What is the dB equation for a power ratio?
dB = 10 × log10(P2 / P1). Doubling the power is +3.01 dB, ten times the power is +10 dB and a hundred times is +20 dB.
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Decibel (dB) calculator
Power uses 10 log, voltage and current use 20 log, and this dB calculator handles both, along with unequal impedances. It converts ratios to decibels and back, adds the gains of stages in series and combines signal levels correctly.
Calculate
Examples
W
Input or reference power. Type 100m for 100 mW.
W
Output or measured power.
Power ratio in dB
3.0103 dB
- Power ratio P₂ / P₁
- 2
- P₂ as a percentage of P₁
- 200 %
- Voltage ratio (equal impedances)
- 1.4142
- In nepers
- 0.34657 Np
Positive dB is gain: the second power is 3.010 dB above the first.
How do you calculate decibels?
For power, dB = 10 × log10(P2 / P1); for voltage or current, dB = 20 × log10(V2 / V1). Doubling the power is +3.01 dB, ten times the power is +10 dB, and ten times the voltage is +20 dB. Negative values are losses. The 20 log form gives the power ratio only when both voltages are across the same impedance.
dB formulas for power, voltage, cascaded gains and added levels
A decibel is a tenth of a bel, the base-10 logarithm of a power ratio. Field quantities such as voltage are squared to get power, which is where the factor of 20 comes from (NIST SP 811 §8.7).
dB = 10 × log10(P₂ ÷ P₁) dB = 20 × log10(V₂ ÷ V₁)
- P₁, P₂
- reference (input) and output power, same units
- V₁, V₂
- reference and output voltage (or current), both RMS or both peak
- R₁, R₂
- impedances the two voltages appear across
When the voltages are across different impedances, the power ratio in dB is:
dB = 20 × log10(V₂ ÷ V₁) − 10 × log10(R₂ ÷ R₁)
Back from dB to a ratio, and the total of stages in series:
P₂ ÷ P₁ = 10^(dB ÷ 10) V₂ ÷ V₁ = 10^(dB ÷ 20) G = G₁ + G₂ + … + Gₙ (dB)
Separate signals add in the linear domain. For uncorrelated signals the powers add; for signals at the same frequency and in phase the voltages add:
L = 10 × log10(Σ 10^(Lᵢ ÷ 10)) L = 20 × log10(Σ 10^(Lᵢ ÷ 20))
How to calculate decibels: worked examples
Example 1: amplifier gain as power and as voltage
An amplifier takes 250 mW and delivers 5 W. The power ratio is 20, so the gain is 10 × log10(20) = 13.01 dB.
An op-amp stage turns 50 mV into 2.5 V: a voltage ratio of 50, or 20 × log10(50) = 33.98 dB of voltage gain. Its power gain is a different number, because the op-amp’s input and output impedances are not equal.
Example 2: total gain of a receiver chain
An LNA (+15 dB), a cable (−2.5 dB), a passive mixer (−7 dB) and an IF amplifier (+20 dB) in series total +25.5 dB, a power ratio of 354.8. A −90 dBm signal at the antenna leaves the IF amplifier at −64.5 dBm.
Example 3: adding two signal levels
Two uncorrelated signals at −60 dBm and −63 dBm add as powers: 1 nW + 501.2 pW = 1.501 nW, which is −58.24 dBm, only 1.76 dB above the larger one. If they were the same frequency and exactly in phase, the voltages would add instead and the total would be −55.35 dBm.
dB to ratio table
Power and voltage ratios for gains (positive dB) and losses (negative dB). Worth knowing by heart: 3 dB ≈ ×2 power, 6 dB ≈ ×2 voltage, 10 dB = ×10 power, 20 dB = ×10 voltage.
| dB | Power ratio (gain) | Power ratio (loss) | Voltage ratio (gain) | Voltage ratio (loss) |
|---|---|---|---|---|
| 0.1 dB | 1.023 | 0.9772 | 1.012 | 0.9886 |
| 0.5 dB | 1.122 | 0.8913 | 1.059 | 0.9441 |
| 1 dB | 1.259 | 0.7943 | 1.122 | 0.8913 |
| 2 dB | 1.585 | 0.631 | 1.259 | 0.7943 |
| 3 dB | 1.995 | 0.5012 | 1.413 | 0.7079 |
| 3.01 dB | 2 | 0.5 | 1.414 | 0.7071 |
| 6 dB | 3.981 | 0.2512 | 1.995 | 0.5012 |
| 6.021 dB | 4 | 0.25 | 2 | 0.5 |
| 10 dB | 10 | 0.1 | 3.162 | 0.3162 |
| 20 dB | 100 | 0.01 | 10 | 0.1 |
| 30 dB | 1000 | 0.001 | 31.62 | 0.03162 |
| 40 dB | 10000 | 0.0001 | 100 | 0.01 |
| 50 dB | 100000 | 0.00001 | 316.2 | 0.003162 |
| 60 dB | 1000000 | 0.000001 | 1000 | 0.001 |
Adding two levels: what to add to the higher one
When two signals differ by Δ dB, their sum is the higher level plus the correction below. The uncorrelated column applies to noise and to carriers at different frequencies; the in-phase column applies to the same signal arriving twice with no phase shift.
