna is the complex impedance the transmitter sees at the antenna’s input terminals:
$$ Z_{\text{in}} = R_{\text{in}} + j X_{\text{in}} $$
where R_in is the resistive part (the real part — power either radiated or dissipated as heat) and X_in is the reactive part (imaginary — stored energy oscillating between electric and magnetic fields, not radiated).
3.1 Resonance
An antenna is resonant at a frequency where X_in = 0 — the reactive part vanishes, and Z_in is purely resistive. Resonance is the design target for most antennas in this series because a resonant antenna:
- Presents a clean, predictable load to the transmitter.
- Doesn’t store reactive energy that would otherwise cause voltage maxima on the feedline (the cause of “RF burns” and arcing).
- Has minimum SWR by definition — Γ is purely real, and the SWR is just R_in / Z_0 (or its inverse if R_in < Z_0).
Below resonance, X_in is capacitive (negative); above resonance, X_in is inductive (positive). An antenna swept across its resonant band traces a near-circular Smith chart loop, passing through the real axis at resonance.
3.2 Feedpoint impedance for canonical antennas
A reference table of free-space (idealized) feedpoint impedances for the antennas in Vols 6–15:
| Antenna | Free-space Z_in (at resonance) | Match to 50 Ω |
|---|---|---|
| Half-wave dipole | 73 + j42 Ω (untrimmed); 73 + j0 Ω at resonance | SWR 1.46:1 → 1:1 with 1.5:1 transformer |
| Folded dipole | 280 + j0 Ω | SWR 5.6:1 → match with 4:1 BALUN |
| Quarter-wave monopole over ideal ground | 36 + j20 Ω (untrimmed); 36 + j0 Ω resonant | SWR 1.39:1 — usable without match |
| Three-element Yagi (driven element only) | 25 + j0 Ω at design freq | SWR 2:1 — match with hairpin or gamma |
| Five-element Yagi | 18 + j0 Ω | SWR 2.7:1 — needs matching |
| Twelve-element Yagi (high-gain) | 12 + j0 Ω | SWR 4:1 — gamma match or LFA driven |
| End-fed half-wave (EFHW) | ~2450 + j0 Ω | SWR 49:1 — match with 49:1 UNUN |
| Small magnetic loop | 0.05 + j(huge) Ω | Series-resonant via tuning capacitor |
| 3λ random wire | ~450 Ω (variable) | 9:1 UNUN brings to ~50 Ω |
The wide variation — from 0.05 Ω for a small loop to 2450 Ω for an EFHW — is why matching networks exist as a major design topic (Vol 16, Vol 17).
3.3 Over-ground vs free-space impedance
The free-space numbers above are theoretical. Real antennas sit near ground, which changes the feedpoint impedance:
- A half-wave dipole at h = λ/2 above ground: Z_in ≈ 73 + j0 Ω (close to free space).
- The same dipole at h = λ/4: Z_in ≈ 60-65 + j0 Ω (ground reflection adds parallel capacitance).
- At h = λ/8: Z_in drops to 50-55 + j0 Ω (a near-50 Ω “natural” match, but with terrible radiation efficiency — see §5).
- At h = 0.05λ: Z_in drops below 40 Ω, mostly because of ground-conduction loss being treated as part of the radiation resistance.
For monopoles, the ground proximity matters even more, because the ground is the antenna’s image plane:
- Quarter-wave monopole on a poor (lossy) radial system: R_in = 36 + R_loss Ω, often appearing as 50-70 Ω total. The antenna “matches better” because the loss resistor is adding to the radiation resistance — but power is going into the ground.
- Quarter-wave monopole with 60+ good radials: R_in ≈ 36 Ω. Power goes to radiation.
This is the unifying theme of Vol 20 (Grounding): a vertical that “matches well” without radials is matching to loss, not to radiation. The SWR is good and the antenna is mostly heat.
3.4 Trimming for resonance
In practice, antennas are cut slightly long, hung in their operational location, and trimmed for resonance. The procedure with a NanoVNA (Vol 24):
- Cut to nominal length (typically 5-7% longer than the theoretical 0.95·λ/2 trim).
