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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.

410

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.

Fig. 212.

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

the generator.

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

Fig. 213.

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

, Leo

tan4>=_.

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

amperes.

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,

1= U/Lco.

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

work done.

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

continuous currents.

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

alternating currents.

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

Fig. 214.

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.

417

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.

Fig. 215.

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

418

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.

419

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

Fig. 219.

420

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'

produce conse-

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

Fig. 220.

DYNAMOS AND MOTORS.

421

toward zero, these broad poles contract

Fig. 221.

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).

their posterior

ends ; and, after

a quarter of a

period, are again

concentrated at

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

worm."

Inside the ring

is mounted a

"rotor," consist-

ing of a lami-

b

422 ELECTRICITY AND MAGNETISM,

CHAPTER XXVI.

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

R<

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

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.

410

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.

Fig. 212.

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

the generator.

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

Fig. 213.

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

, Leo

tan4>=_.

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

amperes.

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,

1= U/Lco.

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

work done.

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

continuous currents.

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

alternating currents.

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

Fig. 214.

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.

417

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.

Fig. 215.

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

418

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.

419

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

Fig. 219.

420

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'

produce conse-

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

Fig. 220.

DYNAMOS AND MOTORS.

421

toward zero, these broad poles contract

Fig. 221.

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).

their posterior

ends ; and, after

a quarter of a

period, are again

concentrated at

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

worm."

Inside the ring

is mounted a

"rotor," consist-

ing of a lami-

b

422 ELECTRICITY AND MAGNETISM,

CHAPTER XXVI.

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

R<

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

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