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Font size with which the current changes. In alternating currents this is
a function of the number of cycles per second, that is, of the
frequency. In Par. 617 it was shown that this E. M. F., which
is also called the reactive E. M. F., is in quadrature with the power
E. M. F. and that its maximum value is

E B = /

500

ELEMENTS OF ELECTRICITY.

The factor, 2irfL, is called the inductive reactance. It obviously
varies with the frequency / and with the inductance L. It is
measured in ohms, as might be inferred from the fact that when
multiplied by current the product is E. M. F. By expressing
it in its dimensional formula, it may be shown to be of the same
dimensions (a velocity) as resistance (Par. 547). It is sometimes
defined as that factor by which the maximum value of an alternat-
ing current in a circuit containing inductance is multiplied in
order to obtain the maximum value of the reactive E. M. F. The
reactance of a circuit for a given frequency is obtained in ohms
by multiplying the inductance in henrys by 2w times the frequency.

620. Impedance. Examination of Fig. 324 will show that od,
the maximum value of the impressed E. M. F., is the hypothe-
nuse of a right-angled triangle whose sides are oa = ImR, the
maximum value of the power E. M. F., and ad = ob = I m .2irfL f
the maximum value of the reactive E. M. F. It follows that

(Fig. 325 a) ^ _ ( r p\ 2 _L/7 o tr\*

& m = (ImJK) -p \JLm3tfFjlj)

whence ^ m

Im= V# 2 + (27r/L) 2

The resemblance of this expression to Ohm's law is obvious.
The denominator of the fraction in the second member is meas-
ured in ohms since it is composed of the resistance and the re-

Fig. 325.

actance, both of which are measured in ohms. It is called the
impedance since it represents the combined effect of the ohmic
resistance and the reactance in impeding the flow of the current.
It is sometimes defined as 'that factor by which the current in an
alternating circuit is multiplied in order to get the corresponding
impressed E. M. F. It will be noted that if /=0, the current
becomes direct and the expression reduces to Ohm's law.

ELECTRO-MECHANICS. 501

Inspection of the expression will show that the impedance is
itself the hypothenuse of a right-angled triangle whose sides
are the resistance and the reactance (Fig. 325 6).

It is also seen that the angle of lag, doc (Fig. 324), is given by;
the relation

da 2-7T/L reactance
tan <b = = ^r =

oa R resistance

and also by the relation

oa Ej^ _ power E. M. F.
od ~ Em ~ impressed E. M. F.

or, the cosine of the angle of

lag is equal to the ratio of the power E. M. F. to the impressed
E. M. F. This may be otherwise expressed by saying that if the
impressed E. M. F. be multiplied by the cosine of the angle of lag,
the result is the E. M. F. required to overcome the ohmic resist-
ance, i. e., the power E. M. F.

621. Choke Coils. The maximum value of an alternating
current in a circuit containing resistance and inductance is shown
in the preceding paragraph to be

T Em

If R, the resistance of the circuit, be small, its value may be
negligible as compared to that of 27T/L, the reactance, and there-
fore the current may depend more upon the reactance of the cir-
cuit than upon its resistance.

The reactance varies directly with the inductance and the
frequency. The inductance varies with the geometric arrange-
ment of the circuit and the proximity of magnetic bodies. The
frequency in currents for commercial purposes ranges from 25 to
130. If the current is to be employed for electric lighting, the
frequency should not fall below 50, otherwise there will be a per-
ceptible vibration or flicker in the lamps.

It is possible to place in an alternating current circuit a coil of
large wire, and hence a small resistance, with a soft iron core
whose position may be varied at will. As the core is inserted in
the coil, the reactance is increased and the current through the
coil is cut down; as the core is withdrawn, the reactance is de-
creased and a greater current passes through.

502 ELEMENTS OF ELECTRICITY.

Such an arrangement is called a reactance coil or a choke coil
and is frequently used for such purposes as regulating the bril-
liancy of the lights in a theatre, or for controlling the current
applied through a starting box to an alternating current motor.
It possesses the great advantage over rheostat control in that it
diminishes a current by setting up an opposing E. M. F. and hence
without loss of energy, while, in the case of the rheostat, power
is reduced by frittering away a portion in heat which waste must
be paid for by the consumer.

