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    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.




CHAPTER XLVII

~ALTERNATING CURRENT DIAGRAMS~


Whenever an alternating pressure is impressed on a circuit, part of it
is spent in overcoming the resistance, and the rest goes to balance the
reverse pressure due to self-induction.

The total pressure applied to the circuit is known as the _impressed
pressure_, as distinguished from that portion of it called the _active
pressure_ which is used to overcome the resistance, and that portion
called the _self-induction pressure_ used to balance the reverse
pressure of self-induction.

The intensity of the reverse pressure induced in a circuit due to
self-induction is proportional to the _rate of change in the current
strength_.

    Thus a current, changing at the rate of one ampere per
    second, in flowing through a coil having a coefficient of
    self-induction of one henry, will induce a reverse pressure of
    one volt.

~Ques. Describe how the rate of change in current strength varies, and
how this affects the reverse pressure.~

Ans. The alternating current varies from zero to maximum strength
in one-quarter period, that is, in one-quarter revolution of the
generating loop or 90° as represented by the sine curve in fig. 1,307.
Now, during, say, the first 10 degrees of rotation (from 0 to A), the
current jumps from zero value to A', or 4 amperes, according to the
scale; during some intermediate 10 degrees of the quarter revolution,
as from B to C, the current increases from B' to C' or 2½ amperes, and
during another 10 degrees as from D to E, at the end of one-quarter
revolution where the sine curve reaches its amplitude, it rises and
falls ½ ampere. It is thus seen that the _rate of change_ varies from
a maximum when the current is least, to zero when the current is at
its maximum. Accordingly, the reverse pressure of self-induction
_being proportional to the rate of change in the current strength_, is
greatest when the current is at zero value, and zero when the current
is at its maximum.

[Illustration: FIG. 1,307.--Sine curve showing that the _rate of
change_ in the strength of an alternating current _is greatest when
the current is least, and zero when the current is at a maximum_. This
is evident from the diagram, since during say the first 10° as OA,
the current increases 4 amperes; during BC, 2½ amperes; during DE it
rises and falls ½ ampere. The reverse pressure of self-induction being
proportional to the _rate of change_ of the current, is a maximum when
the current is zero, and zero when the current is a maximum, _giving a
phase difference of 90° between reverse pressure of self-induction and
current_.]

    This relation is shown by curves in fig. 1,308, and it should
    be noted that _the reverse pressure and current are 90° apart_
    in phase. For this reason many alternating current problems
    may be solved graphically by the use of right angle triangles,
    the sides, drawn to some arbitrary scale, to represent the
    quantities involved, such as resistance, reactance, impedance,
    etc.

~Properties of Right Angle Triangles.~--In order to understand the
graphical method of solving alternating current problems, it is
necessary to know why certain relations exist between the sides of a
right angle triangle. For instance, in every right angle triangle:

_The square of the hypothenuse is equal to the sum of the squares on
the other two sides._

That is, condensing this statement into the form of an equation:

             _hypothenuse² = base² + altitude²_                      (1)

the horizontal side being called the base and the vertical side, the
altitude.

This may be called the equation of the right angle triangle.

[Illustration: FIG. 1,308.--Sine curves showing phase relation between
current and reverse pressure of self-induction. This reverse pressure,
_being proportional to the rate of change in the current strength, is
greatest when the current is at zero value, and zero when the current
is maximum_, and in phase is 90° behind the current.]

[Illustration: FIG. 1,309.--_In a right angle triangle the square
on the hypothenuse is equal to the sum of the squares on the other
two sides._ That is: _hypothenuse² = base² + altitude²_. Draw AB, 4
inches long, and BC, 3 inches long and at right angles to AB. Join AC,
which will be found to be 5 inches long. From the diagram, it must be
clear that the square on AC = sum of squares on AB and BC; that is,
5² = 4² + 3². Further, 4² = 5² - 3²; 3² = 5² - 4²; 5 = √(4² + 3²);
4 = √(5² - 3²); 3 = √(5² - 4²).]

~Ques. Why is the square of the hypothenuse of a right angle triangle
equal to the sum of the squares of the other two sides?~

Ans. This may be explained with the aid of fig. 1,309. Draw a line AB,
4 inches in length and erect a perpendicular BC, 3 inches in height;
connect A and C, giving the right angle triangle ABC. It will be found
that AC the hypothenuse of this triangle is 5 inches long. If squares
be constructed on all three sides of the triangle, the square on the
hypothenuse will have an area of 25 sq. ins.; the square on the base,
16 sq. ins., and the square on the altitude, 9 sq. ins. Then from the
figure 5² = 4² + 3², that is 25 = 16 + 9.

    Repeating equation (1), it is evident from the figure that

  _hypothenuse²_ }   {_base²_ + _altitude²_}
                 } = {                     }
        5²       }   {     4² + 3²         }

    that is,

                              25 = 16 + 9.


In the right angle triangle, the following relations also hold:

         _base²_ = _hypothenuse²_ - _altitude²_                      (2)

  (4² = 5² - 3²)

_altitude²_ = _hypothenuse²_ - _base²_                               (3)

                             (3² = 5² - 4²)

In working impedance problems, it is not the square of any of the
quantities which the sides of the triangle are used to represent that
is required, but the quantities themselves, that is, the sides. Hence
extracting the square root in equations (1), (2) and (3), the following
are obtained:

        _hypothenuse_ = √(_base²_ + _altitude²_)                     (4)

                   (5 = √(4² + 3²))

               _base_ = √(_hypothenuse²_ - _altitude²_)              (5)

                   (4 = √(5² - 3²))

           _altitude_ = √(_hypothenuse²_ - _base²_)                  (6)

                   (3 = √(5² - 4²))

~Representation of Forces by Lines.~--A single force may be represented
in a drawing by a straight line, 1, the point of application of the
force being indicated by an extremity of the line, 2, the intensity
of the force by the length of the line, and 3, the direction of the
force by the direction of the line, an arrow head being placed at an
extremity defining the direction.

    Thus in fig. 1,310, the force necessary to balance the thrust
    on the steam piston may be represented by the straight line
    _f_ whose length measured on any convenient scale represents
    the intensity of the force, and whose direction represents the
    direction of the force.

[Illustration: FIG. 1,310.--Diagram illustrating the representation of
forces by straight lines. If 80 lbs. of steam be applied to a piston
of 4 square inches area, the total pressure acting on the piston is
4 × 80 = 320 lbs. This may be balanced by an equal and opposite force.
To represent the latter by a line, select any convenient scale whose
divisions represent any convenient number of pounds--1, 3, 5 or, as
here taken, 25 lbs. If the scale selected be divided into inches with
¼-inch divisions, then each ¼ inch represents a force of 25 lbs.; or,
as usually stated, 1" = 100 lbs. Strictly speaking 1" is equivalent to
100 lbs. Draw the line _f_ = 3.2 ins., then its length represents the
magnitude of the force or 320 lbs., that is, 3.2 × 100 = 320 lbs.]

~Composition of Forces.~--This is the operation of finding _a single
force whose effect is the same as the combined effect of two or more
given forces_. The required force is called the _resultant_ of the
given forces.

The composition of forces may be illustrated by the effect of the wind
and tide on a sailboat as in fig. 1,311. Supposing the boat be acted
upon by the wind so that in a given time, say half an hour, it would
be moved in the direction and a distance represented by the line AB,
and that in the same time the tide would carry it from A to C. Now, lay
down AB, to any convenient scale, representing the effect of the wind,
and AC that of the tide, and draw BD equal and parallel to AC, and
CD equal and parallel to AB, then the diagonal AD will represent the
direction and distance the boat will move under the combined effect of
wind and tide.

[Illustration: FIG. 1,311.--Parallelogram of forces for boat acted upon
by both wind and tide.]

~Ques. In fig. 1,311 what is the line AD called?~

Ans. The _resultant_, that is, it represents the actual movement of the
boat resulting from the combined forces of wind and tide.

~Ques. What are the forces, AB and AD in fig. 1,311, represented by the
sides of the parallelogram, and which act upon a body to produce the
resultant, called?~

Ans. The _components_.

[Illustration: FIG. 1,312.--Parallelogram of forces; method of
obtaining the resultant of two components acting at right angles.]

    EXAMPLE.--Two forces, one of 3 lbs. and one of 4 lbs. act at a
    point _a_ in a body and at right angles, what is the resultant?

    Take any convenient scale, say 1 in. = 1 lb., and lay off
    (fig. 1312.) AB = 4 ins. = 4 lbs.; also, AC (at right angles
    to AB) = 3 ins. = 3 lbs. Draw CD and BD parallel to AB and AC
    respectively, and join AD. The line AD is the resultant of the
    components AB and AC, and when measured on the same scale from
    which AB and AC were drawn will be found to be 5 inches long,
    which represents 5 lbs. acting in the direction AD.

~Circuits containing Resistance and Inductance.~--In circuits of this
kind where the impressed pressure encounters both resistance and
inductance, it may be looked upon as split up into two components,
as already explained, one of which is necessary to overcome the
resistance, and the other, the inductance. That is, the impressed
pressure is split up into

  1. _Active pressure_, to overcome resistance;
  2. _Self-induction pressure_ to overcome inductance.

[Illustration: FIG. 1,313.--Diagram illustrating the _active_, and
_self-induction_ pressures, or the two components of the impressed
pressure in circuits containing resistance and inductance. The active
pressure is the volts required to overcome the resistance of the
circuit. In the figure only the portion from A to C is considered as
having resistance (the rest being negligibly small) except at R, a
resistance equivalent to that of the inductive coil is inserted next
to the non-inductive coil, so Pₐ will give the total "ohmic drop"
or active pressure, that is, the pressure necessary to force any
equivalent direct current from A to C. This active pressure Pₐ or
component of the impressed pressure is in phase with the current. The
other component or self-induction pressure Pᵢ that is the reactance
drop necessary to overcome the reverse pressure of self-induction
and is at right angles to the current and 90° ahead of the current
in phase. It is registered by a voltmeter between B and C, less the
pressure due to ohmic resistance of the inductive coil. The impressed
pressure Pᵢₘ then or total pressure required to force electricity
around the circuit _not including the resistance_ R, (which is removed
from the circuit when the reading of the impressed pressure is taken),
is equal to the square root of the sum of the squares of the two
components, that is, Pᵢₘ = √(Pₐ² + Pᵢ²).]

    The active pressure is _in phase with the current_.

    The self induction pressure is _at right angles to the current
    and 90 degrees ahead of the current in phase_.

~Ques. Why is the active pressure in phase with the current?~

Ans. The pressure used in overcoming resistance is from Ohm's law,
E = RI. Hence, when the current is zero, E is zero, and when the
current is a maximum E is a maximum. Hence, that component of the
impressed pressure necessary to overcome the resistance must be _in
phase with the current_.

~Ques. Why is this?~

Ans. Since the _reverse pressure of self induction_ is 90° behind the
current, the component of the impressed pressure necessary to overcome
the reverse pressure of self induction, being opposite to this, will be
represented as being 90° ahead of the current.

    The distinction between the reverse pressure of self-induction,
    that is, the induced pressure, and the pressure necessary to
    overcome self-induction should be carefully noted. They are two
    equal and opposite forces, that is, two balancing forces just
    as is shown in fig. 1,310. Here, in analogy, the thrust of the
    piston may represent the induced pressure and the equal and
    opposite force indicated by the arrow _f_, the component of the
    impressed pressure necessary to balance the induced pressure.

[Illustration: FIG. 1,314.--Graphical method of obtaining the impressed
pressure in circuits containing resistance and inductance, having
given the ohmic drop, and reactance drop due to inductance. With any
convenient scale lay off AB = ohmic drop and erect the perpendicular BC
= reactance drop (using same scale). Join AC, whose length (measured
with same scale) will give the impressed pressure. Constructing a
parallelogram with dotted lines AD and CD, it is evident that AC is the
_resultant_ of the two _components_ AB and BC, or its equal AD.]

    ~The Active Pressure or "Ohmic Drop."~--The component of the
    impressed pressure necessary to overcome resistance, is from
    Ohm's law:

         _active pressure = ohmic resistance × virtual current_

    that is

                        ~Eₐ = RₒIᵥ          (1)~

    this is the "ohmic drop" and may be represented by a line AB,
    fig. 1,314 drawn to any convenient scale, as for instance, 1
    in. = 10 volts.

    ~The Self-induction Pressure or "Reactance Drop."~--The
    component of the impressed pressure necessary to overcome the
    induced pressure, is from Ohm's law:

    _inductance pressure = inductance reactance × virtual current;_

    that is,

                       Eᵢ = XᵢIᵥ                                     (2)


[Illustration: FIG. 1,315.--Diagram for impressed pressure on circuit
containing 5 volts ohmic drop and 15 volts reactance drop.]

    Now the reactance Xᵢ, that is the spurious resistance, is
    obtained from the formula

                   Xᵢ = 2π_f_L                                       (3)

    as explained on page 1,038, and in order to obtain the volts
    necessary to overcome this spurious resistance, that is, the
    "reactance drop" as it is called, the value of Xᵢ in equation
    (3) must be substituted in equation (2), giving

                   ~Eᵢ = 2π_f_LI                (4)~

    writing simply ~I~ for the virtual pressure.

    Since the pressure impressed on a circuit is considered as made
    up of two components, one in phase with the current and one at
    right angles to the current, the component Eᵢ or "reactance
    drop" as given in equation (4) maybe represented by the line
    BC in fig. 1,314, at right angles to AB, and of a length BC,
    measured with the same scale as was measured AB, to correspond
    to the value indicated by equation (4).

    EXAMPLE.--In an alternating circuit, having an ohmic drop of 5
    volts, and a reactance drop of 15 volts, what is the impressed
    pressure?

    With a scale of say, ¼ inch = one volt, lay off, in fig.
    1,315, AB = 5 volts = 1¼ in., and, at right angles to it,
    BC = 15 volts = ¹⁵/₄ or 3¾ ins. Join AC; this measures 3.95
    inches, which is equivalent to 3.95 × 4 = 15.8 volts, the
    impressed pressure. By using good paper, such as bristol board,
    a 6H pencil, engineers' scale and triangles or square, such
    problems are solved with precision. By calculation impressed
    pressure = √(5² + 15²) = 15.8 volts. Note that the diagram is
    drawn with the side BC horizontal instead of AB--simply to save
    space.

[Illustration: FIG. 1,316.--Diagram of circuit containing 5 volts ohmic
drop, and 15 volts reactance drop.]

[Illustration: FIG. 1,317.--Diagram for obtaining reactance drop
in circuit containing 5 volts ohmic drop, and 15.8 volts impressed
pressure.]

    EXAMPLE.--In an alternating circuit, having an ohmic drop of
    5 volts and an impressed pressure of 15.8 volts, what is the
    reactance drop?

    In fig. 1,317, draw a horizontal line of indefinite length and
    at any point B erect a perpendicular AB = 5 volts. With A as
    center and radius of length equivalent to 15.8 volts, describe
    an arc cutting the horizontal line at C. This gives BC, the
    reactance drop required, which by measurement is 15 volts.

    [Illustration: FIG. 1,318.--Diagram for obtaining ohmic drop in
    the circuit fig. 1,316 when impressed pressure and reactance
    drop are given. Lay off BC to scale = reactance drop; draw
    AB at right angle and of indefinite length; with C as center
    and radius of length = impressed pressure, describe an arc
    cutting ohmic drop line at A, then AB = ohmic drop = 5 volts by
    measurement.]

    [Illustration: FIG. 1,319.--Graphical method of finding angle
    of lag when the ohmic drop and reactance drop are given. The
    angle of lag φ, is that angle included between the impressed
    pressure and the ohmic drop lines, that is, between AC and AB.]

    EXAMPLE.--An alternating current of 10 amperes having a
    frequency of 60, is impressed on a circuit containing a
    resistance of 5 ohms and an inductance of 15 milli-henrys. What
    is the impressed pressure?

    The active pressure or ohmic drop is 5 × 10 = 50 volts.

[Illustration: FIG. 1,320.--Diagram of circuit containing 5 ohms
resistance, 15 milli-henrys inductance, with 3 ampere 60 frequency
current.]

    The inductance reactance or Xᵢ = is
    2 × 3.1416 × 60 × .015 = 5.66 ohms. Substituting this and
    the current value 10 amperes in the formula for inductance
    pressure or reactance drop (equation 2 on page 1,077) gives
    Eᵢ = 5.65 × 10 = 56.5 volts.

[Illustration: FIG. 1,321.--Diagram for impressed pressure on circuit
containing 5 ohms resistance and inductance of 15 milli-henrys, the
current being 10 amperes with frequency of 60.]

    In fig. 1,321, lay off AB = 50 volts, and BC = 56.6 volts.
    Using a scale of 20 volts to the inch gives AB = 2.5 ins., and
    BC = 2.83 ins. Joining AC gives the impressed voltage, which by
    measurement is 75.4 volts.

In some problems it is required to find the impedance of a circuit in
which the ohmic and spurious resistances are given. This is done in a
manner similar to finding the impressed pressure.

[Illustration: FIG. 1,322.--Graphical method of obtaining the impedance
in circuits containing resistance and inductance, having given the
resistance and reactance, that is, the ohmic resistance and spurious
resistance. With any convenient scale lay off AB = _resistance_, and
erect the perpendicular BC = _reactance_ (using the same scale);
join AC, whose length (measured with the same scale) will give the
_impedance_.]

Ohmic resistance and spurious resistance or inductance reactance
both tend to reduce an alternating current. Their combined action or
impedance is equal to the square root of the sum of their squares, that
is,

             _impedance_ = √(_resistance²_ + _reactance²_)

This relation is represented graphically by the side of a right angle
triangle as in fig. 1322, in which the hypothenuse corresponds to the
impedance, and the sides to the resistance and reactance.

    EXAMPLE.--In a certain circuit the resistance is 4 ohms, and
    the reactance 3 ohms. What is the impedance?

    In fig. 1,323, lay off, on any scale AB = 4 ohms and erect the
    perpendicular BC = 3 ohms. Join AC, which gives the impedance,
    and which is, measured with the same scale, 5 ohms.

    EXAMPLE.--A coil of wire has a resistance of 20 ohms and an
    inductance of 15 milli-henrys. What is its impedance for a
    current having a frequency of 100?

[Illustration: FIG. 1,323.--Diagram for obtaining the impedance of a
circuit containing 4 ohms resistance and 3 ohms reactance.]

    The ohmic value of the inductance, that is, the reactance is

             2π_f_L = 2 × 3.1416 × 100 × .015 = 9.42 ohms.

    In fig. 1,324, lay off, on any scale, AB = 20 ohms, and the
    perpendicular BC to length = 9.42 ohms. Join AC, which gives
    the impedance, which is, measured on the same scale, 22.1 ohms.

    EXAMPLE.--What is the angle of lag in a circuit having a
    resistance of 4 ohms and a reactance of 3 ohms?

    Construct the impedance diagram in the usual way as in fig.
    1,325, then the angle included between the impedance and
    resistance lines (denoted by φ) is the angle of lag, that
    is, the angle BAC. By measurement with a protractor it is 37
    degrees. By calculation the tangent of the angle of lag or

          BC   3
  tan φ = -- = - or .75
          AB   4

    From the table on page 451, the angle is approximately 37°.

[Illustration: FIG. 1,324.--Diagram for impedance of circuit containing
20 ohms resistance, and inductance of 15 milli-henrys, when the
frequency is 100.]

[Illustration: FIG. 1,325.--Diagram showing angle of lag for current
containing 4 ohms resistance and 3 ohms reactance.]

~Circuits containing Resistance and Capacity.~--The effect of capacity
in an alternating current circuit is to cause the current to lead the
pressure, since the reaction of a condenser, instead of tending to
prolong the current, tends to drive it back.

Careful distinction should be made between capacity _in series_ with
a circuit and capacity _in parallel_ with a branch of a circuit. The
discussion here refers to capacity in series, which means that the
circuit is not continuous but the ends are joined to a condenser, as
shown at the right in fig. 1,326, so that no current can flow except
into and out of the condenser.

[Illustration: FIG. 1,326,--Circuit diagram illustrating the
distinction between _capacity in series_ and _capacity in parallel_.
The condition for _capacity in series_ is that the circuit must be
discontinuous as at M; for _capacity in parallel_ the main circuit
must be continuous; this means that the capacity must be inserted in
a branch of the main circuit as at A. In the figure the capacity S is
connected _in series_ with respect to the branch, that is, the branch
is discontinuous, but it is _in parallel_ with respect to the main
circuit, when the latter is continuous, that is, when the switch W is
closed. If W be opened, the main circuit becomes discontinuous and S is
changed from _in parallel_ to _in series_ connection.]

~Ques. In circuits containing resistance and capacity upon what does
the amount of lead depend?~

Ans. Upon the relative values of the resistance and the capacity
reactance.

~Ques. Describe the action of a condenser when current is applied.~

Ans. When the current begins to flow into a condenser, that is, when
the flow is maximum, the back pressure set up by the condenser (called
the _condenser pressure_) is zero, and when the flow finally becomes
zero, the condenser pressure is a maximum.

[Illustration: FIG. 1,327.--Current and pressure curves showing that
the condenser pressure is 90° ahead of the current. A current flowing
into a condenser encounters a gradually increasing pressure which
opposes it, beginning from zero pressure when the current enters at
maximum flow and increasing to the same value as the current pressure,
at which time the current ceases to flow. Hence, since the current
varies from zero to maximum in one quarter period, or 90°, the phase
difference between current and condenser pressure is 90°. The condenser
pressure reaching a positive maximum when the current starts from zero
on the positive wave, is 90° ahead of the current.]

~Ques. What does this indicate?~

Ans. It shows that the phase difference between the wave representing
the condenser pressure and the current is 90°, as illustrated in fig.
1,327.

~Ques. Is the condenser pressure ahead or behind the current and why?~

Ans. It is ahead of the current. The condenser pressure, when the
condenser is discharged being zero, the current enters at maximum
velocity as at A in fig. 1,327, and gradually decreases to zero as the
condenser pressure rises to maximum at B, this change taking place in
one-quarter period. Thus the condenser pressure, which opposes the
current, being at a maximum when the current begins its cycle is 90°
_ahead of the current_, as is more clearly seen in the last quarter of
the cycle (fig. 1,327).

[Illustration: FIG. 1,328.--Current and pressure curves, showing phase
relation between the current, condenser pressure, and impressed or
_capacity_ pressure necessary to overcome the condenser pressure. The
capacity pressure, since it must overcome the condenser pressure,
is equal and opposite to the condenser pressure, that is, the phase
difference is 180°. The condenser pressure being 90° _ahead_ of the
current, the impressed pressure is 90° _behind_ the current.]

~Ques. What is the phase relation between the condenser pressure
and the pressure applied to the condenser to overcome the condenser
pressure?~

Ans. The pressure applied to the condenser to overcome the condenser
pressure, or as it is called, the _capacity pressure_, must be opposite
to the condenser pressure, or 90° _behind the current_.

In circuits containing resistance and capacity, the total pressure
impressed on the circuit, or _impressed pressure_, as it is called, is
made up of two components:

1. The _active pressure_, or pressure necessary to overcome the
resistance;

    The active pressure is in phase with the current.

2. The _capacity pressure_, or pressure ~necessary~ to overcome the
condenser pressure,

    The capacity pressure is 90 degrees behind the current.

[Illustration: FIG. 1,329.--Graphical method of obtaining the impressed
pressure in circuits containing resistance and capacity, having given
the ohmic drop and reactance drop due to capacity. With any convenient
scale, lay off AB = _ohmic drop_, and at right angles to AB draw
BC = _reactance drop_ (using the same scale). Join AC, whose length
(measured with the same scale) will give the _impressed pressure_. The
mathematical expressions for the three quantities are given inside the
triangle, and explained in the text.]

Problems involving resistance and capacity are solved similarly to
those including resistance and inductance.

    ~The Active Pressure or "Ohmic Drop."~--This, as before
    explained is represented, in fig. 1,329, by a line AB, which in
    magnitude equals, by Ohm's law, the product of the resistance
    multiplied by the current, that is,

                           Eₐ = RₒIᵥ                                 (1)


[Illustration: FIG. 1,330.--Diagram of circuit containing a resistance
of 30 ohms and capacity of 125 microfarads. The calculation for
impressed pressure, ohmic drop, and reactance drop for a current of
8 amperes at frequency 60 is given in the example on page 1,089, the
diagram for impressed pressure being given in fig. 1,331.]

    ~The Capacity Pressure or "Reactance Drop."~--This component of
    the impressed pressure, is, applying Ohm's law,

    _capacity pressure_ = _capacity reactance_ × _virtual current_.

                           Ec = XcIᵥ                                 (2)

    That is, the expression for capacity reactance Xc, that is, for
    the value of capacity in ohms is, as explained on page 1,048,

         1
  Xc = ------                                                        (3)
       2π_f_C

    Substituting this value of Xc in equation (2) and writing I for
    virtual current.

         ~I
  Ec = -------                                                       (4)
       2π_f_C~

    CAUTION--The reader should distinguish between the 1 (one) in
    (3) and the letter I in (4); both look alike.

    Since the capacity pressure is 90° _behind_ the current, it
    is represented in fig. 1,329, by a line BC, drawn _downward_,
    at right angles to AB, and of a length corresponding to the
    capacity pressure, that is, to the reactance drop.

    ~The Impressed Pressure.~--Having determined the ohmic and
    reactance drops and represented them in the diagram, fig.
    1,329, by lines AB and BC respectively, a line AC joining A and
    C, will then be the resultant of the two component pressures,
    that is, it will represent the _impressed pressure_ or total
    pressure applied to the circuit.

