Decibel Calculator

A decibel is not a unit, it is a ratio on a logarithmic scale — a tenth of a bel, which is a factor of ten in power. It exists because the quantities engineers care about span many orders of magnitude and because human senses respond roughly logarithmically, so adding decibels corresponds to multiplying the underlying quantity.

10·log or 20·log — the thing everyone gets wrong

There is only one definition of the decibel, and it is in terms of power: dB = 10 · log₁₀(P₂/P₁). The 20 shows up because power goes as the square of an amplitude. Voltage, current, sound pressure and field strength are amplitudes, and power ∝ V², so

dB = 10 · log₁₀(V₂²/V₁²) = 20 · log₁₀(V₂/V₁)

Same law, same decibel, different exponent in the quantity you happened to measure. Use 10·log for watts, milliwatts, acoustic intensity and optical power; 20·log for volts, amps, pascals and microphone signals. Get it backwards and every answer is out by a factor of two in dB — a 6 dB error where it matters most.

This is also why "3 dB" and "6 dB" both mean "double". 3 dB is double the power; 6 dB is double the voltage (which is four times the power). A filter's −3 dB point is where it passes half the power, and therefore 0.707 of the voltage.

Adding decibels

Decibels never add arithmetically. Two 85 dB machines running together are not 170 dB, and not 90 dB either — convert each to a power ratio, add those, and convert back. Two equal incoherent sources always give exactly +3 dB. The consequence in noise control is brutal: silencing one of two equal machines buys you 3 dB, and if one source is 10 dB above another, fixing the quieter one buys you almost nothing. Fix the loudest thing first.

This addition assumes the sources are incoherent — independent noise. Two coherent signals (the same tone through two paths) add as amplitudes and can give +6 dB, or complete cancellation, depending on phase.

Decibels with a reference

Plain "dB" is a ratio and says nothing absolute. Attaching a reference makes it a measurement: dBm is relative to 1 mW, dBW to 1 W, dBV to 1 volt, dBu to 0.7746 V, and dB SPL to 20 µPa, the nominal threshold of hearing. Never subtract a dBm from a dB SPL; check that the references match before comparing two numbers that both end in "dB".

Caveats

• Sound-level meters usually report dB(A), which is weighted to approximate human frequency response. A-weighted and unweighted readings are not interchangeable.
• Loudness is not intensity: +10 dB is ten times the power but is usually described as about twice as loud.
• Very small dB differences are meaningless in the field — 1 dB is around the threshold of perception and is inside most instruments' error bars.