# AllAboutDecibels

## Contents

**AllAboutDecibels**

## The Basics

A decibel (dB) is a logarithmic comparison of two power measurements.

Let P1 and P2 be two measurements of power.

A Bel (as in Alexander Graham Bell) is defined as

B = log10 (P1 / P2)

A Bel is a huge difference, so customarily a tenth of a bel, or deciBel is used.

dB = 10 * log10 (P1 / P2)

(Decibels have never made any sense to me -- John Gilmore -- but Bels make

perfect sense. So just divide by ten and think in Bels, which are just

logarithms base ten. Thus 20dB is 2 Bels which is a factor of 100. Why the radio

world complicated the simplicity of logs by multiplying them all by ten baffles me -- it's like calculating in furlongs per fortnight.)

## dB's vis-a-vis Volts

In the electrical domain, it's sometimes more convenient to measure voltage than power.

Volts can be related to power like this: P = V*V / R, where R is a resistance.

That is, power goes as the square of voltage.

For two voltages V1 and V2, measured across two resistances (impedances) R1 and R2 a little algebra gives you:

P1 = V1^2 / R1 P2 = V2^2 / R2 dB = 10 * log10 ((V1^2 / R1) / (V2^2 / R2)) If R1 == R2, this simplifies to dB = 10 * log10 ((V1/V2)^2) db = 10 * 2 * log10 (V1/V2) db = 20 * log10 (V1/V2)

## dBm's

If you thought that was simple, they've thought of more stuff to complicate

your life. A dBm is a dB value calculated against a 1-milliwatt value.

Yep, not one watt, but a thousandth of a watt. Again the bogons have inserted

random factors into what ought to be simple.

Thus a 5dBm signal is a 0.5 Bel signal times 0.001 watt. 0.5 Bel is 10**0.5

or about 3.16. Thus 5 dBm is an inconvenient way to write 0.00316 watts. Didn't you always want to know

that?

## A few handy numbers

**Power**

Conversion from Power to dB's

3 dB is a factor of 2 [10 * log10 (2) = 3.01] 6 dB is a factor of 4 10 dB is a factor of 10 20 dB is a factor of 100 -3 dB is a factor of 1/2 -6 dB is a factor of 1/4 -10 dB is a factor of 1/10 -20 dB is a factor of 1/100

**Voltage**

Conversion from voltage to dB's

3 dB is a factor of 1.414 6 dB is a factor of 2 12 dB is a factor of 4 20 dB is a factor of 10 40 dB is a factor of 100 -3 dB is a factor of 0.707 -6 dB is a factor of 1/2 -12 dB is a factor of 1/4 -20 dB is a factor of 1/10 -40 dB is a factor of 1/100