September 1930 RadioCraft
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from RadioCraft,
published 1929  1953. All copyrights are hereby acknowledged.

Included in this first of a series of the 'Simple Mathematics for
the Serviceman' series that ran in RadioCraft is another 'cheat
sheet' full of oftused formulas. It begins with basic Ohm's law,
resistance, inductance, and capacitance, then builds from there.
What was valid in 1930 is still valid in 2015. See October 1930
for the
next installment that includes yet another handydandy cheat
sheet of formulas.
Simple Radio Mathematics for the Service Man
By Boris S. Naimark
It is generally believed that mathematics is a highly complicated
science and that formulas, being mathematical expressions, are equally
involved and complicated. There are consequently many radio men
who have acquired a deeprooted, though not very wise, aversion
toward the mathematical end of the radio science. They may be termed
"formula shy."
However, there is no royal road to knowledge, and, since radio
is a mathematical science, the amateur and, particularly, the professional
Service Man must admit that, without a thorough knowledge of at
least the basic formula, he is "licked."
This article is intended primarily for Service Men who take their
calling seriously. It comprises a discussion of formulas most commonly
encountered or useful in the radio shop or practical laboratory.
These are tabulated also in a form that permits quick and convenient
reference.
Use this table as you would a vacuumtube chart. The intelligent
Service Man does not attempt to memorize all the facts found in
the conventional tube chart. He first learns how to use it; he determines
what information the chart contains that may be of interest to him.
Thus, when a question presents itself, he knows that the tube chart
contains the answer; and he consults it as frequently as he may
find desirable.
Follow the same procedure with the formulas given. Of course,
first read this article carefully and familiarize yourself with
the possibilities for more intelligent and efficient application
of theory to practice. Post the table where it can be consulted
when need be; better yet paste it on a stiff cardboard and tack
it to the wall of your workshop.
While it is not recommended that all the data be memorized outright,
those who apply a little effort will be rewarded by a thorough knowledge
of the formulas presented and as time goes on will actually know
them by heart.
To begin with, let us understand that a formula is nothing but
a statement of fact or facts. It is, of course, expressed in symbols
and invariably is the most concise statement of the fact or facts
it deals with.
Current, Resistance, Voltage
Let us consider Ohm's Law  the most fundamental of all electrical
formulas. This law is basic, and finds application wherever potentials
or voltages are involved. It is one of the few formulas that the
Service Man should memorize outright. The basic form of Ohm's Law
is
where I is current in amperes, E the potential difference in
volts, and R the resistance in ohms. This formula simply expresses
the fact that, in a D.C. circuit, the current is directly proportional
to the voltage impressed upon the circuit and is inversely proportional
to the resistance of the circuit. In other words, the current within
a circuit will increase as we increase the E.M.F. impressed upon
it; and will decrease as we increase the resistance of the circuit.
It therefore stands to reason that, if in a given circuit we increase
both the E.M.F. and the resistance in the same ratio, the current
flow remains unchanged.
The first modification of Ohm's Law tells us that the voltage
drop across any two points of a circuit is proportional to just
two factors; current and resistance.
Thus this formula is particularly useful in the workshop.
Fig. 1  In either circuit, multiplying the plate
current of the tube by the value of the cathode  (filament) return
resistor gives the grid bias.
In Fig. 1, two extensivelyused methods of obtaining "C" bias
by the voltage drop across a resistor are shown. In practical service
work, it is often desirable to determine what value of resistor
to use, in order to obtain the desired value of biasing potential.
For this calculation, the third modification of Ohm's Law is employed.
In using the formula
R is the desired resistor value, E the voltage to be obtained,
and I stands for current. This current (in the case of Fig. 1a)
is the normal value in amperes of plate current for that tube. Quite
frequently, as shown in Fig. 2, a single resistor is used to obtain
biasing voltage for several tubes. When such is the case, in the
above formula I represents the sum of the plate currents of the
several tubes to be biased.
