 |
| Home │ Audio
Home Page |
Copyright © 2014 by Wayne Stegall
Created November 21, 2014. See Document History at end for
details.
Decibels
The
calculation and use of decibels in audio
Introduction
Because human senses – including hearing – perceive signal levels exponentially (or logarithmically depending on point of view), it was deemed useful to represent the levels of these signals by a logarithm of their magnitude. As a result the Bell(B) – named after Alexander Graham Bell – was designated to be the unit representing the base ten logarithm of signal power ratio.| (1) |
powerBell = log10(power-ratio) |
These units are rather large for many uses so the smaller unit of decibel (dB) became more commonly used. In this system, level differences of 1dB are just noticeable, 3dB more so, and 10dB is a significant change.
| (2) |
powerdB = 10 ×
log10(power-ratio) |
Because power is proportional to the square of voltage (or current) decibels can represent voltage or current ratios as well.
| (3) |
voltagedB = 20
×
log10(voltage-ratio) |
Used alone dB can represent voltage or power ratios and which is inferred must be determined by context. Additionally, dB also is used to represent absolute values against some fixed reference. Examples used in electronics and audio are
Figure 1: Absolute decibel units
| Unit |
Quantity represented |
Reference value |
Calculation |
|
|
|
|
|
| dBV |
Voltage |
1V |
20 × log10(voltage) |
| dBW |
Power |
1W |
10 × log10(power) |
| dB(SPL) |
Sound pressure level |
20µPa |
20 × log10(sound-pressure / 2×10–5Pa ) |
| dB(SWL) |
Sound power or intensity |
1pW (10–12W) |
10 × log10(sound-power / 10–12W) |
Reference levels for dB(SPL) and dB(SWL) were chosen so that they produced the same decibel values for the same sound level. The following table relates decibel sound levels to their measurements and to some common sound experiences.
Figure 2: Sound pressure level related data1
| Decibels (dB(SPL) or dB(SWL)) | Sound pressure level (Pa or N/m2) | Intensity (W/m2) |
Type of sound |
|
|
|
|
|
| 130 | 63.2456 |
10 | artillery fire at close proximity (threshold of pain) |
| 120 | 20 | 1 | amplified rock music; near jet engine |
| 110 | 6.32456 |
10–1 | loud orchestral music, in audience |
| 100 | 2 | 10–2 | electric saw |
| 90 | 6.32456×10–1 | 10–3 | bus or truck interior |
| 80 | 2×10–1 | 10–4 | automobile interior |
| 70 | 6.32456×10–2 | 10–5 | average street noise; loud telephone bell |
| 60 | 2×10–2 | 10–6 | normal conversation; business office |
| 50 | 6.32456×10–3 | 10–7 | restaurant; private office |
| 40 | 2×10–3 | 10–8 | quiet room in home |
| 30 | 6.32456×10–4 | 10–9 | quiet lecture hall; bedroom |
| 20 | 2×10–4 | 10–10 | radio, television, or recording studio |
| 10 | 6.32456×10–5 | 10–11 | soundproof room |
| 0 | 2×10–5 | 10–12 | absolute silence (threshold of hearing) |
Relating speaker sensitivity and power to resulting sound level
Calculations relating speaker sensitivity, power, and sound level seem the most useful for audio hobbyists. Speaker sensitivity often has to be recalculated because it is not power sensitivity as thought. Because sensitivity specifications are given specifically for 8Ω it instead represents voltage sensitivity referred to the voltage necessary to drive 1W into 8Ω, that is 2.82843V. Before power decibels can be added to speaker sensitivity to give sound level, sensitivity has to be in power decibels.To convert speaker sensitivity to power decibels.
| (4) |
sensitivitypower =
sensitivityvoltage + dB |
 |
speaker impedance
8Ω |
 |
Then sound level can be calculated.
| (5) |
volumedB(SPL) = sensitivitypower + powerdBW |
Example:
A speaker of 90dB/1W sensitivity will produce a sound level of 117dBSPL when driven with 17dB (50W) of power.
