Antennas · Volume 3
dB and dBm
The decibel as a unit, dBm vs dBW vs dBμV, dBi vs dBd vs dBic, return loss vs SWR, link budgets, the +3/+10/-3/-10 shortcuts, and the traps that bite every RF engineer eventually
Contents
1. About this volume
The decibel is the universal currency of RF engineering. Every gain, every loss, every link budget, every noise figure, every spec on every datasheet is in dB or a dB-derived unit (dBm, dBW, dBi, dBd, dBμV, dBc, dBFS). A reader who understands dB fluently can read any RF document at speed; a reader who hesitates on dB lookups will keep getting tripped up.
This volume is the canonical dB reference for the entire series. Every later chapter quietly assumes the reader has the contents of this one cached. Cross-references land here when units need disambiguating (Vol 4 §5 on gain references; Vol 16 §12 on BALUN power-handling specs; Vol 26 §3 on cross-needle wattmeter readings; Vol 29 on per-radio link budgets).
The volume is deliberately long. The decibel is conceptually simple — it’s a log ratio — but its variants are many, the conversion traps are several, and the mental-math shortcuts pay enormous dividends when running a link budget in the field. Read it through once cold; come back to it as a reference any time a unit puzzle appears.
2. What a decibel is — and why we use one
A decibel is a logarithmic ratio, not a quantity:
$$ \text{dB} = 10 \log_{10}!\left( \frac{P_1}{P_2} \right) \quad \text{(for power ratios)} $$
$$ \text{dB} = 20 \log_{10}!\left( \frac{V_1}{V_2} \right) \quad \text{(for voltage / field-strength ratios into the same impedance)} $$
Two important consequences of being a ratio:
- dB by itself is dimensionless. A “10 dB amplifier” multiplies its input power by 10. “10 dB” doesn’t tell you what the absolute output level is — that requires a reference, which is what dBm/dBW/dBμV/dBd are.
- dB values add when ratios multiply. A chain of components multiplies the input by their cascaded gain ratio; their dB values sum. This is the entire reason RF engineers live in dB: a 1500 W amplifier driving 100 ft of LMR-400 into a Yagi at 1 km from a 0.5 μV sensitivity receiver spans 18 orders of magnitude. The dB form sums to a small integer; the linear form is unmanageable.
2.1 The historical “deci-Bel” and why it’s the size it is
The Bel (capital B), named for Alexander Graham Bell, was the original unit defined by Bell Labs in the 1920s for telephone-line attenuation work. One Bel = a factor of 10 in power. The decibel (1/10 of a Bel) was introduced almost immediately because 3 dB is the smallest power change the human ear reliably detects in audio work — the decibel is the natural granularity for engineering and for perceptual systems alike.
The convention has held for a century not because it’s irreplaceable but because it works: 0.1 dB is finer than any practical antenna match, 1 dB is the smallest difference you’d squabble over on a datasheet, 3 dB is the canonical “I noticed that,” 10 dB is “obviously different,” 20 dB is “categorically different.”
2.2 The base-10 thing
The logarithm in dB is base-10, not base-e or base-2. This means:
- A 10× power ratio = 10 dB (exactly)
- A 100× power ratio = 20 dB
- A 1000× = 30 dB
- A 1 000 000× = 60 dB
- A 1 000 000 000 000 000× (a quadrillion) = 150 dB
And the fractional powers of 10:
- A 2× power ratio = 10 log₁₀(2) ≈ 3.010 dB
- A 3× = 10 log₁₀(3) ≈ 4.771 dB
- A 5× = 10 log₁₀(5) ≈ 6.990 dB
- A 7× = 10 log₁₀(7) ≈ 8.451 dB
The 3 dB ≈ ×2 approximation is the workhorse shortcut. The 0.01 dB error in the approximation (rounding 3.010 to 3) is well below anything you’d measure on a real antenna; the approximation is functionally exact for engineering work.
3. Power dB vs voltage dB — the factor-of-two trap
The single most common dB mistake — across all of RF engineering, not just amateur — is confusing the 10·log form (for power) with the 20·log form (for voltage). Get this wrong and every dB value in your calculation is off by a factor of 2.
3.1 Where the factor of 2 comes from
For a constant impedance R, power and voltage are related by:
$$ P = \frac{V^2}{R} $$
Taking the log of both sides:
$$ \log_{10}(P) = 2 \log_{10}(V) - \log_{10}(R) $$
The −log₁₀(R) term is a constant (since R doesn’t change), and 10 log₁₀(P) = 20 log₁₀(V) plus a constant. When the impedance is fixed, the dB value of a power ratio equals the dB value of the corresponding voltage ratio:
$$ 10 \log_{10}(P_1/P_2) = 20 \log_{10}(V_1/V_2) \quad \text{(at constant R)} $$
This is the magic that lets engineers freely switch between power-dB and voltage-dB measurements — both yield the same dB number when the impedance is the same on both sides of the comparison.
