August 1960 Electronics World
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from
Electronics World, published May 1959
 December 1971. All copyrights hereby acknowledged.

Evidently  and
understandably so  a lot of people in the electronics realm have a problem
grasping the concept of a decibel as it pertains to voltage, current, power, et
al. This 1960 Electronics World magazine "The Decibel Without Pain"
article takes a stab at making it all clear. As the c1991 Teen Talk
Barbie famously said, and in some cases like in this YouTube video still says, "Math class is tough."
Believe it or not, that created quite a stir amongst feminists who, BTW, would
have had no problem with Ken saying the same thing. The fact is, for many 
probably most  people math class is tough. It was for me when I was in
elementary school and junior high school. I nearly failed 9th grade algebra
because for the life of me I could not get the hang of factoring polynomials.
Being given the option of enrolling the the
electrical vocational program for high school (grades 10 through 12) was my
salvation. I loved working with electrical and electronic things, so being able
to spend my days in prison (aka school) learning about residential, commercial, and industrial wiring, motors, and control circuits really motivated me. For
some reason the same math that loomed over like the Grim Reaper whilst in a
regular classroom setting somehow made sense when set to Ohm's law, Kirchhoff's
law, Norton's law, magnetic and electric fields, and various sundry other
practical applications. A few years later I breezed through math required in my
USAF radar technician classes. A
few years after that I managed to earn A's in nearly all my engineering level
math classes (calculus, differential equations, linear algebra, complex numbers,
etc.).
The Decibel Without Pain
By Bob Eldridge
Practical hints on understanding and using these strange units. by someone who
actually likes them.
By now, we are all supposed to know that the decibel is a ratio between two quantities
of power (or voltage, or current), not a quantity of power, voltage, or current
itself. Thus we should not express any known electrical quantities as being simply
"so many db"  unless we are in the advertising business. The copywriter may state,
"Sudso is 20 percent faster" and get away with it before anyone asks, "Faster than
what?" However, if a technician should state, "The power at this point is 30 db,"
someone will invariably want to know, "30 db above or below what?"
In other words, there must be a reference level.
Sometimes this level may be understood. In the telephone industry, for example,
this reference is the input level to the switchboard from a subscriber line. Other
levels throughout the system, expressed simply in db, are referred to this input,
with the telephone engineer primarily concerned with the magnitude of the losses
in the various portions of the path between the microphone of one telephone and
the receiver of another.
Fig. 1  Gains and losses in a transmission link are represented as shown to determine adequacy of signal level.
Fig. 2  Two identical amplifiers connected (A) in series yield greater gain than they would (B) when in parallel.
He is also interested in the gain in db of each amplifier in the chain that will
restore the losses: what goes down must be brought back up again if Mom in New York
is to hear Johnny's plea from camp for just another few dollars. In the detailed
engineering of the circuit, he also becomes interested in the actual rather than
relative power present at various points in the path, but that is a matter for db's
twin brother, dbm, of whom more will be heard later.
For the moment, let's consider only that the db is a ratio. An amplifier that
produces 100 watts of output when 1 watt is fed into it is a 20db amplifier.
But so is another that produces 100 milliwatts out from 1 mw. in. From this
comparison, we see that a 20db amplifier is one that has a power gain of 100
times without regard to its actual powerhandling capacity.
What About Logs?
The key to the whole db business, as many technicians unhappily acknowledge,
is a working familiarity with logarithms. A great deal can be done in handling
decibels without using logarithms, as will be noted later. However, those
practical men who have been getting along without logs for years just don't
realize what they are missing. Logs are very easy to usethey make
multiplication and division less of a chore. In fact, logs literally turn
multiplication into addition, and division into subtraction. This same great
advantage is present in the use of db. Although a full discussion of the use of
logs in general calculations will be avoided, it will be useful to review some
facts concerning the logarithm to the base 10 before
getting to the meat of the dog biscuits (db).
The logarithm of a number is the power to which 10 must be raised to obtain that
number. 100 is 10 squared (10^{2}), or 10 to the power of 2. Therefore,
the power to which 10 must be raised to make 100 is 2; so the log of 100 is 2. To
take another example, 2 is 10 to the power of 0.3 (to be precise, 0.3010, but who
wants to be precise at a time like this?). To find this value, or to convert a number
to a log or the reverse, either a table of logarithms or a slide rule may be used.
