THE THOUGHT IS IN THE QUESTION THE INFORMATION IS IN THE ANSWER
[Illustration:
HAWKINS
ELECTRICAL GUIDE
NUMBER
FIVE
QUESTIONS
ANSWERS
&
ILLUSTRATIONS]
A PROGRESSIVE COURSE OF STUDY
FOR ENGINEERS, ELECTRICIANS, STUDENTS
AND THOSE DESIRING TO ACQUIRE A
WORKING KNOWLEDGE OF
ELECTRICITY AND ITS APPLICATIONS
A PRACTICAL TREATISE
by
HAWKINS AND STAFF
[Illustration: THEO. AUDEL & CO. 72 FIFTH AVE. NEW YORK.]
COPYRIGHTED, 1914,
BY
THEO. AUDEL & CO.,
NEW YORK.
Printed in the United States.
TABLE OF CONTENTS
GUIDE NO. 5.
~ALTERNATING CURRENTS~ 997 to 1,066
The word "_alternating_"--advantages of alternating
current--direct current apparatus; alternating current
apparatus--disadvantages of alternating current--~alternating
current principles~--the sine--application and construction of
the sine curve--~illustrated definitions~: cycle, alternation,
amplitude, period, periodicity, frequency--commercial
frequencies--advantages of low frequency--~phase~--phase
difference--phase displacement--synchronism--"in phase"--curves
illustrating "in phase" and "out of phase"--~illustrated
definitions~: in phase; in quadrature, current leading;
in quadrature, current lagging; in opposition--~maximum~
volts and amperes--~average~ volts and amperes--elementary
alternator developing one average volt--~virtual~ volts and
amperes--~effective~ volts and amperes--relation between
shape of wave and ~form factor~--wave form--oscillograph
wave form records--what determines wave form--effect of one
coil per phase per pole--~single phase current~; hydraulic
analogy--~two phase current~; hydraulic analogy--two
phase current distribution--~three phase current~;
hydraulic analogy; distribution--~inductance~--~the
henry~--inductive and non-inductive coils--hydraulic analogy
of inductance--inductance coil calculations--~ohmic value of
inductance~--~capacity~: hydraulic analogy--the farad--specific
inductive capacity--condenser connections--~ohmic value of
capacity~--lag and lead--mechanical analogy of lag--lag
measurement--steam engine analogy of current flow at zero
pressure--~reactance~--examples--choking coil--impedance
curve--~resonance~--critical frequency--skin effect.
~ALTERNATING CURRENT DIAGRAMS~ 1,067 to 1,100
~Definitions~: impressed pressure, active pressure,
self-induction pressure, reverse pressure of
self-induction--rate of change in current strength--properties
of right angle triangles--equations of the right
triangle--representation of forces by wires--parallelogram
of forces; ~the resultant~--~circuits containing resistance
and inductance~--graphical method of obtaining the
impressed pressure--~equations for ohmic drop and reactance
drop~--examples--diagram for impedance, angle of lag,
etc.--~circuits containing resistance and capacity~--capacity
in series, and in parallel--amount of lead--action of
condenser--the condenser pressure--capacity pressure--equation
for impedance--- examples and diagrams--~circuits
containing resistance, inductance, and capacity~--impedance
equation--examples and diagrams--equation for impressed
pressure--examples and diagrams.
~THE POWER FACTOR~ 1,101 to 1,124
Definition of power factor--true watts--- apparent watts--ferry
boat analogy of power factor--limits of power factor--effect
of lag or lead--~how to obtain the power curve~--nature of
the power curve--synchronism of current and pressure; ~power
factor unity~--case of synchronism of current and pressure
with power factor less than ~unity~--steam engine analogy of
power factor--"wattless current;" ~power factor zero~--examples
of phase difference nearly 90 degrees--mechanical analogy
of wattless current--~why the power factor is equal
to cos φ~--graphical method of obtaining the active
component--examples and diagrams--effect of capacity--diagrams
illustrating why the power factor is ~unity~ when there is
no resultant reactance in the circuit--~usual value of power
factor~--power factor test--~how alternators are rated;
kva.~--curves illustrating power factor--~how to keep the
power factor high~--why power factor is important in station
operation--wattmeter method of three phase power measurement.
~ALTERNATORS~ 1,125 to 1,186
Uses of alternators--classes of alternator--~single phase
alternators~; essential features; width of armature
coils--elementary single phase alternator--~polyphase
alternators~--uses for two and three phase current--elementary
three phase alternator--starting difficulty with single phase
motors--six and twelve phase windings--~belt or chain driven
alternators~--sub-base and ratchet device for tightening
the belt--~horse power transmitted by belts~--best speeds
for belts--advantages of chain drive; objections--~direct
connected alternator~--"direct connected" and "direct
coupled" units--~revolving armature alternators~; their
uses--~revolving field alternators~--marine view showing
that motion is purely a relative matter--essential parts
of revolving field alternator--the terms "stator" and
"rotor"--inductor alternators: classes, use, defects--hunting
or surging in alternators--~amortisseur~ windings--~monocyclic
alternators~--diagram of connections--teaser coil--~armature
reaction~--distortion of field--strengthening and
weakening effects--superpositions of fields--three phase
reactions--~magnetic leakage~--~field excitation of
alternators~--self-excited alternator--direct connected
exciter--gear driven exciters--~slow speed alternators~--~fly
wheel alternators~--~high speed alternators~--~water wheel
alternators~--construction of rotor--~turbine driven
alternators~--construction--step bearing--~alternators of
exceptional character~--asynchronous alternators--image current
alternators--extra high frequency alternators--self-exciting
image current alternators.
CONSTRUCTION OF ALTERNATORS~ 1,187 to 1,266
~Essential parts of an alternator~--field magnets--~methods
of excitation~: self-excited, separately excited,
compositely excited--magneto--construction of stationary
magnets--revolving field--slip rings--spider for large
alternator--provision for shifting armature to give access
to field--~armatures~--core construction--advantages of
slotted core armatures--~armature windings~--~classification~:
revolving and stationary windings--half coil and whole coil
windings--~concentrated~ or uni-coil winding; features;
waveform--~distributed~ or multi-coil windings: breadth
of coil, partial and fully distributed coils--the Kapp
coefficient--general equation for voltage--wire, strap, and
bar windings--condition, governing type of inductor--coil
covering--single and double layer multi-wire inductors and
methods of placing them on the core--~insulation~--core
stamping--single and multi-slot windings--- arrangement in
slot of two layer bar winding--table of relative effectiveness
of windings--~single phase windings~--advantage of half coil
winding--~two phase windings~--shape of coil ends--three
phase windings--shape of coil ends--kind of coil used with
three phase windings--~grouping of phases~--two phase
~star connection~--two phase ~mesh connection~--three
phase star connection--winding diagrams with star and
Δ connections--three phase Δ connection--three phase
winding with "short" coils--three phase lap winding star
connection--three phase wave winding star connection--~output
of star and delta connected alternators~--gramme ring
armatures showing three phase star and mesh connections
with direction of currents in the coils--features of star
connection--characteristics of delta connection--proper
ends to connect to star point--determination of path
and value of currents in delta connection--points to be
noted with ~Y~ connection--diagram of ~Y~ connection with
return wire--chain or basket winding--skew winding--fed-in
winding--imbricated winding--spiral winding--mummified
winding--shuttle winding--creeping winding--~turbine alternator
winding~: how the high voltage is obtained with so few poles;
table of frequency and revolutions--~turbine alternator
construction~--form of armature generally used--two pole radial
slot field--parallel slot field--difficulty experienced with
revolving armatures--how the field design is modified to reduce
centrifugal force--examples of revolving fields.
CHAPTER XLVI
ALTERNATING CURRENTS
The word "alternating" is used with a large number of electrical and
magnetic quantities to denote that their magnitudes vary continuously,
passing repeatedly through a definite cycle of values in a definite
interval of time.
As applied to the flow of electricity, an alternating current may be
defined as: _A current which reverses its direction in a periodic
manner, rising from zero to maximum strength, returning to zero, and
then going through similar variations in strength in the opposite
direction_; these changes comprise the cycle which is repeated with
great rapidity.
The properties of alternating currents are more complex than
those of continuous currents, and their behavior more difficult
to predict. This arises from the fact that the magnetic effects
are of far more importance than those of steady currents. With
the latter the magnetic effect is constant, and has no reactive
influence on the current when the latter is once established.
The lines of force, however, produced by alternating currents
are changing as rapidly as the current itself, and they thus
induce electric pressures in neighboring circuits, and even in
adjacent parts of the same circuit. This inductive influence in
alternating currents renders their action very different from
that of continuous current.
~Ques. What are the advantages of alternating current over direct
current?~
Ans. The reduced cost of transmission by use of high voltages and
transformers, greater simplicity of generators and motors, facility of
transforming from one voltage to another (either higher or lower) for
different purposes.
[Illustration: FIGS. 1,206 to 1,212.--Apparatus which operates
successfully on a direct current circuit. The direct current will
operate incandescent lamps, arc lamps, electric heating apparatus,
electro-plating and typing bath, direct current motors; charge storage
batteries, produce electro-chemical action. It will flow through a
straight wire or just as freely through the same wire when wound over
an iron bar.]
[Illustration: FIGS. 1,213 to 1,217.--Apparatus which operates
successfully on an alternating circuit. The alternating current will
operate incandescent lamps, arc lamps, electric heating apparatus,
alternating current motors. It will flow through a straight wire with
slightly increased retarding effect, but if the wire be wound on an
iron bar its strength is greatly reduced.]
The size of wire needed to transmit a given amount of
electrical energy (watts) with a given percentage of drop,
being _inversely proportional to the square of the voltage
employed_, the great saving in copper by the use of alternating
current at high pressure must be apparent. This advantage can
be realized either by a saving in the weight of wire required,
or by transmitting the current to a greater distance with the
same weight of copper.
In alternating current electric lighting, the primary voltage
is usually at least 1,000 and often 2,000 to 10,000 volts.
~Ques. Why is alternating current used instead of direct current on
constant pressure lighting circuits?~
Ans. It is due to the greater ease with which the current can be
transformed from higher to lower pressures.
~Ques. How is this accomplished?~
Ans. By means of simple transformers, consisting merely of two or more
coils of wire wound upon an iron core.
Since there are no moving parts, the attention required and the
likelihood of the apparatus getting out of order are small.
The apparatus necessary for direct current consists of a motor
dynamo set which is considerably more costly than a transformer
and not so efficient.
~Ques. What are some of the disadvantages of alternating current?~
Ans. The high pressure at which it is used renders it dangerous, and
requires more efficient insulation; alternating current cannot be used
for such purposes as electro-plating, charging storage batteries, etc.
[Illustration: FIG. 1,218.--Application and construction of the sine
curve. The sine curve is a wavelike curve used to represent the changes
in strength and direction of an alternating current. At the left of
the figure is shown an elementary alternator, consisting of a loop of
wire ABCD, whose ends are attached to the ring F, and shaft G, being
arranged to revolve in a uniform magnetic field, as indicated by the
vertical arrows representing magnetic lines at equidistances. The
alternating current induced in the loop is carried to the external
circuit through the brushes M and S. The loop, as shown, is in its
horizontal position at right angles to the magnetic field. The dotted
circle indicates the circular path described by AB or CD during the
revolution of the loop. Now, as the loop rotates, the induced electric
pressure will vary in such a manner that _its intensity at any point
of the rotation is proportional to the sine of the angle corresponding
to that point_. Hence, on the horizontal line which passes through the
center of the dotted circle, take any length as 08, and divide into
any number of equal parts representing fractions of a revolution, as
0°, 90°, 180°, etc. Erect perpendiculars at these points, and from
the corresponding points on the dotted circle project lines (parallel
to 08) to the perpendiculars; these intersections give points, on the
sine curve, for instance, through 2 at the 90° point of the revolution
of the loop, and projecting over to the corresponding perpendicular
gives 2'2, whose length is proportional to the electric pressure
at that point. In like manner other points are obtained, and the
curved line through them will represent the variation in the electric
pressure for all points of the revolution. At 90° the pressure is at a
maximum, hence by using a pressure scale such that the length of the
perpendicular 2'2 for 90° will measure the maximum pressure, the length
of the perpendicular at any other point will represent the actual
pressure at that point. The curve lies above the horizontal axis during
the first half of the revolution and below it during the second half,
which indicates that the current flows in one direction for a half
revolution, and in the opposite direction during the remainder of the
revolution.]