3.2 Where the trap happens
The trap is when the comparison crosses an impedance change. Two cases:
Case 1 — Same impedance on both sides. If you measure a 50 Ω circuit with a 50 Ω probe, both sides of the ratio share R. The 10·log of the power ratio equals the 20·log of the voltage ratio. Either form is correct.
Case 2 — Different impedances. If you measure the voltage at a 50 Ω test port and compare to the voltage at a 75 Ω test port (or 100 Ω, or any other), the 20·log form gives the voltage ratio but does not give the power ratio. You must convert via P = V²/R:
$$ \frac{P_1}{P_2} = \frac{V_1^2 / R_1}{V_2^2 / R_2} = \frac{V_1^2}{V_2^2} \cdot \frac{R_2}{R_1} $$
In dB:
$$ 10 \log(P_1/P_2) = 20 \log(V_1/V_2) + 10 \log(R_2/R_1) $$
The 10 log(R_2/R_1) term is the cross-impedance correction.
For 50 Ω ↔ 75 Ω: 10 log(75/50) = 10 log(1.5) = 1.76 dB. A measurement that reads “+20 dB” in voltage between a 50 Ω source and a 75 Ω load is +18.24 dB in power.
3.3 The practical instances
The cross-impedance correction matters in three real-world settings:
- EMI receivers: most calibrated EMI receivers report in dBμV (a voltage reference into the receiver input impedance). When that receiver is fed from a 50 Ω LISN (Line Impedance Stabilization Network), the impedance match is good and dBμV converts cleanly to dBm. When fed via an unmatched probe (high-Z, capacitive), the conversion needs the cross-impedance term.
- Spectrum analyzers reading from non-50 Ω sources: rare in amateur work; common in EMC compliance.
- 75 Ω vs 50 Ω confusion in CATV / SAT installations: a CATV technician’s signal-level meter reads dBμV into 75 Ω; converting to dBm for an RF engineer on a 50 Ω system requires the cross-impedance term.
3.4 The “everything is 50 Ω” simplification
In the Hack Tools envelope (every radio, every test instrument, every coax connector is 50 Ω), the cross-impedance term vanishes. You can freely use 10·log for power ratios and 20·log for voltage ratios; both give the same dB number. Just be sure you’re consistent within a single calculation: pick one form and stay with it.
The dBμV-to-dBm conversion comes up enough to memorize, even in pure 50 Ω work: dBm = dBμV − 107 (for 50 Ω). The full derivation is in §5.1 below.
4. dBm and dBW — absolute power references
Adding a reference to “dB” turns the dimensionless ratio into a quantity. The two universal power references in RF work are dBm and dBW.
4.1 Definitions
dBm: decibels relative to 1 milliwatt. The defining equation is:
$$ P\text{(dBm)} = 10 \log_{10}!\left( \frac{P\text{(mW)}}{1\text{ mW}} \right) $$
So 0 dBm = 1 mW exactly; +10 dBm = 10 mW; +20 dBm = 100 mW; +30 dBm = 1 W; +40 dBm = 10 W; and so on.
dBW: decibels relative to 1 watt:
$$ P\text{(dBW)} = 10 \log_{10}!\left( \frac{P\text{(W)}}{1\text{ W}} \right) $$
0 dBW = 1 W; +10 dBW = 10 W; +30 dBW = 1 kW.
The conversion between the two is fixed at exactly 30 dB:
$$ \text{dBW} = \text{dBm} - 30 $$
4.2 The canonical anchor table
A reference table every RF engineer eventually memorizes:
| Power | dBm | dBW | Where you see it |
|---|---|---|---|
| 1 pW | −90 | −120 | Quiet receiver noise floor (narrowband) |
| 10 pW | −80 | −110 | Faint signal, decoded at threshold |
| 100 pW | −70 | −100 | Typical SDR-receivable signal level |
| 1 nW | −60 | −90 | Comfortable copy on most receivers |
| 100 nW | −40 | −70 | Strong urban signal |
| 1 μW | −30 | −60 | Quiet AM-broadcast in the field |
| 100 μW | −10 | −40 | Half a foot from a Flipper antenna |
| 1 mW | 0 | −30 | The dBm zero — calibration reference |
| 10 mW | +10 | −20 | Flipper Zero sub-GHz TX output |
| 100 mW | +20 | −10 | WiFi Pineapple 2.4 GHz TX |
| 200 mW | +23 | −7 | PWNagotchi Pi Zero W |
| 500 mW | +27 | −3 | Quansheng UV-K5 low-power |
| 1 W | +30 | 0 | Bigger handhelds; small QRP HF |
| 5 W | +37 | +7 | UV-K5 high-power; HT max typical |
| 100 W | +50 | +20 | HF amateur “barefoot” rig |
| 500 W | +57 | +27 | Solid-state HF amp |
| 1.5 kW | +62 | +32 | FCC Part 97 max amateur |
| 50 kW | +77 | +47 | AM broadcast main transmitter |
| 1 MW | +90 | +60 | FM-broadcast effective radiated power |
The pattern: each 10× of power = 10 dB. The whole scale spans 180 dB from quiet-receiver noise to broadcast ERP, which is exactly the dynamic range RF engineering covers in dB-form.
