CIRCUITS WITH MORE THAN ONE SOURCE OF EMF
ACTION OF CONDENSERS IN DIRECT CURRENT CIRCUITS
From our every-day contact with electrical appliances in the home-door bells, electric lights, etc. we all have a general idea of what electricity will do in the simpler forms of circuits. In this Section we shall study the action of electrical currents which flow continuously in a single direction and are known as direct currents. We shall use, insofar as possible, illustrations from the telephone plant with which we are all familiar. Also, to aid us in picturing in our minds some of the things discussed, we shall make use of the simple analogy between the flow of water through pipes and the flow of electricity along conductors.
Fig. 1 shows two tanks, A and B, connected by a pipe C. Tank A has water at a higher level than B; therefore, there is a difference of pressure, and so long as this exists there will be a flow of water through C. The amount of water which will flow through C in a given time will depend upon the difference of pressure between the two tanks and upon the resistance with which the pipe opposes the flow of water through it. As the relation between the driving pressure, the opposing resistance and the resulting flow is definite and exact, this relation can be expressed mathematically if we have units in terms of which to measure the amount of each of these three quantities. For this purpose we can, as is commonly done, express the difference in pressure between the two tanks in pounds per square inch. Also, if a water meter were placed in the pipe, it would record the amount of water through it in terms of cubic feet, or gallons, or some other unit of quantity, so that, by timing the meter with a watch, we could determine the amount of water (in gallons, let us assume) which would flow during some unit of time, say, one second. We should then be able to express the rate of flow, or the current, in terms of gallons per second; i.e., one gallon per second is the unit of current corresponding to unit quantity of one gallon and unit time of one second. Finally, having selected a unit of pressure and a unit of current, we could define the unit of resistance to be that amount of resistance (as determined by the size, length, etc., of the pipe) which would allow a rate of flow of exactly one unit of current under the action of one unit of pressure.
In electrical circuits we have to deal with quantities which are analogous to the pressure, amount, current and resistance just discussed; and for each of these electrical quantities there is a unit to which has been given a name commemorating a famous early investigator of the science of electricity. Electric pressure is called "electromotive force," meaning, of course, the force or pressure which causes electricity to move or flow; but we most commonly refer to it as "voltage," a term which, as will shortly appear, is derived from the name of its unit. The term "emf" (pronounced by spelling out), which is the abbreviation of electromotive force, is also quite commonly used instead of voltage. In the preceding analogy we chose one pound per square inch as the unit in terms of which to express water pressure; the unit of electromotive force, or voltage, is one volt.
We also chose one gallon as the unit in terms of which to express the amount or quantity of water. Quantity of electricity is usually spoken of as the amount of "electric charge"; the unit in terms of which this is ordinarily expressed, i.e., unit electric charge, is one coulomb. With electric charge, however, we shall be but little concerned until we take up the action of condensers in alternating current circuits, in Section II: in our present study of direct currents we shall be interested, not in electric charges themselves, but in their rates of flow, i.e., in electric currents.
There is no commonly used special term for electric current; ordinarily it is spoken of merely as the "current"; but we do have a name for the unit of electric current. Having chosen one gallon as the unit of quantity of water, and one second as the unit of time, we naturally expressed rate of flow or current of water in gallons per second because there is no special name for unit current of water. In like manner, it would be entirely correct to express electric current in terms of coulombs per second. Indeed, this is exactly what is done, except that the unit current of one coulomp per second is called one "ampere." The abbreviation of ampere is amp. For convenience, currents smaller than an ampere are often expressed in milliamperes, commonly abbreviated "mils," one milliampere being one one-thousandth part of one ampere. To convert amperes into milliamperes, multiply by 1,000: e.g., .032 amp = 32 mils.
Electric resistance, the opposition which a conductor offers to the flow of electricity, is simply called "resistance." The unit of resistance to an electric current is one "ohm." By definition, a circuit has a resistance of one ohm if it permits a current of one ampere to flow through it under the action of an emf of one volt. This one-to-one relationship between the volt, the ampere and the ohm is shown graphically by the circuit in Fig. 2.
In the case of the flow of water it was stated that a definite relation existed between the driving pressure, the opposing resistance and the resulting rate of flow, and that this relation could be expressed mathematically. Exactly the same sort of relation exists between the voltage, resistance and current in electrical circuits. We have seen above that the units of these three electrical quantities have been so selected that an emf of one volt, applied to a circuit whose resistance is one ohm, will cause a current of one ampere to flow. Carrying this relationship a step further, we may state that current equals voltage divided by resistance, or
where I is the current in amperes, E is the emf in volts and R is the resistance in ohms. The foregoing general relation is known as "Ohm's Law." Expressed another way, the resistance is equal to the voltage divided by the current, or
The third way of expressing this relation is to say that the voltage equals the current times the resistance, or
We can illustrate these three forms in which Ohm's Law may be stated by examples from the circuit in Fig. 3:-To find the current flowing in this circuit of 3 ohms resistance with a 6-volt battery, we have
To find the resistance of this circuit which has a current of 2 amperes flowing under pressure of 6 volts, we have
To find the voltage which will cause a current of 2 amperes to flow through this circuit of 3 ohms resistance, we have
|E||=||IR||=||I x R||=||2 x 3
The different methods· of expressing Ohm's Law are given in the following table:
I = R
Amperes = --
R = - Ohms = ---
Voltage = Current X Resistance
E = IR Volts = Amperes X Ohms
Current Webmaster comments: (October, 2012)
Now that you are thoroughly confused....
I have always had difficulty in remembering the specific variations of Ohm's law. I have no problem with the concept just the arrangements of formula configurations.
In High School I had a Geometry teacher that solved the problem for me.
Draw a circle.
Draw a line dividing the circle into a top and bottom half.
Now divide the bottom half into quarters with a vertical line.
In the top half, place a E for electromotive force or voltage.
In the bottom left quadrant enter an I for amperage.
In the bottom right quadrant enter an R for resistance.
Now, mentally cover the variable that you are trying to find and the other two variables will fall into the proper positions in your formula.
I think of this in my head as, "E over I R" which I can remember fairly easily.
While it doesn't apply to this course, you can do the same with Watt's law by thinking "P over I E" where the P = power or watts.
SERIES RELATIONS IN DIRECT CURRENT
Let us again refer to our water system analogy to
illustrate a series circuit. We will assume that pipes
a, b and c in Fig. 4 are all of one diameter and are
joined in series by perfectly smooth connections to
form a line through which the water can flow from
a b c B J [ J l::=:j
A to B. If a is three feet long, b three feet long and
c two feet long, we can then consider the total resistance
to the flow of water to be the same as t:t9.t
of one pipe whose length equals the sum of these
three lengths, or 8 feet long. The same relation
holds true for electric circuits where the current
flows through a series of resistances :-The combined
resistance of two or more resistances connected in
series is equal to the sum of the individual resistances.
To illustrate the principle just stated, let us take
an example from one of the simplest of practical telephone
systems. This simple system, shown schematically
in Fig. 5, consists of two magneto telephones
connected together by two wires. In each of these
telephones direct current is supplied through the
transmitter by a battery of dry cells connected in a
local circuit comprised of the battery, the transmitter,
one winding of an induction coil and the connecting
leads, all joined in series as indicated. This
type of circuit is called a simple series circuit. Let
us assume the resistance of the transmitter to be 40
ohms•, the resistance of the coil winding to be 5
•The resistance of transmitters varies over a wide
range, its value depending upon the type of transmitter
and upon the supply current through it.
;Forty ohms is merely a representative value assumed
for purpose of illustration.
·ohms, and the resistance of the connecting leads to
be so small compared with the foregoing values that
it can be neglected in practical computations ; and
let us further assume the transmitter requires a supply
current of .1 ampere. The question we wish to
answer is : What battery voltage will be needed to
furnish this required supply current 1 In applying
Ohm's Law to this question we mu· st first determine
the combined resistance of the local circuit by summing
the resistances of its individual parts ; from
the above assumed values this total resistance 1s
found to be
40 ohms + 5 ohms = 45 ohms.
Then, knowing from Ohm's Law that the voltage
must equal the product of the current multiplied by
the resistance, the voltage needed to supply the required
.1 ampere to the transmitter through the total
local circuit resistance of 45 ohms is
E = I X R = .1 X 45 = 4.5 volts.
A second example illustrating the principle of
combining resistances connected in series is presented
by Fig. 6, in which is shown the path of the
direct current supplied from the central office battery
to a common battery loop and subset. Starting
at the battery, we trace this series circuit through
one winding of the repeating coil, a heat coil, one
side of the cable pair, a winding of the induction
coil, the transmitter, the other side of the cable pair,
another heat coil, the supervisory relay, a second
winding of the repeating coil, and back to the opposite
side of the central office battery. Assuming
the resistances of the various elements in the circuit
to be as shown in the schematic, the combined resistance
of this battery supply path is the sum of
the following :
Repeating coil winding
One mile 22-gauge cable conductor
Induction coil winding
One mile 22-gauge cable conductor
Repeating coil winding
The emf acting upon this circuit whose total resistance
is 300 ohms, is 24 volts. Applying Ohm's
Law, the supply current flowing in this circuit is
found to be
1 =- =- = .080 amp.
Thus far in these examples pertaining to series circuits
we have concerned ourselves solely with the
circuit as a whole; the circuit elements have been
considered only as regards their contribution to the
total resistance; and, in applying Ohm's Law, we
have dealt with total circuit resistance and with the
battery voltage. We shall, however, also have frePage
quent need to apply Ohm's Law to individual circuit
elements. Let us see what sort of information can
be obtained by so doing.
Suppose we were to connect a voltmeter across the
transmitter in Fig. 6, as is indicated in Fig. 7. A
voltage such as would thus be measured is called a
drop in potential or a voltage drop-in this particular
instance, the potential or voltage drop across the
transmitter. By Ohm's Law the voltage drop across
any given circuit element equals the resistance of
that element multiplied by the current through it.
If we were to measure the drop across each of the
circuit elements in Fig. 6, and were to add all these
drops together, the total voltage so obtained would
equal the voltage of the central office battery supplying
the circuit. But the resistance of each circuit
element is known, and the current in the circuit has
already been found to be .080 ampere; consequently
these potential drops and the total voltage can also
be computed by the Ohm's Law relation I X R = E.
The values thus obtained are:
Repeating coil winding
One cable conductor
Induction coil winding
.080 X 21 = 1.68 volts
.080 X 3 = .24 "
.080 X 86 = 6.88 "
.080 X 15 = 1.20 "
One cable conductor
Repeating coil winding
.080 X 56 = 4.48 "
.080 X 86 == 6.88 "
.080 X 3 = .24 "
.080 X 9 = .72 "
.080 X 21 = 1.68 ·'
Likewise, had we obtained the above potential
drops by measurement, and had we also measured
the .080 ampere current, we could have used the
Ohm's Law relation E -:- I = R to determine the resistances
of the circuit elements:
Repeating coil winding
One cable conductor
Induction coil winding
One cable conductor
Repeating coil winding
1.68 -:- .080 = 21 ohms
.24 -:- .080 = 3 "
6.88 -:- .080 = 86 "
1.20 -:- .080 = 15 "
4.48 -:- .080 = 56 "
6.88 -:- .080 = 86 "
.24 -:- .080 = 3 "
.72 -:- .080 = 9 "
1.68 -:- .080 = 21 "
Finally, knowing the resistances of the circuit elements,
and measuring the voltage drop across each,
we could use the Ohm's Law relation E -:- R = I to
demonstrate that the direct current at any instant is
the same in all parts of a series circuit-
Repeating coil winding
One cable conductor
Induction coil winding
1.68 -:- 21 = .080 amp
.24 -:- 3 = " "
6.88 -:- 8£ = " "
1.20 -:- 15 = " "
4.48 -:- 56 = " "
CIRCUITS WITH MORE THAN ONE SOURCE
We shall now consider another important principle
of electric circuits. This is the principle by means
of which can be determined the current resulting
when more than one source of emf is acting in a
Figs. 8 and 9 show simple series circuits in each of
which are two sources of emf. The two sources in
Fig. 8 are poled to act in the same direction around
the circuit, whereas those in Fig. 9 oppose one another.
In each of these circuits the two sources may
be treated as an equivalent single source: the aiding
sources in Fig. 8 are equivalent to a single source
whose voltage is that obtained by adding the two
voltages; the opposing sources in Fig. 9 are equivalent
to a single source whose voltage equals that ob-
.--------------o..o--------------.Jl 3 Ohms
tained by subtracting the smaller of the two voltages
from the larger. Illustrating the foregoing with the
values shown in the schematics, the equivalent voltage
in Fig. 8 is 24 + 6 = 30 volts, while that in Fig. 9
is 24 - 6 = 18 volts. The resistance of each of these
circuits being given as 3 + 3 = 6 ohms, the currents
can be determined by Ohm's Law. In Fig. 8
E 24 + 6 30
I = - = ---= - = 5 amp
and in Fig. 9
R 6 6
E 24 - 6 18
I = - = --= - = 3 amp.
R 6 6
The preceding method, in which the equivalent
voltage is first determined, is a satisfactory means of
computing the current due to more than one source
of emf in simple series circuit arrangements ; but it
does not readily apply to more complex circuits. By
comparing the above values of the currents due to
the two sources acting together, with the currents
which would flow if each of the sources were to act
alone, we are, however, led to a method by which the
current due to more than one source in any type of
circuit can be determined. Referring again- to Figs.
8 and 9, if the 6-volt source were not present the current
due to the 24-volt source alone would be
I = - = - = 4 amp.
Similarly, the current due to the 6-volt source alone
I = - = - = lamp.
.As to the directions of these currents in the two circuits,
the poling of the sources is such that both currents
would flow in the same direction ( clockwise)
around Fig. 8; but in Fig. 9 they would flow in op·
posite directions, the 4 amperes clockwise and the 1
ampere counter-clockwise. Now, where two or more
currents flow along a path at the same time, we
should expect those flowing in the same direction to
add and those in the opposite direction to subtract.
