The Bel is a fundamentally new unit, which was originally established in 1928 by The Bell System to describe sound levels. This scale has since been generalised to relate to the log of the ratio of any two power levels.
The most important principle of the Bel scale to keep in mind is that it is, without exception, by definition, the ratio of powers. It originally related to sound level powers, but has since been generalised to all types of powers.
Sound level power has a huge range. It starts at the threshold of hearing (TOH) with a sound level of about 10-12 W/m². The onset of pain is about 10W/m², and the power level to rupture an eardrum is 1000W/m².
The loudest recorded man-made sound is from rocket launches. The Saturn V produced 100W/m² of sound level power about 1 mile from the launch pad. It only got larger closer to the launch pad.
This amount of sound power reflecting off the launch pad could have easily damaged the Space Shuttle. This is why a 900,000 gallon per minute of water sound suppression system was installed on the launch pad. Even with this system, the sound level power at the position of the Orbiter was about 100W/m².
There are 15 orders of magnitude between the threshold of hearing and the rupture level. When there is such a large dynamic range, it is more convenient to describe the values in terms of the log of the ratio of the power levels to a reference level.
For sound, the TOH is used as the reference. A whisper is a power about 100 times the TOH. This is a sound level of log (100/1) = 2 Bels. A vacuum cleaner is about 10 million times the TOH. Its loudness is log(107/1) = 7 Bels.
But on this scale of Bels, the loudness of a Space Shuttle launch at the orbiter is only 14 Bels. For all its awesome power, 14 just doesn't seem like a very large number. To give it a little larger range, rather than referring to the Bel directly, by convention, we've expanded the scale by 10x and use units of 1/10th a Bel, which is called a deciBel, or dB for short.
A dB is always, without exception, by definition, 10x the log of the ratio of the powers: in units of dB = 10 x log(P1/P0). On the dB scale, loudness ranges from 0 dB as the TOH up to 140 dB for the space shuttle.
Over the past 85 years, the dB scale has been adapted to describe not just the log of the ratio of sound power, but the log of the ratio of any two powers. When we use a reference power of 1W of power, we can measure any other power in dB of power. When we use 1mW as the reference power level, we often designate the dB as dBm to identify that the reference power is 1 mW.
Now comes the subtle part. When we want to compare two quantities that are not powers, we have to somehow convert them into powers so we can use the dB scale.
Voltage isn't power, but amplitude. To use the dB scale to describe a voltage, we have to convert the voltages into powers, and take the ratio of the powers that are related to the voltages.
For example, the power dissipated in a resistor by a voltage, V1, is V1²/R. We can describe the ratio of the power generated by two voltages if they were across the same resistor, as;
dB = 10 x log (P1/P0) = 10 x log (V1²/V0²).
The resistances, of course, cancel out. We can pull the square terms out of the log and get;
10 x log (V1²/V0²) = 10 x 2 x log(V1/V0) = 20 x log (V1/V0).
Now, we see where the factor of 20 comes from. Whenever we describe the ratio of two things that are not powers, but amplitudes, we use a factor of 20 to get back to the original ratio. The value in dB is really about the ratio of the powers of the two quantities we are comparing. The factor of 20 lets us convert to the log of the ratio of the voltages.
An S-parameter is always the ratio of two voltages. This means that to describe an S-parameter in dB, we use the factor of 20 to relate the log of the ratio of the output voltage from some port compared to the input voltage. The power associated with the voltages is V² and the value of the S-parameter in dB is = 20 x log (VOUT/VIN).
On the next page, we look at some of the subtle aspects of the dB scale, such as; how much voltage is 10 dBm; and do we use 10 or 20 when describing an impedance in the dB scale.