4.3 The Hack Tools radio TX power summary in dBm
Quick reference for every TX-capable radio in the hub:
| Radio | TX bands | Output power | dBm |
|---|---|---|---|
| HackRF One | 1 MHz – 6 GHz | varies by band, 0 to +15 dBm typical | 0 to +15 |
| PortaRF | 1 MHz – 6 GHz | same as HackRF One (uses Clifford Heath HackRF silicon) | 0 to +15 |
| Flipper Zero sub-GHz | 300-928 MHz | 10 mW typical (CC1101 limit) | +10 |
| Quansheng UV-K5 | 144 / 430 MHz | 5 W high, 1 W mid, 0.5 W low | +37 / +30 / +27 |
| WiFi Pineapple Mark VII | 2.4 / 5 GHz | up to 100 mW per port | +20 |
| ESP32 (Marauder, Banshee, etc) | 2.4 GHz | up to 100 mW (ESP32 max) | +20 |
| PWNagotchi (Pi Zero W) | 2.4 GHz | ~64 mW typical | +18 |
| DSTIKE Hackheld | 2.4 GHz | up to 25 dBm (with onboard PA) | +25 |
| Bash Bunny / Rubber Ducky / Hak5 USB | (not radio TX) | n/a | n/a |
| Rayhunter | (RX only — cellular detection) | n/a | n/a |
| RTL-SDR V4 | (RX only) | n/a | n/a |
The range across active TX radios is 27 dB — from +10 dBm (Flipper) to +37 dBm (UV-K5 high). That’s a factor of 500 in raw power. The corresponding antenna gain envelope (per Vol 11 §4 for Yagi gains) is roughly +0 to +15 dBi. The link budget conversation in Vol 29 is essentially “how do these two combine.”
4.4 dBuV and the cross-impedance gotcha (preview)
The voltage-reference variant dBμV (decibels relative to 1 microvolt) appears in EMI receivers, in some signal-level meters, and in many high-end spectrum analyzers as an alternate display unit. Conversion to dBm depends on the impedance — full treatment in §5.
5. dBμV, dBμV/m, dBc — the niche references that show up in scopes, EMC, and harmonics
Beyond dBm and dBW, four secondary references appear in instruments and specifications often enough to memorize.
5.1 dBμV — voltage at a reference impedance
dBμV is the voltage equivalent of dBm:
$$ V\text{(dBμV)} = 20 \log_{10}!\left( \frac{V}{1\text{ μV}} \right) $$
0 dBμV = 1 μV; +60 dBμV = 1 mV; +120 dBμV = 1 V.
To convert dBμV to dBm at a known impedance R:
$$ P = \frac{V^2}{R} $$
For 50 Ω:
$$ P\text{(dBm)} = V\text{(dBμV)} - 107 $$
Derivation: 1 μV across 50 Ω = (1×10⁻⁶)² / 50 = 2×10⁻¹⁴ W = 20 pW = −107 dBm. So 0 dBμV at 50 Ω = −107 dBm; the formula follows.
For 75 Ω (CATV / SAT):
$$ P\text{(dBm)} = V\text{(dBμV)} - 108.75 $$
The 1.76 dB difference between 50 and 75 Ω is exactly the cross-impedance term from §3.2.
5.2 dBμV/m — field strength
dBμV/m measures the electric-field strength at a point in space (not the power into a load):
$$ E\text{(dBμV/m)} = 20 \log_{10}!\left( \frac{E}{1\text{ μV/m}} \right) $$
Field-strength meters and EMI receivers with calibrated antennas report in dBμV/m. The conversion to power-density (W/m²) involves the free-space impedance (377 Ω); the conversion to received power at a specific receive antenna involves that antenna’s effective area (cross-link to Vol 4 §8).
Where you see dBμV/m:
- FCC Part 15 emission limits are specified as field strength at a fixed distance (e.g. “200 μV/m at 3 m for the fundamental, harmonics no more than X dB below”).
- OET-65 MPE compliance (Vol 31 §8) specifies maximum permissible exposure in mW/cm², which converts to dBμV/m via the 377 Ω free-space impedance.
5.3 dBc — decibels relative to the carrier
dBc reports the level of a spurious signal relative to the level of the carrier signal it accompanies. The carrier itself is by definition 0 dBc; a harmonic 40 dB below the carrier is −40 dBc; a phase-noise sideband 80 dB below the carrier is −80 dBc.
dBc is the universal way to report:
- Harmonic emissions from transmitters (FCC Part 97.307 requires harmonics −43 dBc below the fundamental for HF amateur, −60 dBc above 30 MHz).