Consequently, when both sources act at the same
time, we should expect the current in Fig. 8 to be
4 + 1 = 5 amperes, and that in Fig. 9 to be 4 - 1 =
3 amperes. These values are seen to agree exactly
with those already found by the equivalent voltage
method, and so our expectations are confirmed.
The manner in which we have just obtained these
values illustrates the application, to simple series
circuits, of a principle which will enable us to determine
the current in a circuit in which two or more
emf sources are acting simultaneously. For our purposes
this principle may be stated as follows : The
current at any point in a circuit in which two or more
sources of emf are acting simultaneously, is equal to
the net sum of the individual currents which would
flow at that point if each of the sources in turn were
to act alone-this net sum being the value obtained
by subtracting the total of such currents flowing in
one direction from the total in the opposite direction.
This easily understood and frequently useful principle
is known by a rather technical sounding name;
it is called the "Superposition Theorem." There is,
however, one restriction to be observed regarding its
application; this principle is not applicable to any
circuit .in which current flows through any element
whose resistance varies with either the magnitude or
the direction of the current through it. A carbon
button transmitter is an example of an element
whose resistance varies with the magnitude of the
current through it, and a varistor is an example of
an element whose resistance depends upon the direction
of the current.
ACTION OF CONDENSERS IN DIRECT
Referring again to Fig. 6, we see another circuit
which, starting from the contact of the switchhook,
passes through the receiver, the induction coil, the
condenser, and thence to the line side of the transmitter.
If an ammeter were placed there, no direct
current would be found in this circuit. This is because
the condenser will not permit direct current
to flow through it. A condenser consists of two conducting
surfaces so separated by an insulating material
that there is no electrical connection between
the two surfaces. The conducting surfaces are commonly
formed of metal plates or of metallic foil,
while mica, glass, paraffined paper, and air are the
common insulating materials.
Fig. 10 shows a condenser, consisting of two metallic
plates separated by air, connected in circuit with
a battery and a sensitive meter. When this circuit is
established electricity will flow from the battery into
the condenser, where it will be stored, thereby giving
rise across the condenser to a potential drop which
increases as the stored charge accumulates, and
which opposes the voltage producing the charging
current. Consequently, as the voltage across the
condenser builds up, the charging current will decrease
from its initial value until, when the condenser
voltage reaches the charging voltage, the current
will cease. The quantity of electricity which
must be stored in the condenser to charge it up to a
given voltage depends upon the capacity of the condenser,
and this, in turn, depends upon the area of
the plates, upon the distance between them, and upon
the kind of insulating material separating them.
Under ordinary conditions the charging of a condenser
occurs so rapidly it may, for most practical
purposes, be thought of as being completed almost
instantly, after which, since the current then ceases,
the circuit may be considered to be open at the condenser.
There are many instances in the telephone plant
where condensers are used primarily to prevent the
flow of direct current. The condensers in subscriber
sets are an example. These prevent direct current
from flowing through the receiver or through the
The action of condensers in alternating current
circuits will be explained in Section II.
In the previous Section we have considered only
the series circuit, i.e., the type of circuit in which all
the current flows in a single path. In the telephone
plant, however, we find many circuits in which the
current divides between two or more paths. Such
paths are known as parallel paths. To begin our
study of these we again consider a water system
Fig. 11 shows a water system in which there are
two similar paths for the flow of water from A to B.
With a constant pressure maintained at A the rate
of flow of water in each pipe will be the same as it
would be if the other pipe were not present. Consequently
the rate of flow with two pipes will be
twice as great as with only one pipe; and the more
pipes we add in parallel between A and B the greater
will be the rate of flow.
We have already learned that rate of flow equals
pressure divided by resistance; hence, in any case
in which the rate of flow is increased, either the pressure
must have been increased or the resistance decreased,
or both. In the above case we assumed the
pressure constant ; the result of placing several pipes
in parallel must, therefore, have been to decrease the
Fig. 12 shows an electrical circuit of the type to
which the water system of Fig. 11 is analogous. We
refer to the two resistances R1 and R2 as being in
"parallel" or in "multiple." With two equal paths in
multiple and with a constant voltage at the battery,
we shall have twice the current flow that we should.
have with only one path. The combined resistance of
the two equal resistances in multiple is then one-half
that of either one; the combined resistance of three
equal resistances in multiple is one-third ; etc. If the
resistances R1 and R2 are not equal we must find
some way in which to calculate the combined re- •
sistance of the two in multiple. There is a simple
rule by which we can compute the combined resistance
of any number of resistances in multiple ; this
combined resistance equals the reciprocal of the sum
of the reciprocals of the individual resistances. The
reciprocal of a number is, of course, "one" divided
by that number. For example, the reciprocal of 3 is
1j3, the reciprocal of 12 is 1j12, etc. If we have only
two resistances in multiple, we can simplify this
rule: the combined resistance of two resistances in
multiple is equal to their product divided by their
To illustrate the preceding, let us assume the value
of the resistances in Fig. 12 to be
R1 = 6 ohms ' and R2 = 12 ohms
as indicated in Fig. 13. Then, applying the first of
the above rules, the sum of the reciprocals of 6 and
12 is found to be
1 1 3
- + - = -
6 12 12
and the reciprocal of 3j12 is
3 12 12
1 +- = 1 X -=-= 4 ohms
12 3 3
The second or simplified rule will, of course, lead to
the same value
6 X 12 72
6 + 12 18
\V e have now determined the combined resistance
of the multiple paths from a to b in Fig. 13 to be 4
..!!.. 24 Volti
6 Ohms f ..
12 Ohms<'.. ..?
ohms. This circuit can, therefore, be treated the
same as it would be if the points a and b were connected
by a single path of 4 ohms as shown in Fig. 14.
If we have to deal with more than two paths in
multiple, we may either use the first rule to deterPage
-. 24 Volts 4 Ohms J,.];
mine their combined resistance, or we may determine
this by simplifying the circuit, step by step, by means ·
of successive applications of the second rule. For
example, Fig. 15 shows a circuit in which there are
a b c
( "'? .. ..24 Volts 7 Ohms .. 9 Ohms<;. 3 Ohms:: ? ?
three paths, a, b, and c, in multiple. As the first step
in simplifying this circuit let us apply the second
rule to determine the combined resistance of paths
a and b; we obtain
---= ---= 3.94 ohms.
a + b 7 + 9
Consequently, we can consider paths a and b to be
replaced by a single path of 3.94 ohms in multiple
with path c. The combined resistance of 3.94 ohms
in multiple with 3 ohms is, therefore, equal to the
combined resistance of all three paths. Again applying
the secontl rule, this value is found to be
3.94 X 3
---= 1.7 ohms.
3.94 + 3
If there were a fourth multiple path in Fig. 15 we
would combine its resistance with the 1.7 ohms, etc.
The term "conductance" is used to express the
facility with which a conductor allows current to
flow; conductance and resistance are, therefore,
merely opposite aspects of the property which determines
how well or how poorly circuit elements
conduct electricity. The unit of conductance is one
mho, a term obtained by spelling ohm backward. By
definition, one mho is the conductance of a one-ohm
resistance. If we halve resistance, conductance is
doubled; if we reduce resistance to one-third, conductance
is tripled; etc. Consequently, since the conductance
of a one-ohm resistance is one mho, the
conductance of a one-half ohm resistance is 2 mhos,
the conductance of a one-third ohm resistance is 3
mhos, etc. In other words the conductance of any
circuit element in mhos is the reciprocal of its resistance
in ohms, and vice versa. Also, just as the
combined resistance of circuit elements connected in
series is the sum of their individual resistances, just
so is the conductance of paths connected in multiple
the sum of the conductances of the individual paths.
To illustrate, the conductances of the three paths in
Fig. 15 are 1j7 mho, 1j9 mho and 1j3 mho, respectively,
and the combined conductance of these
1 1 1 37
- + - + - = -mho.
7 9 3 63
Furthermore, the reciprocal of this combined conPage
ductance must equal the combined resistance of the
paths in multiple-this value we find to be
37 63 63
1 -7- - = 1 X - = - = 1. 7 ohms.
63 37 37
This result agrees, of course, with the value already
obtained by successive applications of the second
rule for evaluating the combined resistance of multiple
paths; and the manner in which it was arrived
at explains the first rule.
A very familiar example of parallel paths in the
plant is the supervisory relay winding which consists
of two windings in multiple-usually represented as
shown in Fig. 16. Winding a is known as a nona
inductive winding and winding b as an inductive
winding. As will be explained more fully in a later
Section, winding a has no effect on the relay action,
but is merely to furnish a path for the voice current.
The resistances of the two windings being approximately
equal, direct current divides about equally
between them. If winding b becomes opened the resultant
failure of the relay to operate makes this
trouble an easy one to locate. If winding a becomes
opened, however, the talking current will be greatly
weakened, but the relay will continue to function,
thus making the cause of this trouble less easy to
detect. One symptom of the latter trouble would be
sticking of the relay because all the current would
then pass through b winding and would, therefore,
produce a very strong magnetic pull.
The multipled cable conductors comprising the
battery feed for private branch exchange boardsknown
to all of us as P.B.X. boards-furnish another
familiar example of parallel paths. P.B.X. boards
require a voltage across the bus bars sufficient to
operate relay and lamp signals and to supply current
to the station transmitters. We can supply this by
putting a storage battery at the P.B.X. (usually not
economical for small boards) , or we can supply the
voltage over cable conductors from the central office
battery. The requirement is that at all times, regardless
of the load on the switchboard, there be sufficient
voltage to operate the board properly. Let us
assume the lowest permissible voltage at a given
P.B.X. to be 14 volts, and the minimum central office
battery voltage to be 22 volts ; this will allow a drop
in voltage over the cable conductors of 8 volts. We
learned in Section I-B that the drop in voltage over
any part of a series circuit is dependent upon the
current flowing iri the circuit. If we know the maximum
current requirement for the P.B.X., we can
easily determine the combined resistance of the
group of cable conductors to give a voltage drop of
8 volts. Let us assume a maximum current drain of
1 ampere ; then
R =- =- = 8 ohms.
Fig. 17 shows the schematic of the battery feed
for a P.B.X. board at B, located 1% miles from the
central office, with only 22-gauge cable available.
The loop resistance of a single pair of 22-gauge
cable conductors 1% miles long is 256 ohms, but as
determined above, the loop resistance of the battery
II 22 Ga. Cable Conductor
..24 Volt• r--" Mil..--1 A
# 22 Ga. Cable Conductor
· Fig. 17
feed must be not more than 8 ohms. From this we
see we shall require a great number of conductors in
We have already learned that two conductors of
the same gauge in mulk,iple have one-half the resistance
of one conductor, three conductors of the same
gauge have one-third, etc. Therefore, the combined
resistance of any number of similar conductors in
multiple equals the resistance of one conductor
divided by the total number. We can reverse this
and say that the resistance of one conductor, divided
by the combined resistance desired, will equal the
number of conductors required.
In the example above we require a lead of 8 ohms
made up of a number of conductors in multiple, each
of which has a resistance of 256 ohms. By the rule
just stated the number of conductors which must be
connected in multiple in this case can be found by
dividing 256 by 8. The result is 32. Hence 32 conductors
will be required in each side of the leada
total of 32 cable pairs for the complete lead.
The cost of 32-cable pairs one and one-half miles
long would, however, be very high ; if possible we
must find some cheaper way to accomplish our purpose.
Fig. 18 is a simplified sket-ch of this circuit with an
equivalent resistance unit of 4 ohms substituted for
the 32 conductors on each side of the line. In Fig. 19
we have replaced one side of the line by the earth
which is assumed to have no resistance. By this
means we not only save all the conductors in the side
of the lead replaced by the ground circuit, but also,
as we can now allow twice as much resistance in the
other side, we shall save half the conductors in that
side as well. The scheme in Fig. 19 will, therefore,
require but one-quarter as many conductors as that
in Fig. 18, viz., 8 cable pairs instead of 32.
The "A-board" cord circuits are a good example
of a large number of circuits in multiple. One cord
circuit requires only a small current, say, .15 ampere
; but when we have a large number of cords in
service simultaneously, as there would be in a large
office, the total current may amount to several hundred
In the telephone man's work it often becomes
necessary to build up a circuit which will have a
specified resistance, and in many cases a single resistance
unit of the desired value is not available.
If, however, we have an assortment of resistance
units we may be able to combine two or more of them
to give the value required.
l ..!!!.24 Volts 25 Ohms
Let us suppose, for example, that we wish to build
up the circuit in Fig. 20, but that the lowest resistance
unit in the assortment at hand is a 30-ohm unit
-how are we to obtain the required 25 ohms f' Noting
that the desired resistance is less than the smallest
unit available, we remember that the combined resistance
of paths in multiple is less than that of any
of the individual paths ; consequently we know we
shall have to connect some resistance in multiple
with the 30-ohm unit to secure the needed l ower
value. We also know it is easier to deal with parallel
paths in terms of their conductances than in terms
of their resistances: Hence we say we wish to determine
the value of the conductance which it will
be necessary to connect in multiple with (i.e., add
to) the available 1j30 mho to obtain the required
1j25 mho. This added conductance we find to be
1 1 1
- - - = -mho.
· 25 30 150
But 1j150 mho is the conductance of a 150-ohm resistance.
Consequently, a resistance of 150 ohms conPage
nected in multiple with the 30-ohm unit will provide
the required 25 ohms. If we do not have a 150-ohm
unit we shall have to combine other units-for example,
one 30-ohm and two 60-ohm units in seriesto
obtain the 150 ohms.