Fig. 2  The rule is the same here, except that
the current of more than one tube passes through R. The sum of their
currents is therefore used.
In the practical application of Ohm's Law in any of the basic
forms, it should be understood that it can be applied equally well
to the whole circuit or to any definite portion of a circuit. Care
must be exercised that the known values are those of that particular
section of the circuit, and not those of the entire circuit. Fig.
3 illustrates this point graphically.
Fig. 3  While the voltage drop across any resistance
in a circuit is equal to its value multiplied by the current passed,
the wattage (which measures the heat generated) is equal to the
resistance times the square of the current. Different wattage ratings
may be necessary in a series of resistors.
Wattage Formulas
In electrical circuits it is not sufficient to know that, for
a given purpose a resistor of a certain ohmage must be employed.
We must also know what wattage rating the resistor must have, in
order to carry safely the required current. The wattage may be easily
determined from the expression
where W is the power in watts, E the E.M.F. in volts, I in the
current in amperes, and R the resistance in ohms. The same precautions
must be observed when using this formula as in the use of Ohm's
Law; that is, with regards to its use in entire or partial circuits.
It is quite customary, in radio design, to employ resistors of
a wattage rating several times greater than the calculations call
for. This is a precaution against resistor burnouts, where facilities
for ventilation, are poor and the danger of burnout through overheating
is correspondingly high.
The above formulas are fundamental in radio work, and the Service
Man who preserves them for reference will find them frequently useful.
The wattage rating of a resistor applies to the entire resistor
only, and not to any section of the resistor. It is apparent that,
in order to pass a certain amount "of current through a given resistor
of definite resistance and wattage, it is necessary to apply a certain
definite potential across the resistor. If this potential, however,
is applied across a section of the unit, the decrease in the resistance
will cause a corresponding increase in the current, through the
section employed, which will naturally be beyond the permissible
safe value and the useful life of that particular section will be
materially shortened if not suddenly terminated. Thus wattage ratings
of units are applicable to the entire units only, and not to any
section of them.
Commercial resistors are commonly marked as follows: "1,000
ohms  10 watts." It is frequently desirable to determine from such
a rating the value of current that may be safely passed through
the resistor without overloading. This (1/10ampere, in the instance
above) is obtained from the relation
Series and Parallel Circuits
We need not devote much space to the formulas devoted to the
determination of the resultant value of a combination of resistors
or condensers, in series or in parallel. The formulas themselves
are quite clear and require no special explanation. The wattage
of the individual resistors is not affected by the series connection.
It is clear, therefore, that the resulting resistor, consisting
of several resistances in series, should not be permitted to carry
more current than is permissible for the resistor of the series
having the lowest wattage rating. When two or more resistors are
connected in parallel, the resultant value of resistance is less
than that of the smallest of the resistors used in the parallel
connection. While the currentcarrying capacity of the individual
resistors remains unaffected the current permitted to pass through
the parallel circuit may be as great as the sum of the currentcarrying
capacities of the individual resistors.
With regard to condensers, when two or more are connected in
parallel, the resultant capacity is the sum of the individual capacities.
When two or more condensers, however, are connected in series, the
resultant capacity is smaller than the smallest of the individual
capacities used in the series arrangement. Condensers of the higher
capacities are, as a rule, rated for their working voltage. When
voltages in excess of the rating are applied across the terminals
of anyone condenser its useful life is materially shortened, and,
if the excess is great, its usefulness may be instantly terminated.
The question naturally presents itself, how this working voltage
of the individual condensers is affected when two or more of them
are connected either in series or in parallel. For a parallel connection,
the voltage impressed across the parallel bank should not exceed
the voltage rating of the condenser having the lowest working voltage
rating in the parallel circuit. When condensers are connected in
a series circuit the voltage applied across the entire series may
be greater than the working voltage specified for the individual
condensers. When all of the individual condensers are of equal capacity
and of equal voltage rating we may apply across the series a voltage
equal to the sum of the individual working voltages.
