Back to Basics: The dB (decibel) value

Let’s talk about decibels (commonly abbreviated as dB)!

What is dB? The simplest way to explain it is “decibel (dB) is just the RATIO of 2 numbers.” That’s all there is to it, it’s that simple.

To be more exact, dB allows us to compare the RATIO of 2 numbers (or values) in a logarithmic manner. And the ratio between Number2 and Number1 may be very, very, very small… or very, very, very large!

“I hate math! What is this math and ratio you’re talking about?”

Well, if you look at a preamp’s front panel, and you see it has a maximum gain of 60dB, or 66dB or 72dB… what does that really mean? How do you relate that to the real world?

As I said, the dB is the ratio of 2 numbers. In the case of a preamp, the 2 numbers we’re interested in is the voltage level of the input signal (Vin) and the output signal (Vout).

And the ratio between these 2 numbers is also called the Gain (or more exactly, the Voltage Gain of the preamp).

VGain = Vout / Vin

Since both the numerator and the denominator have the same units (Volts or millivolts), then it cancels out and we’re left with a a simple number.

So for example you feed your preamp an input of 10 milliVolts, and you get 20 milliVolts output at the end, what is the Gain (i.e. voltage gain)?

VGain = 20mV / 10mV = 2 ;the preamp has a voltage gain of two (2).

Another example, if you have an input of 10milliVolts, and your output is 20Volts, what is Gain of the preamp?

VGain = 20,000 mV / 10mV = 2,000 ;the preamp has a voltage gain of two-thousand (2,000).

And that’s when this plain number RATIO becomes awkward to use… when we’re talking about very large numbers (or very small numbers)!

We’ll hear something like this in recording studios.

Bruce: “Hey Chuck! I’m using this vintage ribbon mic. Can you turn up the gain on the preamp two-thousand times?”

Chuck: “Two-thousand what?” (brain explodes!)

We don’t want that to happen so enter dB (decibels) so we can express the ratio of 2 numbers in a logarithmic manner.

The mathematical formula is:

dB = 20 log (Vout/Vin) ; where Vout = voltage output, and Vin = voltage input
or
dB = 20 log (VGain)

We get the RATIO between Vout and Vin. Using the example above

Gain = 20,000 mV / 10mV = 2,000

dB = 20 log (2000)

So, first enter 2000, then press LOG.

Multiply by 20.

And the answer is 66.

66dB is the same as saying 2,000x gain.

Bruce: “Hey Chuck! We need more gain for this vintage ribbon mic! Turn preamp to 66dB gain.”

Chuck: “You got it!”

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Note: There are a few things you can remember when dealing with ratios and decibels.

For example: Doubling the voltage (i.e. 2x) is 6dB increase. Easy enough.

– Note: the word “signal” below refers to the voltage levels at the input and output of your preamp

Increasing the signal 10x is a 20dB increase.

Increasing the signal 50x is approx 34dB increase.

Increasing the signal 100x is a 40dB increase.

Increasing the signal 1,000x is a 60dB increase.

Increasing the signal 2,000x is a 66dB increase.

Increasing the signal 10,000x is a 80dB increase.

As you can see, even as the ratio between the 2 numbers becomes orders of magnitude larger, the dB value is still an easier/friendlier 2-digit number.

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ADDENDUM: How did we derive the formula Gain dB = 20 log (Vout / Vin)?

For power quantities, the formula is dB = 10 log (Pout / Pin)
and Power = V x I (volts x current)
and current is I = V / R (volts divided by resistance)
let’s express current I in terms of Voltage and Resistance only.
so Power = V^2 / R
Let’s assume resistance or impedance is held constant… then we can cancel R in the equation.

(1) Gain dB = 10 log (Vout^2 / Vin^2)

simplify to remove the square(^2) in Vout and Vin, we get

Gain dB = 2 * 10 log (Vout / Vin)
(2) Gain dB = 20 log (Vout / Vin)

Both the above formulas (1) and (2) will return the same answer (try it in your calculator).

But the (2) formula is “easier” since we don’t have to square the Vout values, and we avoid having to work with very large numbers. (just imagine, 20,000^2 = 400,000,000, then divide by 10^2 (or 100), get log, then multiply by 10!)