SERIES AND PARALLEL RESISTANCE COMBINATIONS
IN DIRECT CURRENT CIRCUITS
SERIES AND PARALLEL RESISTANCE
In previous discussions we have considered only
the simpler forms of circuits, viz., circuits in which
the resistance is either all in series in a single path
or is in two or more parallel paths. The circuits in
the telephone plant, however, are usually made up
of combinations both of series paths and of parallel
paths. Such circuit arrangements are called "networks."
There are some types of networks which
must be so designed that one part may be varied
without appreciably affecting the currents in other
parts. In a common battery system, for instance, we
have a vast network connected to a single source of
voltage (the central office battery) , and here it is
essential that we be able to change any branch or
any combination of _branches without disturbing conditions
at other points in the system. As the first step
in our study of how this can be accomplished we
shall again consider a water system analogy.
Fig. 21 shows a water system which illustrates a
combination of series and parallel paths. This system
consists of a supply tank A which maintains a
constant pressure, and a conducting pipe B and
header F which supply water to a system of water
motors C, D and E. If the operation of each of the
water motors is to be practically independent of the
others, the pressure at F must be maintained substantially
constant, regardless of how many of the
motors are operated simultaneously, otherwise the
operation of the motor C, for example, when operated
singly will be different from its operation when
the other two motors are also running. To meet this
requirement we would make pipe B large enough
so that, if we desire to run all three motors at the
same time, the pressure in F will be sufficient to insure
proper speed of all three, and also so that, if
motors D and E were shut down, the pressure in F
would not increase enough to seriously alter the
speed of motor C.
The foregoing water system is a very close analogy
to the P.B.X. battery feed sho..n in Fig. 22 : the supply
tank A is analo-gous to the central office battery ;
the pipe line B corresponds to the group of cable
conductors from the central office to the P.B.X. ; the
disttibuting head F plays the role of the P.B.X. busbars
; and the motors C, D and E represent the circuits
we wish to operate. The circuits C, D and E,
shown as simple resistances, may represent the simple
line lamp circuit ; or they may represent the complete
circuit from one P.B.X. extension to another
P.B.X. extension or trunk, including the cord circuit.
We know from experience that, if we do not have
sufficient cable pairs to make up B, the voltage at F,
when all the cords are busy, will be so low that the
signals will be dim and the battery supply current
will be insufficient to make the stations "talk up."
To make clear the reason for this we shall work out
Example : Let us assume the elements making up
the battery feed arrangement in Fig. 22 to have the
following values :
( 1 ) Let the voltmeter reading for the central
office battery be A = 22 volts.
( 2 ) Let the resistance of the supply lead from
the central office be B = 12 ohms.
(3) Let the resistance of the path connected
across the P.B.X. bus-bars by each busy
cord circuit be C = D = E = 70 ohms.
( 4) Also, assume the lowest bus-bar potential
permissible for satisfactory operation to be
i 70 Ohms .... C
With one cord busy, as shown in Fig. 23, the total
resistance of the circuit is
B + C = 12 + 70 = 82 ohms.
In this case the current in C is
I = EjR = 22 -7- 82 = .268 amp
and the potential across C ( same as from F to
ground, see Fig. 22) is
E = I X R = .268 X 70 = 18.8 volts.
..!!. 22 Volts
..1\ 1\1\' .
t 70 Ohms C
D j 70 Ohms
Next, with two cords busy, as shown in Fig. 24, the
equivalent resistance of the two equal paths then
bridged across the bus-bars is half that of one path,
or 35 ohms ; and the combined resistance of the circuit
12 + 35 = 47 ohms.
Under this condition the current fed from the central
office battery is
I = E/R = 22 + 47 = .468 amp.
This total current divides-equally, of course-between
the equal paths C and D, so that the current
in each path is
.468 + 2 = .234 amp.
The potential across the P.B.X. bus-bars may be
computed either from the total current and the combined
resistance of the two parallel paths, or from
the resistance of one of the paths and the current
E = IR = .468 X 35 = 16.4 volts
or .234 X 70 = 16.4 volts.
In like manner,· with three cords busy, as shown in
Fig. 25, the equivalent resistance of paths C, D and
E in multiple is 23.3 ohms ; the combined resistance
of the circuit is 35.3 ohms ; the current fed from the
battery is .623 ampere and that through each of the
three equal paths C, D, and E is .208 ampere ; and
the potential across the bus-bars is 14.5 volts. '
From the foregoing example we ·see that the voltage
across the P.B.X. bus-bars drops as the number
of busy cords is increased. Tabulating the results of
the above computations, we have-
The relation between the number of busy cords and
the corresponding bus-bar potential embodied in
these tabulated values, can be shown graphically by
what is known as a curve ; and as much information
in bulletins and text books is presented by means of
curves it will be well worth while, for those who have
not already done so, to take this opportunity to learn
to read a curve.
In Fig. 26 is shown a curve which portrays the relation-
just tabulated-between the voltage across
the P.B.X. bus-bars and the number of busy cords.
As this illustrates, curves are drawn upon a gridwork
background which most commonly consists of
uniformly spaced horizontal and vertical lines. The
first step in laying out a curve upon such a grid is
to assign suitable values ( of the quantities to be represented
by the curve) to the divisions into which
the vertical and horizontal grid lines divide one anPage
22 !::! 20 .. ID"' "" "
B .. '-
.. .. 1 2.
Number of Busy Cords
.... ........ 4
other. Thus, in Fig. 26, as designated by the scale of
values written along border line A, each division of
the horizontal lines represents one busy cord-the
value assigned to denote the number of busy cords
increases by one for each division we advance along
line A from the point representing zero cords. Similarly,
as designated by the scale of values written
·along border line B, each division of the vertical
lines in Fig. 26 represents 2 volts-the values assigned
start at 14 volts and increase by 2 volts per
division up to 22 volts. This range of voltages was
chosen here because it is the range in which we are
interested in the example we are now considering :
14 volts is the lowest bus-bar voltage at which operation
is satisfactory ; at the other extreme, with zero
busy cords there will be no current drain, the drop
over the feeder pairs will be zero, and hence the
voltage across the P.B.X. bus-bars will equal the assumed
central office battery voltage of 22 volts.
Having assigned suitable scales of values to the
grid lines, we next proceed to locate the points
through which the curve itself is to be drawn. We
have j ust seen that, with zero cords, the voltage
across the bus-bars is 22 volts ; hence, on the vertical
line representing zero cords, we mark the point corresponding
to 22 volts. Next, referring to the tabulation
above, we see that, with one cord busy, we shall
have 18.8 volts across the bus-bars ; consequently, on
the vertical line representing one busy cord, we mark
the point corresponding to 18.8 on the voltage scale.
In like manner, on the line representing two busy
cords, we mark the point corresponding to 16.4 on
the voltage scale ; and on the line representing three
busy cords we mark the point corresponding to 14.5
volts. We have now located four points through
which can be drawn a curve C. If we like, we can
extend this curve to the point at which it will cross
the vertical line representing four cords, and from
this crossing point we can estimate what the voltage
across the bus-bars would be with four cords busy.
We find it would be about 13 volts, which is lower
than required for proper operation.
The preceding example illustrates the method of
constructing simple curves and of reading them. A
more detailed description of the method of drawing
curves will be found in Appendix "A" at the end of
In the above discussion we assumed a constant resistance
for the battery supply lead from the central
office, and determined the values of the P.B.X. busbar
voltage with different numbers of busy cords.
It is also often necessary to ascertain the maximum
resistance of the battery supply feeder which satisfactory
operation of the P.B.X. will permit. Let us,
therefore, next consider the values to which, with
P.B.X.'s having various numbers of cords, the feeder
resistance must be limited in order that, when all
cords are busy, the bus-bar voltage shall not drop
below the value required for satisfactory operation.
Inasmuch as the P.RX. will not operate satisfactorily
on a voltage lower than 14, and as the voltage
of the central office battery is 22 volts, the voltage
drop in the battery supply conductors must not exceed
8 volts. Referring to Fig. 22, the highest resistance
we may have in B without having the voltage
drop between the central office battery and the
P.B.X. exceed 8 volts, can be figured by Ohm's Law ;
applying the relation R =Eji, we see we have merely
to divide 8 by the total current fed to the P.B.X.
In these computations the average current per cord
circuit will be assumed to be .2 ampere. The following
table shows the computations and results for
each of various numbers of cords.
Allowable resistance with
2 cords, 8 + .4 = 20.0 ohms
4 " 8 + .8 = 10.0 "
6 " 8 + 1.2 = 6.7 "
8 " 8 + 1.6 = 5.0 "
10 " 8 + 2.0 = 4.0 "
12 " 8 + 2.4 = 3.3 "
We now have the values of two variable quantities
one of which depends upon the other. The relation
between these quantities is represented graphically
by the curve in Fig. 27. This curve, drawn from the
information in Table I, shows us the allowable feeder
resistance for any number of busy cords. We can
now use the curve to find the resistance of a lead
required for a P.B.X. board with any number of
·..5 . 12
g 10 I : 4
4 6 8
Number of Busy Cords
cords. For example, if we have a board equipped
with five cords we select the point midway between
4 and 6 on the horizontal ..cale, and follow a vertical
line until it intersects the curve ; then, following a
horizontal line from this point, we find it intersects
the vertical scale at 8. Our lead must, therefore, be
so made up that its resistance is not more than 8
Fig. 28 shows the sleeve circuit of a cord circuit.
This is another good example of a combination of
series and parallel paths. When the contact of the
supervisory relay is open no current will flow
through the 40-ohm resistance ; we then have a single
path with 30, 83 and 120 ohms (a total of 233 ohms)
in series. With 24-volt battery the current in this
circuit will be
24 + 233 = .103 ampere.
When the supervisory relay operates, however, a resistance
of 40 ohms is connected in parallel with the
120-ohm lamp. The combined resistance of the lamp
and this 40-ohm shunt is
40 X 120 ·
= 30 ohms.
40 + 1..0
Consequently the total ..resistance of the circuit becomes
30 + 83 + 30 = 143 ohms
and the current drawn from the battery then is
24 + 143 = .168 amp.
The object of the 40-ohm shunt is to reduce the
current through the lamp, thereby dimming the
light so that it does not show through the lamp cap.
Let us see how ·much this reduction figures out to be.
In the first case the current in the circuit, all of it
flowing through the lamp, was .103 ampere. The .168
ampere in the second case divides, p art of it flowing
through the lamp and part through the shunt. There
are two methods by which we can compute how
· much of the total current flows in each path. The
first is a short-cut method : we note that the conductance
of the 40-ohm shunt is three times that of
the 120-ohm lamp, and consequently the current
through the lamp is one-third of that through the
shunt, or one-fourth of the .168 ampere through both
paths ; i.e., the lamp current is .042 ampere and the
current through the shunt is .126 ampere. The second
method is a general one which can be applied
to determine the current in any or all of any number
of parallel paths. This general method is based upon
the following line of reasoning. We have seen that
the voltage across any group of parallel paths is
equal to the total current through the group ( sum of
the currents in the individual paths) multiplied by
_ the equivalent resistance of the group. This, however,
is the same as the voltage across each path in
the group, and it must, therefore, equal the product
of the current in any one path multiplied by the resistance
of that path. Consequently the current in
any particular path of the group must equal the
voltage across the group divided by the resistance of
that particular path. In the example we have just
been considering, we found the combined resistance
of the lamp and the shunt to be 30 ohms and the
total current through the two paths to be .168 ampere
; the voltage across these parallel paths must,
E = I X R = .168 X 30 = 5.04 volts.
Hence, the current through the 120-ohm lamp is
I = EjR = 5.04 + 120 = .042 amp
and the current through the 40-ohm shunt is
5.04 + 40 = .126 amp.
These results are, of course, identical with those already
found by the short-cut method. The steps to
be carried out in this method of computing the current
in any one of a group of paths in multiple are
as follows :
( 1) Determine the combined resistance of the
group of parallel paths.
( 2 ) Compute the total current through the
( 3 ) Multiply the current in (2) by the resistance
in ( 1 ) . This gives the voltage across
the group of parallel paths.
( 4) Divide the voltage found in (3) by the resistance
of the particular path in which it is
desired to find the current. The result is
the current required.
MEASURING INSTRUMENTS-CURRENT AND
CURRENT AND VOLTAGE MEASUREMENTS
In our studies we have learned how to apply Ohm's
Law to determine either the voltage, the current, or
the resistance, when any two of these quantities are
known and it is desired to find the third. It is assumed
that the two which are known have been previously
determined, either by calculating their values
from known conditions or by measuring them with
Current can be measured directly with an ammeter,
and voltage, with a voltmeter. Each of these
instruments consists essentially of a meter element
which moves a pointer over a scale. This scale has
previously been calibrated by comparison with a
standard, and it is only necessary to note the scale
reading indicated by the pointer to learn the value
of current flowing through an ammeter or the voltage
impressed across the terminals of a voltmeter.
Galvanometer : In the Weston model direct cur-
rent instruments, which are the ones most commonly
used, the meter element is the same for either a voltPage
meter or an ammeter. This meter element is known
as a D'.A.rsonval galvanometer. When associated with
the proper series resistance it acts as a voltmeter,
and when associated with proper current carrying
shunts it acts as an ammeter. For some types of
measurements it is necessary, as we shall shortly see,
that the instrument be equipped with leads whose
resistance is taken into account in calibrating the
meter. We shall refer to these as calibrated leads.
Fig. 29 shows a simple galvanometer of the D'.A.rsonval
type. Current flowing through the movable
coil sets up a magnetic field which reacts with the
field between the poles of N and S of a permanent
magnet, causing the coil to rotate against the spring
tension. This movement of the coil, which is proportional
to the current through it, carries the pointer
across the scale.
