Bels and decibels

We saw that absolute power levels are measured in watts. However, radio engineers often work with relative measures. Rather than saying a radio has a 100 watt transmitter, they would tell us how many times more (or less) powerful it is than a reference radio.

The decibel (dB) is used to measure relative power. (A decibel is 1/10 of a bel (B), and the bel was named after Alexander Graham Bell).

Lets take an example. How much more powerful is a 200 watt radio than a 2 watt radio?

The simple ratio is 100:1, but a bel is not a simple ratio. It is the log of the ratio. Therefore, one bel corresponds to a power ratio of 10:1; two bels is 100:1, three bels is 1,000:1, etc.

Continuing our example:

     B = log (power of radio 1/power of radio 2
     B = log (100)
     B = 2
     dB = 20

(We multiplied by 10 in the last step because a bel is 10 decibels).

So, we can describe our 200 watt radio in two ways. We can say: "it has a transmission power of 200 watts" or "its power is 20 decibels above that of a 2 watt radio." Radio engineers typically use decibels.

If the power ratio is less than 1:1, then the number of decibels will be negative.

We can summarize the above calculation with this formula:

     dB = 10*log(P1/P2)

We can also solve the above equation for the ratio of the absolute power levels given relative dB:

     log(P1/P2) = dB/10

Since the log of a number is the power to which ten must be raised to get that number, this is the same as saying:

     P2/P1 = 10^(dB/10)

These formulas are used in these calculators which convert between relative power levels and decibels.

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