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

varies inversely with the load and for each particular load there

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

be discarded.

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

such currents are graduated to read the virtual amperes and volts

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

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

varies inversely with the load and for each particular load there

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

be discarded.

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

such currents are graduated to read the virtual amperes and volts

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46