The above is correct only if the D.C. resistances of the condensers
are of like value. Such a. condition very seldom obtains in practice
and, in order to prevent uneven distribution of voltage, a resistor
of 100,000 ohms is customarily shunted across each individual condenser
of the series. This shunt resistance is sufficiently great and can
not interfere with the normal functions of the condensers. When
the condensers in the series are of unequal capacities, the voltage
distribution is uneven; being affected by difference of capacity
as well as by difference in the D.C. resistance of the individual
condensers.
Voltages Across Condensers
Ignoring the effect of D.C. resistance it may be stated, with
sufficient accuracy (especially where a reasonable factor of safety
is allowed), that the voltage across the individual condensers in
the series is inversely proportional to their capacities. The smallest
of the condensers will have the highest voltage drop across its
terminals. Thus if we have connected in series three filter condensers
rated as follows  2 mf.  350 volts; 3 mf.  350 volts; 4 mf. 
350 volts  and the voltage impressed across the series is 900 volts,
the 4mf. condenser will have approximately 225 volts drop through
it. The 3mf. condenser will have approximately 300 volts; and the
2mf. condenser, approximately 450 volts. It can be seen at once
that the 2mf. 350volt condenser is not suitable for the series;
since the voltage drop through it is considerably greater than the
specified working voltage,
It is frequently asked why it is that the A.C. rating of a condenser
is lower than its D.C. rating. The explanation lies in the nature
of alternating current, which does not possess the steady characteristics
of direct current. The form of alternating current, usually assumed
in A.C. theory is that of a "sine curve," shown in Fig. 4. It is
apparent, then, that the value of either current or voltage in an
A.C. circuit is continually changing; assuming successively all
values from its positive maximum to its negative maximum. A vacuumtube
voltmeter indicates the maximum or peak values of an alternating
voltage. A.C. voltmeters and ammeters, however, indicate the effective
values; which, as the formula chart shows, are equal to only 0.707
of the maximum values.
A.C. voltages or currents generally employed are the effective
or "rootmeansquare" (R.M.S.) values. Thus, if an electric receiver
is said to operate from 110 volts A.C., the effective value is referred
to. Similarly, if a condenser is rated for 300 volts A.C., the effective
value is meant. The very same condenser is probably rated for 450
volts D.C. The peak value of A.C. that may be impressed upon the
condenser is also approximately 450 volts. Thus the A.C. voltage
rating of a condenser is lower than its D.C. rating because the
A.C. rating is in effective, and not maximum, value. If the maximum
value were used for A.C. condenser rating, it would be of the same
value as the D.C. rating. If a condenser is rated for direct current
only, and it is desired to determine the A.C. rating by calculation,
assume the A.C. rating (in effective volts R.M.S.) to be equal to
approximately seventenths of the D.C. rating.
Fig. 4  The familiar "waveform" at the right
is not the picture of a real wave; it is merely a spreading out
of the circle at the left, so that the time of each current value
can be measured along the base line, instead of in an angle around
the center of the circle.
A.C. Circuits
We now come to the subject of Ohm's Law for alternating current.
When an A.C. circuit has resistance only, the formulas given for
D.C. are equally well applicable to A.C. circuits. Where inductance
and capacity are present, the term "impedance" takes the place of
"resistance" in the formula. The value of impedance depends upon
how much capacity and inductance are present in an A.C. circuit;
and it can be easily determined from the formulas given in the chart.
As with all other formulas, the terms of the expression may be transposed,
to solve for either the voltage or the impedance.
Both capacity and inductance retard the current in an A.C. circuit.
This opposition to the current flow is dependent upon the inductive
and capacitive reactances, of the inductance and capacity respectively.
Inductive reactance increases with the inductance value and the
frequency. Capacitive reactance, however, decreases with both the
capacity value and the frequency.
(To be
continued)
November 25, 2015
