EE/Mu 107 "Projects in Music & Science" California Institute of Technology
Thanks to DB, DSB, JMG, GJS and PES.
October 15, 1999; October 5, 2000.

Decibel (dB) Measurements In Audio

Decibels in Power Comparisons

The decibel begins by being a relative measure comparing two powers. It's one tenth of a Bel, named for Alexander Graham Bell. (Note that one "l" is omitted from the name of the unit.) The number of Bels difference between two powers is simply the log (base 10) of the power ratio. The number of decibels difference is 10 times this. (The abbreviation for decibel or decibels is dB. Note that the "b" is capitalized in the abbreviation only.)

Number of decibels difference = 10 ū log (P2/P1)

Bearing in mind that log 2 is 0.3 and the square root of 10 is 3.16, and that decibels can be used for any kind of power, answer the following:

Q 1.  By how many decibels is P2 greater than P1 when... (careful of signs!)
P2 = P1 ? _________
P2 = 2 ū P1 ? _________
P2 = 10 ū P1 ? _________
P2 = 20 ū P1 ? _________
P2 = 31.6 ū P1 ? _________
P2 = 100 ū P1 ? _________
P2 = 0.1 ū P1 ? _________

Q 2.  A car engine of 150 horsepower is how many decibels more powerful than one of 75 horsepower? _________

Q 3.  If the power of a 100 Watt light bulb were reduced by 20 dB, it would be equivalent to a bulb of what power? _________

NB: I don't think decibels really are used for any kind of power except that which bears a signal: acoustic, audio and radio powers, for instance. Questions 2 and 3 thus do not represent typical usage; they're given just to make a point.

Q 4.  Imagine that, to play music at the volume you like, your audio amplifier is producing an average of 5W (five watts; the watt is a unit of power) into each of your speakers. If the momentary peaks of the music (for instance, the "attacks," the noises at the beginning of each note) are 20 dB louder than the average (the extreme figure I've seen quoted), how many watts must the amplifier produce into each speaker during those peaks, assuming that you want to reproduce the peaks accurately? _________
How many watts would be needed during the peaks if they were 15 dB greater than the average? _________
If 10 dB (as on a rather distantly-miked piano recording I've measured)? _________

Q 5.  Your old audio amplifier is rated at 40W. You buy a new one identical in every respect except for being rated at 80W. The new one is how many decibels more powerful than the old one? _________ dB
(Guess!) Will this increase in loudness be significant?

Q 6.  A change of +1 dB increases power by what percentage? _________

Q 7.  A change of -1 dB decreases power by what percentage? _________

Decibels in Voltage Comparisons
We often use dB to compare voltages rather than powers. Power goes as the square of voltage, which suggests this derivation:

Number of decibels difference = 10 ū log (P2/P1) = 10 ū log (V22/V12) = 10 ū log (V2/V1)2 = 20 ū log (V2/V1)

Over time, the relation of decibels to voltage has floated free of its origins in power comparisons, and we now simply speak of comparing voltages independent of the impedances involved. We still use the same formula.

Q 8.  By how many decibels is V2 greater than V1 when... (careful of signs!)
V2 = V1 ? _________
V2 = 2 ū V1 ? _________
V2 = 10 ū V1 ? _________
V2 = 0.1 ū V1 ? _________

Q 9.  A transformer has a 1:5 turns ratio, so the voltage across the secondary winding is five times the voltage across the primary winding. The secondary voltage is how many decibels greater than the primary? _________

Q 10.  In the previous question, one voltage was five times another. By what multiple is one voltage greater than another when that multiple is numerically the same as the number of decibels difference between the two voltages? (Give to three significant figures.) _________ Is there more than one such multiple? If so, give any other(s) to three significant figures.
(In other words, we're looking for a ratio of two voltages that is equal to 20 times its own log. Call the ratio R; then we're seeking all values of R for which R = 20 ū log R.)

