Decibel in all its appearances:


There is often a lot of confusion about how to calculate with different types of dB expressions like dBm, dBv etc. This chapter tells us the most relevant information about the decibel in all her appearances. Most of the expressions are related to electronics, witch this site is specially written for.

dB at a glance:
History of Abraham Graham Bell:
The bel:
The decibel:
Calculating with dB's:
dB definitions:
The Neper:

History of Abraham Graham Bell:
In 1876 the telephone was invented by Scottish-born speech therapist and inventor Alexander Graham Bell (1847-1922). His device used a thin diaphragm to convert the vibrations of the human voice into electrical signals, then reconverted them into sound waves. Bell experimented on this system while developing his father's work on a method of teaching deaf people to speak, in which symbols represented the position of the lips and tongue. In March of 1876, the first intelligible words transmitted by telephone were: 'Mr Watson, come here, I want to see you!' spoken by Bell to his assistant, Thomas Watson.

Alexander Graham Bell

But he did more than that:  He transmitted sound over wires.  This event is much more significant to modern recording than the transmission of sound vibrations to a cylinder with a recording horn (as accomplished by Edison). Mr. Bell did much more.  Through his efforts to market his invention, Bell caused a research laboratory (Bell Telephone Laboratories) to become established.  This laboratory team has come up with almost every significant development in sound recording (and electronics in general) used today.  This is why I call Mr. Bell the "Grandfather" of modern recording and this is why I have named my new online advanced recording magazine by the name of Alexander. 

Without thinking a lot about it, every audio manufacturer and every recording engineer acknowledges Bell's contributions to recording.  It was Bell Labs that came up with the unit that is used to measure and compare sound recording levels, the decibel.  The "bel" in "decibel" is a shortening of the name of this famous inventor.  Every manufacture uses the decibel (or "dB") in the design and specification of performance; every recording engineer uses controls and meters calibrated in decibels.

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The Bel:
As a unit, the bel was actually devised as a convenient way to represent power loss in telephone system wiring rather than gain in amplifiers. The unit's name is derived from Alexander Graham Bell, the famous American inventor whose work was instrumental in developing telephone systems. Originally, the bel represented the amount of signal power loss due to resistance over a standard length of electrical cable. In formula:

: Ratio notation
: Bel notation

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The decibel:
It was later decided that the bel was too large of a unit to be used directly, and so it became customary to apply the metric prefix deci (meaning 1/10) to it, making it decibels, or dB. Now, the expression "dB" is so common that many people do not realize it is a combination of "deci-" and "-bel," or that there even is such a unit as the "bel." To put this into perspective, here is another table contrasting power gain/loss ratios against decibels:

Formula: Po/Pi Po/Pi Po/Pi Po/Pi Po/Pi Po/Pi Po/Pi  
1000 100 10 1 0.1 0.01 0.001  
3dB 2dB 1dB 0dB -1dB -2dB -3dB  
30dB 20dB 10dB 0dB -10dB -20dB -30dB  

Human hearing is highly nonlinear: in order to double the perceived intensity of a sound, the actual sound power must be multiplied by a factor of ten. Relating telephone signal power loss in terms of the logarithmic "bel" scale makes perfect sense in this context: a power loss of 1 bel translates to a perceived sound loss of 50 percent. A power gain of 1 bel translates to a doubling in the perceived intensity of the sound. By the decibel the bel is multiplied by a factor of ten. So a loss of 50 percent of the power will decrease with a factor of 10 and this correspond with -10 decibel.

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Calculating with dB's:
Converting decibels into unit less ratios for power gain is much the same, only a division factor of 10 is included in the exponent term:

If then

There are several rules how to calculate with dB's. To keep the formulas as simple as possible, I've used alphanumeric values X and Y. The examples here below demonstrates the simplicity of it.

Because the bel is fundamentally a unit of power gain or loss in a system, voltage or current gains and losses don't convert to bels or dB in quite the same way. When using bels or decibels to express a gain other than power, be it voltage or current, we must perform the calculation in terms of how much power gain there would be for that amount of voltage or current gain. The following formulas are used to express voltage or current into power:

The correlation between power and voltage with the same dB value is given by the following table:

Decibel: Relative Power ratio: Relative Voltage ratio:
0 1 1
1 1.26 1.12
3 2 1.41
6 3.98 2
10 10 3.16
20 100 10
30 1000 31.16
40 10,000 100
50 100,000 316.23
60 1,000,000 1,000
70 10,000,000 3,162.28
80 100,000,000 10,000
90 1,000,000,000 31,622.78
100 10,000,000,000 100,000

In the table above the difference between the Power doubling (3dB) and voltage doubling (6dB) is clear, by the same dB value (3dB) the voltage will increase with a factor 1.41. By 100dB the difference is exact the double in the numbers of zero's. Most of the dB definitions are related to a Power factor. The most power factors are showed here below:

