denote the power expended on the motor and the power
given out by the motor respectively, then the electrical
efficiency, or conversion-factor, is
W = IE' = &
W~ ~ IE E'
or the ratio of the counter E.M.F. to the applied E.M.F.
If the applied E.M.F. is a constant, the efficiency increases
with the counter E.M.F. Now the effective E.M.F. pro-
ducing the current is E E', and the larger E' the smaller
is this difference and the smaller the current. When the
current is small work is done at a slow rate, but a larger
fraction of the power applied is spent in useful work. It
is necessary to point out that this relation assumes an
electrically perfect motor. Since a certain current is
needed to make the motor run at the required speed
without doing any useful work, the useful current is the
difference between the whole current and the current
required to run the motor up to speed without load. It is
therefore evident that a practical motor does not have its
highest commercial efficiency when working under the
smallest loads, for then a large fraction of the current does
not contribute to the useful work done.
The work done by a motor per second is
/>r. \AMOS AND MOTORS. 409
Since R is constant the work done will be a maximum
when the product E'^EE') is a maximum. Now the
sum of the two factors of this product is the applied
K.M.F., E ; and when the sum of two factors is a con-
stant their product is greatest when they are equal to
each other. The condition for maximum activity is then
E' = EE',GtE' = E.
A motor does work at the greatest rate when the current
is reduced by the counter electromotive force to half the
value it would have if the motor were standing still. The
efficiency is then only 50 per cent.
358. Efficiency of Transmission. - When power is
transmitted to a distance electrically, high efficiency re-
quires high electromotive force. This is equally true
whether the energy is used for lighting or for power. The
energy lost in the line as heat is I-R watts, where R is
the resistance of the line. To keep this waste small while
the power transmitted is increased, the voltage must be
raised. The current depends on the difference between
the applied and the counter electromotive forces .Z? E',
while the power put into the circuit is IE watts and
the power given out by the motor IE' watts. If the
difference E E' is kept constant, the current and the
waste in heat will remain constant, while the power trans-
mitted will be proportional to the applied E.M.F. The
factor that determines the heat waste is controlled by
keeping the current small; while the other factor that
enters into the measure of the power transmitted, that is,
the electromotive force, is raised. The other way of re-
ducing the energy lost in the line is to reduce the resist-
ance ; but this method involves the use of a quantity of
copper the cost of which is prohibitive.
ELECTRICITY AND MAGNETISM.
359. Alternators. The armatures already described
generate alternating electromotive forces that follow the
law of variation of a sine curve more or less closely. A
complete series of changes in the electromotive force or
current represented by this curve is called a period, and
the number of periods in a second is the frequency of
the alternations. In two-pole machines the frequency
is the same as the number of revolutions per second.
1 1 | When the alternating cur-
rent is utilized in the exter-
nal circuit, the frequency
is restricted to a lower
limit of about 25 and a
higher one of about 150.
If the frequency is less than
25 per second the eye can
detect the variations in the
brightness of an incandes-
cent lamp ; while for fre-
quencies much above 130
or 140 the effects of self-induction are greatly exaggerated.
Within the above limits multipolar machines must be used
to avoid excessive speed of revolution. The frequency n
is then the speed of rotation multiplied by the number of
pairs of poles.
The circuit through the armature of an alternator is of
the simplest kind. The field is separately excited so that
the polarity of the poles remains fixed. It will readily be
seen that the successive armature coils must be so con-
nected that the circuit reverses in direction around the
coils from one to the next (Fig. 212). For high voltage
they are all joined in series. A complete period is the
time required for a coil to pass from one pole to the next
one of the same sign.
DYNAMOS AND MOTORS. 411
360. Lag of Current behind the Electromotive Force.
- When an alternating electromotive force is applied to a
circuit possessing inductance one of the novel and essential
facts is that the current reaches its maximum value later
than the electromotive force ; and, as a consequence, Ohm's
law is no longer adequate to give its value. The effect of
self-induction is not only to introduce an additional electro-
motive force, but to produce a lag of the current in phase
behind the electromotive force impressed on the circuit by
Let an alternating current, following the simple har-
monic law, be represented by the heavy sine curve / of
Fig. 204. Then, since the induced electromotive force is
proportional to the rate of
change of the current when
there is no iron in or about
the circuit, the induced E.