1 622. Explanation of Operation of Choke Coil. If a more physi-
cal conception of the operation of a choke coil be desired, it may
perhaps be obtained from the following. In Par. 436 it was shown
how induction retards the growth of a current. It could have been
shown in a similar manner that inductance also retards the decay
of a current, a dying current being represented by a logarithmic
curve also whose ordinates are complementary to those of the
curve representing the growing current. Fig. 207 shows that under
the conditions given, the -current in the circuit whose inductance
was one henry required, after the E. M. F. was impressed, about
one second to reach the value of six amperes. Suppose this to
have been an alternating current of a frequency of 50. In one- two-
hundredth of a second after the current started to rise, or when it
had reached a value of about .03 ampere, the E. M. F. would be
reversed and the current would be beaten back. It would die
down as slowly as it rose and would then start to rise in the op-
posite direction but in one-hundredth of a second after it had been
beaten down it would encounter a reversed E. M. F. and would
be checked and driven back, and so on, or, figuratively, it would
be a shuttle cock at the mercy of the alternating E. M. F.s. We
thus see that inductance makes the changes in the current slug-
gish and that increase of frequency causes the rising current to
be driven back more promptly.

623. Inductance and Resistance in Series. The fact that in
alternating current circuits containing inductance and resistance
alone the current always lags behind the impressed E. M. F. (Par.
617) affords an explanation of some of the peculiarities of alternat-
ing currents to which reference was made in Par. 607. As an
illustration, Fig. 326 represents a switch A by which an alternat-
ing E. M. F. may be thrown upon a circuit including in series a

ELECTRO-MECHANICS.

503

coil BC with an iron core, and therefore of considerable inductance,
and a rheostat whose resistance is assumed to be non-inductive,
or purely ohmic. Suppose the switch to be closed and that with
a voltmeter we read first the drop across the inductance BC, then
the drop across the resistance DE, and finally the total drop
between B and E. This total drop will be found to be less than
the sum of the partial drops. The explanation is that the volt-
meter takes no account of phase but indicates the virtual volts

RHEOSTAT

Fig. 326.

between its terminals as if the E. M. F. remained constantly at
this value. The current through the circuit at any one instant
is of course the same at every point, but while it is in phase with
the E. M. F. across DE, it lags 90 behind the E. M. F. across BC.
The maximum E. M. F. across BC occurs therefore one-quarter
of a period in advance of the maximum E. M. F. across DE. The
total drop is therefore not the sum of the partial drops, since the
maximum values of these do not occur simultaneously, but is
represented by the hypothenuse of a right triangle whose remain-
ing sides are the partial drops.

624. Inductance and Resistance in Parallel. Fig. 327 repre-
sents an inductive resistance BC and a non-inductive resistance

^W^

Fig. 327.

DE connected in parallel in an alternating current circuit. A, F
and G are ammeters arranged to read the currents in the main

504 ELEMENTS OF ELECTRICITY.

circuit and in the branches of the divided circuit respectively.
It will be found that the sum of the currents indicated by F and G
is greater than the current indicated by A. This happens because,
as explained in the preceding paragraph, the ammeters take no
heed of phase but indicate the virtual values of the separate cur-
rents as if these currents were constant, or as if their maxima
occurred simultaneously. The drop over the two branches is
always the same, being the difference of potential between M and
N, but the current through DE is in phase with this E. M. F.
while the current through EC lags 90. When the current through
DE is a maximum, the current through EC is still one-quarter
period removed from its maximum and is correctly the difference
between the current through A and that through G. The am-
meter F, however, indicates the virtual value of the current
through EC, as if the current were constantly of this strength.
The currents through F and G are in quadrature and therefore
the total current is represented by the hypothenuse of a right
triangle whose remaining sides are the currents as indicated by
F and G.

From the foregoing we see that the current through resistance
and reactance in series is the same at every point, but the voltage
across the combination is the vectorial resultant of the separate
drops as given by a voltmeter. On the other hand, the voltage
across resistance and reactance in parallel is the same over each,
but the total current is the vectorial resultant of the separate
currents as given by an ammeter.

625. Capacity. The subject of capacity was discussed in
Chapter 10 and it was there shown that the capacity of a con-
denser is not measured by the quantity of electricity which it
can contain but by the quantity which must be imparted to it
in order to raise its potential unity.