    In the diagram it should be noted that the active pressure
    is called the _ohmic drop_, and the capacity pressure, the
    _reactance drop_.

    EXAMPLE.--A circuit as shown in fig. 1,330 contains a
    resistance of 30 ohms, and a capacity of 125 microfarads. If an
    alternating current of 8 amperes with frequency 60 be flowing
    in the circuit, what is the ohmic drop, the reactance drop, and
    the impressed pressure?

    The ohmic drop or active pressure is, substituting in formula
    (1) on page 1,087,

                        Eₐ = 30 × 8 = 240 volts

    which is the reading of voltmeter A in fig. 1,330.

[Illustration: FIG. 1,331.--Diagram for obtaining the impressed
pressure of the circuit shown in fig. 1,330.]

    The reactance drop or

            I                8
    Ec = ------ = ------------------------- = 170 volts
         2π_f_C   2 × 3.1416 × 60 × .000125

    in substituting, note that the capacity C of 125 microfarads is
    reduced to .000125 farad.

    Using a scale of say 1 inch = 80 volts, lay off in fig. 1,331,
    AB equal to the ohmic drop of 240 volts; on this scale AB = 3
    inches. Lay off at right angles, BC = reactance drop = 170
    volts = 2.125 inches. Join AC, which gives the impressed
    voltage, (that is the reading of voltmeter I in fig. 1,330,)
    which measures 294 volts.

    By calculation, impressed pressure = √(240² + 170²) = 294
    volts.

    EXAMPLE.--In the circuit shown in fig. 1,330, what is the angle
    of lead?

    The tangent of the angle of lead is given by the quotient of
    the reactance divided by the resistance of the circuit. That is,

              _reactance_    _reactance drop_
    _tan φ_ = ------------ = -----------------
              _resistance_   _resistance drop_

            Ec     I
    tan φ = -- = ------ ÷ Eₐ                                         (1)
            Eₐ   2π_f_C

    The tangent is given a negative sign because lead is opposed
    to lag and because the positive value is assigned to lag.
    Substituting in (1)

          170    2.125"
  tan φ = --- or ------ = -.71
          240    3"

    the angle corresponding is approximately 35¼° (see table page
    451).

[Illustration: FIGS. 1,332 and 1,333.--Diagrams for circuits containing
inductance and capacity. Since inductance and capacity act 180° apart,
their reactances, or their ohmic drops may be represented by oppositely
directed lines. These may be drawn above and below a reference line, as
in fig. 1,332, and their algebraic sum taken, or both may be drawn on
the same side of the reference line and their difference in lengths,
as CD, fig. 1,333, measured. Recourse to a diagram for obtaining the
resultant reactance in circuits containing inductance and capacity is
unnecessary as it is simply a matter of taking the difference of two
quantities.]

~Circuits Containing Inductance and Capacity.~--The effect of capacity
in a circuit is exactly the opposite of inductance, that is, one tends
to neutralize the other. The method of representing each graphically
has been shown in the preceding figures. Since they act oppositely,
that is 180° apart, the reactance due to each may be calculated and
the values thus found, represented by oppositely directed vertical
lines: the inductance resistance upward from a reference line, and
the capacity resistance downward from the same reference line. The
difference then is the resultant impedance. This method is shown in
fig. 1,332, but it is more conveniently done as in fig. 1,333.

[Illustration: FIG. 1,334.--Diagram of circuit containing 30
milli-henrys inductance and 125 microfarads capacity, with current of
20 amperes, 100 frequency.]

    EXAMPLE.--In a circuit, as in fig. 1,334, containing
    an inductance of 30 milli-henrys and a capacity of 125
    microfarads, how many volts must be impressed on the circuit to
    produce a current of 20 amperes having a frequency of 100.

    The inductance reactance is

           Xᵢ = 2π_f_L = 2 × 3.1416 × 100 × .03 = 18.85 ohms.

    Substituting this and the current value of 20 amperes in the
    formula for inductance pressure

                   Eᵢ = RᵢI = 18.85 × 20 = 377 volts.

    Reducing 125 microfarads to .000125 farad, and substituting in
    the formula for capacity pressure

          I                20
  Ec = ------ = -------------------------- = 255 volts.
       2π_f_C   2 × 3.1416 × 100 × .000125

    A diagram is unnecessary in obtaining the impressed pressure
    since it is simply the difference between inductance pressure
    and capacity pressure (the circuit being assumed to have no
    resistance), that is

         impressed pressure = Eᵢ - Ec = 377 - 255 = 122 volts.

    EXAMPLE.--A circuit in which a current of 20 amperes is flowing
    at a frequency of 100, has an inductance reactance of 18.25
    ohms, and a capacity of 125 microfarads. What is the impedance?

    The reactance due to capacity is

         1                 1
  Xc = ------ = -------------------------- = 12.76 ohms.
       2π_f_C   2 × 3.1416 × 100 × .000125

    The impedance of the circuit then is the difference
    between the two reactances, that is impedance = inductance
    reactance - capacity reactance, or

                Z = Xᵢ - Xc = 18.25 - 12.76 = 5.49 ohms.


[Illustration: FIG. 1,335.--Impedance diagram for circuit (of above
example) containing inductance and capacity. With any convenient scale,
erect a perpendicular AB = 18.25 ohms, and CD = 12.76 ohms. Continue CD
by dotted line to D' so that CD' = AB, then DD' = AB - CD = inductance
reactance - capacity reactance, which is equal to the impedance.
Expressed by letters Z = Xᵢ - Xc = DD', which by measurement = 5.49
ohms.]

~Circuits Containing Resistance, Inductance, and Capacity.~--When the
three quantities resistance, inductance, and capacity, are present in
a circuit, the combined effect is easily understood by remembering
that inductance and capacity always act oppositely, that is, they
tend to neutralize each other. Hence, in problems involving the three
quantities, the resultant of inductance and capacity is first obtained,
which, together with the resistance, is used in determining the final
effect.

Capacity introduced into a circuit containing inductance reduces the
latter and if enough be introduced, inductance will be neutralized,
giving a resonant circuit which will act as though only resistance were
present.

[Illustration: FIG. 1,336.--Impedance diagram for circuit containing
resistance, inductance and capacity. The symbols correspond to those
used in equation (1) below. In constructing the diagram from the given
values, lay off AB = resistance; at B, draw a line at right angles, on
which lay off above the resistance line, BC = inductive reactance, and
below, BD = capacity reactance, then the resultant
reactance = BC - BD = BD'. Join A and D', then AD' = impedance.]

~Ques. What is the expression for impedance of a circuit containing
resistance, inductance and capacity?~

Ans. It is equal to _the square root of the sum of the resistance
squared plus the square of inductance reactance minus capacity
reactance._

    This is expressed plainer in the form of an equation as follows:

_impedance_ =
    √(_resistance²_ + (_inductance reactance_ - _capacity reactance_)²)

    or, using symbols,

                   ~Z = √(R² + (Xᵢ - Xc)²)      (1)~

~Ques. If the capacity reactance be larger than the inductance
reactance, how does this affect the sign of (Xᵢ-Xc)²?~

Ans. The sign of the resultant reactance of inductance and capacity
will be negative if capacity be the greater, but since in the formula
the reactance is squared, the sign will be positive.

[Illustration: FIG. 1,337.--Impedance diagram of a circuit containing
25 ohms resistance, 30 ohms inductance, and 40 ohms capacity. The
resultant reactance being due to excess of capacity, the impedance line
AC' falls _below_ the horizontal line AB, indicating that the current
_leads_ pressure.]

    EXAMPLE.--What is the impedance in a circuit having 25 ohms
    resistance, 30 ohms inductance reactance, and 40 ohms capacity
    reactance?

    To solve this problem graphically, draw the line AB, in fig.
    1,337, equal to 25 ohms resistance, using any convenient scale.

    At B draw upward at right angles BC = 30 ohms; draw from
    C downward CC' = 40 ohms. This gives -BC' (= BC - CC')
    showing the capacity reactance to be 10 ohms in excess of
    the inductance reactance. Such a circuit is equivalent to
    one having no inductance but the same resistance and 10 ohms
    capacity reactance.

    The diagram is completed in the usual way by joining AC giving
    the required impedance, which by measurement is 26.9 ohms.

    By calculation, Z = √(25² + (30 - 40)²) = √(25² + (-10)²) = 26.9.

~Form of Impedance Equation without Ohmic Values.--~

Using the expressions 2π_f_L for inductance reactance and 1 / (2π_f_C)
for capacity reactance, and substituting in equation (1) on page 1,093
gives the following:

             Z = √(R² + (2π_f_L - 1 / (2π_f_C))²)                    (2)

which is the proper form of equation (1) to use in solving problems in
which the ohmic values of inductance and capacity must be calculated.

[Illustration: FIG. 1,338.--EXAMPLE: A resistance of 20 ohms and
an inductance of .02 henry are connected in parallel as in the
diagram. What is the impedance, and how many volts are required
for 50 amperes, when the frequency is 78.6? SOLUTION: The time
constants are not alike, hence the geometric sum of the reciprocals
must be taken as the reciprocal of the required impedance. That
is, the combined conductivity will be the hypothenuse of the
right triangle, of which the ohmic conductivity and the reactive
conductivity are the two sides, respectively. Accordingly:
1 / R = 1 / 20 = .05, and 1 / (2π_f_L) = 1 / 10 = .1, from which,
1 / R = √((1 / R₁)² + (1 / (2π_f_L))²) = .111. Whence Z = 1 / .111 = 9
ohms.]

[Illustration: FIG. 1,339.--Diagram of circuit containing 23 ohms
resistance, 41 milli-henrys inductance, and 51 microfarads capacity,
with current supplied at a frequency of 150.]

    EXAMPLE.--A current has a frequency of 150. It passes through
    a circuit, as in fig. 1,339, of 23 ohms resistance, of 41
    milli-henrys inductance, and of 51 microfarads capacity. What
    is the impedance?

    The inductance reactance or

           Xᵢ = 2π_f_L = 2 × 3.1416 × 150 × .041 = 38.64 ohms

    (note that 41 milli-henrys are reduced to .041 henry before
    substituting in the above equation).

    The capacity reactance, or

         1                 1
  Xc = ------ = -------------------------- = 20.8 ohms
       2π_f_C   2 × 3.1416 × 150 × .000051

    (note that 51 microfarads are reduced to .000051 farad before
    substituting in the above equation).

    Substituting the values as calculated for 2π_f_L and
    1 / (2π_f_C) in equation (2)

               Z = √(23² + (38.64 - 20.8)²) = 29.1 ohms.

    To solve the problem graphically, lay off in fig. 1,340, the
    line AB equal to 23 ohms resistance, using any convenient
    scale. Draw upward and at right angles to AB the line
    BC = 38.64 ohms inductance reactance, and from C lay off
    downward CC' = 20.8 ohms capacity reactance. The resultant
    reactance is BC' and being above the horizontal line AB shows
    that inductance reactance is in excess of capacity reactance by
    the amount BC'. Join AC' which gives the impedance sought, and
    which by measurement is 29.1 ohms.

In order to obtain the ~impressed pressure in circuits containing
resistance, inductance and reactance~, an equation similar to (2) on
page 1,095 is used which is made up from the following:

  Eₒ = RI                                                            (3)

  Eᵢ = 2π_f_LI                                                       (4)

         I
  Ec = ------                                                        (5)
       2π_f_C

[Illustration: FIG. 1,340.--Impedance diagram for the circuit shown in
fig. 1,339. Note that the resultant reactance being due to excess of
inductance, the impedance line AC' falls _above_ the horizontal line
AB. This indicates that the current _lags_ behind the pressure.]

When all three quantities, resistance, inductance, and capacity are
present, the equation is as follows:

_impressed pressure = √(ohmic drop² + (inductive drop - capacity drop)²)_

                   Eᵢₘ = √(Eₒ² + (Eᵢ - Ec)²)                         (6)

Substituting in this last equation (6), the values given in (3), (4)
and (5)

  Eᵢₘ = √(R²I² + (2π_f_LI - (I / (2π_f_C)))²)

     = I √(R² + (2π_f_L - (1 / (2π_f_C)))²)                          (7)

[Illustration FIG. 1,341.--Diagram of circuit containing 25 ohms
resistance, .15 henry inductance, and 125 microfarads capacity, with
current of 8 amperes at 60 frequency.]

~Ques. What does the quantity under the square root sign in equation
(7) represent?~

Ans. It is the impedance of a circuit possessing resistance,
inductance, and capacity.

~Ques. Why?~

Ans. Because it is that quantity which multiplied by the current gives
the pressure, which is in accordance with Ohm's law.

[Illustration: Fig. 1,342.--Diagram for finding the pressure necessary
to be impressed on the circuit shown in fig. 1,341, to produce a
current of 8 amperes.]

    EXAMPLE.--An alternator is connected to a circuit having, as in
    fig. 1,341, 25 ohms resistance, an inductance of .15 henry, and
    a capacity of 125 microfarads. What pressure must be impressed
    on the circuit to allow 8 amperes to flow at a frequency of 60?

    The ohmic drop is

                     Eₒ = RI = 25 × 8 = 200 volts.

    The inductance drop is

          Eᵢ = 2π_f_LI = 2 × 3.1416 × 60 × .15 × 8 = 452 volts

    The capacity drop is

         I                 8
  Ec = ------ = ------------------------- = 170 volts.
       2π_f_C   2 × 3.1416 × 60 × .000125

    Substituting the values thus found,

  impressed pressure = √(Eₒ² + (Eᵢ - Ec)²)

                     = √(200² + (452 - 170)²)

                     = √(200² + 282²)

                     = √(119524)

                     = 345.7 volts.





CHAPTER XLVIII

THE POWER FACTOR


The determination of the power in a direct current circuit is a simple
matter since it is only necessary to multiply together the volts and
amperes to obtain the output in watts. In the case of alternating
current circuits, this holds true only when the current is in phase
with the pressure--a condition rarely found in practice.

When the current is not in phase with the pressure, the product of
volts and amperes as indicated by the voltmeter and ammeter must be
multiplied by a coefficient called the _power factor_ in order to
obtain the _true watts_, or actual power available.

There are several ways of defining the power factor, any of which
requires some explanation. The power factor may be defined as: _The
number of watts indicated by a wattmeter, divided by the apparent
watts_, the latter being the _watts as measured by a voltmeter and
ammeter_.

The power factor may be expressed as being equal to

    _true power_       _true watts_        _true watts_
  ---------------- = ---------------- = -------------------
  _apparent power_   _apparent watts_   _volts_ × _amperes_

~Ques. What are the true watts?~

Ans. The watts as measured by a wattmeter.

~Ques. What are the apparent watts?~

Ans. The watts obtained by multiplying together the simultaneous
voltmeter and ammeter readings.

~Ques. What is usually meant by power factor?~

Ans. The multiplier used with the apparent watts to determine how much
of the power supplied is available.

[Illustration: FIG. 1,343.--Marine analogy of power factor. A ferry
boat in crossing a river to a slip C would head for some point B
up stream from C to allow for the effect of the tide. Under such
conditions the actual motion (referred to the water) would be from A
to B, and the apparent motion, from A to C. Accordingly, the energy
expended in propelling the boat from A to B in still water, will
propel it from A to C when the tide is running in the direction of
the arrow. The effect of the tide is the same as that of inductance
or capacity in an alternating circuit, that is, it puts the applied
force or thrust (impressed volts) out of phase with the motion of the
boat (amperes), this phase difference being indicated by the angle
BAC or φ. Now, _work_ (watts) is the product of two factors, pressure
(volts) and distance (amperes); accordingly, the apparent work done in
propelling the boat from A to C is the product of the _thrust of the
paddle wheels multiplied by_ AC, which in analogy corresponds to the
product of voltmeter and ammeter readings at the alternator, called
"kva." Actually, however, the power is only applied from A to B, the
boat being carried sidewise by the tide, as it crosses, a distance BC
which represents no energy expended by the paddle wheels. In analogy,
the actual power, expended in propelling boat from A to B corresponds
to the wattmeter reading in an alternating current power circuit. To
obtain the actual work done on the boat, the product of its apparent
motion × thrust must be multiplied by a coefficient or _power factor_
because the thrust is applied at an angle to the apparent motion,
the power factor being equal to the cosine of this angle, (φ) or
AB ÷ AC. Similarly, when there is phase difference between pressure
and alternating current, the voltmeter and ammeter readings must be
multiplied by the power factor or cos φ to give the output of an
alternator available for external work, the excess power indicated
by ammeter and voltmeter readings, performing no external work, but
causing objectionable heating of the alternator.]

~Ques. Upon what does the power factor depend?~

Ans. Upon the relative amounts of resistance inductance and capacity
contained in the circuit.

~Ques. How does the power factor vary in value?~

Ans. It varies from one to zero.

    The power factor, as will be shown later, is equal to _the
    cosine of the angle of phase difference_; its range then is
    from one to zero because these are the limiting values of the
    cosine of an angle (neglecting the + or-sign).

~Ques. What is the effect of lag or lead of the current on the power
factor?~

Ans. It causes it to become less than one.

[Illustration: FIG. 1,344.--Method of drawing the power curve from the
pressure and current curves. As shown, the same scale is used for all
curves. This as a rule, makes the power curve inconveniently high,
hence it is usually drawn to smaller scale as in fig. 1,345.]

~How to Obtain the Power Curve.~--Since under any phase condition, the
power at any instant is equal to the product of the pressure multiplied
by the current at that instant, a curve may be easily plotted from the
pressure and current curves, giving the instantaneous values of the
power through a complete cycle.

    In fig. 1,344, from the zero line of the current and pressure
    curves, draw any ordinate as at F cutting the current curve at
    G and the pressure curve at G'. The values for current and
    pressure at this point are from the scale, 2 amperes and 3.7
    volts. Since watts = amperes × volts, the ordinate FG is to be
    multiplied by ordinate FG' that is,

                             2 × 3.7 = 7.4.

    Project up through F the ordinate FG" = 7.4, and this will
    give one point on the power curve.

    Similarly at another point, say M, where the current and
    pressure are maximum

                        MS × MS' = MS", that is
                         3 × 5   = 15

    giving S" another point on the curve. Obtaining several points
    in this way the power curve is then drawn through them as shown.

[Illustration: FIG. 1,345.--Usual method of drawing power curve from
the pressure and current curves. A smaller scale is employed for the
power curve in order to reduce its height.]

~Ques. Why is the power curve positive in the second half of the period
when there are negative values of current and pressure?~

Ans. Because the product of two negative quantities is positive.

~Ques. Does fig. 1,344 represent the usual way of drawing a power
curve?~

Ans. Since ordinates of the power curve are products of the current and
pressure ordinates, they will be of inconvenient length if drawn to
the same scale; it is therefore customary to use a different scale for
the power ordinates, as in fig. 1,345.

    The illustration is lettered identical with fig. 1,344, with
    which it should be compared.

~Synchronism of Current and Pressure; Power Factor Unity.~--The current
and pressure would be in phase as represented in fig. 1,346 were it
possible to have a circuit containing resistance only. In actual
practice all circuits contain at least a small amount of reactance.

[Illustration: FIG. 1,346.--Synchronism of current and pressure. Power
curve showing that the power factor is unity. This is indicated by the
fact that the power curve does not project below the base or zero line.]

A circuit supplying nothing but incandescent lamps comes very nearly
being all resistance, and may be so considered in the discussion here.
Fig. 1,347 illustrates a circuit containing only resistance. In such a
circuit the pressure and current (as shown in fig. 1,346) pass through
zero and through their maximum values together.

Multiplying instantaneous values of volts and amperes will give the
power curve, as before explained, whose average value is half-way
between the zero line and the maximum of the curve; that part of the
power curve above the line of average power WW, exactly filling the
open space below the line WW. That is,

  average power = maximum power ÷ √2̅

                = maximum voltage × maximum current / √2̅

                = virtual voltage × virtual current.

This latter is simply the product of the voltmeter and ammeter readings
which gives the watts just the same as in direct current.

[Illustration: FIG. 1,347.--Diagram of circuit containing only
resistance; in such a circuit the power factor is unity.]

~Ques. What should be noticed about the power curve?~

Ans. Its position with respect to the zero line; it lies wholly
above the zero line which denotes that all the power delivered to
the circuit except that dissipated by friction is useful, that is,
the power factor is unity. Hence, _to keep the power factor as near
unity as possible is one of the chief problems in alternating current
distribution_.

~Ques. Can the power factor be less than unity if the current and
pressure be in phase?~

Ans. Yes, if the waves of current and voltage be distorted as in fig.
1,348.

~Effect of Lag and Lead.~--In an alternating circuit the amount of
power supplied depends on the phase relationship of the current and
pressure. As just explained, when there is synchronism of current and
pressure, that is, when they are in phase (as in fig. 1,346) the power
factor is unity, assuming no distortion of current and pressure waves.
In all other cases the power factor is less than unity that is, _the
effect of lag or lead is to make the power factor less than unity_.

[Illustration: FIG. 1,348.--Case of synchronism of current and pressure
with power factor less than unity. Suppose the waves of current and
voltage to be in phase, but distorted in form, and not symmetrical, so
that they do not run uniformly together, as shown in the figure. Then
the real power factor may not be unity, although indicated as such by
the power factor meter. However, the switchboard instruments are made
to show the angle of lag as the power factor, because the error due to
wave distortion is generally too small to be considered.]

[Illustration: FIG. 1,349.--Effect of lag on the power factor. When
the current lags behind the pressure the power factor becomes less
than unity. It will be seen that the power curve projects below the
zero line giving the shaded area which represents negative power which
must be subtracted from the + areas above the zero line to get the net
power. In the figure the line WW' is drawn at a height corresponding
to the average power, and HN at a height corresponding to the average
power that would be developed if the current were in phase with the
pressure. The power factor then is represented by M ÷ S, and by
inspection of the figure it is seen that this is less than unity.]

[Illustration: FIG. 1,350.--Effect of lead on the power factor. When
the current is in advance of the pressure the power factor becomes
less than unity. The curve, as shown, projects below the zero line,
giving the shaded area which represents negative power which must
be subtracted from the + areas above the zero line to get the net
power. As in fig. 1,349, the line WW' at a height M represents the
average power, and HN the average power for synchronism of current and
pressure. The power factor then is M ÷ S which is less than unity.]

The effect of lag on the power factor may be illustrated by fig. 1,349,
in which the angle between the pressure and current, or the angle of
lag is taken as 40°, corresponding to a power factor of .766. Plotting
the power curve from the products of instantaneous volts and amperes
taken at various points, the power curve is obtained, a portion of
which lies below the horizontal line. The significance of this is that
at certain times, the current is flowing in the opposite direction to
that in which the impressed pressure would send it. During this part
of the period conditions are reversed, and the power (indicated by the
shaded area), instead of being supplied by the source to the circuit,
is being supplied by the circuit to the source.

[Illustration: FIG. 1,351.--Steam engine analogy of power factor. The
figure represents an indicator card of an engine in which the steam
distribution is such that the steam is expanded below the back pressure
line, that is below the pressure of the exhaust. This results in
_negative work_ which must be overcome by the _momentum_ or _kinetic
energy_ previously stored in the fly wheel, and which is represented on
the diagram by the shaded loop S. If the exhaust valve had opened at G,
the amount of work done during the revolution would be represented by
the area M, but continuing the expansion below the back pressure line,
the work done is M - S. This latter case as compared with the first
when expansion does not continue below the back pressure line gives an
efficiency (power factor) of (M - S) ÷ M, the shaded area representing
so much loss.]

    This condition is exactly analogous to the case of a steam
    engine, expanding the steam below the back or exhaust pressure,
    a condition sometimes caused by the action of the governor in
    considerably reducing the cut off for very light load. An
    indicator diagram of such steam distribution is shown in fig.
    1,351. This gives a negative loop in the diagram indicated by
    the shaded section.

    It must be evident that the average pressure of the shaded loop
    portion of the diagram must be subtracted from that of the
    other portion, because during the expansion below the exhaust
    pressure line, the back pressure is in excess of the forward
    pressure exerted on the piston by the expanding steam, and the
    engine would accordingly reverse its motion, _were it not for
    the energy previously stored up in the fly wheel_ in the form
    of _momentum_, which keeps the engine moving during this period
    of back thrust. Evidently the shaded area must be subtracted
    from the positive area to obtain the net work done during the
    stroke. Hence following the analogy as far as possible if M
    work (watts) be done during each revolution (cycle) when steam
    does not expand below back pressure (when current and pressure
    are in phase), and S negative work (negative watts) be done
    when steam expands below back pressure (when there is lag), the
    efficiency (power factor) is (M - S) ÷ M.

[Illustration: FIG. 1,352.--Power curve illustrating the so-called
wattless current in which case the power factor is zero. By noting that
the curve projects equally on each side of the zero line, the + power
areas equal the negative power areas, hence the summation of these
areas for the period is zero, that is, the two + areas minus the two
shaded areas equal zero. It should be noted that the line of average
power WW', which is visible in the other figures, here coincides
with the zero line, and the average power then is zero, since the
positive part above the zero line is equal to and offsets the negative
(shaded) part below the line. This is the case of "wattless" current
and (considering a circuit with resistance so small that it may be
considered as zero) shows plainly the possibility of having full load
current and voltage on a circuit yet delivering no power, the current
simply surging to and fro without an actual transfer of power.]

~"Wattless Current;" Power Factor Zero.~--When the power factor is
zero, it means that the phase difference between the current and the
pressure is 90°.