M.F. curve may be repre-
sented by the light line //.
This is also a sine curve, ^
since the differential coef- a
ficient of a sine function
is itself a sine function. But the latter curve reaches
its maximum value a quarter of a period later than the
former. When the current is a maximum at A its rate
of change is zero, and when it diminishes through its zero at
B its rate of change is a maximum. The induced electro-
motive force and the current are said to be in quadrature.
The effective electromotive force producing the cur-
rent by Ohm's law must correspond in phase with the
current itself. The maximum induced and effective elec-
.tromotive forces may therefore be represented by the two
adjacent sides of a right triangle (Fig. 213), where be
412 ELECTRICITY AND MAGNETISM.
is the induced E.M.F. and ab the effective E.M.F. ; the
hypotenuse ac is therefore the maximum impressed E.M.F.
(I., 31). But the current agrees in phase with ab ; it
therefore lags behind the impressed electromotive force by
the angle </>. In the absence of capacity in the circuit, this
angle becomes zero only when the inductance is zero.
The instantaneous values of the several electromotive
forces may be found by revolving the triangle around a as
a centre, and projecting the three sides upon some straight
line through a, as in Part I., Fig. 18.
361. Value of an Alternating 1 Current. The instan-
taneous value of an alternating current following the law
of sines is
i = I sin JTsin cot,
where I is its maximum value and co the angular velocity
2 (I., 33).
If the induced electromotive force is proportional to the
change-rate of the current (338), then
L - di/dt = Lcol cos o>,
since the rate of change of the sine is the cosine. This is
the expression for the instantaneous value of the induced
electromotive force. Its maximum value is Z/col, the
maximum value of the cosine of an angle being unity.
Therefore in the triangle of electromotive forces (Fig.
213), the side be equals Lcol. Also ab equals RI, because
it is the effective electromotive force, and by Ohm's law it
is the product of the resistance and the current. There-
fore ac equals I (12 2 + L~ar)k ; but the hypotenuse is the
maximum impressed electromotive force. Then
DYNAMOS AND MOTORS. 413
The expression (R 2 + ZV)^ is called the impedance. The
impedance shows that the effecj of inductance on the
value of the current is equivalent to additional resistance.
Also from the figure
It is evident, therefore, that the angle of lag increases with
the coefficient of self-induction L and with the frequency
(co= 2?). In these equations I and E denote the max-
imum current and impressed electromotive force. The
current lags as if the angle in the auxiliary circle of refer-
ence were &> $ instead of cot. We may therefore write
for the instantaneous current
where the term </> is added to show that the current lags
behind the electromotive force E.
The effect of capacity in series is to produce a lead
instead of a lag of the current, and the one offsets the
other Avhen L(o= \/Ca>. 1
362. Virtual Volts and Amperes. All practical in-
struments for measuring alternating currents and pressures
take account of the "square root of the mean square"
values and not the arithmetical mean. Thus the electro-
dynamometer (301), the Kelvin balances (302), and the
electrostatic voltmeter (147) all integrate the forces oper-
ating them, and these are proportional to the squares of
the current and of the electric pressure. If the current
and the electromotive force follow the sine law, the mean
given by these instruments is 0.707 of the maximum
1 Carhart and Patterson's Electrical Measurements, p. 239.
414 ELECTRICITY AND MAGNETISM.
values. When a voltmeter on an alternating circuit reads
70.7, the voltage alternately rises to +100 and sinks to
100 as positive and negative maxima. The values
given by these instruments are virtual volts and virtual
The virtual values exceed the arithmetical mean values
by 10 per cent. 1 A continuous current and an alternating
current of equal virtual value have the same heating
effect ; but a continuous current equal to the arithmetical
mean of the alternating one will have a smaller heating
effect in the ratio of 1 to 1.23 (or .637 2 to .707 2 )-
363. Choking Coils. Consider a circuit with small
resistance and large inductance. The current will then
depend largely on the latter ; or, if R is negligible,
This formula holds either for maximum or for virtual
values. Coils with a divided iron core, having small
resistance and large self-induction, are called choking coils.
