# Decibel dB: Formulas Equations & Calculations

### The decibel, dB is a logarithmic scale used for comparing two physical quantities especially in electronics. There are several easy to remember formulas that enable the values to be calculated

The decibel, dB utilises a logarithmic scale based to compare two quantities. It is a convenient way of comparing two physical quantities like electrical power, intensity, or even current, or voltage.

The decibel uses the base ten logarithms, i.e. those commonly used within mathematics. By using a logarithmic scale, the decibel is able to compare quantities that may have vast ratios between them.

The decibel, dB or deci-Bel is actually a tenth of a Bel - a unit that is seldom used.

The abbreviation for a decibel is dB - the capital "B" is used to denote the Bel as the fundamental unit.

## Decibel applications

The decibel, dB is widely used in many applications. It is used within a wide variety of measurements in the engineering and scientific areas, particularly within electronics, acoustics and also within control theory.

Typically the decibel, dB is used for defining amplifier gains, component losses (e.g. attenuators, feeders, mixers, etc), as well as a host of other measurements such as noise figure, signal to noise ratio, and many others.

In view of its logarithmic scale the decibel is able to conveniently represent very large ratios in terms of manageable numbers as well as providing the ability to carry out multiplication of ratios by simple addition and subtraction.

## Decibel formula for power comparisons

The most basic form for decibel calculations is a comparison of power levels. As might be expected it is ten times the logarithm of the output divided by the input. The factor ten is used because decibels rather than Bels are used.

The decibel formula or equation for power is given below:

${N}_{\mathrm{dB}}=10{\mathrm{log}}_{10}\left(\frac{{P}_{2}}{{P}_{1}}\right)$

Where:
Ndb is the ratio of the two power expressed in decibels, dB
P2 is the output power level
P1 is the input power level

If the value of P2 is greater than P1, then the result is given as a gain, and expressed as a positive value, e.g. +10dB. Where there is a loss, the decibel equation will return a negative value, e.g. -15dB. In this way a positive number of decibels implies a gain, and where there is a negative sign it implies a loss.

Use our decibel power calculator

## Decibel formulas for voltage & current

Although the decibel is used primarily as comparison of power levels, decibel current equations or decibel voltage equations may also be used provided that the impedance levels are the same. In this way the voltage or current ratio can be related to the power level ratio.

When using voltage measurements it is easy to make the transformation of the decibel formula because power = voltage squared upon the resistance:

${N}_{\mathrm{dB}}=10{\mathrm{log}}_{10}\left(\frac{{V}_{2}^{2}}{{V}_{1}^{2}}\right)$

And this can be expressed more simply as

${N}_{\mathrm{dB}}=20{\mathrm{log}}_{10}\left(\frac{{V}_{2}}{{V}_{1}}\right)$

Where:
Ndb is the ratio of the two power expressed in decibels, dB
V2 is the output voltage level
V1 is the input voltage level

It is possible to undertake a similar transformation for the formula to use current. Power = current squared upon the resistance, and therefore the decibel current equation becomes:

${N}_{\mathrm{dB}}=10{\mathrm{log}}_{10}\left(\frac{{I}_{2}^{2}}{{I}_{1}^{2}}\right)$

And this can be expressed more simply as

${N}_{\mathrm{dB}}=20{\mathrm{log}}_{10}\left(\frac{{I}_{2}}{{I}_{1}}\right)$

Where:
Ndb is the ratio of the two power expressed in decibels, dB
I2 is the output current level
I1 is the input current level

## Voltage & current decibel formulas for different impedances

As a decibel, dB is a comparison of two power or intensity levels, when current and voltage are used, the impedances for the measurements must be the same, otherwise this needs to be incorporated into the equations.

${N}_{d}=20{\mathrm{log}}_{10}\left(\frac{{V}_{2}}{{V}_{1}}\right)+10{\mathrm{log}}_{10}\left(\frac{{Z}_{1}}{{Z}_{2}}\right)$

Where:
Ndb is the ratio of the two power expressed in decibels, dB
V2 is the output voltage level
V1 is the input voltage level
Z2 is the output impedance
Z1 is the input impedance

In this way it is possible to calculate the power ratios in terms of decibels between signals on points that have different impedance levels using either voltage or current measurements. This could be very useful when measuring power levels on an amplifier that may have widely different impedance levels at the input and output. If the voltage or current readings are taken then this formula can be sued to provide the right power comparison in terms of decibels.

More Basic Concepts:
Voltage     Current     Resistance     Capacitance     Power     Transformers     RF noise     Decibel, dB     Q, quality factor