The term _wattless current_, as understood, does not indicate an
absence of electrical energy in the circuit; its elements are there,
but not in an available form for external work. The false power due
to the so called wattless current pulsates in and out of the circuit
without accomplishing any useful work.

    An example of wattless current, showing that the power factor
    is zero is illustrated in fig. 1,353. Here the angle of lag is
    90°, that is, the current is 90° behind the pressure.

    The power curve is constructed from the current and pressure
    curves, and, as shown in the diagram, it lies as much below the
    zero line as above, that is, the two plus power areas which
    occur during each period are equal to the two negative (shaded)
    power areas, showing that the circuit returns as much energy
    as is sent out. Hence, the total work done during each period
    is zero, indicating that although a current be flowing, this
    current is not capable of doing external work.

[Illustration: FIG. 1,353.--Example of wattless current showing that
the power factor is zero when the phase difference between current and
pressure is 90°. For zero power factor the current may _lead_ 90° as in
fig. 1,352, or _lag_ 90° as here shown. Since the shaded or negative
areas = the plus areas, the average power (indicated by WW' which
coincides with the zero line) is zero, that is the circuit is carrying
current under pressure yet delivering no power, hence, the power factor
is zero.]

~Ques. Is the condition as just described met with in practice?~

Ans. No.

~Ques. Why not?~

Ans. The condition just described involves that the circuit have no
resistance, all the load being reactance, but it is impossible to have
a circuit without some resistance, though the resistance may be made
very small in comparison to the reactance so that a close approach to
wattless current is possible.

~Ques. Give some examples where the phase difference is very nearly
90°.~

Ans. If an alternator supplies current to a circuit having a very small
resistance and very large inductance, the current would lag nearly 90°
behind the pressure. The primary current of a transformer working with
its secondary on an open circuit is a practical example of a current
which represents very little energy.

[Illustration: FIG. 1,354.--Performance curves of General Electric
single phase repulsion induction motor.]

~Ques. When the phase difference between the current and pressure is
90°, why is the current called "wattless"?~

Ans. Because the product of such a current multiplied by the pressure
does not represent any watts _expended_.

    A man lifting a weight, and then allowing it to descend the
    same distance to its initial position, as shown in figs. 1,355
    to 1,357, presents a mechanical analogy of wattless current.

    Let the movement of the weight represent the current and the
    weight the pressure. Then calling the weight 10 pounds (volts),
    and the distance two feet (amperes). The work done by the man
    (alternator) on the weight in lifting it is

        10 pounds × 2 feet    = 20 foot pounds                       (1)
        (10 volts × 2 amperes = 20 watts.)

    The work done on the man by the weight in forcing his hand down
    as his muscles relax is

        10 pounds × 2 feet    = 20 foot pounds                       (2)
        (10 volts × 2 amperes = 20 watts.)

    From (1) and (2) it is seen that the _work done by the man
    on the weight is equal to the work done by the weight on the
    man_, hence no useful work has been accomplished; that is, the
    potential energy of the weight which it originally possessed
    has not been increased.

[Illustration: FIGS. 1,355 to 1,357.--Mechanical analogy of wattless
current. If a man lift a weight any distance, as from the position of
fig. 1,355 to position of fig. 1,356, he does a certain amount of work
on the weight giving it potential energy. When he lowers it to its
original position, as in fig. 1,357, the weight loses the potential
energy previously acquired, that is, it is given back to the man, the
"system" (man and weight) having returned to its original condition as
in fig. 1,355. During such a cycle, the work done by the man on the
weight is equal to the work done by the weight on the man and no useful
external work has been accomplished.]

~Why the Power Factor is equal to Cos φ.~--In the preceding figures
showing power curves for various phase relations between current and
pressure, the curves show the instantaneous values of the fluctuating
power, but what is of more importance, is to determine the average
power developed.

When the current is in phase with the pressure, it is a simple matter,
because the power or

                      _watts = amperes_ × _volts_

that is, the product of the ammeter and voltmeter readings will give
the power. However, the condition of synchronism of current and
pressure hardly ever exists in practice, there being more or less phase
difference.

[Illustration: FIG. 1,358.--Method of obtaining the _active component_
of the current; diagram illustrating why the power factor is equal to
cos φ. If AB and AC be respectively the given current and pressure,
or readings of the ammeter and voltmeter, and φ the angle of phase
difference between current and pressure, then drawing from B, BD
perpendicular to AC will give AD the active component. Now, true
power = AC × AD, but AD = AB cos φ, hence true power = AC × AB cos
φ. Again, apparent power = AC × AB, and since true power = apparent
power × power factor, the power factor = cos φ.]

When the current is not in phase with the pressure, it is considered as
made up of two components at right angles to each other.

1. _The active component_, in phase with the pressure;

2. _The wattless component_, at right angles to the pressure.

With phase difference between current and pressure the product of
ammeter and voltmeter readings do not give the true power, and in
order to obtain the latter, the _active component_ of the current in
phase with the pressure must be considered, that is,

           _true power_ = _volts_ × _active amperes_                 (1)

    The active component of the current is easily obtained
    graphically as in fig. 1,358.

    With any convenient scale draw AB equal to the current as given
    or read on the ammeter, and AC, equal to the pressure, making
    the angle φ between AB and AC equal to the phase difference
    between the current and pressure.

    From B, draw the line BD perpendicular to AC, then BD will be
    the wattless component, and AD (measured with the same scale as
    was used for AB) the active component of the current, or that
    component in phase with the pressure.

Hence from equation (1)

                     true power = AC × AD                            (2)

Now in the right triangle ABD

  AD
  -- = cos φ
  AB

from which

                         AD = AB cos φ                               (3)

Substituting this value of AD in equation (2) gives

                  true power = AC × AB cos φ                         (4)

Now the power factor may be defined as: _that quantity by which the
apparent watts must be multiplied in order to give the true power_.
That is

        _true power = apparent watts_ × _power factor_               (5)

Comparing equations (4) and (5), AC × AB in (4) is equal to the
apparent watts, hence, the power factor in (5) is equal to cos φ. That
is, _the power factor is numerically equal to the cosine of the angle
of phase difference between current and pressure_.

    EXAMPLE I.--An alternator supplies a current of 200 amperes at
    a pressure of 1,000 volts. If the phase difference between the
    current and pressure be 30°, what is the true power developed?

    In fig. 1,359, draw AB to scale, equal to 200 amperes, and
    draw AC of indefinite length making an angle of 30° with AB.
    From B, draw BD perpendicular to AC which gives AD, the active
    component, and which measured with the same scale as was used
    in laying off AB, measures 173.2 amperes. The true power
    developed then is

                 true watts = 173.2 × 1,000 = 173.2 kw.

    The true power may be calculated thus:

    From the table cos 30° = .866, hence

              true watts = 200 × 1,000 × .866 = 173.2 kw.


[Illustration: FIG. 1,359.--Diagram for obtaining the active component
of the current in a circuit having a current of 200 amperes and angle
of lag of 30°.]

    EXAMPLE II.--If in an alternating current circuit, the
    voltmeter and ammeter readings be 110 and 20 and the angle of
    lag 45°, what is the apparent power and true power?

    The apparent power is simply the product of the current and
    pressure readings or

                apparent power = 20 × 110 = 2,200 watts

    The true power is the product of the apparent power multiplied
    by the cosine of the angle of lag. Cos 45° = .707, hence

               true power = 2,200 × .707 = 1,555.4 watts.


~Ques. Does the power factor apply to capacity reactance in the same
way as to inductance reactance?~

Ans. Yes. The angles of lag and of lead, are from the practical
standpoint, treated as if they lay in the first quadrant of the circle.
Even the negative sign of the tangent φ when it occurs is simply
used to determine whether the angle be one of lag or of lead, but in
finding the value of the angle from a table it is treated as a positive
quantity.

[Illustration: FIG. 1,360.--Diagram for obtaining the power factor for
example II. With convenient scale, lay off AB = 20 amperes. From A draw
AC at 45° to AB, and from B, draw BD perpendicular to AC. Then, the
power factor which is equal to cosine of angle of lag, = AD ÷ AB = (by
measurement) 14.15 ÷ 20 = .707.]

~Ques. In introducing capacity into a circuit to increase the power
factor what should be considered?~

Ans. The cost and upkeep of the added apparatus as well as the power
lost in same.

~Ques. How is power lost in a condenser?~

Ans. The loss is principally due to a phenomenon known as _dielectric
hysteresis_, which is somewhat analogous to magnetic hysteresis. The
rapidly alternating charges in a condenser placed in an alternating
circuit may be said to cause alternating polarization of the
dielectric, and consequent heating and loss of energy.

~Ques. When is inductance introduced into a circuit to increase the
power factor?~

Ans. When the phase difference is due to an excess of capacity.

    EXAMPLE.--A circuit having a resistance of 3 ohms, and a
    resultant reactance of 4 ohms, is connected to a 100 volt line.
    What is: 1, the impedance, 2, the current, 3, the apparent
    power, 4, the angle of lag, 5, the power factor, and 6, the
    true power?

    1. _The impedance of the circuit._

        Z = √(3² + 4²) = 5 ohms.

2. _The current._

        current = volts ÷ impedance = 100 ÷ 5 = 20 amperes.

3. _The apparent power._

        apparent power = volts × amperes = 100 × 20 = 2,000 watts.

4. _The tangent of the angle of lag._

        tan φ = reactance ÷ resistance = 4 ÷ 3 = 1.33. From table
        of natural tangents (page 451) φ = 53°.

5. _The power factor._

        The power factor is equal to the cosine of the angle of
        lag, that is, power factor = cos 53° = .602 (from table).

6. _The true power._

        The true power is equal to the apparent watts multiplied by
        the power factor, or

  true power = volts × amperes × cos φ
             = 100 × 20 × .602 = 1,204 watts.


~Ques. Prove that the power factor is unity when there is no resultant
reactance in a circuit.~

Ans. When there is no reactance, tan φ which is equal to
reactance ÷ resistance becomes 0 ÷ R = 0. The angle φ (the phase
difference angle) whose tangent is 0 is the angle of 0 degrees. Hence,
the power factor which is equal to cos φ = cos 0° = 1.

[Illustration: FIGS. 1,361 to 1,365.--Diagrams illustrating why the
power factor is unity or one when there is no resultant reactance
in the circuit, that is, when the circuit is resonant or has only
resistance. The power factor is equal to the cosine of the angle of
lag (or lead). In the figures this angle is BAC or φ and the value of
the _natural cosine_ AC gives the power factor. By inspection of the
figures, it is evident that decreasing the reactance decreases the
angle φ and increases cos φ or the power factor. The circular arc in
each figure being at unity distance from the center A, the power factor
with decreasing reactance evidently approaches unity as its limit, this
limit being shown in fig. 1,365 where the reactance B'C' = 0.]

~Ques. What is the usual value of the power factor in practice?~

Ans. Slightly less than one.

~Ques. Why is it desirable to keep the power factor near unity?~

Ans. Because with a low power factor, while the alternator may be
carrying its full load and operating at a moderate temperature, the
consumer is paying only for the actual watts which are sent over the
line to him.

[Illustration: FIG. 1,366.--Diagram illustrating power factor test,
when on non-inductive and inductive circuits. The instruments are
connected as shown and by means of the double throw switch can
be put on either the non-inductive or inductive circuit. First
turn switch to left so that current passes through the lamps;
for illustration, the following readings are assumed: ammeter
10, voltmeter 110, and wattmeter 1,100. The power factor then is
wattmeter reading ÷ volts × amperes = 1,100 actual watts ÷ 1,100
apparent watts = 1, that is, on non-inductive circuit the power
factor is unity. Now throwing the switch to the right connecting
instruments with the inductive circuits, then for illustration
the following readings may be assumed: ammeter 8, voltmeter 110,
and wattmeter 684. Now, as before, power factor = wattmeter
reading ÷ volts × amperes = 684 ÷ (8 × 110) = 684 ÷ 880 = .78.]

    For instance, if a large alternator supplying 1,000 kilowatts
    at 6,600 volts in a town where a number of induction motors are
    used on the line be operating with a power factor of say .625
    during a great portion of the time, the switchboard instruments
    connected to the alternator will give the following readings:

    Voltmeter 6,600 volts; ammeter 242.4 amperes; power factor
    meter .625.

    The apparent watts would equal 1,600,000 watts or 1,600
    kilowatts, which, if multiplied by the power factor .625 would
    give 100,000,000 watts or 1,000 kilowatts which is the actual
    watts supplied. The alternator and line must carry 242.4
    amperes instead of 151 amperes and the
    difference 242.4 - 151 = 91.4 amperes represents a _wattless
    current_ flowing in the circuit which causes useless heating of
    the alternator.

    The mechanical power which is required to drive the alternator
    is equivalent to the actual watts produced, since that portion
    of the current which lags, is out of phase with the pressure
    and therefore requires no energy.

~Ques. How are alternators rated by manufacturers in order to avoid
disputes?~

Ans. They usually rate their alternators as producing so many kilovolt
amperes instead of kilowatts.

[Illustration: FIG. 1,367.--Ayrton and Sumpner method of alternating
current power measurement. Three voltmeters are required, and
accordingly the method is sometimes called the three voltmeter method.
It is a good method where the voltage can be regulated to suit the
load. In the figure, let the non-inductive resistance R be placed in
series with the load AB. Measure the following voltages: V across the
terminals of R, V₁ across the load AB, and V₂ across both, that is from
A to C. Then, true watts = (V₂² - V₁² - V²) ÷ 2R. The best conditions
are when V = V₁, and, if R = ½ ohm, then W = V₂² - V₁² - V².]

~Ques. What is a kilovolt ampere (kva)?~

Ans. A unit of apparent power in an alternating current circuit which
is equal to one kilowatt when the power factor is equal to one.

    The machine mentioned on page 1,120 would be designed to carry
    151 amperes without overheating and also carry slight overloads
    for short periods. It would be rated as 6.6 kilo volts and 151
    amperes which would equal approximately 1,000 kilowatts when
    the power factor is 1 or unity, and it should operate without
    undue heating. Now the lower the power factor becomes, the
    greater the heating trouble will be in trying to produce the
    1,000 actual kilowatts.

[Illustration: FIG. 1,368--Curves illustrating _power factor_. In a
circuit having no capacity or inductance, the power is given by the
product of the respective readings of the voltmeter and ammeter, as in
the case of a direct current. In the case of a circuit having capacity
or inductance, this product is higher than the true value as found by
a wattmeter, and is known as the _apparent watts_. The ratio _true
watts_ ÷ _apparent watts_ is known as the power factor. The current
flowing in an inductive circuit, such as the primary of a transformer,
is really made up of two components, as already explained, one of which
(the load or active component), is in phase with the pressure, while
the other the magnetizing component, is at right angles to it, that
is, it attains its crest value when the other is at zero, and vice
versa. To illustrate, take a complete cycle divided into 360 degrees
and lay out on it the current required to correspond to a given load
on the secondary of a transformer, say a crest value of 100 amperes,
and at right angles to this lay out the current required for exciting
the magnetic circuit of the transformer, giving A, merely for purposes
of illustration, a crest value of 25 amperes. Combining these curves,
the dotted curve in the figure is obtained and which represents the
resultant current that would be indicated by an ammeter placed in the
primary circuit of the transformer. It will be noted that this current
attains its maximum at a point 14° 2´ later than the load current,
giving the angle of lag. Multiplying the apparent watts by the cosine
of the angle of lag gives the true watts. Now assuming the diagram to
show the full load condition of the transformer, the angle of lag being
14° 2´, the power factor at full load is .97 (.97 being the value of
the natural cosine of 14° 2´ as obtained from table, such as on page
451). With no external load on the transformer, the load component of
the current is that necessary to make up the core losses. For instance,
at 5 amperes, while the magnetizing current remains as before at 25
amperes, the angle of lag becomes 78° 41´ and the power factor .196. It
is thus seen that in transformers, induction motors, etc., the power
factor is a function of the load.]

~Ques. How can the power factor be kept high?~

Ans. By carefully designing the motors and other apparatus and even
making changes in the field current of motors which are already
installed.

~Ques. How is the power factor determined in station operation?~

Ans. Not by calculation, but by reading a meter which forms one of the
switchboard instruments.

[Illustration: FIG. 1,368.--Fleming's combined voltmeter and ammeter
method of measuring power in alternating current circuits. It is
quite accurate and enables instruments in use to be checked. In the
figure, R is a non-inductive resistance connected in shunt to the
inductive load. The voltmeter V measures the pressure across the
resistance XY. A and A₁ are ammeters connected as shown. Then, true
watts = (A₁² - A² - (V/R)²) × R / 2. If the volt meter V take an
appreciable amount of current, it may be tested as follows: disconnect
R and V at Y, and see that A and A₁ are alike; then connect R and V at
Y again, and disconnect the load. A₁ will equal current taken by R and
V in parallel.]

~Ques. When is the power factor meter of importance in station
operation, and why?~

Ans. When rotary converters are used on alternating current lines for
supplying direct currents and the sub-station operators are kept busy
adjusting the field rheostat of the rotary to maintain a high power
factor and prevent overheating of the alternators during the time of
day when there is the maximum demand for current or the peak of the
load.

    EXAMPLE.--An alternator delivers current at 800 volts pressure
    at a frequency of 60, to a circuit of which the resistance is
    75 ohms and .25 henry.

    Determine: _a_, the value of the current, _b_, angle of lag,
    _c_, apparent watts, _d_, power factor, _e_, true power.

    _a. Value of current_

            pressure            E
  current = --------- = -----------------
            impedance   √(R² + (2π_f_L)²)

                              800
          = --------------------------------- = 6.7 amperes
            √(75² + (2 × 3.1416 × 60 × .25)²)

    _b. The angle of lag_

          reactance    2π_f_L   2 × 3.1416 × 60 × .25
  tan φ = ---------- = ------ = --------------------- = 1.25
          resistance     R                75

      φ = angle of lag = 51° 15´ (from table, page 451).

    _c. The apparent power_

  apparent power = volts × amperes = 800 × 6.7 = 5,360 watts
                                               = 5.36 kva.

    _d. The power factor_

  power factor = cosine of the angle of lag
               = cos 51° 15´ = .626.

    _e. The true power_

  true power = apparent power × power factor
             = 5,360 × .626 = 3,355 watts.

[Illustration: FIG. 1,369.--Wattmeter method of three phase power
measurement. Two wattmeters are required in unbalanced systems as shown
in the illustration. The total power transmitted is then the algebraic
sum of the readings of the two wattmeters. If the power factor be
greater than .5, the power is the arithmetical sum, and if it be less
than .5, the power is the arithmetical difference of the readings.]




CHAPTER XLIX

ALTERNATORS


~Use of Alternators.~--The great increase in the application of
electricity for supplying power and for lighting purposes in
industry, commerce, and in the home, is due chiefly to the economy of
distribution of alternating current.

Direct current may be used to advantage in densely populated districts,
but where the load is scattered, it requires, on account of its low
voltage, too great an investment in distributing lines. In such cases
the alternator is used to advantage, for while commutators can be built
for collecting direct current up to 1,000 volts, alternators can be
built up to 12,000 volts or more, and this voltage increased, by step
up transformers of high economy, up to 75,000 or 100,000 volts. Since
the copper cost is inversely as the square of the voltage, the great
advantage of alternating current systems is clearly apparent.

The use of alternating current thus permits a large amount of energy to
be economically distributed over a wide area from a single station, not
only reducing the cost of the wiring, but securing greater economy by
the use of one large station, instead of several small stations.

The higher voltages generated by alternators enables the transmission
of electrical energy to vastly greater distances than possible by a
direct current system, so that the energy from many waterfalls that
otherwise would go to waste may be utilized.

~Classes of Alternator.~--There are various ways of classifying
alternators. They may be divided into groups, according to: 1, the
nature of the current produced; 2, type of drive; 3, method of
construction; 4, field excitation; 5, service requirements, etc.

From these several points of view, alternators then may be classified:

1. With respect to the current, as:

  _a._ Single phase;
  _b._ Polyphase.

2. With respect to the type of drive, as:

  _a._ Belt or chain driven;
  _b._ Direct connected.

3. With respect to construction, as:

  _a._ Revolving armature;
  _b._ Revolving field;
  _c._ Inductor.
          Homopolar and heteropolar.

4. With respect to mode of field excitation, as:

  _a._ Self-exciting;
  _b._ Separately excited;
            Exciter direct connected, or gear driven.
  _c._ Compositely excited.

5. With respect to service requirements, as:

  _a._ Slow speed;
  _b._ Fly wheel;
  _c._ High speed;
  _d._ Water wheel type;
  _e._ Turbine driven.

~Single Phase Alternators.~--As a general rule, when alternators
are employed for lighting circuits, the single phase machines are
preferable, as they are simpler in construction and do not generate the
unbalancing voltages often occurring in polyphase work.

[Illustration: FIG. 1,370.--Elementary four pole single phase
alternator. It has four "inductors" whose pitch is the same as the pole
pitch. They are connected in series and terminate at the two collector
rings as shown. The poles being alternate N and S, it is evident that
there will be two cycles of the current per revolution of the armature.
For any number of poles then the number of cycles equals the number of
poles divided by two. Applying Fleming's rule for induced currents,
the direction of the current induced in the inductors is easily found
as indicated by the arrows. The field magnets are excited by coils
supplied with direct current, usually furnished from an external
source; for simplicity this is not shown. The magnets may be considered
as of the permanent type.]

~Ques. What are the essential features of a single phase alternator?~

Ans. Fig. 1,370 shows an elementary single phase alternator. It
consists of an armature, with single phase winding, field magnets, and
two collector rings and brushes through which the current generated in
the armature passes to the external circuit.

~Ques. In what respect do commercial machines differ mostly from the
elementary alternator shown in fig. 1,370, and why?~

Ans. They have a large number of poles and inductors in order to obtain
the desired frequency, without excessive speed, and electromagnets
instead of permanent magnets.

[Illustration: FIG. 1,371.--Developed view of elementary single phase
four pole alternator and sine curve showing the alternating current or
pressure generated during one revolution. The armature is here shown as
a flat surface upon which a complete view of the winding is seen. If M
be any position of an inductor, by projecting up to the curve gives N,
the corresponding value of the current or pressure. Magnetic lines are
shown at the poles representing a field decreasing in intensity from
a maximum at the center to zero at points half way between the poles,
this being the field condition corresponding to the sine form of wave.
In actual machines the variation from the sine curve is considerable in
some alternators. See figs. 1,247 and 1,248.]

~Ques. In actual machines, why must the magnet cores be spaced out
around the armature with considerable distance between them?~

Ans. In order to get the necessary field winding on the cores, and also
to prevent undue magnetic leakage taking place, laterally from one limb
to the next of opposite sign.

~Ques. Is there any gain in making the width of the armature coils any
greater than the pole pitch, and why?~

Ans. No, because any additional width will not produce more voltage,
but on the contrary will increase the resistance and inductance of the
armature.

[Illustration: FIG. 1,372.--Elementary four pole two phase alternator.
The winding consists of one inductor per phase per pole, that is, four
inductors per phase, the inductors of each phase being connected in
series by the "connectors" and terminating at the collector rings.
This arrangement requires four collector rings, giving two independent
circuits. The pitch of the inductors of each phase is equal to the
pole pitch, and the phase difference is equal to one-half the pole
pitch, that is, phase B winding begins at B, a point half-way between
inductors A and A' of phase A winding. Hence when the current or
pressure in phase A is at a maximum, in the ideal case, when inductor A
for instance is under the center of a pole, the current or pressure in
B is zero, because B is then half-way between the poles.]

~Polyphase Alternators.~--A multiphase or polyphase alternator is one
which delivers two or more alternating currents differing in phase by a
definite amount.

    For example, if two armatures of the same number of turns each
    be connected to a shaft at 90 degrees from each other and
    revolved in a bipolar field, and each terminal be connected to
    a collector ring, two separate alternating currents, differing
    in phase by 90 degrees, will be delivered to the external
    circuit. Thus a two phase alternator will deliver two currents
    differing in phase by one-quarter of a cycle, and similarly
    a three phase alternator (the three armatures of which are
    set 120 degrees from each other) will deliver three currents
    differing in phase by one-third of a cycle.

    In practice, instead of separate armatures for each phase, the
    several windings are all placed on one armature and in such
    sequence that the currents are generated with the desired phase
    difference between them as shown in the elementary diagrams
    1,372 and 1,373 for two phase current, and figs. 1,374 and
    1,375 for three phase current.

[Illustration: FIG. 1,373.--Developed view of elementary two phase four
pole alternator and sine curves, showing the alternating current or
pressure generated during one revolution of the armature. The complete
winding for the three phases are here visible, the field magnets being
represented as transparent so that all of the inductors may be seen. By
applying Fleming's rule, as the inductors progress under the poles, the
directions and reversals of current are easily determined, as indicated
by the sine curves. It will be seen from the curves that four poles
give two cycles per revolution. Inductors A, and B are lettered to
correspond with fig. 1,372, with which they should be compared.]

~Ques. What use is made of two and three phase current?~

Ans. They are employed rather for power purposes than for lighting, but
such systems are often installed for both services.

~Ques. How are they employed in each case?~

Ans. For lighting purposes the phases are isolated in separate
circuits, that is, each is used as a single phase current. For driving
motors the circuits are combined.