If two points between which there exists a difference of poten-
tial be connected by a conductor, there will be produced a current
which will vary directly with this difference of potential and which
will continue to flow so long as a difference exists. If, therefore,
a source of E. M. F. be connected to the terminal of a condenser,
a current will flow into the condenser so long as the potential of
the source is higher than that of the condenser. This current,
however, will not be of constant strength, for as the condenser
becomes charged its potential rises, hence the difference of poten-

ELECTRO-MECHANICS.

505

tial between it and the source grows smaller, and it is to this
difference of potential that the current is proportional. We may
regard the potential of the condenser as a counter E. M. F. which
opposes the charging E. M. F. and thereby diminishes the current.

Fig. 328.

As an analogy, Fig. 328 represents a large tank A of water of
unvarying head connected through a pipe closed by a stop cock
to the smaller tank B. The difference of the level of the water in
A and B determines whether there shall be a flow when the stop
cock is opened. If at first B be empty, the flow is urged by the
full head of water in A and the current is a maximum. When,
however, B has been partly filled, as shown in the figure, its head
is opposed to that of the water in A and the flow is determined
by the diminishing difference in level e; therefore, as B fills up,
the current dwindles to zero.

626. A Condenser in an Alternating Current Circuit. Suppose
a condenser to be connected in series in an alternating current
circuit, as shown in Fig. 329. So long as the potential of the brush

Fig. 329.

A is higher than that of the terminal B, a current will flow into
the condenser, and when the potential of A is a maximum, the
condenser will contain a charge Q, the maximum under the given
conditions. As the potential of A diminishes, Q flows out and
B is entirely discharged when the potential of A is zero. As the

506 ELEMENTS OF ELECTRICITY.

potential of A continues to sink to a negative maximum, a charge
Q flows into the coating D of the condenser and finally flows out
again as the potential of A returns to zero. It is thus seen that
although the circuit is broken at the condenser, a charge Q flows
through the circuit four times in each cycle. If the capacity of
the condenser and the frequency be sufficiently great, an incan-
descent lamp, connected as shown, may be made to glow by this
oscillating charge.

627. E. M. F. and Current Curves in Case of Capacity. The

foregoing may be shown graphically as follows. The sine curve
EMFGH, Fig. 330, represents the impressed E. M. F., or the
potential of the brush A. The current from A to the condenser
B is determined by the difference of potential between A and B
and is a maximum when the potential A is increasing most rapidly.
This maximum rate of increase occurs at M when the potential
of A is zero. It is here that the tangent to the curve is steepest
or the curve climbs up most rapidly. The current therefore,
represented by the curve JKLNO, reaches a maximum value
MKat this point. When the potential of A reaches its maximum
at L, the condenser is fully charged and the current no longer
flows into it. At this point the current curve is at zero. As

the potential of A falls, the condenser discharges, or the cur-
rent is now negative. As before, the negative current is a maxi-
mum when the potential of A is falling most rapidly, and this is
the case at G where the potential of A is again zero. Finally,
the current is again zero at where the potential of A is a negative
maximum. It is thus seen that in the case of capacity, the cur-
rent curve leads the E. M. F. curve and is in quadrature with it.

ELECTRO-MECHANICS. 507

628. Capacity Reactance. The instantaneous value of the
E. M. F. in an alternating current circuit is

E = Em . sin at

If this circuit contains capacity alone, the current leads the
E. M. F. by 90 and is given by

/ = Im COS co

The instantaneous value of the power developed is (Par. 494)

IE = ImEm. sin u>t . cos coZ
The work done in a time dt is

dw = ImEm . sin at . cos wt . dt

Since the condenser is charged in one-fourth of a period (Par. 626)
if this expression be integrated between the limits t = and

t = = - (Par. 612), it will give the work expended in

4 \ co / Zco

charging the condenser.
Performing the integration

IE 1

w = m m 5 (sin 2 coO -f a constant

co Z

Taking this between the above limits

But in Par. 97 it was shown that the work spent in charging a
condenser of capacity K is

Equating (I) and (II) and solving for Em

I

Substituting for co its value 27r/ (Par. 617)

Em = Im *

508 ELEMENTS OF ELECTRICITY.

Whence also, since E m = E v ^/2 and Im = IvV2 (Par. 612)

1
Ev = Iv '2rfK

E v and Iv being the virtual
E. M. F. and current respectively.