Quantity: Unit:   Conversion factor: Unit:  
1 Watt = 1E-18 Exa-Watt  
1 Watt = 1E-15 Peta-Watt  
1 Watt = 1E-12 Tera-Watt  
1 Watt = 1E-9 Giga-Watt  
1 Watt = 1E-6 Mega-Watt  
1 Watt = 1E-3 Kilo-Watt  
1 Watt = 1E-2 Hecto-Watt  
1 Watt = 1E-1 Deca-Watt  
1 Watt = 1E-0 Watt  
1 Watt = 1E-1 Deci-Watt  
1 Watt = 1E+2 Centi-Watt  
1 Watt = 1E+3 Milli-Watt  
1 Watt = 1E+6 Micro-Watt  
1 Watt = 1E+9 Nano-Watt  
1 Watt = 1E+12 Pico-Watt  
1 Watt = 1E+15 Femto-Watt  
1 Watt = 1E+18 Atto-watt  

The next table is the relation between dB(1watt) and dBm(1mWatt)

Quantity: Unit:   numerical value: 0dB = 1Watt 0dBm = 1mWatt  
1 Exa-Watt = 1,000,000,000,000,000,000 +180dB +210dBm  
1 Peta-Watt = 1,000,000,000,000,000 +150dB +180dBm  
1 Tera-Watt = 1,000,000,000,000 +120dB +150dBm  
1 Giga-Watt = 1,000,000,000 +90dB +120dBm  
1 Mega-Watt = 1,000,000 +60dB +90dBm  
1 Kilo-Watt = 1000 +30dB +60dBm  
1 Hecto-Watt = 100 +20dB +50dBm  
1 Deca-Watt = 10 +10dB +40dBm  
1 Watt = 1 0dB +30dBm  
1 Deci-Watt = 0.1 -10dB +20dBm  
1 Centi-Watt = 0.01 -20dB +10dBm  
1 Milli-Watt = 0.001 -30dB 0dBm  
1 Micro-Watt = 0.000,001 -60dB -30dBm  
1 Nano-Watt = 0.000,000,001 -90dB -60dBm  
1 Pico-Watt = 0.000,000,000,001 -120dB -90dBm  
1 Femto-Watt = 0.000,000,000,000,001 -150dB -120dBm  
1 Atto-watt = 0.000,000,000,000,000,001 -180dB -150dBm  

 

Most common power formula: and
Most common voltage formula: and

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dB definitions:
In practice we used several dB definitions for many occasions, like Telephone industry, Power handling, Voltage handling, Sound pressure levels. For instance the dBm, used in Telephone sector. This unit has his reference by 1mW into 600
W, where audio signals are transported in symmetrical signal lines and terminated with 600W transformers. The following formula is given:

Where:
P1 = Reference of 1mW
P2 = Derived Power formula to Voltage and resistor, where R = 600
W.

I would like to make one remark on this, dBm is also used with different kind of impedances, therefore I suggest to define at the beginning of your document the impedance where the dBm is referred. To complete this sub-chapter I've defined a list with the most common dB(x) references in the electronic sector.

 

Expression: Related reference: Formula definition:
Bel Logarithm relation between two Power units
dB Proportional scaling (dB=1/10bel) of the bel
dBa Stand for decibel adjusted. It is the weighted absolute noise power, calculated in dB referenced to
P2 = 3.16 picowatts (-85
dBm), which is 0 dBa.
dBA Weighted filter curve, according to the Weighted-A standard, similar to dBSPL (Sound Pressure Level)
only with the difference that dBA is measured with a special filter curve that's compensate the human ear.
 
dBc (decibels relative to carrier) The ratio, in dB, of the sideband power of a signal, measured in a given bandwidth at a given frequency offset from the same signal, to the total power of the signal. P0=unmodulated-carrier power, P=modulation signal power
dBd Power gain in dBd (arial gain against a dipole radiator in free
space with a gain of 0dB)
dBf Power reference to 1 femto Watt into 75W, used as criteria
for measuring to radio sensitivity.
dBFS Decibel full scale. Reference level for full scale of a measuring instrument.  
dBi Power gain in dBi (arial gain in dBi against a isotrope radiator
in free space with a gain of 0dB)
dBK Power, referred to 1kiloWatt
dBm Power, (dB referred to 1mWatt over 600W), could also be a
referred to other different impedances.
dBmV Power, (dB referred to 1mVolt over 75W)
dB_SPL Voltage, (dB referred by 20μPascal)
dBu Power, (dB referred to 775mVolt over 600W)
dBv Alternative for dBu notation.
dBV Voltage, (dB referred to 1Volt rms)
dBμV Voltage, (dB referred to 1μVolt rms)
dBW Power, (dB referred to 1Watt)
dBx dB above reference coupling (crosstalk), normally used to express crosstalk in telephone lines.
dBZ A logarithmic expression for reflectivity factor, referenced to (1 mm6 / 1 m3). Reflectivity
(designated by the letter Z) covers a wide range of signals from very weak to very strong.
Neper The Neper is the natural logarithm of the power ratio

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The Neper:


John Napier

The neper is named for John Napier (15501617), the Scottish mathematician who invented logarithms to the base e. Like the decibel, which performs the same functions, the neper came into use in the middle 1920's, replacing the mile of standard cable (m.s.c.). The decibel, however, became popular in Britain and America and is based on log to the base ten, while the neper was used in continental Europe and the base of its logarithm was e. One neper = 8.686 decibels. The neper is a dimensionless unit used in telecommunications to express the ratio between two measurements of power. The neper is the natural logarithm of the power ratio. In formula:

If the input and output impedances are the same, we can use the following formulas for two voltages or two currents:
   

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