Thus \in were 134, L 100 henrys, and E 1,000 volts, the
current through the coil of negligible resistance would be
only 0.012 ampere. A current of about this value flows
through the primary of a transformer on a thousand-volt
circuit when the secondary is open. It is approximately
independent of the resistance.
364. Wattmeters. - - The measurement of power in
circuits conveying alternating currents cannot be made
in the same way as when continuous currents are employed,
i The mean of the squares of the sines throughout a half-period is 1/2. The
square root of the mean square value is therefore 1 A/ 2 of the maximum, or
0.707. The mean value of the sines throughout a hall-period, on the contrary, is
2/7T, or 0.637.
DYNAMOS AND MOTORS. 415
where the energy spent on any part of the circuit is
measured by finding the current through it and the poten-
tial difference between its extreme points ; for the potential
difference and the alternating current are not in step
unless the circuit is non-inductive. Thus in the example
of Art. 341, the energy expended on the coil with the
alternating current was apparently 100 / watts, while in
reality it was only 27 I watts. When the electromotive
force and current differ in phase, one of them is sometimes
positive while the other is negative ; hence a part of their
instantaneous products are positive and part negative.
During that part of the period when this product is nega-
tive the circuit is restoring power to the source. The
integrated difference between the two products is the
Power on alternating circuits may be measured by a
wattmeter. If the movable coil of an electrodynamometer,
consisting of several turns of wire, be disconnected from
the field coil and be connected in series with sufficient non-
inductive resistance as a shunt to the circuit in which the
power is to be measured, while the fixed coil is connected
in series with this circuit, the indications of the instrument
will be proportional to the integrated sum of the instan-
taneous products of the electric pressure and the current.
When the instrument, which is then called a wattmeter,
has been properly calibrated, it measures the power ex-
pended in watts. It is of course equally applicable to
365. Transformers (J. J. T., 4O5). A transformer
is an induction coil with a primary of many turns, a second-
ary of a smaller number, and a closed magnetic circuit.
It is employed with alternating currents as a " step-down "
416 ELECTRICITY AND MAGNETISM.
instrument for the purpose of reducing the high electro-
motive force on the transmitting line to a low electromotive
force for lighting and power. It is entirely reversible and
can be used equally well for the " step-up " process with
The primary and secondary coils are wound round an
iron core (Fig. 214), but are insulated from each other as
perfectly as possible. In
practical transformers the
iron encloses the wire
rather than the reverse.
The iron serves as a path
for the flux of magnetic
induction. The student
should notice that the re-
lation of the current and
the flux is a reciprocal
one, so that they may always exchange places. With
either relative arrangement of the iron and the coils, nearly
all the lines of induction produced by the primary pass
also through the secondary, and vice versa.
When the secondary is open the transformer acts simply
as a " choking coil ; " the current passing through the
primary is then only the very small one required to mag-
netize the iron for the generation of the counter E.M.F.,
which is then nearly equal to the impressed E.M.F. When
the secondary is closed the currents in the primary and
secondary are nearly in the inverse ratio of the turns of
wire on the two, or N Z /N^ , where JVj denotes the turns on
the primary and N. 2 the number on the secondary. The
electromotive forces generated in them, when there is no
magnetic leakage, is directly as the ratio of transformation
The energy in the secondary circuit is therefore
DYNAMOS AND MOTORS.
nearly the same as that expended on the primary. The
small difference is chargeable to loss in the copper of the
primary and to losses in heating the iron on account of
hysteresis and Foucanlt currents.
The secondary current is nearly opposite in phase to the
primary, and causes a diminution in the apparent self-in-
duction of the primary coil, so that the larger the second-
ary current the larger the primary. The transformer is
therefore nearly self-governing. The power absorbed by
the primary increases as the resistance of the secondary
decreases; but it reaches a maximum for a particular
value of the secondary resistance, below which the energy
absorbed by the transformer decreases. This critical value
of the resistance is larger the higher the frequency.
