Decibels formulas and equations
Decibel Tutorial: Formulas & Equations
The decibel is used within the electronics and associated industries to provide a method of indicating the ratio of a physical quantity - often electrical power, intensity, current, or voltage.
The decibel uses the base ten logarithms, i.e. those commonly used within mathematics.
As it can be seen from the name, a 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 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 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 he 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.
The decibel formula or equation for power is given below:
Where:
Ndb is the ratio of the two power expressed in decibels
P2 is the output power level
P1 is the input power level
Ndb is the ratio of the two power expressed in decibels
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.
Decibel equations for voltage and 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.
In the first instance for voltage because power = voltage squared upon the resistance:
Where:
Ndb is the ratio of the two power expressed in decibels
V2 is the output voltage level
V1 is the input voltage level
Ndb is the ratio of the two power expressed in decibels
V2 is the output voltage level
V1 is the input voltage level
Similarly because power = current squared upon the resistance, the decibel current equation becomes:
Where:
Ndb is the ratio of the two power expressed in decibels
I2 is the output current level
I1 is the input current level
Ndb is the ratio of the two power expressed in decibels
I2 is the output current level
I1 is the input current level
Voltage & current decibel equations for different impedances
As a decibel 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.
Where:
Ndb is the ratio of the two power expressed in decibels
V2 is the output voltage level
V1 is the input voltage level
Z2 is the output impedance
Z1 is the input impedance
Ndb is the ratio of the two power expressed in decibels
V2 is the output voltage level
V1 is the input voltage level
Z2 is the output impedance
Z1 is the input impedance
Table of decibel values
The decibel is widely used in radio and electronics design calculations.
When undertaking electronics or RF design and development it is often necessary to compare the value of two parameters.
It is then possible to alter the electronics design to match the requirements.
As values in electronics circuits may vary quite widely, it is often convenient to use a logarithmic scale such as the decibel (dB).
| DECIBELS (DB) | POWER RATIO | CURRENT OR VOLTAGE RATIO |
|---|---|---|
| 0.1 | 1.023 | 1.012 |
| 0.2 | 1.047 | 1.023 |
| 0.3 | 1.072 | 1.035 |
| 0.4 | 1.096 | 1.047 |
| 0.5 | 1.122 | 1.059 |
| 0.6 | 1.148 | 1.072 |
| 0.7 | 1.175 | 1.084 |
| 0.8 | 1.202 | 1.096 |
| 0.9 | 1.230 | 1.109 |
| 1.0 | 1.259 | 1.122 |
| 2.0 | 1.585 | 1.259 |
| 3.0 | 1.995 | 1.413 |
| 4.0 | 2.512 | 1.585 |
| 5.0 | 3.162 | 1.778 |
| 6.0 | 3.981 | 1.995 |
| 7.0 | 5.012 | 2.239 |
| 8.0 | 6.310 | 2.512 |
| 9.0 | 7.943 | 2.818 |
| 10 | 10.0 | 3.162 |
| 20 | 10^2 | 10.0 |
| 30 | 10^3 | 31.62 |
| 40 | 10^4 | 100.0 |
| 50 | 10^5 | 316.2 |
| 60 | 10^6 | 1000.0 |
| 70 | 10^7 | 3162.3 |
| 80 | 10^8 | 10000 |
| 90 | 10^9 | 31623 |
| 100 | 10^10 | 100000 |
dBm - dBw Watts Conversion Table
The decibel is widely used in radio and electronics design calculations in particular it is used in many RF or radio frequency associated specifications and details.
When measuring radio frequency, or RF power, it is often easier to have a measurement made in a way that it is easy to compare the two levels.
As a result many power levels are specified in dBm or dBW, and much RF test equipment including power meters, spectrum analysers, signal generators and the like have calibrations in dBm or dBW. Often RF components such as mixers, oscillators and the like, as well as the interfaces between modules in RF equipment have their levels specified in dBm or dBW. Radio transmitters may also have their output levels expressed in this way.
What are dBm and dBW?
In itself a decibel is not an absolute level. It is purely a comparison between two levels, and on its own it cannot be used to measure an absolute level. As a result of this the quantities of dBm and dBW are used:
- dBm - This is a power expressed in decibels relative to one milliwatt.
- dBW - This is a power expressed in decibels relative to one watt.
From this it can be seen that a level of 10 dBm is ten dB above one milliwatt, i.e. 10 mW. Similarly a power level of 20 dBW is 100 times that of one watt, i.e. 100 watts.