- Hang the antenna in its final location, attach feedline and BALUN.
- Sweep S11 across the band of interest.
- Identify the resonance — the frequency where reactance crosses zero (Smith-chart trace crosses the real axis).
- If resonance is low of target, trim each leg by 2-3% and re-sweep.
- Iterate until resonance lands at target frequency.
For an HF dipole, the trim amount per kHz of resonance shift is roughly:
- 80 m: 4 cm per 10 kHz (per leg)
- 40 m: 2 cm per 10 kHz
- 20 m: 1 cm per 10 kHz
- 10 m: 0.5 cm per 10 kHz
These are starting points; real antennas vary by ±50% from these numbers depending on wire diameter and nearby objects.
4. SWR, reflection coefficient, and the Smith chart
The reflection coefficient Γ, SWR, and mismatch loss were covered as units in Vol 3 §8. This section is the geometric / visualizable version of the same content: the Smith chart.
4.1 What the Smith chart is
The Smith chart is a polar plot of the complex reflection coefficient Γ. The real axis runs horizontally, the imaginary axis vertically; |Γ| = 1 is the outer boundary (the unit circle); |Γ| = 0 is the centre. Every passive impedance Z (positive resistance, any reactance) maps to a single point inside the unit circle.
The chart’s superpower is that it overlays two grids:
- Constant-resistance circles — vertical curves (actually circles tangent to the rightmost point of the unit circle) where R/Z₀ is constant.
- Constant-reactance arcs — curves running from the leftmost point of the unit circle to the rightmost, where X/Z₀ is constant.
A given impedance Z = R + jX maps to the intersection of its R/Z₀ circle with its X/Z₀ arc. The position on the chart immediately reveals: where the resistance lies relative to 50 Ω; whether reactance is inductive (top half) or capacitive (bottom half); how far from a perfect match the impedance is.
Smith chart (50 Ω normalized) — schematic view
+jX (inductive)
┃
╭────────╋────────╮
╭─┘ ┃ └─╮
╱ ┃ ╲
│ X=+50 ┃ X=+150 │
│ ┃ │
│ R=20 ┃ R=100 │
───────●━━━━━━━━━━━●━━━━━━━━━━━●─────── Real (R)
short │ ┃ │ open
(R=0) R=50 ┃ R=200 (R=∞)
│ ┃ │
│ X=−50 ┃ X=−150 │
╲ ┃ ╱
╰─┐ ┃ ┌─╯
╰────────╋────────╯
┃
−jX (capacitive)A perfect 50 Ω match is the centre of the chart. An open circuit (R = ∞) is the rightmost point on the real axis. A short circuit (R = 0) is the leftmost point. Pure inductance falls on the upper unit-circle boundary; pure capacitance on the lower.
4.2 Reading an antenna’s behaviour from the Smith chart
A well-behaved resonant antenna swept across its band traces a clean small loop on the Smith chart:
- Below resonance: the trace sits in the lower (capacitive) half.
- At resonance: the trace crosses the real axis. If at the centre, the antenna is perfectly matched to 50 Ω; if to the right of centre, R > 50 Ω; if to the left, R < 50 Ω.
- Above resonance: the trace sits in the upper (inductive) half.
The loop’s size indicates bandwidth: a tight loop near the centre = wide bandwidth (the antenna stays close to a good match over many MHz); a large loop that swings far from the centre = narrow bandwidth.
A bad antenna shows characteristic shapes:
- A loop that misses the centre entirely — resonant but at the wrong R. Needs a matching network to translate to 50 Ω.
- A loop that doesn’t close — antenna isn’t resonant in the swept range; need to extend the sweep up or down.
- A trace that wanders unpredictably — common-mode currents on the feedline, or multiple unintended resonances. Vol 5 §10 and Vol 16 §3 for cures.