The factor 5-7^ * s called the capacity reactance of the circuit.

It is quite analogous to the inductive reactance discussed in Par.
619. It is measured in ohms and its dimensional formula (Par.
547) shows it to be of the same dimensions, a velocity, as resist-
ance. It is that factor by which the maximum value of an alter-
nating current in a circuit containing capacity must be multiplied
in order to obtain the value of the reactive E. M. F. due to
capacity.

629. Alternating E. M. F. in Circuit Containing Resistance and
Capacity. If the circuit contains both resistance and capacity,
in order to drive a current Im through it, the impressed E. M. F.
must be sufficient to overcome both the ohmic resistance and the
capacity reactance. The E. M. F. to overcome the ohmic resist-
ance is ImR, that to overcome the capacity reactance is Im . /gr >

^TT/A

and these being in quadrature (Par. 627)

whence

an expression analogous to

the one deduced in Par. 620, the denominator being the impedance.
It will be noted that inductive reactance varies directly with
the frequency /, while capacity reactance varies inversely with
this factor. Changes in the frequency therefore produce diametri-
cally opposite results in the reactances. As the frequency increases,
the current through an inductive circuit decreases while that
through a capacity increases. On the other hand, as the frequency
decreases, the current through an inductive circuit increases and
that through a capacity decreases.

ELECTRO-MECHANICS. 509

630. Alternating E. M. F. in Circuit Containing Resistance,
Inductance and Capacity. In the most general case, in order to
drive a current Im through an alternating current circuit contain-
ing resistance, inductance and capacity, the impressed E. M. F.
must be sufficient to overcome the ohmic resistance and the com-
bined reactance of the inductance and capacity. It has been shown
(Par. 617) that in the case of inductance the current lags 90; on
the other hand, in the case of capacity (Par. 627) the current
leads by 90. The E. M. F.s to overcome these separate react-
ances therefore differ in phase by 180 and are combined by simple
subtraction, hence the resultant reactance is

and the most general ex-
pression for the current is

631. Electric Resonance. Fig. 331 represents an alternating
current circuit in which there are connected in series a coil of
resistance R and inductance L and a condenser of capacity K.

Fig. 331.

From the preceding paragraph, the current through the combina-
tion is

E

If in this expression we assign a regular series of values to /, the
frequency, the remaining factors being kept constant, and plot
the corresponding values of the current, it will be seen that at a
certain value of /, which may be called the critical frequency, the

510

ELEMENTS OF ELECTRICITY.

current jumps abruptly to a maximum. Inspection- will show
that this maximum is reached when 2w/L = , in which case

the above expression reduces to Ohm's law. In this case also

i

/ = - Tf^' an( * the periodic time= !//= 2wVLK seconds. The

above is shown in Fig. 332, the curve representing the values for

60
50
40

n
ui
a: 30

*0

30 40 .50 60
FREQUENCY

70

80 90 JOO

Fig. 332.

different frequencies of the current in a circuit in which E= 110
w volts, jR = 2 ohms, L = 0.5 henry and K= 25 microfarads. At a
frequency of about 45, the current mounts suddenly to 55 amperes,
while at a frequency of 5 more or 5 less it is but little greater than
three amperes.

If a heavy pendulum be given a series of slight impulses, no
especial effect will be produced unless these impulses be timed
at the natural period of vibration of the pendulum, in which case
their effect is cumulative and it may be made to swing through a
wide arc.

Again, if various tuning forks be caused to vibrate near the open
end of a closed organ pipe, no effect will be produced until a fork
is used whose period of vibration corresponds to the natural period
of vibration of the column of air within the pipe, and when this
happens the column of air will vibrate in unison with the fork and
the total volume of sound emitted will be greatly increased. This
phenomenon is called resonance.

In the case of the alternating current circuit under considera-
tion, the E. M. F. is not applied steadily but in a series of impulses

ELECTRO-MECHANICS. 51 1

following each other at regular intervals. These impulses produce
no very marked effect until the critical frequency is reached, at
which time the current rises abruptly to its maximum value.
From analogy, the circuit is now said to possess electric resonance.
Resonance exists in an alternating current circuit whenever

27r/L= > r when the inductive reactance is exactly counter-

balanced by the capacity reactance.