~Alternating Current Principles.~--In the operation of a direct current
generator or _dynamo_, as explained in Chapter XIII, alternating
currents are generated in the armature winding and are changed into
direct current by the action of the commutator. It was therefore
necessary in that chapter, in presenting the basic principles of the
dynamo, to explain the generation of alternating currents at length,
and the graphic method of representing the alternating current cycle
by the sine curve. In order to avoid unnecessary repetition, the
reader should carefully review the above mentioned chapter before
continuing further. The diagram fig. 168, showing the construction and
application of the sine curve to the alternating current, is however
for convenience here shown enlarged (fig. 1,218). In the diagram the
various alternating current terms are graphically defined.
[Illustration: FIG. 1,219--Diagram illustrating the sine of an angle.
In order to understand the sine curve, it is necessary to know the
meaning of the sine of an angle. This is defined as the _ratio of the
perpendicular let fall from any point in one side of the angle to the
other side divided by the hypotenuse of the triangle thus formed_.
For instance, in the diagram, let AD and AE be the two sides of the
angle φ, and DE a perpendicular let fall from any point D of the side
AD to the other side AE. Then, the sine of the angle (written sin
φ) = DE ÷ AD. It is evident that if the perpendicular be let fall at a
unit's distance from the apex A, as at B,
BC BC
Sin φ = -- = -- = BC
AB 1
This line BC is called the natural sine of the angle, and its values
for different angles are given in the table on page 451.]
[Illustration: FIG. 1,220.--Diagram illustrating the equation of
the sine curve: _y_ = sin φ. _y_ is any ordinate, and φ, the angle
which the coil makes with the horizontal line, corresponding to the
particular value of _y_ taken.]
The alternating current, as has been explained, _rises from zero to
a maximum, falls to zero, reverses its direction, attains a maximum
in the new direction, and again returns to zero_; this comprises the
_cycle_.
This series of changes can best be represented by a curve, whose
abscissæ represent time, or degrees of armature rotation, and whose
ordinates, either current or pressure. The curve usually chosen for
this purpose is the sine curve, as shown in fig. 1,218, because it
closely agrees with that given by most alternators.
The equation of the sine curve is
_y_ = sin φ
in which _y_ is any ordinate, and φ, the angle of the
corresponding position of the coil in which the current is
being generated as illustrated in fig. 1,220.
~Ques. What is an alternation?~
Ans. The changes which the current undergoes in rising from zero to
maximum pressure and returning back to zero; that is, a single positive
or negative "wave" or half period, as shown in fig. 1,221.
[Illustration: FIG. 1,221.--Diagram showing one _alternation_ of the
current in which the latter varies from zero to maximum and back to
zero while the generating loop ABCD makes one half revolution.]
~Ques. What is the amplitude of the current?~
Ans. The greatest value of the current strength attained during the
cycle.
The foregoing definitions are also illustrated in fig. 1,218.
[Illustration: FIG. 1,222.--Diagram illustrating _amplitude_ of
the current. The current reaches its amplitude or maximum value in
one quarter period from its point of zero value, as, for instance,
while the generating loop moves from position ABCD to A'B'C'D'. At
three-quarter revolution, the current reaches its maximum value in the
opposite direction.]
~Ques. Define the term "period."~
Ans. This is the time of one cycle of the alternating current.
~Ques. What is periodicity?~
Ans. A term sometimes used for _frequency_.
~Frequency.~--If a slowly varying alternating current be passed through
an incandescent lamp, the filament will be seen to vary in brightness,
following the change of current strength. If, however, the alternations
take place more rapidly than about 50 to 60 per second, the eye cannot
follow the variations and the lamp appears to burn steadily. Hence it
is important to consider the rate at which the alternations take place,
or as it is called, the _frequency_, which is defined as: _the number
of cycles per second_.
[Illustration: FIG. 1,223.--Diagram of alternator and engine,
illustrating _frequency_. The frequency or _cycles per second_
is equal to _the revolution of armature per second multiplied by
one-half the number of poles per phase_. In the figure the armature
makes 6 revolutions to one of the engine; one-half the number of
poles = 8 ÷ 2 = 4, hence frequency = (150 × 4 × 6) ÷ 60 = 60. The
expression in the parenthesis gives the cycles per minute, and dividing
by 60, the cycles per _second_.]
In a two pole machine, the frequency is the same as the number of
revolutions _per second_, but in multipolar machines, it is greater in
proportion to the number of _pairs_ of poles per phase.
Thus, in an 8 pole machine, there will be four cycles per
revolution. If the speed be 900 revolutions per minute, the
frequency is
8 900
- × --- = 60 ~
2 60
The symbol ~ is read "cycles per second."
~Ques. What frequencies are used in commercial machines?~
Ans. The two standard frequencies are 25 and 60 cycles.
[Illustration: FIG. 1,224--Diagram answering the question: Why are
alternators always built multipolar? They are made multipolar because
it is desirable that the frequency be high. It is evident from the
figure that to obtain high frequency would require too many revolutions
of the armature of a bipolar machine for mechanical safety--especially
in large alternators. Moreover a double reduction gear in most cases
would be necessary, adding complication to the drive. Comparing the
above illustration with fig. 1,223, shows plainly the reason for
multipolar construction.]
~Ques. For what service are these frequencies adapted?~
Ans. The 25 cycle frequency is used for conversion to direct current,
for alternating current railways, and for machines of large size; the
60 cycle frequency is used for general distribution for lighting and
power.
The frequency of 40 cycles, which once was introduced as a
compromise between 25 and 60 has been found not desirable, as
it is somewhat low for general distribution, and higher than
desirable for conversion to direct current.
[Illustration: FIG. 1,225.--Diagram illustrating "phase." In wave,
vibratory, and simple harmonic motion, phase may be defined as: _the
portion of one complete vibration, measured either in angle or in time,
that any moving point has executed_.]
~Ques. What are the advantages of low frequency?~
Ans. The number of revolutions of the _rotor_ is correspondingly low;
arc lamps can be more readily operated; better pressure regulation;
small motors such as fan motors can be operated more easily from the
circuit.
~Phase.~--As applied to an alternating current, phase denotes _the
angle turned through by the generating element reckoned from a given
instant_. Phase is usually measured in degrees from the initial
position of zero generation.
If in the diagram fig. 1,225, the elementary armature or loop
be the generating element, and the curve at the right be the
sine curve representing the current, then the phase of any
point _p_ will be the angle φ or angle moved through from the
horizontal line, the starting point.
~Ques. What is phase difference?~
Ans. The angle between the phases of two or more alternating current
quantities as measured in degrees.
~Ques. What is phase displacement?~
Ans. A change of phase of an alternating pressure or current.
[Illustration: FIGS. 1,226 and 1,227.--Diagram and sine curves
illustrating _synchronism_. If two alternators, with coils in parallel
planes, be made to rotate at the same speed by connecting them with
chain drive or equivalent means, they will then be "in synchronism"
that is, the alternating pressure or current in one will vary in step
with that in the other. In other words, the cycles of one take place
with the same frequency and at the same time as the cycles of the other
as indicated by the curves, fig. 1,226. It should be noted that the
maximum values are not necessarily the same but the maximum and zero
values must occur at the same time in both machines, and the maximum
value must be of the same sign. If the waves be distorted the maximum
values may not occur simultaneously. See fig. 1,348.]
~Synchronism.~--This term may be defined as: _the simultaneous
occurrence of any two events_. Thus two alternating currents or
pressures are said to be "in synchronism" _when they have the same
frequency and are in phase_.
~Ques. What does the expression "in phase" mean?~
Ans. Two alternating quantities are said to be in phase, when there is
no phase difference between; that is when the angle of phase difference
equals zero.
Thus the current is said to be in phase with the pressure when
it neither lags nor leads, as in fig. 1,228.
A rotating cylinder, or the movement of an index or trailing
arm is brought into synchronism with another rotating cylinder
or another index or trailing arm, not only when the two are
moving with exactly the same speed, but when in addition they
_are simultaneously moving over similar portions of their
respective paths_.
[Illustration: FIG. 1,228--Pressure and current curves illustrating the
term "in phase." The current is said to be _in phase_ with the pressure
when it _neither_ lags nor leads.]
When there is phase difference, as between current and pressure, they
are said to be "out of phase" the phase difference being measured as in
fig. 1,229 by the angle φ.
[Illustration: FIG. 1,229--Pressure and current curves illustrating
the term "out of phase." The current is said to be _out of phase_ with
the pressure when it _either_ lags or leads, that is when the current
is not in synchronism with the pressure. In practice the current and
pressure are nearly always out of phase.]
When the phase difference is 90° as in fig. 1,231 or 1,232,
the two alternating quantities are said to be _in quadrature_;
when it is 180°, as in fig. 1,233, they are said to be _in
opposition_.
When they are in quadrature, one is at a maximum when the other
is at zero; when they are in opposition, one reaches a positive
maximum when the other reaches a negative minimum, being at
each instant opposite in sign.
~Ques. What is a departure from synchronism called?~
Ans. Loss of synchronism.
[Illustration: FIGS. 1,230 to 1,233.--Curves showing some phase
relations between current and pressure. Fig. 1,230, _synchronism of
current and pressure_, expressed by the term "in phase," meaning
simultaneous zero values, and simultaneous maximum values of the same
sign; fig. 1,231, _in quadrature_, current _leading_ 90°; fig. 1,232
_in quadrature_, current lagging 90°; fig. 1,233, _in opposition_,
meaning that the phase different between current and pressure is 180°.]
~Maximum Volts and Amperes.~--In the operation of an alternator, the
pressure and strength of the current are continually rising, falling
and reversing. During each cycle there are two points at which the
pressure or current reaches its greatest value, being known as the
_maximum value_. This maximum value is not used to any great extent,
but it shows the maximum to which the pressure rises, and hence, the
greatest strain to which the insulation of the alternator is subjected.
[Illustration: FIG. 1,234.--Elementary alternator developing one
average volt. If the loop make one revolution per second, and the
maximum number of lines of force embraced by the loop in the position
shown (the zero position) be denoted by N, then each limb will cut 2N
lines per second, because it cuts every line during the right sweep
and again during the left sweep. Hence each limb develops an average
pressure of 2N units (C.G.S. units), and as both limbs are connected
in series, the total pressure is 4N units _per revolution_. Now, if
the loop make _f_ revolutions _per second_ instead of only one, then
_f_ times as many lines will be cut _per second_, and the average
pressure will be 4N _f_ units. Since the C.G.S. unit of pressure is so
extremely small, a much greater practical unit called the _volt_ is
used, which is equal to 100,000,000, or 10⁸ C.G.S. units is employed.
Hence average voltage = 4N_f_ ÷ 10⁸. The value of N in actual machines
is very high, being several million lines of force. The illustration
shows one set of conditions necessary to generate one average volt. The
maximum pressure developed is 1 ÷ .637 = 1.57 volts; virtual pressure =
1.57 × .707 = 1.11 volts.]
