Using the Decibel - Part 1: Introduction and underlying concepts

10log(E₁²/E₂²) = 10log(P₁/P₂)                             (2-5)

which is only true if both values are measured at an identical point in their circuit. A common usage has been to remove the exponent from the ratio and apply it to the multiplier.

2 x 10log(E₁/E₂) = 20log(E₁/E₂)                             (2-6)

Bear in mind that the decibel is always and only based upon a power ratio. Any other kind of ratio (i.e., voltage, current, or sound pressure) must first be turned into a power ratio by squaring and then converted into a power level in decibels.

Converting Voltage Ratios to Power Ratios
Many audio technicians are confused by the fact that doubling the voltage results in a 6 dB increase while doubling the power only results in a 3 dB increase. Fig. 2-1 demonstrates what happens if we simultaneously check both the voltage and power in a circuit where we double the voltage. Note that for a doubling of the voltage, the power increases four times.

10log(P₁/P₂) = 10log(40 W/10 W) = 6.02 dB

Figure 2-1. Voltage and power relationships in a circuit.

10log(E₁²/E₂²) = 20log(20 V/10 V) = 6.02 dB

The dBV
One of the most common errors when using the decibel is to regard it as a voltage ratio (i.e., so many decibels above or below a reference voltage). To compound the error, the result is referred to as a "level." The word "level" is reserved for power; an increase in the voltage magnitude is properly referred to as "amplification."

However, the decibel can be legitimately used with a voltage reference. The reference is 1.0 V. When voltage magnitudes are referenced to it logarithmically, they are called dBV (i.e., dB above or below 1.0 V). This use is legitimate because all such measurements are made open circuit and can easily be converted into power levels at any impedance interface.

The following definition is from the IEEE Standard Dictionary of Electrical and Electronics Terms, Second Edition:

244.62
Voltage Amplification (1) (general). An increase in signal voltage magnitude in transmission from one point to another or the process thereof. See also: amplifier. 210 (2) (transducer). The scalar ratio of the signal output voltage to the signal input voltage. Warning: By incorrect extension of the term decibel, this ratio is sometimes expressed in decibels by multiplying its common logarithm by 20. It may be currently expressed in decilogs. Note: If the input and/or output power consist of more than one component, such as multifrequency signal or noise, then the particular components used and their weighting must be specified. See also: Transducer.

239.210
Decilog (dg). A division of the logarithmic scale used for measuring the logarithm of the ratio of two values of any quantity. Note: Its value is such that the number of decilogs is equal to 10 times the logarithm to the base 10 of the ratio. One decilog therefore corresponds to a ratio of 10^0.1 (that is 1.25829+).