The moving coil of a galvanometer is very light
and is mounted on j eweled bearings so that a very
small current will cause the pointer or "needle" to
travel the full length of the scale. For example, let
us assume it takes a current of .025 ampere to cause
full scale deflection of the needle ; and let us also
assume the resistance of the coil and calibrated leads
to be 2 ohms. By applying Ohm's Law the voltage
which must be impressed across the terminals to
cause a full scale deflection is determined to be
E = IR = .025 X 2 = .05 volt.
Fractions of volts are spoken of in terms of millivolts,
j ust as we express fractions of amperes in
milliamperes. To convert volts into millivolts, multiply
by 1000, e.g., .05 volt = 50 millivolts. Consequently,
depending upon whether we are interested
in measuring current or voltage, we can consider
that full scale deflection is caused by a current of
25 milliamperes through the galvanometer or by an
emf of 50 millivolts across its terminals. We can,
therefore, calibrate the galvanometer for use either
as a milliammeter or as a millivoltmeter. If we divide
the scale into one hundred parts, a full scale reading
of 100 divisions will indicate a current" of 25 milliamperes,
and each division will equal .25 milliampere.
Likewise, full scale reading will indicate an
emf of 50 millivolts, and each division will equal .5
millivolt. Fig. 30 shows the scale as it would look
if it were calibrated as we have just described. The
meter can now be used to measure either voltage
drop in millivolts or current flow in milliamperes.
Millivoltmeter : A good example of the use as a
millivoltmeter is in electrolysis measurements where
we measure the millivolt drop over a section of cable
sheath, as in Fig. 31. If we know the resistance per
unit length of a certain size cable sheath, we can
determine the amount of current flowing in it by
measuring the voltage drop across a known length,
and calculating by Ohm's Law. In the example in
Fig. 31 let us assume the meter reading to be 20
millivolts when the instrument is connected, by
means of its calibrated leads, across a length of cable
sheath whose resistance is .005 ohm. First converting
the 20 millivolts into volts, and then applying
Ohm's Law, the current in the cable sheath is found
I = EjR = .020 + .005 = 4 amp.
Use of the galvanometer as a millivoltmeter is one
of the cases, recently referred to, in which it is important
that the instrument be equipped with calibrated
leads. It will be well at this time to examine
A voltage mea-surement is a measurement of a difference
between two points which may be more or
less widely separated. Leads of some sort must,
therefore, be provided to connect the meter terminals
to the points between which the voltage is to be
measured. To see why calibrated leads must be used
if small voltages are to ..e measured accurately, let
us again refer to the example in Fig. 30 ; and let us
assume that the value of 20 millivolts which we
measured using calibrated leads is the true voltage.
Now let us repeat the measurement using leads
whose resistance is .05 ohm less than the calibrated
leads-will the meter still measure the voltage correctly
Y It will not. The resistance of the meter
element plus the calibrated leads was 2 ohms ; hence,
with 20 millivolts across the leads, the current
through the meter was
1000 X ( .020 voltj2 ohms) = 10 milliamperes.
'The meter measures this current, its needle deflect..
ing to 10 on its milliampere scale. As will be seen in
Fig. 30, this corresponds to 20 on the millivolt scale,
and so we read the voltage as 20 millivolts. The total
resistance of the meter element plus the new leads,
2 - .05 = 1.95 ohms.
Consequently the current through the meter element
when 20 millivolts is impressed across the new leads
1000 X ( .020 voltj1.95 ohms) = 10.25 milliamperes.
Again, the meter measures the current through it ;
its needle deflects to 10.25 on its milliampere scale.
But this, we see from Fig. 31, corresponds to 20.5
on the millivolt scale. Hence we would now read the
voltage to be 20.5 millivolts. The change in the meter
leads has, of course, not altered the actual value of
the voltage drop across the cable sheath in Fig. 31.
What has happened is merely that the meter reading
of the voltage has changed from the correct value
of 20 millivolts with the calibrated leads to an incorrect
indication of 20.5 millivolts. We thus see
that the calibrated leads must be used with the instrument
when it is employed as a millivoltmeter.
Milliammeter : Fig. 32 shows a meter connected in
a circuit as a milliammeter to measure the current
through a relay. We shall assume this to be the same
meter we have j ust been discussing. The values of
the current in this circuit under the following con-
ditions would be as shown below : ( 1 ) before the
meter is connected into it ; ( 2 ) when the meter with
calibrated leads-2 ohms total resistance-is connected
; and ( 3 ) when the meter with the same uncalibrated
leads as previously-1.95 ohms total resistance-
I = 1000 X
I = 1000 X
1000 + 198
1000 + 2 + 198
= 20.033 milliamps
= 20.000 milliamps
I = 1000 X = 20.001 milliamps
1000 + 1.95 + 198
The question o£ relative accuracy in this case depends
upon what current it is we wish to know. In
any event, when the meter is included in the circuit
it measures the current then flowing through the circuit.
Connecting the meter in the circuit, however,
introduces additional resistance and hence reduces
the current. Consequently i£, as is usual, we wish
to learn as closely as we can the current when the
meter is not included in the circuit, then the lower
the resistance of the meter and leads the less will its
reading differ from the value we wish to know. The
values computed above bear out the foregoing : the
value in ( 3 ) -obtained with the lower resistance, uncalibrated
leads-is slightly nearer the desired value
in ( 1 ) than is the value in ( 2 ) obtained with the
calibrated leads ; and the reading would more closely
approach the desired value if meter leads were
omitted. Actually, of course, the difference between
the value in (2) or (3) and that in ( 1 ) is far less.
than the accuracy with which the meter could be
read, so that the resistance of the leads in the example
in Fig. 32 is not a matter of practical importance.
In general, however, we conclude that, although
the meter must be used with its calibrated
leads when employed as a millivoltmeter, it is preferable
to omit the leads when it is employed as a
Ammeter : One of the most common uses to which
a millivoltmeter is put is to make it serve as an ammeter
by connecting it across an external shunt.
Fig. 33 shows how the shunt S and meter G are connected
to measure the current in a circuit. It is important
that the meter's calibrated leads be the leads
used to connect it across the shunt. Let us now see
how we can figure out what the resistance of the
shunt must be in' order that the meter may be read
as an ammeter whose full scale deflection indicates
any desired maximum value of current in the circuit.
The first step in designing a shunt is to decide
what circuit current--whether 1, 5, 10, 50 or 100
amperes or other convenient value-is to be indicated
by full scale meter reading. We then compute
the ratio of this maximum circuit current to the
maximum meter current ( current through the meter
which will cause full scale deflection) . If the maximum
meter current is not known we can apply Ohm's
Law to determine its value from the resistance of
the meter plus leads and the full scale reading of the
meter in volts. We next note that, if the current in
the circuit in Fig. 33 is, for example, to be 10 times
the current through the meter, then the current
through the shunt will be 9 times the current through
the meter ; and the resistance of the shunt will have
to be 1j9th the resistance of the meter and leads.
J:i-,rom this observation we see that the resistance of
the required shunt can always be obtained by dividing
the resistance of the meter and leads by one less
than the above ratio of the desired maximum circuit
current to the maximum meter current.
To illustrate the foregoing, let us suppose we wish
to measure currents up to a maximum of 1 ampere
by means of a millivoltmeter whose resistance (including
leads) is 2 ohms and whose full scale reading
is 100 millivolts. Maximum meter current in this
I = EjR = .100 -7- 2 = .050 amp.
The ratio of maximum circuit current ( 1 ampere ) to
maximum meter current then is
1 -7- .050 = 20.
Consequently the resistance of the 1-ampere shunt
2j ( 20-1) = 2j19 = .1053 ohm.
In like manner we could determine the resistance
of a 10-ampere shunt, a 100-ampere shunt, or a shunt
which would enable us, with a given millivoltmeter,
to measure currents up to any other full scale value
we desire. If the scale of a meter has 100 divisions,
then each division is read as .01 ampere when the
meter is used with a 1-ampere shunt, as .1 ampere
when used with a 10-ampere shunt, as .5 ampere
when used with a 50-ampere shunt, etc. There is an
advantage in having a wide range of shunts available
; it enables us to select one whose rating is such
that the reading for the current we wish to measure
will be well up on the scale. An example will show
how important this is from the standpoint of accuracy.
If a meter whose scale has 100 divisions were
used with a 100-ampere shunt to measure a current
of 2 amperes, the needle would register but two
divisions above zero, so that an error of one-tenth
division would be 5% error. A higher degree of
accuracy could be obtained by using a 10-ampere
shunt. The needle would then register twenty divisions
above zero, and an error of one-tenth division
would be only a .5% error.
When meters of the type which we have been referring
to as millivoltmeters are primarily intended
to be used with external shunts for measuring currents,
their scales are usually graduated to read amperes
directly. Such meters are called ammeters. In
some of the older types of ammeters the shunt was
located in the base of the instrument, with heavy
lugs on the meter for connecting it into a circuit.
Because of the heating of the shunt, and the magnetic
field which is set up in the meter element, this
type did not prove satisfactory. The use of external
shunts eliminates these difficulties and has the added
advantage that one meter can be used with any one
of several shunts of whatever ratings are desired.
Fig. 34 shows a common type of external shunt for
use in measuring large currents. A is a wooden base.
BB are heavy copper lugs joined together by strips
of resistance metal 0. These strips are made of an
alloy which has a higher resistance than copper and
which will not appreciably change resistance when
heated. DD are heavy bolts for connecting into the
circuit, and EE are small screws for connecting the
Voltmeter : Let us now consider how the same
galvanometer or moving element which we first
called a milliammeter and then a millivoltmeter, can
be used to measure higher voltages. Suppose we wish
to measure values up to 30 volts by means of a meter
element whose resistance is 2 ohms and which deflects
full scale when .025 ampere flows through it.
By Ohm's Law 30 volts will cause .025 ampere to flow
through a resistance of
R = Ejl = 30 -;- .025 = 1200 ohms.
Hence, if we place 1198 ohms in series with the 2
ohms of the meter element as shown in Fig. 35, the
deflection will be full scale when 30 volts are impressed
across the terminals. Similarly, 150 volts
will cause .025 ampere to flow through
150 -;- .025 = 6000 ohms.
Consequently, if we wish to measure up to 150 volts
we can do so by placing 5998 ohms in series with
thP- met-er element.
1198 Ohms [-fl50..-------..=,:..: ______ _
Fig. 36 shows how we can connect a single meter
element to measure values up to 30 volts or up to 150
volts. In the modern voltmeter the resistance units
would be mounted inside the meter with the terminals
on the instrument case, somewhat as shown
'in Fig. 37. Fig. 38 shows how the scale of the double
range voltmeter would look.
We have now seen that the galvanometer type
meter element serves several purposes. Used without
accessories, it acts as a milliammeter ; provided
with calibrated leads, it becomes a millivoltmeter ;
with calibrated leads and a calibrated shunt it serves
as an ammeter ; and a resistance in series with it
( called a "multiplier") converts it into a voltmeter.
The "volt-milliammeter" in the No. 14 local test desk
is an example of a meter element arranged for use
either as a milliammeter or as a voltmeter. In this
"' .s:E 0
case, instead of being connected to terminals as in
portable meters, the leads are wired to switch keys
by means of which the arrangement of shunt and
series resistances is controlled. Fig. 39 shows the
netw ork of resistances used with the meter element
in the No. 14 local test desk, and Fig. 40 shows the
simplified circuit for each connection. By calculating
the distribution of current in the first two circuits
we find that only .0006 ampere flows through the
V'-o 1.2 M . A.
.....,-----oo480 M. A.
meter element to give a full scale deflection of the
needle. In the third circuit we find the 1600-ohm
meter element shunted by 1600 ohms resistance, so
that only half the current will flow through the
meter ; and as the total resistance is 20,000 ohms, the
total current will be .0012 ampere when 24 volts is
impressed. But as only half the current flows
through the meter element, the current causing full
scale deflection is again seen to be .0006 ampere. The
same is true of the 120-volt, 100,000-ohm circuit in
the fourth arrangement.
The 1600 ohms designated as meter element is
made up of a m.eter element of about 400 ohms with
resistances in series and parallel to give the total
combined resistance of 1600 ohms. This network is
adjusted for each individual meter element. Meters
of this type are much more sensitive than the usual
ELEMENTARY PRINCIPLES DIRECT
Measurement of Resistance : In considering methods
of measuring resistance it is desirable to know
the accuracy required in any particular case in order
that we may suit the method of measurement to the
requirements. For instance, if we desired to know
the effect of placing a temporary relay in a signal
circuit, we would probably not need to measure the
resistance of the relay more accurately than to within
5 or 10% ; but if we were locating a fault in a
cable conductor for the purpose of making repairs
which involve the expense of open-ing a street pavement,
we would be justified in taking considerable
time in getting a very accurate measurement.
From the standpoint of the accuracy attainable
with the instruments employed in routine work, the
methods of measuring resistance which we shall consider
here fall into two classes. In the first class are
a group of methods, useful where a high degree of
accuracy is not required, in which resistance is computed
from measurements made with voltmeters and
ammeters. The more accurate method which we shall
consider is measurement by means of a sensitive galvanometer
in combination with a network of known
resistances arranged in what is known as a Wheatstone
Volt-Ammeter Method : In this method a voltmeter
V, an ammeter A, and the unknown resistance
R, are connected in either of the ways shown schematically
in Fig. 41. For most practical purposes the
result obtained by applying Ohm's Law to the voltage
and current read from the meters is accepted as
the resistance of R. We can see from the schematics,
however, that this is somewhat in error, and that
where greater accuracy is required an appropriate
.... __ --lllt---.....J
correction must be applied to the resistance indicated
by the meter readings.
In Fig. 41A. the ammeter measures the combined
current through the parallel paths R and V, and the
voltmeter measures the voltage across them ; consequently
the result obtained by applying Ohm's Law
to these meter readings is the combined resistance of
R and V in parallel. This result will differ fr..m R
by less than 1 % if the voltmeter resistance is one
hundred or more times the resistance of R. If we
desire to correct this indicated resistance we have
merely to convert it into its equivalent conductance,
subtract the conductance of the voltmeter, and convert
the remaining conductance into equivalent resistance.
