ditional tips on making decibels intuitive.) While decibels are defined as a power ratio, it is common to convert this formula to handle voltage. For RMS voltage,
A similar formula can be derived for current:
In constant impedance (resistance) systems, the ratio R1/R2 equals one and drops out of the equation, leaving the familiar decibel equation for voltage:
Because so many of the signals in electrical engineering are measured in voltage, it is handy to be able to apply decibels to voltage values, instead of power.
Use with caution
Ignoring that R1/R2 term can lead to serious confusion, so caution is in order. As an example, let’s consider Figure 1, which shows an amplifier with a high input impedance (R1, 100 kΩ) and a low output impedance driving a load impedance of 100 Ω (R2). The input voltage is 0.1 VRMS and the output voltage is 1 VRMS, for a voltage gain of 10. In decibels, the voltage gain is 20log(10) = 20 dB.
Now let’s look at the power situation. The input power is (0.1)²/100 k = 0.1 µW, while the power delivered to the load is (1.0)²/100 = 10 mW. Using the power values to calculate the gain, we get 10log(10 mW / 0.1 µW) = 50 dB. That causes a different amplifier gain in dB depending on whether we use voltage or power. The decibel formula for voltage ratios was derived from the power formula, so we would expect them to be consistent.
Because we ignored the differences in impedance, we effectively assumed that the impedances are always the same. That’s not true in this case. Going back to the R1/R2 term in the voltage equation, we can calculate the missing factor for the impedance change in Fig. 1. That is, 10log(R1/R2) = 10log(100k/100) = 30 dB, which explains the difference between the two gain numbers.
Many systems have consistent impedances. For example, many RF systems are built consistently around a 50 Ω system impedance, with all input, output and transmission line impedances the same. In such a scenario, the power and voltage decibel equations can be used interchangeably, and the decibel numbers track.
In other systems, the impedances vary. You can compensate for this with the R1/R2 correction factor, but more commonly engineers just choose to work exclusively with either voltage or power to avoid any problems. Now, a purist could argue that decibels are defined strictly in terms of power (that’s true) and that all decibel calculations must be done with that in mind. I might agree except that I’ve seen engineers work strictly in terms of voltage and 20log(V2/V1), without a problem. They get the benefits of working in decibels without having to adjust for impedance, but they have to be careful not to mix in power calculations. It is a good example of how engineers use mathematically-impure methods that have practical value.
We’ve just touched on a few decibel concepts, emphasizing the potential problem with dB calculations using voltage. Back in 2008, EDN published a series on decibels that was an excerpt from Sound System Engineering, by Don Davis and Eugene Patronis (see Refs. 6 through 8).
Next time, we’ll take a look at how decibels are used to refer to absolute values. For example, a common power reference is 1 mW, resulting in dBm (decibels relative to 1 mW).