Gain in Decibels
Linear ↔ dB conversion
Datasheets, filters and cascaded stages usually speak dB; resistor ratios give linear gain. Convert between the two in either direction.
Converted gain
How to Use This Calculator
Amplifier gain can be written two ways: as a linear ratio (×10, ×0.5) or in decibels (+20 dB, −6 dB). Datasheets, filter specifications and cascaded system budgets usually use dB, while resistor ratios from the gain calculators give linear values — so converting between them is everyday work. For voltage (or current) gains the relation is G[dB] = 20·log10|Av| and its inverse |Av| = 10^(G/20). Pick a direction with the toggle, enter the gain — a positive magnitude in linear mode, any real number (positive or negative) in dB mode — and read the converted value. The result panel also reminds you which way each common landmark points: +6.02 dB doubles an amplitude, +20 dB multiplies it by 10, −20 dB divides by 10.
Input fields
| Mode | Field | Unit | What to enter |
|---|---|---|---|
| Linear → dB | Voltage gain |Av| | × | A non-zero magnitude. Enter the absolute value of an inverting stage's gain — dB describes size, and the 180° phase is a separate fact. |
| dB → Linear | Gain in decibels | dB | Any real number. Negative values are attenuation (−20 dB → ×0.1). |
Step by step
- Choose the direction that matches where your number came from: resistor ratios and the other calculators here give linear gain (use Linear → dB); datasheet specs like "gain of +40 dB" or filter attenuation "−3 dB" come in dB (use dB → Linear).
- Enter the value. With the defaults, ×10 converts to +20.0 dB — the most common landmark: one decade of amplitude is 20 dB.
- Use the result in your budget: cascading a ×10 stage (+20 dB) with a ×4 stage (+12.04 dB) gives +32.04 dB total, which converts back to ×126 — multiplication becomes addition, which is the whole point of the unit.
Two conventions worth internalizing. First, 0 dB means ×1 (unity), not "no gain exists" — a voltage follower sits at exactly 0 dB. Second, dB values are relative: "gain in dB" implicitly means "relative to the input amplitude", which is why the formula needs no reference resistance — that only enters for power quantities (see FAQ).
What It Means in a Real Circuit
The decibel exists because real circuits are chains. An audio path might run through a microphone preamp (+40 dB), a tone control (−6 dB at the top end), a line driver (+10 dB) and a power stage (+30 dB); in linear terms that is 100 × 0.5 × 3.16 × 31.6 ≈ 5,000, but in dB it is simply 40 − 6 + 10 + 30 = +74 dB. Engineers budget systems this way because additions are easy to check and errors are easy to spot, and because component specifications — op-amp open-loop gain (≈100 dB for a 741 at low frequency), filter passband ripple (±0.5 dB), amplifier noise figure — all arrive in the same units. When you design with the calculators in this suite, the workflow is: set gains with resistors (linear), check each stage's contribution in dB, sum them for the system, and convert back to linear only when a specific voltage or current number is needed.
The factor of 20 (instead of 10) has a physical reason that shows up everywhere. Power scales with the square of voltage (P = V²/R), so a power ratio in dB is 10·log10(P2/P1); substituting P ∝ V² gives 10·log10((V2/V1)²) = 20·log10(V2/V1). That is why voltage and current gains use 20 while power gains use 10, and why a "+3 dB" statement means different things for the two: +3.01 dB doubles power but only raises amplitude by √2 ≈ 1.41×. This is also why "−3 dB" marks the bandwidth edge in the Bandwidth Analysis calculator — it is the frequency where the power has halved, i.e. the amplitude has fallen to 0.707 of its passband value.
Negative gains in dB are ordinary and useful: a passive divider at ×0.1 is −20 dB, an attenuator pad might specify −15 dB, and filter stopbands are described by how many dB below the passband they reach (a 3rd-order RC low-pass falls off at 60 dB/decade = 3 poles × 20 dB). In op-amp work you will also meet dB in less obvious places: CMRR of a difference amplifier is expressed in dB (the Difference Amplifier calculator reports it), open-loop gain on a Bode plot is in dB, and "gain of 26 dB" on a datasheet row converts to ×20 — handy to know before you wire Rf/Rg. One caution: dB never carries a sign of phase or polarity. A +20 dB inverting stage and a +20 dB non-inverting stage are identical in this unit; the 180° difference lives outside it, which is why this calculator always works with magnitudes.
Frequently Asked Questions
The decibel was defined for power ratios: dB = 10·log10(P2/P1). Since electrical power scales with the square of voltage (P = V²/R), a voltage ratio becomes 10·log10((V2/V1)²) = 20·log10(V2/V1). Same unit, different quantity — which is why "+3 dB" doubles power but only multiplies amplitude by √2.
Attenuation: the output is smaller than the input. −20 dB means ×0.1, −6 dB means ×0.5 (half amplitude). Attenuators, dividers and filter stopbands are all specified this way; nothing is "broken" about a negative gain — it just describes shrinkage on the same logarithmic scale.
Strictly, the decibel is a dimensionless ratio — it compares two quantities of the same kind. In practice engineers write "dB" as if it were a unit and anchor it to references: dBm (relative to 1 mW), dBV (relative to 1 V). A bare "+20 dB gain" means "20 dB relative to the input", which is exactly what this calculator converts.
+20 dB — enter the magnitude 10. The minus sign in "−10×" is a phase statement (180° inversion), not a size statement; decibels measure only how much bigger or smaller the output amplitude is. If you need to record both facts, write "+20 dB, inverted".
Three reasons: cascades multiply, and multiplication becomes addition; human hearing and most sensor ranges span decades, which a log scale compresses naturally; and every specification in the chain (gain, loss, ripple, noise figure, CMRR) arrives in the same unit, so budgets can be summed directly.