Decibels
In its simplest form, an amplifier's
gain is a ratio of output over input. Like all ratios, this form of
gain is unitless. However, there is an actual unit intended to represent
gain, and it is called 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. Now, it is defined in terms of the
common (base 10) logarithm of a power ratio (output power divided by
input power):
Because the bel is a logarithmic unit, it
is nonlinear. To give you an idea of how this works, consider the
following table of figures, comparing power losses and gains in bels
versus simple ratios:
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:
As a logarithmic unit, this mode of power
gain expression covers a wide range of ratios with a minimal span in
figures. It is reasonable to ask, "why did anyone feel the need to
invent a logarithmic unit for electrical signal power loss in a
telephone system?" The answer is related to the dynamics of human
hearing, the perceptive intensity of which is logarithmic in nature.
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, or 1/2. A power gain of 1 bel translates to a
doubling in the perceived intensity of the sound.
An almost perfect analogy to the bel
scale is the Richter scale used to describe earthquake intensity: a 6.0
Richter earthquake is 10 times more powerful than a 5.0 Richter
earthquake; a 7.0 Richter earthquake 100 times more powerful than a 5.0
Richter earthquake; a 4.0 Richter earthquake is 1/10 as powerful as a
5.0 Richter earthquake, and so on. The measurement scale for chemical pH
is likewise logarithmic, a difference of 1 on the scale is equivalent to
a tenfold difference in hydrogen ion concentration of a chemical
solution. An advantage of using a logarithmic measurement scale is the
tremendous range of expression afforded by a relatively small span of
numerical values, and it is this advantage which secures the use of
Richter numbers for earthquakes and pH for hydrogen ion activity.
Another reason for the adoption of the
bel as a unit for gain is for simple expression of system gains and
losses. Consider the last system example where two amplifiers were
connected tandem to amplify a signal. The respective gain for each
amplifier was expressed as a ratio, and the overall gain for the system
was the product (multiplication) of those two ratios:
If these figures represented power
gains, we could directly apply the unit of bels to the task of
representing the gain of each amplifier, and of the system altogether:
Close inspection of these gain figures in
the unit of "bel" yields a discovery: they're additive. Ratio gain
figures are multiplicative for staged amplifiers, but gains expressed in
bels add rather than multiply to equal the overall system
gain. The first amplifier with its power gain of 0.477 B adds to the
second amplifier's power gain of 0.699 B to make a system with an
overall power gain of 1.176 B.
Recalculating for decibels rather than
bels, we notice the same phenomenon:
To those already familiar with the
arithmetic properties of logarithms, this is no surprise. It is an
elementary rule of algebra that the antilogarithm of the sum of two
numbers' logarithm values equals the product of the two original
numbers. In other words, if we take two numbers and determine the
logarithm of each, then add those two logarithm figures together, then
determine the "antilogarithm" of that sum (elevate the base number of
the logarithm -- in this case, 10 -- to the power of that sum), the
result will be the same as if we had simply multiplied the two original
numbers together. This algebraic rule forms the heart of a device called
a slide rule, an analog computer which could, among other things,
determine the products and quotients of numbers by addition (adding
together physical lengths marked on sliding wood, metal, or plastic
scales). Given a table of logarithm figures, the same mathematical trick
could be used to perform otherwise complex multiplications and divisions
by only having to do additions and subtractions, respectively. With the
advent of high-speed, handheld, digital calculator devices, this elegant
calculation technique virtually disappeared from popular use. However,
it is still important to understand when working with measurement scales
that are logarithmic in nature, such as the bel (decibel) and Richter
scales.
When converting a power gain from units
of bels or decibels to a unitless ratio, the mathematical inverse
function of common logarithms is used: powers of 10, or the antilog.
Converting decibels into unitless ratios
for power gain is much the same, only a division factor of 10 is
included in the exponent term:
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. For a
constant load impedance, a voltage or current gain of 2 equates to a
power gain of 4 (22); a voltage or current gain of 3 equates
to a power gain of 9 (32). If we multiply either voltage or
current by a given factor, then the power gain incurred by that
multiplication will be the square of that factor. This relates back to
the forms of Joule's Law where power was calculated from either voltage
or current, and resistance:
Thus, when translating a voltage or
current gain ratio into a respective gain in terms of the bel
unit, we must include this exponent in the equation(s):
The same exponent requirement holds true
when expressing voltage or current gains in terms of decibels:
However, thanks to another interesting
property of logarithms, we can simplify these equations to eliminate the
exponent by including the "2" as a multiplying factor for the
logarithm function. In other words, instead of taking the logarithm of
the square of the voltage or current gain, we just multiply the
voltage or current gain's logarithm figure by 2 and the final result in
bels or decibels will be the same:
The process of converting voltage or
current gains from bels or decibels into unitless ratios is much the
same as it is for power gains:
Here are the equations used for
converting voltage or current gains in decibels into unitless ratios:
While the bel is a unit naturally scaled
for power, another logarithmic unit has been invented to directly
express voltage or current gains/losses, and it is based on the
natural logarithm rather than the common logarithm as bels
and decibels are. Called the neper, its unit symbol is a
lower-case "n."
For better or for worse, neither the
neper nor its attenuated cousin, the decineper, is popularly used
as a unit in American engineering applications.
- REVIEW:
- Gains and losses may be expressed in
terms of a unitless ratio, or in the unit of bels (B) or decibels
(dB). A decibel is literally a deci-bel: one-tenth of a bel.
- The bel is fundamentally a unit for
expressing power gain or loss. To convert a power ratio to
either bels or decibels, use one of these equations:
-
- When using the unit of the bel or
decibel to express a voltage or current ratio, it must
be cast in terms of the an equivalent power ratio. Practically,
this means the use of different equations, with a multiplication
factor of 2 for the logarithm value corresponding to an exponent of 2
for the voltage or current gain ratio:
-
- To convert a decibel gain into a
unitless ratio gain, use one of these equations:
-
- A gain (amplification) is expressed as
a positive bel or decibel figure. A loss (attenuation) is expressed as
a negative bel or decibel figure. Unity gain (no gain or loss; ratio =
1) is expressed as zero bels or zero decibels.
- When calculating overall gain for an
amplifier system composed of multiple amplifier stages, individual
gain ratios are multiplied to find the overall gain ratio. Bel
or decibel figures for each amplifier stage, on the other hand, are
added together to determine overall gain.
|