A more extensive conversion table of dBm, dBW and power is given below:
| DBM | DBW | WATTS | TERMINOLOGY |
|---|---|---|---|
| +100 | +70 | 10 000 000 | 10 Megawatts |
| +90 | +60 | 1 000 000 | 1 Megawatt |
| +80 | +50 | 100 000 | 100 kilowatts |
| +70 | +40 | 10 000 | 10 kilowatts |
| +60 | +30 | 1 000 | 1 kilowatt |
| +50 | +20 | 100 | 100 watts |
| +40 | +10 | 10 | 10 watts |
| +30 | 0 | 1 | 1 watt |
| +20 | -10 | 0.1 | 100 milliwatts |
| +10 | -20 | 0.01 | 10 milliwatts |
| 0 | -30 | 0.001 | 1 milliwatt |
| -10 | -40 | 0.0001 | 100 microwatts |
| -20 | -50 | 0.00001 | 10 microwatts |
| -30 | -60 | 0.000001 | 1 microwatt |
| -40 | -70 | 0.0000001 | 100 nanowatts |
| -50 | -80 | 0.00000001 | 10 nanowatts |
| -60 | -90 | 0.000000001 | 1 nanowatt |
The use of the values dBm and dBW is widespread. They are found as direct calibration scales on many items of RF test equipment often being used in preference to the more elementary basic units of watts or milliwatts. Items of RF test equipment including power meters, signal generators and RF spectrum analyzers in particular use these units. Accordingly to be able to understand the RF test equipment specifications it is necessary to have an understanding of dBm and dBW. Also many RF components are also specified in terms of dBm or dBW.
dBm & Voltage Conversion Table
When working with RF power, it is often useful to know the voltage level for a given power.
The table below provides a chart to convert between dBm, watts and voltage - peak to peak in a 50Ω system.
Although voltage levels are unlikely to rise to significant levels which might cause damage for power levels measured in dBm, the voltages are often used in other calculations.
Three tables have been included. These have been chosen because the voltages move from readings measured in millivolts to those in volts. Also as the milliwatts change to watts, the change in the table is made.
dBm / millivolts / milliWatts conversion table
This conversion table charts the values for dBm against milliwatts and the relevant voltage expressed in millivolts.
It is applicable to many lower power applications.
| DBM | MILLIWATTS | VOLTAGE MILLIVOLTS (P-P) | VOLTAGE MILLIVOLTS (RMS) |
|---|---|---|---|
| -30 | 0.0010 | 20 | 7.1 |
| -28 | 0.0016 | 25.2 | 8.9 |
| -26 | 0.0025 | 31.7 | 11.2 |
| -24 | 0.0040 | 40.0 | 14.2 |
| -22 | 0.0063 | 50.2 | 17.8 |
| -20 | 0.010 | 63.2 | 22.4 |
| -18 | 0.016 | 79.6 | 28.2 |
| -16 | 0.025 | 100 | 35.5 |
| -14 | 0.040 | 126 | 44.7 |
| -12 | 0.063 | 159 | 56.4 |
| -10 | 0.100 | 200 | 71.0 |
| -8 | 0.16 | 252 | 89.4 |
| -6 | 0.25 | 317 | 112 |
| -4 | 0.40 | 399 | 142 |
| -2 | 0.63 | 502 | 178 |
| 0 | 1.00 | 632 | 224 |
| 2 | 1.58 | 796 | 282 |
| 4 | 2.51 | 4000 | 1420 |
dBm - milliwatts - Volts conversion table:-
This conversion table charts the values for dBm against milliwatts and the relevant voltage expressed in volts.
It is applicable to many medium power applications.
| DBM | MILLIWATTS | VOLTAGE VOLTS (P-P) | VOLTAGE VOLTS (RMS) |
|---|---|---|---|
| 0 | 1.00 | 0.632 | 0.224 |
| 2 | 1.58 | 0.796 | 282 |
| 4 | 2.51 | 4.00 | 1.42 |
| 6 | 3.98 | 1.26 | 0.45 |
| 8 | 6.31 | 1.59 | 0.56 |
| 10 | 10 | 2.00 | 0.71 |
| 12 | 15.8 | 2.52 | 0.89 |
| 14 | 25.1 | 3.17 | 1.12 |
| 16 | 39.8 | 3.99 | 1.41 |
| 18 | 63.1 | 5.02 | 1.78 |
| 20 | 100 | 6.32 | 2.24 |
| 22 | 158 | 7.95 | 2.82 |
| 24 | 25.1 | 10.0 | 3.55 |
| 26 | 398 | 12.6 | 4.48 |
| 28 | 631 | 15.9 | 5.64 |
| 30 | 1000 | 20.0 | 7.10 |
| 32 | 1585 | 25.2 | 8.94 |
| 34 | 2510 | 31.7 | 11.2 |
dBm - Watts - Volts conversion table
This conversion table charts the values for dBm against milliwatts and the relevant voltage expressed in volts.