~Average Volts and Amperes.~--Since the sine curve is used to represent
the alternating current, the _average value_ may be defined as: _the
average of all the ordinates of the curve for one-half of a cycle_.
~Ques. Of what use is the average value?~
Ans. It is used in some calculations but, like the maximum value, not
very often. The relation between the average and virtual value is of
importance as it gives the form factor.
~Virtual Volts and Amperes.~--The virtual[1] value of an alternating
pressure or current _is equivalent to that of a direct pressure or
current which would produce the same effect_; those effects of the
pressure and current are taken which are not affected by rapid changes
in direction and strength,--in the case of pressure, the reading of an
electrostatic voltmeter, and in the case of current, the heating effect.
[1] NOTE.--"I adhere to the term _virtual_, as it was in use before the
term _efficace_ which was recommended in 1889 by the Paris Congress
to denote the _square root of mean square_ value. The corresponding
English adjective is _efficacious_; but some engineers mistranslate
it with the word _effective_. I adhere to the term _virtual_ mainly
because the adjective _effective_ is required in its usual meaning
in kinematics to represent the resolved part of a force which acts
obliquely to the line of motion, the effective force being the whole
force multiplied by the cosine of the angle at which it acts with
respect to the direction of motion. Some authors use the expression
'R.M.S. value' (meaning 'root mean square') to denote the virtual or
quadratic mean value."--_S. P. Thompson._
[Illustration: FIG. 1,235.--Maximum and average values of the sine
curve. The average value of the sine curve is represented by an
ordinate MS of such length that when multiplied by the base line FG,
will give a rectangle MFSG whose area is equal to that included between
the curve and base line FDGS.]
[Illustration: FIG. 1,236.--Diagram illustrating "virtual" volts
and amperes. The word _virtual_ is defined as: _Being in essence or
effect, not in fact; not actual, but equivalent, so far as effect
is concerned_. As applied to the alternating current, it denotes an
imaginary direct current of such value as will produce an effect
equivalent to that of the alternating current. Thus, a _virtual
pressure_ of 1,000 volts is one that would produce the same deflection
in an electrostatic voltmeter as a direct pressure of 1,000 volts: a
_virtual current_ of 10 amperes is that current which would produce the
same heating effect as a direct current of 10 amperes. Both pressure
and current vary continually above and below the virtual values in
alternating current circuits. Distinction should be made between the
virtual and "effective" values of an alternating current. See fig.
1,237. The word _effective_ is commonly used erroneously for _virtual_.
See note page 1,011.]
The attraction (or repulsion) in electrostatic voltmeters is
proportional to the square of the volts.
The readings which these instruments give, if first calibrated by using
steady currents, are not true means, but are _the square roots of the
means of the squares_.
Now the mean of the squares of the sine (taken over either one quadrant
or a whole circle) is ½; hence the _square root of mean square_ value
of the sine functions is obtained by multiplying their maximum value by
1 ÷ √2̅, or by 0.707.
The arithmetical mean of the values of the sine, however, is 0.637.
Hence an alternating current, if it obey the sine law, will produce
a heating effect greater than that of a steady current of the same
average strength, by the ratio of 0.707 to 0.637; that is, about 1.11
times greater.
If a Cardew voltmeter be placed on an alternating circuit in which the
volts are oscillating between maxima of +100 and -100 volts, it will
read 70.7 volts, though the arithmetical mean is really only 63.7; and
70.7 steady volts would be required to produce an equal reading.
[Illustration: FIG. 1,237.--Diagram illustrating _virtual_ and
_effective_ pressure. If the coil be short circuited by the switch and
a constant virtual pressure be impressed on the circuit, the whole of
the impressed pressure will be effective in causing current to flow
around the circuit. In this case the virtual and effective pressures
will be equal. If the coil be switched into circuit, the reverse
pressure due to self induction will oppose the virtual pressure; hence,
the effective pressure (which is the difference between the virtual and
reverse pressures) will be reduced, the virtual or impressed pressure
remaining constant all the time. A virtual current _is that indicated
by an ammeter regardless of the phase relation between current and
pressure_. An effective current _is that indicated by an ammeter when
the current is in phase with the pressure_. In practice, the current
is hardly ever in phase with the pressure, usually lagging, though
sometimes leading in phase. Now the greater this phase difference,
either way, the less is the power of a given virtual current to do
work. With respect to this feature, effective current may be defined
as: _that proportion of a given virtual current which can do useful
work_. If there be no phase difference, then effective current is equal
to virtual current.]
The matter may be looked at in a different way. If an
alternating current is to produce in a given wire the same
amount of effect as a continuous current of 100 amperes, since
the alternating current goes down to zero twice in each period,
it is clear that it must at some point in the period rise to
a maximum greater than 100 amperes. How much greater must the
maximum be? The answer is that, if it undulate up and down
with a pure wave form, its maximum must be √2̅ times as great
as the virtual mean; or conversely the virtual amperes will
be equal to the maximum divided by √2̅. In fact, to produce
equal effect, the equivalent direct current will be a kind of
mean between the maximum and the zero value of the alternating
current; but it must not be the arithmetical mean, nor the
geometrical mean, nor the harmonic mean, but the _quadratic_
mean; that is, it will be the _square root of the mean of the
squares_ of all the instantaneous values between zero and
maximum.
~Effective Volts and Amperes.~--Virtual pressure, although already
explained, may be further defined as the pressure _impressed_ on a
circuit. Now, in nearly all circuits the impressed or virtual pressure
meets with an opposing pressure due to inductance and hence the
_effective_ pressure is something less than the virtual, being defined
as _that pressure which is available for driving electricity around
the circuit, or for doing work_. The difference between virtual and
effective pressure is illustrated in fig. 1,237.
~Ques. Does a given alternating voltage affect the insulation of the
circuit differently than a direct pressure of the same value?~
Ans. It puts more strain on the insulation in the same proportion as
the maximum pressure exceeds the virtual pressure.
[Illustration: FIG. 1,238.--Current or pressure curve illustrating
_form factor_. It is simply _the virtual value divided by the average
value_. For a sine wave the virtual value is 1 / √2̅ times the maximum,
and the average is 2 / π times the maximum, so that the form factor is
π / 2√2̅ or 1.11. The induction wave which generates an alternating
pressure wave has a maximum value proportional to the area, that
is, to the average value of the pressure wave. Hence the induction
values corresponding to two pressure waves whose virtual values are
equal, will be inversely proportional to their form factors. This is
illustrated by the fact that a _peaked_ wave causes less hysteresis
loss in a transformer core than a flat topped wave, owing to the higher
form factor of the peaked wave. See wave forms, figs. 1,245 to 1,248.]
~Form Factor.~--This term was introduced by Fleming, and denotes the
ratio of the virtual value of an alternating wave to the average value.
That is
virtual value .707
form factor = ------------- = ---- = 1.11
average value .637
~Ques. What does this indicate?~
Ans. It gives the relative heating effects of alternating and direct
currents, as illustrated in figs. 1,239 and 1,240.
That is, the alternating current will have about 11 per cent.
more heating power than the direct current which is of the same
_average_ strength.
If an alternating current voltmeter be placed upon a circuit
in which the volts range from +100 to -100, it will read 70.7
volts, although the arithmetical average, irrespective of +
or-sign, is only 63.7 volts. If the voltmeter be connected to a
direct current circuit, the pressure necessary to give the same
reading would be 70.7 volts.
[Illustration: FIGS. 1,239 and 1,240.--Relative heating effects of
alternating and direct currents. If it takes say five minutes to
produce a certain heating effect with alternating current at say 63.7
_average_ volts, it will take 33 seconds longer with direct current at
the same pressure, that is, the alternating current has about 11 per
cent. more heating power than the direct current of the same _average_
pressure. The reader should be careful not to get a wrong conception of
the above; it does not mean that there is a saving by using alternating
current. When both voltmeters read the same, that is, when the
_virtual_ pressure of the alternating current is the same as the direct
current pressure, the heating effect is of course the same.]
~Ques. What is the relation between the shape of the wave curve and the
form factor?~
Ans. The more peaked the wave, the greater the value of its form factor.
A form factor of units would correspond to a rectangular wave;
this is the least possible value of the form factor, and one
which is not realized in commercial machines.
[Illustration: FIGS. 1,241 to 1,244.--Various forms of pressure or
current waves. Figs. 1,241 to 1,243 show the general shape of the waves
produced by some alternators used largely for lighting work and having
toothed armatures. The effect of the slots and shape of pole pieces
is here very marked. Fig. 1,244 shows a wave characteristic of large
alternators designed for power transmission and having multi-slot or
distributed windings.]
~Wave Form.~--There is always more or less irregularity in the shape of
the current waves as met in practice, depending upon the construction
of the alternator.
The ideal wave curve is the so called _true sine wave_, and is obtained
with a rate of cutting of lines of force, by the armature coils,
equivalent to the swing of a pendulum, which increases in speed from
the end to the middle of the swing, decreasing at the same rate after
passing the center. This swing is expressed in physics, as "simple
harmonic motion".
[Illustration: FIGS. 1,245 and 1,246.--Resolution of complex curves
into sine curves. The heavy curve can be resolved into the simpler
curves A and B shown in No. 1, the component curves A and B have in the
ratio of three to one; that is, curve B has three times as many periods
per second as curve A. All the curves, however, cross the zero line at
the same time, and the resultant curve, though curiously unlike either
of them, has a certain symmetry. In No. 2 the component curves, besides
having periods in the ratio of three to one, cross the zero line at
different points. The resultant curve produced is still less similar to
its components, and is curiously and unsymmetrically humped. At first
sight it is difficult to believe that such a curious curve could be
resolved into two such simple and symmetrical ones. In both figures the
component curves are sine curves, and as the curves for sine and cosine
functions are exactly similar in form, the simplest supposition that
can be made for the variation of pressure or of current is that both
follow a _sine law_.]
[Illustration: FIG. 1,247.--Reproduction of oscillograph record of wave
form of alternator with one coil per phase per pole. Here the so-called
"super-imposed harmonic" is clearly indicated.]
[Illustration: FIG. 1,248.--Reproduction of oscillograph record of
Wagner alternator having three coils per phase per pole.]
The losses in all secondary apparatus are slightly lower with the so
called _peaked_ form of wave. For the same virtual voltage, however,
the top of the peak will be much higher, thereby submitting the
insulation to that much greater strain. By reason of the fact that the
losses are less under such wave forms, many manufacturers in submitting
performance data on transformers recite that the figures are for sine
wave conditions, stating further that if the transformers are to be
operated in a circuit more peaked than the sine wave, the losses will
be less than shown.
The slight saving in the losses of secondary apparatus, obtained with
a peaked wave, by no means compensates for the increased insulation
strains and an alternator having a true sine wave is preferred.
~Ques. What determines the form of the wave?~
Ans. 1. The number of coils per phase per pole, 2, shape of pole faces,
3, eddy currents in the pole pieces, and 4, the air gap.
~Ques. What are the requirements for proper rate of cutting of the
lines of force?~
Ans. It is necessary to have, as a minimum, two coils per phase per
pole in three phase work.
~Ques. What is the effect of only one coil per phase per pole?~
Ans. The wave form will be distorted as shown in fig. 1,247.
~Ques. What is the least number of coils per phase per pole that should
be used for two and three phase alternators?~
Ans. For three phase, two coils, and for two phase, three coils, per
phase per pole.
~Single or Monophase Current.~--This kind of alternating current is
generated by an alternator having a single winding on its armature. Two
wires, a lead and return, are used as in direct current.
An elementary diagram showing the working principles is illustrated in
fig. 1,249, a similar hydraulic cycle being shown in figs. 1,250 to
1,252.