[Illustration: FIG. 1,374.--Elementary four pole three phase
alternator. There are three sets of inductors, each set connected in
series and spaced on the drum with respect to each other two-thirds
pole pitch apart. As shown, six collector rings are used, but on actual
three phase machines only three rings are employed, as previously
explained. The inductors have distinctive coverings for the different
phases. The arrows indicate the direction in which the induced
pressures tend to cause currents.]

~Ques. Why are they combined for power purposes?~

Ans. On account of the difficulty encountered in starting a motor with
single phase current.

    Ferarris, of Italy, in 1888 discovered the important principle
    of the production of a rotating magnetic field by means of two
    or more

[Illustration: FIG. 1,375.--Elementary four pole three-phase alternator
and sine curves showing current or pressure conditions for one
revolution. Six collector rings are shown giving three independent
circuits. The pitch of the inductors for each phase is the same as the
pole pitch, and the phase difference is equal to two-thirds of the pole
pitch, giving the sequence of current or pressure waves as indicated by
the sine curves. The waves follow each other at ⅓ period, that is, the
phase difference is 120 degrees. Inductors A, B, and C, the beginning
of each phase winding, are lettered to correspond with fig. 1,374, with
which they should be compared.] /# alternating currents displaced
in phase from one another, and he thus made possible by means of the
induction motor, the use of polyphase currents for power purposes.

~Ques. What is the difficulty encountered in starting a motor with
single phase current?~

Ans. A single phase current requires either a synchronous motor to
develop mechanical power from it, or a specially constructed motor of
dual type, the idea of which is to provide a method of getting rotation
by foreign means and then to throw in the single phase current for
power.

[Illustration: FIG. 1,376.--Diagram of six phase winding with star
grouping, being equivalent to a three phase winding in which the three
phases are disconnected from each other and their middle points united
at a common junction.]

[Illustration: FIG. 1,377.--Diagram of six phase winding with mesh
grouping.]

~Six Phase and Twelve Phase Windings.~--These are required for the
operation of rotary converters. The phase difference in a six phase
winding is 60 degrees and in a twelve phase winding 30 degrees. A six
phase winding can be made out of a three phase winding by disconnecting
the three phases from each other, uniting their middle points at a
common junction, as shown by diagram fig. 1,376. This will give a star
grouping with six terminals.

In the case of a mesh grouping, each of the three phases must be cut
into two parts and then reconnected as shown in fig. 1,377.

[Illustration: FIG. 1,378.--Diagram of twelve phase winding star
grouping.]

[Illustration: FIG. 1,379.--Diagram of six phase winding consisting of
combination of mesh and star grouping.]

As the phase difference of a twelve phase winding is one-half that of a
six phase winding, the twelve phases may be regarded as a star grouping
of six pairs crossed at the middle point of each pair as shown in fig.
1,378, or in mesh grouping for converters they may be arranged as a
twelve pointed polygon. They may also be grouped as a combination of
mesh and star as shown in fig. 1,379, which, however, is not of general
interest.

~Belt or Chain Driven Alternators.~--The mode in which power is
transmitted to an alternator for the generation of current is governed
chiefly by conditions met with where the machine is to be installed.

In many small power stations and isolated plants the use of a belt
drive is unavoidable. In some cases the prime mover is already
installed and cannot be conveniently arranged for direct connection,
in others the advantage to be gained by an increase in speed more than
compensates for the loss involved in belt transmission.

[Illustration: FIG. 1,380.--Belt-driven alternator. By use of a belt,
any desired speed ratio is obtained, enabling the use of a high speed
alternator which, being smaller than one of slow speed, is cheaper. It
affords means of drive for line shaft and has other advantages, but
requires considerable space and is not a "positive" drive. Belting
exerts a side pull which results in friction and wear of bearings.
Means for tightening the belt as shown in fig. 1,381, or equivalent,
must be provided.]

There are many places where belted machines may be used advantageously
and economically. They are easily connected to an existing source of
power, as, for instance, a line shaft used for driving other machinery,
and for comparatively small installations they are lower in first cost
than direct connected machines. Moreover, when connected to line shaft
they are run by the main engine which as a rule is more efficient than
a small engine direct connected.

Where there is sufficient room between pulley centers, a belt is a
satisfactory medium for power transmission, and one that is largely
used. It is important that there be liberal distance between centers,
especially in the case of generators or motors belted to a medium or
slow speed engine, because, owing to the high speed of rotation of the
electric machines, there is considerable difference in their pulley
diameters and the drive pulley diameter; hence, if they were close
together, the arc of contact of the belt with the smaller pulley would
be appreciably reduced, thus diminishing the tractive power of the belt.

[Illustration: FIG. 1,381.--Sub-base and ratchet device for moving
alternator to tighten belt. A ratchet A, operated by lever B, works the
block C by screw connection, causing it to move the block. The latter,
engaging with the frame, causes it to move, thus providing adjustment
for belt. After tightening belt, the bolts D, which pass through the
slots in the sub-base, are tightened, thus securing the machine firmly
in position.]

~Ques. What provision should be made in the design of an alternator to
adapt it to belt drive?~

Ans. Provision should be made for tightening the belt.

[Illustration: FIG. 1,382.--Allis-Chalmers pedestal type, belted
alternator. The bearings are of the ring oiling form with large oil
reservoirs. The bearings have spherical seats and are self aligning.]

~Ques. How is this done?~

Ans. Sometimes by an idler pulley, but usually by mounting the machine
on a sub-base provided with slide rails, as in fig. 1,381, the belt
being tightened by use of a ratchet screw which moves the machine along
the base.

[Illustration: FIG. 1,383.--Diagram illustrating rule for horse power
transmitted by belts. _A single belt travelling at a speed of 1,000
feet per minute will transmit one horse power; a double belt will
transmit twice that amount_, assuming that the thickness of a double
belt is twice that of a single belt. This is conservative practice, and
a belt so proportioned will do the work in practically all cases. The
above rule corresponds to a pull of 33 lbs. per inch of width. Many
designers proportion single belts for a pull of 45 lbs. For double
belts of average thickness, some writers say that the transmitting
efficiency is to that of single belts as 10 is to 7. This should not
be applied to the above rule for single belts, as it will give an
unnecessarily large belt.]

~Ques. Give a rule for obtaining the proper size of belt to deliver a
given horse power.~

Ans. _A single belt travelling at a speed of one thousand feet per
minute will transmit one horse power; a double belt will transmit twice
that amount._

    This corresponds to a working strain of 33 lbs. per inch of
    width for single belt, or 66 lbs. for double belt.

    Many writers give as safe practice for single belts in good
    condition a working tension of 45 lbs. per inch of width.

~Ques. What is the best speed for maximum belt economy?~

Ans. From 4,000 to 4,500 feet per minute.

    EXAMPLE.--What is the proper size of double belt for an
    alternator having a 16 inch pulley, and which requires 50 horse
    power to drive it at 1,000 revolutions per minute full load?

    The velocity of the belt is

  circumference in feet × revolutions = feet per minute

  16
  -- × 3.1416 × 1,000 = 4,188.
  12

    Horse power transmitted per inch width of double belt at 4,188
    feet speed

        4,188
    2 × ----- = 8.38.
        1,000

[Illustration: FIG. 1,384.--Fort Wayne revolving field belt driven
alternator. It is designed for belted exciter, having a shaft extension
at the collector ring end for exciter driving pulley.]

    Width of double belt for 50 horse power

                     50 ÷ 8.38 = 5.97, say 6 inch.


~Ques. What are the advantages of chain drive?~

Ans. The space required is much less than with belt drive, as the
distance between centers may be reduced to a minimum. It is a positive
drive, that is, there can be no slip. Less liability of becoming
detached, and, because it is not dependent on frictional contact, the
diameters of the sprockets may be much less than pulley diameter for
belt drive.

[Illustration: FIGS. 1,385 and 1,386.--Diagram showing the distinction
between _direct connected_ and _direct coupled_ units. In a direct
connected unit, fig. 1,385, the engine and generator are permanently
connected on one shaft, there being one bed plate upon which both are
mounted. An engine and generator are said to be direct coupled when
each is independent, as in fig. 1,386 being connected solely by a jaw
or friction clutch or equivalent at times when it is desired to run the
generator. At other times the generator may be disconnected and the
engine run to supply power for other purposes.]

~Ques. What are some objections?~

Ans. A lubricant is required for satisfactory operation, which causes
more or less dirt to collect on the chain, requiring frequent
cleaning; climbing of teeth when links and teeth become worn; noise and
friction.

[Illustration: FIG. 1,387.--Engberg direct connected, or "engine type"
alternator. In many places direct connected units are used, owing to
the great saving in floor space, convenience of operation, and absence
of belts.]

~Direct Connected Alternators.~--There are a large number of cases
where economy of space is of prime importance, and to meet this
condition the alternator and engine are direct connected, meaning, that
there is no intermediate gearing such as belt, chain, etc., between
engine and alternator.

    One difficulty encountered in the direct connection of engine
    and alternator is the fact that the most desirable rotative
    speed of the engine is less than that of the alternator.
    Accordingly a compromise is made by raising the engine speed
    and lowering the alternator speed.

    The insistent demand for direct connected units in the small
    and medium sizes, especially for direct current units, was the
    chief cause resulting in the rapid and high development of what
    is known as the "high speed automatic engine."

    Increasing the engine speed means that more horse power is
    developed for any given cylinder dimensions, while reducing
    the speed of the generator involves that the machine must be
    larger for a given output, and in the case of an alternator
    more poles are required to obtain a given frequency, resulting
    in increased cost.

    The compactness of the unit as a whole, simplicity, and general
    advantages are usually so great as to more than offset any
    additional cost of the generator.

[Illustration: FIG. 1,388.--Crocker-Wheeler 2,000 kva. 2,400 volt
coupled type alternator. The coupled type of alternator is desirable
for use with steam, gas, and oil engines, and water wheels where it
is inconvenient to mount the alternator on the engine shaft or to
extend the engine base to accommodate a bearing. This type consists
of alternator complete with shaft and bearings similar to belt type
machines, but with bearings not necessary designed for the side pull of
belts.]

~Ques. What is the difference between a direct connected and a direct
coupled unit?~

Ans. A direct connected unit comprises an engine and generator
permanently connected; direct coupling signifies that engine and
generator are each complete in itself, that is, having two bearings,
and are connected by some device such as friction clutch, jaw clutch,
or shaft coupling.

~Revolving Armature Alternators.~--This type of alternator is one
which has its parts arranged in a manner similar to a dynamo, that is,
the armature is mounted on a shaft so it can revolve while the field
magnets are attached to a circular frame and arranged radially around
the armature, as shown in fig. 1,389. It may be single or polyphase,
belt driven, or direct connected.

[Illustration: FIG. 1,389.--Revolving armature alternator. Revolving
armatures are suitable for machines generating current at comparatively
low pressure, as no difficulty is experienced in collecting such
current. Revolving armature alternators are also suitable for small
power plants, isolated lighting plants, where medium or small size
machines are required.]

~Ques. When is the revolving type of armature used and why?~

Ans. It is used on machines of small size because the pressure
generated is comparatively low and the current transmitted by the
brushes small, no difficulty being experienced in collecting such a
current.

[Illustration: FIG. 1,390.--Ring wound dynamo arranged as alternator by
replacing commutator with collector rings connected to the winding at
points 180° apart.]

~Ques. Could a dynamo be converted into an alternator?~

Ans. Yes.

~Ques. How can this be done?~

Ans. By placing two collector rings on one end of the armature and
connecting these two rings to points in the armature winding 180°
apart, as shown in fig. 1,390.

~Ques. Would such arrangement as shown in fig. 1,390 make a desirable
alternator?~

Ans. No.

    Alternating current windings are usually different from those
    used for direct currents. One distinction is the fact that
    a simple open coil winding may be, and often is, employed,
    but the chief difference is the intermittent action of the
    inductors.

    In a direct current Gramme ring winding a certain number of
    coils are always active, while those in the space between the
    pole pieces are not generating. In this way a practically
    steady pressure is produced by a large fraction of the coils.

    In the case of an alternator all of the coils are either active
    or inactive at one time. Hence, the winding need cover only as
    much of the armature as is covered by the pole pieces.

[Illustration: FIG. 1,391.--Engberg alternating current generating set;
shown also in cross section in fig. 1,387. The set comprises a vertical
engine and alternator, direct connected and placed on one base. The
lubrication system comprises an oil pump situated in the base of the
engine, pumping the oil from an oil reservoir up into a sight feed
oil cup which leads to a distributing oil trough on the inside of the
engine frame, from here oil pipes lead to all movable bearings, which
are grooved to insure proper distribution of oil. The oil is drained
from bearings into the base, filtered and re-pumped. A water shed
partition is provided in the engine frame, preventing any water passing
from the cylinder down into the engine base and mixing with the oil,
consequently leaving good, clean oil in the oil reservoir at all times.
The details of the lubrication system are shown in fig. 1,387.]

~Revolving Field Alternators.~--In generating an electric current by
causing an inductor to cut magnetic lines, it makes no difference
whether the cutting of the magnetic lines is effected by moving an
inductor across a magnetic field or moving the magnetic field across
the inductor.

[Illustration: FIG. 1,392.--Allis-Chalmers revolving field
self-contained belted type alternator.]

_Motion is purely a relative matter_, that is, an object is said to
move when it changes its position with some other object regarded as
stationary; it may be moving with respect to a second object, and at
the same time be at rest with respect to a third object. Thus, a dory
has a speed of four miles per hour in still water; if it be run up
stream against a current flowing four miles per hour it would move at
that speed with respect to the water, yet remain at rest with respect
to the earth.

It must be evident then that motion, as stated, being a purely relative
matter, it makes no difference whether the armature of a generator
move with respect to the field magnets, or the field magnets move with
respect to the armature, so far as inducing an electric current is
concerned.

[Illustration: FIG. 1,393.--Marine view, showing that motion is purely
a relative matter. In order that there may be motion something must be
regarded as being stationary. In the above illustration a catboat is
shown at anchor in a stream which is flowing at a rate of four miles
per hour in the direction of the arrow. The small dory running at a
speed of four miles per hour against the current is _moving_ at that
velocity _relative_ to the current, yet is at a standstill relative to
the catboat. In this instance both catboat and dory are moving with
respect to the water if the latter be regarded as stationary. Again if
the earth be regarded as being stationary, the two boats are at rest
and the water is moving relative to the earth.]

For alternators of medium and large size there are several reasons why
the armature should be stationary and the field magnets revolve, as
follows:

1. By making the armature stationary, superior insulation methods may
be employed, enabling the generation of current at very much higher
voltage than in the revolving armature type.

2. Because the difficulty of taking current at very high pressures from
collector rings is avoided.

    The field current only passes through the collector rings.
    Since the field current is of low voltage and small in
    comparison with the main current, small brushes are sufficient
    and sparking troubles are avoided.

[Illustration: FIG. 1,394.--Diagram showing essential parts of
a revolving field alternator and method of joining the parts in
assembling.]

3. Only two collector rings are required.

4. The armature terminals, being stationary, may be enclosed
permanently so that no one can come in contact with them.

~Ques. What names are usually applied to the armature and field magnets
with respect to which moves?~

Ans. The "stator" and the "rotor."

    The terms armature and field magnets are to be preferred to
    such expressions. An armature is an armature, no matter whether
    it move or be fixed, and the same applies to the field magnets.
    There is no good reason to apply other terms which do not
    define the parts.

~Ques. Explain the essential features of a revolving field alternator.~

Ans. The construction of such alternators is indicated in the diagram,
fig. 1,394. Attached to the shaft is a field core, which carries the
latter, consisting of field coils fitted on pole pieces which are
dovetailed to the field core. The armature is built into the frame and
surrounds the magnets as shown. The field current, which is transmitted
to the magnets by slip rings and brushes, consists of direct current of
comparatively low pressure, obtained from some external source.

[Illustration: FIG. 1,395.--Western Electric stationary armature and
frame of engine driven alternator. It is of cast iron and surrounds the
laminated iron core in which the armature windings are embedded. Heavy
steel clamping fingers hold the core punchings in place and numerous
ventilating ducts are provided in the core at frequent intervals to
allow free circulation of cool air. The armature coils are form wound,
insulated, and retained in the core slots by means of wedges.]

~Inductor Alternators.~--In this class of alternator both armature and
field magnets are stationary, a current being induced in the armature
winding by the action of a so called inductor in moving through the
magnetic field so as to periodically vary its intensity.

[Illustration: FIGS. 1,396 and 1,397.--Elementary inductor alternator;
diagram showing principle of operation. It consists of a field magnet,
at the polar extremities of which is an armature winding both being
stationary as shown. Inductors consisting of iron discs are arranged
on a shaft to rotate through the air gap of the magnet poles. Now in
the rotation of the inductors, when any one of them passes through
the air gap as in fig. 1,396, the reluctance or magnetic resistance
of the air gap is greatly reduced, which causes a corresponding
increase in the number of magnetic lines passing through the armature
winding. Again as an inductor passes out of the air gap as in fig.
1,397, the number of magnetic lines is greatly reduced; that is, when
an inductor is in the air gap, the magnetic field is dense, and when
no inductor is in the gap, the field is weak; a variable flux is thus
made to pass through the armature winding, inducing current therein.
The essential feature of the inductor alternator is that iron only is
revolving, and as the design is usually homopolar, the magnetic flux
in its field coils is not alternating, but undulating in character.
Thus, with a given maximum flux through each polar mass, the total
number of armature turns required to produce a given voltage is just
twice that which is required in an alternator having an alternating
instead of an undulating flux through its field windings. The above
and the one shown in figs. 1,398 and 1,399 are examples of real
inductor alternators, those shown in the other cuts are simply so
called inductor alternators, the distinction being that, as above, the
inductor constitutes no part of the field magnet.]

~Ques. What influence have the inductors on the field flux?~

Ans. They cause it to undulate; that is, the flux rises to a maximum
and falls to a minimum value, but does not reverse.

~Ques. How does this affect the design of the machine as compared with
other types of alternator?~

Ans. With a given maximum magnetic flux through each polar mass, the
total number of armature turns necessary to produce a given pressure
is twice that which is required in an alternator having an alternating
flux through its armature windings.

[Illustration: FIGS. 1,398 and 1,399.--A low tension ignition system
with an inductor magneto of the oscillating type. The inductor E is
rotated to and fro by means of a link R, one end of which is attached
to the inductor crank, and the other to the igniter cam C. Two views
are shown: immediately before and after sparking. S is the grounded
electrode of the igniter; T an adjustable hammer which is secured in
position by a lock nut N.]

~Ques. Is the disadvantage due to the necessity of doubling the number
of armature turns compensated in any way?~

Ans. Yes, the magnetic flux is not reversed or entirely changed in
each cycle through the whole mass of iron in the armature, the abrupt
changes being largely confined to the projections on the armature
surface between the coils.

~Ques. What benefit results from this peculiarity?~

Ans. It enables the use of a very high magnetic flux density in the
armature without excessive core loss, and also the use of a large flux
without an excessive increase in the amount of magnetic iron.

    The use of a large flux permits a reduction in the number
    of armature turns, thus compensating, more or less, for the
    disadvantage due to the operation of only one-half of the
    armature coils at a time.

[Illustration: FIGS. 1,400 and 1,401.--One form of inductor alternator.
As shown, the frame carries the stationary armature, which is of
the slotted type. Inside of the armature is the revolving inductor,
provided with the projections built up of wrought iron or steel
laminations. The circular exciting coil is also stationary and
encircles the inductor, thus setting up a magnetic flux around the path
indicated by the dotted line, fig. 1,401. The projecting poles are all,
therefore, of the same polarity, and as they revolve, the magnetic
flux sweeps over the coils. Although this arrangement does away with
collector rings, the machines are not so easily constructed as other
types, especially in the large sizes. The magnetizing coil becomes
large and difficult to support in place, and would be hard to repair
in case of breakdown. Inductor alternators have become practically
obsolete, except in special cases, as inductor magnetos used for
ignition and other purposes requiring a very small size machine. The
reasons for the type being displaced by other forms of alternator are
chiefly because only half as great a pressure is obtained by a flux of
given amount, as would be obtained in the ordinary type of machine. It
is also more expensive to build two armatures, to give the same power,
than to build one armature. This type has still other grave defects,
among which may be mentioned enormous magnetic leakage, heavy eddy
current losses, inferior heat emissivity, and bad regulation.]

~Classes of Inductor Alternator.~--There are two classes into which
inductor alternators may be divided, based on the mode of setting of
their polar projections:

1. Homopolar machines;

2. Heteropolar machines.

    ~Homopolar Inductor Alternators.~--In this type the positive
    polar projections of the inductors are set opposite the
    negative polar projections as shown in fig. 1,402. When the
    polar projections are set in this manner, the armature coils
    must be "staggered" or set displaced along the circumference
    with respect to one another at a distance equal to half the
    distance from the positive pole to the next positive pole.

[Illustration: FIGS. 1,402 and 1,403.--Homopolar and heteropolar
"inductors". Homopolar inductors have their N and S poles opposite each
other, while in the heteropolar type, they are "staggered" as shown.]

    ~Heteropolar Inductor Alternators.~--Machines of this class are
    those in which the polar projections are themselves staggered,
    as shown in fig. 1,403, and therefore, do not require the
    staggering of the armature coils. In this case, a single
    armature of double width may be used, and the rotating inductor
    then acts as a _heteropolar magnet_, or a magnet which presents
    alternatively positive and negative poles to the armature,
    instead of presenting a series of poles of the same polarity as
    in the case of a _homopolar magnet_.

    ~Use of Inductor Alternators.~--Morday originally designed
    and introduced inductor alternators in 1866. They are not the
    prevailing type, as their field of application is comparatively
    narrow. They have to be very carefully designed with regard to
    magnetic leakage in order to prevent them being relatively
    too heavy and costly for their output, and too defective with
    respect to their pressure regulation, other defects being heavy
    eddy current losses and inferior heat conductance.

~Hunting or Singing in Alternators.~--Hunting is a term applied to
the state of two parallel connected alternators running out of step,
or not synchronously, that is, "see sawing." When the current wave of
an alternator is peaked and two machines are operated in parallel it
is very difficult to keep them in step, that is in synchronism. Any
difference in the phase relation which is set up by the alternation
will cause a local or synchronizing current to flow between the
two machines and at times it becomes so great that they must be
disconnected.

[Illustration: FIG. 1,404.--Revolving field of Fort Wayne alternator
equipped with _amortisseur winding_. The object of this winding is to
check any tendency toward _hunting_ when the alternator is to be run
as a synchronous motor, either for rotary condenser or power service.
The amortisseur winding consists of heavy copper bars, placed around
and through the pole faces and short circuited at the ends by heavy
copper rings; it serves as a starting winding to bring the rotor up to
speed as an induction motor, and also serves as a damping device to
neutralize any tendency toward "hunting" caused by variation in speed
of the generator supplying the current.]

Alternators which produce a smooth current wave and are maintained at
uniform speed by properly designed governors, operate fairly well in
parallel, but are not entirely free from hunting, and other means are
provided to overcome the difficulty.

When heavy copper flanges, called dampers, are put over the polar
projections or copper bars laid in grooves on the pole face and short
circuited by connecting rings (called amortisseur winding), the
powerful induced currents which are produced when the alternators get
out of step tend to quickly re-establish the phase relation.

[Illustration: FIG. 1,405.--Westinghouse field with amortisseur or
"damper" winding for 75 kva. and larger belted alternators, which
prevents hunting and reduces eddy currents in the pole pieces. The
copper bars of the amortisseur cage winding are arranged in partially
closed slots in the pole pieces.]

Two examples of a field provided with amortisseur[4] winding are shown
in figs. 1,404 and 1,405.

[4] NOTE.--Amortisseur windings are often erroneously called "squirrel
cage" windings on account of similarity of construction. The latter
term should be reserved for its proper significance as being the name
of the type of armature winding generally used for induction motors,
the name being suggested by the resemblance of the finished armature
to the wheel of a squirrel cage. A comparison of figs. 1,405 and 1,746
will show the distinction. In a squirrel cage winding there is a large
number of bars uniformly spaced; an amortisseur winding consists of
a comparatively small number of bars, usually unevenly spaced, that
is they are divided into groups with considerable space between the
groups, as in fig. 1,405, and less pronounced in fig. 1,404. The bars
are short circuited by rings the same as in squirrel cage winding.

[Illustration: FIG. 1,406.--Diagram of monocyclic system, showing
monocyclic armature and transformer connections. The monocyclic system
is a single phase system primarily intended for the distribution of
lights with an incidental load of motors. The lighting load is entirely
connected to one single phase circuit, and the motors are started and
operated from this circuit with the assistance of the teazer wire.
The long coil indicates the main winding of the armature, which is
similar in its arrangement and size to the ordinary armature winding
of a single phase alternator. The short coil which connects at one
end to the middle point of the coil above mentioned, and at the other
to a third collector ring is called the "teazer" coil. Its use is to
generate a pressure in quadrature with that of the main coil. This
pressure is combined with the main pressure of the alternator by
transformers, so as to give suitable phase relations for operating
induction motors. In the diagram the voltage has been assumed to be
2,080 volts, and the voltages marked to correspond with the generated
pressure. The coils of the alternator armature are connected, as
shown, to two main leads and to a teazer wire. Between each end of the
main coil and the end of the teazer coils, a resultant pressure is
generated. These resultants are about 12 per cent. larger than half the
main pressure. They also have a phase difference.]

~Monocyclic Alternators.~--This type of alternator was designed
prior to the introduction of the polyphase systems, to overcome the
difficulties encountered in the operation of single phase alternators
as motors. A single phase alternator will not start from rest as a
motor, but must first be started and brought up to the proper speed
before being connected with single phase mains. This condition
constituted a serious difficulty in all cases where the motor had to be
stopped and started at comparatively frequent intervals.

