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Decibel & S-Readings

Serge Stroobandt, ON4AA

Copyright 2015–2021, licensed under Creative Commons BY-NC-SA

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Base-10 logarithms: log10 x

\(\log_{10}{x}=\dots\quad\) means: “To what power do I need to raise 10, in order to obtain x?”

\[\log_{10}{x}=y\quad\Leftrightarrow\quad 10^y=x\]

dB as a power ratio

The decibel (dB) is a logarithmic unit used to express the ratio of two values of a physical quantity.1 For power ratios the decibel unit is defined as follows:

\[L_{dB}=10\cdot\log_{10}{\frac{P_{out}}{P_{in}}}\]

dB as a field amplitude ratio

For intensity ratios the decibel unit is defined as follows:

\[G_{dB}=20\cdot\log_{10}{\frac{A_{out}}{A_{in}}}\]

Decibel conversion table

Mnemonic decibel conversion table
dB \(\frac{P_{out}}{P_{in}}\) \(\frac{A_{out}}{A_{in}}\)
40 10000 100
30 1000 ≈31.62
20 100 10
10 10 ≈3.162
6 ≈ 4 ≈ 2
3 ≈ 2 \(\sqrt{2}\) ≈ 1.414
1 ≈ 1.25 ≈ 1.125
0 1 1
-1 ≈0.8 ≈0.9
-3 \(\frac{1}{2}\) = 0.5 \(\frac{1}{\sqrt{2}}\) ≈ 0.707
-6 \(\frac{1}{4}\) = 0.25 \(\frac{1}{2}\) = 0.5
-10 0.1 ≈0.3162
-20 0.01 0.1
-30 0.001 ≈0.03162
-40 0.0001 0.01

dBm as a power level

dBm is a logarithmic unit of power level, expressed in decibel (dB) and referenced to a power level of one milliwatt (mW).2

dBm as a power level
dBm \(P_{out}\) typical for
60 1kW typical radiated RF power of a microwave oven
50 100W typical maximum output RF power from a ham radio HF transceiver
40 10W
37 ≈ 5W typical maximum output RF power from a handheld ham radio VHF/UHF transceiver
33 ≈ 2W maximum output from a GSM 850/900 mobile phone
30 1W DCS or GSM 1 800/1 900 MHz mobile phone
20 100mW EIRP for a IEEE 802.11b/g 20 MHz-wide channel in the 2.4 GHz ISM band (5 mW/MHz)
10 10mW
0 1mW Bluetooth class 3 radio with 1 m range
-10 100µW IEEE 802.11 maximal signal strength
-60 1nW power received per m2 of a magnitude +3.5 star
-73 ≈ 50pW S9 signal strength on S-meter
-100 100fW IEEE 802.11b/g minimal signal strength
-101 ≈ 83fW noise floor of a IEEE 802.11b/g 20 MHz channel at 300 K
-134 ≈ 41aW noise floor of a 10 kHz wide FM signal at 300 K
-140 ≈ 12aW noise floor of a 2.7 kHz wide SSB signal at 300 K

In this table, the term noise floor refers to the calculated thermal noise, also known as the Johnson–Nyquist noise.3

HF S-meter

Many amateur radio and shortwave broadcast receivers feature a signal strength meter (S‑meter).4 In 1981, the International Amateur Radio Union (IARU) Region 1 agreed on a technical recommendation for S‑meter calibration of HF and VHF/UHF transceivers.5,6

IARU Region 1 Technical Recommendation R.1 defines S9 for the HF bands to be a receiver input power of -73 dBm. This is a level of 50 µV at the receiver’s antenna input assuming the input impedance of the receiver is 50 Ω.

The recommendation defines a difference of one S-unit corresponds to a difference of 6 dB, equivalent to a voltage ratio of two, or a power ratio of four. Signals stronger than S9 are given with an additional dB rating, thus “S9 + 20 dB”, or, verbally, “20 decibel over S9”, or simply “20 over 9” or even the simpler “20 over.”