[Illustration: FIG. 1,249.--Elementary one loop alternator and sine
curve illustrating single phase alternating current. There are three
points during the revolution at which there is no current: at 0° the
position shown, 180°, and 360°; in other words, at the beginning,
middle point and end of the cycle. The current reaches a maximum at
90°, reverses at 180°, and reaches a maximum in the reverse direction
at 270°.]
~Two Phase Current.~--In most cases two phase current actually consists
of two distinct single phase currents flowing in separate circuits.
There is often no electrical connection between them; they are of equal
period and equal amplitude, but differ in phase by one quarter of a
period. With this phase relation one of them will be at a maximum when
the other is at zero. Two phase current is illustrated by sine curves
in fig. 1,253, and by hydraulic analogy in figs. 1,254 and 1,255.
[Illustration: FIGS. 1,250 to 1,252.--Hydraulic analogy illustrating
the difference between _direct_ (continuous) and _alternating_ current.
In fig. 1,250 a centrifugal pump C forces water to the upper pipe,
from which it falls by gravity to the lower pipe B and re-enters the
pump. The current is continuous, always flowing in one direction,
that is, it does not reverse its direction. Similarly a direct
electric current is constant in direction (does not reverse); though
not necessarily constant in value. A direct current, constant in
both value and direction as a result of constant pressure, is called
"continuous" current. Similarly in the figure the flow is constant,
and a gauge D placed at any point will register a constant pressure,
hence the current may be called, in the electrical sense, "continuous."
The conditions in fig. 1,251 are quite different. The illustration
represents a double acting cylinder with the ends connected by a
pipe A, and the piston driven by crank and Scotch yoke as shown. In
operation, if the cylinder and pipe be full of water, a current of
water will begin to flow through the pipe in the direction indicated
as the piston begins its stroke, increasing to maximum velocity at
one-quarter revolution of the crank, decreasing and coming to rest
at one-half revolution, then reversing and reaching maximum velocity
in the reverse direction at three-quarter revolution, and coming to
rest again at the end of the return stroke. A pressure gauge at G
will register a pressure which varies with the current. Since the
alternating electric current undergoes similar changes, the sine curve
will apply equally as well to the pump cycle as to the alternating
current cycle.]
[Illustration: FIG. 1,253.--Elementary two loop alternator and sine
curves, illustrating two phase alternating current. If the loops be
placed on the alternator armature at 90 magnetic degrees, a single
phase current will be generated in each of the windings, the current
in one winding being at its maximum value when the other is at zero.
In this case four transmission conductors are generally used, two for
each separate circuit, and the motors to which the current is led have
a double winding corresponding to that on the alternator armature.]
If two identical simple alternators have their armature shafts
coupled in such a manner, that when a given armature coil on
one is directly under a field pole, the corresponding coil on
the other is midway between two poles of its field, the two
currents generated will differ in phase by a half alternation,
and will be two phase current.
~Ques. How must an alternator be constructed to generate two phase
current?~
Ans. It must have two independent windings, and these must be so spaced
out that when the volts generated in one of the two phases are at a
maximum, those generated in the other are at zero.
In other words, the windings, which must be alike, of an equal
number of turns, must be displaced along the armature by an
angle corresponding to one-quarter of a period, that is, to
half the pole pitch.
[Illustration: FIGS. 1,254 and 1,255.--Hydraulic analogy illustrating
two phase alternating current. In the figure two cylinders, similar to
the one in fig. 1,251, are shown, operated from one shaft by crank and
Scotch yoke drive. The cranks are at 90° as shown, and the cylinders
and connecting pipes full of water. In operation, the same cycle of
water flow takes place as in fig. 1,251. Since the cranks are at 90°,
the second piston is one-half stroke behind the first; the flow of
water in No. 1 (phase A) is at a maximum when the flow in No. 2 (phase
B) comes to rest, the current conditions in both pipes for the entire
cycle being represented by the two sine curves whose phase difference
is 90°. Comparing these curves with fig. 1,253, it will be seen that
the water and electric current act in a similar manner.]
The windings of the two phases must, of course, be kept
separate, hence the armature will have four terminals, or if it
be a revolving armature it will have four collector rings.
As must be evident the phase difference may be of any value
between 0° and 360°, but in practice it is almost always made
90°.
~Ques. In what other way may two phase current be generated?~
Ans. By two single phase alternators coupled to one shaft.
~Ques. How many wires are required for two phase distribution?~
Ans. A two phase system requires four lines for its distribution;
two lines for each phase as in fig. 1,253. It is possible, but not
advisable, to reduce the number to 3, by employing one rather thicker
line as a common return for each of the phases as in fig. 1,256.
[Illustration: FIG. 1,256.--Diagram of three wire two phase current
distribution. In order to save one wire it is possible to use a common
return conductor for both circuits, as shown, the dotted portion of
one wire 4 being eliminated by connecting across to 1 at M and S. For
long lines this is economical, but the interconnection of the circuits
increases the chance of trouble from grounds or short circuits. The
current in the conductor will be the resultant of the two currents,
differing by 90° in phase.]
If this be done, the voltage between the A line and the B line
will be equal to √2̅ times the voltage in either phase, and the
current in the line used as common return will be √2̅ times as
great as the current in either line, assuming the two currents
in the two phases to be equal.
~Ques. In what other way may two phase current be distributed?~
Ans. The mid point of the windings of the two phases may be united in
the alternator at a common junction.
[Illustration: FIGS. 1,257 to 1,259.--Various two phase armature
connections. Fig. 1,257, two separate circuit four collector ring
arrangement; fig. 1,258, common middle connection, four collector
rings; fig. 1,259, circuit connected in armature for three collector
rings. In the figures the black winding represents phase A, and the
light winding, phase B.]
This is equivalent to making the machine into a four phase
alternator with half the voltage in each of the four phases,
which will then be in successive quadrature with each other.
~Ques. How are two phase alternator armatures wound?~
Ans. The two circuits may be separate, each having two collector rings,
as shown in fig. 1,257, or the two circuits may be coupled at a common
middle as in fig. 1,258, or the two circuits may be coupled in the
armature so that only three collector rings are required as shown in
fig. 1,259.
[Illustration: FIG. 1,260.--Elementary three loop alternator and sine
curves, illustrating three phase alternating current. If the loops
be placed on the alternator armature at 120 magnetic degrees from
one another, the current in each will attain its maximum at a point
one-third of a cycle distant from the other two. The arrangement here
shown gives three independent single phase currents and requires
six wires for their transmission. A better arrangement and the one
generally used is shown in fig. 1,261.]
[Illustration: FIG. 1,261.--Elementary three wire three phase
alternator. For the transmission of three phase current, it is not
customary to use six wires, as in fig. 1,260, instead, three ends (one
end of each of the loops) are brought together to a common connection
as shown, and the other ends, connected to the collector rings, giving
only three wires for the transmission of the current.]
~Three Phase Current.~--A three phase current consists of three
alternating currents of equal frequency and amplitude, but differing
in phase from each other by one-third of a period. Three phase current
as represented by sine curves is shown in fig. 1,260, and by hydraulic
analogy in fig. 1,262. Inspection of the figures will show that when
any one of the currents is at its maximum, the other two are of half
their maximum value, and are flowing in the opposite direction.
[Illustration: FIGS. 1,262 and 1,263.--Hydraulic analogy illustrating
three phase alternating current. Three cylinders are here shown with
pistons connected through Scotch yokes to cranks placed 120° apart. The
same action takes place in each cylinder as in the preceding cases, the
only difference being the additional cylinder, and difference in phase
relation.]
~Ques. How is three phase current generated?~
Ans. It requires three equal windings on the alternator armature, and
they must be spaced out over its surface so as to be successively ⅓
and ⅔ of the period (that is, of the double pole pitch) apart from one
another.
~Ques. How many wires are used for three phase distribution?~
Ans. Either six wires or three wires.
Six wires, as in fig. 1,260, might be used where it is desired
to supply entirely independent circuits, or as is more usual
only three wires are used as shown in fig. 1,261. In this case
it should be observed that if the voltage generated in each one
of the three phases separately E (virtual) volts, the voltage
generated between any two of the terminals will be equal to
√3̅ × E. Thus, if each of the three phases generate 100 volts,
the voltage from the terminal of the A phase to that of the B
phase will be 173 volts.
[Illustration: FIG. 1,264.--Experiment illustrating _self-induction_ in
an alternating current circuit. If an incandescent lamp be connected
in series with a coil made of one pound of No. 20 magnet wire, and
connected to the circuit, the current through the lamp will be
decreased due to the self-induction of the coil. If now an iron core
be gradually pushed into the coil, the self-induction will be greatly
increased and the lamp will go out, thus showing the great importance
which self-induction plays in alternating current work.]
~Inductance.~--Each time a direct current is started, stopped or varied
in strength, the magnetism changes, and induces or tends to induce a
pressure in the wire which always has a direction opposing the pressure
which originally produced the current. _This self-induced pressure
tends to weaken the main current at the start and prolong it when the
circuit is opened._
The expression _inductance_ is frequently used in the same
sense as _coefficient of self-induction_, which is a quantity
pertaining to an electric circuit depending on its geometrical
form and the nature of the surrounding medium.
If the direct current maintains the same strength and flow steadily,
_there will be no variations in the magnetic field surrounding the wire
and no self-induction_, consequently the only retarding effect of the
current will be the "_ohmic resistance_" of the wire.
If an alternating current be sent through a circuit, there will be two
retarding effects:
1. The _ohmic_ resistance;
2. The _spurious_ resistance.
[Illustration: FIG. 1,265.--Non-inductive and inductive resistances.
Two currents are shown joined in parallel, one containing a lamp
and non-inductive resistance, and the other a lamp and inductive
resistance. The two resistances being the same, a sufficient direct
pressure applied at T, T' will cause the lamps to light up equally.
If, however, an alternating pressure be applied, M will burn brightly,
while S will give very little or no light because of the effect of the
inductance of the inductive resistance.]
~Ques. Upon what does the ohmic resistance depend?~
Ans. Upon the length, cross sectional area and material of the wire.
~Ques. Upon what does the spurious resistance depend?~
Ans. Upon the frequency of the alternating current, the shape of the
conductor, and nature of the surrounding medium.
[Illustration: FIG. 1,266.--Inductance test, illustrating the
self-induction of a coil which is gradually increased by moving an iron
wire core inch by inch into the coil. The current is kept constant with
the adjustable resistance throughout the test and readings taken, first
without the iron core, and again when the core is put in the coil and
moved to the 1, 2, 3, 4, etc., inch marks. By plotting the voltmeter
readings and the position of the iron core on section paper, a curve is
obtained showing graphically the effect of the self-induction. A curve
of this kind is shown in fig. 1,302.]
~Ques. Define inductance.~
Ans. It is the total magnetic flux threading the circuit per unit
current which flows in the circuit, and which produces the flux.
In this it must be understood that if any portion of the flux
thread the circuit more than once, this portion must be added
in as many times as it makes linkage.
Inductance, or the coefficient of self-induction is the
capacity which an electric circuit has of producing induction
within itself.
Inductance is considered as the ratio between the total
induction through a circuit to the current producing it.
~Ques. What is the unit of inductance?~
Ans. The henry.
~Ques. Define the henry.~
Ans. A coil has an inductance of one henry when the product of the
number of lines enclosed by the coil multiplied by the number of turns
in the coil, when a current of one ampere is flowing in the coil, is
equal to 100,000,000 or 10⁸.
An inductance of one henry exists in a circuit _when a current
changing at the rate of one ampere per second induces a
pressure of one volt in the circuit_.
~Ques. What is the henry called?~
Ans. The coefficient of self-induction.