[Illustration: FIG. 1,407.--Monocyclic system diagram showing
transformer connections.]

[Illustration: FIG. 1,408.--Diagram showing section of monocyclic
alternator armature illustrating the armature winding. The main coils
are wound on every other tooth, and the teazer coils are placed in
quadrature with them, as shown.]

The monocyclic alternator is a single phase machine provided with
an additional coil, called a _teaser coil_, wound in two phase
relationship with, and connected to the center of the main single
phase coil. It is provided with three collector rings; two for the
single phase coil, and one for the free end of the teaser coil.

[Illustration: FIG. 1,409.--Diagram showing connections of General
Electric Monocyclic alternator. For 2,300 volt machine, connect as
shown by solid lines. For 1,150 volt machine, omit connections A to
B, C to D, E to F, and G to H, and connect as shown by dotted lines.
The armature of a standard monocyclic alternator rotates in a counter
clockwise direction facing the commutator. When the alternator is
loaded, the voltage between the teazer coil and the two terminals of
the main coil may be different; therefore, it is necessary to have the
commutator connected in corresponding ends of the main coil. If the
machine has not been arranged for clockwise rotation, the following
change in the connections on the commutator-collector must be made if
the machine is to be run in parallel with another. Fig. 1,410 shows the
connections of monocyclic alternators. In fig. 1,409, the studs on the
commutator-collector marked 1 and 6 are the terminals of the main coil.
These should be reversed. The numbers are stamped on the ends of the
stud and may be seen with the assistance of a mirror. By reference to
this diagram it is a simple matter to trace out the connections with
a magneto, after the armature leads are disconnected and the brushes
raised.]

By this arrangement ordinary single phase incandescent lighting can be
accomplished by means of a single pair of wires taken from the single
phase coil. Where three phase motors have to be operated, however, a
third wire, called the power wire, which is usually smaller than the
main single phase wires is carried to the point at which the motor is
located, and by the use of two suitably connected transformers three
phase currents are obtained from the combined single phase and power
wires for operating the motors.

[Illustration: FIG. 1,410.--Diagram showing connections of General
Electric monocyclic alternator. The solid lines show standard
connections for counter-clockwise rotation; the broken lines show
connection changed for clockwise rotation.]

Fig. 1,406 shows the connections of the monocyclic system and it is
only necessary to carry the teaser wire into buildings where motors are
to be used.

~Armature Reaction.~--Every conductor carrying a current creates a
magnetic field around itself, whether it be embedded in iron or lie
in air. Armature inductors, therefore, create magnetic fluxes around
themselves, and these fluxes will, in part, interfere with the main
flux from the poles of the field magnet. The effect of these fluxes is:

1. To distort the field, or

2. To weaken the field.

    These disturbing fluxes form, in part, stray fluxes linked
    around the armature inductors tending to choke the armature
    current.

[Illustration: FIGS. 1,411 and 1,412.--Section of armature and field
showing _distorting effect_ of armature reaction on the field. When
a coil is opposite a pole as in fig. 1,411, no current is flowing
(assuming no self induction) and the field is undisturbed, but, as the
inductors pass under a pole face as in fig. 1,412, current is induced
in them, and lines of force are set up as indicated by the dotted
lines. This distorts the main field so that the lines of force are
crowded toward the forward part of the pole face as shown.]

~Ques. Explain how the field becomes distorted by armature reaction.~

Ans. Considering a slotted armature and analyzing the electrical
conditions as the inductors move past a pole piece, it will be
observed: 1, when the coil is in the position shown in fig. 1,411, the
current will be zero, assuming no armature self-induction, consequently
for this position the armature coil has no disturbing effect upon the
field set up by the field magnet; 2, when the inductors have moved
under the pole face, as in fig. 1,412, currents will be induced in
them, and they will tend to set up a magnetic field as indicated by the
dotted lines, and in direction, by the arrow heads. The effect of this
field will be to distort the main field, strengthening one side of the
pole and weakening the other side.

[Illustration: FIG. 1,413.--Section of armature and field showing
_weakening effect_ of armature reaction in the field. Self-induction
being present (as it almost always is), the current lags more or less
behind the pressure, so that when the coil is in the position of zero
induction, as shown, the current has not yet come to rest. Accordingly,
lines of force (indicated by the dotted lines) are set up by the
current flowing through the coils which are in opposition to the field,
thus weakening the latter. The dots and crosses in inductor sections,
have their usual significance in defining the direction of current,
representing respectively the heads and tails of arrows.]

~Ques. Explain how the field becomes weakened by armature reaction.~

Ans. In all armatures there is more or less inductance which causes the
current to lag behind the pressure a corresponding amount. Accordingly,
the current does not stop flowing at the same instant that the pressure
becomes zero, therefore, when the coil is in the position of zero
pressure, as in fig. 1,413, the current is still flowing and sets up a
magnetic field which opposes the main field as indicated by the dotted
arrows, thus weakening the main field.

~Ques. In what kind of armature is this effect especially pronounced?~

Ans. In slotted armatures provided with coils of a large number of
turns.

[Illustration: FIG. 1,414.--Section of armature and field showing
_strengthening effect_ of armature reaction when the current leads the
pressure. If the circuit contain an excess of capacity the current will
lead the pressure, so that when the coil is in the position of zero
induction, as shown, the current will have come to rest and reversed.
Accordingly, lines of force (indicated by the dotted lines) are set
up by the current flowing through the coil and which are in the same
direction as the lines of force of the field, thus strengthening the
latter.]

~Ques. What would be the effect if the current lead the pressure?~

Ans. It would tend to strengthen the field as shown in fig. 1,414.

    The value of the armature ampere turns which tend to distort
    and to diminish or augment the effect of the ampere turns on
    the field magnet is sometimes calculated as follows:

      .707 × I × T × P
  A = ----------------
          _s_

    in which

  A = armature ampere turns;
  I = current per phase;
  T = turns per pole per phase;
  P = number of phases;
  _s_ = product of the distribution and pitch factors of the winding.

    This value of ampere turns, combined at the proper phase angle
    with the field ampere turns gives the value of the ampere turns
    available for producing useful flux.

[Illustration: FIG. 1,415.--Fort Wayne separately excited belt driven
alternator, a form adapted for installation in small plants where low
power factor is to be encountered. This condition exists in a line
where power is supplied to induction motors, transformers or other
inductive apparatus. The type here shown is built in sizes from 37½
kw. to 200 kw., 60 cycles, two or three phases and voltages of 240,
480, 600, 1,150 or 2,300 volts. They may be operated as single phase
alternators by using two of the phases and may then be rated at 70
per cent. of the polyphase rating. The field is excited by direct
current at a pressure of 125 volts. These alternators may be used
as synchronous motors and for this duty are fitted with amortisseur
winding in the pole faces which does not interfere with their use as
alternators.]

    ~Single Phase Reactions.~--Unlike three phase currents, a
    single phase current in an alternator armature produces a
    periodic disturbance of the flux through the machine. In
    the magnet system this disturbance is of twice the normal
    frequency, while in the armature core it is the

[Illustration: FIGS. 1,416 to 1,425.--Diagrams illustrating
superposition of fields. In the figures magnetic curves representing
the effect of the armature currents in several different cases are
superposed upon the magnetic curves assumed to be due to the field
magnet. The uppermost line shows the primary field due to the exciting
coils on the magnet poles. They are shown passing into the armature
teeth in two principal positions, where the middle of a pole is: 1,
opposite a tooth, and 2, opposite a slot. In the second line is shown
the field due to the armature currents assuming no lag, and that the
magnets are not excited. If there be no lag, the places of strongest
current will be opposite the poles. As shown in the right hand figure
when the current in one phase C, is at its maximum, those in the other
phases A and B will be of half strength. In the left hand figure when
the current in one phase B, is at its zero value, those in the other
phases will be of equal value, or 87 per cent. of the maximum. In the
third line is shown the effect of superposing these fields due to the
current upon those due to the magnets as depicted in the first line.
Inspection of this resultant field shows how the armature current
distorts the field without altering the total number of lines per
pole. In the fourth and fifth lines are shown the effects of a lagging
current. A lag of 90° is assumed; and in that case the maximum current
occurs in any inductor one quarter period after the pole has passed,
or at a distance of half a pole pitch behind the middle point of the
pole, as in the fourth line. When these armature fields are superposed
on those of the magnets in the first line the resultant fields are
those depicted in the fifth line. On inspection it will be seen that
in this case there is no distortion, but a diminution of the flux from
each pole, as the lines due to the armature currents, tending to pass
through the pole cores in the sense opposite to those of the primary
magnetism, must be deducted from the total. The twelve lines per pole
are correspondingly reduced to eight; and, of these eight, four go
astray constituting a leakage field. This illustrates the effect of a
lagging current in demagnetizing the field magnets and in increasing
the dispersion.] /# same as the normal frequency. In both cases the
eddy currents which are set up, produce a marked increase in the load
losses, and thus tend to give the machine a higher temperature rise on
single phase loading.

Designers continue to be singularly heedless of these single phase
reactions, resulting in many cases of unsatisfactory single phase
alternators. Single phase reactions distort the wave form of the
machine.

~Three Phase Reactions.~--The action of the three phase currents in
an alternator is to produce a resultant field which is practically
uniform, and which revolves in synchronism with the field system. The
resultant three phase reaction, because of its uniformity, produces no
great increase in the load losses of the machine, the small additional
losses which are present being due to windings not being placed
actually in space at 120°, and to the local leakage in the teeth.

[Illustration: FIG. 1,426.--Diagram showing lateral field between
adjacent poles.]

~Magnetic Leakage.~--In the design of alternators the drop of voltage
on an inductive load is mainly dependent upon the magnetic leakages,
primary and secondary. They increase with the load, and, what is of
more importance, they increase with the fall of the power factor of the
circuit on which they may be working. This is one reason why certain
types of alternator, though satisfactory on a lighting circuit, have
proved themselves unsatisfactory when applied to a load consisting
chiefly of motors.

The designer must know the various causes which contribute to leakage
and make proper allowance.

In general, to keep the leakage small, the pole cores should be short,
and of minimum surface, the pole shoes should not have too wide a span
nor be too thick, nor present needless corners, and the axial length
of the pole face and of the armature core should not be too great in
proportion to the diameter of the working face.

[Illustration: FIGS. 1,427 and 1,428.--Diagram showing respectively
the character of stray field between adjacent straight poles, and
between adjacent poles with shoes. Across the slightly V-shaped spaces
the stray field passes in lines that, save near the outer part, are
nearly straight. Quite straight they would not be, even were the sides
parallel, because the difference of magnetic pressure increases from
the roots towards the pole ends. At the roots, where the cores are
attached to the yoke, the magnetic pressure difference is almost zero.
It would be exactly zero if there were not a perceptible reluctance
offered by the joints and by the metal of the yoke. The reluctance of
the joint causes a few of the lines to take paths through the air by a
leakage which adds to the useful flux. At the tops of the cores there
is a difference of magnetic pressure equal to the sum of the ampere
turns on the two cores, tending to drive magnetic lines across. This
difference of magnetic pressure increases regularly all the way up
the cores from root to top; hence, the average value may be taken as
equal to the ampere turns on one core. The stray field, therefore, will
steadily increase in density from the bottom upwards. In addition to
this stray field between the pole cores there is also a stray field
between the projecting tips or edges of the pole shoes, as shown in
fig. 1,428. In some machines the dispersion due to the pole shoes is
greater than that between the flanks of the cores.]

To keep the increase of leakage between no load and full load from
undue magnitude, it is required that armature reactions shall be
relatively small, that the peripheral density of the armature current
(ampere-conductors per inch) be not too great, and that the pole cores
be not too highly saturated when excited for no load.

[Illustration: FIG. 1,429.--Lincoln revolving field alternator. The
frame has openings for ventilation, the fanning action of the pole
pieces causing a current of air to pass not only over the end of the
windings, as is usual with other designs, but also through ventilating
slots in the windings themselves. The armature core laminations are
annealed after punching and before assembling to guard against the
crystalizing effect of the punching. The armature coils are form
wound and insulated before being placed in the slot. There is also
slot insulation which is put in the slot previous to inserting the
coil. When the winding is completed, it is tested with a pressure of
4 to 10 times the normal voltage of the machine. The bearings are
self-aligning. The machine is normally designed to operate at a power
factor of approximately 70 per cent., which means that at that power
factor, the armature and fields at full load will heat equally. If it
have a higher power factor than 70 per cent. it means that the field
windings will run considerably cooler than the armature windings with
full load. If the power factor be lower than this, it will mean that
the field windings will run hotter than the armature on full load;
however, the machine is designed so that harmful heating does not occur
on full load with greater power factor than 40 per cent.]

The general character of the stray field between adjacent poles is
shown in figs. 1,427 and 1,428 for straight poles and those having
shoes.

~Field Excitation of Alternators.~--The fields of alternators require
a separate source of direct current for their excitation, and this
current should be preferably automatically controlled. In the case of
alternators that are not self exciting, the dynamo which generates the
field current is called the _exciter_.

    The excitation of an alternator at its rated overload and .8
    power factor would not, in some cases, if controlled by hand,
    exceed 125 volts, although, in order to make its armature
    voltage respond quickly to changes in the load and speed, the
    excitation of its fields may at times be momentarily varied by
    an automatic regulator between the limits of 70 and 140 volts.

[Illustration: FIG. 1,430.--Western Electric armature for self-excited
alternator. The main winding is placed at the bottom of the slots, each
coil being surrounded by an armour of horn fibre. The exciter winding
occupies a very small portion of the slot, being placed on top of the
main winding, and connected to the commutator immediately in front of
the core and between core and collector rings as shown.]

    The exciter should, in turn, respond at once to this demand
    upon its armature, and experience has shown that to do this
    its shunt fields must have sufficient margin at full load
    to deliver momentarily a range from 25 to 160 volts at its
    armature terminals.

    It is obvious from the above that an exciter suitable for use
    with an automatic regulator must commutate successfully over a
    wide range in voltage, and, if properly designed, have liberal
    margins in its shunt fields and magnetic circuits.

    Alternator fields designed for and operated at unity power
    factor have often proved unsatisfactory when the machines were
    called upon to deliver their rated kva. at .8 power factor or
    lower. This is due to the increased field current required at
    the latter condition and results, first, in the overheating of
    the fields and, second, in the necessity of raising the direct
    current exciting voltage above 125 volts, which often requires
    the purchase of new exciters.

~Ques. What is a self-excited alternator?~

Ans. One whose armature has, in addition to the main winding, another
winding connected to a commutator for furnishing direct field exciting
current, as shown in fig. 1,430.

[Illustration: FIG. 1,431.--Frame, bed plate and armature winding for
Westinghouse bracket bearing polyphase alternator.]

~Ques. How is a direct connected exciter arranged?~

Ans. The exciter armature is mounted on the shaft of the alternator
close to the spider hub, or in some cases at a distance sufficient to
permit a pedestal and bearing to be mounted between the exciter and
hub. In other designs the exciter is placed between the bearing and
hub.

    Figs. 1,432 and 1,433 are examples of direct connected exciter
    alternators, in fig. 1,432 the exciter being placed between the
    field hub and bearing, and in fig. 1,433, beyond the bearing.

~Ques. What is the advantage of a direct connected exciter?~

Ans. Economy of space.

    This is apparent by comparing figs. 1,432 and 1,433 with fig.
    1,434, which shows a belted exciter.

[Illustration: FIG. 1,432.--General Electric alternator with direct
connected exciter mounted on shaft between field hub and bearing. In
the smaller sizes, the magnet frame is bolted to the bearing bracket,
but in the larger sizes special construction is used depending upon
the conditions to be met. The exciters are capable of furnishing the
desired excitation for low power factors.]

~Ques. What is the disadvantage of a direct connected exciter?~

Ans. It must run at the same speed as the alternator, which is slower
than desirable, hence the exciter must be larger for a given output
than the gear driven type, because the latter can be run at high speed
and accordingly be made proportionally smaller.

~Ques. What form of gear is generally used on gear driven exciters?~

Ans. Belt gear.

[Illustration: FIG. 1,433.--Fort Wayne alternator with direct connected
exciter mounted on the field shaft at such distance as to permit a
pedestal and bearing to be mounted between the exciter and revolving
field. In the view, the bearing is hidden by the exciter, only the foot
of the pedestal being visible.]

~Ques. What are the advantages of gear driven exciters?~

Ans. Being geared to run at high speed, they are smaller and therefore
less costly than direct connected exciters. In large plants containing
a number of alternators one exciter may be used having sufficient
capacity to excite all the alternators, and which can be located at any
convenient place.

~Ques. What is the disadvantage of gear driven exciters?~

Ans. The space occupied by the gear.

[Illustration: FIG. 1,434.--Diagram showing a Westinghouse 50 kva.,
2,400 volt, three phase, 60 cycle revolving field separately excited
alternator direct connected to a steam engine. The exciter is belted to
the alternator shaft, the driving pulley being located outside the main
bearing. The small pulley on the exciter gives an indication of its
high speed as compared with that of the alternator.]

    In the case of a chain drive very little space is required, but
    for belts, the drive generally used, there must be considerable
    distance between centers for satisfactory transmission.

~Slow Speed Alternators.~--By slow speed is here understood relatively
slow speed, such as the usual speeds of reciprocating engines. A slow
speed alternator is one designed to run at a speed slow enough that
it may be direct connected to an engine. Such alternators are of
the revolving field type and a little consideration will show that
they must have a multiplicity of field magnets to attain the required
frequency.

In order that there be room for the magnets, the machine evidently must
be of large size, especially for high frequency.

[Illustration: FIG. 1,435.--Crocker-Wheeler 350 kva., slow speed
alternator direct connected to a Corliss engine. In front is seen a
belted exciter driven from a pulley on the main shaft between the
alternator and the large band wheel. The latter serves to give the
additional fly wheel effect needed for close speed regulation.]

    EXAMPLE.--How many field magnets are required on a two
    phase alternator direct connected to an engine running 240
    revolutions per minute, for a frequency of 60?

    An engine running 240 revolutions _per minute_ will turn

    240 ÷ 60 = 4 revolutions _per second_.

    A frequency of 60 requires

    60 ÷ 4 = 15 cycles per phase per revolution, or

    15 × 2 = 30 poles per phase. Hence for a two phase alternator
    the total number of poles required is

                              30 × 2 = 60.

    It is thus seen that a considerable length of spider rim is
    required to attach the numerous poles, the exact size depending
    upon their dimensions and clearance.

[Illustration: FIG. 1,436.--Three Crocker-Wheeler 75 kva., slow speed
alternators direct connected to high speed engines. The alternator is
styled slow speed although connected to a high speed engine, because
what is considered high engine speed is slow speed for alternator
operation. The alternators have direct connected exciters which are
plainly seen in the illustration placed on an extension of bearing
pedestal. Direct connected exciters on units of this kind do not, as a
rule, assume too bulky proportions, because of the high engine speed.]

~Fly Wheel Alternators.~--The diameter of the revolving fields on
direct connected alternators of very large sizes becomes so great that
considerable fly wheel effect is obtained, although the revolutions be
low. By giving liberal thickness to the rim of the spider, the rotor
then answers the purpose of a fly wheel, hence no separate fly wheel
is required. In fact, the revolving element resembles very closely an
ordinary fly wheel with magnets mounted on its rim, as illustrated in
fig, 1,437.

[Illustration: FIG. 1,437.--General Electric 48 pole 750 kw., three
phase fly wheel type alternator. It runs at a speed of 150 revolutions
per minute, giving a frequency of 60 cycles per second and a full load
pressure of 2,300 volts. The slip rings and leads to the field winding
are clearly shown in the figure. The field magnets are mounted directly
on the rim of the spider, which resembles very closely a fly wheel, and
which in fact it is--hence the name "fly wheel alternator."]

~High Speed Alternators.~--Since alternators may be run at speeds far
in excess of desirable engine speeds, it must be evident that both size
and cost may be reduced by designing them for high speed operation.

Since the desired velocity ratio or multiplication of speed is so
easily obtained by belt drive, that form of transmission is generally
used for high speed alternators, the chief objection being the
space required. Accordingly where economy of space is not of prime
importance, a high speed alternator is usually installed, except in the
large sizes where the conditions naturally suggest a direct connected
unit.

[Illustration: FIG. 1,438.--Allis-Chalmers high speed belted type
alternator. The small pulley at the right and the angle of the belt
suggest the high speed at which such alternators are run, a 50 kva.
machine turning 1,200 revolutions per minute.]

    An example of high speed alternator is shown in fig. 1,438.
    Machines of this class run at speeds of 1,200 to 1,800 or more,
    according to size.

    No one would think of connecting an alternator running at any
    such speed direct to an engine, the necessary speed reduction
    proper for engine operation being easily obtained by means of a
    belt drive.

~Water Wheel Alternators.~--In order to meet most successfully the
requirements of the modern hydro-electric plant, the alternators
must combine those characteristics which result in high electrical
efficiency with a mechanical strength of the moving elements which
will insure uninterrupted service, and an ample factor of safety when
operating at the relatively high speeds often used with this class of
machine.

[Illustration: FIG. 1,439.--Allis-Chalmers 5,000 kva., 450 R. P. M.,
6,600 volt, 60 cycle, 3 phase, horizontal water wheel alternator. The
shaft is extended for the reception of a flange coupling for direct
connection to water wheel. Owing to the wide range in output of the
generating units and also in the speed at which they must operate
to suit varying conditions of head, types of wheels used, and other
features pertaining to water power developments, it has been necessary
to design a very complete line of machines for this work. The bearings
are of the ring oiling type with large oil reservoirs.]

When selecting an alternator for water wheel operation a careful
analysis of the details of construction should be made in order to
determine the relative values which have been assigned by the designers
to the properties of the various materials used. Such analysis will
permit the selection of a type of machine best adapted to the intended
service and which possesses the required characteristics of safety,
durability and efficiency.

[Illustration: FIG. 1,440.--Stator of 500 k.w. Allis-Chalmers
alternator for direct connection to vertical shaft hydraulic turbine.]

The large use of electric power transmitted by means of high pressure
alternating current has led to the development of a large number
of water powers and created a corresponding demand for alternators
suitable for direct connection to water wheels.

~Ques. Name two forms of water wheel alternator.~

Ans. Horizontal and vertical.

    Examples of horizontal and vertical forms of water wheel
    alternator are shown in figs. 1,439 and 1,440.

~Ques. How should the rotor be designed?~

Ans. It should be of very substantial construction.

~Ques. Why?~

Ans. Because water wheel alternators are frequently required to operate
safely at speeds considerably in excess of normal.

[Illustration: FIG. 1,441.--Allis-Chalmers revolving field for water
wheel alternator. In this type of alternator it is essential that the
rotating part be designed to have a liberal factor of safety not only
at the ordinary operating speed, but also at speeds much in excess of
normal. Frequently machines are required to operate safely at a speed
50 to 75 per cent. in excess of normal, so that there may be no danger
in case the water wheel races. In most machines the field spider is of
steel cast in a single piece for the smaller alternators and in two
or more parts for the larger sizes. For alternators running at high
peripheral speed, the rim is built up of steel laminations supported by
a cast steel spider; the latter serves merely to rotate the rim, which
is in itself able to withstand all stresses due to the high speed.
The field poles are laminated, being built up of steel punchings held
between malleable iron or bronze end plates, the latter being used
on high speed machines. With but very few exceptions the poles are
attached by dovetail projections that fit into corresponding slots.
Steel tapered keys are driven in alongside the dovetails, and the pole
pieces cannot become loose. All field coils, except on a few of the
smallest machines, are of edgewise wound copper strip. This style of
coil is essential for revolving field alternators where the pressure
on the insulation, due to centrifugal force, is so great that cotton
insulation on round wire will not stand. Current is led into the rings
by means of carbon brushes, the number of brushes being such that the
current density at the rubbing contact is kept within conservative
limits. At least two brushes per ring are always provided so that
one can be removed for inspection without interrupting the exciting
current. In large machines the brush holder studs are mounted on a
stand supported from the base; in small alternators they are usually
fastened to the cap of one of the bearing pedestals.]

[Illustration: FIGS. 1,442 to 1,444.--Diagram of turbine alternator
windings for revolving armature. Fig. 1,442 illustrates a two pole
design in which all overlapping is avoided. It has 72 slots of which
only 48 are filled, giving 8 slots per phase. The projecting claws
from the brass end shield which hold the coils in position are shown
in section. Fig. 1,443 shows a four pole design having 48 slots or 4
slots per phase per pole, the coils being made up of 8 inductors per
slot taped together, the end bends forming two ranges. Fig. 1,444 shows
a two pole design for a two phase armature with 18 slots per pole per
phase. The core discs are spaced out as for 108 slots, but of these,
4 lots of 7 each are not stamped out, and 8 of those stamped are left
empty, so that there are 72 slots filled.]

~Ques. What special provision is made for cooling the bearings?~

Ans. They are in some cases water cooled.

~Turbine Driven Alternators.~--Although the principle of operation of
the steam turbine and that of the reciprocating engine are decidedly
unlike, the principle of operation of the high speed turbine driven
alternator does not differ from that of generators designed for
being driven by other types of engine or by water wheels. There are,
therefore, with the turbine driven alternator no new ideas for the
operator who is familiar with the older forms to acquire.

It must be obvious that the proportions of such extra high speed
machines must be very different from those permissible in generators of
much slower speeds.