4.3 Smith-chart matching — the geometric operations
The Smith chart’s other superpower: matching-network design becomes geometric, not algebraic. The four primitive moves on the Smith chart:
- Series capacitor — moves along a constant-resistance circle, downward (toward more capacitive reactance).
- Series inductor — moves along a constant-resistance circle, upward (toward more inductive reactance).
- Shunt capacitor — flip to admittance (rotate 180° around the centre), move along a constant-conductance circle downward (more capacitive susceptance), flip back.
- Shunt inductor — flip to admittance, move along constant-conductance upward, flip back.
A complete L-network match is two moves: one series + one shunt, in some combination. The geometric procedure:
- Plot the antenna’s complex impedance Z_in on the Smith chart.
- Decide whether to use series-first or shunt-first.
- Trace a constant-resistance arc to one of the two intersection points with the 1.0 normalized-conductance circle (the “Smith circle for matching”).
- The arc traversed gives the series element value (L or C).
- Move along the constant-conductance circle to the centre — this gives the shunt element value.
The Smith chart turns matching-network design into “draw lines and read off values” — which is why every RF engineer eventually loves it. Modern NEC modeling (Vol 28) and NanoVNA software (Vol 24) automate the geometry, but the reasoning remains geometric.
4.4 Time-domain readout from the Smith chart
A frequency sweep of S11 across a wide band carries information beyond the impedance at each individual frequency. The Smith chart trace also tells you, in time-domain terms:
- The location of impedance discontinuities along the feedline. A discontinuity at distance d down a coax produces a Smith-chart trace that rotates around the centre with frequency — the rotation rate gives d. NanoVNA’s time-domain reflectometry mode (TDR, Vol 24 §10) computes this directly via inverse FFT.
- Multiple reflections — a coax + connector + antenna chain with two discontinuities produces beats in the frequency-domain trace.
This is the basis of cable-fault location (find the loose connector 23 m down the coax run) and of impedance discontinuity diagnosis.
5. Gain, directivity, and efficiency — the three numbers and how they relate
The three quantities are often conflated but are distinct. Get this section right and most marketing-spec confusion evaporates.
5.1 Directivity D
Directivity is a geometric quantity: it measures the concentration of radiation in the antenna’s peak direction relative to an isotropic radiator. Defined as:
$$ D = \frac{P_{\text{peak}}}{P_{\text{average over sphere}}} $$
Directivity is lossless — it’s purely about the shape of the radiation pattern, not how much of the input power makes it out as radiation. A perfectly-modeled half-wave dipole has D = 2.15 dBi by geometry alone, regardless of conductor loss.
Directivity has a useful approximation in terms of beamwidth:
$$ D \approx \frac{32400}{\Theta_E \cdot \Theta_H}\ \text{(linear)} $$
where Θ_E and Θ_H are the half-power beamwidths in the E-plane and H-plane (degrees). A Yagi with 30° E-plane beamwidth × 35° H-plane beamwidth has D ≈ 32400/(30·35) ≈ 30.9 (15 dBi).
5.2 Radiation efficiency η
Radiation efficiency is the fraction of input power that actually radiates as electromagnetic energy (vs being dissipated as heat in conductors, the ground, the matching network, etc.):
$$ \eta = \frac{P_{\text{radiated}}}{P_{\text{input}}} $$
Where the loss goes:
- Conductor losses: I²R heating in wires, especially at MF/LF where skin depth is large. Typically <1% for HF wire dipoles; 2-5% for thin VHF/UHF wires.
- Dielectric losses: in insulators, loaded coils, traps. <1% for ceramic insulators; 5-15% for loaded mobile whips.
- Ground losses: the dominant loss for verticals over real ground. 30-70% loss without a radial system; 5-15% with adequate radials. The driver of Vol 20.
- BALUN / matching-network losses: 0.2-1 dB typical for HF BALUNs; can be 3-6 dB for badly-designed tuners forcing extreme impedance ratios.
