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A Practical Guide To Key Audio Calculations
Thanks to modern technology, you can do dB calculations without knowing a thing about the mathematics of logs, anti-logs, ratios, exponents, or even much about math
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This is a practical guide to doing audio calculations, particularly dB (decibel) calculations, covering most common situations.

You see dB numbers all the time in audio, and probably understand that 3 dB is considered a just noticeable change in volume level.

You may also be aware that dB calculations involve “logs” (logarithms). But perhaps you’re not quite so clear on how to figure out what 24 dB from your mixing console means to your amplifier rated for 1.4V input sensitivity.

Thanks to modern technology, you can do dB calculations without knowing a thing about the mathematics of logs, anti-logs, ratios, exponents, or even much about math. This article is necessarily long in order to cover many situations where you might need to use dB.

But, if you understand the calculations, you will find that most of them repeat the same things in different ways. If you begin to see this, it means you are beginning to understand how to calculate dB.

INTRODUCTION
First, if you don’t have one, you need to buy yourself a cheap scientific calculator. It MUST have several specific functions on it. One is a “Log” function. There are two common Log functions. One is “Log base e” or “natural Log” which you DON’T want. The other is “Log base 10” which is what you need. You can check if it is “Log base 10” by simply entering 10 and hitting the “Log” key. The display should read 1. If it doesn’t, don’t buy it.

Another necessity is a 10x function (also called 10 to the x function or the anti-log function). You can check if this is the proper function by entering 2 then hitting the 10x key. The display should read 100. You also need a +/- (change sign) key. It must also have plus, minus, divide, multiply, and equal (=) keys. You can basically ignore the other functions on it to do dB calculations.

Once armed with this tool, you can now learn to do dB calculations. Actually, you don’t really have to learn much except to push the right buttons on the calculator.

In each example, you will be told exactly what calculator key to hit and what your answer should be. Once you get the right answers as shown, you can substitute your own numbers in the various examples to figure out your own things.

Remember, a calculator is a DUMB device. It will only do what you tell it to do. So you MUST use your brain a bit to see if your answers make sense. This means if you come up with a number like 23841 dB or 20,000,000 watts, it is wrong. Nothing in audio has 23841 dB of anything and 20,000,000 watts is out of the question, unless you are providing sound reinforcement for Space Shuttle launches.

All the answers given in this article only show the first two digits (numbers) to the right of the decimal point. Because fractions of a dB or a watt rarely have little practical significance in audio, these two digits are shown ONLY so you know you got the right answer.

The answer on your calculator may be 15.84893192 but the answer given here is 15.84. As you will see in one of the examples, chopping off digits like this can lead to slight, but not significant, errors. Note again the last digit to the right in the examples is NOT a rounded off value in the examples, but chopped off. If it was rounded off, the answer would be shown as 15.85.

Most calculators can be set to display only 2 digits after the decimal point. It rounds off numbers to do this so that 15.84893192 becomes 15.85. So, if you set the calculator to do this for figuring out the examples, your last digit to the right may be 1 larger or smaller than in the examples.

For actual audio work, you should only be concerned with dB numbers to the left of the decimal point. Thus an answer of either 15.84 or 15.85 dB should be rounded off to 16 dB when stating the result.

For each example calculation, the actual formula being used for the calculation is also shown in red. The actual calculation procedure is in blue.

The most common “Log” calculations you need are: dB to voltage, voltage to dB, voltage gain to dB, dB to voltage gain, calculating SPL for distances, and converting amplifier watts to SPL changes or SPL changes to amplifier watts. Therefore, these are the only examples given. With a little brainwork, you may be able to apply the example calculations to figure out other things.

DEFINITIONS
Some definitions to know:

Any dB value is a RATIO, meaning it represents one number divided by another. If you simply state something in dB then you are only stating the ratio in between one thing and another. So you might say the difference in two voltages is 6 dB but that only means one voltage is twice the other (6 dB = 2 times voltage). It doesn’t tell you anything about the actual voltages.

If you want to state the actual value of something in dB, most common audio calculations have a 0 dB reference value that is indicated by suffix (a letter or letters following dB). The 0 dB reference value is always used as one of the numbers for the ratio. You should always use a suffix when stating the dB of something as an actual value so that anyone else will know what 0 dB reference is.

So when you say your mixer is putting out + 6 dB, you really need to say + 6 dBu or whatever the 0 dB reference is. Thus +6 dB means you have twice as much voltage while 0 dBu means you have 0.775 volts or 1.55 volts.

For electronic calculations (voltage and wattage) the 0 dB references are: 0 dBu (or dBv) = 0.775 volts
0 dBV = 1 volt
0 dBm = 1 milliwatt (0.001 watts). The standard reference value for 0dBm is 0.775 volts into a load of 600 Ohms. For most audio calculations, simply assume you are NOT using a 600 Ohm load and thus dBm can equal dBu.

For example, a +24 dBm output specification on a mixing console can be used as +24 dBu to calculate voltages. The reason for this is that modern audio equipment will put out the same voltage whether there is a “load” on it or not. Thus knowing the power in a line level audio circuit is of little value and simply complicates what you need to know.

Another example: 6 dBu is the ratio of some voltage divided by 0.775 volts. That voltage is 1.55 volts.
For SPL (Sound Pressure Level) calculations the 0 dB reference is:
0 dB SPL = 0.00002 pascals (a pascal is a measure of pressure, in this case air pressure, just like a meter is for distance).

Another example: 100 dB SPL is the ratio of some sound pressure to 0.00002 Pascals. That pressure is 2 Pascals
These are the symbols used in the formulas:
”x” means multiply
”/” means divide
”^” This symbol indicates what follows is an exponent of the number preceding it. An exponent means “raised to the power of”, as in 10^2 is 10 raised to the power of 2 or more simply stated as “10 squared”. With dB calculations you get “funny” powers like 10^(34/20). Spelled out this is “ten to the power of thirty four divided by twenty”. Don’t be afraid of this. The instructions and your trusty calculator will get you through it without having to fully understand it.

”( )” in the formulas means that everything inside the parenthesis is calculated first to come up with a single number. In the first example below, (24/20) is calculated first to come up with 1.2. Then 10 is raised to the power of 1.2.


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