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Analog scales tend to be very large. As an example, the hearing threshold of a young adult is 20 micro-Pascal; the maximum level without damaging the ear can be more than 20 Pascal, a ratio of one to 1 million. Particularly because of the widely varying sound levels, a need for a logarithmic measure appeared early on in electronics.
The Neper, Bel, and Decibel
There are two logarithmic measures: the neper and the bel. The neper is based on the natural logarithm and named after John Napier, a 16th-century Scottish mathematician who came up with the logarithmic table. “Neper” is the Latinized spelling of his name.
In the 1920s, a measure based on logarithm with base 10 began to be used at Bell Laboratories. At first called the "transmission unit," it was subsequently re-christened the bel, after Alexander Graham Bell.
The idea of this unit is simple. A bel is the logarithm of two power levels:
$$\text{bel} ~=~ \log \frac{P_1}{P_2}$$
However, the bel turned out to be a bit coarse. One-tenth of a bel suited the Bell Labs people better, hence the decibel (dB):
$$\text{dB} ~=~ 10\log \frac{P_1}{P_2}$$
This is the ratio of two power levels. Since power is related to the square of the voltage, we get:
$$\text{dB} ~=~ 20 \log \frac{V_1}{V_2}$$
The neper has more or less disappeared as a measure, decibels having proved more convenient. When using decibels, however, always keep in mind there’s a 2:1 difference between the ratios of power and those of voltages or currents (or pressure, for that matter).
Even though a logarithmic ratio is very convenient, it’s helpful to picture the actual voltage ratios as described in Table 1.
Table 1. Decibel levels and their corresponding voltage ratios.
| Decibel level | Voltage ratio |
| –60 dB | 1/1,000 |
| –40 dB | 1/100 |
| –20 dB | 1/10 |
| –6 dB | Approximately 0.5, or exactly 0.5012 |
| –3 dB | 0.707 |
| 0 dB | 1 |
| 20 dB | 10 |
| 40 dB | 100 |
| 60 dB | 1,000 |
Variations on the Decibel
Fundamentally, decibels are relative, expressing only a ratio. But there are many modifications in which one of the two levels is a fixed, absolute value. Here are a few important examples:
- dB SPL: Sound pressure level where the hearing threshold (0 dB) is based on 20 μPA.
- dBμV: Voltage ratio relative to 1 μV.
- dBm: Power ratio where 0 dB = 1 mW.
Originally, dBm was based on an impedance of 600 Ω (that of a telephone line). It’s now used with any impedance, which is fine as long as we calculate power ratios and not voltage ratios.