Turning now to Fig. 41B, here the ammeter
measures the current through A and R, and the voltmeter
measures the drop across these two elements
in series ; hence the result obtained by applying
Ohm's Law to the meter readings in this case is the
resistance of R plus the resistance of A. This indicated
resistance will differ from R by less than 1 %
i f the ammeter resistance i s less than 1j100 o f the
resistance of R. To correct this indicated resistance
we have merely to subtract from it the resistance of
To illustrate the two methods in Fig. 41, and the
v. 15,000.. V· 15,000..
E .. 24 ' '"----i
E = 24v
degree by which the result obtained with each differs
from the true value of R, let us first compute what
the meters will read if we assume the battery to be
of 24 volts and the resistance of each element to be
as shown in the corresponding networks in Fig. 42.
In Fig. 42A the combined resistance of R and V in
(500 X 15,000) / ( 500 + 15,000) = 484 ohms.
Hence the reading of the ammeter in Fig. 41A would
24/ ( 484 + 2 ) = .0494 amp
and the voltmeter would read
.0494 X 484 = 23.9 volts.
The meter readings in Fig. 41A would, therefore, indicate
the resistance of R to be
23.9 ...;--.0494 = 484 ohms.
This is 16 ohms, or 3.2%, less than the actual value
of R. In Fig. 41B the voltmeter would read 24 volts
and the reading of the ammeter would be
24/ (500 + 2 ) = .0478 amp.
These readings would indicate the resistance of R
24j.04 78 = 502 ohms.
In this case the indicated resistance is 2 ohms, or
.±%, higher than the actual resistance of R.
It is not to be infe:-red from the above comparison
that the resistance indicated by the meter readings
in Fig. 41B is always closer to the actual value of R
than is the resistance indicated by the meter readings
in Fig. 41A. With the 2-ohm ammeter and the
15,000-ohm voltmeter assumed in the above examples,
computations like those just carried out will show
that, if R were 174 ohms, the correction would be
1.15 % for either set of meter readings ; that if R
were less than 174 ohms the correction for Fig. 41A
would be smaller than with Fig. 41B ; and that, as
we found in the above examples, the correction with
Fig. 41B will be smaller than with Fig. 41A if R is
greater than 174 ohms.
Where values more accurate than those obtained
directly from the meter readings are required, the
corrections already mentioned can be applied. Thus,
in the example of Fig. 41A worked out above, the
indicated resistance of 484 ohms may be corrected
by converting it into its equivalent conductance, and
from that, subtracting the conductance of the voltmeter.
The conductance of R so obtained is
( 1j484) - ( 1j15,000) = 1j500 mho.
Hence the resistance of R is 500 ohms.
The method in Fig. 41B is much easier to correct.
To correct the value of 502 ohms indicated in the
above example of this method, we merely deduct
from it the resistance of the ammeter. Thus, for the
corrected value of R we again obtain
502 - 2 = 500 ohms.
Voltmeter Method : If the resistance to be measured
is high, and we have a voltmeter whose resistance
is known, the measurement can be made
using only the voltmeter. Fig. 43 shows how the
voltmeter would be connected for this test. The
switch is first thrown to the position indicated by
solid lines, and the battery voltage E read on the
Rv • l5,000"'
E • 24V
voltmeter. Then the switch is thrown to the position
indicated by dotted lines and the voltmeter again
read. Let this second voltage-which is, of course,
the drop across the voltmeter-be denoted by V. Inasmuch
as the voltmeter of resistance Rv and the
unknown resistance Rx are now in series across the
battery, the voltage drop across Rx at this time must
be (E - V) ; and as the same current is flowing
through both of these elements, the ratio of their
resistances must be the same as the ratio of the voltage
drops across them, that is
Rx E-V E
- = -- = --1.
Rv V V
Hence the value of the unknown resistance is
Rx = R. (- -1) ohms.
This relation enables us to determine the value of
Rx from the two voltmeter readings E and V and
the known resistance R. of the voltmeter.
To illustrate this method let us apply it to find the
value of Rx in Fig. 43. We shall assume the resistance
of the voltmeter to be R. = 15,000 ohms, the
voltage of the battery (reading of the voltmeter with
switch in solid line position) to be E = 24 volts, and
the voltmeter reading with the switch thrown to the
dotted line position to be V = 20 volts ; then
Rx = 15,000 X (- - 1) = 3,000 ohms.
In the above discussion we have ignored the resistance
of the power supply used in the measurement.
This neglect, however, will not increase the
probable error if the resistance of the voltmeter is a
thousand or more times as great as the power supply
resistance. Assuming a voltmeter of high enough resistance
to eliminat_e any error from this source, the
accuracy attainable with this method is greatest
when the unknown resistance equals the resistance
of the voltmeter. It follows, therefore, that this
method is best adapted to the measurement of quite
v Rx = Leakage Resist
A familiar example of the measurement of high
resistances by this voltmeter method is its use by test
desks in routine measurements of the insulation resistance
of outside plant. As the leakage resistance
is usually several megohms ( one megohm equals one
million ohms) greatest accuracy will be attained by
employing a voltmeter of very high resistance. The
100,000-ohm arrangement of the test desk voltmeter
is the one now used for this work. The voltmeter is
first thrown across the testing battery and the voltage
E read ; then it is thrown in circuit with the bat-
tery, line, and ground, as shown in Fig. 44, and the
voltage V which it then indicates is read. As an illustration,
if the values of E and Rv be as shown in
Fig. 44, and the reading V is 10 volts, then, by means
of the above formula for computing the unknown
resistance from the two readings of the voltmeter,
the insulation resistance is found to be
Rx = 100,000 (- - 1) = 1,400,000 ohms
Rx = 1.4 megohms.
If the line is 10 miles long and the leakage is
evenly distributed we can assume the total resistance
to be made up of ten equal resistances in multiple,
as in Fig. 45. Each of these resistances will be equal
to ten times the combined resistance of the group
Wheatstone Bridge : Of the several methods for
measuring electrical resistance, the Wheatstone
bridge method is in general the most accurate, and
is the one most often used. The principles governing
its operation will therefore be explained in detail.
The Wheatstone bridge method of measurement
is very simple. The forms of the Wheatstone bridge
ordinarily used, however, seem quite confusing until
the connections are traced out and set down in a
diagram. When this is done it is found that however
much different forms of the bridge may differ in
mechanical details or in the arrangement of parts,
the connections of all can be reduced to the very
simple circuit which will be discussed here in explaining
the principle on which these bridges
To understand the operation of the Wheatstone
bridge it is only necessary to keep in mind the fundamental
relations governing current, voltage and
resistance in a direct current circuit. These relations
are now quite familiar to us under the name of
In Fig. 46 is a simple circuit in which, between
points A and B, are four resistance elements arranged
in two parallel paths ACB and ADB. Let the
voltage drop across each of these resistances be denoted
as shown on the diagram. Since the potential
drop from A to C is V" and the drop from A to D is
V2, we can see that the potential difference V between
points C and D is
V = V2 - V1•
We can also see that, by varying the resistances, we
can make V1 and V2-each independently of the
other-whatever portion we please of the total volt-·
age E impressed across points A and B. Consequently-
and this is the important thing-the resistances
can always be so adjusted as to make
V1 = V2 and V3 = V4•
When the resistances are thus adjusted the bridge is
said to be "balanced."
There are two aspects of this particular adjustment
which make it the basis of Wheatstone bridge
measurements. Firstly, since V1 = V2, then V2 = V1
= 0, so that V = 0. In other words, because the
drop from A to C equals the drop from A to D the
potential difference between points C and D is zero.
If, therefore, we connect or "bridge" a galvanometer
across points C and D as indicated in Fig. 47', no
current will flow through it when the bridge is balanced,
and so its needle will show no deflection. Conversely,
if we so adjust the arms of the bridge that
the galvanometer needle shows no deflection, the
bridge will be balanced. We are thus provided with
a very sensitive visual indication as to when an accurate
balance has been attained. Secondly, when
V1 = V2, then also, of course, V3 = V4. Hence we
can see that, when the bridge is in balance, the drops
across the two elements in the path ACB are proportioned
in the same ratio as are the drops across
the two elements in the path ADB. This, stated symbolically,
This relation, it will now be shown, leads directly to
the fundamental formula of the Wheatstone bridge.
Since no current flows through the galvanometer
when the bridge is balanced, the current from A to
C is the same as from C to B and the current from
A to D is the same as from D to B. If, therefore, we
denote the currents as shown in Fig. 47, and designate
the four resistances as shown in Fig. 48, then,
by Ohm's Law, the voltage drops in the above ratios
can be expressed as ·
V, = IuR ..
v. = IDR2
Consequently the above relation among the voltage
drops in the balanced bridge, written in terms of IR
But in this the currents cancel out, leaving
Rx = --Ra.
The diagram in Fig. 48 is a simplified schematic of
the Wheatstone bridge circuit. This diagram should
be memorized because, as previously stated, regardless
of how greatly various forms of the Wheatstone
bridge differ in appearance, their connections will be
found to be the same in effect as this simple network.
The final one of the above equati9ns should also be
memorized because it shows the relation among the
four groups of resistances or "arms" of the bridge
after a balance has been attained.
Referring to Fig. 48, the arms R1 and R2 of a
Wheatstone bridge are known as the ratio arms ; R3
is known as the rheostat arm, or balance arm ; and Rx
is the unknown resistance that is being measured.
The measurement is performed by setting the ratio
arms as discussed later, and then adjusting the rheostat
arm until the galvanometer shows no deflection .
when the two keys, first the battery key and then the
galvanometer key, are closed. The value of the measured
resistance is then computed by the formula
above from the settings of the ratio arms and the
resistance to which the ratio arm was set -in balancing
The settings of the ratio arms depend upon the
range of resistances provided for adjusting the rheostat
arm and upon the order of magnitude of the
resistance to be measured. A common arrangement
is for the rheostat arm to be adjustable, in steps of
one ohm, to any resistance from zero to 9999 ohms.
With such an arrangement, the ratios for which arms
R1 and R2 would be set when measuring resistances
in various ranges of magnitude are as shown in the
Range of Rx
.001 to ' 9.999 ohms
.01 " 99.99 "
. 1 " 999.9 "
" 9999 "
" 99990 "
" 999900 "
" 9999000 "
The resistances of the Wheatstone bridge must be
accurate, non-inductive, readily cut in or out of circuit
within the limits needed to get a proper balance
for the range of resistances to be measured, and
must be moistureproof and protected from injury.
There are three principal means of varying the resistance
in the bridge arms :
( 1 ) Connecting the individual resistances between
massive brass blocks which are joined by removable
solid brass plugs. With the plug in
place the resistance is short-circuited. Removing
the plug cuts the resistance in circuit. A variation
of this method provides two bus-bars for
each group of resistances, by means of which the
resistances desired are cut into circuit by the
plugs, the unused resistances remaining ''legged
on," but not in circuit. This type is known as
the post office type or plug bridge.
(2) Using massive terminals to the resistances over
which a dial switch is moved. For the "resistance"
arm or rheostat arm, three or more dial
switches are needed, one for units, one for tens,
one for hundreds, etc. This type is known as a
( 3 ) Having a movable arm in direct contact with
the resistance wire, the position of the arm determines
the amount of resistance in circuit.
This type is known as a slide wire bridge.
Magnetism is of great importance to the telephone
man. We use this property in relays, subscriber
bells, receivers, generators, repeating coils, induction
coils, and many other types of apparatus common
in our telephone work. It is, therefore, necessary
for us to have a good working knowledge of
magnets and magnetism. The actual operation of our
relays or of any other apparatus which depends upon
magnetism may seem far removed from the ideas
that are to be brought out by describing experiments
in which iron filings sprinkled on glass are made to
move by a magnet ; nevertheless we must have these
ideas in our minds, for they are the most concrete
means of explaining the action of the force 'which
does so much work for us.
The subject of magnetism naturally divides itself
into several parts, the first of which deals with permanent
Permanent magnets are made of steel or of such
alloys as cobalt-steel, realloy (iron-cobalt-molybdenum
) , etc. They are called permanent because,
with proper treatment, they retain their magnetism
indefinitely. Heat or sharp jarring, however, causes
them to lose their strength. Magnets possess the
property of attracting iron ; this is the property
which we call magnetism. The two end areas of a
magnet over which the magnetic force is most pronounced
are called the poles. If a bar magnet be
freely suspended it will align itself lengthwise with
its poles in a north-south direction like a compass
needle. The pole which points north is called the ·
north or positive pole and the pole which points
south is called the south or negative pole. If we
bring a north pole and a south pole close together
we find they attract each other, but if we bring two
north poles or two south poles close together we find
they repel one another. A fundamental law of the
action of magnetic poles which should be remembered
is, therefore, unlike poles attract each other,
like poles repel one another. The substation ringer,
which will be described later, is a good example of
the practical application of this principle of the attraction
and repulsion of magnetic bodies.
Any region in which a magnetic force acts is called
a magnetic field. Hence a magnetic field exists in the
region surrounding a magnet. To get a clear idea of
a magnetic field of force let us consider Fig. 49.
Here we have a thin plate of glass placed over an
ordinary bar magnet which is resting in a horizontal
position. On the glass has been sprinkled a light covering
of iron filings which, when the glass is tapped,
arrange themselves, as the illustration indicates, in
curved lines extending from one pole of the magnet
to the other. The direction of these lines is, at every
point in the field, the direction in which the magnetic
force at that point acts ; and, as will be discussed
later, the greater the magnetic force in any region,
the closer together are the lines in that region. We
are led by this to think of the magnetic field in terms
of these lines, and to look upon these "lines of force"
as representing both the direction of the magnetic
forces in the field and the strength of the field as
well. These "lines of force" are also commonly
known as "lines of magnetic induction," and the
lines collectively are frequently spoken of as the
":O.ux." We shall now consider certain attributes of
the lines of force which will help us understand the
behavior of magnets and magnetic fields.
