Decibels explained: 10 log or 20 log, dBm, dBu and the rules of thumb
The decibel is a way of writing a ratio as a logarithm, so that the huge ranges in electronics (a microvolt at an aerial, tens of volts at a speaker) become small numbers, and so that the gains of stages in a chain can be added instead of multiplied. dB = 10 log10(P2/P1) for power, 20 log10(V2/V1) for voltage or current. Add a reference (1 mW, 1 W, 1 V, 0.775 V) and the ratio becomes a level: dBm, dBW, dBV, dBu. This calculator converts every way, works out the output of a stage from its input, and sums a chain.
How to use the decibel calculator
- Type a gain and say how it is given: decibels (negative for a loss), a power ratio, or a voltage/current ratio. The other two forms appear at once; switching the selector converts the value you typed.
- Add more stages as a comma-separated list of dB figures to sum a chain.
- Give an input level in dBm, dBW, mW, W, V, dBV or dBu, and the impedance that links power to voltage (50 Ω RF, 600 Ω audio lines, 8 Ω speakers). The output level appears in every unit.
- The drawing shows the input, the amplifier with its gain and the output, the two ratios, and a dB ruler. Click any of them for its own notes; the Reference table tab lists the common figures.
Why 10 log for power and 20 log for voltage
dB = 10 log10(P2/P1) = 20 log10(V2/V1)
The bel was defined on power: one bel is ten times the power, and a decibel a tenth of that. Power in a resistance is V²/R, so when the voltage doubles the power quadruples; log(V²) = 2 log(V), and the 2 turns the 10 into a 20. The two formulas give the same decibels for the same change as long as the impedance is the same at both ends. If it is not (an amplifier with a 1 MΩ input driving 8 Ω), the voltage gain in dB and the power gain in dB are different numbers, and you must say which you mean.
Rules of thumb
| dB | Power ratio | Voltage ratio | Remember as |
|---|---|---|---|
| 1 dB | 1.259× | 1.122× | about 26 % more power, 12 % more voltage |
| 3 dB | 1.995× | 1.413× | double the power |
| 6 dB | 3.981× | 1.995× | double the voltage |
| 10 dB | 10× | 3.162× | ten times the power |
| 20 dB | 100× | 10× | ten times the voltage |
| 30 dB | 1,000× | 31.623× | a thousand times the power |
| −3 dB | 0.5012× | 0.7079× | half the power: the cut-off point |
| −20 dB | 0.01× | 0.1× | a tenth of the voltage |
Everything else is built from these: 13 dB is 10 + 3, twenty times the power; 26 dB is 20 + 6, twenty times the voltage; 43 dB is 40 + 3, 20 000× the power.
Levels: dBm, dBW, dBV and dBu
A ratio against a fixed reference is a level. dBm is power against 1 mW: 0 dBm = 1 mW, 30 dBm = 1 W, −30 dBm = 1 µW; radio people live in dBm. dBW is against 1 W, always 30 less. dBV is voltage against 1 V RMS. dBu is voltage against 0.7746 V, the voltage that 1 mW makes across 600 Ω, the old telephone line impedance; professional audio's +4 dBu line level is 1.23 V RMS, consumer −10 dBV is 316 mV. To go between a power level and a voltage you need the impedance: 0 dBm is 224 mV in 50 Ω but 775 mV in 600 Ω.
Gain chains
Ratios multiply along a chain; their logarithms add. A receiver front end with a 20 dB low-noise amplifier, a 6 dB filter loss and a 30 dB IF amplifier has 20 − 6 + 30 = 44 dB of gain, and a −100 dBm signal at the aerial comes out at −56 dBm. The same with ratios would be 100 × 0.25 × 1000 = 25 000, which is why nobody does it that way. The "More stages" field adds any list of figures to the main gain.
Your values, step by step
- Power ratio: 1020/10 = 100×.
- Voltage ratio: 1020/20 = 10×.
- Input: 0 dBm = 1 mW = 224 mV RMS in 50 Ω.
- Output: 0 dBm + 20 dB = 20 dBm = 100 mW = 2.24 V RMS.
Worked example: a 20 dB amplifier with 0 dBm in
20 dB is 1020/10 = 100× the power and 1020/20 = 10× the voltage. An input of 0 dBm is 1 mW, which across 50 Ω is √(0.001 × 50) = 223.6 mV RMS. The output is 0 + 20 = 20 dBm = 100 mW = 2.236 V RMS, which is 6.99 dBV or 9.21 dBu. Put a −3 dB cable after it and the level is 17 dBm, 50.1 mW: the power halves, the voltage falls to 1.58 V.
Questions
Is 3 dB exactly double?
No: 100.3 = 1.995×. Exactly double is 3.01 dB. For engineering the difference never matters.
Can decibels be negative?
Yes: a ratio below one gives a negative figure, a loss. −3 dB is half the power; −20 dB is a tenth of the voltage; 0 dB is no change at all.
What is a −3 dB point?
The frequency at which a filter or amplifier's output power has fallen to half (the voltage to 70.7 %). It is the usual definition of bandwidth and cut-off.
Why does my voltmeter not agree with the dBm figure?
dBm is power, and the voltage it means depends on the impedance. Most "dBm" scales on meters assume 600 Ω; across 50 Ω the same power is 224 mV rather than 775 mV.