To multiply several numbers together, we merely find their logs, add these together,
and translate the new log thus obtained back to a number again. Thus, using the
two logs we have already examined, let us multiply 100 x 100 x 100 x 2 x 2. With logarithms,
this becomes 2 + 2 + 2 + 0.3 + 0.3, which quickly adds up to 6.6, the new logarithm. An
antilog table or a slide rule tells us that the number of which is 6.6 is the logarithm
is 4x10^{6}, or 4,000,000.
Now for tying up logarithms with decibels  in formula: db=10 log P_{1}P_{2}.
"Which means?" as the straight man might say to his partner.
"A decibel is ten times the log of the ratio between two power levels."
"Why ten times?"
"Because the bel, the first unit for measuring the relative level between two
sounds, was too large to handle; so they chopped it into ten parts and called each
a decibel (1/10 bel)."
From the formula it is obvious that the db is logarithmic. In this fact lies
many great advantages. Power ratios of millions to one can be expressed in simple
numbers and vast products can be obtained by merely adding simple numbers together.
If you think that this is rarely necessary, consider that a difference of only 60
db represents a power ratio of a million to one.
Table 1 shows some of the easily remembered and frequently used power ratios,
expressed as numerical ratios and in equivalent db. With these figures, many complicated
calculations can be reduced to mental arithmetic. For example, whenever a given
power level is multiplied by 10, we add 10 db (or, when it is divided by 10, subtract
10 db). To double any power, add 3 db. Since quadrupling a power is doubling it
twice, this comes to 6 db (3+3) . Just by manipulating this table, many other ratios
can be deduced. For example, to increase a power ratio 20 times, we must increase
it 10 times (10 db) and then double it (3 db). Calculation will show that a power
increase of 20 times is indeed a change of 10+3, or of 13 db.
Other examples: To obtain 5 times a power, we can find 10 times that power and
then halve it. In db, this would be +103, or +7 db. To increase by 50 times, we
could increase by 10 times (+10 db) and then by 5 times (+7 db). The answer would
be +17 db. To find 25 times (half of 50 times), we know that +173=+14 db.
The Dbm
Fig. 3. Doubling voltage (B) applied across a circuit (A) also
doubles current. Thus the power drop quadruples.
The standard test tone used in the telephone industry is a 1000cps signal with
a power of 1 milliwatt. Since this is a convenient reference level, the term dbm,
meaning "db with reference to 1 milliwatt," is now in widespread use. Since there
is a specific reference point (1 milliwatt is always zero dbm) , any figure given
in dbm is not a ratio. It is a definite quantity of power. Thus 1 watt (1000 mw.)
can be written as +30 dbm; and 1 microwatt is 30 dbm.
The dbm is very useful in system calculations. For example, let us take an imaginary
u.h.f. link that must be set up. We have the following: 1. a 50watt transmitter,
2. a receiver that requires an input of 50 microvolts of signal for efficient operation,
3. antenna feeders and connectors that entail a loss of 3 db at each end, 4. a required
safety margin of 20 db to allow for fading, and 5. attenuation of the signal in
the air path between transmitter and receiver shown to be 111 db. We draw a block
diagram, as in Fig. 1.
A 50microvolt signal across the 50ohm input impedance of our receiver tells
us we need a power here of 73 dbm. This is another way of stating the receiver's
sensitivity. The transmitter output, already given in watts, can be restated as
+47 dbm. If we forget everything else (losses) in between transmitter and receiver,
the ratio between the transmitter output and the power required at the receiver
input is 120 db (not dbm) . This is the difference between transmitter output and
receiver input, which is +47(73).
This 120 db is called the system gain, and represents the maximum amount of attenuation
that could be tolerated between transmitter and receiver without degrading performance.
Now, on the debit side, we have: 3db loss in the transmitter antenna feeder system,
111db loss in the air path between these stations, and 3db loss in the receiver
antenna feeder system. These losses add up to 117 db.
Thus, if the antennas used show unity gain (such as conventional dipoles), the
system will work with 3 db to spare. However, we are looking for a 20db safety
margin to allow for fading. We must therefore use antennas that will give us a total
gain of 17 db. Antennas at each end with a gain of 8.5 db each will do the trick.