A bar magnet and the magnetic field surrounding
it are shown schematically in Fig. 50. As this diagram
indicates, the lines of force are always closed
loops, their positive direction being as indicated by
the arrowheads ; they are considered to pass through
a magnet from the south pole to the north pole, leaving
the magnet at its north pole and reentering the
magnet at its south pole. We may think of each
closed loop as acting like a stretched rubber band in
that it tends to shorten itself as much as possible ;
and we may also look upon each line as having a
repelling effect upon all neighboring lines, thereby
tending to make the lines spread apart. Fig. 51 indicates
the action of the lines of force when two
Magnets tend to draw together.
Unlike poles attract.
Magnets tend to push
apart. Like poles repel.
magnets are brought near each other, and illustrates
how the behavior of the magnets can be explained in
terms of the above attributes of the lines. When the
adjacent poles of the two magnets are unlike, the
relative directions of the lines from them are such
that the two sets of lines combine to form a single
set of closed loops, as shown in Fig. 51A ; and the
tendency of these loops to shorten evinces itself in
the mutual attraction between the unlike poles. On
the other hand, when the adjacent poles are alike, as
shown in Fig. 51B, the relative directions of the two
sets of lines are seen to be such that the tendency
of the lines to spread apart accounts for the repulsion
between the like poles.
The path which the flux takes is called the "magnetic
circuit." The amount of flux or number of lines
of force depends upon the opposition offered by the
various materials making up the magnetic circuit.
Iron forms a very easy path for the flux while air
and copper form a difficult path. For this reason the
magnetic circuit of relays is so designed as to be
mostly of iron, the air-gaps being made very short.
This opposition which the magnetic circuit offers to
the flux is called "reluctance." Reluctance contrQls
the amount of flux in the magnetic circuit in a manner
analogous to that in which resistance controls the
amount of current in the electric circuit.
It was stated above that we may look upon the
lines of force as representing, not only the direction
of the magnetic forces in the field, but the strength
of the field as well. In considering the field strength
aspect of this statement further, we shall again make
use of an analogy. Where a stream of water is con·
fined to a narrow channel the current is strong, but
where the stream spreads out the current is weak.
Just so with the magnetic circuit. It is an easily verified
experimental fact that the magnetic force increases
as we approach the poles of a magnet, and
decreases as we recede from the poles. Likewise, as
we have seen in Figs. 49 and 50, and as is shown
more clearly in Fig. 52, as we approach the poles of
a magnet the flux density increases ; but as we move
away from the poles the lines spread out in all direcPage
tions, and, of course, as the area over which they
are spread becomes greater, the number of lines per
square inch decreases. Thus field strength and flux
density are seen to vary together-the greater the
field strength the greater the flux density, and conversely.
Hence, wherever the path of the flux is contracted
to a small area, the flux density is reloatively
high and the field is strong. Where the flux .spreads
over a wide area, however, the flux density is, of
course, relatively low and the field weak.
The magnets we use in hand generators are designed
to take advantage of the principles set forth
in the previous paragraphs. Fig. 53A illustrates the
"U" shaped magnets referred to. Compare this type
of magnet with the bar magnet shown in Figs. 49 and
50, and note the shorter air path taken by the lines
of force from one pole to the other as compared with
that in the case of the bar magnet. If we fill the air
space between the poles with iron-e.g., by placing
(A) ( 8 )
the pole pieces and armature of the hand generator
between the magnet poles, as shown in Fig. 53Bwe
further improve the magnetic circuit of the "U"
magnet. It is of interest to note that, by using a
number of "U" bars placed side by side in the hand
generator, the total amount of the flux through the
armature is increased ; and this, by virtue of a property
later described, enables the magneto to generate
sufficient voltage to ring down the drops on
the rural lines in our magneto switchboards, to ring
subscriber bells, and to do other work for us. Care
must be taken, however, that all like poles are placed
side by side as shown in Fig. 54A ; otherwise the
magnetic effect of one bar will neutralize the effect
of another, as indicated in Fig. 54B. This would
result in decreasing the number of lines of force
passing through the armature, and the magneto
would then not generate enough voltage for satisfactory
service-in other words we would have a
case of trouble.
End view of magnets of End view of magnets of
a 5-bar generator with a 5-bar generator with
bars placed ..- bars placed "\1aliil8ttg. .. .. r .., t·/t (A) ,_; .p-/11( (B) I I .. .. I
(;- Fig. 54
While discussing magnets it may be well to mention
that in some cases, such as in the core of repeating
coils, we have closed magnetic circuits of
laminated iron. Here there are no poles set up .(see
Fig. 55) ; but the lines of force, nevertheless, are
present in the core, and have a definite direction
· Fig. 55
In Section I-G we discussed permanent magnets.
We now come to a consideration of another type of
magnet known as the electromagnet. Electromagnets
are very common in telephone equipment.
Field about a Straight Wire End View
Magnetism and electricity are so interrelated that
magnetism can be created and controlled by an electric
current. This enables us to make electricity perform
a great deal of useful work. Wherever a current
of electricity flows there is set up a magnetic
field whose lines of force encircle the conductor
which carries the current. "Fig. 56 depicts the magnetic
field set up about a straight wire when an electric
current flows through it. These lines of force
are circular loops lying in planes at right angles to
the wire. There is a definite relation between the
direction of the current flow and the positive direction
of the lines of force : if we grasp the wire in the
right hand, the thumb pointing along the wire in the
direction of the current, the fingers will then point
around the wire in the direction of the lines of force.
The relative directions of the current and of the
lines of force, indicated by the arrowheads in Fig. 56,
accord with the foregoing rule.
The magnetic field set up about a single turn of
wire when an electric current flows through it is
shown in Fig. 57. We see that all lines of force encircle
the wire in the direction stipulated by the rule
just cited, and that all pass through the turn in the
same direction. Fig. 58 shows what happens when
we place a number of turns of wire together. The
effects of the several turns combine to set up lines of
Fig. 58 Cross Section
force, a very small proportion of which encircle the
turns individually, but most of which link themselves
through the whole group of turns. We immediately
recognize the field represented by these lines to be
similar to that set up by the bar magnet in Fig. 50,
and we rightly conclude that, when current flows in
it, the coil acquires the properties .of a magnet. The
direction of the flux through the coil and the consequent
polarity may be inferred, by the rule given
above, from the direction in which the current flows
around the coil ; but, for this purpose, the rule is
more 00nvenient when restated in slightly different
form. Place the right hand on the coil, the fingers
pointing around the coil in the direction in which
the current flows and the thumb extended parallel
with the axis ; the thumb then points in the direction
in which the lines of induction pass through the coil,
i.e., towards the north pole of the coil.
We have already seen how the current, the resistance,
and the electromotive force (em£) in an
electric circuit are related by Ohm's Law. IIi a magnetic
circuit there is an analogous relation between
the flux, the reluctance, and the force which sets up
the flux, or "magnetomotive force" (mmf) as it is
called-the flux equals the magnetomotive force
divided by the reluctance. The magnetomotive force
due to an electric current flowing around a coil is
directly proportional both to the number of turns in
the coil and to the number of amperes through them ;
in other words, the mmf depends upon the number
of amperes multiplied by the number of turns. Consequently,
magnetomotive force · is commonly expressed
in terms of "ampere-turns." The reluctance
of a magnetic circuit in which the path of the flux is
wholly through air or other non-magnetic substances,
is a constant ; no matter how the mmf is varied, the
flux varies in like proportion. If, however, the path
of the flux is partly or wholly through iron or other
magnetic materials, the reluctance is not constant.
Experiment shows that, if the mmf acting upon a
magnetic substance be increased in a succession of
equal steps, the corresponding increments in the resulting
flux density eventually tend to become successively
smaller-i.e., the reluctance increases-until
a point is reached beyond which further increases
in the acting ampere-turns produce practically no
further increase in the flux density. When this condition
exists the magnetic material is said to be
"saturated," and the point at which saturation is
reached is called the "saturation point."
From what we have now learned we can see that
the magnetic effect of a given current flowing through
a coil can be increased in two ways : ( 1) by increasing
the number of turns in the coil, thereby increasing
the mmf ; and (2) by winfl.ing the coil on an iron
core, thereby greatly reducing the reluctance of the
magnetic circuit. A very strong magnet can, therefore,
be obtained by passing a current through a coil
of many turns wound on an iron core. Such magnets
are called "electromagnets." The core of these magnets
is usually made of a special soft iron or steel
which has been carefully prepared and treated so
that it will give up its magnetism as completely as
possible immediately the energizing current ceases to
flow through the winding. It should be borne in
mind, however, that a small amount of magnetism,
called "residual magnetism," is normally retained in
the core of electromagnets. Fig. 59 shows a crude
electromagnet made by winding a coil of wire about
an iron nail. To demonstrate the action of the magnet,
a piece of iron shaped like a relay armature is
shown below the nail. When the key is closed curPage
rent flows through the coil and the nail becomes a
magnet having the power to attract the piece of iron
as shown in Fig. 60 ; but when the circuit is opened
the nail loses its magnetism, the piece of iron falls
off, and we again have the condition shown in Fig.
Our relays consist essentially of an electromagnet
and an armature, together with such accessory parts
as contact springs, etc. By taking the coil just described
and mounting it as indicated in Fig. 61, we
can make a crude relay which will demonstrate the
action of our standard relays. When the energizing
'-==:l-ft j l--c======:=.--b
electric circuit is closed the current through the
winding sets up a flux in the magnetic circuit, making
the core a magnet. For a definite number of
ampere turns the amount of flux set up depends
upon the reluctance of the magnetic circuit. The
greater part of the reluctance in the magnetic circuit
of our relays is in the air-gap. If the current
is in such a direction as to make the end of the
core near the armature a north pole, the flux will
pass from the core into the air-gap, cross the gap,
enter the armature, and then pass through the iron
part of the relay structure to the other end of the
core and back to the starting point. The north pole
at the end of the core adjacent the armature is,
as we have already seen in Figs. 50, 51, 53 and 54,
the region in which the flux passes from the core into
the air-gap. Likewise, the region in which the flux
passes from the air-gap into the armature is a south
pole. These unlike poles will attract each other just
as the north pole of one magnet will attract the south
pole of another ; hence, if the magnetic attraction is
great enough, the armature will be pulleP. towards
the core as Fig. 61 shows. Thus, through the agency
of the magnetic effects which accompany it, the electric
current is made to perform the work of moving
the armature. If the current is reversed the effect
will be the same, save that the armature will be the
north pole and the end of th.e core adjacent the
armature will be the south pole.
When the armature pulls up the air-gap is shortened.
This decreases the reluctance of the magnetic
circuit, thereby causing the flux to be increased ; and
since the magnetic attraction depends upon the
amount of flux, the armature will be held up with a
greater force than that which initiated its motion
towards the core. Were the flux allowed to become
too great the core might become saturated, a condition
which would cause some relays to release too
slowly. This condition may be prevented in several
ways. Some relays have the cross-section of the magnetic
material in the flux path greatly reduced for a
portion of its length. The increase in reluctance
which occurs when this constricted portion reaches
saturation, prevents too high a flux density in the
remainder of the magnetic circuit, no matter how
closely the armature approaches the core. Other relays
have stop pins of copper or German Silver which
prevent the armature from striking the core. Still
others have non-freezing discs.
From our discussion of magnetomotive force it will
be appreciated that determination of the proper
number of ampere-turns is an important factor in
relay design. The ampere-turns for a given relay are
carefully figured out to meet the requirements imposed
by the load to be put upon the relay, the
length of the air-gap, the distance through which the
load must be operated, etc. Thi.. load may be a single
heavy armature, or it may be a group or "pile-up"
of springs whose tension has to be overcome. Fig. 62
shows the manner in which experimental methods
may be employed to determine the number of ampere-
turns needed to overcome known test loads applied
to the relay at substantially the point on the
Back Stop Screw
armature at which the actual working load will come
on the finished relay. From the results of these
measurements can be determined the number of ampere-
turns required to overcome the load which gravity,
or the exact pile-up of springs to be used, will
impose upon the armature of the relay whose design
is being worked out. It should, therefore, be clear
why we use current flow test sets in the field and
why we adjust our relays to a given current flowthe
very basis of the design is that the relay must
get a proper amount of current in order to do its
Where two windings are wound on the same core,
care must be taken to insure that the magnetism
produced by one winding is properly poled relative
to the magnetism produced by the other. Whether
· the currents in the two windings flow around the
core in the same direction or in opposite directions,
will determine whether the magnetic effects of the
two windings aid each other as indicated in Fig.
63A, or oppose one another as indicated in Fig. 63B.
r..It l It l Armature is pulled up.
Flux is strong.
Armature does not pull
up. Flux is weak.
In some cases, such, for example, as the winding a
referred to in discussing Fig. 16, it is required to
obtain a non-magnetic winding, i.e., to wind a length
of wire in the form of a coil without producing a
magnetic effect. This can be done by arranging the
wire, as indicated in Fig. 64A, so that the current
flows in one direction around the core in passing
through the turns made by one half the length of the
wire, and in the opposite direction around the core
in passing through the equal number of turns made
by the other half of the wire. Since the direction of
Coil wound non · inductively. .. Nail has no magnetism. ..
Non . inductive Winding ]i;tft.n .. .. ( B)
the magnetic field set up by each turn depends upon
the direction of the current around the core, it can
be seen that the magnetic effects of the two equal
and opposite sets of turns must oppose and neutralize
each other. Such a coil is said to be wound noninductively.
Fig. 64B indicates the arrangement of
a relay which, like the relay in Fig. 16, has an energizing
winding and a non-inductive winding connected
We frequently have use for special relays such as
slow acting, polarized, and alternating current relays.
