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Font size Fig. 312.

602. Series Motors. In a series motor, shown diagrammati-
cally in Fig. 312, the same current passes through both the field
coils and the armature. As was seen in the discussion of the

486

ELEMENTS OF ELECTRICITY.

magnetization curve (Par. 393), at first and when remote from
saturation, the field H increases nearly in proportion to the ex-
citing current, hence, at starting, the torque of a series motor,
I Hlx2r, varies practically as the square of the current. These
motors are therefore especially valuable where great torque is
needed at starting, for example in trolley cars, hoists, etc.

603. Speed of Series Motors. The speed of series motors
is a corresponding speed. This renders them unsuitable for many
kinds of machines which require a constant speed under varying
loads, but well adapted for street railways where the speed is of
necessity constantly varied.

Consider a generator supplying a series motor M (Fig. 313).
The power developed by the motor must be equal to that sup-

Fig. 313.

plied by the generator, less the heat loss. This last is small, hence
the back E. M. F. must be nearly equal to the impressed E. M. F.
As the back E. M. F. increases, the current through the motor,
and hence the current through the field coils, grows smaller. The
field grows correspondingly weaker and to maintain the back
E. M. F. the speed of the motor must increase. This tendency to
race under diminished loads is an objectionable feature of a series
motor.

604. Change of Direction of Rotation. It may sometimes be
desirable to change the direction of rotation of a motor. Suppose
a, Fig. 314, to represent a shunt motor, the current flowing as

b

'* ; - '

Fig. 314.

indicated. The lines of force of the coil will run upwards and the
rotation will therefore be clockwise. If the direction of the cur-
rent in the mains be reversed, as shown in 6, the lines of force of

ELECTRO-MECHANICS. 487

the coil will run downward, but the polarity of the field magnets
is also reversed, and the rotation will as before be clockwise.
Hence, reversing the current in the mains does not change the
direction of rotation. If, however, the direction of the current
be reversed in either the field or the armature, but not in both,
the direction of rotation will be changed.

605. Motor Generators. Alternating currents are readily
stepped up or down in voltage by means of a transformer (Par.
431), but this method is not applicable to direct currents. Where
such transformation is required, the direct current may be em-
ployed to operate a motor and this motor in turn operates a gener-
ator whose armature is so wrapped, or whose field is of such
strength, as to develop a current of the desired voltage. Instead
of having the motor rotate the generator by means of a belt or
gearing, they may both be mounted upon a common shaft. This
combination is called a motor generator, but electrically it is the
same as two separate machines.

A step further may be taken and two sets of coils may be
wrapped upon the same armature and rotate in a common field.
Each set has its own commutator, current being delivered to the
motor commutator and drawn from the generator commutator.
Transformation is effected by varying the ratio of the number
of coils or of the number of turns in the two sets of wrappings.
This machine is called a dynamotor.

488 ELEMENTS OF ELECTRICITY.

CHAPTER 43.

ALTERNATING CURRENTS.

606. Alternating E. M. F. and Current. We have seen (Par.
552), that if a coil rotates at a uniform rate in a uniform field it
will generate an E. M. F. which varies as the sine of the angle
through which the coil has turned from its primary position at
right angles to the field. If the coil is a closed circuit, or forms a
part of such a circuit, there will be produced in it a current which
will vary in the same manner. At every revolution of the coil,
therefore, the E. M. F. and current pass through a complete cycle
of values, positive and negative. The term alternating is applied
to an E. M. F., or to a current, which thus undergoes these periodic
reversals.

607. Why Considered Separately. The mere fact that a cur-
rent reverses its direction at regular intervals might not of itself
warrant special discussion. There are, however, two properties,
induction and capacity, which are common to all electric circuits
and whose effects are conspicuously revealed in varying currents.
Alternating currents vary continually and with such currents the
above factors give rise to certain peculiar phenomena, some of
which appear to contradict the principles which have been developed
in the preceding pages. Among such we may mention

(a) The current through a circuit is not always equal to the
E. M. F. divided by the resistance.

(b) The sum of the partial drops between two points is not
always the same as the total drop.

(c) The sum of the currents in the branches of a divided circuit
is not always equal to the total current.