Now here is a point worth thinking about: If we left the unitygain antenna at
the transmitter and stacked two 8.5db antennas at the receiver without losses,
we would fall short of the gain requirement. We would get, not 17 db, but only 11.5
db! Let's consider this statement in a more familiar form  the case of the two
amplifiers in Fig. 2. In series, with one amplifier multiplying the output of the
other, the logarithmic increase is such that, expressed in db, we get twice the
gain of one amplifier. In parallel (Fig. 2B), we only double the power. This (Table
1) is a gain of only 3 db over the use of one amplifier  or antenna. In the case
of the amplifiers, this could easily be measured.
To return to the dbm  it is so useful precisely because it indicates a definite
power and yet can be manipulated in a logarithmic manner along with ordinary decibels.
Transmitters, amplifiers, receivers, microphones, and loudspeakers can all be rated,
as to sensitivity or output, in dbm, even though the impedance of each type of unit
may be different. Other units similar to the dbm include dbw (referred to a level
of 1 watt) , dbk (in reference to a kilowatt), and dbx, a unit of crosstalk between
two circuits. There are also others.
Voltage, Current Ratios
So far, we have talked about the ratio between two powers. The ratio between
two voltages, or between two currents, can also be expressed in db, where these
are the quantities that are knownbut the formula is different. This stems from
the fact that, when you relate a given power to the voltage or current that is involved
with it, the relationship is not linear. There is always a square involved (power,
in watts, is I^{2}R or E^{2}/R). Again, we look at the matter in
practical terms, with the help of Fig. 3.
If we put a 10ohm resistor across a 10volt supply, 10 watts of power (Fig.
3A) will be dissipated in the resistor. If the voltage is doubled (Fig. 3B), the
power in the resistor is not doubled  it is quadrupled. This is because, following
the rule made public by Mr. Ohm, the current is doubled as well as the voltage.
This brings us to another principle in the use of logarithms: to square any number,
multiply its log by 2, then find the antilog of this product. Since a square is
involved when we go from a power to a voltage or a current, we go back to our old
db formula (Ugh! There he goes again!) for power ratios and multiply the logarithm
by 2. Thus, for converting two voltages to a db ratio, the formula is db=20 log
E_{1}E_{2}. However, a db is a db. If you compare two specific signals,
the respective powers will be different from the respective voltages, but the db
ratio will always be the same.
The point may be further illustrated by reference to. Table 2, where the db equivalents
for many voltage ratios are given. If the voltage across a given load is doubled,
there is an increase of 6 db. Note that doubling the voltage across the load will
have quadrupled the power. Thus a 4time increase in power (Table 1) is a gain of
6 db.
Reading a Db Meter
Table 1  Common power ratios in db.
Table 2  Common power ratios in db.
A word of caution is necessary when reading db directly on a meter. The latter
is actually a voltmeter, not a power meter, and is therefore calibrated for readings
across a specific impedance. For relative readings, this is not much of a problem.
For instance, if one should start out with a reading of +5 db across any circuit,
and this reading should change to +3 db during tests across the same impedance,
the change will always be a difference of 2 db.
For reading dbm however, the impedance for which the meter was calibrated must
be known. In most cases, this is "0 db1 mw. in 600 ohms," usually printed on the
meter face. However some older meters are calibrated to make zero db equal to 6
mw. in 500 ohms. In either case, this direct reading is obtained on one range, and
direct readings on other ranges are corrected by adding or subtracting so many db,
as specified by the manufacturer, where separate db scales are not used.
If the impedance at which the db scales have been calibrated is in doubt, a check
can be made against the voltage scales. For example, if 0.775 volt corresponds to
zero db, the meter has been calibrated to the modern standard, and powers in a 600ohm
circuit can be read directly off the meter in dbm. Such readings can be made across
other impedances too, simply by applying a correction factor. For example, let us
suppose the meter is calibrated for 600 ohms. In reading across a 1200ohm circuit,
one would simply subtract 3 db from the scale reading to obtain the power in dbm.
Across a 300ohm circuit, add 3 db to the reading; across 150 ohms, add 6 db; and
across 75 ohms, add 9 db. Correction factors can be worked out for any impedance.
The important thing is to start using db wherever possible, even when they are
not needed. The familiarity thus acquired will simplify matters when db becomes
essential.
Posted January 10, 2023