366. Polyphase Currents. It has long been known
that two or more alternating currents of the.-same frequency,
but differing in phase by any desired quantity, may be
ELECTRICITY AND MAGNETISM.
obtained from one generator. If, instead of a commutator,
four insulated rings on the shaft be connected to four
equidistant points of either a drum armature or a Gramme
ring, the currents in
A B A the externally sepa-
rate circuits will differ
in phase by a quarter
of a period. In the
small laboratory ma-
chine of Fig. 215 the
exciting current flows
through the revolving
field-magnet by way
of the brushes bearing on the two rings. The armature
is a stationary ring wound continuously on a laminated
iron core, with four con-
ductors leading from
points 90' apart. Each
pair, 180 apart, compose
an alternating circuit.
It is obvious that one
current passes through
its maximum at the same
instant that the other
passes through its mini-
mum value (Fig. 216). In a similar way three-phase cur-
rents will pass through conductors 120 apart. If there
are but three conductors, each one serves as a return for
the other two, since the algebraic sum of either two cur-
rents is at any instant equal to the third (Fig. 217).
367. The Rotatory Field. When an alternating cur-
rent passes through a coil of wire without iron it produces
DYNAMOS AND MOTORS.
an alternating magnetic field along its axis. If the current
follows the sine law, the magnetic flux will follow the sine
law also. Let two such coils be set with their axes at
right angles, and let the equal alter-
nating currents through them differ
in phase by a quarter of a period.
Two simple harmonic motions of
equal amplitude, at right angles,
and differing in phase by a quarter
of a period, combine to produce
uniform circular motion (I., 29).
Hence the two coils, AA and BB
(Fig. 218), will produce in a simi-
lar way a rotatory magnetic field near their common centre.
Ferraris (1888) mounted within them a hollow copper
cylinder on pivots at top and
bottom. When the two-phase
currents from the small machine
(Fig. 215) are sent through the
Ferraris apparatus, the copper
cylinder is set rotating. The
rotation of the field produces
currents in the copper, as in
Arago's rotations. By Lenz's
law the motion of the cylinder
is in a direction to check the
action going on ; hence the cyl-
inder is dragged around in the same direction as the
rotation of the field ; for, if the speed of the cylinder were
the same as that of the field, no current would be induced.
If one current is reversed with respect to the other, that is,
if its phase is changed by 180, the direction of rotation of
both the field and the cylinder is reversed. The cylinder
ELECTRICITY AND MAGNETISM.
tends to run up to synchronism with the field, but never
reaches it ; the difference in their speeds is just sufficient
to produce currents to supply the requisite torque. If
the rotation of the field produces a direct E.M.F., the
rotation of the cylinder, which is equivalent to the rota-
tion of the field in the other direction, produces a counter
E.M.F., and the latter is always smaller than the former.
368. Induction Motor. A rotation of the field may
also be produced by winding the coils of the two circuits
on an iron ring
(Fig. 219). The
coils A and A'
are wound so as
to make conse-
quent poles at B
and B', while the
coils B and B'
quent poles at A
and A'. When
one of these cur-
rents is a maxi-
mum, the poles in
the ring are con-
centrated as in
Fig. 220, which
was made from a
photograph. Fig. 221 shows the field an eighth of a
period later, when the two currents have the same instan-
taneous value. Both poles have spread out uniformly a
quarter of the way around the ring in the direction of
the rotation. As the first current diminishes further
DYNAMOS AND MOTORS.
toward zero, these broad poles contract
nated iron cylinder with heavy
conductors embedded in its
periphery and running parallel
witli its axis of rotation. They
are connected together at the
ends of the cylinder so as to
form a " squirrel-cage " of cop-
per. The induced currents
through this cage produce a
torque which drags the cylin-
der after the rotating field.
Three-phase induction motors
are constructed on a similar
plan (Fig. 222).
ends ; and, after
a quarter of a
period, are again
points 90 in ad-
vance of the
s t a r t i n g-point.
The poles thus
move round the
ring by a motion
which may be
compared to that
of a " measuring
Inside the ring
is mounted a
ing of a lami-
422 ELECTRICITY AND MAGNETISM,
ELECTRIC OSCILLATIONS AND WAVES.