Typical efficiencies for antennas in this series:
| Antenna | Typical η |
|---|---|
| Half-wave dipole at λ/2 above good ground | 95-99% |
| Half-wave dipole at λ/8 above ground | 60-80% |
| Trap dipole | 70-85% (trap loss is the main cost) |
| Full-size vertical with 60 ground radials | 80-90% |
| Same vertical with 4 ground radials | 40-60% |
| Loaded mobile whip on 80 m | 5-15% |
| Yagi (well-built) | 95-98% |
| Small magnetic transmitting loop | 30-90% (depends on conductor quality and capacitor Q) |
| Rubber-duck HT antenna | 5-15% |
| EFHW with quality 49:1 UNUN | 90-95% |
| EFHW with cheap saturating UNUN | 50-70% |
5.3 Gain G
Gain is what the link budget cares about. It’s directivity times efficiency:
$$ G = \eta \cdot D $$
Or in dB:
$$ G\text{(dBi)} = D\text{(dBi)} + 10 \log_{10}(\eta) $$
A perfectly-efficient antenna with 10 dBi directivity has 10 dBi gain. A 70%-efficient antenna with the same directivity has G = 10 + 10·log(0.7) = 10 − 1.55 = 8.45 dBi.
This is the foundation of one of the great vendor-marketing tricks: quote directivity instead of gain. A short loaded mobile whip might have 6 dBi directivity (the same pattern shape as a quarter-wave whip) but η = 0.1 (10% efficiency, the loss being in the loading coil). Its actual gain is 6 + 10 log(0.1) = 6 − 10 = −4 dBi — 1/4 the radiation of the lossless geometric prediction. Vendor specs that say “6 dBi gain” are quoting the upper number.
5.4 The “real” gain question
For any antenna you’re considering buying or building, the operationally-relevant gain isn’t the spec-sheet number; it’s the measured gain into a calibrated reference at a known range. Most vendors don’t measure; they model in NEC (Vol 28) and report the model’s predicted directivity. Real-world losses (ground, feed asymmetry, weather, construction tolerance) eat 1-3 dB below the model.
A reasonable rule for skeptical reading of vendor specs:
- For a clean wire dipole or vertical: trust the spec to within 0.5 dB.
- For a multi-band trap antenna: trust the spec to within 1.5 dB on the design band; expect 2-4 dB worse off-band.
- For any “miracle multi-band gain antenna”: expect the spec to be 3-6 dB optimistic.
- For a Yagi from a reputable manufacturer with measured (not modeled) gain: trust to within 0.5 dB.
The remedy for spec inflation is measurement: NanoVNA for SWR (Vol 24) + a calibrated antenna range for pattern (Vol 25 §9) tells you the truth.
6. Radiation patterns — azimuth, elevation, beamwidth, HPBW, front-to-back
The radiation pattern is the spatial distribution of radiated power. Most antennas of interest have approximately separable patterns: an azimuth pattern (looking down from above, the “footprint” in the horizontal plane) and an elevation pattern (looking from the side, the “shape” in vertical-cut planes).
6.1 Pattern presentations
Three common forms:
- Polar plots, log magnitude (dB) — the standard engineering form. The pattern is plotted in dB relative to the peak; concentric circles mark −3 dB, −10 dB, −20 dB, −30 dB rings. The −3 dB ring shows the half-power beamwidth.
- Polar plots, linear magnitude — appears in some textbooks; less useful because the −3 dB point falls at 70.7% of the peak radius, not visually distinctive.
- 3-D radiation surface — modern NEC modeler output. Visually compelling but harder to read off specific numbers.
The engineering convention: always use the log-magnitude polar plot when characterizing or comparing antennas. The linear form lies — it visually emphasizes the main beam at the expense of the side lobes and nulls that determine whether an antenna works in a specific environment.
6.2 Half-power beamwidth (HPBW)
HPBW is the angular width of the main beam at the −3 dB point — the angle inside which the antenna delivers at least 50% of its peak radiated power density. The number every Yagi datasheet quotes.