- Phase noise of oscillators (e.g. “−110 dBc/Hz at 10 kHz offset”).
- Spurious emissions from any signal source.
- IMD products in amplifier specifications (e.g. “third-order IMD products −30 dBc at full rated power”).
The advantage of dBc over absolute dBm: a 100 W transmitter and a 1 W transmitter both have the same “−43 dBc” requirement, and the absolute spur level scales appropriately (the 100 W transmitter’s spur is at +57 − 43 = +14 dBm; the 1 W transmitter’s is at +30 − 43 = −13 dBm). The relative level captures what regulators actually care about.
5.4 dBFS — decibels relative to full scale (digital)
dBFS appears in digital RF instruments — SDRs, digital oscilloscopes, audio interfaces. 0 dBFS = the largest signal the ADC can encode without clipping; everything else is negative.
For HackRF / RTL-SDR / BladeRF work in GNU Radio: the IQ stream is in dBFS. A signal at −20 dBFS is 100× lower in power than a clipping signal. The conversion to dBm requires knowing the ADC’s full-scale calibration (rarely documented for amateur SDRs; well-documented for lab-grade SDRs like USRPs).
5.5 dBsm — decibels relative to one square metre
dBsm is the unit for radar cross-section (RCS). 0 dBsm = a 1 m² perfectly-reflecting target; a typical fighter aircraft is ~0 dBsm; a stealthed aircraft is around −20 dBsm; an insect is −40 dBsm; the moon is roughly +60 dBsm.
dBsm shows up in radar systems work and occasionally in passive-radar amateur experiments (cross-link to Vol 27 §10 for the spectrum-analyzer-as-radar variant). Out of scope for primary antenna work but worth recognising.
6. Gain references — dBi vs dBd vs dBic vs dBq
Antenna gain is reported in dB but always relative to some reference. The three references in common use, and one obsolete:
6.1 dBi — gain relative to isotropic
The reference is a theoretical isotropic radiator — a point source radiating equally in all directions. This is a mathematical fiction (no physical antenna radiates isotropically) but it’s the cleanest mathematical reference and the universal default in modeling software.
A half-wave dipole in free space has 2.15 dBi gain — this is the “extra” radiation in its peak direction (broadside) versus a point that radiates uniformly to all directions. A perfect 100% efficient isotropic radiator has 0 dBi gain by definition.
NEC modeling output (Vol 28) is always in dBi.
6.2 dBd — gain relative to a half-wave dipole
The reference is a half-wave dipole in free space. dBd = dBi − 2.15.
A “10 dBd” Yagi has the same actual gain as a “12.15 dBi” Yagi; the numbers describe the same antenna with different references.
Why dBd exists: a half-wave dipole is the easiest gain antenna to measure against in practice (you can build one and put it on a turntable). Vendors used dBd in the 1950s-1970s when range measurements against a physical dipole were the only practical way to characterize.
Today: dBi is more common, but dBd persists in older vendor specs and in some amateur radio publications. The 2.15 dB difference matters: a “9 dBd” antenna sounds like a “9 dBi” antenna in casual conversation but is 11.15 dBi — a factor of 1.6 in power.
6.3 dBic — gain relative to isotropic of the same polarization
The reference is a theoretical isotropic radiator of the same polarization as the antenna under test — typically circular for satellite antennas.
dBic shows up in GPS, satellite, and circular-polarization specifications. A GPS antenna with “5 dBic” gain has 5 dB peak gain in the right-hand circularly-polarized component versus an isotropic RHCP point source.
For linearly-polarized antennas, dBic equals dBi (the polarization specification is implicit). For circularly-polarized antennas, dBic ≠ dBi: a circular antenna with 3 dBic gain has 3+3 = 6 dBi gain if you’re measuring the linear field component of one axis, but only 3 dBic gain if you correctly use a circularly-polarized reference. Most circular antennas spec in dBic to avoid this 3 dB ambiguity.
6.4 dBq — historic, obsolete
dBq referenced gain to a quarter-wave monopole over a perfect ground plane. dBq = dBi − 5.15 (since a quarter-wave monopole over ideal ground has 5.15 dBi gain).
Used in some 1960s-1970s land-mobile literature (Larsen, Antenex, GE Mobile Radio) for vehicle-mounted whip antennas. Nearly extinct today; if you see it on an old datasheet, convert and forget.
6.5 The vendor-marketing gotcha
Antenna gain is the single most-inflated number in vendor literature. Two specific tricks:
- Quote in dBi without saying so — a “9 dB gain” antenna is implicitly dBi (the bigger number); the dBd value is 2.15 dB smaller.
- Quote peak directivity instead of gain — directivity is the geometric concentration; gain is directivity times efficiency. A short loaded mobile whip might have 6 dBi directivity and 70% loss in the loading coil, making the actual gain −2.4 dBi. Vendor specs quote 6 dBi.