[Illustration: FIG. 1,267.--Diagram illustrating the henry. By
definition: _A circuit has an inductance of one henry when a rate of
change of current of one ampere per second induces a pressure of one
volt._ In the diagram it is assumed that the internal resistance of the
cell and resistance of the connecting wires are zero.]
The henry is the coefficient by which the time rate of change
of the current in the circuit must be multiplied, in order to
give the pressure of self-induction in the circuit.
The formula for the henry is as follows:
magnetic flux × turns
henrys = ---------------------
current × 100,000,000
or
N × T
L = ----- (1)
10⁸
where
L = coefficient of self induction in henrys;
N = total number of lines of force threading a coil when the
current is one ampere;
T = number of turns of coil.
If a coil had a coefficient of self-induction of one henry, it
would mean that if the coil had one turn, one ampere would set
up 100,000,000, or 10⁸, lines through it.
[Illustration: FIGS. 1,268 to 1,270.--Various coils. The inductance
effect, though perceptible in an air core coil, fig. 1,268, may be
greatly intensified by inserting a core made of numerous pieces of iron
wire, as in fig. 1,269. Fig. 1,270 shows a non-inductive coil. When
wound in this manner, a coil will have little or no inductance because
each half of the coil neutralizes the magnetic effect of the other.
This coil, though non-inductive, will have "capacity." It would be
useless for solenoids or electromagnets, as it would have no magnetic
field.]
The henry[2] is too large a unit for use in practical computations,
which involves that the millihenry, or ¹/₁₀₀₀th henry, is the accepted
unit. In pole suspended lines the inductance varies as the metallic
resistance, the distance between the wires on the cross arm and the
number of cycles per second, as indicated by accepted tables. Thus,
for one mile of No. 8 B. & S. copper wire, with a resistance of 3,406
ohms, the coefficient of self-induction with 6 inches between centers
is .00153, and, with 12 inches, .00175.
[2] NOTE.--The American physicist, Joseph Henry, was born in 1798
and died 1878. He was noted for his researches in electromagnetism.
He developed the electromagnet, which had been invented by Sturgeon
in England, so that it became an instrument of far greater power
than before. In 1831, he employed a mile of fine copper wire with an
electromagnet, causing the current to attract the armature and strike a
bell, thereby establishing the principle employed in modern telegraph
practice. He was made a professor at Princeton in 1832, and while
experimenting at that time, he devised an arrangement of batteries and
electromagnets embodying the principle of the telegraph relay which
made possible long distance transmission. He was the first to observe
magnetic self-induction, and performed important investigations in
oscillating electric discharges (1842), and other electrical phenomena.
In 1846 he was chosen secretary of the Smithsonian Institution at
Washington, an office which he held until his death. As chairman of
the U. S. Lighthouse Board, he made important tests in marine signals
and lights. In meteorology, terrestrial magnetism, and acoustics,
he carried on important researches. Henry enjoyed an international
reputation, and is acknowledged to be one of America's greatest
scientists.
[Illustration: FIG. 1,271.--Hydraulic-mechanical analogy illustrating
_inductance_ in an alternating current circuit. The two cylinders are
connected at their ends by the vertical pipes, each being provided with
a piston and the system filled with water. Reciprocating motion is
imparted to the lower pulley by Scotch yoke connection with the drive
pulley. The upper piston is connected by rack and pinion gear with a
fly wheel. In operation, the to and fro movement of the lower piston
produces an alternating flow of water in the upper cylinder which
causes the upper piston to move back and forth. The rack and pinion
connection with the fly wheel causes the latter to revolve first in
one direction, then in the other, in step with the upper piston. The
inertia of the fly wheel causes it to resist any change in its state,
whether it be at rest or in motion, which is transmitted to the upper
piston, causing it to offer resistance to any change in its rate or
direction of motion. Inductance in the alternating current circuit
has precisely the same effect, that is, _it opposes any change in the
strength or direction of the current_.]
~Ques. How does the inductance of a coil vary with respect to the core?~
Ans. It is least with an air core; with an iron core, it is greater in
proportion to the permeability[3] of the iron.
[3] NOTE.--The permeability of iron varies from 500 to 1,000 or more.
The permeability of a given sample of iron is not constant, but
decreases in value as the magnetizing force increases. Therefore the
inductance of a coil having an iron core is not a constant quantity as
is the inductance of an air core coil.
The coefficient L for a given coil is a constant quantity so
long as the magnetic permeability of the material surrounding
the coil does not change. This is the case where the coil is
surrounded by air. When iron is present, the coefficient L is
practically constant, provided the magnetism is not forced too
high.
In most cases arising in practice, the coefficient L may be
considered to be a constant quantity, just as the resistance R
is usually considered constant. The coefficient L of a coil or
circuit is often spoken of as its _inductance_.
[Illustration: FIG. 1,272.--Experiment showing effect of inductive and
non-inductive coils in alternating current circuit. The apparatus is
connected up as shown; by means of the switch, the lamp may be placed
in parallel with either the inductive or non-inductive coil. These
coils should have the same resistance. Pass an alternating current
through the lamp and non-inductive coil, of such strength that the
lamp will be dimly lighted. Now turn the switch so as to put the lamp
and inductive coil in parallel and the lamp will burn with increased
brilliancy. The reason for this is because of the opposition offered
by the inductive coil to the current, less current is shunted from
the lamp when the inductive coil is in the circuit than when the
non-inductive coil is in the circuit. That is, each coil has the same
ohmic resistance, but the inductive coil has in addition the spurious
resistance due to inductance, hence it shunts less current from the
lamp than does the non-inductive coil.]
~Ques. Why is the iron core of an inductive coil made with a number of
small wires instead of one large rod?~
Ans. It is laminated in order to reduce eddy currents and consequent
loss of energy, and to prevent excessive heating of the core.
~Ques. How does the number of turns of a coil affect the inductance?~
Ans. The inductance varies as the square of the turns.
That is, if the turns be doubled, the inductance becomes four
times as great.
The inductance of a coil is easily calculated from the following
formulæ:
L = 4π²_r²n²_ ÷ (_l_ × 10⁹) (1)
for a thin coil with air core, and
L = 4π²_r²n²_μ ÷ (_l_ × 10⁹) (2)
for a coil having an iron core. In the above formulæ:
L = inductance in henrys;
π = 3.1416;
_r_ = average radius of coil in centimeters;
_n_ = number of turns of wire in coil;
μ = permeability of iron core;
_l_ = length of coil in centimeters.
EXAMPLE.--An air core coil has an average radius of 10
centimeters and is 20 centimeters long, there being 500 turns,
what is the inductance?
Substituting these values in formula (1)
_L_ = 4 × (3.1416)² × 10² × 500² ÷ (20 × 10⁹) = .00494 henry
~Ques. Is the answer in the above example in the customary form?~
Ans. No; the henry being a very large unit, it is usual to express
inductance in thousandths of a henry, that is, in _milli-henrys_. The
answer then would be .04935 × 1,000 = 49.35 milli-henrys.
[Illustration: FIGS. 1,273 to 1,275.--General Electric choke coils.
Fig. 1,273, hour glass coil, 35,000 volts; fig. 1,274, 4,600 volt
coil; fig. 1,275, 6,600 volt coil. A choke coil is _a coil with
large inductance and small resistance, used to impede alternating
currents_. The choke coil is used extensively as an auxiliary to the
lightning arrester. In this connection the primary objects of the
choke coil should be: 1, to hold back the lightning disturbance from
the transformer or generator until the lightning arrester discharges
to earth. If there be no lightning arrester the choke coil evidently
cannot perform this function. 2, to lower the frequency of the
oscillation so that whatever charge gets through the choke coil will
be of a frequency too low to cause a serious drop of pressure around
the first turns of the end coil in either generator or transformer.
Another way of expressing this is from the standpoint of wave front:
a steep wave front piles up the pressure when it meets an inductance.
The second function of the choke coil is, then, to smooth out the
wave front of the surge. The principal electrical condition to be
avoided is that of resonance. The coil should be so arranged that if
continual surges be set up in the circuit, a resonant voltage due to
the presence of the choke coil cannot build up at the transformer or
generator terminals. In the types shown above, the hour glass coil
has the following advantages on high voltages: 1, should there be any
arcing between adjacent turns the coils will re-insulate themselves,
2, they are mechanically strong, and sagging is prevented by tapering
the coils toward the center turns, 3, the insulating supports can be
best designed for the strains which they have to withstand. Choke coils
should not be used in connection with cable systems.]
EXAMPLE.--An air core coil has an inductance of 50
milli-henrys; if an iron core, having a permeability of 600 be
inserted, what is the inductance?
The inductance of the air core coil will be multiplied by the
permeability of the iron; the inductance then is increased to
50 × 600 = 30,000 milli-henrys, or 30 henrys.
~Ohmic Value of Inductance.~--The rate of change of an alternating
current at any point expressed in degrees is equal to the product of
_2π multiplied by the frequency, the maximum current, and the cosine of
the angle of position θ_; that is (using symbols)
rate of change = 2π_f_Iₘₐₓ_cos θ_.
The numerical value of the rate of change is independent of its
positive or negative sign, so that the sign of the cos φ is disregarded.
[Illustration: FIG. 1,276.--Inductance experiment with intermittent
direct current. A lamp S is connected in parallel with a coil of
fairly fine wire having a removable iron core, and the terminals T, T'
connected to a source of direct current, a switch M being provided to
interrupt the current. The voltage of the current and resistance of
the coil are of such values that when a steady current is flowing, the
lamp filament is just perceptibly red. _At the instant of making the
circuit, the lamp will momentarily glow more brightly than when the
current is steady; on breaking the circuit the lamp will momentarily
flash with great brightness._ In the first case, the reverse pressure,
due to inductance, as indicated by arrow _b_, will momentarily oppose
the normal pressure in the coil, so that the voltage at the lamp will
be momentarily increased, and will consequently send a momentarily
stronger current through the lamp. On breaking the main circuit at
M, the field of the coil will collapse, generating a momentary much
greater voltage than in the first instance, in the direction of arrow
_a_, the lamp will flash up brightly in consequence.]
The period of greatest rate of change is that at which cos φ has the
greatest value, and the maximum value of a cosine is when the arc has a
value of zero degrees or of 180 degrees, its value corresponding, being
1. (See fig. 1,037, page 1,068.)
The pressure due to inductance is equal to the product of the rate
of change by the inductance; that is, calling the inductance L, the
pressure due to it at the point of maximum value or
Eₘₐₓ = 2π_f_Iₘₐₓ × L (1)
Now by Ohm's law
Eₘₐₓ = RIₘₐₓ (2)
for a current Iₘₐₓ, hence substituting equation (2) in equation (1)
RIₘₐₓ = 2π_f_Iₘₐₓ × L
from which, dividing both sides by Iₘₐₓ, and using Xᵢ for R
~Xᵢ = 2π_f_L~ (3)
which is the ~ohmic equivalent of inductance~.
[Illustration: FIG. 1,277.--Diagram showing alternating circuit
containing inductance. Formula for calculating the ohmic value
of inductance or "inductance reactance," is Xᵢ = 2π_f_L in which
Xᵢ = inductance reactance; π = 3.1416; _f_ = frequency; L = inductance
_in henrys_ (not milli-henrys). L = 15 milli-henrys = 15 ÷ 1000 = .015
henrys. Substituting, Xᵢ = 2 × 3.1416 × 100 × .015 = 9.42 ohms.]