It is applicable to many high power applications.
| DBM | WATTS | VOLTAGE MV (P-P) | VOLTAGE MV (RMS) |
|---|---|---|---|
| 30 | 1.00 | 20 | 7.10 |
| 32 | 1.58 | 25.2 | 8.94 |
| 34 | 2.51 | 31.7 | 11.3 |
| 36 | 3.98 | 40.0 | 14.1 |
| 38 | 6.31 | 50.2 | 17.8 |
| 40 | 10.0 | 63.2 | 22.4 |
| 42 | 15.9 | 79.6 | 28.2 |
| 44 | 25.1 | 100 | 35.5 |
| 46 | 39.8 | 126 | 44.7 |
| 48 | 63.1 | 159 | 56.4 |
| 50 | 100 | 200 | 71.0 |
| 52 | 159 | 252 | 89.4 |
| 54 | 251 | 317 | 112 |
| 56 | 398 | 399 | 142 |
| 58 | 631 | 502 | 178 |
| 60 | 1000 | 632 | 224 |
| 62 | 1585 | 796 | 282 |
Decibel Calculator
The calculation of a power ratio in decibels requires the use of logarithms.
Our simple to use decibel calculator provides the calculation on your computer, be it a desktop, laptop or mobile device.
Simply enter the values for the input and output levels into the decibel calculator, press calculate, and the calculated answer will be provided.
Decibel calculator for power levels
Decibel Calculator for Power Levels
Nepers & Neper to dB Conversion Table
When comparing power levels it is normal to use the decibel. However on some occasions nepers may be used.
Accordingly it is sometimes necessary to convert nepers to dB.
Nepers differ from decibels in that they use logarithms to the base "e" rather than to the base 20.
Accordingly it is sometimes necessary to convert between Nepers and decibels or vice versa.
What are Nepers
The neper is a logarithmic unit for ratios of measurements. It is very similar to more familiar decibel and can be used for the comparison of physical quantities such as gain or loss in electronic circuits, or other physical quantities.
The neper has the symbol Np, and it derives its name from John Napier, the inventor of logarithms.
In the same way that the decibel is not a unit that has been incorporated into the SI International System of units, it is accepted for use alongside it.
While the decibel and the bel use the decadic or base-10 logarithm to compute ratios, the neper uses the natural logarithm, based on Euler's number, e. This equal to 2.71828 . . . .
The equation for calculating nepers is given by:
Neper to dB conversion
The neper and dB are related by the following relationships:
Using these equations it is easy to convert from nepers to dB and dB to nepers. It should be remembered that the figures for the conversion are not exact, but the number of significant figures given should be sufficient for most engineering applications.
Decibel, dB to Neper conversion table
The table below gives some of the more popular conversion points for nepers to dB and vice versa.
| DECIBEL, DB TO NEPER CONVERSION | ||
|---|---|---|
| DECIBELS, DB | NEPERS | POWER RATIO |
0.1
|
0.01
|
1.023
|
0.2
|
0.02
|
1.047
|
0.3
|
0.03
|
1.071
|
0.4
|
0.05
|
1.096
|
0.5
|
0.6
|
1.122
|
0.6
|
0.07
|
1.148
|
0.7
|
0.08
|
1.175
|
0.8
|
0.09
|
1.202
|
0.9
|
0.10
|
1.230
|
1.0
|
0.12
|
1.259
|
2.0
|
0.23
|
1.585
|
3.0
|
0.35
|
1.995
|
4.0
|
0.46
|
2.512
|
5.0
|
0.58
|
3.162
|
6.0
|
0.69
|
3.981
|
7.0
|
0.81
|
5.012
|
8.0
|
0.92
|
6.310
|
9.0
|
1.04
|
7.943
|
10
|
1.15
|
10.000
|
15
|
1.73
|
31.62
|
20
|
2.30
|
100.00
|
30
|
3.45
|
1000.0
|
40
|
4.60
|
10000
|
50
|
5.76
|
100 000
|
While nepers are not nearly as widely used as dB, they nevertheless occur in some applications because they use natural logarithms rather than use a base ten which is a far more arbitrary figure, but convenient for us as we use base ten figures..
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