~Ques. How does a turbine rotor differ from the ordinary construction?~

Ans. It is made very small in diameter and unusually long.

~Ques. Why?~

Ans. To reduce vibration and centrifugal stresses.

~Ques. What are the two classes of turbine driven alternators?~

Ans. They are classed as vertical or horizontal.

~Ques. How do they compare?~

Ans. The vertical type requires less floor space than the horizontal
design, and while a step bearing is necessary to carry the weight of
the moving element, there is very little friction in the main bearings.

The horizontal machine, while it occupies more space, does not require
a step bearing.

~Ques. Describe a step bearing.~

Ans. It consists of two cylindrical cast iron plates bearing upon each
other and having a central recess between them into which lubricating
oil is forced under considerable pressure by a steam or electrically
driven pump, the oil passing up from beneath.

~Ques. What auxiliary is generally used in connection with a step
bearing?~

Ans. A weighted accumulator is sometimes installed in

[Illustration: FIG. 1,445.--5,000 kw. Curtis turbine alternators
installed for the New York Central R. R. at Yonkers, N. Y. The
illustration shows also the auxiliary apparatus consisting of
condensers, vacuum augmenters, circulating pumps, air pumps, etc. The
augmenter of the first machine is plainly seen between the condenser
and centrifugal circulating pump.] connection with the oil pipe as a
convenient device for governing the step bearing pumps, and also as a
safety device in case the pumps fail.

    ~Alternators of Exceptional Character.~--There are a few types
    of alternator less frequently encountered than those already
    described. The essentials of such machines are here briefly
    given.

    ~Asynchronous Alternators.~--In these machines, the rotating
    magnet, which, with definite poles, is replaced by a rotor
    having closed circuits. In general construction, they are
    similar to asynchronous induction motors having short circuited
    rotors; for these alternators, when operating as motors, run at
    a speed slightly below synchronism and act as generators when
    the speed is increased above that of synchronism. Machines of
    this class are not self-exciting, but require an alternating or
    polyphase current previously supplied to the mains to which the
    stationary armature is connected.

    Asynchronous alternators may be advantageously used in central
    stations that may be required to sustain a very sudden increase
    of load. In such cases, one or more asynchronous machines might
    be kept in operation as a non-loaded motor at a speed just
    below synchronism until its output as a generator is required;
    when by merely increasing the speed of the engine it will be
    made to act as a generator, thus avoiding the delays usually
    occurring before switching in a new alternator.

    ~Image Current Alternators.~--When the generated frequency of
    alternators excited by low frequency currents is either the sum
    or the difference of the excitation and rotation frequencies,
    any load current flowing through the armature of the machine
    is exactly reproduced in its field circuit. These reproduced
    currents are characteristic of all types of asynchronous
    machines, and are called "image currents," as they are actually
    the reflection from the load currents delivered by the armature
    circuit.

    As the exciter of a machine of this type carries "_image
    currents_" proportional to the generated currents, its size
    must be proportional to the capacity of the machine multiplied
    by the ratio of the excitation and generated frequencies;
    therefore, in the commercial machines, the excitation frequency
    is reduced to the minimum value possible; from two to five
    cycles per second being suitable for convenient employment.

    These machines as heretofore constructed are not self-exciting,
    but as the principle of image current enables the construction
    of self-exciting alternators, it will be of advantage to have a
    general understanding of the separately excited machine under
    different conditions of excitation.

[Illustration: FIG. 1446.--Diagram of constant pressure image
current alternator connections. The image or reproduced currents are
characteristic of all types of asynchronous machines, and are called
image currents because they are actually the reflection from the load
currents delivered by the armature circuit. The principle of operation
is explained in the accompanying text.]

    When the generated frequency of the machine is equal to the
    difference of the excitation and rotation frequencies, the
    magnetization of the machine is higher under a non-inductive
    load than under no load. This is principally due to the ohmic
    resistance of the field circuit, which prevents the image
    current from entirely neutralizing the magnetomotive force
    of the armature current. In other words, the result of the
    magnetomotive force of the armature and image currents not only
    tends to increase the no load magnetization of the machine at
    non-inductive load, but depresses the original magnetization
    at inductive load, so that the terminal voltage of the machine
    increases with non-inductive load, and decreases with inductive
    load.

    Again, the generated frequency is equal to the sum of the
    excitation and rotation frequencies, the resistance of the
    field circuit reacts positively; that is, it tends to decrease
    the magnetization, and consequently the terminal voltage of the
    machine at both inductive and non-inductive loads.

    In the constant pressure machine, the two effects are combined
    and opposed to one another.

    The connections of two alternators with diphase excitation are
    shown by fig. 1,446.

    ~Extra High Frequency Alternators.~--Alternators generating
    currents having a frequency up to 10,000 or 15,000 cycles per
    second have been proposed several times for special purposes,
    such as high frequency experiments, etc. In 1902 Nikola Tesla
    proposed some forms of alternators having a large number of
    small poles, which would generate currents up to a frequency of
    15,000 cycles per second.

    Later, the Westinghouse Company constructed an experimental
    machine of the inductor alternator type for generating currents
    having a frequency of 10,000 cycles per second. This machine
    was designed by Samms. It had 200 polar projections with a pole
    pitch of only 0.25 inch, and a peripheral speed of 25,000 feet
    per minute. The armature core was built up of steel ribbon 2
    inches wide and 3 mils thick. The armature had 400 slots with
    one wire per slot, and a bore of about 25 inches. The air gap
    was only 0.03125 inch. On constant excitation the voltage
    dropped from 150 volts at no-load to 123 volts with an output
    of 8 amperes.

    ~Self-Exciting Image Current Alternators.~--The type of machine
    described in the preceding paragraph can be made self-exciting
    by connecting each pair of brushes, which collect the current
    from the armature, with a field coil so located that the
    flux it produces will be displaced by a predetermined angle
    depending on the number of phases required, as shown by fig.
    1,447. The direction of the residual magnetism of the machine
    is shown by the arrows A, A.

[Illustration: FIG. 1,447.--Diagram of connections of self-exciting
image current alternator.]

    When the armature is rotated, a pressure will be generated
    between the brushes 2 and 4, and a current will flow from
    C through the coils XX to B, producing a flux through the
    armature at right angles to the residual magnetism and
    establishing a resultant magnetic field between D, B, and D,
    C. This field will generate a pressure between the brushes 1
    and 3, and a current will flow D through XX to E in such a
    direction that it will at first be opposed to the residual
    magnetism, and afterward reverse the direction of the latter.
    At the moment the residual magnetism becomes zero, the only
    magnetism left in the machine will be due to the currents
    from the brushes 2 and 4, and their field combining with the
    vertical reversed field will produce a resultant polar line
    between B and E. As these operations are cyclic, they will
    recur at periodic intervals, and the phenomena will become
    continuous. The negative field thus set up in the air gap of
    the machine will cut the conductors of the stator and will
    be cut by the conductors of the rotor in such a manner that
    the electromotive forces generated between the brushes of
    the armature will be equal and opposite to those between the
    terminals of the stator.




CHAPTER L

CONSTRUCTION OF ALTERNATORS


The construction of alternators follows much the same lines as dynamos,
especially in the case of machines of the revolving armature type.
Usually, however, more poles are provided than on direct current
machines, in order to obtain the required frequency without being
driven at excessive speed.

The essential parts of an alternator are:

  1. Field magnets;
  2. Armature;
  3. Collector rings;

and in actual construction, in order that these necessary parts may be
retained in proper co-relation, and the machine operate properly there
must also be included:

  4. Frame;
  5. Bed plate;
  6. Pulley.

~Field Magnets.~--The early forms of alternator were built with
permanently magnetized steel magnets, but these were later discarded
for electromagnets.

Alternators are built with three kinds of electromagnets, classed
according to the manner in which they are excited, the machines being
known as,

  1. Self-excited;
  2. Separately excited;
  3. Compositely excited.

[Illustration: FIGS. 1,448 and 1,449.--Westinghouse laminated hub
and laminated pole piece for revolving field having squirrel cage
winding. Thin steel is used for the laminations of both hub and pole
piece; these are assembled and firmly riveted together under hydraulic
pressure. The laminations are of the same thickness in both hub and
spider.]

[Illustration: FIGS. 1,450 to 1,452.--Views of Triumph pole pieces.
These consist of laminated punchings securely clamped between two cast
steel end plates. The laminations are shaped with polar horns or shoes
as shown, and which serve to keep the field coils securely wedged in
position. In some designs the horns are separate. The two holes in
each pole piece are for through bolts which secure the pole piece and
coil to the spider run. Dovetail joints are sometimes used instead of
through bolts, as in figs. 1,448 and 1,449.]

~Ques. What is a self-excited alternator?~

Ans. One in which the field magnets are excited by current from one
or more of the armature coils, or from a separate winding (small in
comparison with the main winding), the current being transformed into
direct current by passing it through a commutator.

[Illustration: FIG. 1,453.--Fort Wayne armature for self-excited
alternator. There are two independent windings, one for the main
current, and one for the exciting current. The winding for the latter
current occupies a very small amount of space, and is placed in the
slots on top the main winding. The commutator to which the exciter
winding is connected, is located between the collector rings and the
core. It is of standard construction with end clamps holding the bars
in place on the insulated commutator drum. The armature coils are form
wound and the core is built of sheet steel laminations, annealed and
japanned to prevent hysteresis and eddy current losses. Ventilated
openings are provided to allow a free circulation of air both around
the ends of the windings and through ducts in the laminated core. The
core is clamped by bolts between the flanges of the armature spider
which is keyed to the shaft. These flanges have cylindrical extensions
with ribbed surfaces, which form a support for the ends of the armature
coils. The ribbed surfaces form air passages from the core outward
around the ends of the coils, thus ventilating both core and coils.]

    Fig. 1,453 shows an armature of a self-excited machine, the
    exciting current being generated in a separate winding and
    passed through a commutator.

~Ques. For what class of service are self-exciting alternators used?~

[Illustration: FIG. 1,454.--Allis-Chalmers three bearing type
alternator with exciter direct connected. The bearing pedestals are
bolted to a substantial cast iron base having, in the large sizes,
sufficient length to permit shifting frame sideways along the base
to give access to the field and armature coils. The field coils are
designed for 120 volt excitation, and are wound edgewise with copper
strip. There is a liberal margin of field excitation to take care
of overloads or for operation on loads of low power factor. The
regulating qualities are as good as can be obtained without making
the machine unnecessarily large and expensive. By regulation is meant
the percentage rise in voltage when full load is thrown off, field
excitation and speed being held constant; the percentage is referred to
normal full load voltage. An alternator with poor regulation will show
large variations in voltage with changes in load, the pressure falling
whenever a load is thrown on and rising when it is thrown off. These
changes will be especially pronounced if the load be inductive. A badly
designed alternator might show very fair regulation on non-inductive
load and yet be unable to give full voltage on inductive load.]

Ans. They are employed in small power plants and isolated lighting
plants where inductive loads are encountered.

~Ques. What is a separately excited alternator?~

Ans. One in which the field magnets are excited from a small dynamo
independently driven or driven by the alternator shaft, either direct
connected or by belt as shown in fig. 1,455.

[Illustration: FIG. 1,455.--Diagram of separately excited alternator.
The field winding is supplied with direct current, usually at 125 volts
pressure by a small dynamo called the "exciter." The latter may be
driven by independent power, or by belt connection with the main shaft,
and in some cases the exciter is directly connected to the alternator
shaft.]

~Ques. What is a compositely excited alternator?~

Ans. A composite alternator is similar to a compound wound dynamo in
that it has two field windings. In addition to the regular field coils
which carry the main magnetizing current from the exciter, there is a
second winding upon two or upon all of the pole pieces, carrying a
rectified current from the alternator which strengthens the field to
balance the losses in the machine, and also if so desired, the losses
on the line as shown in fig. 1,456.

[Illustration: FIG. 1,456.--Diagram of compositely excited alternator.
The current for exciting the field magnets is obtained, partly from
an exciter and partly from the windings of the alternator, being
transformed into direct current by the rectifier. The connections are
as shown. One end of the armature winding is connected to one of the
collector rings; the other end, to the light part of the rectifier,
as shown, the solid black part of the rectifier being connected to
the other collector ring. Two brushes bear on adjacent teeth of the
rectifier and are connected to the compensating winding circuit across
which is a shunt. These connections are shown more clearly in fig.
1,457. In operation the separately excited coils set up the magnetism
necessary for the generation of the voltage at no load. The main
current coming from the armature is shunted, part going through the
shunts and the remainder around the compensating winding, furnishing
the additional magnetism necessary to supply the voltage to overcome
the armature impedance. This composite method of field excitation is
very similar to that used on a compound wound dynamo. As shown, both
field windings encircle every pole, but in some machines the rectified
current will traverse a few poles only, the current from the exciter
traversing the remainder.]

~Ques. What is a magneto?~

Ans. A special form of alternator having permanent magnets for its
field, and used chiefly to furnish current for gas engine ignition and
for telephone call bells.

    Details of construction and operation are shown in figs. 1,458
    to 1,461.

[Illustration: FIG. 1,457.--Diagram showing construction of rectifier
and connections of compositely excited alternator. The rectifier
consists of two castings M and S with teeth which fit together as
shown, being insulated so they do not come in contact with each other.
Every alternate tooth being of the same casting is connected together,
the same as though joined by a conducting wire. There are as many teeth
as there are poles. One end of the armature winding is connected direct
to one of the collector rings, while the other is connected to M of the
rectifier, the circuit being through brushes P and Q, the shunt, and
compensating winding to the other collector ring. The brushes P and
Q contact with adjacent teeth, when one is in contact with the solid
black casting the other touches the light casting. The principle of
action is the same as a commutator, briefly: to reverse the connections
terminating at the brushes P and Q in synchronism with the reversals of
the alternating current induced in the armature winding, thus obtaining
direct current for the compensating field winding. The shunt resistance
placed across the compensating winding circuit permits adjusting the
compounding of the machine to the circuit on which it is to work, since
by varying the resistance the percentage of the total current passing
through the compensating winding can be changed. It will be seen by
tracing the path of the current for each direction in the armature
winding that while the rectifier causes the current to flow in the
same direction in the compensating field winding, it still remains
alternating in the external circuit.]

[Illustration: FIG. 1,458.--Connecticut magneto; view showing permanent
magnets in dotted lines. It consists of three permanent U shape
magnets, between the poles of which is a shuttle type armature. The
latter is geared to a hand crank in sufficient velocity ratio to give
the desired speed without too rapid turning of the crank. This type of
magneto is used to generate current for operation of telephone call
bells.]

[Illustration: FIGS. 1,459 to 1,461.--Diagram illustrating the
operation of a magneto. The shuttle shaped armature is wound from
end to end with insulated wire, so that when rotated, a powerful
alternating current is produced in the windings by cutting the magnetic
lines, whose varying strength is shown by the shaded portions in the
two views. When in the position shown in the first diagram, the lines
of force mostly converge at the top and bottom, finding a direct path
through the metal end flanges of the shuttle. When in the position
shown in the second diagram, the lines are converged so as to pass
through the armature core. Fig. 1,460 shows detail of the armature
core.]

~Ques. What are the two principal types of field magnet?~

Ans. Stationary and revolving.

~Ques. What is the usual construction of stationary field magnets?~

Ans. Laminated pole pieces are used, each pole being made up of a
number of steel stampings riveted together and bolted or preferably
cast into the frame of the machine. The field coils are machine wound
and carefully insulated. After winding they are taped to protect them
from mechanical injury. Each coil is then dipped in an insulating
compound and afterwards baked to render it impervious to moisture.

[Illustration: FIG. 1,462.--Stationary field of Fort Wayne multiphase
revolving armature alternator; view showing brass girds on pole pieces
for synchronous motor operation. When designed for this use the machine
is provided with amortisseur winding on the poles. As shown in the
illustration this winding consists of a brass collar around the pole
tip with a cross rib integral with the collar, fitting in a slot in the
pole face parallel to the shaft. This construction assists in bringing
the machine up to synchronous speed as an induction motor, ordinarily
checks any tendency toward hunting and does not in any way affect the
operation of the machine as an alternator. The main field winding
should be connected through switches on the field frame in order that
the field circuit may be broken up to eliminate any danger that might
arise from induced voltage. It is not advisable to throw on a full
rated voltage and a compensator should, therefore, be provided to
reduce the pressure.]

[Illustration: FIG. 1,463.--Triumph 36 pole fly wheel type revolving
field. The spider has the form of a fly wheel having spokes and rim
to which the field magnets are attached by through bolts. The field
coils are of copper strap bent on end, the kind generally used on large
machines. The series connection of the coils is plainly shown, also the
two cables leading via one of the spokes to the slip rings.]

[Illustration: FIG. 1,464.--Wagner cast steel hub with dovetail grooves
for attaching the revolving field magnets. Such construction is
generally used on machines of small and medium size.]

[Illustration: FIG. 1,465.--Wagner laminated pole piece with horns
stamped in one piece. The laminations are held together between two end
pieces by through rivets, as shown.]

~Ques. Describe the construction of a revolving field.~

Ans. The entire structure or rotor consists of a shaft, hub or spider,
field magnets and slip rings. The magnet poles consist of laminated
iron stampings clamped in place by means of through bolts which, acting
through the agency of steel end plates, force the laminated stampings
into a uniform, rigid mass. This mass is magnetically subdivided into
so many small parts that the heating effect of eddy currents is reduced
to a minimum. The cores are mounted upon a hub or spider either by
dovetail construction or by means of through bolts, according to the
centrifugal force which they must withstand in operation, either method
permitting the easy removal of any particular field pole if necessary.
The field coils are secured upon the pole pieces either by horns in one
piece with the laminations, or separate and bolted. All the coils are
connected in series, cable leads connecting them to slip rings placed
on the shaft.

[Illustration: FIG. 1,466.--Wagner revolving field of 300 kilowatt
alternator during construction, illustrating the method of attaching
the field magnets to the hub by dovetail joints. After the notched ends
of the pole pieces are slid into the grooves in the hub, tapered keys,
which are plainly seen, are driven in, thus making a tight joint which
will not shake loose.]

~Ques. What are slip rings?~

Ans. Insulated rings mounted upon the alternator shaft to receive
direct current for the revolving field, as distinguished from collector
rings which collect the alternating currents generated in an alternator
of the revolving armature type.

    In construction provision is made for attaching the field
    winding leads. The rings are usually made of cast iron and are
    supported mechanically upon the shaft, but are insulated from
    it and from one another.

    The current is introduced by means of brushes as with a
    commutator. Carbon brushes are generally used.

    A good design of slip ring should provide for air circulation
    underneath and between the rings.

[Illustration: FIG. 1,467.--General Electric field coil, showing one
method of winding. In the smaller machines the wire is wound on spools
which are slipped over the pole pieces, which are built of sheet iron,
spreading at the pole face so as to secure not only a wide polar arc
for the proper distribution of the magnetic flux, but also to hold the
field windings in place.]

~Ques. What form of spider is used on large alternators?~

Ans. It is practically the same form as a fly wheel, consisting of hub,
spokes, and rim to which the magnets are bolted.

[Illustration: FIG. 1,468.--General Electric field coil showing another
method of winding. The field coils on the larger machines consist of
a single strip of flat copper, wound on edge as shown, so that the
surface of every turn is exposed to the air for cooling. The flat sides
of the copper strip rest against each other and the entire coil forms a
structure of great solidity which can be easily removed for inspection
and repair.] [Illustration: FIG. 1,469.--Allis-Chalmers 60 kva. belted
two bearing alternator on base arranged so the armature can be shifted
sideways as shown, to give access to the field and armature coils.] On
alternators of the fly wheel type the spider rim is made of sufficient
weight to obtain full fly wheel effect, thus making a separate fly
wheel unnecessary.

[Illustration: FIG. 1,470.--Revolving field of Fort Wayne 10 pole
alternator. In construction, the cores of the field poles are built up
from punchings of laminated steel, and assembled under considerable
pressure between malleable iron or steel end plates and riveted
together. Substantial insulation is placed on the pole cores and over
this is wound the field coils of cotton covered wire. After the wire
is in place, the completed poles are baked to expel any moisture and
are then treated with insulating varnish. They are then assembled on
a laminated spider, being held in place by dovetail joints made tight
by the use of taper keys. Special casting plates are finally fastened
in place over the dove tails effectually closing them. The assembly of
the field is completed by the insertion of the shaft into the field
spider under heavy hydraulic pressure. All the coils are connected in
series, cable leads connecting them to slip rings placed on the shaft.
Each slip ring is provided with a double type brush holder, making it
possible to clean brushes while the alternator is in service, by simply
removing one brush at a time.]

[Illustration: FIG. 1,471.--General Electric slip rings; view showing
construction and attachment of cable leads to field winding. They are
so designed that all surfaces of the rings have easy access to the
air, in order to obtain good ventilation. _Slip rings_, through which
current is transmitted to a revolving field, are to be distinguished
from _collector rings_ whose function it is to "collect" or transmit
the alternating currents induced in the armature to the brushes.]

~Armatures.~--In construction, armatures for alternators are similar
to those employed on dynamos; they are in most cases simpler than
direct current armatures due to the smaller number of coils, absence of
commutator with its multi-connections, etc. Alternator armatures may be
classified in several ways:

1. With respect to operation, as

  _a._ Revolving;
  _b._ Stationary.

[Illustration: FIG. 1,472.--Allis-Chalmers brush holder and slip rings.
The latter are made of cast copper, which the builders claim to be more
satisfactory than cast iron. On some of the large low speed machines
the collector rings are split, but on the majority of alternators they
are in one piece. Current is led into the rings by means of carbon
brushes, the number of brushes being such that the current density
at the rubbing contact is kept within conservative limits. At least
two brushes per ring are provided, so that one can be removed for
inspection without interrupting the exciting current. In large machines
the brush holder studs are mounted on a stand supported from the base;
on small alternators they are usually fastened to the cap of one of the
bearing pedestals.]

[Illustration: FIG. 1,473.--Fort Wayne multiphase revolving armature
alternator, designed for use in small power plants and isolated
lighting plants where inductive loads are encountered. Built for
pressures of 120, 240, 480, and 600 volts. These voltages have been
recommended by the American Institute of Electrical Engineers, and will
cover the needs of any set of conditions ordinarily met with. These
standard voltages not only permit economical distribution, but they are
such that no transformers are necessary to reduce the line pressure
for ordinary cases. For transmitting power relatively long distances,
600 volts is usually employed. Where there is a demand for 480 volt
service, a 480 volt alternator should be selected and if lower voltages
are also desired an auto-transformer may be furnished by means of which
240 volts can be obtained. When 120 volt circuits are necessary for
lighting, etc., the 240 volt pressure can be still further reduced to
120 volts by means of another auto-transformer. However, this double
reduction will rarely be found necessary.]

[Illustration: FIG. 1,474.--Western Electric stationary armature. In
this type of armature, the core upon which the winding is placed, is
built into the frame as shown, the core teeth projecting inwardly like
internal gear teeth, forming a cylindrical chamber for the revolving
field. The core is built up of iron, laminated and japanned to prevent
eddy currents and hysteresis losses. The laminations are rigidly bolted
between two heavy end plates. The armature coils are of copper bar
impregnated with insulating compound. They are held in the slots by
wedges which allow their ready removal for inspection or repairs.]

[Illustration: FIGS. 1,474 to 1,477.--Various types of armature;
fig. 1,474 ring armature; fig. 1,475 disc armature; fig. 1,476 drum
armature. The latter type is now almost universally used, the others
being practically obsolete. A Gramme ring wound and connected to
collector rings as in fig. 1,474, will yield an alternating current.
In a multipolar field, the ring will need multipolar connections
alternated at points corresponding to the pitch of the poles. Fig.
1,475 illustrates the so-called "Siemens" disc armature. The armature
coils are arranged around the periphery of a thin disc. The field
magnets consist of two crowns of fixed coils, with iron cores arranged
so that their free poles are opposite one another. This type was
created in 1878 by Herr von Hefner, engineer to Messrs. Siemens and
Halske. Fig, 1,476 shows a modern drum armature of a three phase
machine. It is similar in appearance to a direct current armature
except for the absence of the commutator and its connections. The drum
armature is the prevailing type.]

2. With respect to the core, as

  _a._ Ring;
  _b._ Disc;
  _c._ Drum.

    Ring and disc armatures are practically obsolete and need not
    be further considered. A ring armature has the inherent defect
    that the copper inside the ring is inactive.

    Disc armatures were employed by Pacinotti in 1878, and
    afterwards adopted by Brush in his arc lighting dynamos.

    The design failed for mechanical reasons, but electrically it
    is, in a sense, an improvement upon the Gramme ring, in that
    inductors on both sides of the ring are active, these being
    connected together by circumferential connectors from pole to
    pole, thus, corresponding to the end connectors on modern drum
    armatures.

3. With respect to the core surface, as

  _a._ Smooth core;
  _b._ Slotted core.

    In early dynamos the armature windings were placed upon an
    iron core with a smooth surface. A chief disadvantage of this
    arrangement is that the magnetic drag comes upon the inductors
    and tends to displace them around the armature. To prevent

[Illustration: FIG. 1,478.--A style of disc largely used for
armature cores. The teeth are provided with dovetail grooves near
the circumference. After the coil is inserted in a groove, a wooden
wedge is driven in the groove which encloses the coil and secures it
firmly in position. This obviates the necessity of bands to resist the
centrifugal force acting on the inductors.]