Typical HPBW values:
- Isotropic radiator: undefined (uniform sphere)
- Half-wave dipole (free space): 78° in the E-plane (axis perpendicular to wire), omnidirectional in the H-plane (azimuth around the wire)
- Quarter-wave vertical: omnidirectional in azimuth, ~80° in elevation
- 3-element Yagi: ~60° azimuth × ~60° elevation
- 5-element Yagi: ~45° × ~50°
- 9-element Yagi: ~30° × ~35°
- 15-element Yagi: ~20° × ~25°
- 60 cm dish at 2.4 GHz: ~12° × 12°
- 1 m parabolic at 10 GHz: ~2° × 2°
The narrower the HPBW, the higher the directivity, but the harder the antenna is to aim. A 15-element Yagi mis-aimed by 25° is a 5 dB-down beam, not the 14 dBi nominal.
6.3 Front-to-back (F/B) ratio
For directional antennas, F/B ratio is the gain in the peak forward direction minus the gain in the 180° backward direction:
$$ F/B = G_{\text{forward}} - G_{\text{backward}} $$
A Yagi optimized for high F/B (using NEC optimization or LFA techniques) hits 25-35 dB. A standard 3-element Yagi: 15-20 dB. An LPDA: 12-18 dB.
F/B matters in:
- Direction-finding (the back lobe is what makes an antenna “bidirectional” in a DF context — you need it to be small to localize a TX cleanly).
- Operating in a high-noise environment where the noise comes from the back direction (point the antenna at the signal; the back-side null rejects local noise sources).
- Stacking antennas (rear lobes of stacked antennas combine constructively in unwanted directions if F/B is poor).
6.4 Side lobes
A directional antenna’s main beam is accompanied by side lobes — secondary peaks 30-60° off the main beam axis, typically 8-15 dB below the main beam.
Side lobes matter because:
- They radiate power in unwanted directions — TX side lobes hit unintended receivers; RX side lobes pick up signal from unwanted directions.
- They limit how cleanly an antenna can null a specific direction — moving an interfering source out of the main beam puts it in a side lobe, not in a null.
- They are the price of high gain — narrowing the main beam concentrates power and sharpens the side-lobe structure.
NEC optimization can suppress side lobes at the cost of gain or F/B. Commercial high-end Yagis (M2, InnovAntennas, Force 12) often choose configurations that trade 0.5 dB of gain for 8-10 dB lower side lobes.
6.5 Pattern conventions for the specific antennas
A pattern reference for the wire-and-air cluster, summarized for quick recall:
| Antenna | Azimuth | Elevation | Pol |
|---|---|---|---|
| Half-wave dipole, horizontal, high | Figure-8 broadside | Lobes split, peak elevation 30° | H |
| Half-wave dipole, horizontal, low (NVIS) | Figure-8 broadside | Peak near zenith | H |
| Quarter-wave vertical, ground plane | Omnidirectional | Peak 25-30°, null at zenith | V |
| Vertical dipole | Omnidirectional | Peak ~30°, null at zenith | V |
| End-fed half-wave | Figure-8 if horizontal; modified if sloped | Like the half-wave dipole | depends on orientation |
| 3-el Yagi | 60° HPBW main + 15 dB F/B | 60° HPBW with peak ~elev. of mount | matches element orientation |
| Discone | Omnidirectional | Peak 25-30°, broad | V |
| LPDA | Wide forward beam | Wide | V or H depending on orientation |
| Small magnetic loop, vertical plane | Figure-8 in the loop plane | Mixed depending on height | mixed |
| Beverage, terminated | Cardioid in azimuth, broadside null | Low-angle peak | H (typical) |
The depth on each is in the per-antenna chapter; this is a quick-recognition table only.
7. Bandwidth and Q — why high-gain narrow-band antennas are a thing
An antenna’s bandwidth is the frequency range over which it remains operationally usable. The 2:1 SWR window (per Vol 3 §8.5) is the standard operational definition; outside this window, the SWR climbs into territory where transmitters fold back power and matching networks struggle.