The way to defend against both: ask for the antenna’s measured pattern on a calibrated range. If the vendor can’t produce a polar plot and a calibrated gain measurement, treat the number as marketing.
6.6 The 50 Ω caveat for dB-referenced gain
Antenna gain in dBi/dBd is independent of impedance — it’s a pure radiation-pattern quantity. The impedance question (is the antenna a good match to 50 Ω coax) is the return-loss/SWR question (§8 below), which is separate from gain. A perfectly-matched 0 dBi antenna and a high-SWR 0 dBi antenna both produce the same field at the same distance; the latter just wastes some incident power into reflection.
The conflation of “gain” and “return-loss/match quality” is the second most common dB confusion after the 10·log/20·log trap. The two are independent.
7. Mental-math shortcuts — +3, +10, −3, −10, and the dB-to-ratio cheat-table
Running a link budget on paper or in your head requires fluent dB-to-ratio conversion. Three core shortcuts handle 95% of the work.
7.1 The +3 / +10 / −3 / −10 rules
| dB change | Power ratio | Voltage ratio |
|---|---|---|
| +10 | ×10 | ×3.16 |
| +3 | ×2 | ×1.41 |
| 0 | ×1 | ×1 |
| −3 | ÷2 | ÷1.41 |
| −10 | ÷10 | ÷3.16 |
These four numbers compose freely:
- +6 dB = +3 +3 = ×4
- +13 dB = +10 +3 = ×20
- +20 dB = +10 +10 = ×100
- +23 dB = +10 +10 +3 = ×200
- +30 dB = +10 +10 +10 = ×1000
- +43 dB = +10 +10 +10 +10 +3 = ×20000
The 3 dB ≈ ×2 approximation has 0.27% error (3.010 dB is exact; 3 is approximate). Cumulative error over a long calculation: 10 dB worth of “+3 +3 +3” steps = +9 dB exactly, +9.030 dB precise — 0.030 dB error, well below anything measurable on a real antenna.
7.2 The 1/2/3/5/7/10 cheat table — every integer dB
When the +3/+10 shortcuts aren’t sufficient (you have a 7 dB amp and need the exact power ratio), this table maps every integer dB from 0 to 10 to a memorable ratio:
| dB | Power ratio | Voltage ratio | Memorable as |
|---|---|---|---|
| 0 | 1.000 | 1.000 | unity |
| 1 | 1.259 | 1.122 | ~5/4 power |
| 2 | 1.585 | 1.259 | ~8/5 power |
| 3 | 1.995 | 1.413 | ”×2” |
| 4 | 2.512 | 1.585 | ~5/2 power |
| 5 | 3.162 | 1.778 | ”×π” |
| 6 | 3.981 | 1.995 | ”×4” |
| 7 | 5.012 | 2.239 | ”×5” |
| 8 | 6.310 | 2.512 | ”×6.3” |
| 9 | 7.943 | 2.818 | ~×8 |
| 10 | 10.000 | 3.162 | ”×10” |
The “5 dB = ×π” approximation is exact to 0.6% and pleasant to remember (1 dB = 1.26 ≈ ⁵√10; 5 dB = ⁵√10⁵ = √10 ≈ π).
7.3 Octaves vs decades
Some specifications report ratios in octaves (factor of 2) or decades (factor of 10). The conversions:
- 1 octave = 3.010 dB (power) = 6.020 dB (voltage)
- 1 decade = 10.000 dB (power) = 20.000 dB (voltage)
These show up in filter specifications (“Bessel filter, −20 dB/decade rolloff above the corner”) and in oscillator phase-noise plots.
7.4 The natural log version that almost no one uses
In some academic literature you’ll see nepers (Np) instead of dB:
$$ \text{Np} = \ln!\left( \frac{V_1}{V_2} \right) $$
1 Np ≈ 8.686 dB. Outside transmission-line theory papers, you almost never see this in practice; mention it because it’ll occasionally surface in 1950s telephone-line literature and in some Russian / German RF papers.
8. Return loss, mismatch loss, and SWR — three views of the same problem
When an antenna doesn’t perfectly match the 50 Ω characteristic impedance of the feedline, some signal reflects back from the antenna toward the transmitter. The fraction that reflects is the reflection coefficient Γ (Greek capital gamma):
$$ \Gamma = \frac{Z_{\text{load}} - Z_0}{Z_{\text{load}} + Z_0} $$
A perfect match (Z_load = Z₀ = 50 Ω) gives Γ = 0. An open circuit (Z_load = ∞) gives Γ = +1. A short circuit (Z_load = 0) gives Γ = −1. Anything in between gives Γ somewhere on the complex plane with magnitude between 0 and 1.
This single quantity Γ generates three different specifications of “how bad is the match” — each commonly used in different communities, and each easily confused with the others.