(d) Finally, there may be a flow of current in a broken circuit.
In the following pages it will be shown that these contradictions

are only apparent and that Ohm's law is as true of alternating
currents as it is of direct. In order, however, to be able to explain
these peculiarities, the subject of alternating currents must be
considered in detail. We shall therefore begin with certain pre-
liminary definitions and principles.

ELECTRO-MECHANICS. 489

608. Cycle, Period and Frequency. In Par. 555 it was shown
that an alternating E. M. F. and current can be represented
graphically by a sine curve (Fig. 315), the ordinates corresponding
to the instantaneous values (values at any instant) of the E. M. F.
or current and the abscissae to the angle through which the coil

Fig. 315.

has rotated, or, if the scale of time be used, to the time elapsed
since the coil moved from its primary position in the neutral
plane.

If E m be the maximum instantaneous value of the E. M. F. and
if the abscissae represent the angle through which the coil has
rotated, the equation of the E. M. F. curve is

E = E m . sin 6
If the abscissae represent elapsed time, the equation is

E = Em . sin ut

in which o> is the angular

velocity of the coil and t is the time in seconds since the coil lay
in the neutral plane.

With every revolution of the coil, the portion of the curve be-
tween A and B (Fig. 315) is repeated, and the complete set of
values, positive and negative, between A and B is therefore called
a cycle. The more rapid the motion of the coil, the greater the
number of cycles in a given time. The lengths of time of one cycle
is called a period and the number of cycles per second is the
frequency. The word "revolution," as used above, must be inter-
preted in an electric sense. Thus, in a four pole generator one
revolution of the armature corresponds to two electric revolutions.

An additional term, sometimes encountered in books treating
of this subject, is alternation, an alternation being a reversal of
direction of E. M. F. or current. There are therefore two alter-
nations per cycle. The number of alternations is usually given as
so many per minute. It is recommended that the use of this term

490

ELEMENTS OF ELECTRICITY.

609. Phase. For purposes of descriptive location, a cycle is
considered to be divided into 360 degrees. Any point of the cycle
is designated as a certain phase, as, for example, the thirty degree
phase, etc.

Fig. 316 represents diagrammatically a ring-wound, bipolar,
alternating current generator. Consider in either half of the

Fig. 316.

armature any two adjacent coils, as, for example, B and C. In
each an E. M. F. is being induced and since in every complete
revolution of the armature each coil travels around the same path
and returns to its starting point, the cycle, the period and the
frequency must be the same for each. At the instant shown how-
ever, the E. M. F. being induced in C is proportional to the sine

Fig. 317.

of the angle CO A, while that being induced in B is proportional to
the sine of BOA, and will not reach the value of that now in C
until sufficient time has elapsed for B to move through the angle
0= BOC. The E. M. F. in C therefore has reached a value which

ELECTRO-MECHANICS. 491

will not be reached by that in B for a time corresponding to the
angle <. This is shown graphically in Fig. 317. The sine curve
CCCCC represents the E. M. F. of the coil C; the sine curve
BBBBB represents the E. M. F. of the coil B.

Two sine curves whose periods are the same and which reach
their maximum and minimum values simultaneously (see Fig. 265)
are said to be in phase, otherwise they are said to differ in phase.
The phase difference may be expressed in time but more frequently
in angular measure. Thus, the curves in Fig. 317 differ in phase
by the angle <f> which is represented by the horizontal distance
CB. If the phase difference is 90, the curves are said to be in
quadrature; if it be 180, they are in opposition.

It will be shown shortly that an alternating current generally
differs in phase from its corresponding E. M. F. If the current
reaches a maximum value after the E. M. F. has passed through
its maximum, the current is said to lag; on the other hand, if it
reaches its maximum in advance of the E. M. F., it is said to lead.
In these cases, the corresponding phase difference is spoken of as
the angle of lag or as the angle of lead.

610. Vector Diagrams. Let the vector OA (Fig. 318), whose
length represents the maximum value
Em of an alternating E. M. F. (or cur-
rent), rotate about the point in a
counter-clockwise direction and at the
same uniform angular velocity co as
the armature. The instantaneous value

of the E. M. F. (or current) is repre- Fi - 318 -

sented by the line A B, for AB = E m . sin at. But DO, the pro-
jection of OA upon the vertical axis, is equal to AB, hence, when
the vector makes the phase angle with the horizontal axis, the
corresponding instantaneous value of the E. M. F. (or current)
is represented by the projection of the vector upon the vertical
axis.