Well-designed S-meter on the DRS WJ-8711A HF transceiver. Source: N9EWO

Conversion between power and HF S-units
S-reading \(P_{out}\) @50Ω \(V_{out}\) @50Ω \(\frac{V_{out}}{\left[1\,\text{µV}\right]}\) @50Ω
S9 + 40 dB -33 dBm 5.0 mV 74 dBµV
S9 + 30 dB -43 dBm 1.6 mV 64 dBµV
S9 + 20 dB -53 dBm 0.50 mV 54 dBµV
S9 + 10 dB -63 dBm 0.16 mV 44 dBµV
S9 -73 dBm 50 µV 34 dBµV
S8 -79 dBm 25 µV 28 dBµV
S7 -85 dBm 12.6 µV 22 dBµV
S6 -91 dBm 6.3 µV 16 dBµV
S5 -97 dBm 3.2 µV 10 dBµV
S4 -103 dBm 1.6 µV 4 dBµV
S3 -109 dBm 800 nV -2 dBµV
S2 -115 dBm 400 nV -8 dBµV
S1 -121 dBm 200 nV -14 dBµV

The noise floor for a \(B=2700\) Hz wide SSB signal at \(T=300\) K is:3

\(P=k_B\cdot T\cdot B=k_B\cdot300\cdot2700=11.8\cdot10^{-18}\,\text{W}=11.8\text{aW}=-139.5\,\text{dBm}\)

where \(k_B=1.3806488\cdot10^{-23}\,\text{J/K}\) is Boltzmann’s constant.

VHF/UHF S-meter

The same IARU Region 1 recommendation defines S9 for VHF/UHF to be a receiver input power of -93 dBm. This is the equivalent of 5 µV in 50 Ω. Again, one S-unit corresponds to a difference of 6 dB, equivalent to a voltage ratio of two, or a power ratio of four.

Conversion between power and VHF/UHF S-units
S-reading \(P_{out}\) @50Ω \(V_{out}\) @50Ω \(\frac{V_{out}}{\left[1\,\text{µV}\right]}\) @50Ω
S9 + 40 dB -53 dBm 0.50 mV 54 dBµV
S9 + 30 dB -63 dBm 0.16 mV 44 dBµV
S9 + 20 dB -73 dBm 50 µV 34 dBµV
S9 + 10 dB -83 dBm 16 µV 24 dBµV
S9 -93 dBm 5.0 µV 14 dBµV
S8 -99 dBm 2.5 µV 8 dBµV
S7 -105 dBm 1.26 µV 2 dBµV
S6 -111 dBm 630 nV -4 dBµV
S5 -117 dBm 320 nV -10 dBµV
S4 -123 dBm 160 nV -16 dBµV
S3 -129 dBm 80 nV -22 dBµV
S2 -135 dBm 40 nV -28 dBµV
S1 -141 dBm 20 nV -34 dBµV

The noise floor for a 10 kHz wide FM signal at 300 K is:3

\(P=k_B\cdot T\cdot B=k_B\cdot300\cdot10^{4}=41\cdot10^{-18}\,\text{W}=41\text{aW}=-134\,\text{dBm}\)

where \(k_B=1.3806488\cdot10^{-23}\,\text{J/K}\) is Boltzmann’s constant.

References

1.
2.
3.
Wikipedia. Johnson–Nyquist noise. https://en.wikipedia.org/wiki/Johnson–Nyquist_noise
4.
5.
IARU Region 1 Technical Recommendation R.1. International Amateur Radio Union Region I; 1981. http://hamwaves.com/decibel/doc/iaru.region.1.s-meter.pdf
6.
Ulrich Mueller, DK4VW. IARU Region 1 HF Manager Handbook v8.1. IARU; 1994. http://www.iaru-r1.org/index.php/downloads/Documents/HF/IARU-Region-1-HF-Manager-Handbook-V.8.1/
5
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