A deciBel (dB) is a tenth of a Bell. It makes life very simple in probably all disciplines of engineering and technology. There are two fundamental forks in the road for the deciBel.

- Power
- Voltage

You may have noticed that sometimes a dB is

- dB = 20*log(V
_{o}/V_{i}) and sometimes it is - dB = 10*log(P
_{o}/P_{i}).

Do you see the distinctions above? One is a power (Watts) unit and the other is a voltage (V) unit.

## What’s the dB Advantage?

DeciBells are a form of logarithms. They facilitate convenient viewing and manipulation of data-sets that transcend decades. We can use a logarithm as an example. What is the mid-point frequency between 1 Hz and 1 kHz? Should you average the two? If so it would be as:

(1 + 10e3)/2 = 5 kHz

The mid-frequency between 1 Hz and 10 kHz is not 5 kHz. But let us reformulate applying log functions:

( log_{10}(1) + log_{10}(10k) ) / 2 = ( 0 + 4 ) / 2 = 2

We obtained the answer with the application of a log function. The next step is to “un-log” that function, more commonly called applying an anti-log.

10^{2} = 100 Hz

This is also called a geometric-mean which can also be expressed as:

( 1 Hz * 10 kHz ) ^{0.5} = 100 Hz ==> the geometric mean

If the span of numbers was LESS THAN a decade, we could resort to ordinary averaging.

What is the mean of the two frequencies 50 Hz and 52 Hz? By inspection, we could say 51 Hz. But a geometric mean would still work for that as well.

( 50 Hz * 52 Hz )^{0.5} = 51 Hz

We have used units of frequency to illustrate the usefulness of decibels but their use transcends all disciplines of engineering. For the present application, we are talking about Watts and Volts as well as unitless ratios. Decibels make a comparison of 1 milliWatt with 1 KilloWatt convenient. Decibels are your friend. You are shooting yourself in the foot if you let them intimidate you.

## What’s the Diff between V and P Regarding dB?

Why do you multiply the result of a log function of one by ten (10) and the other by twenty (20)?

After going through a lot of trouble to try to explain where these numbers came from, it was clear to me that the medicine was worse than the cure. We recommend remembering that it derives from whether or not a squaring function is in play.

Where does the 2 of the “20” come from? Recall from your high school algebra and transcendental functions that logarithm rules say:

20 * log_{10} (X) = 10 * log10(X^{2})

We will need to use 20 instead of 10 when the quantity is a “squarable” quantity. Power already contains a square (i.e. Power = i^{2}e) and thus, we always multiply by 10 for power ratios. Voltage, however, is linear and does not already contain a square function so you need to use 20. Just remember that you square voltage ratios and you don’t square power ratios. This is one of the exceptions where we tell you to stick with the cookbook.

We found a nice discussion of this subject for those wanting to punish themselves more.