611. Composition of Alternating E. M. F.s. During the rota-
tion of the armature of the generator shown in Fig. 316, the coils
in series combine in producing a resultant E. M. F. Thus, in Fig.
317 the broken and dotted curve is the resultant E. M. F. curve
obtained by adding the ordinates representing the corresponding
simultaneous values of the E. M. F. in the separate coils. By an

492 ELEMENTS OF ELECTRICITY.

application of trigonometry, it can be shown that this resultant
curve is also a sine curve and is of the same periodicity as the
component curves, although differing from them in phase. The
trigonometric process is somewhat tedious and it is thought that

the following explanation will be
more easily followed. In Fig. 319,

, w . / , the vectors OB and OC represent the

' / maximum values of the E. M. F. in

the coils B and C of Fig. 316, and
is the angle of phase difference. The
instantaneous value of the E. M. F.
in B is, from the preceding para-
Fig. 319. graph, OB'', the instantaneous value
of the E. M. F. in C is OC', and the resultant E. M. F. is the sum
of OB' and OC'. Complete the parallelogram CDBO and project
its diagonal OD upon the vertical axis. C'D' is equal to OB',
hence OD' is equal to the sum of OB' and OC', or is the desired
resultant. Therefore, the resultant E. M. F. of the coils B and C
is always given by the projection upon the vertical axis of the
vector OD, the diagonal of a parallelogram of which the adjacent
sides represent the maximum values of the E. M. F. in the corre-
sponding coils and the included angle represents the difference in
phase. The length of OD represents the maximum value of the
resultant E. M. F. Since <, the difference in phase, is constant,
the vector OD does not vary in length or in position relative to
OB and OC. Its projections are therefore the ordinates of a sine
curve of the same periodicity as the E. M. F. curves of the separate
coils.

From the foregoing we see that alternating E. M. F.s which
differ in phase are not compounded by simple addition but in a
similar manner to that employed in the parallelogram of forces in
mechanics.

612. Value of an Alternating Current. During each cycle, an
alternating current passes through the entire range of values from
zero to the positive maximum, thence through zero to the mini-
mum (negative maximum), thence back to zero. Which of all
these values should be taken as a measure of the current? The
logical agreement is reached that such a current is equal to that
direct current which performs the same amount of work in the
same length of time. Of the three classes of work which a current

ELECTRO-MAGNETICS.

493

may perform (Par. 444), only one, the heating effect, is inde-
pendent of the direction of the current, and this is accordingly
selected as the basis of comparison.

A

M N

Fig. 320.

Let the curve AB (Fig. 320) represent an alternating current
produced by a coil rotating with an angular velocity co. If the
maximum value of the current be Im, the equation of this curve is

I = Im . sin ut (I)

Consider any ordinate of this curve as /. The instantaneous
value of the power being developed at this point is PR (Par. 494),
R being the resistance of the circuit through which the current is
flowing. Let M N represent a minute interval of time dt. Since
work = power Xtime, the work done by I during this interval is

dw^PR .dt

Substituting in this the value of / from (I)
dw = f m R . sin 2 o> . dt

The integral of this between the proper limits will give the total
work performed by the current during the cycle.

Jsin'W. (co dt)

CO

( \ cos u>t . sin

a constant

Taking this between the limits ut = and ut =

CO

(ID

Hence 7= = 0.707 I

494 ELEMENTS OF ELECTRICITY.

The work performed by a direct current flowing through the
same resistance for the same length of time is

w = PRt

A

Since at = 2ir, t, the time of one cycle =

CO

Substituting, we have

w=/ 2 #. (Ill)

CO

Equating the second members of (II) and (III)
/ 2 #. =I 2 m R.-

CO CO

m

that is, the alternating current is

equivalent to a direct current whose value is only .707 of the maxi-
mum value of the alternating current. This may be otherwise
expressed by saying that the effective or virtual value of the alter-
nating current is only .707 of its maximum value. The same
relation exists between the effective and the maximum voltage of
an alternating current, and ammeters and voltmeters for use with
respectively.