7.1 Antenna Q
The figure of merit relating bandwidth and resonance behavior is the quality factor Q:
$$ Q = \frac{f_{\text{resonance}}}{\text{bandwidth at 2:1 SWR}} $$
Roughly: Q = 50 means 2:1 SWR bandwidth is f/50 (e.g. a 50 MHz antenna with Q = 50 has 1 MHz bandwidth). Q = 5 means 10 MHz bandwidth. Q = 500 means 0.1 MHz bandwidth.
7.2 The bandwidth-vs-size tradeoff
Antenna Q scales inversely with electrical size. A “small” antenna (one that’s much smaller than its operating wavelength) has high Q — narrow bandwidth, sharp resonance. A “large” antenna (multiple wavelengths in linear size) has low Q — wide bandwidth, lazy resonance.
Examples:
| Antenna | Approx. size | Q | 2:1 BW |
|---|---|---|---|
| Small magnetic loop, 1 m dia, 14 MHz | λ/15 perimeter | 200-1000 | 0.1-0.5% |
| Quarter-wave vertical, 14 MHz | λ/4 | 30-50 | 2-4% |
| Half-wave dipole, 14 MHz | λ/2 | 25-40 | 3-6% |
| Folded dipole, 14 MHz | λ/2 | 12-20 | 6-12% |
| Discone, 100-1000 MHz | spans λ/4 to 5λ | <2 | >100% |
| LPDA, 100-1000 MHz | spans 0.5λ to 5λ | <3 | >100% |
| Yagi, 5-el, 144 MHz | ~λ × 1 | 30-50 | 2-4% |
The unifying physics: a small antenna stores a lot of reactive energy relative to what it radiates per cycle. Stored reactive energy means high Q. High Q means narrow bandwidth.
7.3 The Chu-Harrington limit
There’s a hard theoretical floor on how small an antenna can be at a given Q. The Chu-Harrington limit (Lan Jen Chu, 1948; Harrington, 1960) gives:
$$ Q_{\min} = \frac{1 + 3(ka)^2}{(ka)^3 \cdot [1 + (ka)^2]} $$
where a is the radius of the smallest sphere enclosing the antenna and k = 2π/λ.
For an antenna with ka = 0.1 (the antenna fits in a sphere of radius λ/63), Q_min ≈ 800. The antenna cannot have lower Q regardless of cleverness — the small dimensions force the bandwidth.
This is why the “miracle small antenna” claims of advertising are physically impossible: you cannot escape Chu-Harrington with geometry tricks. You can only get close to the limit (the small magnetic loop, the meanderline antenna, the chip antenna in a Wi-Fi card are all near-Chu-limit designs at their respective frequencies).
7.4 Why high-gain narrow-band antennas exist
High-gain antennas trade bandwidth for directivity. A Yagi with 15 dBi gain is narrowband (typically 3-5% 2:1 SWR window) because:
- The driven element’s feedpoint impedance changes rapidly across frequency (the constant-impedance match holds only over a small band).
- The element-to-element coupling that produces the gain enhancement is frequency-dependent — at the design frequency, parasitics enhance forward radiation; at other frequencies, the same parasitics distort the pattern.
The OWA (Optimized Wideband Antenna) and LFA (Loop-Feed Antenna) Yagi designs sacrifice 0.5-1.5 dB of peak gain to widen the 2:1 SWR window from 3% to 8-12% — useful for hams covering an entire ham band on one antenna.
7.5 Practical implications for antenna selection
- For a wideband receiver (HackRF One, RTL-SDR scanning, EMI work): use a wideband antenna (discone, LPDA). Even at moderate gain, the wideband performance beats a “high gain” antenna across the receiver’s actual scan range.
- For a single-band station with TX: use a resonant antenna (dipole, vertical, Yagi). The narrower bandwidth maps onto the band cleanly.
- For multi-band TX: pick several resonant antennas (fan, trap, multi-band Yagi) or a tuner-fed long-doublet. Don’t try to use one wideband antenna across multiple bands — the SWR will be tolerable but radiation efficiency will be terrible off the design band.
