lesson
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Human ears and electronic sensors can detect signals ranging from a few microvolts up to hundreds of volts. If we plotted these vast ranges on a regular linear scale, small changes would completely vanish.
To solve this, engineers use the decibel (dB), a logarithmic unit that compresses massive voltage ratios into clean, manageable numbers that can simply be added together.
Where did this unit come from, and how do we convert simple voltage measurements into decibels?
The Decibel and Voltage Ratios
In the 1920s, Bell Telephone Laboratories developed the Bel (named after Alexander Graham Bell) to quantify audio signal loss across telephone lines. One decibel (dB) is one-tenth of a Bel.
Decibels were originally defined for power gain as GaindB=10log10(PinPout). Because electric power across a fixed resistance is proportional to voltage squared (P=RV2), substituting voltage gives GaindB=10log10((VinVout)2).
Using the logarithmic power rule log(x2)=2log(x), we bring the exponent of 2 to the front, giving our primary formula for voltage gain in decibels:
Av(dB)=20log10(VinVout)
📊Interactive diagram
What happens to the sign of the decibel value when a filter reduces a signal compared to when an amplifier boosts it?
Gain, Unity, and Attenuation
When Vout=Vin, the ratio is 1. Since log10(1)=0, a ratio of 1 corresponds exactly to 0 dB (called unity gain).
If an active circuit amplifies the signal (Vout>Vin), the ratio is greater than 1, yielding a positive dB value called gain. If a passive filter weakens the signal (Vout<Vin), the ratio is less than 1, yielding a negative dB value called attenuation.