632. Resonance with Inductance and Capacity in Series.

When resonance exists in a circuit containing inductance and
capacity in series (Fig. 331), the current follows Ohm's law and the
impressed E. M. F. is simply the IR drop. The fact, however,
that the inductive and the capacity reactances neutralize each
other, or that their sum is zero, does not mean that they are
separately zero. On the contrary, the difference of potential across
the terminals of the inductance is J.27T/L (Par. 619) and that

across the terminals of the condenser is / . fv (Par. 628) and

these may very greatly exceed the impressed E. M. F. For ex-
ample, in the numerical example given in the preceding paragraph,
while the impressed E. M. F. is 110 volts, the drops across the
terminals of the inductance and of the condenser are each 7778
volts.

633. Resonance with Inductance and Capacity in Parallel. A

particular case of resonance is where the inductance and capacity
are in parallel as shown in Fig. 333. The current on arriving at A

Fig. 333.

divides, but in the branch L it is retarded while in the branch K
it is advanced an equal amount. The result is that the loop
AKBL acts as a short circuit, the current surging around it in
one direction during one-half of a period and in the other direction
during the remaining half. Although the current in the main cir-
cuit may be small, that in this loop may be very large. This can
be shown graphically, for the current in the main circuit is the
resultant of the currents in L and K (Par. 624), that is, it is the

512 ELEMENTS OF ELECTRICITY.

other an angle of very nearly 180.

634. Power in an Alternating Current Circuit. In an alternat-
ing current circuit the instantaneous value of the power is the
product of the corresponding simultaneous instantaneous values
of the E. M. F. and the current (Par. 494). Two cases may arise:
(a) the E. M. F. and current may be in phase, or, (b) they may
differ in phase.

If the E. M. F. and current are in phase, at any one instant
they are either both positive or both negative and therefore their
product, the power, is always positive. This is shown in Fig. 334

Fig. 334.

in which the broken curve representing the instantaneous values
of the power lies always above the horizontal axis. The power
curve is seen to be periodic and of twice the frequency of the
E. M. F. and current curves. Since its ordinates represent rate
of doing work and its abscissae represent time (Par. 608), the
area included between the curve and the horizontal axis represents
work performed by the current. The work is positive, for whether
the current flow in or out it performs work in overcoming the
resistance of the circuit.

If the E. M. F. and current differ in phase, their simultaneous
values must at times differ in sign and at these times their product,
the power, must be negative. The power curve, therefore, as

ELECTRO-MECHANICS.

513

shown in Fig. 335, extends below the horizontal axis. The areas
of the loops below this axis represent negative work, or energy
imparted to the field about the circuit and restored by this field
to the system (Par. 616).

Fig. 335.

635. Power Factor. In Par. 613 it was shown that the work
done by an alternating current in one cycle is I 2 m Rt, I m being
the maximum value of the current, R the resistance of the cir-
cuit and t the time of one cycle. By dividing this by t we get the
average rate of doing work, in other words, the average power,
hence

which may be written

p =i

watts

W2/

In the same paragraph it was shown that I v , the virtual current,
is equal to I m /V2, hence

P_ j2 r> T 7 r>
1 y/l V J- V-ft

But I V R is that component of the virtual E. M. F. which is in
phase with the current, hence (Par. 620)

IvR Ev . COS

hence

P = IvEv . cos watts

or the average power in
an alternating current circuit is equal to the product of the virtual

514 ELEMENTS OF ELECTRICITY.

current, the virtual E. M. F., and the cosine of the angle of lag (or

The power in an alternating current circuit must be read by a
wattmeter, for, except when the E. M. F. and current are in phase,
it can not be determined by taking simultaneous measurements
with an ammeter and a voltmeter and multiplying these readings
together. The product of these readings, I V E V , is called the
apparent power, and cos </> is called the power factor, since, as shown
above, it is that factor by which the apparent power must be
multiplied in order to obtain the true power.

If <f> becomes 90, that is, if there is no resistance in the circuit
so that the E. M. F. and current are in quadrature, cos = and
the power as given above reduces to zero. In this case, the area
of the negative loops of the power curve (Fig. 335) equals that of
the positive loops.

Online LibraryWirt RobinsonThe elements of electricity → online text (page 39 of 46)