369. Oscillatory Discharges. Allusion lias already
been made to the oscillatory character of the discharge of
a Ley den jar. It was discovered by Joseph Henry in 1842
by studying the singular phenomena of the magnetic effects
produced by it in small steel needles, which were not
always found to be magnetized in the expected direction.
In 1853 Lord Kelvin gave the mathematical theory of
electric oscillations, and in 1858 Fedderson analyzed the
spark of a small discharge into a number of images by a
revolving mirror. Such a discharge consists of electric
surges first in one direction and then the other. The
charge deports itself as if it possessed inertia ; when the
condenser is suddenly discharged through a low resistance,
the first rush surges beyond the condition of equilibrium,
and the condenser is charged in the opposite sense ; a
reverse discharge follows, and so on, each successive
oscillation being weaker than the preceding, till after a
few surges the oscillations cease. That such is the char-
acter of the discharge of a Leyden jar has been abundantly
demonstrated by experiment.
When the coatings are connected by a discharger of self-
induction L and negligible resistance, the electrostatic
energy, %Q- / C, disappears and becomes the electromagnetic
energy of the discharge current, J LP. This in turn is re-
converted into the electrostatic energy of a reverse charge
ELECTRIC OSCILLATIONS AND WAVES. 423
of the jar; a second conversion into the electromagnetic
form follows, and so on. Each conversion of the energy
from the potential form to the kinetic or the reverse is ac-
companied by a loss of heat, till the energy is all expended.
The oscillations of a small Leyden jar, charged by con-
necting its two coatings with the secondary terminals of
an induction coil, can be readily exhibited to a large
number of persons. It is convenient, though not essential,
to elose and open the primary circuit by means of a seconds
pendulum. A pointed strip of tin foil must be brought
over from the inner coating of the jar so as to leave a small
spark gap between it and a point connected with the outer
coating. At every break of the primary circuit a spark
will leap across this gap if the adjustments are properly
made. If it is viewed in a four-square mirror rotating
with moderate speed, it is found to consist of from about
four to twelve successive images. A single observer may
view it by a telescope after reflection from a mirror on the
end of a tuning-fork making about 100 vibrations a second.
The rate of oscillation in this case is comparatively slow
on account of the large self-induction of the secondary
coil, but the whole series of oscillations takes place in the
4 * incredibly short space of time occupied by a spark."
37O. Period of an Oscillation. Whether a discharge
is oscillatory or only intermittent depends on the relation
between the resistance and self-induction of the discharge
circuit and the capacity of the condenser.
If R denotes the resistance in ohms, L the self-induction
in henrys, and O the capacity in farads, the discharge will
be oscillatory when
Phil. Mag. (4) 5, p. 393.
424 ELECTRICITY AND MAGNETISM.
When R is small the period of the oscillations is
This formula corresponds with the condition required
for capacity to neutralize self-induction (361), when
La) = l/(7a>. Since co^^Trn and T= I/ft, if we solve the
equation Lw = ~L/Ca) for T, we obtain the expression
above for the period, 27rv OL.
When the jar is discharged through a low resistance,
oscillations take place because the choking reactions due
to self-induction are neutralized by the capacity. The
oscillations then continue, like the vibrations of a tuning-
fork, till their energy is expended partly in heat and partly
in a manner to be described presently.
371. Electrical Resonance. If the period of oscilla-
tion of a Leyden jar is determined by its capacity and
self-induction, it should be pos-
sible to apply to the phenomenon
the principle of resonance in
Sound (I., 151), provided the
inductive effects of discharge
currents are conveyed to other
condensers. This has been done.
The oscillatory character of a
condenser discharge is demon-
strated by its power of evoking
oscillations of the same period
in neighboring condensers. The
following instructive experiment
F . 223 is due to Lodge : l Two similar
Leyden jars are connected to
discharge circuits of equal size (Fig. 223) ; but while that
1 Modem Views of Electricity, p. 338.
ELECTRIC OSCILLATIONS AM) WAVES. 425
of A is interrupted by a spark gap, that of B is complete
and is adjustable by means of the slider S.
If now the coatings of A are connected to the two elec-
trodes of an influence machine, this jar discharges across