Last time we were noting how the humble dB has been used and abused, and how it has spread like a computer virus throughout engineering circles. It’s now used to measure so many things (other than audio power) in so many ways (often incorrectly) that much of its meaning can be lost.
Let me get back to basics before I confuse anyone (including self) any further. It is possible to make sense of this bedlam, by rigidly enforcing two simple rules:
(1) The decibel is always used to express a ratio of power levels. Quantities that are not powers must be made proportional to power. Since power is proportional to the square of voltage for a fixed impedance, output and input voltages may be squared before calculating their ratio. Most people just remember to multiply the log of the linear quantities (e.g. Volts) by 20 instead of 10, which can accomplish the same thing in fewer steps, but you must remember this only works if the impedance remains the same throughout. If the impedance changes, you must work out the output and input power some other way (like figuring out the resistance at each point and calculating the powers from there), before logging and multiplying by 10.
(2) The units used to measure the input and output power levels must be the same, so they will cancel out. Example measurements could be in watts, volts, cubits, or pints of beer. As a result decibels themselves have no dimension as such, and so technically speaking they are not units. The dB (without any suffix attached) is used to measure power ratios, so it can measure gain or loss, but there is no reference to either the input or output level, just the ratio. The statement “I set the level to 0 dB” is meaningless.
Make these two simple rules into your dB religion, and you’ll find you won’t have nearly as much trouble working with decibels.
An additional note concerning dB’s and audio: because modern audio systems don’t worry much about impedance matching (all sources are very low impedance, and all inputs are very high impedance, so “open-circuit” conditions prevail), there has been a transition from using dB with power references (“dBm”) towards voltage references (“dBu”). A few months ago in this very space, I was referring to older transmission practices in broadcast, and used “dBu.” Laverne Siemens of Golden West quite properly pounced, and reminded me that in the old days nobody used the expression “dBu.” Transformer coupled audio equipment would be specified using dBm, since impedance was still vitally important then. Today we use dBu (0 dBu = 0.775 V, the voltage across a 600 ohm load at 0 dBm) extensively, and sometimes carelessly, and pretend that impedance doesn’t matter. And it doesn’t, really, as long as it doesn’t change.
But try always to remember the power origins of the decibel, and you’ll avoid a lot of confusion, and have guaranteed happy karma.
dB = 10 log(PO/PI)
= 10 log(VO/VI)2* = 20 log(VO/VI)*
*but only if the impedance remains the same!