The many slow acting relays in our dial system
offices and the a-c relays in our four-party subsets
are typical examples of two of these types. Polarized
relays find considerable use in telegraph practice
; and we also find them in our telephone circuits,
both in manual and in dial systems.
Slow acting relays are designed so that the flux
builds up slowly when the winding is energized and
decays slowly when the current stops flowing
through the winding of the relay. This condition is
obtained by using a secondary short-circuited winding,
which may be either a regular winding or a
single turn of very low resistance. In some cases this
winding takes the form of a copper tube around the
full length of the core or around the core for a portion
of its length ; in other cases it is a copper collar
placed at one end of the core ; and in still other cases
it is a closed coil. With any of these arrangements,
when the circuit through the main winding is first
closed and the magnetic field begins to build up, or
when the circuit is first opened and the magnetic
field. begins to die out, this increase or decrease in
the main field "induces" in the closed secondary
winding a voltage which causes a current to flow in
it, just as the changing current through the primary
winding of a transformer would cause a current to
flow in a closed secondary circuit. This secondary
current tends, of course, to set up a magnetic field
which, depending upon the direction of this current,
will oppose or will aid the main field. The direction
of this secondary current is always such that the secondary
flux will be in whichever direction opposes
the causative change in the main flux. Hence, ,when
the circuit through the main winding is closed, the
secondary flux opposes ( and so retards) the building
up of the main flux, thereby delaying the instant at
which the field becomes strong enough to pull up the
armature. Likewise, when the circuit through the
main winding is broken, the secondary flux opposes
and slows down the dying off of the main flux, and
consequently the release of the armature is delayed.
Relays having this action delaying feature may be
either slow operating or slow releasing, depending
upon certain other factors of the design. If the
winding for the relay is so chosen that the relay gets
but little more than just enough current to operate
it, the flux will not become strong enough to move
the armature until the above effect of the shortcircuited
secondary has been overcome and the flux
has reached almost its full strength. We thus obtain
a slow operating relay. To secure a slow releasing
relay, the winding must be so chosen that the final
value of the flux produced by the current through
the relay is much greater than will permit release.
In this case when the circuit is opened the flux will
not decrease enough to allow the armature to fall
back until the retarding effect of the short-circuited
winding has been overcome.
The polarized relay differs from the regular direct
current types in that a permanent magnet is used in
its construction in addition to the electromagnet.
Relays of this type utilize two principles with which
we have already become familiar. We know that
like poles repel and unlike poles attract, and we have
also seen that the polarity of an electromagnet can
be reversed by reversing the direction of the current
energizing it. Consequently the fixed polarity permanent
magnet and the controlled polarity electromagnet
in the polarized relay can be made to repel
or to attract one another at will. The polarized
ringer described later also uses a permanent magnet
and an electromagnet.
Alternating current relays are of three general
types. As will be shown later, an alternating current
passes through zero value twice during each cycle,
and at each such instant there is no current flowing
to produce a magnetic force. The pull on the armature
of an ordinary d-e type of relay would, therefore,
come in spurts, and the relay would hum or
chatter, if we attempted to operate it on alternating
current. Prevention of this vibration or chatter is
an important matter in the design of alternating current
relays. One means employed to overcome this
effect is to make the armature heavy enoug:ll that it
will not fall back during the instants of nu- magnetic
pull. This type of a-c relay is known as the
inertia type ; it is used in some of the four-party
subscriber sets. Two other general types make use
of means by which, so long as the winding is excited,
the magnetic pull on the armature is prevented from
dropping to zero.
In discussing Figs. 50 and 51 it was pointed out
that lines of induction exhibit the property of exerting
forces in an effort to shorten themselves and in
an effort to spread apart. The operation of electric
motors and of innumerable other electrical devicesincluding
the D'Arsonval galvanometer and the
Weston type meters whose uses were considered in
Section I-E-depends upon the force set up by this
property of the lines of induction when, as represented
in Fig. 65, the circular field produced by an
(A) ( 8 )
electric current flowing through a wire (see Fig. 56)
is combined with the field produced by the poles of
a magnet (see Fig. 54.A ) . Comparing Fig. 65 with
Fig. 54.A, we note that the flux produced by the current
strengthens the field above the wire and weakens
it below, the result of this being that the lines
above the wire are crowded together, and lines passing
from one pole to the other are lengthened from
straight lines into curves. Here the efforts of the
lines to shorten and to spread apart will both tend
to force the wire downward as indicated by the
arrow in Fig. 65B. To illustrate this action we shall
consider a specific application.
Fig. 66 is a sketch of a crude meter . .A coil A-B
is so mounted that it can rotate in the field between
the poles N and S of a U-shaped permanent magnet .
.Attached to the coil are flat spiral springs (see Fig.
29) which resemble the spring on the balance wheel
of a watch. When no current flows through the coil,
these springs maintain the coil in the position indicated
in Fig. 66.A, with the pointer on the zero of
the seale ; and they oppose rotation of the coil away
from this zero position with a restoring force which
is proportional to the amount of rotation. Now suppose
the current we desire to measure is led into the
coil so that it goes in at B and out at A. The current
in wire B strengthens the field above B and weakens
the field below it, thereby forcing the wire B downward
as indicated in Fig. 66B. Likewise the current
in wire A strengthens the field below A and weakens
it above, thereby forcing wire A upward. This action
rotates the coil so that the needle swings across
the scale. The arc through which it swings depends
upon the strength of the field of the permanent magnet,
and also upon that set up about wires A and B ;
the latter, in turn, depends upon the amount of current
flowing in the coil. Therefore, if the scale were
calibrated in units of current, e.g., in milliamperes,
we would have a milliammeter. Or, since the current
through the coil would be proportional to the voltage
across its terminals, the scale could be graduated in
volts and the meter used as a voltmeter.
The part which magnetism plays in the operation
of the telephone plant has been illustrated by brief
descriptions of its use in relays, generators, and
measuring instruments. Other important illustrations
are to be found in the receiver and in the
'--- -----ljti-+;... ____ --J
Fig. 67 shows the structure of a crude telephone
receiver which embodies the same principles as the
standard types. In front of a U-shaped permanent
magnet about whose pole pieces are wound coils of
wire, is mounted a thin, sheet iron diaphragm. The
diaphragm moves in response to variations in the
magnetic pull which, in turn, are controlled by the
current flowing through the coils. The permanent
magnet exerts a steady pull on the diaphragm. For
the reason which will be explained in Section II-D
this is necessary to make the receiver work properly.
Fig. 68 is a sketch of a biased ringer. Here again
we find the combination of a permanent magnet and
an electromagnet. The permanent magnet M-N esPage
tablishes two south poles S-S, one at the end of the
core of each of the coils ; and, by induction, north
poles N-N are set up at the ends of the armature,
and a south pole S' is set up at its center. In other
words, the armature, which is a piece of soft iron,
has three poles, one in the center and one at each
end, induced in it by the action of the permanent
magnet. The biasing spring holds the armature in a
fixed position. Now suppose an impulse of current
flows through the coils in the direction indicated by
the arrowheads. The magnetism set up by the current
will strengthen the pole at the end of coil L,
and will either weaken the strength of the pole at
the end of coil R or reverse its polarity, depending
upon the strength of the energizing current. The
armature will, therefore, be attracted more strongly
by the pole of coil L, and it will either be attracted
less strongly or it will be repelled by the pole of coil
R, with the result that the effect of the biasing spring
will be overcome and the bell clapper will be moved
to the opposite side and will strike the r gong. When
the current stops flowing the magnetic relations in
the ringer will be reestablished as they -were prior
to the impulse, and the biasing spring will pull the
armature back to its original position thereby causing
the clapper to strike gong 1. We can see that a
series of impulses recurring in rapid succession will
ring the bell for so long as they continue.
Now let us assume there are two ringers on one
line, connected as indicated in Fig. 69. If a series of
current impulses be sent from line to ground through
both ringers as denoted by the arrowheads, the magnetic
effects in ringer ( A ) will be seen to be exactly
as just described, so that (A) will ring. In ringer
( B ) , ho..ever, the pole at the end of R becomes a
stronger\wuth pole, and the pole at the end of L
changes to north. Consequently these magnetic effects
aid the biasing spring to hold the armature
stationary, thus preventing ( B ) from ringing.
The Principle of Electromagnetic Induction
We have seen that electricity and magnetism are
very closely akin. In particular we have learned
that, whenever an electric current flows in a conductor,
a magnetic field is set up whose lines of induction
are closed loops which link with the electric
circuit in which the exciting current flows. That an
electric current produces a magnetic field is an important
principle, but is only one aspect of th.e relationship
between electricity and magnetism ; equally
important is the converse principle that a magnetic
field can produce an electric current.
In studying the latter principle it will be helpful
to have it stated in two ways :
( 1 ) If a conductor in a magnetic field is so moved as
to cut across the lines of induction, an emf is
set up in the conductor.
(2) If the number of lines of induction linked with
an electric circuit be increased or decreased, an
emf is set up in the circuit.
The action by which emf's are thus generated is
known as electromagnetic induction. Although the
foregoing two statements of this action may appear
superficially to differ somewhat, they are fundamentally
equivalent in that both state that the emf generated
is a result of a change in a magnetic field
with respect to a conductor in the field. Voltages
produced in this manner are called induced voltages,
and the currents to which they give rise are called
induced currents. An induced emf exists only while
t..e change producing it is taking place.
We shall shortly direct our attention to illustrations
of the above two ways of stating the principle
of electromagnetic induction. Before doing so, however,
let us become acquainted with the rules relatPage
ing the magnitude and the direction of an induced
emf to the change which generates it. We shall then
be in a better position to understand without difficulty
the points to be brought out in discussing the
The Magnitude of Induced Voltages
The magnitude at any instant of an emf induced
by a conductor moving across a magnetic field, depends
upon the rate (number of lines per second) at
which the conductor is cutting the lines of induction
at that instant. Similarly, the magnitude of an
emf induced by a change in the linkages between a
magnetic field and an electric circuit depends upon
the rate at which the number of linkages is changing.
An emf of one volt is induced when lines are cut or
linkages are changed at the rate of 100,000,000 per
The Direction of Induced Voltages
.An induced emf will, of course, tend to produce a
current ; but unless the conductor in which the emf
is induced forms part of a closed electric circuit, no
induced current can flow. The direction of an induced
emf is always such that, if the current which
the emf tends to produce can flow, the magnetic
field set up by this induced current (referred to in
the remainder of this discussion as the secondary
field) will be in the direction to oppose the change
by which the emf is induced. This rule for determining
the direction of induced, emf's will be easier
to apply if restated in the following two ways corresponding
to the two ways in which the principle
of electromagnetic induction was stated above :
( 1 ) The direction of an emf induced by a moving
conductor cutting the lines of a magnetic field,
is such that the secondary field would be in the
direction for its reaction with the original field
to oppose the motion of the conductor.
(2) The direction of an emf induced by a change in
the number of linkages between a magnetic field
and an electric circuit, is such that the secondary
field would be in the direction for it to
tend to maintain the number of linkages unchanged.
Induction by Motion
We are now prepared to take up some illustrations
of the principle of electromagnetic induction. The
first of the two ways in which this principle was
stated is illustrated in Fig. 70. A conductor AB is so
located with respect to the poles N and S of a permanent
magnet or electromagnet that by moving the
conductor up or down it will cut across the lines of
force. The ends of the conductor are connected to
a galvanometer. When the conductor is moved
downward or upward through the field a voltage is
generated in the conductor, causing the galvanometer
to deflect because of the induced current
through it. From what we have already learned, we
know that the direction along the conductor, of the
emf induced when the motion of the, conductor is
upward, must be from A to B. This conclusion is
reached by the following line of reasoning :-According
to the rule for determining the direction of an
induced emf, the secondary field in this case must
be in the direction for its reaction with the original
field to oppose the upward motion of the conductor.
The force exerted upon the conductor by this reaction
must, therefore, be downward. We know,
however, that the circular lines of force set up by a
current flowing through the conductor from A to B
would, as in Fig. 65, strengthen the field above the
conductor and weaken the field below it ; and this,
as was explained in discussing Fig. 65, we know
would exert a downward force upon the conductor.
The direction of the emf induced by the upward
motion of the conductor in Fig. 70 must, therefore,
be from A to B along the conductor, thus impressing
a voltage directed from B to A in the circuit connecting
to the galvanometer.
By sjmilar reasoning we also learn that the direction
along the conductor, of the emf induced when
the conductor is moved downward across the field,
is from B to A. The polarity of the voltage generated
when the conductor is moved downward is thus seen
to be the opposite of that generated when the conductor
is moved upward. With the galvanometer
poled as indicated in Fig. 70, upward motion of the
conductor would cause the needle to deflect to the
right, and downward motion would cause a deflection
to the left. The same effects as produced by
moving the conductor across the stationary field can
be obtained by holding the conductor stationary and
moving the magnet up or down in such a manner
that the lines of force will be cut by the conductor.
In whichever manner the relative motion between
the conductor and the magnetic field occurs, however,
it is particularly to be noted that the lines of
force must be cut by the conductor ; if this motion is
in a direction parallel with the lines of force no lines
will be cut and there will be no voltage generated.
The voltage of electrical generators is produced
in the ma..ner just described. For example, in the
hand generator (magneto) a coil of wire forming
the armature winding rotates in the magnetic field
set up by the U-shaped permanent magnets. This
coil is the equivalent of many conductors connected
in series ; and since each of these conductors cuts the
magnetic field and thereby has a voltage induced in
it, the result is similar to connecting a group of batteries
in series. In discussing Fig. 54 the statement
was made that if the strength of the magnetic field
be cut down by improperly assembled or defective
magnets, the voltage of the generator will be reduced.
