Thank Alexander Graham Bell (1847-1922) for the introduction of the dB. Though best known for the invention of the telephone, he was even better know in his day for his research in studying the deaf and quantifying the sensation of hearing.

He noticed that our sensation of hearing is not linear with the power in a wave, but scales with the log of the power of the wave. Bell created a perceived loudness scale based on the log of the acoustic power in a sound wave.

For historical reasons, when we take the log of the ratio of powers, we refer to the units as Bels. Even though this is in reference to Alexander Bell, we drop one of the ”l”s and just call it the Bel scale. A Bel is ALWAYS the log of the ratio of two powers.

The threshold of hearing (TOH) is about 10^{-12} W/m^{2} of sound intensity. A normal conversation is about 10^{-6} W/m^{2}. On the Bel scale, the conversation would be rated as log(10^{-6}/10^{-12}) = 6 Bels.

On the Bel scale, a value of 1 means a power is 10^{1} or 10x higher than the reference base. A value of 3 Bels means the power is 10^{3} or 1000x higher than the reference base . Historically, we have come to use the Bel scale to measure all powers, such as light intensity and radio power, relative to some baseline value.

In general, the Bel scale is not very large. For example, the entire range of hearing goes from the TOH to about 10^{4} W/m^{2}, where the ear drum is perforated, or a total of 16 Bels. For such a large range of sensations, 16 is just not a very large number.

This is why it has become conventional to use not Bels but deciBels as the scale. A deci means 1/10th, so there are 10 deciBels in 1 Bel. We abbreviate this as dB. This means that we can write any power in terms of is dB value, relative to a reference level as: Power_in_dB = 10 x log(P/P_{0}). The factor of 10 is to convert the value in Bels into deciBels.

This sets the range of hearing to start at 0 dB at the TOH to 160 dB as damaging. More examples of sound levels can be found here.

A Saturn V Apollo rocket launch generated sound levels of 135 dB 1 mile away from the launch pad.

But, if we want to measure a quantity that is NOT a power, such as the amplitude of a wave, like a voltage or current, we can’t use the dB scale. It is only used for the log of the ratio of powers.

The work around is that if we want to measure the ratio of two voltages in dB, we actually measure the ratio of the powers in the waves. The power in a wave is the square of the amplitude. So, when we measure the ratio of two voltages in dB, we are really measuring the ratio of the powers in the voltages.

For example, an S-parameter is really the ratio of two amplitudes, not powers. When we calculate the magnitude of the S-parameter in dB, we use the factor of 20:

Sometimes it is confusing to figure out is the quantity we are looking at a power or an amplitude. For example, we often will see impedance measured in dBOhms. Is impedance an amplitude or a power? It turns out it is an amplitude. If you want to convert an impedance from dBOhms into Ohms, you need to use the factor of 20: Z = 10^{(Z_dB/20)} .

As long as you keep in mind that dB is ALWAYS the log of the ratio of powers, it’s pretty clear how to interpret the results.

For handy reference, keep in mind that a –3 dB drop in power means the power decreased by 50%. The amplitude decreased to only 70% of its initial value.

When you see a drop in amplitude of 50%, this is a drop in power of –6 db.

A signal that drops off by a factor of 10 in amplitude with each decade, linearly decreasing with frequency, for example, has a drop of a factor of 100 in power per decade, or –20 dB per decade.

Why impedance turn out to be amplitude?