8. Aperture, effective area, and the receive-side picture
Antennas on receive are described by a capture area — the equivalent geometric area the antenna presents to the incoming wave. For a perfectly-efficient antenna with gain G, the effective area is:
$$ A_e = \frac{G \cdot \lambda^2}{4\pi} $$
8.1 Effective area for canonical antennas
A reference table at canonical frequencies:
| Antenna | f | G (dBi) | λ | A_e (m²) |
|---|---|---|---|---|
| Isotropic | 14 MHz | 0 | 21.4 m | 36.5 |
| Isotropic | 144 MHz | 0 | 2.1 m | 0.35 |
| Isotropic | 2.4 GHz | 0 | 0.125 m | 0.0012 |
| Half-wave dipole | 14 MHz | 2.15 | 21.4 m | 59.8 |
| Half-wave dipole | 144 MHz | 2.15 | 2.1 m | 0.575 |
| Half-wave dipole | 2.4 GHz | 2.15 | 0.125 m | 0.00203 |
| 12 dBi Yagi | 144 MHz | 12 | 2.1 m | 5.59 |
| 15 dBi patch | 2.4 GHz | 15 | 0.125 m | 0.0395 |
| 60 cm dish | 2.4 GHz | ~20 | 0.125 m | 0.124 (per gain formula) |
| 60 cm dish | 10 GHz | ~32 | 0.030 m | 0.179 (per gain formula) |
The same dish at the same physical aperture but at higher frequency: same physical area, but the “effective” capture in λ-relative terms changes because λ shrinks. This is why microwave links use parabolic dishes — the physical aperture is what counts, and you can build big physical apertures at microwave that would be impossible at HF.
8.2 The implication for HF vs microwave
At HF (14 MHz), even a small isotropic antenna captures 36 m² of “incoming wave equivalent area” because λ is huge. The atmospheric noise field is large too; SNR is set by atmospheric noise, not by antenna capture area.
At 2.4 GHz, an isotropic captures 1.2 cm² — practically nothing. The atmospheric noise is small too, so the receiver’s noise floor (set by the LNA noise figure, Vol 19) sets the SNR.
Consequence: at HF, the limit on receive performance is atmospheric noise; antenna gain helps a little but external noise dominates. At microwave, the limit is the receiver itself; antenna gain (effectively, capture area) helps a lot.
8.3 Aperture efficiency for apertures larger than λ/2
For apertures larger than λ/2 — parabolic dishes, horns, large patch arrays — the directivity comes from the physical aperture rather than from a resonant geometry. The maximum directivity is:
$$ D_{\max} = \frac{4\pi A_p}{\lambda^2} $$
where A_p is the physical aperture area. Actual directivity is reduced by aperture efficiency η_ap, typically 0.4-0.7 for parabolic dishes (the rest is illumination spillover, edge taper, blockage):
$$ D = \eta_{ap} \cdot \frac{4\pi A_p}{\lambda^2} $$
For a 60 cm parabolic at 2.4 GHz with η_ap = 0.55:
$$ D = 0.55 \cdot \frac{4\pi (0.30)^2}{(0.125)^2} = 0.55 \cdot 72.4 = 39.8\ (16\text{ dBi}) $$
Larger dishes at the same frequency get more gain; the same dish at higher frequency gets more gain. Beyond ~50 dBi the apertures get tens of metres across (radio astronomy and satellite ground stations), out of the Hack Tools envelope.
9. Reciprocity — when “TX antenna” = “RX antenna” and when it doesn’t
The reciprocity theorem states that for any passive, linear antenna, the radiation pattern and feedpoint impedance are identical on transmit and receive. The antenna doesn’t “know” which direction the signal is flowing; the same field patterns, the same impedance, the same gain.
9.1 The implication for measurement
This is why antenna characterization is universally done in transmit mode: drive the antenna under test with a known source, measure the far-field pattern. The receive-pattern result follows from reciprocity, no separate measurement needed.
It’s also why NanoVNA-based antenna sweeps (transmit a signal into the antenna, measure the reflection) are valid: the SWR seen on transmit is the same SWR you’d see if you reversed the signal direction.