The frequency of a current being the number of periods or waves per
second, then, if T = the time of a period, the frequency of a current
may be obtained by dividing 1 second by the time of a period; that is
one second 1
frequency = ------------------ = --- (4)
time of one period T
substituting 1 / T for _f_ in equation (3)
L
Xᵢ = 2π -
T
[Illustration: FIG. 1,278.--- Diagram illustrating effect of capacity
in an alternating circuit. Considering its action during one cycle of
the current, the alternator first "pumps," say from M to S; electricity
will be heaped up, so to speak, on S, and a deficit left on M, that
is, S will be + and M-. If the alternator be now suddenly stopped,
there would be a momentary return flow of electricity from S to M
through the alternator. If the alternator go on working, however, it
is obvious that the electricity heaped up on S helps or increases the
flow when the alternator begins to pump from S to M in the second half
of the cycle, and when the alternator again reverses its pressure,
the + charge on M flows round to S, and helps the ordinary current.
The above circuit is not strictly analogous to the insulated plates
of a condenser, but, as is verified in practice, that with a rapidly
alternating pressure, the condenser action is not perceptibly affected
if the cables be connected across by some non-inductive resistance as
for instance incandescent lamps.]
~Capacity.~--When an electric pressure is applied to a condenser,
the current plays in and out, charging the condenser in alternate
directions. As the current runs in at one side and out at the other,
the dielectric becomes charged, and tries to discharge itself by
setting up an opposing electric pressure. This opposing pressure rises
just as the charge increases.
A mechanical analogue is afforded by the bending of a spring, as in
fig. 1,279, which, as it is being bent, exerts an opposing force equal
to that applied, provided the latter do not exceed the capacity of the
spring.
~Ques. What is the effect of capacity in an alternating circuit?~
Ans. It is exactly opposite to that of inductance, that is, it assists
the current to rise to its maximum value sooner than it would otherwise.
[Illustration: FIG. 1,279.--Mechanical analogy illustrating effect of
capacity in an alternating circuit. If an alternating twisting force
be applied to the top R of the spring S, the action of the latter
may be taken to represent capacity, and the rotation of the wheel W,
alternating current. The twisting force (impressed pressure) must
first be applied _before_ the rotation of W (current) will begin. The
resiliency or rebounding effect of the spring will, in time, cause the
wheel W to move (amperes) in advance of the twisting force (voltage)
thus representing the current _leading in phase_.]
~Ques. Is it necessary to have a continuous metallic circuit for an
alternating current?~
Ans. No, it is possible for an alternating current to flow through a
circuit which is divided at some point by insulating material.
~Ques. How can the current flow under such condition?~
Ans. Its flow depends on the capacity of the circuit and accordingly
a condenser may be inserted in the circuit as in fig. 1,286, thus
interposing an insulated gap, yet permitting an alternating flow in the
metallic portion of the circuit.
[Illustration: FIG. 1,280.--Hydraulic analogy illustrating capacity
in an alternating current circuit. A chamber containing a rubber
diaphragm is connected to a double acting cylinder and the system
filled with water. In operation, as the piston moves, say to the left
from the center, the diaphragm is displaced from its neutral position
N, and stretched to some position M, in so doing offering increasing
resistance to the flow of water. On the return stroke the flow is
reversed and is assisted by the diaphragm during the first half of
the stroke, and opposed during the second half. The diaphragm thus
acts with the flow of water one-half of the time and in opposition to
it one-half of the time. This corresponds to the electrical pressure
at the terminals of a condenser connected in an alternating current
circuit, and it has a maximum value when the current is zero and a zero
value when the current is a maximum.]
~Ques. Name the unit of capacity and define it.~
Ans. The unit of capacity is called the _farad_ and its symbol is C. A
condenser is said to have a capacity of one farad if one coulomb (that
is, one ampere flowing one second), when stored on the plates of the
condenser will cause a pressure of one volt across its terminals.
The farad being a very large unit, the capacities ordinarily
encountered in practice are expressed in millionths of a farad,
that is, in _microfarads-_-a capacity equal to about three
miles of an Atlantic cable.
It should be noted that the microfarad is used only for convenience,
and that _in working out problems, capacity should always be expressed
in farads before substituting in formulæ_, because the farad is chosen
with respect to the volt and ampere, as above defined, and hence must
be used in formulæ along with these units.
[Illustration: FIG. 1,281--Diagram illustrating a _farad_. A condenser
is said to have a capacity of one farad if it will absorb one coulomb
of electricity when subjected to a pressure of one volt. The farad is
a very large unit, and accordingly the microfarad or one millionth of
a farad is often used, though _this must be reduced to farads before
substituting in formulæ_.]
For instance, a capacity of 8 microfarads as given in a problem
would be substituted in a formula as .000008 of a farad.
The charge Q forced into a condenser by a steady electric pressure E is
Q = EC
in which
Q = charge in coulombs;
E = electric pressure in volts;
C = capacity of condenser in farads.
~Ques. What is the material between the plates of a condenser called?~
Ans. The _dielectric_.
~Ques. Upon what does the capacity of a condenser depend?~
Ans. It is proportional to the area of the plates, and inversely
proportional to the thickness of the dielectric between the plates, a
correction being required unless the thickness of dielectric be very
small as compared with the dimensions of the plates.
The capacity of a condenser is also proportional to the
_specific inductive capacity_ of the dielectric between the
plates of the condenser.
[Illustration: FIG. 1,282.--Condenser of one microfarad capacity. It is
subdivided into five sections of .5, .2, .2, .05 and .05 microfarad.
The plates are mounted between and carried by lateral brass bars which
are fastened to a hard rubber top. Each pair of condenser terminals is
fastened to small binding posts mounted on hard rubber insulated posts.]
~Specific Inductive Capacity.~--Faraday discovered that different
substances have different powers of carrying lines of electric force.
Thus the charge of two conductors having a given difference of pressure
between them depends on the medium between them as well as on their
size and shape. The number indicating the magnitude of this property of
the medium is called its _specific inductive capacity_, or _dielectric
constant_.
The specific inductive capacity of air, which is nearly the same
as that of a vacuum, is taken as unity. In terms of this unit the
following are some typical values of the dielectric constant: water 80,
glass 6 to 10, mica 6.7, gutta-percha 3, India rubber 2.5, paraffin wax
2, ebonite 2.5, castor oil 4.8.
In underground cables for very high pressures, the insulation,
if homogeneous throughout, would have to be of very great
thickness in order to have sufficient dielectric strength. By
employing material of high specific inductive capacity close
to the conductor, and material of lower specific inductive
capacity toward the outside, that is, by _grading_ the
insulation, a considerably less total thickness affords equally
high dielectric strength.
[Illustration: FIG. 1,283.--Parallel connection of condensers. Like
terminals are joined together. The joint capacity of such arrangement
is equal to _the sum of the respective capacities_, that is C =
c + c' + c".]
~Ques. How are capacity tests usually made?~
Ans. By the aid of standard condensers.
~Ques. How are condensers connected?~
Ans. They may be connected in parallel as in fig. 1,283, or in series
(cascade) as in fig. 1,284.
Condensers are now constructed so that the two methods of
arranging the plates may conveniently be combined in one
condenser, thereby obtaining a wider range of capacity.
~Ques. How may the capacity of a condenser, wire, or cable be tested?~
Ans. This may be done by the aid of a standard condenser, trigger key,
and an astatic or ballistic galvanometer.
In making the test, first obtain a "constant" by noting the
deflection _d_, due to the discharge of the standard condenser
after a charge of, say, 10 seconds from a given voltage. Then
discharge the other condenser, wire, or cable through the
galvanometer after 10 seconds charge, and note the deflection
_d'_. The capacity C' of the latter is then
_d'_
C' = C × ----
_d_
in which C is the capacity of the standard condenser.
[Illustration: FIG. 1,284.--Series or cascade connection of condensers.
Unlike terminals are joined together as shown. The total capacity
of such connection is equal to _the reciprocal of the sum of the
reciprocals of the several capacities_, that is,
C = 1 ÷ (1 / c + 1 / c' + 1 / c")]
~Ohmic Value of Capacity.~--The capacity of an alternating current
circuit is the measure of the amount of electricity held by it when its
terminals are at unit difference of pressure. Every such circuit acts
as a condenser.
If an alternating circuit, having no capacity, be opened, no current
can be produced in it, but if there be capacity at the break, current
may be produced as in fig. 1,286.
The action of capacity referred to the current wave is as follows:
As the wave starts from zero value and rises to its maximum value,
the current is due to the discharge of the capacity, which would
be represented by a condenser. In the case of a sine current, the
period required for the current to pass from zero value to maximum is
one-quarter of a cycle.
[Illustration: FIGS. 1,285 and 1,286.--Diagrams showing effect of
condenser in direct and alternating current circuits. Each circuit
contains an incandescent lamp and a condenser, one circuit connected to
a dynamo and the other to an alternator. Since the condenser interposes
a gap in the circuit, evidently in fig. 1,285 no current will flow.
In the case of alternating current, fig. 1,286, the condenser gap
does not hinder the flow of current in the metallic portion of the
circuit. In fact the alternator produces a continual surging of
electricity backwards and forwards from the plates of the condenser
around the metallic portion of the circuit, similar to the surging of
waves against a bulkhead which projects into the ocean. It should be
understood that the electric current ceases at the condenser, there
being no flow between the plates.]
At the beginning of the cycle, the condenser is charged to the maximum
amount it receives in the operation of the circuit.
At the end of the quarter cycle when the current is of maximum value,
the condenser is completely discharged.
The condenser now begins to receive a charge, and continues to receive
it during the next quarter of a cycle, the charge attaining its maximum
value when the current is of zero intensity. Hence, the _maximum charge
of a condenser_ in an alternating circuit is equal to the average value
of the current multiplied by the time of charge, which is one-quarter
of a period, that is
maximum charge = average current × ¼ period (1)
[Illustration: FIG. 1,287.--Diagram showing alternating circuit
containing capacity. Formula for calculating the ohmic value of
capacity or "capacity reactance" is Xc = 1 ÷ 2π_f_C, in which Xc =
capacity reactance; π = 3.1416; _f_ = frequency; C = capacity _in
farads_ (not microfarads). 22 microfarads = 22 ÷ 1,000,000 = .000022
farad. Substituting, Xc = 1 ÷ (2 × 3.1416 × 100 × .000022) = 72.4 ohms.]
Since the time of a period = 1 ÷ frequency, the time of one-quarter of
a period is ¼ × (1 ÷ frequency), or
¼ period = ¼_f_ (2)
_f_, being the symbol for frequency. Substituting (2) in (1)
maximum charge = Iₐᵥ × ¼_f_ (3)
The pressure of a condenser is equal to the quotient of the charge
divided by the capacity, that is
charge
condenser pressure = -------- (4)
capacity
Substituting (3) in (4)
1 Iₐᵥ
condenser pressure = (Iₐᵥ × ----) ÷ C = ----- (5)
4_f_ 4_f_C
But, Iₐᵥ = Iₘₐₓ × 2 / π, and substituting this value of Iₐᵥ in equation
(5) gives
Iₘₐₓ × 2 / π Iₘₐₓ
condenser pressure = ----------- = ------ (6)
4_f_C 2π_f_C
This last equation (6) represents the condenser pressure due to
capacity at the point of maximum value, which pressure is opposed to
the impressed pressure, that is, it is the maximum reverse pressure due
to capacity.
Now, since by Ohm's law
E
I = ---, or E = I × R
R
and as
Iₘₐₓ 1
------ = Iₘₐₓ × ------
2π_f_C 2π_f_C
it follows that 1 / (2π_f_C) is the _ohmic value_ of capacity, that is
it expresses the resistance equivalent of capacity; using the symbol Xc
for capacity reactance
~1
Xc = ------- (7)
2π_f_C~
EXAMPLE.--What is the resistance equivalent of a 50 microfarad
condenser to an alternating current having a frequency of 100?
Substituting the given values in the expression for ohmic value
1 1 1
Xc = ------ = -------------------------- = ------- = 31.8 ohms.