[Illustration: FIG. 1,479.--Large revolving armature construction with
segmental discs dovetailed to spider spokes.]

[Illustration: FIG. 1,480.--Construction of large stationary armature;
view showing section of core and frame. The core discs are in segments
and are attached to the frame by dovetail joints as shown. The joints
are staggered in building up the core, that is, they are overlapped
so as not to unduly increase the reluctance of the magnetic circuit.
Dovetail joints obviate the use of through bolts which, if not
insulated, are liable to give rise to eddy currents by short circuiting
the discs.] /# this, projecting metal pieces called _driving horns_
were fixed into the core so as to take the pressure, but they proved
unsatisfactory. This defect together with the long air gap necessary
in smooth core construction resulted in the type being displaced by
slotted core armatures.

A slotted core is one whose surface is provided with slots or teeth
which carry the inductors, as shown in the accompanying illustrations,
and is the type almost universally used. The inductors are laid in the
slots, the sides and bottoms of which are first carefully insulated by
troughs of mica-canvas, micanite or other suitable insulating material.

~Ques. What are the advantages of slotted core armatures?~

Ans. The teeth protect the inductors, retain them in place against the
electrical drag and centrifugal force, and the construction permits a
reduction of air gap to a minimum, thus reducing the amount of copper
required for the field.

[Illustration: FIG. 1,481.--General Electric revolving field and
exciter armature. This is an example of direct connected exciter
construction. In this arrangement the armature of the exciter is
carried on the alternator shaft at the end farthest from the pulley. In
the smaller sizes the magnet frame is bolted to the bearing bracket,
but in the larger sizes special construction is used depending upon
the conditions to be met. On all alternators of standard design,
the field is built for 125 volts excitation and on account of the
increased danger from induced voltage, in case the machine is used
as a synchronous motor, the builders consider any higher voltage
undesirable.]

[Illustration: FIG. 1,482.--Section of General Electric Alternator
showing method of dovetailing core laminations to frame. The latter
is made in two general styles, known as the _box type_ and _skeleton
type_. ~The box type~ consists of a single casting for the smaller
sizes, but for large capacity alternators the frame castings are
usually divided into upper and lower sections. ~The skeleton type~
consists of two side castings between which substantial spacing rods
are set at regular intervals. The core consists of the usual sheet
iron lamination slotted and assembled; they are mounted on the inner
periphery of the frame, making lap joints (that is "staggered" as in
fig. 1,480), each section being dovetailed to the frame. Heavy clamping
rings or end plates are mounted on both sides of the core by means of
bolts, and supporting fingers extend along the slot projections. The
design is such as to provide for air circulation as shown in figs.
1,483 and 1,484.]

~Armature Windings.~--In general, the schemes for armature windings
for alternators are simpler than those for direct current machines,
as in the majority of cases the inductors are an even multiple of the
number of poles, and the groupings are usually symmetrical with respect
to each pole or each pair of poles. Furthermore, as a general rule,
all the inductors of any one phase are in series with one another;
therefore, there is only one circuit per phase, and this is as it
should be, since alternators are usually required to generate high
voltages. These general principles establish the rule, that in the
circuit in a single phase armature, and in the individual circuits
in a polyphase armature, the winding is never re-entrant, but the
circuits have definite endings and beginnings. In exceptional cases,
as those of polyphase converters, re-entrant circuits are employed,
and the armature windings are so constructed that a commutator can
be connected to them exactly as in direct current machines. These
armatures are usually of the lap wound drum type.

[Illustration: FIG. 1,483.--Section of General Electric alternator
frame showing air ducts and supporting fingers extending along the slot
projections. The air circulation is provided for by means of ducts
formed by suitable spacing blocks inserted at intervals between the
laminations, as shown here and in fig. 1,484. The armature coils are
form wound and designed so they can be readily replaced in case of
injury. They are taped and treated with an impregnating compound, in
the usual way, then inserted in the armature slots in an armour of horn
fibre and retaining wedges of wood are dovetailed into the slot walls.]

Alternator windings are usually described in terms of the number of
slots per phase per pole. For instance, if the armature of a 20 pole
three phase machine have 300 slots, it has 15 slots per pole or 5
slots per each phase per pole, and will be described as a five slot
winding. Therefore, in order to trace the connections of a winding,
it is necessary to consider the number of slots per pole for any one
phase on one of the following assumptions: 1, that each slot holds one
inductor; 2, that there is one side of a coil in each slot; and 3, that
one side of a coil is subdivided so as to permit of its distribution in
two or more adjacent slots.

The voltage depends upon the number of inductors in a slot, but the
breadth coefficient and wave form are influenced by the number of slots
per pole, and not by the number of inductors within the slots.

[Illustration: FIG. 1,484.--Section of General Electric stationary
armature showing method of assembling the coils. These are form
wound and are held in the slots by suitable wedges, the open slot
construction permitting the use of form wound coils that can be
easily removed and replaced in case of damage. Where heavy windings
project beyond the laminations, an additional support is provided
by means of an insulated metal ring, to which the outer ends of the
coils are fastened; the coils are thereby protected from mechanical
displacement, or distortion due to the magnetic disturbances caused
by violent fluctuations of the load or short circuits. The figure
shows a section of a supporting ring of this type and indicates the
method of connecting the coils to it. In order to admit of the prompt
replacement of damaged coils, sufficient space is usually provided
between the alternator bearings to allow ample movement of the armature
to permit of ready access to both armature and field coils. Where space
necessitates the use of a short shaft, access to the windings may be
had by disconnecting some of the coils and lifting the upper half of
the armature.]

~Classification of Windings.~--The fact that alternators are built
in so many different types, gives rise to numerous kinds of armature
winding to meet the varied conditions of operation. In dividing these
forms of winding into distinctive groups, they may be classified,
according to several points of view, as follows:

1. With respect to the form of the armature, as:

  _a._ Revolving;
  _b._ Stationary.

2. With respect to the mode of progression, as:

  _a._ Lap winding;
  _b._ Wave winding.

3. With respect to the relation between number of poles and number of
coils, as:

  _a._ Half coil winding;
  _b._ Whole coil winding.

4. With respect to the number of slots, as:

  _a._ Concentrated or uni-coil winding;
  _b._ Distributed or multi-coil winding.
          Partially distributed;
          Fully distributed.

5. With respect to the form of the inductors, as:

  _a._ Wire winding;
  _b._ Strap winding;
  _c._ Bar winding.

6. With respect to the number of coils per phase per pole, as:

  _a._ One slot winding;
  _b._ Two slot winding;
          etc.

7. With respect to the kind of current delivered, as:

  _a_. Single phase winding;
  _b_. Two phase winding;
  _c_. Three phase winding.

[Illustration: FIG. 1,485.--Section of Western Electric stationary
armature core showing laminations clamped in place, and ventilating
ducts. The stator or stationary armature consists of soft iron
laminations assembled in the magnet frame with stator coils embedded
in the core slots. The laminations are punched separately and then
carefully annealed to reduce hysteresis losses. After annealing, a coat
of japan is applied, effectively preventing the flow of eddy currents
in the assembled core. The frame is cast iron and of the box type
construction. The frames of the smaller sizes are cast in one piece,
while frames of the larger sizes are split to facilitate installation.
Large openings are provided in the box type frame, in order to improve
the ventilation. The laminations are securely held in place in the
frame by heavy end rings and by steel clamping fingers which are firmly
bolted to the frame. The outer circumference of the core is dovetailed
to the frame, and the inner circumference is slotted to receive the
windings. The alignment of the slots is insured by means of metal
wedges, and no filing is done on the slots, so that each lamination is
always insulated from the next one. Numerous ventilating ducts allow
the free circulation of cool air through and around the coils. The open
slot construction is employed and the coils are fitted into insulating
troughs which offer excellent mechanical and electric protection. The
coils are held in place by suitable wedges.]

8. With respect to the shape of the coil ends, as:

  _a_. Single range;
  _b_. Two range;
          etc.

In addition to these several classes of winding, there are a number of
miscellaneous windings of which the following might be mentioned:

  _a._ Chain or basket winding;
  _b._ Skew coil winding;
  _c._ Fed-in winding;
  _d._ Imbricated winding;
  _e._ Mummified winding;
  _f._ Spiral winding;
  _g._ Shuttle winding;
  _h._ Creeping winding;
  _i._ Turbine alternator winding.

[Illustration: FIG. 1,486.--Method of assembling form wound coils. The
picture shows a section of a General Electric armature with part of the
coils in place. A layer of insulating material is first placed in the
slots, before inserting the coils as seen at the left. When the coils
are in place and surrounded by this layer of insulating material the
retaining wedges are inserted in the notches, thus closing the slots
and protecting the coils from mechanical injury. A few wedges are seen
in position at the right.]

~Ques. Define a revolving and a stationary winding.~

Ans. The words are self-defining; a winding is said to be revolving or
stationary according as the armature forms the rotor or stator of the
machine.

~Ques. What is the significance of the terms lap and wave as applied to
alternator windings?~

Ans. They have the same meaning as they do when applied to dynamo
windings.

[Illustration: FIG. 1,487.--Section of General Electric stationary
armature ventilating ducts and winding in position.]

    These are described in detail in Chapter XVIII. Briefly a lap
    winding is one composed of lap coils; a wave winding is one
    which roughly resembles in its diagram, a section of waves.

~Half Coil and Whole Coil Windings.~--The distinction as to whether the
adjacent sides of consecutive coils are placed together under one pole
or whether they are separated a distance equal to the pole pitch, gives
rise to what is known as half coil and whole coil windings.

A half coil or hemitropic winding is _one in which the coils in any
phase are situated opposite every other pole_, that is, _a winding in
which there is only one coil per phase_ ~per pair of poles~, as in fig.
1,488.

_A whole coil winding is one in which there is one coil per phase_
~per pole~, as in fig. 1,489, the whole (every one) of the poles being
subtended by coils.

[Illustration: FIGS. 1,488 and 1,489.--Elementary bipolar alternators
with _half coil_ and _whole coil_ windings. In a half coil winding
there is one coil per phase _per pair of poles_; in a whole coil
winding there is one coil per phase _per pole_.]

~Concentrated or Uni-Coil Winding.~--Fig. 1,492 shows the simplest type
of single phase winding. It is a one slot winding and is sometimes
called "monotooth" or "uni-coil" winding. The surface of the armature
is considered as divided into a series of large teeth, one tooth to
each pole, and each tooth is wound with one coil, of one or more turns
per pole. Since all the turns of the coil are placed in single slots,
the winding is called "concentrated."

[Illustration: FIG. 1,490.--Multi-polar revolving armature alternator
with half coil winding, shown in radially developed diagram to clearly
indicate the path of the winding. A half coil or hemitropic winding has
a slightly higher reactance than a winding in which two distinct coils
are used in the same slot, one going forward and the other backward.
The most usual three phase windings are of the half coil type as the
three sets of coils are equispaced over a pair of poles.]

[Illustration: FIG. 1,491.--Multi-polar revolving armature alternator
with whole coil winding shown in radially developed diagram to clearly
indicate the path of the winding.]

~Ques. What are the features of concentrated windings?~

Ans. Cheap construction, maximum voltage for a given number of
inductors. Concentrated windings have greater armature reaction and
inductance than other types hence the terminal voltage of an alternator
with concentrated winding falls off more than with distributed winding
when the current output is increased. An alternator, therefore,
does not have as good regulation with concentrated winding as with
distributed winding.

[Illustration: FIGS. 1,492 and 1,493.--Concentrated windings. A
concentrated winding is one in which the armature has only _one tooth
per phase per pole_, that is, the number of teeth equals the number of
poles. A concentrated winding of the half coil type has only one side
of a coil in each slot as in fig. 1,492. In the whole coil variety,
each slot contains neighboring sides of adjacent coils, as in fig.
1,493. In construction, wedges are generally used for retaining the
half coils, and with whole coils the teeth have projecting horns for
this purpose.]

~Ques. What should be noted with respect to concentrated windings?~

Ans. A concentrated winding, though giving higher voltage than the
distributed type with no load, may give a lower voltage than the latter
at full load.

[Illustration: FIG. 1,494.--Laminated core with two coils in position;
type of punchings used on some machines having concentrated whole coil
windings. The manner of assembling the coils is shown in fig. 1,495.]

~Ques. What is the wave form with a concentrated winding?~

Ans. The pressure curve rises suddenly in value as the armature slots
pass under the pole pieces, and falls suddenly as the armature slots
recede from under the pole pieces.

[Illustration: FIG. 1,495.--Westinghouse single phase concentrated coil
armature; view showing method of placing coils. The coils are machine
wound on formers and after being taped, varnished and baked, are spread
out slightly so as to pass over the teeth and are then forced into
place in the deep slots by means of wooden wedges, being securely held
in place by retaining wedges, as shown in fig. 1,494.]

~Distributed or Multi-Coil Windings.~--Instead of winding an armature
so it will occupy only one slot per phase per pole, it may be spread
out so as to fill _several slots per phase per pole_. This arrangement
is called a distributed winding.

    To illustrate, fig. 1,496 represents a coil of say fifteen
    turns. This could be placed on an armature just as it is, in
    which case only one slot would be required for each side, that
    is, two in all. In place of this thick coil, the wire could be
    divided into several coils of a lesser number of turns each,
    arranged as in fig. 1,497; it is then said to be _partially
    distributed_, or it could be arranged as in fig. 1,498, when it
    is said to be _fully distributed_.

[Illustration: FIGS. 1,496 to 1,498.--Alternator coils, showing
difference between the concentrated, partially distributed, and fully
distributed forms. Fig. 1,496 shows a concentrated coil in which all
the wire is wound in one large coil; in the partially distributed type
fig. 1,497, the wire of fig. 1,496, is wound in two or more coils or
"sections" connected as shown, leaving some space inside not taken up
by the subdivisions. In fig. 1,498 the wire of fig. 1,496 is _fully
distributed_, being wound in a series of coils, so that all the
interior space is taken up by the wire, that is to say, the spaces not
occupied by the wire (the teeth when placed on the armature) are of
equal size.]

A partially distributed winding, then, is one, as in fig. 1,499,
in which the coil slots do not occupy all the circumference of the
armature; that is, the core teeth are not continuous.

A fully distributed winding is one in which the entire surface of the
core is taken up with slots, as in fig. 1,500.

~Ques. In a distributed coil what is understood by the breadth of the
coil?~

Ans. The distance between the two outer sides, as B in figs. 1,497 and
1,498.

[Illustration: FIG. 1,499.--Partially distributed winding. Each coil
unit is here divided into two concentric coils of different dimensions
and connected in series, as shown in detail in fig. 1,497. This being
a "whole coil" winding the several units are so connected that the
winding of adjacent units proceeds in opposite directions, that is, one
coil is wound clockwise, and the next counter clockwise, etc., so that
the induced currents flow in a common direction as indicated by the
arrows for the position shown.]

[Illustration: FIG. 1,500.--Fully distributed winding. In this type
of winding each coil consists of so many sub-coils that the winding
occupies the entire surface of the armature core; that is, there are no
extensive spaces unoccupied, the spacing being uniform as shown.]

~Ques. How far is it advisable to spread distributed coils of a single
phase alternator?~

Ans. There is not much advantage in reducing the interior breadth
much below that of the breadth of the pole faces, nor is there much
advantage in making the exterior breadth greater than the pole pitch.

    Undue spreading of distributed coils lowers the value of the
    Kapp coefficient (later explained) by reducing the breadth
    coefficient and makes necessary a larger number of inductors to
    obtain the same voltage.

    The increase in the number of inductors causes more armature
    self-induction. From this point of view, it would be preferable
    to concentrate the winding in fewer slots that were closer
    together. This, however, would accentuate the distorting and
    demagnetizing reactions of the armature. Accordingly, between
    these two disadvantages a compromise is made, as to the extent
    of distributing the coils and spacing of the teeth, the
    proportions assigned being those which experience shows best
    suited to the conditions of operation for which the machine is
    designed.

[Illustration: FIG. 1,501.--Developed diagram of single phase
concentrated whole coil winding in two slot stamping for six pole
alternator. If the sides of adjacent whole coils be slightly separated
by placing the winding in a two slot stamping the electrical result
will not differ materially from the monotooth whole coil winding, but
if the winding be hemitropic, as in fig. 1,502, and has coils of two
sizes as shown, it will be suitable for high voltages.]

~The Kapp Coefficient.~--A volt or unit of electric pressure is defined
as the pressure induced by the cutting of 100,000,000 or 10⁸ lines
of force per second. In the operation of an alternator the maximum
pressure generated may be expressed by the following equation:

        π_f_ZN
  Eₘₐₓ = ------                                                      (1)
         10⁸

in which

  _f_ = frequency;
  Z   = number of inductors in series in any one magnetic circuit;
  N   = magnetic flux, or total number of magnetic lines in one
        pole or in one magnetic circuit.

The maximum value of the pressure, as expressed in equation (1), occurs
when θ = 90°.

[Illustration: FIG. 1,502.--Developed diagram of single phase partially
distributed half coil winding for six pole alternator in two slot
stamping, same as in fig. 1,501. In this arrangement the direction of
rotation is not reversed. It is a question as to how far the coils of a
single-phase armature may be spread with advantage. There is not much
advantage in reducing the interior breadth of the coils below that of
the pole face, nor in widening the exterior breadth beyond that of the
pole pitch.]

The virtual value of the volts is equal to the maximum value divided by
√2̅, or multiplied by ½ √2̅, hence,

          ½ √2̅ × π_f_ZN   2.22_f_ZN
  Eᵥᵢᵣₜ = -------------- = ---------                                 (2)
              10⁸            10⁸

This is usually taken as the fundamental equation in designing
alternators. It is, however, deduced on the assumptions that the
distribution of the magnetic flux follows a sine law, and that the
whole of the loops of active inductors in the armature circuit acts
simultaneously, that is to say, the winding is concentrated.

[Illustration: FIG. 1,503.--Developed diagram of single phase winding
with fully distributed coils. As explained, excessive spreading lowers
the value of the "Kapp" coefficient, and consequently the voltage; also
the use of a larger number of inductors to obtain the same voltage
results in an increase of armature self-induction. On the other hand,
if the winding were concentrated in fewer slots and these slots were
closer together, the result will be an increase in distorting and
demagnetizing reactions of the armature. Therefore, a compromise
between these two disadvantages must be made. The common practice is to
wind in two or three slots per pole per phase.]

[Illustration: FIG. 1,504.--Allis-Chalmers lap wound coils forming a
three slot distributed coil unit. In construction, after the coils have
been covered with insulating materials and treated with insulating
compound, the parts that lie in the slots are pressed to exact size
in steam-heated moulds. This runs the insulating material into all
the small spaces in the coil, excluding moisture and rendering the
insulation firm and solid. The ends of the coils, where they project
beyond the slots, are heavily taped.]

[Illustration: FIG. 1505.--Allis-Chalmers armature construction; view
showing section of frame and two layer winding.]

In practice, the coils are often more or less distributed, that is,
they do not always subtend an exact pole pitch; moreover, the flux
distribution, which depends on the shaping and breadth of the poles, is
often quite different from a sine distribution. Hence, the coefficient
2.22 in equation (2) is often departed from, and in the general case
equation (2) may be written

          ~_kf_ZN
  Eᵥᵢᵣₜ = -------                                                    (3)
             10⁸~

where _k_ is a number which may have different values, according to the
construction of the alternator. This number _k_ is called the _Kapp
coefficient_ because its significance was first pointed out by Prof.
Gisbert Kapp.

[Illustration: FIGS. 1,506 and 1,507.--Effect of breadth of coils in
distributed windings. In the section of the alternator shown in fig.
1,506 the directions of the pressures induced as the armature rotates
clockwise are represented by dots for those which act towards the
reader, and by crosses for those which act from the reader (the dots
and crosses representing respectively the heads and tails of arrows).
Since the field is not uniform but maximum at the center and gradually
weakening towards the extremities, it is obvious that the maximum
pressure is induced in any inductor as it passes the center of the
pole, this variation being indicated by the heavier dots and crosses
toward the center. Now if a number of these inductors be connected up
to form a distributed coil as in fig. 1,507, the pressures induced in
each will be added, but all the maximum pressure will not be induced
in all at the same time, hence the total pressure induced in the
distributed coil is less than it would be if the coil were concentrated
as in fig. 1,509.]

[Illustration: FIG. 1,508.--Diagram of distributed coil whose
inner breadth is less than the breadth of the pole face, showing
the disadvantage of such arrangement. The pressures induced in the
inner windings of such a coil are opposing each other at the instant
depicted, that is, while the inductors are under the pole face, such
action of course being objectionable.]

The value of _k_ is further influenced by a "breadth coefficient" or
"winding factor."

The effect of breadth in distributed windings is illustrated in figs.
1,506 to 1,508.

~Wire, Strap, and Bar Windings.~--In the construction of alternators,
the windings may be of either wire, strap, or bar, according to which
is best suited for the conditions to be met.

~Ques. What conditions principally govern the type of inductor?~

Ans. It depends chiefly upon the current to be carried and the space in
which the inductor is to be placed.

[Illustration: FIG. 1,509.--Simple form of alternator coil, consisting
of numerous turns of insulated wire wound around a form, then covered
with a tape winding, varnished and baked.]

~Ques. What kind of inductors are used on machines intended for high
voltage and moderate current?~

Ans. The winding is composed of what is called _magnet wire_, with
double or triple cotton insulation.

~Ques. Where considerable cross section is required how is a wire
inductor arranged?~

Ans. In order that the coil may be flexible several small wires in
multiple are used instead of a single large wire.

~Ques. How is the insulation arranged on inductors of this kind?~

Ans. Bare wire is used for the wires in parallel, insulation being
wrapped around them as in fig. 1,510.

    This construction reduces the space occupied by the wires, and
    the insulation serves to hold them in place.

[Illustration: FIGS. 1,510 and 1,511.--Multi-wire inductors. When the
cross section of inductor necessary to carry the current is large, the
use of a single wire would present difficulties in winding on account
of its stiffness. Accordingly two or more smaller wires are used in
parallel to secure the required cross section. Bare wire is used and
the several sections encased in insulation as shown, the combination
being more flexible than an equivalent single wire.]

[Illustration: FIG. 1,512.--Two coil slot for whole coil winding. The
slot has two recesses A and B for the reception of separate coils. In
assembling the winding, the inner wedge is first placed in position and
then the slot line with the insulating material. This usually consists
of alternate layers of mica and pressboard. The coils composed of
several turns of wire or copper strip are wound in place, and after
covering with a layer of insulation, the outer wedge is pushed in place
to retain the inductors in position.]

~Ques. What precaution is taken in insulating a wire wound coil
containing a large number of turns?~

Ans. On account of the considerable difference of pressure between
layers, it is necessary to insulate each layer of turns as well as the
outside of the coil, as shown in fig. 1,513.

[Illustration: FIG. 1,513.--Method of winding a coil containing a large
number of turns, when there is considerable difference of pressure
between the layers. In such cases to guard against short circuits or
breakdown of the insulation, each layer of turns is insulated from the
next layer by the insulating strips A, B, C, in addition to the regular
insulation around each wire. After the coil is made up it is wound with
insulating tape, varnished and baked.]

~Ques. Do distributed coils require insulation between the separate
layers?~

Ans. Since they are subdivided into several coils insulation between
layers is usually not necessary.

~Ques. How is a coil covered?~

Ans. It is wound with a more or less heavy wrapping of tape depending
upon the voltage.

[Illustration: FIGS. 1,514 and 1,515.--Single and double layer
multi-wire inductors and methods of placing them on the core. Here the
term layer means unit, in fact each unit is made up of several "layers"
of wires. In fig. 1,514, where so many wires are bunched together in
one unit, each layer of turns is separated from those adjacent by
insulating strips on account of the considerable difference of pressure
between layers. This insulation is not necessary in fig. 1,515 where
there are two units or so called layers. In both cases the inductors
are held in place by wedges driven into dovetail grooves.]

    Linen tape of good quality, treated with linseed oil, forms a
    desirable covering. Where extra high insulation is required the
    tape may be interleaved with sheet mica.

~Ques. Is the insulation placed around the coils all that is necessary?~

Ans. The slots into which the coils are placed, are also insulated.

[Illustration: FIG. 1,516.--Copper strap or ribbon with insulation.
These are generally from ¹/₃₂ to ¹/₁₆ inch thick with rounded edges as
shown to avoid cutting the insulation.]

[Illustration: FIG. 1,517.--Bar inductor. Its shape enables putting the
maximum cross section of copper into the slot and is used to advantage
on machines which generate large currents.]

[Illustration: FIG. 1,518.--Style of armature core stamping used with
bar wound machines. This construction, since there are no indentations
in the teeth for wedges, makes it necessary to provide bands to hold
the bars in place.]

~Ques. How are bar windings sometimes arranged?~

Ans. In two layers, as in fig. 1,523.

~Single and Multi-Slot Windings.~--These classifications correspond to
_concentrated_ and _distributed windings_, previously described. In
usual modern practice, only two-thirds of the total number of slots
(assuming the spacing to be uniform)

[Illustration: FIGS. 1,519 and 1,520.--Bent bar inductor and method of
connection with soldered joint. Fig. 1,519 shows one bar and shape of
bent ends. The portion from C to D is placed in the slot; B to C and D
to E, bent or connector sections; A to B and E to F, ends bent parallel
to slot for soldering. Fig. 1,520 shows two bar inductors connected.]

