Decibels made (as) ultra-simple (as possible): Part 4

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The complications

If you just want a general idea of how decibels work, you need read no further. Unfortunately, though, decibels have proved so useful that the basic idea has become somewhat subverted. It's turned out that we can produce a handy tool by a bit of creative rule breaking!

I have insisted up to now that decibels are to do with power or energy, but quite often the power is not what matters. For example, with electrical signals in audio systems, quite often what we want to compare is not the power but the voltage. And our ears have no way to tell the sound power, but the eardrum moves in and out as the air pressure goes up and down.

Why does this complicate matters? Well, to double the size of the pressure fluctuations, it takes, not twice the power, but four times as much. This idea takes a bit of getting used to, but maybe it will make more sense if you think of a mill stream driving a waterwheel. If the water pressure doubles, the water will push the wheel round twice as hard. And because it is being pushed twice as hard, it will also go twice as fast. Because it is pushing twice as hard as well as twice as fast, the miller can grind, not just twice as much flour, but four times.

Since you don't get something for nothing in nature, we can put it equally well the other way round. Imagine that the water was not from a natural stream but being supplied by a pump. Then to generate twice the water pressure, the pump will have to be four times as powerful.

In fact the power is proportional to the square of the pressure. If the pressure doubles, the power increases by four times, and if the pressure triples, then the power increases by nine times, and so on.

In the case of electricity, voltage is analogous to pressure, so electrical power is proportional to the square of the voltage.

Decibels and pressure (or voltage)

We saw in the previous pages that a power ratio of 10:1 corresponded to what is called a level difference of one bel, or 10 dB. But what if we're not interested in power, but sound pressure? (That's the fluctuating pressure that causes audible sound: a sound level meter detects this, and ignores the average air pressure that is indicated by a barometer.)

So, what level difference corresponds to a voltage or pressure ratio of 10:1?

Well I have insisted all along that decibels are to do with power, and I'm not going to change my mind: we must convert to a power ratio. If the pressure changes by 10 times, then the power must change by the square of that, which is 100 times. Now we have a power ratio, so we can convert quite easily to bels (by counting the zeroes) and then to decibels (by multiplying by 10). Try it: you should get the answer 20 dB.

In fact the level difference in decibels, starting from a voltage or pressure ratio, will be double what it would be from the same ratio, if it was referring to power.

The great pretence

In our analogy of the watermill a few paragraphs back, there was one assumption that I made, when I said that if the pressure doubled, there would be four times as much power. This was quite simply, that everything else was the same: the same stream and the same mill. For voltage ratios, this is equivalent to making measurements in the same electrical circuit. For sound, it corresponds to making measurements with the microphone in the same position, in the same physical surroundings. But decibels are far too useful to be so restricted!

So we pretend that we can relate voltage (or pressure) to power even when the measurement conditions are quite different. We convert voltage (or pressure) ratios to decibels exactly as if we were converting power ratios, and then just double the answer. We do this even when the things being compared are measured in different electrical circuits or different surroundings. So, we multiply the bels by 10 to convert to decibels, and also by two because it is a voltage or pressure ratio. In short, to get a level difference in decibels, we multiply the logarithm of a pressure ratio by 20.

But we pay a penalty for the great pretence. We can no longer convert the level difference (dB) back to a power ratio, unless we know that the signals being compared were measured in identical circuits (or, for sound, in identical physical situations). In many situations this doesn't matter a scrap, but it mustn't be forgotten.

Reference levels

Once we agree to ignore system differences, we can specify a reference value of voltage or pressure. Once you have a standard reference to compare with, decibels become an absolute measure. For example, if the reference is one volt, then a voltage level of 20 dB corresponds to 10 volts. And this is true whatever the circuit, so we now have a very useful way to translate directly between volts and decibels. We then refer to "voltage level" or "sound pressure level" (often abbreviated to "sound level").

Of course it is important - in fact vital - to know what the reference is, as without it the decibel levels are meaningless. In audio and video engineering, special symbols are sometimes used to indicate that levels in decibels refer to a specific reference value. For example, dBV is used when the reference value is 1 volt.

Another way to show the reference is to put it in brackets: for example (mV) indicates that the signal is measured relative to one millivolt, or 1/1000 volt. How many volts is 60 dB (mV)? Because we are dealing with voltage not power, we must first divide by two, which gives 30. Then we knock off a zero to deal with the "deci" bit. This gives 3. Looking back at section 2, on logarithms, we see that the number whose log is 3 is 1000 (count the zeroes!) So the signal in question is 1000 times the reference, i.e. 1000 millivolts, which is one volt.

In the case of ordinary sound, we're normally interested in pressure because that's what ears respond to. A standard reference of pressure is always used, and except in scientific work it is very rarely stated (naughty but that's the way it is.) So for example, we might be told that the sound level of a road drill is 100 dB. Usually we just stick to decibels and don't try to work out the size of the air pressure fluctuations. But if you should ever want to do this, you would need to know that the standard reference is 20 micropascals, or 20 millionths of a pascal.

Of course you can use a reference level for decibels that relate to power as well as to pressure. For example, in electronics, a reference power of 1 mW (1/1000 watt) is often used. In acoustics, as with pressure, the reference power is standardised (it's rather small at 1/1,000,000,000,000 watt) and, once again, this is often left un-stated.

Threshold of hearing

The reference sound pressure is also approximately the quietest sound that a healthy young person can hear, at a standard pitch (1000 Hz - about 2 octaves above middle C), a region of pitch where the ear is particularly sensitive. The log of a ratio of 1 is zero, so sound pressure that is equal to the reference has a level of 0 dB. In practice we rarely come across quieter sounds than this, and it is also difficult to measure them, so it is rare to see negative decibel values for sound levels.

What do you think?

I hope this dispelled some of the mystery of decibels. I'd be interested in feedback - please use the contact form.

Tony Woolf