2π_f_C 2 × 3.1416 × 100 × .000050 .031416
If the pressure of the supply be, say 100 volts, the current
would be 100 ÷ 31.8 = 3.14 amperes.
[Illustration: FIG. 1,288.--Pressure and current curves, illustrating
_lag_. The effect of inductance in a circuit is to retard the current
cycle, that is to say, if the current and pressure be in phase, the
introduction of inductance will cause a phase difference, the current
wave "lagging" behind the pressure wave as shown. In other words,
inductance causes the current wave, indicated in the diagram by
the solid curve, to lag behind the pressure wave, indicated by the
dotted curve. Following the curves starting from the left end of the
horizontal line, it will be noted that the current starts after the
pressure starts and reverses after the pressure reverses; that is, _the
current lags in phase behind the pressure_, although the frequency of
both is the same.]
~Lag and Lead.~--Alternating currents do not always keep in step
with the alternating volts impressed upon the circuit. If there
be inductance in the circuit, the current will _lag_; if there be
capacity, the current will _lead_ in phase. For example, fig. 1,288,
illustrates the lag due to inductance and fig. 1,289, the lead due to
capacity.
~Ques. What is lag?~
Ans. Lag denotes the condition where the phase of one alternating
current quantity lags behind that of another. The term is generally
used in connection with the effect of inductance in causing the current
to lag behind the impressed pressure.
[Illustration: FIG. 1,289.--Pressure and current curves illustrating
_lead_. The effect of capacity in a circuit is to cause the current to
rise to its maximum value sooner than it would otherwise do; capacity
produces an effect exactly the opposite of inductance. The phase
relation between current and pressure with current leading is shown
graphically by the two armature positions in full and dotted lines,
corresponding respectively to current and pressure at the beginning of
the cycle.]
~Ques. How does inductance cause the current to lag behind the
pressure?~
Ans. It tends to prevent changes in the strength of the current. When
two parts of a circuit are near each other, so that one is in the
magnetic field of the other, any change in the strength of the current
causes a corresponding change in the magnetic field and sets up a
reverse pressure in the other wire.
This induced pressure causes the current to reach its maximum
value a little later than the pressure, and also tends to
prevent the current diminishing in step with the pressure.
~Ques. What governs the amount of lag in an alternating current?~
Ans. It depends on the relative values of the various pressures in the
circuit, that is, upon the amount of resistance and inductance which
tends to cause lag, and the amount of capacity in the circuit which
tends to reduce lag and cause lead.
~Ques. How is lag measured?~
Ans. In degrees.
[Illustration: FIG. 1,290.--Mechanical analogy of lag. If at one end
force be applied to turn a very long shaft, having a loaded pulley at
the other, the torsion thus produced in the shaft will cause it to
twist an appreciable amount which will cause the movement of the pulley
to _lag_ behind that of the crank. This may be indicated by a rod
attached to the pulley and terminating in a pointer at the crank end,
the rod being so placed that the pointer registers with the crank when
there is no torsion in the shaft. The angle made by the pointer and
crank when the load is thrown on, indicates the amount of lag which is
measured in degrees.]
Thus, in fig. 1,288, the lag is indicated by the distance
between the beginning of the pressure curve and the beginning
of the current curve, and is in this case 45°.
~Ques. What is the physical meaning of this?~
Ans. In an actual alternator, of which fig. 1,288 is an elementary
diagram showing one coil, if the current lag, say 45° behind the
pressure, it means that the coil rotates 45° from its position of zero
induction before the current starts, as in fig. 1,288.
EXAMPLE I.--A circuit through which an alternating current is
passing has an inductance of 6 ohms and a resistance of 2.5
ohms. What is the angle of lag?
[Illustration: FIG. 1,291.--Diagram of circuit for example I.]
Substituting these values in equation (1), page 1,053,
6
tan φ = --- = 2.4
2.5
Referring to the table of natural sines and tangents on page
451 the corresponding angle is approximately 67°.
EXAMPLE II.--A circuit has a resistance of 2.3 ohms and an
inductance of .0034 henry. If an alternating current having a
frequency of 125 pass through it, what is the angle of lag?
[Illustration: FIG. 1,292.--Diagram of circuit for example II.]
Here the inductance is given as a fraction of a henry; this
must be reduced to ohms by substituting in equation (3),
page 1,038, which gives the ohmic value of the inductance;
accordingly, substituting the above given value in this equation
inductance in ohms or Xᵢ = 2π × 125 × .0034 = 2.67
Substituting this result and the given resistance in equation
(1), page 1,053,
2.67
tan φ = ---- = 1.16
2.3
the nearest angle from table (page 451) is 49°.
~Ques. How great may the angle of lag be?~
Ans. Anything up to 90°.
The angle of lag, indicated by the Greek letter φ(phi), is the
angle whose tangent is equal to the quotient of the inductance
expressed in ohms or "spurious resistance" divided by the ohmic
resistance, that is
reactance 2π_f_L
tan φ = ---------- = ------ (1)
resistance R
[Illustration: FIG. 1,293.--Steam engine analogy of current flow at
zero pressure (see questions below). When the engine has reached the
dead center point the full steam pressure is acting on the piston, the
valve having opened an amount equal to its lead. The force applied at
this instant, indicated by the arrow is perpendicular to the crank
pin circle, that is, the tangential or _turning_ component is equal
to zero, hence there is no pressure tending to turn the crank. The
latter continues in motion past the dead center because of the momentum
previously acquired. Similarly, the electric current, which is here
analogous to the moving crank, continues in motion, though the pressure
at some instants be zero, because it acts as though it had weight, that
is, it cannot be stopped or started instantly.]
~Ques. When an alternating current lags behind the pressure, is there
not a considerable current at times when the pressure is zero?~
Ans. Yes; such effect is illustrated by analogy in fig. 1,293.
~Ques. What is the significance of this?~
Ans. It does not mean that current could be obtained from a circuit
that showed no pressure when tested with a suitable voltmeter, for no
current would flow under such conditions. However, in the flow of an
alternating current, the pressure varies from zero to maximum values
many times each second, and the instants of no pressure may be compared
to the "dead centers" of an engine at which points there is no pressure
to cause rotation of the crank, the crank being carried past these
points by the momentum of the fly wheel. Similarly the electric current
does not stop at the instant of no pressure because of the "momentum"
acquired at other parts of the cycle.
~Ques. On long lines having considerable inductance, how may the lag be
reduced?~
Ans. By introducing capacity into the circuit. In fact, the current may
be advanced so it will be in phase with the pressure or even lead the
latter, depending on the amount of capacity introduced.
There has been some objection to the term _lead_ as used in
describing the effect of capacity in an alternating circuit,
principally on the ground that such expressions as "lead of
current," "lead in phase," etc., tend to convey the idea that
the effect precedes the cause, that is, the current is in
advance of the pressure producing it. There can, of course, be
no current until pressure has been applied, but if the circuit
has capacity, it will lead the pressure, and this peculiar
behavior is best illustrated by a mechanical analogy as has
already been given.
~Ques. What effect has lag or lead on the value of the effective
current?~
Ans. As the angle of lag or lead increases, the value of the effective
as compared with the virtual current diminishes.
~Reactance.~--The term "reactance" means simply _reaction_. It is
used to express certain effects of the alternating current other than
that due to the ohmic resistance of the circuit. Thus, _inductance
reactance_ means the reaction due to the spurious resistance of
inductance expressed in ohms; similarly, _capacity reactance_, means
the reaction due to capacity, expressed in ohms. It should be noted
that the term _reactance_, alone, that is, unqualified, is generally
understood to mean _inductance reactance_, though ill advisedly so.
The resistance offered by a wire to the flow of a direct current is
expressed in ohms; this resistance remains constant whether the wire
be straight or coiled. If an alternating current flow through the
wire, there is in addition to the ordinary or "_ohmic_" resistance of
the wire, a "_spurious_" resistance arising from the development of
a reverse pressure due to induction, which is more or less in value
according as the wire be coiled or straight. _This spurious resistance
as distinguished from the ohmic resistance is called the reactance, and
is expressed in ohms._
Reactance, may then be defined with respect to its usual significance,
that is, _inductance reactance_, as _the component of the impedance
which when multiplied into the current, gives the wattless component of
the pressure._
Reactance is simply inductance measured in ohms.
[Illustration: FIG. 1,294.--Diagram of the circuit for example I. Here
the resistance is taken at zero, but this would not be possible in
practice, as all circuits contain more or less resistance though it may
be, in some cases, negligibly small.]
EXAMPLE I.--An alternating current having a frequency of 60 is
passed through a coil whose inductance is .5 henry. What is the
reactance?
Here _f_ = 60 and L = .5; substituting these in formula for
inductive reactance,
Xᵢ = 2π_f_L = 2 × 3.1416 × 60 × .5 = 188.5 ohms
The quantity 2π_f_L or reactance being of the same nature
as a resistance is used in the same way as a resistance.
Accordingly, since, by Ohm's law
E = RI (1)
an expression may be obtained for the volts necessary to
overcome reactance by substituting in equation (1) the value of
reactance given above, thus
E = 2π_f_LI (2)
[Illustration: FIG. 1,295.--Diagram of circuit for example II. As in
example I, resistance is disregarded.]
EXAMPLE II.--How many volts are necessary to force a current of
3 amperes with frequency 60 through a coil whose inductance is
.5 henry? Substituting in equation (2) the values here given
E = 2π_f_LI = 2π × 60 × .5 × 3 = 565 volts.
The foregoing example may serve to illustrate the difference in
behaviour of direct and alternating currents. As calculated,
it requires 565 volts to pass only 3 amperes of alternating
current through the coil on account of the considerable spurious
resistance. The ohmic resistance of a coil is very small, as
compared with the spurious resistance, say 2 ohms. Then by Ohm's law
I = E ÷ R = 565 ÷ 2 = 282.5 amperes.
Instances of this effect are commonly met with in connection
with transformers. Since the primary coil of a transformer
has a high reactance, very little current will flow when an
alternating pressure is applied. If the same transformer were
placed in a direct current circuit and the current turned on
it would at once burn out, as very little resistance would be
offered and a large current would pass through the winding.
[Illustration: FIG. 1,296.--Diagram of circuit for example III.]
EXAMPLE III.--In a circuit containing only capacity, what is
the reactance when current is supplied at a frequency of 100,
and the capacity is 50 microfarads?
1
50 microfarads = 50 × --------- = .00005 farad
1,000,000
capacity reactance, or
1 1
Xc = ------ = ------------------------- = 31.84 ohms
2π_f_C 2 × 3.1416 × 100 × .00005
~Impedance.~--This term, strictly speaking, means the _ratio of any
impressed pressure to the current which it produces in a conductor_. It
may be further defined as _the total opposition in an electric circuit
to the flow of an alternating current_.
All power circuits for alternating current are calculated with
reference to impedance. The impedance may be called the combination of:
1. Ohmic resistance;
2. Inductance reactance;
3. Capacity reactance.
The impedance of an inductive circuit which does not contain capacity
is equal to _the square root of the sum of the squares of the
resistance and reactance_, that is
_impedance_ = √(_resistance²_ + _reactance²_) (1)
[Illustration: FIG. 1,297.--Diagram showing alternating circuit
containing resistance, inductance, and capacity. Formula for
calculating the impedance of this circuit is Z = √(R² + (Xᵢ - Xc)²)
in which, Z = impedance; R = resistance; Xᵢ = inductance
reactance; Xc = capacity reactance. Example: What is the
impedance when R = 4, Xᵢ = 94.2, and Xc = 72.4? Substituting
Z = √(4² + (94.2 - 72.4)²) = 22.2 ohms. Where the ohmic values of
inductance and capacity are given as in this example, the calculation
of impedance is very simple, but when inductance and capacity are given
in milli-henrys and microfarads respectively, it is necessary to first
calculate their ohmic values as in figs. 1,295 and 1,296.]