8.1 Return loss
Return loss is the power ratio between forward and reflected wave, expressed in dB:
$$ RL\text{(dB)} = -20 \log_{10}(|\Gamma|) $$
Return loss is positive for a real load (it’s a loss). A perfect match has infinite return loss (no reflection); a complete mismatch has 0 dB return loss (all power reflects).
Note the convention: some engineering communities write return loss as a negative number (S11 in dB is the magnitude of the reflection coefficient in dB, which is negative for any real reflection — e.g. “S11 = −20 dB”). VNAs typically report S11 in dB as a negative number; antenna analyzers typically report return loss as a positive number. They mean the same thing: |−20 dB S11| = 20 dB return loss.
8.2 Mismatch loss
Mismatch loss is the power actually lost due to mismatch — the fraction of incident power that doesn’t make it into the load:
$$ ML\text{(dB)} = -10 \log_{10}(1 - |\Gamma|^2) $$
Mismatch loss is the link-budget-relevant number. Return loss describes the reflection ratio; mismatch loss describes what’s lost to reflection.
8.3 SWR — the operator-readable form
Standing-wave ratio (SWR, sometimes called VSWR for “voltage standing-wave ratio”) is the ratio of maximum to minimum voltage standing on the feedline:
$$ SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$
A perfect match has SWR = 1.0 (no standing wave). A complete mismatch has SWR = ∞.
SWR is the number that shows up on every cross-needle meter, every tuner display, every transceiver SWR indicator. The whole amateur ecosystem is calibrated to read in SWR.
8.4 The canonical conversion table
The three quantities convert via Γ, so they convert directly to each other:
| SWR | |Γ| | RL (dB) | ML (dB) | Reflected power |
|---|---|---|---|---|
| 1.00 | 0.000 | ∞ | 0.00 | 0.00% |
| 1.05 | 0.024 | 32.3 | 0.003 | 0.06% |
| 1.10 | 0.048 | 26.4 | 0.010 | 0.23% |
| 1.15 | 0.070 | 23.1 | 0.022 | 0.49% |
| 1.20 | 0.091 | 20.8 | 0.036 | 0.83% |
| 1.25 | 0.111 | 19.1 | 0.054 | 1.23% |
| 1.30 | 0.130 | 17.7 | 0.075 | 1.70% |
| 1.40 | 0.167 | 15.6 | 0.122 | 2.78% |
| 1.50 | 0.200 | 14.0 | 0.177 | 4.00% |
| 1.60 | 0.231 | 12.7 | 0.238 | 5.33% |
| 1.70 | 0.259 | 11.7 | 0.302 | 6.72% |
| 1.80 | 0.286 | 10.9 | 0.370 | 8.16% |
| 2.00 | 0.333 | 9.5 | 0.512 | 11.11% |
| 2.25 | 0.385 | 8.3 | 0.696 | 14.81% |
| 2.50 | 0.429 | 7.4 | 0.881 | 18.37% |
| 2.75 | 0.467 | 6.6 | 1.062 | 21.78% |
| 3.00 | 0.500 | 6.0 | 1.249 | 25.00% |
| 3.50 | 0.556 | 5.1 | 1.602 | 30.86% |
| 4.00 | 0.600 | 4.4 | 1.938 | 36.00% |
| 5.00 | 0.667 | 3.5 | 2.554 | 44.44% |
| 7.00 | 0.750 | 2.5 | 3.590 | 56.25% |
| 10.0 | 0.818 | 1.7 | 4.812 | 66.94% |
| ∞ | 1.000 | 0.0 | ∞ | 100.0% |
8.5 The point of the 2:1 SWR convention
The “2:1 SWR is the band edge” convention has two roots:
- 0.5 dB mismatch loss is negligible. A 2:1 SWR loses 0.5 dB to reflection — less than the loss of a single coax connector and far less than what feedline loss eats in any nontrivial run. The signal still gets to the antenna; the operational impact is invisible.
- Modern transmitters fold back power above 2:1. The reflected power, on its way back to the transmitter, looks like reduced load impedance to the PA stage. Solid-state PAs are intolerant of off-impedance loading (they were designed for 50 Ω); the protection circuit reduces drive until SWR is back inside tolerance. Above 2:1 the rig delivers reduced power; above 3:1 most rigs deliver 0 W.
Some operators chase 1.1:1 SWR aesthetically. The mismatch-loss penalty between 1.1:1 and 1.5:1 is 0.17 dB — well below anything you’d measure on a real antenna in field conditions. Polish to 1.1 if you enjoy polish; for radiated effectiveness, 1.5:1 is as good as 1.1:1.
8.6 Why “antenna SWR” and “system SWR” can disagree
A common scenario: an operator measures 1.2:1 SWR at the antenna feedpoint (with NanoVNA on a portable battery, Vol 24) but reads 2.5:1 at the rig end of the coax. What happened?
