# 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.

The deciBel is widely used for measuring sound intensity or sound pressure level. For this the sound is referred to a pressure of 0.0002 microbars which equates to the standard for the threshold of hearing.

## How the deciBel arrived

Since the beginning of telecommunications there has been the need to measure the levels of relative signal strengths so that loss and gain can be seen.

Original telecommunications systems used the loss that occurred in a mile of standard cable at a frequency of 800Hz.

However this was not a particularly satisfactory method of determining loss levels, or relative signal strengths and as radio and other electronics based applications started to need to use some form of standard unit for comparison, the Bel was introduced in the 1920s. This gained its name from the Scot, Alexander Graham Bell who was originally credited with the invention of the telephone.

With this system, one Bel equalled a tenfold increase in signal level. Once it was introduced the Bel was found to be too large for most suers and so the deciBel was used instead. This is now the standard that has been adopted universally.

## 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 used to provide the right power comparison in terms of deciBels.

## DeciBel abbreviations

The deciBel is used in many areas from audio to radio frequency scenarios. In all of these it provides a very useful means of comparing two signals.

Accordingly there are many variations onto e deciBel abbreviation and it may not always be obvious what they mean. A table of deciBel abbreviations is given below:

DeciBel abbreviation Meaning / usage
dBA "A" weighted sound pressure or sound intensity measurement.
dBc Level of a signal with reference to the carrier being measured - normally used for giving the levels of spurious emissions and noise
dBd Gain of an antenna with reference to a half wave dipole in free space
dBFS Level with reference to full scale reading
dBi Gain of an antenna with reference to an isotropic source, i.e. one that radiations equally in all directions.
dBm Power level with reference to 1 mW
dBV Level with reference to 1 volt
dBµV Level with reference to 1 microvolt
dBW Power level with reference to 1 watt

The deciBel is widely used in many areas of electronics and sound measurement. It provides a very useful means of comparing different levels that may vary over a huge range. Being logarithmically based, the deciBel is able to accommodate variations of many orders of magnitude without getting lost in a huge number of zeros. In this way it is an ideal way of comparing different values.

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