Head math
Scientific calculators are not always handy when you want to do decibel calculations. However because decibels are logarithms, the laws of logarithms simplify calculations.Figure 3: Some basic laws of logarithms in decibel form
| dB(x × y) |
= |
dB(x) + dB(y) |
| dB(x / y) |
= |
dB(x) – dB(y) |
| dB(1 / x) |
= |
–dB(x) |
| dB(xy) |
= |
dB(x) × y |
From a few known decibel relations, the remainder can be calculated in your head or on paper without undue difficultly, so long as calculations are short enough not to accumulate error approaching 1dB.
Figure 4: A table of decibels and now they would have been derived from head math
| Ratio |
|
dBW |
dBV |
Calculating decibels from
estimated ratio (you can infer how to calculate in the other direction) |
|||
|
|
|
|
|
||||
| 1 |
0 |
0 |
A given |
||||
| 1.25893 |
1 |
2 |
dB(1.25) = dB(10/23) = dB(10) – (dB(2) × 3) = 10dB – (3dB × 3) = 1dB | ||||
| 1.58489 |
2 |
4 |
dB(3.16228 / 2) = dB(3.16228) – dB(2) = 5dB – 3dB = 2dB | ||||
| 1.99526 |
3 |
6 |
Round to 2. This is a given
you would expect to know and begin with. |
||||
| 2.51189 |
4 |
8 |
dB(2.5) = dB(10/(2 × 2)) = dB(10) – dB(2) – dB(2) = 10dB – 3dB – 3dB = 4dB | ||||
| 3.16228 |
5 |
10 |
|
||||
| 3.98107 |
6 |
12 |
dB(4) = dB(2 × 2) = dB(2) + dB(2) = 3dB + 3dB = 6dB | ||||
| 5.01187 |
7 |
14 |
dB(5) = dB(10 / 2) = dB(10) – dB(2) = 10dB – 3dB = 7dB | ||||
| 6.30957 |
8 |
16 |
dB(3.16228 × 2) = dB(3.16228) + dB(2) = 5dB + 3dB = 8dB | ||||
| 7.94328 |
9 |
18 |
dB(23) = dB(2) × 3 = 3dB × 3 = 9dB | ||||
| 10 |
10 |
20 |
A given |
||||
| 100 |
20 |
40 |
A given |
||||
| 1000 |
30 |
60 |
A given |
Example Calculations
Example 1: Determine the sound pressure level to which a 30W amp
will drive a 6Ω speaker of 90dB sensitivity.From equation 4 calculate speaker sensitivity.
| (6) |
sensitivitypower =
90dB
+ dB |
|
6Ω
8Ω |
|
= 90dB + dB |
|
3 4 |
|
= 90dB + dB | |
3 2 × 2 |
|
| (7) |
sensitivitypower ≈ 90dB + dB(3.16228) – (dB(2) + dB(2)) = 90dB + 5dB – (3dB + 3dB) |
| (8) |
sensitivitypower ≈ 89dB/W (88.7506dB by calculator) |
Calculate power using head math and table of decibels in figure 4.
| (9) |
powerdBW = dB(30W) = dB(10W × 3) = dB(10) + dB(3.16228) = 10dB + 5dB |
| (10) |
powerdBW = 15dB |
Finally calculate sound pressure level using equation 5.
| (11) |
volumedB(SPL) = 89dB + 15dBW = 104dB(SPL) |
Example 2: Determine power required to drive a 4Ω speaker of 88dB sensitivity to 110dB.
From equation 4 calculate speaker sensitivity.
| (12) |
sensitivitypower =
88dB
+ dB |
|
4Ω
8Ω |
|
= 88dB + dB |
|
1 2 |
|
| (13) |
sensitivitypower = 88dB – dB(2) = 88dB – 3dB = 85dB/W |
Calculate power required by reversing equation 5.
| (14) |
powerdBW = volumedB(SPL) – sensitivitypower = 110dB – 85dB = 25dBW |
Convert power in watts to decibels.
| (15) |
power = dB-1(25dBW) = dB-1(20dBW + 5dBW) = dB-1(20dBW) × dB-1(5dBW) |
| (16) |
power = (100 x 3.16)W = 316W |
|
|
1Table partially based on
Encyclopedia Britannica, "Table 1: Sound Levels for Nonlinear (Decibel)
and Linear (Intensity) Scales," 2002.
Document History
November 21, 2014 Created.