Two possible answers:
- Lossy coax masks high antenna SWR. Coax loss is a one-way attenuation; reflected power goes back through the coax and is attenuated again. A 3 dB lossy run between rig and antenna shows half the actual mismatch at the rig end. A 6 dB lossy run shows a quarter. This makes the rig “happy” but does not improve the antenna — and it costs you the round-trip loss in actual radiated power.
- Coax common-mode currents add length-dependent SWR. If the antenna feed isn’t properly choked (Vol 16 §3), the coax shield becomes part of the antenna, and the SWR seen at the rig depends on where in the standing-wave pattern the rig sits. Moving the rig changes the SWR.
The cure for both is to measure with the NanoVNA at the antenna feedpoint, not at the rig end. The antenna’s SWR is the antenna’s SWR; what the rig sees is the antenna’s SWR modified by the feedline.
9. Link budgets — worked examples
Composing the pieces of this volume into actual link calculations. Each example uses the form:
$$ P_{rx}\text{(dBm)} = P_{tx}\text{(dBm)} + G_{tx}\text{(dBi)} - L_{tx_feed}\text{(dB)} - L_{fs}\text{(dB)} + G_{rx}\text{(dBi)} - L_{rx_feed}\text{(dB)} - L_{pol}\text{(dB)} - L_{mismatch}\text{(dB)} $$
FSPL values come from Vol 2 §7; feedline loss numbers from Vol 5 §5; antenna gains from the per-antenna chapters (Vols 6-15).
9.1 Flipper Zero to discone, 100 m
A Flipper Zero transmitting at 433 MHz, +10 dBm output, into a HackRF One on a discone at 100 m. Both are essentially 0 dBi (Flipper PCB whip is approximate; discone is omnidirectional). Coax to the HackRF: 3 m of LMR-240 at 433 MHz, ~0.3 dB.
$$ P_{rx} = +10 + 0 - 0 - 65.2 + 0 - 0.3 - 0 - 0.5 = -56.0\text{ dBm} $$
Where the −0.5 dB is the canonical SWR-mismatch budget for a hand-tuned dipole.
HackRF noise floor at 50 kHz bandwidth (Flipper’s typical signal bandwidth): roughly −85 dBm. Link margin: 29 dB. Solid copy.
9.2 HackRF + 11 dBi Yagi to RTL-SDR + discone, 10 km
A HackRF One at +15 dBm (146 MHz, full output) into a 5-element 2 m Yagi at 11 dBi. 5 m of LMR-400 feedline (0.11 dB at 146 MHz). RTL-SDR V4 on a discone (0 dBi) at 10 km with 3 m of LMR-400 (0.07 dB). FSPL at 10 km, 146 MHz = 95.8 dB. Polarization match assumed (both horizontal); −0.5 dB SWR.
$$ P_{rx} = +15 + 11 - 0.11 - 95.8 + 0 - 0.07 - 0 - 0.5 = -70.48\text{ dBm} $$
RTL-SDR V4 noise floor at narrow bandwidth: roughly −115 dBm. Link margin: 44 dB. Strong copy. The Yagi is doing the heavy lifting; without its 11 dB, the link is −81 dBm — still copyable, but with thinner margin.
9.3 WiFi Pineapple to client at 1 km
A Pineapple Mark VII + AC at +20 dBm at 2440 MHz into a 15 dBi panel antenna. 1 m of LMR-240 (0.55 dB at 2.4 GHz). Stock Wi-Fi laptop at 1 km, internal antenna ~0 dBi. FSPL at 1 km, 2.4 GHz = 100.2 dB. Polarization match: assume vertical panel into vertical laptop antenna = −0.5 dB margin.
$$ P_{rx} = +20 + 15 - 0.55 - 100.2 + 0 - 0 - 0.5 - 0.5 = -66.75\text{ dBm} $$
Laptop Wi-Fi sensitivity at 1 Mbps is ~−95 dBm; at full 802.11n MCS 7 (65 Mbps), the requirement climbs to ~−70 dBm. The link works at low data rates with margin; at full rate it’s marginal. Cross-link to Vol 29 §7 for the Pineapple-specific antenna matrix and the “panel-vs-Yagi for sector audit” trade.
9.4 The link-budget format you’ll write a hundred times
In practice, write link budgets vertically — one row per element, with running totals:
Pineapple TX output +20.0 dBm
Pineapple feedline (1m LMR-240) −0.55 dB
Pineapple panel antenna +15.0 dBi
─────
EIRP +34.45 dBm
─────
FSPL @ 1km @ 2.44 GHz −100.2 dB
Polarization mismatch −0.5 dB
Antenna SWR mismatch (TX side) −0.5 dB
─────
Signal at receiver antenna −66.75 dBm
Client antenna gain +0.0 dBi
─────
Signal into client receiver −66.75 dBm
vs sensitivity at MCS7 −70 dBm
─────
Margin at full rate 3.25 dBThe vertical form makes it instantly visible what’s eating the margin. In this example: −100.2 dB FSPL is the dominant term, and the only places where engineering can recover margin are antenna gain (already at +15 dBi panel; bigger means more sectored beam), TX power (capped by Part 15 / FCC certification), or reduced data rate (which lowers the sensitivity threshold).