EXAMPLE I.--If an alternating pressure of 100 volts be
impressed on a coil of wire having a resistance of 6 ohms and
inductance of 8 ohms, what is the impedance of the circuit and
how many amperes will flow through the coil? In the example
here given, 6 ohms is the resistance and 8 ohms the reactance.
Substituting these in equation (1)
Impedance = √(6² + 8²) = √(100) = 10 ohms.
The current in amperes which will flow through the coil is, by
Ohm's law using impedance in the same way as resistance.
volts 100 volts
current = --------- = --------- = 10 amperes.
impedance 10 ohms
The reactance is not always given but instead in some problems
the frequency of the current and inductance of the circuit. An
expression to fit such cases is obtained by substituting 2π_f_L
for the reactance as follows: (using symbols for impedance and
resistance)
Z = √(R² + (2π_f_L)²) (2)
[Illustration: FIG. 1,298.--Diagram of circuit for example II.]
EXAMPLE II.--If an alternating current, having a frequency of
60, be impressed on a coil whose inductance is .05 henry and
whose resistance is 6 ohms, what is the impedance?
Here R = 6; _f_ = 60, and L = .05; substituting these values in
(2)
Z = √(6² + (2π × 60 × .05)²) = √(393) = 19.8 ohms.
[Illustration: FIG. 1,299.--Diagram of circuit for example III.]
EXAMPLE III.--If an alternating current, having a frequency of
60, be impressed on a circuit whose inductance is .05 henry,
and whose capacity reactance is 10 ohms, what is the impedance?
Xᵢ = 2π_f_L = 2 × 3.1416 × 60 × .05 = 18.85 ohms
Z = Xᵢ - Xc = 18.85 - 10 = 8.85 ohms
When a circuit contains besides resistance, _both inductance and
capacity_, the formula for impedance as given in equation (1), page
1,058, must be modified to include the reactance due to capacity,
because, as explained, inductive and capacity reactances work in
opposition to each other, in the sense that the reactance of inductance
acts in direct proportion to the quantity 2π_f_L, and the reactance
of capacity in inverse proportion to the quantity 2π_f_C. The net
reactance due to both, when both are in the circuit, is obtained by
subtracting one from the other.
[Illustration: FIG. 1,300.--Diagram of circuit for example IV.]
To properly estimate impedance then, in such circuits, the following
equation is used:
_impedance_ =
√(_resistance²_ + (_inductance reactance_ - _capacity reactance_)²)
or using symbols,
Z = √(R² + (Xᵢ - Xc)²) (3)
EXAMPLE IV.--A current has a frequency of 100. It passes
through a circuit of 4 ohms resistance, of 150 milli-henrys
inductance, and of 22 microfarads capacity. What is the
impedance?
_a. The ohmic resistance_ R, is 4 ohms.
_b. The inductance reactance_, or
Xᵢ = 2π_f_L = 2 × 3.1416 × 100 × .15 = 94.3 ohms.
(note that 150 milli-henrys are reduced to .15 henry before
substituting in the above equation).
[Illustration: FIG. 1,301.--Simple choking coil. There is an important
difference in the obstruction offered to an alternating current
by ordinary resistance and by reactance. Resistance obstructs the
current by dissipating its energy, which is converted into heat.
Reactance obstructs the current by setting up a reverse pressure,
and so reduces the current in the circuit, _without wasting much
energy_, except by hysteresis in any iron magnetized. This may be
regarded as one of the advantages of alternating over direct current,
for, by introducing reactance into a circuit, the current may be cut
down with comparatively little loss of energy. This is generally
done by increasing the inductance in a circuit, by means of a device
called variously a _reactance coil, impedance coil, choking coil_,
or "_choker_." In the figure is a coil of thick wire provided with a
laminated iron core, which may be either fixed or movable. In the first
case, the inductance, and therefore also the reactance of the coil, is
invariable, with a given frequency. In the second case, the inductance
and consequent reactance may be respectively increased or diminished by
inserting the core farther within the coil or by withdrawing it, as was
done in fig. 1,266, the results of which are shown in fig. 1,302.]
[Illustration: FIG. 1,302.--Impedance curve for coil with variable iron
core. The impedance of an inductive coil may be increased by moving
an iron wire core into the coil. In making a test of this kind, the
current should be kept constant with an adjustable resistance, and
voltmeter readings taken, first without the iron core, and again with
1, 2, 3, 4, etc., inches of core inserted in the coil. By plotting the
voltmeter readings and the positions of the iron core on section paper
as above, the effect of inductance is clearly shown.]
_c. The capacity reactance_, or
1 1
Xc = ------ = -------------------------- = 72.4 ohms
2π_f_C 2 × 3.1416 × 100 × .000022
(note that 22 microfarads are reduced to .000022 farad before
substituting in the formula. Why? See page 1,042).
Substituting values as calculated in equation (3), page 1,060.
Z = √(4² + (94.2 - 72.4)²) = √(491) = 22.2 ohms.
[Illustration: FIG. 1,303.--Diagram of a resonant circuit. A
circuit is said to be resonant when the inductance and capacity
are in such proportion that the one neutralizes the other, the
circuit then acting as though it contained only resistance. In the
above circuit Xᵢ = 2π_f_L = 2 × 3.1416 × 100 × .01 = 6.28 ohms;
Xc = 1 ÷ (2 × 3.1416 × 100 × .000253) = 6.28 ohms whence the resultant
reactance = Xᵢ - Xc = 6.28 - 6.28 = 0 ohms. Z = √(R² + (Xᵢ - Xc)²) =
√(7² + 0²) = 7 ohms.]
~Ques. Why is capacity reactance given a negative sign?~
Ans. Because it reacts in opposition to inductance, that is it tends to
reduce the spurious resistance due to inductance.
In circuits having both inductance and capacity, the tangent of
the angle of lag or lead as the case may be is the algebraic
sum of the two reactances divided by resistance. If the sign be
positive, it is an angle of lag; if negative, of lead.
~Resonance.~--The effects of inductance and capacity, as already
explained, oppose each other. If inductance and capacity be present in
a circuit in such proportion that the effect of one neutralizes that of
the other, the circuit acts as though it were purely non-inductive and
is said to be in a state of _resonance_.
For instance, in a circuit containing resistance, inductance,
and capacity, if the resistance be, say, 8 ohms, the inductance
30, and the capacity 30, then the impedance is
√(8² + (30² - 30²)) = √(8²) = 8 ohms.
[Illustration: FIG. 1,304.--Application of a choking coil to a lighting
circuit. The coil is divided into sections with leads running to
contacts similar to a rheostat. Each lamp is provided with an automatic
short-circuiting cutout, and should one, two, or more of them fail, a
corresponding number of sections of the choking apparatus is put in
circuit to take the place of the broken lamp or lamps, and thus keep
the current constant. It must not be supposed that this arrangement of
lamps, etc. is a general one; it being adopted to suit certain special
conditions.]
The formula for inductance reactance is Xᵢ = 2π_f_L, and for capacity
reactance, Xc = 1 ÷ (2π_f_C); accordingly if capacity and inductance in
a circuit be equal, that is, if the circuit be resonant
1
2π_f_L = ------ (1)
2π_f_C
from which
1
_f_ = ------- (2)
2π√(CL)
~Ques. What does equation (1) show?~
Ans. It indicates that by varying the frequency in the proper way as by
increasing or decreasing the speed of the alternator, the circuit may
be made resonant, this condition being obtained when the frequency has
the value indicated by equation (2).
~Ques. What is the mutual effect of inductance and capacity?~
Ans. One tends to neutralize the other.
~Ques. What effect has resonance on the current?~
Ans. It brings the current in phase with the impressed pressure.
[Illustration: FIG. 1,305.--Curve showing variation of current by
increasing the frequency in a circuit having inductance and capacity.
The curve serves to illustrate the "critical frequency" or frequency
producing the maximum current. The curve is obtained by plotting
current values corresponding to different frequencies, the pressure
being kept constant.]
It is very seldom that a circuit is thus balanced unless
intentionally brought about; when this condition exists, the
effect is very marked, the pressure rising excessively and
bringing great strain upon the insulation of the circuit.
~Ques. Define "critical frequency."~
Ans. In bringing a circuit to a state of resonance by increasing the
frequency, the current will increase with increasing frequency until
the critical frequency is reached, and then the current will decrease
in value for further increase of frequency. The critical frequency
occurs when the circuit reaches the condition of resonance.
~Ques. How is the value of the current at the critical frequency
determined?~
Ans. By the resistance of the circuit.
~Skin Effect.~--This is the tendency of alternating currents to avoid
the central portions of solid conductors and to flow or pass mostly
through the outer portions. The so-called skin effect becomes more
pronounced as the frequency is increased.
[Illustration: FIG. 1,306.--Section of conductor illustrating "skin
effect" or tendency of the alternating current to distribute itself
unequally through the cross section of the conductor as shown by the
varied shading flowing most strongly in the outer portions of the
conductor. For this reason it has been proposed to use hollow or flat
conductors instead of solid round wires. However with frequency not
exceeding 100 the skin effect is negligibly small in copper conductors
of the sizes usually employed. Where the conductor is large or the
frequency high the effect may be judged by the following examples
calculated by Professor J. J. Thomson: In the case of a copper
conductor exposed to an electromotive force making 100 periods per
second at 1 centimetre from the surface, the maximum current would be
only .208 times that at the surface; at a depth of 2 centimetres it
would be only .043; and at a depth of 4 centimetres less than .002 part
of the value at the surface. If the frequency be a million per second
the current at a depth of 1 millimetre is less than one six-millionth
part of its surface value. The case of an iron conductor is even more
remarkable. Taking the permeability at 100 and the frequency at 100
per second the current at a depth of 1 millimetre is only .13 times
the surface value; while at a depth of 5 millimetres it is less than
one twenty-thousandth part of its surface value. The disturbance of
current density may be looked upon as a self-induced eddy current in
the conductor. It necessarily results in an increase of ohmic loss;
as compared with a steady current: proportional to the square of the
total current flowing and consequently gives rise to an apparent
increase of ohmic resistance. The coefficient of increase of resistance
depends upon the dimensions and the shape of the cross section, the
frequency and the specific resistance. A similar but distinct effect is
experienced in conductors due to the neighborhood of similar parallel
currents. For example in a heavy multicore cable the non-uniformity
of current density in any core may be considered as partly due to
eddy currents induced by the currents in the neighboring cores and
partly to the self-induced eddy current. It is only the latter effect
which should rightly be considered as comprised under the term _skin
effect_.]
~Ques. What is the explanation of skin effect?~
Ans. It is due to eddy currents induced in the conductor.
Consider the wire as being composed of several small insulated
wires placed closely together. Now when a current is started
along these separate wires, mutual induction will take place
between them, giving rise to momentary reverse pressures. Those
wires which are nearer the center, since they are completely
surrounded by neighboring wires, will clearly have stronger
reverse pressures set up in them than those on or near the
outer surface, so that the current will meet less opposition
near the surface than at the center, and consequently the flow
will be greater in the outer portions.
~Ques. What is the result of skin effect?~
Ans. It results in an apparent increase of resistance.
The coefficient of increase of resistance depends upon the
dimensions and the shape of the cross section, the frequency,
and the specific resistance.
Hughes, about 1883, called attention to the fact that the
resistance of an iron telegraph wire was greater for rapid
periodic currents than for steady currents.
In 1888 Kelvin showed that when alternating currents at
moderately high frequency flow through massive conductors,
the current is practically confined to the skin, the interior
portions being largely useless for the purpose of conduction.
The mathematical theory of the subject has been developed by
Kelvin, Heaviside, Rayleigh, and others.