| Difference Δ | Uncorrelated (powers add) | In phase (voltages add) |
|---|---|---|
| 0 dB | +3.01 dB | +6.02 dB |
| 1 dB | +2.54 dB | +5.53 dB |
| 2 dB | +2.12 dB | +5.08 dB |
| 3 dB | +1.76 dB | +4.65 dB |
| 4 dB | +1.46 dB | +4.25 dB |
| 5 dB | +1.19 dB | +3.88 dB |
| 6 dB | +0.97 dB | +3.53 dB |
| 7 dB | +0.79 dB | +3.21 dB |
| 8 dB | +0.64 dB | +2.91 dB |
| 9 dB | +0.51 dB | +2.64 dB |
| 10 dB | +0.41 dB | +2.39 dB |
| 12 dB | +0.27 dB | +1.95 dB |
| 15 dB | +0.14 dB | +1.42 dB |
| 20 dB | +0.04 dB | +0.83 dB |
Where dB arithmetic goes wrong
The most common mistake is using 20 log on voltages measured across different impedances. The same 1 V across 50 Ω carries 10.79 dB more power than across 600 Ω, even though the voltage ratio is 0 dB. Op-amp voltage gain in dB is a useful number, but it is not a power gain. The second mistake is adding levels in dB: gains in series add, levels of separate signals combine in watts or volts. Finally, keep the precision honest. A change of 0.01 dB is 0.23 % in power, below what most measurements can resolve, so two decimal places are almost always enough.
Glossary
- Decibel (dB)
- One tenth of a bel: 10 lg of a power ratio, or 20 lg of a voltage, current or field ratio.
- Bel (B)
- The base-10 logarithm of a power ratio. Rarely used directly; the decibel is the practical unit.
- Neper (Np)
- The natural logarithm of a field ratio. 1 Np ≈ 8.686 dB.
- Power quantity
- A quantity proportional to power or energy (power, power density, noise power). Uses 10 lg.
- Field quantity
- A quantity whose square is proportional to power (voltage, current, field strength). Also called a root-power quantity. Uses 20 lg.
- Level
- A quantity in dB relative to a stated reference, such as dBm (1 mW), dBV (1 V) or dBµV (1 µV).
- Gain and attenuation
- Output relative to input in dB: positive is gain, negative is attenuation (loss).
Related guides
- dBm to watts converter: absolute power levels and voltages in 50 Ω.
Frequently asked questions
Why do voltage ratios use 20 log instead of 10 log?
Power is proportional to voltage squared (P = V² / R), and 10 log of a squared ratio is 20 log of the ratio. So ten times the voltage is +20 dB, the same as a hundred times the power. This holds only when both voltages are across the same impedance.
How do you calculate gain in dB?
Divide the output by the input, take log10 and multiply by 10 for power or 20 for voltage. An amplifier that turns 250 mW into 5 W has 13.01 dB of power gain.
How do I convert dB to linear and linear to dB?
For dB to linear, use 10^(dB/10) for a power ratio and 10^(dB/20) for a voltage ratio. For linear to dB, use 10 × log10 of a power ratio or 20 × log10 of a voltage ratio, so ×100 in power and ×10 in voltage are both +20 dB.
Is 3 dB exactly double?
No. 3 dB is a power ratio of 1.995; exactly double is 3.0103 dB. Likewise 6 dB is ×3.981 in power and ×1.995 in voltage, close to but not exactly 4 and 2.
How do you add decibels?
Gains and losses of stages in series add directly in dB. Levels of separate signals do not: two uncorrelated 30 dBm signals make 33.01 dBm (1 W + 1 W = 2 W), not 60 dBm. Two equal signals exactly in phase make 36.02 dBm.
What is the difference between dB and dBm?
dB is a ratio between two quantities. dBm is a level: a power in dB relative to 1 mW, so 0 dBm is 1 mW and 30 dBm is 1 W. Adding a gain in dB to a level in dBm gives a level in dBm.
How do I convert dB to a percentage?
For power, percent = 100 × 10^(dB/10); for voltage, percent = 100 × 10^(dB/20). A −3 dB point passes 50.1 % of the power and 70.8 % of the voltage.
What does a negative dB value mean?
A loss, or attenuation: the output is smaller than the input. −10 dB leaves 10 % of the power and −20 dB leaves 10 % of the voltage.
How many decibels is one neper?
One neper is 8.68589 dB, and 1 dB is 0.115129 Np. The neper uses the natural logarithm of a field ratio, the decibel the base-10 logarithm; transmission-line attenuation constants are often given in nepers per meter.
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Sources
- NIST SP 811 (2008), Guide for the Use of the SI, §8.7: Logarithmic quantities and units: level, neper, bel
- Rohde & Schwarz Application Note 1MA98, dB or not dB? (decibel arithmetic, adding levels, Table 6-2)
- Keysight AN 1449-1, Fundamentals of RF and Microwave Power Measurements, Part 1 (dB and dBm definitions)
- NIST SP 330 (2019), The International System of Units: Table 8, units accepted for use with the SI (neper, bel, decibel)
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