9.2 Where reciprocity appears to break down
In practical engineering, three cases look like reciprocity failures but are actually system-level effects, not antenna-level:
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Active receive antennas with built-in preamps. The antenna is passive and reciprocal, but the LNA is a unidirectional element. The “active antenna” (Wellbrook ALA1530, LZ1AQ, etc., Vol 19) has different effective gain on TX (the LNA doesn’t pass; you can’t TX through it) vs RX (LNA amplifies). The antenna part is reciprocal; the system isn’t.
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Lossy receive antennas — beverages, terminated loops, EWE antennas. These antennas are intentionally lossy. The beverage’s 470 Ω terminating resistor absorbs the back-lobe energy, giving directionality. As an antenna, the beverage radiates very inefficiently; as a receiver, its SNR is excellent because the directional pattern rejects local noise. The system breaks reciprocity in the sense that “good receive antenna” ≠ “good transmit antenna” for this geometry. Vol 15 is the full story.
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Electronically-steered phased arrays with time-modulated beamforming. The pattern is steered by switching elements on and off in time; the resulting “average” pattern differs between TX and RX modes because the switching mode and timing are usually different. Not relevant to the Hack Tools envelope but worth knowing about.
9.3 The reciprocity-of-noise question
A subtle point: an antenna with high gain has high antenna temperature if pointed at a noisy source (the Sun, a city, a noisy patch of sky). The receiver’s noise floor is set by:
$$ T_{\text{system}} = T_{\text{antenna}} + T_{\text{receiver}} $$
A good receive antenna isn’t always the highest-gain antenna; it’s the antenna whose pattern minimizes coupling to noise sources while still capturing the desired signal. This is the entire premise of receive-only beverage farms, K9AY loops, and active receive loops — the antenna sacrifices gain to gain SNR. Vol 15 is the detailed treatment.
9.4 The “TX antenna is also the RX antenna” practical conclusion
For most Hack Tools applications, the same antenna is used for both TX and RX. Reciprocity guarantees this works. The cases where you’d separate TX and RX antennas:
- High-power TX with sensitive RX nearby — separate antennas + T/R switching to protect the RX from leakage.
- Low-band DX work — TX with a vertical for low-angle radiation, RX with a beverage for direction-finding sensitivity.
- Wi-Fi pentest setups — TX from a Pineapple antenna while monitoring with a separate SDR + discone for spectrum awareness.
In all these cases, the antennas themselves are reciprocal; the design choice is at the system level (which antenna goes where in the signal chain).
10. Resources
- Balanis, Antenna Theory: Analysis and Design (4th ed., Wiley) — Ch. 2 (fundamental parameters), Ch. 4 (linear wire antennas), Ch. 11 (frequency-independent antennas) — the canonical academic reference
- Stutzman & Thiele, Antenna Theory and Design (3rd ed.) — more concise than Balanis, often clearer on practical questions
- Kraus, Antennas (3rd ed., McGraw-Hill) — historical and foundational; original derivation of dBi as a unit
- ARRL Antenna Book (25th ed.+) — Ch. 2 (fundamental parameters), Ch. 4 (impedance and matching), Ch. 5 (radiation patterns)
- Cebik, W4RNL — Basic Antenna Modeling: A Hands-On Tutorial — the canonical NEC modeling guide (archived at the antenneX library)
- IEEE Std 145 — Definitions of Terms for Antennas — the canonical unit and definition reference
- Pozar, Microwave Engineering (4th ed.), Ch. 6-7 — Smith chart and matching networks at depth
- Chu, “Physical Limitations of Omni-Directional Antennas,” J. Appl. Phys. 19, 1163 (1948) — the Chu-Harrington limit
- Harrington, “Effect of Antenna Size on Gain, Bandwidth, and Efficiency,” J. Res. NBS 64D, 1-12 (1960)
- Friis, “A Note on a Simple Transmission Formula,” Proc. IRE 34, 254-256 (1946)