613. Second Deduction. The foregoing deduction may be
made without the use of the calculus, as follows:

Let AA and BB (Fig. 321) be two
coils at right angles to each other, both
rotating at a uniform rate in a uniform
field and each sending current through a
resistance #. In one complete revolution
the work done by the currents from both
/^''" \ / is twice the work done by the current

\ \ /' from one. The current from A being

X^ v * .S' ?m sm that from B is I m cos 0. The

"p"~T~ B power developed at any instant by the

current from A is I 2 m sin^&R; that de-

veloped at the same instant by the current from B is I 2 m cos 2 R.
The total instantaneous power is the sum of these two, or

Jj, (sin 2 6 + cos 2 0) R = f m R

ELECTRO-MECHANICS. 495

The total work done during the time t of one complete revolu-
tion is f m Rt, hence the work done in this time by the current from
one coil is \ 1^ Rt. A direct current / flowing for a time t through
the resistance R does work P Rt.

= ^ I 2 m Rt

Hence

=
V2

/ = ^ as before.

614. Self-Induction. Self-induction was explained in detail
in Pars. 432-436 and it was shown that its characteristic effect is
to oppose any change in a current-produced field and that it does
this by setting up a counter E. M. F. which opposes any change in
the current in the circuit involved. Since alternating currents are
always changing, it is in dealing with such currents that the con-
sideration of induction assumes the greatest importance.

Fig. 322.

If an alternating E. M. F. be applied to a circuit of a simple loop
of wire (Fig. 322 a), the effect of induction may be so slight as to
be negligible and the current may be considered to follow Ohm's
law.

If the same piece of wire be wrapped into a coil of 100 turns and
the E. M. F. be applied so as to produce the same number of lines
of force in the field in the same time as before, these are now cut
one hundred times instead of once and the effect of induction is
one hundred times as great.

Finally, if there be inserted in this coil a soft iron core (Fig.
322 b) and the E. M. F. be applied, the same change of current
will produce about 2000 times as many lines of force (Par. 394)
and the effect of induction will be 200,000 times as great as in the
first case. These examples show that self-induction is developed
by the cutting of the lines of force in the embraced field rather
than by changes in the current in the embracing circuit.

615. Inductance. Self-induction is measured by the cutting
of lines of force produced when the current in the circuit is varied
one unit. The practical unit, the henry, is the self-induction of

496 ELEMENTS OF ELECTRICITY.

that circuit in which a change of one ampere produces a cutting of
10 8 lines of force. When the self-induction of a circuit is expressed
numerically, as so many henrys, it is called inductance. The in-
ductance of a given circuit is constant provided the circuit is
distant from magnetic bodies. If it be not distant from such
bodies, owing to their saturation, the field does not vary uniformly
with the current.

Although the inductance is thus constant, the counter E. M. F.
which it is instrumental in producing, and whose effect is so im-
portant, is not at all constant but varies with the rate of change
of the current (Par. 432) and has therefore a different value at
every different phase and for every different frequency employed.
This will be shown more clearly later on.

616. Inductance and Resistance. Inductance and resistance
agree in that they oppose the flow of current in a circuit, but here
the similarity ends. The following will bring out the difference
between the two.

(a) The resistance of a circuit is constant and does not vary
with changes in the current. The inductance of a circuit appears
only when the current is changing and the counter E. M. F. which
it sets up is proportional to the rate of this change.

(b) Resistance does not vary with the geometric form of the
circuit nor with the proximity of magnetic bodies. Inductance
depends essentially upon these factors.

(c) The energy spent in overcoming resistance is lost in the
form of heat. That spent in overcoming the induction counter
E. M. F. (Par. 359) is periodically absorbed in the field about the
conductor as the current rises and is restored to the circuit as the
current falls. As an analogy, the energy spent upon a fly-wheel
does two things: (1) it overcomes the friction of the bearings and
is thus lost as heat, and (2) it is absorbed by the wheel which,
after the power is shut off, continues to turn and thus restores
the absorbed energy.

All circuits contain resistance, inductance and capacity, but
one or more may be so small as to be negligible. For the sake of
simplicity we shall first consider a circuit in which the capacity
may be disregarded.

