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MarkdownDecibel & S-Readings
Serge Stroobandt, ON4AA
Copyright 2015–2021, licensed under Creative Commons BY-NC-SA
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
| 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 | \(P_{out}\) | 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
| 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.
| 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
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