The vertical format is the standard for any serious link planning — RF link planners (Pathloss, RadioMobile, Cellular Mapper), satellite link budgets, microwave repeater planning all produce this format.
10. The most common dB mistakes — a triage list
A field reference for diagnosing dB-related confusion in others’ work — or your own. Each error has a characteristic symptom and a 30-second test to identify it.
10.1 Confusing dBm with dBW (30 dB error)
Symptom: numbers either too large by 30 dB (saying you have +50 dBm when you meant +50 dBW = 1500 kW), or too small by 30 dB. Numbers are absurd on inspection.
Test: stop and ask “is this in mW or W?” If a “+50 dBm amplifier” outputs 100 W, fine; if it outputs 100 kW, you mean dBW.
10.2 Confusing dBi with dBd (2.15 dB error)
Symptom: antenna gain numbers differ from vendor spec by 2.15 dB. Compounded across multiple gain numbers in a link budget = several dB error.
Test: check the vendor’s spec sheet for the reference. If it’s not stated, default to dBi (it’s the higher number, hence the marketing default). Convert all to one reference before the link-budget arithmetic.
10.3 Using 10·log when measuring voltage (3 dB per stage error)
Symptom: dB values from voltage measurements are half what they should be. A “10 dB gain stage” measured by voltage ratio reads 20 dB.
Test: every dB value either uses 10·log (power) or 20·log (voltage). If you’re working with voltage measurements and the dB values look small, suspect this.
10.4 Forgetting impedance when converting dBμV to dBm (1.76 dB error)
Symptom: signal level calculations between 75 Ω and 50 Ω systems are off by 1.76 dB.
Test: if the calculation crosses a 50 ↔ 75 Ω boundary (CATV ↔ ham, EMI receiver ↔ spectrum analyzer), include the cross-impedance term 10 log(75/50) = 1.76 dB.
10.5 Adding return loss when you meant mismatch loss (large error at high SWR)
Symptom: a “2:1 SWR antenna” allegedly loses 9.5 dB of power. The actual mismatch loss is 0.5 dB.
Test: if a dB number represents how much power doesn’t get to the load, it’s mismatch loss, not return loss. Return loss is the ratio of reflected to forward; mismatch loss is what’s actually dissipated.
10.6 Treating dBi as additive without subtracting feed mismatch and polarization (several dB error)
Symptom: a link budget overshoots reality by 3-10 dB. The antenna is “10 dBi” but the feed has 2:1 SWR (−0.5 dB) and the polarization is mismatched (3 dB), so the effective gain into the link is more like 6.5 dB.
Test: each link-budget term must be honest. Use the full formula from §9.4 — TX-side gain minus TX-feedline loss minus FSPL plus RX-side gain minus RX-feedline loss minus polarization mismatch minus SWR mismatch.
10.7 Negative versus positive convention on return loss / S11
Symptom: a NanoVNA reports “S11 = −20 dB” and an antenna analyzer reports “return loss = 20 dB”. The same antenna; different signs.
Test: |S11 in dB| = return loss in dB. They mean the same magnitude of reflection; the sign is a convention.
10.8 The “everything is fine, but the link doesn’t work” diagnostic
When a link budget says you should have 20 dB margin but the link is failing, suspect (in order of probability):
- Antenna polarization mismatch — 20 dB swings happen here.
- Pattern misalignment — a 15 dBi Yagi aimed 30° off the path is a 6 dB Yagi.
- Multipath / multipath fading — instantaneous nulls of 20-40 dB in the field, especially at microwave.
- Common-mode currents distorting the radiation pattern — pattern looks fine on a tower but actually radiates 90° off-axis from where the model says.
- Loss in connectors and adapters — three SMA-N adapters in series at 5 GHz is easily 2 dB.
Cross-link to the relevant volume for each: Vol 4 §6 for pattern; Vol 5 §10 for common-mode; Vol 16 for choke balun deployment; Vol 22 for connector failures.
11. Resources
- ARRL Antenna Book Ch. 1-2 (units, fundamentals)
- IEEE Std 145 — Definitions of Terms for Antennas — the canonical unit reference
- Pozar, Microwave Engineering (4th ed.), Ch. 1-2 (transmission lines, network parameters)
- Sevick, Building and Using Baluns and Ununs — the cross-impedance derivations
- ITU-R Recommendation P.525 — free-space attenuation
- FCC Office of Engineering and Technology Bulletin 65 — RF exposure / dB-to-MPE conversions
- Everything RF calculator collection: https://www.everythingrf.com/rf-calculators
- RF Cafe dB/dBm calculator: https://www.rfcafe.com/references/calculators/db-calculator.htm
- Mini-Circuits app notes (AN-10-005, AN-40-018, AN-95-010) — dB in the practical RF-design context