617. Alternating E. M. F. in a Circuit Having Resistance and
Inductance. The instantaneous value of the current produced

ELECTRO-MECHANICS. 497

in a coil rotating at a uniform rate in a uniform field is (Par. 612)

I = Im .sin coZ

In this expression, co is the angular velocity of the moving coil,
whence (d is the angular distance through which the coil rotates
in t seconds. In one revolution the coil turns through the angle
27T. If the frequency be /, that is, if the coil makes / revolutions
per second, the angular distance through which it travels in one
second is 2?r/ and in t seconds is 2irft. We may therefore sub-
stitute 2irjt for ut in the above expression, whence

I = Im . sin 2irft

If the resistance of the circuit be R, the E. M. F. required to
drive the current / through this resistance is, from Ohm's law,
ER = IR, or, substituting the above value of I

This E. M. F., which is variously called the active, the efficient,
or the power E. M. F., reaches its maximum value ImR when
sin 2irft= 1, that is, at the 90 and the 270 phases, or at B and D
(Fig. 323), and may be represented by the sine curve AFCGE,
the corresponding current being in phase with it and being repre-
sented by the sine curve ASCTE.

Should there be in the circuit an inductance of L henry, there
will be produced a counter E. M. F. whose value is (Par. 434)

F T dl

EB = ~ L df

Since from above

I = Im . sin 2irft

whence

E B = -L.Im. 2irf. COS

This counter E. M. F. may therefore be represented by a sine
curve. It reaches its maximum value Im . 2wfL when cos 2irft = 1,
that is, at the and the 180 phases, or at A, C and E (Fig. 323).
It is therefore in quadrature with the E. M. F. represented by
the curve AFCGE. Also, since this E. M. F. opposes any change
in the existing current, it is positive as the latter falls and negative
as the latter rises. It is a maximum when the current passes

498

ELEMENTS OF ELECTRICITY.

through zero, since at this moment the rate of change of the cur-
rent is greatest, and it is zero when the current is a maximum,
either positive or negative. It may therefore be represented by
the sine curve HBJDK. .

Fig. 323.

In order to drive the current / through the circuit, the im-
pressed E. M. F. must be greater than that required by Ohm's
law of a direct current, for it must not only be sufficient to over-
come the ohmic resistance but also to counterbalance the back
E. M. F. due to self-induction.

The curve LBMD N represents the E. M. F. required to over-
come the counter E. M. F. Its ordinates are equal but of opposite
sign to the corresponding ordinates of the curve HBJDK. The
impressed E. M. F. is the resultant obtained by compounding
the E. M. F. represented by the curve AFCGE and that repre-
sented by the curve LBMD N. The curve LPFMQG N, obtained
by adding the corresponding ordinates of these two curves,
represents this resultant. It will be noted that it reaches its
maximum at P before the current reaches its maximum at S,
that is, it leads the current by a difference of phase correspond-
ing to RE. In alternating current circuits containing resistance
and inductance alone, the current always lags behind the im-
pressed E. M. F.

618. Graphic Construction of E. M. F. and Current Curves.

The power E. M. F., or E. M. F. required to overcome the ohmic
resistance, and the E. M. F. required to counterbalance the
E. M. F. of self-induction, may be compounded as just explained.
They may also be compounded according to the method described
in Par. 611. Thus, to find the instantaneous values of the various

ELECTRO-MECHANICS.

499

E. M. F.s and current at any phase, such as x, Fig. 324. Lay off
oa to represent the maximum value, I m R, of the power E. M. F.
and making with the horizontal axis an angle 6 corresponding to
the phase angle Ex. On this same line, since the current is in
phase with this E. M. F., lay off oc to represent the maximum
value Im of the current. Lay off ob at right angles to oa (the
two E. M. F.s being in quadrature) and of a length to represent
the maximum value, ImZirfL, of the E. M. F. to overcome the
E. M. F. of self-induction. The diagonal od of the parallelogram
constructed upon oa and ob is the vector corresponding to the
required impressed E. M. F. The projection of oa upon the

Fig. 324.

ordinate at x locates the point A of the curve of power E. M. F.,
that of ob locates the point B of the curve of E. M. F. to counter-
balance the induced E. M. F., and that of od locates the point D
of the curve of impressed E. M. F. Finally, the projection of oc
upon the ordinate at x locates the point C of the current curve.

619. Inductive Reactance. The counter E. M. F. due to self-
induction varies with the rate at which the lines of force are cut.
It therefore varies not only with the inductance, or number cut
when the current is varied one ampere, but also with the rapidity

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