Q 11.  Imagine that, to play music at the volume you want, your audio amplifier must produce an average of 6.32 Volts across the terminals of each of your speakers. If the peaks of the music are 20 dB louder than the average, how much voltage does the amplifier need to produce across each speaker for those peaks, assuming that you want to reproduce the peaks accurately? _________

Q 12.  How much voltage would be needed for the attacks if they were 15 dB greater than the average? _________
If 10 dB? _________

Q 13.  My microphone preamplifier has a voltage gain of 60 decibels. One millivolt at the input results in an output voltage of what? _________

Absolute Decibel Scales
So far, our discussion has explored using the decibel as a relative measurement, a means of comparing two things. If one establishes a 0 dB reference point, then the decibel becomes an absolute measure.

dBW means power referenced to 1 watt. That is, 1 watt is 0 dBW; so the power rating of a 10W amplifier, for instance, can be written as 10 dBW.

Q 14.  What is the dBW rating of a 20W amplifier? _________

dBm means power referenced to 1 milliwatt. (1 mW is 0 dBm.)

Q 15.  0 dBW = _________ dBm

dBV means voltage referenced to 1 V. (1 V is 0 dBV.)

dBu means voltage referenced to 774.6 mV. (774.6 mV is 0 dBu.)
Where does 774.6 mV come from? In the old days, many circuits had impedances of 600 ohms at input and output. When you have 1 milliwattŚ0 dBmŚacross a resistance of 600 ohms, the voltage is 774.6 mV (by the relationship P = E2/R, which gives 0.001 W = E2/600 ohms). After a while, this voltage floated free of the original context, and is now a reference on its own-dBu-without regard to impedance or power.

Q 16.  0 dBV = _________ dBu

dB SPL means sound pressure level referenced to 0.0002 dyne/cm2, which is 20 microPascals (uPa). (Important: From looking at the units, decide whether SPL will use the "10 log" dB formula like voltage or the "20 log" formula like power.)
This reference was established long ago as the "threshold of hearing" at 1000 Hz (Hertz, or "cycles per second") of a group of college students. SPL differences of 1 dB are sometimes referred to as the smallest perceptible differences under ideal conditions. It doesn't seem that the dB could have been chosen this way, because the Bel seems to precede it logically, and the Bel was chosen as being a simple mathematical relationship. Perhaps it was simply discovered that a 1 dB difference in SPL was about the smallest difference one could perceive.
The "threshold of pain" is usually given as about 120 dB SPL. Our neighbor, the Jam Room, sometimes has levels of 124 dB, according to a former student who took a meter in there (while himself wearing hearing protection). So high a level can, in the opinion of experts, instantaneously damage one's hearing.

Q 17.  124 dB SPL = _________ dyne/cm2 = _________ Pa

dBA, also written dB(A):
This is dB SPL measured after a weighting filter. The purpose of the filter is to simulate the effects of varying frequencies on the ear. For instance, in soft sound, we hear bass pitches less well than middle ones. Measurements in dBA were intended to incorporate this sort of knowledge and thus to reflect more accurately the subjective loudness of various sounds. (This approach has serious flaws, however; so it's unfortunate that it is enshrined in noise-related legislation at every level of government!)

Why Logs?
It's odd but true that we have a perception of "twice as loud." If you listen to a sound through your speakers and turn up the volume control, you will find that at some point the sound seems "twice as loud." This is already noteworthy! And if you raise the volume 'til the music sounds twice as loud again, you will have taken the music through two increases of about the same power ratio. Each will have been a change of roughly 10 dB. Thus, our perception of loudness works logarithmically, so it's appropriate that we use a logarithmic measure for it. (A deeper question is why our perception should work logarithmically, as it does for both loudness and pitch.)

Q 18.  You have a 50W amplifier and you wish to get a new amp which can supply twice as much power. Other things being equal, what will be the power rating of your new amp? _________

Q 19.  You have a 50W amplifier and you wish to get a new amp which can play the music twice as loud. Other things being equal, what will be the power rating of your new amp? _________

Q 20.  Getting close to the realm of fantasy, what would be the power rating of an amp which could play the music twice as loud again—that is, four times as loud as the original sound? _________ End