April 1974 Popular Electronics
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from
Popular Electronics,
published October 1954  April 1985. All copyrights are hereby acknowledged.

Since new
people are constantly entering the field of electronics, there is a constant need to post articles covering
some of the basics of the craft. Just as the seasoned practitioner looked to currently published magazines
and books for guidance, so too do contemporary technician and engineer fledglings. Decibels have long
been a cause of confusion for many  even some who have been in the field for many years. I have seen
on many occasions engineers who are way smarter than me routinely mix units of dB (dimensionless) with
units of dBm and dBV (power and volts, respectively) when writing. Author George Board does as good
of a job as anyone I've read at explaining the basics.
Understanding Decibels
Here Are Some Rules of Thumb that Make it Easy to Use and Understand
dB's
By George Board
Learning to understand the decibel  really understanding it  can be very confusing. Actually, a
decibel is nothing more than the logarithm of the ratio of two power levels; but the problem gets complex
when the term is used in such diverse areas as antenna gain or microphone voltage. As a result, the
only people who toss off decibel ratings with apparent confidence are salesmen and others who have memorized
the values without knowing what they really mean.
Actually, understanding decibels requires just a slight amount of general knowledge and two simple
rules of thumb. There is no "higher" math involved. The basic expression is
G_{dB} = 10 log_{10} (P_{2}/P_{1})
The power ratio has no units since the two powers cancel each other.
The output power, P_{2}, is some multiple of the input power, P_{1}. For example,
with a power ratio of 2 and an input power of 5 watts, the output power is twice the input, or 10 watts.
This power ratio of 2 expressed in decibels is 3 dB. (The output is 3 dB greater than the input.) The
input power, then, is usually the starting point or reference level to which the output power is compared.
Knowing the reference level defines what the dB measures.
The concept of reference level is not found only in the use of decibels. Voltages are usually referenced
to one volt. Multiples of one volt can then be used to define all voltage levels. Similarly, the power
ratio in decibels defines the power level in multiples of the reference level. The difference is that
the reference for voltage is one volt, while the reference for dB could be any defined power or just
an arbitrary test input.
These references are usually understood for each type of measurement. Typically, in measuring acoustical
energy, 10^{6} watt/cm^{2} is used as a reference. In antenna measurements, everything
is compared to a simple dipole. Hopefully, if the reference is not standard, it is abbreviated as a
letter following the dB value. Audio power, dBm, is referenced to 1 milliwatt. Other common references
are dBW (1 watt reference) and dBn (thermal noise). There is no set rule for the derivation of these
abbreviations; they are originated for special cases.
Why Are They Used? The beauty of using the decibel system is that decibel figures
can be added (as can any logarithm) rather than multiplied, as must be done with regular gain figures.
The circuit gains in dB from one stage to the next can simply be added or subtracted from the input
reference level. For example, assume that a microphone has an output of 0.00001 mW (50 dBm) which is
attenuated by a factor of 1/10 ( 10 dB) in the cable to the amplifier. The amplifier has a power gain
of 10,000,000,000 (100 dB). The output of the amplifier can be found in one of two ways:
(1) 0.00001 mW X 1/10 X 10,000,000,000 = 10,000 mW = 10 W
(2) 50 dBm  10 dB + 100 dB = 40 dBm = 10 W
Using the dB figures requires an additional step to convert the answer to watts, but the rest of
the calculation is much easier. The numbers are of a more manageable size and addition is easier than
multiplication.
Any signal level or known characteristic can be used as a reference as long as it is related to power.
The power measurements, however, are seldom taken directly. That would require a calorimeter or a directreading
wattmeter. Usually the voltage or current through a known resistance is measured and the power is calculated
from
P = V^{2}/R = I^{2}R = VI
Input and Output Resistances. The power ratio is simplified when both the input
and output resistances are equal. By cancelling the like resistance terms, it is only necessary to measure
voltage or resistance and square it to obtain the power ratio. The logarithm of the voltage ratio squared
is twice the logarithm of the voltage ratio unsquared. Thus,
G_{dB} = 20 log_{10} V_{2}/V_{1}
The special case of having equal input and output resistances is not unusual. Equipment requiring
many signal interconnections usually has input and output resistances of 50 ohms. In such a system,
the formula
20 log_{10} V_{2}/V_{1} is always usable in finding the dB gain.
Instant chaos develops if the input and output resistances are not equal. The voltage ratio squared
would then not be equivalent to the power ratio. In this case the power ratio must be calculated using
the proper resistance values.
Rules of Thumb. Knowing what decibel ratings are, the next thing to learn is how
they are used. When a salesman says, "Instead of the beam, why not get this new kW rig," he knows that
the beam antenna gain in dB cannot be directly compared to the power ratio between your old transmitter
and a kilowatt. To figure out what he means, using algebra and logarithm tables, would take too long
and making a guess is difficult with decibels. Being logarithmic, they don't increase in a normal fashion.
A power ratio of 5, for instance, represents 7 dB, but increasing the ratio to 10 only increases the
decibels by 3. So some rules of thumb are useful.
The first rule of thumb pertaining to the decibel scale is that adding 3 dB doubles the power ratio.
(The factor is actually 1.9953 ... , but doubling is close enough.) Thus the gain represented by 6 dB
is twice that of 3 dB just as a gain of 58 dB is twice that of 55 dB. Similarly, a gain of 55 dB is
half that of 58 dB and 3 dB is a power ratio of one half.
If powers of two were so easy to remember, things would be easy. But it wouldn't be worth the effort
to figure out that 30 dB is 10 times 3 dB for a power ratio of 2^{10}.
The second rule of thumb is that each 10 dB represents a poweroften change in the power ratio.
Or, add 10 dB for each zero in the power ratio. If the power ratio is less than one, subtract 10 dB
for each zero, including the zero to the left of the decimal. A ratio of 1000 is 30 dB (3 zeros), and
a ratio of 0.001 is 30 dB (3 zeros).
Combining the counting of zeros and the multiples of two, the simplified system expands to provide
a handy reference for all decibel levels. A power ratio of 20,000 is 40 dB (4 zeros) plus 3 dB for the
factor of two, totaling 43 dB. Half (3 dB) of 100 (20 dB) is 50; so 50 is 20 dB minus 3 dB or 17 dB.
Interpolating, 0.0001875 is between 0.0001 (40 dB) and 0.0002 (37 dB). But the number is much closer
to 0.0002, so it must be 38 dB. A closer figure could only be obtained by using logarithm tables and
many calculations.
The two rules of thumb for power ratios can also be used for voltage ratios. Remember that the voltage
ratio in dB is 20 log_{10} V_{2}/V_{1}, and the power ratio is the square of
the voltage ratio. Thus a power ratio of 16 (12 dB) is the same as a voltage ratio of 4. Now, 4 would
be 6 dB as a power ratio  one half of the 12 dB for 16. So, the voltage ratio in dB is twice what the
rules for the power ratio would make it. In other words, a voltage ratio of 2 is 6 dB, 4 is 12 dB, and
0.01 is 40 dB.
Summary. (1) Adding decibels is like multiplying power ratios. (2) The reference
level must be known. (3) The voltage ratio in dB (resistances equal) is twice what the power ratio would
be. (4) Doubling the power ratio adds 3 dB (6 dB for the voltage ratio). (5) Multiplying the power ratio
by 10 adds 10 dB (20 dB For the voltage ratio) .
Posted May 3, 2017
