dB From Voltages

Although dB is technically a ratio of power levels, it is commonly used with voltages by assuming that both voltages are driving a constant resistive load, such that:

Power = Voltage^2 / Resistance

So if we have 2 different voltages V1 and V0 and resistance R, the R drops out:

 
dB = 10 * log((V1^2 / R) / (V0^2 / R))

   = 10 * log(V1^2 / V0^2)

   = 10 * log(V1 / V0)^2
 

Then using the property that the square of a log is equal to twice the log:

 
dB = 10 * 2 * log(V1 / V0)

   = 20 * log(V1 / V0)
 

We can rearrange the dB formula to get the voltage ratio from a specified dB value:

 
V1 / V0 = 10^(dB / 20)
 

This assumption of constant resistance may not always be true, such as when the voltage is driving a complex load like a loudspeaker whose impedance changes with frequency. But it is still proper, for example, for a manufacturer to assume constant resistance when discussing sound card amplifier performance, since there are no standards for loudspeaker impedance versus frequency... and there are a lot of different loudspeakers on the market.

See also dB, Typical dB Applications, Working With dB, Absolute dB (SPL, etc.)