dB Gain Calculator

What the dB gain calculator does

Gain is how much a system boosts a signal, and it is usually expressed in decibels. This calculator finds the decibel gain from two power levels or two voltage levels, an initial value and a final value, doing the logarithm for you.

Below is what dB gain is, the equations behind it, why power and voltage use different factors, and a worked example.

How to use it

1. Enter the initial and final power, or the initial and final voltage.
2. Choose the units for each value.
3. Press Calculate for the gain in decibels, or Reset to clear it.

What dB gain is

Gain describes how much a circuit or system increases the strength of a signal, comparing the output to the input. An amplifier with a large gain takes a weak signal and makes it much stronger, while a gain of one leaves it unchanged. Gain is almost always expressed in decibels, a logarithmic unit, because the signal levels involved can span an enormous range and because decibels make a chain of amplifiers easy to work with. A positive decibel gain means the signal got stronger, and a negative value means it got weaker, which is really a loss.

The decibel is a way of expressing a ratio, in this case the ratio of the output to the input. Rather than saying an amplifier multiplies the power by a hundred, you can say it has a gain of twenty decibels, which is often more convenient and is how amplifiers, antennas, and signal paths are routinely specified. This calculator takes the two signal levels, the starting one and the resulting one, and works out the gain in decibels between them, handling both power and voltage and doing the logarithm automatically.

The equations it uses

For a gain measured from two power levels, the decibel value is ten times the logarithm of their ratio:

gain (dB) = 10 × log₁₀(P₂ ÷ P₁)

For a gain measured from two voltage levels, the factor is twenty:

gain (dB) = 20 × log₁₀(V₂ ÷ V₁)

Here P₁ and V₁ are the initial values and P₂ and V₂ are the final ones, so the ratio is output over input. A ratio greater than one gives a positive gain, and a ratio less than one gives a negative value, a loss. The calculator takes whichever pair you enter and applies the right factor, so you never compute a logarithm by hand.

Why power and voltage use different factors

The factor is ten for power but twenty for voltage, and the reason is the relationship between the two. Power is proportional to the square of the voltage, so doubling a voltage quadruples the power. When you express a voltage ratio in decibels, you are really describing the power change that accompanies it, and squaring the voltage ratio inside the logarithm is equivalent to doubling the factor in front, turning ten into twenty.

This means the two formulas agree rather than conflict: a given change in signal works out to the same number of decibels whether you measure it as a power ratio or a voltage ratio, as long as you use the matching factor. The factor of twenty for voltage simply accounts for the squaring that links voltage to power. Using the wrong factor would give an answer off by a factor of two, so it matters which quantity you are working with. The calculator keeps this straight by asking whether you are entering powers or voltages and applying the correct factor for each.

Gain, loss, and adding stages

One of the great conveniences of expressing gain in decibels is how it behaves when signals pass through several stages. Because decibels are logarithmic, the gains of successive stages simply add up instead of multiplying. An amplifier of twenty decibels followed by another of ten decibels gives a total of thirty decibels, found by addition, where the underlying power ratios would have to be multiplied. This makes designing and analysing signal chains far easier.

The same additive property handles losses naturally, since a loss is just a negative gain. A signal that gains thirty decibels in an amplifier and then loses ten decibels in a cable ends up twenty decibels above where it started, again by simple addition. This is why engineers track signal levels through a system in decibels, adding the gains and subtracting the losses stage by stage to find the overall result, a process called a link or gain budget. This calculator gives the gain of a single stage, the building block from which those budgets are assembled.

Units and precision

The calculator takes the initial and final power in watts and its multiples, or the initial and final voltage in volts and its multiples, and returns the gain in decibels. It applies a factor of ten for power and twenty for voltage, with the base-ten logarithm. A final value larger than the initial gives a positive gain, while a smaller one gives a negative value, a loss. The conversion is exact.

A worked example

Suppose an amplifier raises a signal's power from 1 watt to 100 watts.

The gain is 10 × log₁₀(100 ÷ 1) = 10 × 2 = 20 decibels, since the logarithm of 100 is 2. Measured as a voltage change instead, raising a signal from 1 volt to 10 volts gives 20 × log₁₀(10 ÷ 1) = 20 decibels, the same gain, because a tenfold voltage increase corresponds to a hundredfold power increase.

Questions people ask

How do you calculate gain in decibels from power?

Use gain = 10 × log₁₀(P₂ ÷ P₁), the ratio of output power to input power. Take the logarithm and multiply by ten.

How do you calculate gain in decibels from voltage?

Use gain = 20 × log₁₀(V₂ ÷ V₁). The factor is twenty for voltage because power is proportional to voltage squared.

What does a negative gain mean?

A loss. The signal came out weaker than it went in, so the ratio is less than one and the decibel value is negative.

Why do decibel gains add up?

Because decibels are logarithmic, and adding logarithms corresponds to multiplying the underlying ratios. So gains of successive stages are simply summed.

References

A quick note on where this comes from. The decibel and its use for gain are standard across engineering, described by NIST and in Georgia State University's HyperPhysics. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. HyperPhysics, Decibels. http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/db.html
2. National Institute of Standards and Technology (NIST), SP 811, Guide for the Use of the International System of Units. https://www.nist.gov/pml/special-publication-811
3. Wikipedia, Decibel. https://en.wikipedia.org/wiki/Decibel