We can now see this would be true, because
it would result in fewer lines of force being cut by
the armature. We can also see why the speed at
which the armature turns is an important consideration
; it is one of the factors which, by determining
the number of lines cut per second, determine the
voltage generated. The hand generator will be further
explained in connection with the discussion of
alternating currents in Section II-A.
The aspect of the principle of electromagnetic induction
presented by the first of the two statements
of the principle, and discussed in the three preceding
paragraphs, is of great practical importance because
it enables us to convert mechanical energy into electrical
energy. The second statement of the principle
is likewise of extreme practical interest because the
aspect which it presents affords lm explanation of
certain important properties of electric circuits. We
shall need to have a clear understanding of these
properties and shall, therefore, devote the remainder
of this Section to a study of them.
We are all familiar with every-day examples of
inertia. Take, for instance, a person seated in a car
which has seats running lengthwise along its sides :
when the car starts, his body sways away from the
direction of the car's motion ; and when the car stops,
he sways in the direction in which the car was going.
This is because his body, due to its inertia, resists
being set in motion from a state of rest, but once
moving, it resists being stopped. Note, however, that
inertia does not prevent a change of speed, but
merely restricts the rate at which the change occurs.
It is because of inertia that time is required to build
up or to reduce the speed of a moving object.
Electric circuits exhibit a property which is quite
closely analogous to inertia. We call this property
self-inductance. Let us see how self-inductance comes
about and how it affects the behavior of circuits.
From what we have already learned we know that
the lines of force set up when a current flows through
an electric circuit are closed
loops which link with
the circuit, and that the number of these linkages
will increase or decrease as the current increases or
decreases. Consequently, according to the principle
of electromagnetic induction, an emf will be induced
in a circuit whenever the current in the circuit
changes. The direction of any emf so induced can be
determined by applying .the rule already stated :
when a change in the current in a circuit causes a
change in the number of magnetic flux linkages with
that circuit, the emf thereby induced must be in the
same direction as an induced current whose field
would tend to offset the clJ.ange in linkages. We can
see, however, that to offset an increase in linkages
the secondary field would have to link with the circuit
in the reverse direction from the original field ;
and that to offset a decrease in linkages the secondary
field would have to link with the circuit in the
same direction as the original field. It follows, therefore,
that the increase in flux linkages accompanying
an increase in current will induce an emf in the opposite
direction from the current, and that the decrease
in linkages accompanying a decrease in current
will induce an emf in the same direction as the
The foregoing may be summarized by saying that
the change in magnetic flux linkages accompanying
a change in the current in a circuit induces in that
circuit an emf in the direction to oppose the change.
To supplement this statement, we recall that the
greater the change in the current and the more rapidly
it occurs, the faster will be the rate at which
the number of linkages changes, and hence the
greater the induced emf will be. We must also remember
that the induced emf exists only while the
change producing it is taking place.
One thing in particular stands out in the above discussion-
the self-inductance of a circuit comes into
play only during a change in the current. In directcurrent
circuits this property has no effect upon the
value arrived at when the current is changed from
one steady value to another ; but, as we shall see in
a moment, it does prevent the transition from occurring
instantaneously. It is, therefore, as earlier
stated, analogous to inertia.
· For example, when a voltage is applied to a circuit,
the emf induced by the flux linkages created
while the current is being established, is in the opPage
posite direction from the current, so that part of the
applied voltage is counteracted by this induced emf,
and only the remainder is available to overcome the
resistance of the circuit. Initially-the current and
hence also the IR drop then being zero-the current
starts to build up at a rapid enough rate to make the
induced counter emf equal to the applied voltage.
But as the current increases, thus increasing the portion
of the applied voltage required to overcome the
IR drop, the portion left over to further increase the
current must decrease. The rate at which the current
builds up towards the steady value I = EjR must,
therefore, continuously diminish. Thus, during the
establishment of a current, the self-inductance of a
eircuit prevents the current from jumping instantly
. to its steady value, but constrains it instead to require
time to build up.
On the other hand, when a circuit is opened the
current is broken and the magnetic flux linkages
die out, thereby inducing in the ..ircuit an emf which,
we have learned, must be in the direction tending to
maintain the current flow. This induced voltage is
commonly high enough to force a flow of current to
continue momentarily across the short initial length
of the air-gap formed as the circuit is opened, thereby
causing a spark at the point of break ; and in
many circuits this voltage is so high as to necessitate
protective measures to prevent destructive burning
of contact points. Certain types of spark coils are
designed to take practical advantage of this effect
of self-inductance by utilizing it to obtain voltages
high enough for various ignition purposes.
We now know what self-inductance is-how it
manifests itself. Let us next consider what factors
determine the magnitude of this property of an electric
From what we have already learned we know that
the greater the emf induced in a circuit when the
current in it changes at a given rate, the greater is
its self-inductance. Self-inductance is expressed in
terms of a unit called the henry : by definition the
self-inductance of a circuit in henrys equals the number
of volts induced in the circuit when the current
in it is changing at the rate of one ampere per second.
This induced voltage, however, we know to be
proportional to the change per second in the number
of flux linkages. Consequently the self-inductance
of a circuit is proportional to the change in the number
of flux linkages, per ampere change in the current.
Likewise, the self-inductance of any portion
of a circuit is proportional to the change in the number
of flux ltnkages with that portion, per ampere
change in the current.
In the case of the two straight wires comprising an
open-wire pair the flux lines linking with the pair
are only those which, as indicated in Fig. 7 1, pass
between the two wires ; and each such flux line links
with the circuit but once. The self-inductance of an
open-wire pair is, therefore, quite small. And the.
self-inductance of a cable pair is smaller still, because
the two wires are separated only by the thickness
of the insulation, and hence relatively fewer
lines pass between the wires to link with the pair.
In the case of a coil of wire, however, the number
of flux linkages, per line of induction threading the
coil, depends upon the number of turns; and the
number of these lines is directly dependent upon the
number of turns ( which determines the magnetomotiveforce
per ampere) , and inversely dependent
upon the reluctance of the associated magnetic circuit.
The change in the number of flux linkages per
ampere change in the current, and hence also the
self-inductance of a coil, is thus seen to have a twofold
dependence upon number of turns and a reciprocal
dependence upon reluctance. Where--as is
quite closely approached in coils in which this is a
design objective--the flux set up by each turn links
all turns, the magnitude of the self-inductance of a
coil is directly proportional to the square of the number
of turns and is inversely proportional to the reluctance
of the associated magnetic circuit. The selfinductance
of relay windings, loading coils, retardation
coils, etc., may, therefore, be made as high as
As has already been noted, self-inductance comes
into play only during a change in current. It then
tends to retard the transition ; but, in direct-current
circuits ( d-e circuits) , it has no effect upon the value
arrived at when the current is changed from one
steady value to another. Moreover, even where selfinductance
is large, the interval during which its retarding
effect causes the transition current in a d-e
circuit to differ appreciably from the succeeding
steady value is exceedingly brief. Ordinarily, therefore,
in dealing with d-e circuits, we are not concerned
with transition currents, but only with steady
values ; and so we can ignore self-inductance. In
alternating current circuits (a-c circuits) , however,
the current is continuously changing, so that, as we
shall find in Section II, self-inductance may be an
important factor in determining the current under
Every line of induction in the magnetic field set
up by an electric current links at least once with the
exciting circuit, and a portion of these lines, or even
substantially all of them, may also link with other
electric circuits. According to the principle of electromagnetic
induction, a change in current will,
therefore, not only induce an emf of self-induction
in the exciting circuit, as we have just learned, but
it will also induce an emf in any circuit which is
magnetically linked with the exciting circuit. The
property, by virtue of which a change in the current
in either of two circuits induces an emf in the other,
is called mutual inductance. This property, like selfinductance,
is measured in henrys. The mutual inductance
of two circuits, expressed in henrys, equals
the number of volts induced in either circuit when
the current in the other is changed at the rate of one
ampere per second.
The direction of an emf produced by mutual induction
can readily be determined. As we shall see
in a moment, the emf which a change in the current
in either of two magnetically linked electric circuits
induces in the other, is always in the same direction
(relative to the linking magnetic circuit) as the emf
of self-induction which the same change in current
simultaneously induces in the exciting circuit. By
saying that these two induced emf's are ip. the same
direction relative to the linking magnetic circuit, we
mean that the induced currents which these emf's
tend to produce would both encircle the magnetic
circuit in the same direction. This, we know, must be
true, because the rule governing the direction of all
induced voltages tells us that the secondary fields of
the emf's we are now considering are both required
to be in the direction to oppose the change in the
original magnetic field. Inasmuch, therefore, as we
already know how to determine the direction of an
emf induced by self-induction, we can also determine
the direction of any emf produced by mutual induction.
Two coils arranged to demonstrate the action of
mutual inductance are shown schematically in Fig.
72. These coils are insulated from each other and
are wound on the same core so that, of the total flux
set up by a current in either coil, the major portion
links both coils (mutual flux), and only a minor portion
merely links the excited . coil (leakage flux ). A
relay with two inductive windings, and the two
windings of a repeating coil, are familiar examples
of such an arrangement. Across the terminals of one
coil is placed a voltmeter, and the other coil is connected
to a battery through a key. Let us examine
into what happens when the k..y is closed and when
it is later opened.
When the key is closed current starts to build up
in coil A, flowing in the direction indicated by the
arrowhead. As this current and the field produced
by it build up, the rising total flux induces an emf
of self-induction in coil A, and the rising mutual flux
induces an emf in coil B. The emf of self-induction
will, of course, be in the direction to oppose the
building up of the current in A. Its direction around
the core will, therefore, be opposite that in which
the core would be encircled in traversing coil A in
the direction pointed by the arrowhead in circuit A.
Both the emf of self-induction in A and the emf of
mutual induction in B will, therefore, be in the direction
( around the core) in which the core would
be encircled in traversing coil B in the direction
pointed by the arrowhead in circuit B. The current
in B will also be in this direction. It is, of course,
only while the current in A is building up that the
induced emf in B exists. Hence, at the instant the
key in Fig. 72 is closed, the meter will indicate a
current in B which lasts but momentarily and then
When the key in Fig. 72 is subsequently opened,
the current in coil A is stopped and the field dies out.
The decrease in linkages while the field is dying,
again induces an emf in each of the coils. This time
the emf of self-induction in A will be in the direction
tending to keep the exciting current flowing. Its
direction around the core will, therefore, be that in
which the core would be encircled in traversing coil
A in the direction pointed by the arrowhead in circuit
A. The emf of mutual induction in B will be in
this same direction ( around the core) , i.e., opposite
the direction pointed by the arrowhead in circuit B.
We shall, therefore, again note a momentary deflection
of the meter, but in the opposite direction from
the deflection when the key was closed.
Referring further to Fig. 72, we have now seen
that, because of mutual inductance, changes of the
current in coil A induce momentary emf's in coil B.
These momentary induced voltages set up induced
currents in B which, in ordinary circuits, also build
up and die out quickly. As has already been implied,
however, the mutual inductance of circuit A with
respect to circuit B is the same as that of circuit B
with respect to circuit A. The building up and the
dying out of the momentary pulses of current induced
in B will, therefore, in turn, induce emf's in
A which affect the rate at which the current in A
builds up when the switch is closed, and the rate at
which it dies out when the switch is opened. This
effect of mutual inductance in d-e circuits depends
somewhat upon the nature of the circuit to which
the secondary coil B is connected, and is far too complex
for us to attempt to delve into in detail here.
We can, however, quite easily get a useful picture
of what is usually the predominant manifestation of
this effect. A change in the current in A induces in
B a current which, while it is building up, in turn
induces in A an emf in the direction to aid ( speed
up) the change of the current in A. This result of
the current induced in B is seen to be partially to
counteract the retarding action which, were coil B
open circuited or not present, the self-inductance of
A would exercise upon changes in the exciting
Like self-inductance, mutual inductance has no
effect upon the steady current values in circuits
under d-e excitation, but is important in a-c circuits
and will be studied further in Section II. Repeating
coils, induction coils and various input transformers
and output transformers, are familiar examples of
apparatus in the telephone plant, whose operation
depends upon mutual inductance.
For the purpose of bringing out more clearly a
number of the ideas concerning mutual induction
discussed in the preceding paragraphs, a group of
experiments to demonstrate those points are outlined
in Figs. 73 to 79. These experiments can easily
be performed with the usual apparatus at hand in
many of the offices. The apparatus required, the
manner of connecting it and manipulating it, and the
points to be observed in carrying out the various
steps, are all indicated in detail by the schematics
and by the legends below them.
.. f--- X
Description:-Any two-winding relay and the api)aratus
shown, to demonstrate the effects of mutual induction. At
the start, with switch open, no current 11ows in either circuit.
When switch is closed armature pulls up. Momentary current
11ows in secondary in direction shown.
While switch remains closed armature stays operated and
ammeter shows steady current 11.owing in primary, but there
1s no current in secondary.
- :h -
Pri. t Sec.
t ! - 4-
When switch is opened armature falls back. Momentary
current 11.ows in secondary in direction shown.
Use a rheostat in primary circuit instead of switch. By varying
current in primary a current can be induced in secondary.
Note with little or no current change in primary there is no
current in secondary.
Move rheostat to right and note current 1low in secondary in
direction shown. When rheostat comes to rest at "0" note
that current in secondary returns to zero.
Pri. t .. -
- - ,. + - t -
Move rheostat to left and note current 1lows in secondary in
direction shown. When rheostat comes to rest no current
1lows in secondary. Repeat the above experiments using a
No. 20 induction coil in place of relay.
Elements of Electricity
Elementary Electricity and
Magnetism . . .
Practical Electricity . . . .
Theory and Elements of
. . . . . W. H. Timbie
. . . . . . Jackson and Black
. . . . . .T errell Croft
Telephony . . . . . . . . . . . . . . . . . . . . .K empster B. Miller
Lessons in Practical Electricity . . C. W. Swoope