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 Scottishborn speech therapist and inventor Alexander
Graham Bell (18471922). 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.
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 
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.
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  =  1E18  ExaWatt  
1  Watt  =  1E15  PetaWatt  
1  Watt  =  1E12  TeraWatt  
1  Watt  =  1E9  GigaWatt  
1  Watt  =  1E6  MegaWatt  
1  Watt  =  1E3  KiloWatt  
1  Watt  =  1E2  HectoWatt  
1  Watt  =  1E1  DecaWatt  
1  Watt  =  1E0  Watt  
1  Watt  =  1E1  DeciWatt  
1  Watt  =  1E+2  CentiWatt  
1  Watt  =  1E+3  MilliWatt  
1  Watt  =  1E+6  MicroWatt  
1  Watt  =  1E+9  NanoWatt  
1  Watt  =  1E+12  PicoWatt  
1  Watt  =  1E+15  FemtoWatt  
1  Watt  =  1E+18  Attowatt 
The next table is the relation between dB(1watt) and dBm(1mWatt)
Quantity:  Unit:  numerical value:  0dB = 1Watt  0dBm = 1mWatt  
1  ExaWatt  =  1,000,000,000,000,000,000  +180dB  +210dBm  
1  PetaWatt  =  1,000,000,000,000,000  +150dB  +180dBm  
1  TeraWatt  =  1,000,000,000,000  +120dB  +150dBm  
1  GigaWatt  =  1,000,000,000  +90dB  +120dBm  
1  MegaWatt  =  1,000,000  +60dB  +90dBm  
1  KiloWatt  =  1000  +30dB  +60dBm  
1  HectoWatt  =  100  +20dB  +50dBm  
1  DecaWatt  =  10  +10dB  +40dBm  
1  Watt  =  1  0dB  +30dBm  
1  DeciWatt  =  0.1  10dB  +20dBm  
1  CentiWatt  =  0.01  20dB  +10dBm  
1  MilliWatt  =  0.001  30dB  0dBm  
1  MicroWatt  =  0.000,001  60dB  30dBm  
1  NanoWatt  =  0.000,000,001  90dB  60dBm  
1  PicoWatt  =  0.000,000,000,001  120dB  90dBm  
1  FemtoWatt  =  0.000,000,000,000,001  150dB  120dBm  
1  Attowatt  =  0.000,000,000,000,000,001  180dB  150dBm 
Most common power formula:  and  
Most common voltage formula:  and 
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 600W,
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 = 600W.
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 subchapter 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 WeightedA 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=unmodulatedcarrier 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 mm^{6} / 1 m^{3}).
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 
John Napier 
The neper is named for John Napier (1550–1617),
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: