For some reason even a few really good technicians and engineers
have problems with decibels. Ever since learning about, and truly
understanding, logarithms, I have appreciated the convenience of
being able to use addition and subtraction to perform multiplication
and division, respectively. Decibels, being logarithms, have always
made perfect sense to me. Even the difference between voltage dBs
and power dBs has been easy to remember because of the power rule
of logarithms, log (A^{B}) = B·log (A).
Calculators offer little help when you don't comprehend the basics
of decibels. In 1955 when this article was written, people used
tables of logarithms rather than punching calculator keys. Mathematicians
and appointed underlings spent their lives generating tomes of logarithms.
Historians have found many errors in those tables while doing research,
but no ships ever sailed off the edge of the Earth due to a computational
error based on them.
On a side note, I recall a day when I was a teenager where I
was listening to a distinguished lady who was the wife of a former
assistant attorney general of the U.S. (both he and his wife were
Rhodes Scholars). My mother typed legal transcripts for him. I sometimes
made a few bucks doing odd jobs around their sprawling waterfront
property on the Chesapeake Bay. The woman knew of my interest in
astronomy and mechanics and began talking about celestial navigation
and how logarithms were of importance in precision excursions like
cartography (map making); hers were the days before electronic navigational
aids. I was mesmerized by the ease at which she conversed on the
subject. After her somewhat lengthy dissertation, she pointed to
a picture on the fireplace mantle (one of about 5 in the house)
and asked me to hand it to her (she was bedridden). As I picked
it up I immediately recognized the two people in it  her as a much
younger woman, and none other than Albert Einstein!
Using the Decibel
In any listening situation, the smallest increase in the volume
of any sound that can be detected by the human ear is onefourth,
or 25 percent, over a previous sound. In other words, if any two
sounds have a power ratio of at least 1.25 to 1, we will detect
that the former is louder.
This ratio holds true for a wide range of power regardless of
the absolute power of a particular sound. If we hear two sounds
whose powers are respectively 12.5 and 10 watts, we would still
hear the same difference in their loudness as we heard between the
sounds at 1.25 and 1 watt, since the ratio is still the same (1.25).
This is because we hear approximately in proportion to the logarithm
of the intensity, rather than in direct linear response to it. The
decibel has been developed as a convenient unit for expressing and
measuring. intensity logarithmically. Mathematically, "1 decibel"
is approximately 10 multiplied by the common logarithm of the ratio,
1.25 to 1.
The factor of 10 enters the picture because the original unit
used was the "bel" (named for Alexander Graham Bell), which is the
logarithm of 10 to the base 10. The decibel is actually onetenth
of a "bel" and is used in preference to the bel inasmuch as a change
of sound intensity of 1 decibel approximates very closely the ratio
of 1.25 to 1, which is the minimum change in sound intensity human
ears can detect.
The decibel is used widely in audio work because it represents
accurately the response of the ear to different intensities and
because it can be used over a wide range of intensities. Decibels
are used for expressing power ratios, voltage ratios, current ratios,
amplifier gain, hum level, loss due to negative feedback, network
loss, and loss in attenuator circuits and in transmission lines.
Gain is expressed as plus dB; loss as minus dB. Ratios between
currents and voltages across the same or equal resistors are also
expressed in decibels. In the case of voltages or currents, the
logarithm of the ratio must be multiplied by 20. This is because
the decibel is basically an expression of power (wattage) which
is always a function of the square of either current or voltage.
To square a number, you double its logarithm. Thus, in the case
of values already expressed as powers (wattage), we multiplied the
logarithms of the ratio by 10. But in the case of values not yet
expressed as powers, such as voltage or current, we multiply the
logarithm of their ratio by 10 doubled, or 20.
We now can state all the above in terms of these simple formulas:
dB = 10 log
when P is known in watts.
dB = 20 log
when E is known in volts.
dB = 20 log
when I is known in amps.
The value of the "common logarithm" (sometimes written as log_{10})
is easily obtained from standard tables that are included in most
mathematics and technical textbooks. From then on it's a case of
simple arithmetic.
The table on the opposite page is a shortcut aid in determining
dB gain or loss. It has, in effect, already computed the logarithms
of the power (and voltage and current) ratios for you. Notice that
the righthand side (4th and 5th columns) expresses ratios in which
there is a gain (1 or higher). The lefthand side (1st and 2nd columns)
expresses ratios in which there is a loss (1 or lower). The center
column gives you the number of decibels of either gain or loss for
a given ratio.
Let us now work a few problems using both the formulas and the
table.
Example: What will be the gain in dB of an amplifier whose output
power rises to 5 times its input?
The formula tells us that for power (in wattage),
dB =
In this case, P_{2}
over P_{1} is
given; it is known to be 5. (In other words, the input might be
2, the output 10, resulting in a ratio of 5 to 1). The log of 5
is approximately 0.7. Multiplying this by 10, we get 7, which is
the solution. In other words, this amplifier has a gain of 7 decibels.
In practical terms this means that the difference in sound intensity
between the input to the amplifier and the output from it would
be heard by the ear as seven times the minimum change in loudness
that we could detect.
Now, let us use the table to work this problem. Since there is
a gain involved, we refer to the righthand portion of the table.
Since the values are in terms of power (watts), we use the 5th column.
The nearest figure in this column to our power ratio of 5 happens
to be 5.012. This corresponds to plus 7 in the dB column. Again,
our answer is plus 7 dB.
Let us work a problem using voltages. Example: What will be the
gain in dB of an amplifier whose output voltage rises to 9 times
its input (across equal resistances) ?
Here we must multiply the logarithm of the ratio by 20, since
we are dealing with a voltage value rather than a wattage value.
The common log of 9 is 0.95. Multiplying this by 20 we get 19 dB.
Again, the same answer could be obtained directly from our table.
Since a gain is involved we again confine ourselves to the righthand
side of the table. Since our ratio is expressed in voltage, we check
down the 4th column. We find that the number of decibels that corresponds
most closely to a voltage ratio of about 9 happens also to be 19
dB.
As long as this table is available, there is no need for the
formulas or for logarithmic values of the ratios. If the table is
not handy, though, the formulas and a table of common logarithms
will solve any problem.
Let us now take a situation in which there is a decibel loss
to be calculated. For example, an amplifier has a negative voltage
feedback loop which is intended to reduce distortion at the output.
This feedback voltage also reduces the overall gain of the amplifier.
But by how much? Assume that we measure 1.2 volts at the output
of the amplifier with its feedback loop in operation. Then we disconnect
the feedback loop and find the output measures 12 volts.
Our ratio in this case is 1.2 over 12, or 0.1. We now consult
the lefthand side of our table for decibel loss. Since these are
voltages we check down the column so headed. We discover that a
voltage ratio of 0.1 indicates a 20 dB loss. Thus we express the
feedback value in this amplifier as minus 20 dB.
Conversely, if an amplifier's specifications claim that the circuit
incorporates a minus 20 dB feedback loop (or "negative feedback,
20 dB"), this means that the output of the amplifier should measure
onetenth the voltage with the loop that it does without the loop.
Another example of decibel loss: Assume that an amplifier has
a rated output of 20 watts. We want to determine what its hum level
is because in order not to hear the objectionable hum, its level
should be very low  maybe 50 dB below the rated output of 20 watts.
Here's how this is done: We apply a signal to the input of the amplifier
and connect a voltmeter across its output terminals, say the 8ohm
terminals. Next we turn up the gain of the amplifier to the point
necessary to produce its rated 20 watts output. Since we are using
a voltmeter at the output terminals, we must translate watts into
volts. From Ohm's Law we know that power in watts is equal to the
square of the voltage divided by the resistance. (P = E^{2}/R)
Therefore, E equals the square root of P x R. P is 20 and R is 8.
Thus E equals the square root of 160 which is approximately 12.7
volts.
Consequently, when our voltmeter  connected across the 8ohm
output terminals  reads 12.7 volts, we have reached the amplifier's
rated output of 20 watts. We now disconnect the input signal and
short the input. Naturally, the voltage to be expected with no input
signal should be quite small. But whatever is present will be noise
and hum within the amplifier circuit itself. Again, consulting our
voltmeter (still connected at the 8ohm terminals) we discover that
it reads 3 millivolts (0.003 volts).
To determine the number of "minus decibels" the hum level is
with respect to the 20 watts output, we must first get our voltage
ratio, which is 0.003 over 12.7. This comes to approximately 0.00024.
Since we are dealing with a loss in voltage, we consult the 1st
column of our table, and we find there is no figure like our 0.00024!
Therefore. we must interpolate. The nearest significant figure
to our ratio of 0.00024 happens to be 0.251. This gives us minus
12 dB. But our ratio is about onethousandth, or 10^{3},
of 0.251. We, therefore, consult the 10^{3} value in the
same column and discover we must add another minus 60 dB to the
minus 12 we already have. Thus our final answer is minus 72 dB.
This means the hum level of the amplifier is 72 decibels below its
rated output, which puts it well below the level at which it could
be heard.
Conversely, this means that if an amplifier is rated at 20 watts
output with a hum level of minus 72 dB, the actual voltage measured
across its 8ohm output terminals with no signal input should not
exceed 0.003 volts.
Three main types of meters are used for measuring dB directly,
without the need for calculating values by the use of logarithms
or the table. The simplest and possibly the most familiar type is
the "output meter" or the decibel scale found on many multimeters.
This is actually an a.c. voltmeter calibrated to read the number
of dB that expresses a ratio between the power being fed into the
meter and some fixed reference level, usually 6 milliwatts. The
meter calibration assumes that the voltage is measured across 500
ohms resistance. This type of meter is used in determining the relative
outputs of various audio circuits and is also used in receiver alignment.
The "VU meter" is similar to the output meter, except the reference
level is 1 milliwatt in 600 ohms resistance. In addition, the VU
meter has timeconstant characteristics which determine its response
to voltage peaks, such as "sound bursts" or other short time interval
peaks. It is widely used in broadcasting and recording studios to
monitor the output levels of programs.
A third type of decibel meter is the sound level indicator. This
is actually an assembly of a microphone, an amplifier, and an a.c.
voltmeter calibrated to provide a dB reading which corresponds to
human hearing levels. On this meter, zero dB represents the threshold
of hearing. This meter is used by acoustics technicians to determine
hearing conditions in auditoriums and theaters.
In summary, the decibel is used to express any ratio of power,
voltage, current, acoustic energy, etc. whether it be a gain relationship
or a loss. It can be used to express the range of a symphony orchestra
and then to determine how much amplification is needed to carry
the music across lines of certain distance in order to fill a hall
of a certain size or cut a particular recording. Any type of gain
or loss in any circuit may be expressed in decibels which provide
a quick and accurate key to the operating conditions of the circuit.
The advantage of using decibels is that it permits the simple addition
of ratios to obtain complete gain and loss data whereas using E,
I, or P ratios would involve multiplication and division. For example,
it is easier to add 25 dB and 36 dB than it is to multiply the corresponding
gain figures of 316.2 and 4000, to get the total gain of two amplifiers
in cascade.
Decibel Conversion Table
Posted April 16, 2013