[Illustration: FIGS. 1,521 and 1,522.--Method of avoiding a soldered
joint at one end of a bar inductor by using a bar of twice the length
shown in fig. 1,519, and bending it into a long U form, as in fig.
1,521, after which it is spread out forming two inductors, as in
fig. 1,522.] of a single phase armature are wound with coils. The
reason for this may be explained by aid of fig. 1,524, which shows an
armature with six slots per pole, four of which are wound. Owing to the
different positions of, say, coils A and B, there will be a difference
in phase between the pressure generated in them and consequently the
resultant pressure of the two coils joined in series will be less than
the sum of the pressure in each coil.

[Illustration: FIG. 1,523.--Arrangement in slot of two layer bar
winding. With bar inductors, as must be evident from the illustration,
the maximum cross section of copper can be placed in a slot of given
dimension, hence a bar winding is used to advantage for alternators
designed to carry a large current. Bar inductors, on account of the
shape of their ends, must be placed in the slots from the top, because
the bent ends do not admit of pushing them in. Straight slots are
therefore necessary, the inductors being held in place by wooden strips
and tie bands as shown.]

Fig. 1,525 shows the pressure plotted out as vector quantities, and the
table which follows gives the relative effectiveness of windings with
various numbers of slots wound in series.

The figures in the last column of the table show that a large increase
in the weight of active material is required if the inductors in a
single phase machine are to be distributed over more than two-thirds
the pole pitch. Again, if much less than two-thirds of the surface be
wound, it is more difficult to provide a sine wave of pressure.

[Illustration: FIG. 1,524.--Diagram of single phase multi-coil or
distributed winding to show characteristic differences in action and
construction from single coil or concentrated winding.]

[Illustration: FIG. 1,525.--Vector diagram of pressures induced in the
single phase multi-coil or distributed winding shown in diagram in fig.
1,524.]

  ~TABLE OF RELATIVE EFFECTIVENESS OF WINDINGS~

   ----------------------------------------------------------------
  |            |                 |             |                   |
  |Slots wound | Pressure across |   Winding   |    Quantity of    |
  | in series  |      coils      | coefficient | copper to produce |
  |            |                 |             |   same pressure   |
  +------------+-----------------+-------------+-------------------+
  |            |                 |             |                   |
  |     1      |      1          |   1         |       1           |
  |     2      |      1.93       |    .97      |       1.03        |
  |     3      |      2.73       |    .91      |       1.10        |
  |     4      |      3.34       |    .84      |       1.19        |
  |     5      |      3.72       |    .74      |       1.35        |
  |     6      |      3.86       |    .64      |       1.56        |
  +------------+-----------------+-------------+-------------------+

~Ques. What other advantage besides obtaining a sine wave is secured by
distributing a coil?~

Ans. There is less heating because of the better ventilation.

[Illustration: FIG. 1,526.--Developed diagram of a single phase
monotooth or one slot bar winding; it is suitable only for operation at
low voltage.]

~Single Phase Windings.~--There are various kinds of single phase
winding, such as, concentrated, distributed, hemitropic, etc. Fig.
1,527 shows the simple type of single phase winding. It is a "one slot"
winding, that is, concentrated coils are used.

The armature has the same number of teeth as there are poles, the
concentrated coils being arranged as shown. In designing such a
winding, the machine, for example, may be required to generate, say,
3,000 volts, frequency 45, revolutions 900 per minute.

    These conditions require 720 inductors in series in the
    armature circuit, and as the armature is divided into six slots
    corresponding to the six poles, there will be 120 inductors
    per slot, and the coil surrounding each of the six teeth on
    the surface of the armature will consist of 60 turns. The
    connections must be such as to give alternate clockwise and
    counter-clockwise winding proceeding around the armature.

[Illustration: FIG. 1,527.--Diagram of six pole single phase revolving
armature alternator, with monotooth or concentrated whole coil
winding. For 3,000 volts at 900 revolutions per minute, 120 inductors
are required. And in the case of a concentrated or monotooth winding
they may be arranged in "whole coils" as above or in "half coils"
(hemitropic) as in fig. 1,528.]

~Ques. In what other way could the inductors be arranged in
concentrated coils?~

Ans. They could be grouped in three coils of 120 turns each, as shown
in fig. 1,528.

    When thus grouped the arrangement is called a hemitropic
    winding, as previously explained.

[Illustration: FIG. 1,528.--Diagram of six pole single phase alternator
with concentrated half coil or hemitropic winding of same capacity as
in fig. 1,527. There are an equal number of inductors, but in this case
arranged in three instead of six coils. In this winding the direction
of winding is alternately reversed so that the induced pressures do not
oppose one another.]

~Ques. What is the advantage, if any, of a half coil winding?~

Ans. In single phase machines a half coil winding is equivalent,
electrically, to a monotooth winding, and, therefore, is not of any
particular advantage; but in three phase machines, it has a decided
advantage, as in such, a concentrated winding yields a higher pressure
than a distributed winding.

[Illustration: FIG. 1,529.--Two phase concentrated whole coil winding.
In this style winding the total number of slots is twice the number
of poles, or one slot per pole per phase. It comprises two windings
identical with fig. 1,527, being spaced 90 polar degrees as shown. The
two circuits are independent, the windings terminating at the four
collector rings.]

[Illustration: FIG. 1,530.--Two phase winding in two slots per pole
per phase. This stamping distributes the coils of each phase into two
sections, as A and B. The coils are of the "whole" type and with six
poles the total number of slots is 4 × 6 = 24, uniformly spaced as
shown.]

~Two Phase Armature Windings.~--This type of winding can be made from
any single phase winding by providing another set of slots displaced
along the surface of the armature to the extent of one-half the pole
pitch, placing therein a duplicate winding.

[Illustration: FIGS. 1,531 and 1,532.--Developed diagram of the single
phase monotooth windings shown in figs. 1,527 and 1,528.]

    For instance: If the six pole monotooth, single phase winding,
    shown in fig. 1,527, be thus duplicated, the result will be the
    one slot two-phase winding shown in fig. 1,529, which will have
    twelve slots, and will require four slip rings, or two rings
    for each phase.

    By connecting up the two windings in series, the machines could
    be used as a single phase, with an increase of voltage in the
    ratio of 1.41 to 1.

[Illustration: FIG. 1,533.--Two phase winding in three slots per pole
per phase. The coils of each phase are of the partially distributed
type, each coil being made up of three sections as shown. The direction
of winding is alternately reversed.]

[Illustration: FIG. 1,534.--Section of two phase winding showing
shaping of the coil ends. Every other coil is flat, while the
alternates have their ends bent down as shown. With respect to the
shaping of the coil ends, it is called a _two range winding_.]

~Ques. How must the coils be constructed for two phase windings?~

Ans. They must be made of two different shapes, one bent up out of the
way of the other, as in fig. 1,534.

    There are numerous kinds of two phase windings; the coils may
    be concentrated or distributed, half coil or whole coil, etc.
    Fig. 1,530 shows a two phase winding with four slots per pole,
    and fig. 1,533 one with six slots per pole.

[Illustration: FIG. 1,535.--Section of Triumph armature showing method
of arranging the three phase winding.]

~Three Phase Armature Windings.~--On the same general principle
applicable to two phase windings, a three phase winding can be made
from any single phase winding, by placing three identical single phase
windings spaced out successively along the surface of the armature
at intervals _equal to one-third and two-thirds, respectively, of
the double pole pitch_, the unit in terms of which the spacing is
expressed, being that pitch, which corresponds to one whole period.

[Illustration: FIG. 1,536.--Three phase winding with distributed
coils--wound in four slots per pole per phase; diagram showing
placement of the coils.]

[Illustration: FIG. 1,537.--Treatment of coil ends in two phase, two
range windings. In this arrangement _straight out_ (B) and _bent up_
(A) coils are used which are placed on the armature as is clearly shown
in the illustration.]

[Illustration: FIG. 1,538.--Three phase, 10 pole, 30 slot winding in
two ranges. In this winding perfect symmetry occurs after every four
poles. Accordingly in the case of an odd number of pairs of pole, one
of the coils must necessarily be askew going from the inner to the
outer range as at M.]

Each of the three individual windings must be concentrated into narrow
belts so as to leave sufficient space for the other windings between
them. This limits the breadth or space occupied by the winding of any
one phase to one-third of the pole pitch.

[Illustration: FIG. 1,539.--Three phase 10 pole 30 slot winding in
three ranges. The coils of each phase are alike, those of the A phase
being all in the straight out range, those in the B phase, in a bent up
range, and those in the C phase in a bent down range. This arrangement
has the disadvantage, that by reason of the third range, the field
magnet cannot be withdrawn. This treatment of the coil ends is more
clearly shown in fig. 1,540.]

~Ques. How are three phase coil ends treated?~

Ans. They may be arranged in two ranges, as in fig. 1,538, or in three
ranges, as in fig. 1,539.

~Ques. What kind of coil must be used for three phase windings in order
that the ends may be arranged in only two ranges?~

Ans. Hemitropic or half coils; that is, the number of coil per phase
must be equal to one-half the number of pole.

[Illustration: FIGS. 1,540 and 1,541.--Treatment of coil ends in three
phase, three range windings. Fig. 1,540, inadmissible arrangement in
which the field magnet cannot be withdrawn; fig. 1,541, admissible
arrangement in which the armature segments can be divided. This enables
the top half of armature to be removed by disconnection without
unwinding any coil.]

~Grouping of Phases.~--In the preceding diagrams, the general
arrangement of the coils on the armature surface are shown for the
numerous classes of winding. In polyphase alternators the separate
windings of the various phases may be grouped in two ways:

1. Star connection;

2. Mesh connection.

[Illustration: FIG. 1,542.--Three phase winding with half coils. The
advantage of employing half coils is that the ends may be arranged in
two ranges as shown. There is one slot per phase per pole, that is,
total number of slots = 3 × number of poles.]

[Illustration: FIG. 1,543.--Three phase winding with whole coils.
Two sides of adjacent coils come in one slot. Number of coils per
phase = number of poles per phase. Total number of slots = 3 multiplied
by number of poles per phase. Whole coils require the ends arranged in
three ranges as indicated. The coils are concentrated.]

~Ques. Describe the two phase star connection.~

Ans. In this method of grouping, the middle points of each of the
two phases are united to a common junction M, and the four ends are
brought out to four terminals _a_, _a'_, _b_, _b'_, as shown in fig.
1,544, or in the case of revolving armatures, to four slip rings.

[Illustration: FIG. 1,544.--Diagram of two phase star grouping.]

~Ques. What does this arrangement give?~

Ans. It is practically equivalent to a four phase system.

[Illustration: FIG. 1,545.--Diagram of two phase mesh grouping.]

~Ques. How is the two phase mesh connection arranged?~

Ans. In this style of grouping, the two phases are divided into two
parts, and the four parts are connected up in cyclic order, the end
of one to the beginning of the next, so as to form a square, the four
corners of which are connected to the four terminals _a_, _b_, _a'_,
_b'_, as shown in fig. 1,545, or in the case of revolving armatures, to
four slip rings.

~Ques. Describe a three phase star connection?~

Ans. In three phase star grouping, one end of each of the three
circuits is brought to a common junction M, usually insulated, and the
three other ends are connected to three terminals _a_, _b_, _c_, as
shown in fig. 1,546, or in the case of revolving armatures to three
slip rings.

[Illustration: FIG. 1,546.--Diagram of three phase star grouping,
commonly called ~Y~ grouping owing to its resemblance of the letter
~Y~. The current in each main is obviously equal to the current in
each phase winding, but the terminal pressure is the vector sum of the
pressures in the component phase windings, that is, √3̅ multiplied by
the pressure in one phase.]

~Ques. What other name is given to this connection, and why?~

Ans. It is commonly called a ~Y~ connection or grouping owing to the
resemblance of its diagrammatic representation to the letter ~Y~.

[Illustration: FIG. 1,547.--Radial diagram of three phase, one slot
winding with ~Y~ connection.]

[Illustration: FIG. 1,548.--Radial diagram of three phase one slot
winding with delta connection.]

~Ques. How is a three phase mesh connection arranged?~

Ans. The three circuits are connected up together in the form of a
triangle, the three corners are connected to the three terminals, _a_,
_b_, _c_, as shown in fig. 1,549, or in the case of revolving armatures
to three slip rings.

[Illustration: FIG. 1,549.--Diagram of three phase mesh grouping,
commonly called delta grouping owing to its resemblance to the Greek
letter Δ. The voltage at the terminals is equal to the voltage in one
phase, and the current in each line is equal to the vector sum of the
currents in two phases, that is, it is equal to √3̅ multiplied by the
current in one phase.]

~Ques. What other name is given to this style of connection, and why?~

Ans. It is commonly called a _delta_ grouping on account of the
resemblance of its diagrammatic representation to the Greek letter Δ.

[Illustration: FIG. 1,550.--Three phase winding with short coils. The
use of short coils as here shown, in which the coil breadth = ⅔ pole
pitch, avoids the necessity of overlapping.]

    In polyphase working, it is evident that by the use of four
    equal independent windings on the armature, connected to eight
    terminals or slip rings, a two phase alternator can be built to
    supply currents of equal voltage to four independent circuits.
    Likewise, by the use of three equal independent windings,
    connected to six terminals or slip rings, a three phase
    alternator can be made to supply three independent circuits.

    This is not the usual method employed in either case, however,
    as the star grouping or mesh grouping methods of connection
    not only gives the same results, but also, in star grouping,
    a greater plurality of voltages for the same machine, and a
    higher voltage between its main terminals.

    Radial diagrams of the arrangement and connections of ~Y~
    grouping of lap windings and wave windings for three phase
    alternators are shown by figs. 1,551 and 1,552.

[Illustration: FIG. 1,551.--Radial diagram of three phase _lap_ winding
with star connection.]

~Ques. In three phase star grouping, what is the point where the phases
join, called?~

Ans. The star point.

~Ques. In a three phase star connected alternator what is the voltage
between any two collector rings?~

Ans. _It is equal to the voltage generated per phase multiplied by √3̅
or 1.732._

[Illustration: FIG. 1,552.--Radial diagram of three phase _wave_
winding with star connection.]

~Ques. In a three phase star connected alternator what is the value of
the current in each line?~

Ans. The same as the current in each phase winding.

[Illustration: FIGS. 1,553 and 1,554.--Gramme ring armatures showing
three phase star and mesh connections, respectively, with direction of
currents in the coils. In the figures, the coils A, B, C, are spaced at
equidistant positions on the ring core. The arrow heads represent the
directions of the induced pressures or currents for the position shown,
the rotation being clockwise. In coil A the pressure is increasing, in
coil B it is diminishing, but is in the same direction as in A, whereas
in coil C it is also diminishing, but is in the opposite direction to
what it is in coils A and B. As the rings rotate the three coils have
similar alternations of pressure induced in them, but differ in phase.
If _a_, _b_ and _c_ be joined to collector rings three phase currents
can be supplied to the outer circuits. In fig. 1,553 at the instant
represented _a_ and _b_ are giving their current to their lines, while
_c_ is receiving from its line a current equal to the sum of _a_ and
_b_. In fig. 1,554, at the instant represented, the currents sent out
from _a_ will be equal to the sum of the currents in _x_ and _y_, and
intermediate between them in phase. The current from _b_ will be equal
to the difference of the currents in _z_ and _y_, and of intermediate
phase, while similarly the current received by _c_ will be equal to the
sum of the currents in _x_ and _z_.]

~Ques. What is the value of the total output in watts of a star
connected alternator?~

Ans. It is equal to the sum of the outputs of each of the three phases.
When working on a non-inductive load, the total output of a star
connected alternator is equal to √3̅ multiplied by the product of the
line current and line voltage.

~Ques. What is the value of the line voltage in a three phase delta
connected alternator?~

Ans. It is equal to the voltage generated in each phase.

~Ques. What is the value of the line current in a three phase delta
connected alternator?~

Ans. It is equal to the current in each phase multiplied by √3̅.

~Ques. What is the total output of a three phase delta connected
alternator working on a non-inductive load?~

Ans. The total watts is equal to √3̅ multiplied by the product of the
line current and the line voltage.

[Illustration: FIGS. 1,555 to 1,557.--Separate coils, and section
of Allis-Chalmers alternator with coils in place. Numerous openings
are provided in the frame through which air currents, set up by the
revolving field, can pass freely and carry off heat. Shields are
provided to protect the armature coils where they project beyond the
core. In assembling the core spacing segments are placed at intervals
to form ventilating ducts. After the coils have been covered with
insulating materials and treated with insulating compound, the parts
that are to lie in the slots are pressed to exact size in steam heated
moulds. This runs the insulating material into all the small spaces in
the coil so as to exclude moisture, it also makes the coil structure
firm and solid. The projecting ends of the coils are heavily taped,
suitable supports being provided for the coil connections so that they
cannot become displaced on account of stresses due to short circuits
or other causes. On high pressure machines the armature terminals are
arranged so that it is impossible for an attendant to make accidental
contact with them.]

~Ques. What are the features of the star connection?~

Ans. It gives a higher line voltage than the delta connection for the
same pressure generated per phase, hence it is suited for machines of
high voltage and moderate current.

    The delta connection gives a lower line voltage than the
    star[5] connection for the pressure generated per phase, and
    cuts down the current in the inductors; since the inductors, on
    this account, may be reduced in size, the delta connection is
    adapted to machines of large current output.

[5] NOTE.--In the star connected armature the proper ends to connect
to the common terminal or star point are determined as follows: Assume
that the inductor opposite the middle of a pole is carrying the maximum
current, and mark its direction by an arrow. Then the current in the
inductors on either side of and adjacent to it will be in the same
direction. As the maximum current must be _coming from_ the common
terminal, the end toward which the arrow points must be connected
to one of the rings, while the other end is connected to the common
terminal. The current in the two adjacent inductors evidently must
be flowing into the common terminal, hence the ends toward which the
arrows point must be connected to the common terminal, while their
other ends are connected to the remaining two rings.

[Illustration: FIG. 1,558.--Diagram of Westinghouse two phase composite
wound alternator, showing connections between two phase armature and
a single phase rectified and composite field winding. The arrangement
makes use of a series transformer, mounted on the spokes of the
armature. By means of this series transformer the voltage delivered
to the rectifying commutator and the fields is much less than that
generated by the machine. The armature of this machine is of the closed
coil single winding type, all the armature inductors being connected
with each other to form a closed circuit which resembles to a certain
extent the ordinary drum winding of a multipolar direct current
machine. This winding is tapped out at two points per pole just as is
the continuous winding of a two phase rotary converter, these taps
running to collector rings through which the currents are delivered to
the outside circuits. On account of this connection of both phases to
one winding there is a definite voltage set up between the inductors
of phase A, and of phase B, this voltage being shown by the figures
given in the diagram. The arrangement is adapted for two phase work
by fitting the series transformer for the auxiliary field excitation
with two primaries connected respectively in one leg of each of the
two phases; thus the transformer is excited by two currents normally ¼
period out of phase with each other. The result upon the secondary is
a combination of the effects of the two primary currents, the voltage
delivered by the secondary being intermediate in phase between those
pressures which would be separately set up by the two primaries. This
combination effect is shown in the small diagram in the upper right
hand corner of the illustration. If OA be the effect set up in the
secondary of the series transformer by the primary current of phase
A, and OB be the effect set up by the primary current of phase B, OC
represents in magnitude and phase relation the resultant effect upon
the secondary. It is readily seen that this resultant is not equal to
the arithmetical sum of the two components since, to a certain extent,
they work at cross purposes. However, if either one of them increase
the resultant effect increases, although not in exact proportion. If
the load remain balanced, the two components remaining equal to each
other, the resultant OC varies in exact proportion to any changes in
the components. If the load become unbalanced, the resultant swings
around more nearly into phase with the larger load; thus if OB become
greater, OA remaining the same, OC swings around, becoming more nearly
horizontal. This requires a readjustment of the position of the brushes
on the commutator to set them properly for minimum sparking, an
adjustment exactly similar to that required when the power factor of
the load changes.]

[Illustration: FIG. 1,559.--Diagram of Westinghouse three phase
composite wound alternator. The armature inductors are of the closed
coil or delta connected type, but are tapped at three points per
pair or poles to the three collector rings. All three connections
between the armature coils and the collector rings run through primary
circuits of the series transformer within the armature, these three
primaries each giving their own effect upon the secondary. Since the
resultant of three equal alternating electromotive forces 120° apart
is zero, so that some special arrangement must be adopted to make
these electromotive forces act with instead of against each other. The
arrangement is a reversal of the connections of one of the primaries
of the series transformer. This is shown in the case of the lowest
primary indicated in the diagram. The combination of the effects of
the three primaries is again indicated in the small vector diagram in
the upper right hand corner. Here OA is the effect of one primary,
OB that of another ⅓ of a period displaced from the former in phase,
and OC that which the third would exert were it not reversed, but the
reversal brings the effect of this third coil into the phase relation
OD, so that the three are only 60° apart. The combination of OA and OB
is equal to OC, which combined again with OD gives a resultant effect,
OE. In this case, as in the other, the effect upon the series field
does not remain exactly proportional to the load unless the latter is
balanced; in fact, an increased current through the one leg represented
by OD, affects the series field as much as an equal increase in each
of the other legs put together. Practically, however, any increase of
the load--distributed as it must be in two legs at least--increases the
field excitation so that proper regulation is secured.]

~Ques. How is the path and value of currents in a delta connected
armature determined?~

Ans. Starting with the inductors of one phase opposite the middle of
the poles, assume the maximum current to be induced at this moment;
then but one-half of the same value of current will be induced at the
same moment in the other two phases, and its path and value will best
be shown by aid of fig. 1,560, in which X may be taken as the middle
collector ring, and the maximum current to be flowing from X toward Z.
It will be seen that no current is coming in through the line Y, but
part of the current at Z will have been induced in the branches _b_ and
_c_.

[Illustration: FIG. 1,560.--Diagram showing determination of path and
value of current flowing in delta connected armature.]

~Ques. Since most three phase windings can be connected either Y or
delta, what should be noted as to the effects produced?~

Ans. With the same winding, the delta connection will stand 1.732 as
much current as the ~Y~ connection, but will give only 1 ÷ 1.732
or .577 as much voltage.

[Illustration: FIG. 1,561.--Triumph brushes and brush holder. The
holder is of the box type provided with an adjustable tension spring,
making the brushes self-feeding. Each holder is carried on insulated
studs attached to a cast iron yoke which is mounted on the bearing.]

[Illustration: FIG. 1,562.--Diagram of ~Y~ connection with a common
return wire. When the three lines leading from _a_, _b_ and _c_ are
equal in resistance and reactance, or in other words when the system
is _balanced_, the currents of the three phases are equal and are
120° apart in phase (each current lagging behind its pressure by the
same amount as the others) and their sum is at each instant equal to
zero. In this case the resultant current being equal to zero there is
no need of a common return wire. However, in some cases, where power
is distributed from transformers or three wire systems, the different
branches are liable to become unbalanced. Under such circumstances the
common return wire is sometimes used, being made large enough to take
care of the maximum unbalancing that may occur in operation. The return
wire is used sometimes on alternators that furnish current mostly for
lighting work.]

~Chain or Basket Winding.~--One disadvantage in ordinary two-range
windings is that two or three separate shapes of coil are required.
The cost of making, winding, and supplying spares would be less if
one shape of coil could be made to do for all phases. One way of
accomplishing this is by the method of chain winding, in which the two
sides of each coil are made of different lengths, as shown in fig.
1,563, and bent so that they can lie behind one another.

[Illustration: FIG. 1,563.--Diagram showing chain winding. In this
method of winding the coils are all similar with long and short sides.
It obviates the extra cost of making coils of several different shapes.
The diagram represents a winding for one slot per pole per phase.]

[Illustration: FIG. 1,564.--General Electric terminal board showing
cables leading to three phase winding.]

In the case of open slots the coils may be former wound and afterwards
wedged into their places.

In chain winding the adjacent coils link one another as in a chain
(hence, the name); the winding is similar to a skew coil winding. This
plan of winding is supposed to have some advantage in keeping coils of
different phases further separated than the two range plan.

[Illustration: FIG. 1,565.--Section of armature winding of
Allis-Chalmers 500 kw. three phase water wheel alternator. The coils
are of the concentrated "half" type. Each coil is completely insulated
before being placed on the core and no insulation is placed in the
slot itself. The ends of the coils where they project beyond the slots
are heavily taped. Where necessary suitable supports are provided for
the coil connections so that they cannot become displaced on account
of stresses due to short circuits or other causes. The winding is of
the "chain" type. This is shown by the way the coils are connected
together at the right. The armature terminals are either provided with
insulated connectors or are led to a marble terminal board on which
the terminals are so mounted and protected that it is impossible for
an attendant to make accidental contact with them. The position of the
illustration would indicate a horizontal alternator but the machine is
of the vertical type; the lug on the right shows this, being used for
adjusting the alternator on the foundation.]

~Skew Coil Winding.~--In this type of winding the object is to shape
the coils so that all may be of one pattern. This is accomplished by
making the ends skew shape as shown in figs. 1,566 to 1,568.

[Illustration: FIGS. 1,566 to 1,568.--Views of a section of skew coil
winding; so called on account of the skew shape given to the coil ends
in order that all the coils may be of one shape.]

~Fed-in Winding.~--This name is given to a type of winding possible
with open or only partially closed slots, in which coils previously
formed are introduced, only a few inductors at a time if necessary.
They are inserted into the slots from the top, the slot being provided
with a lining of horn fibre or other suitable material, which is
finally closed over and secured in place by means of a wedge, or by
some other suitable means. An example of a fed-in winding is shown in
figs. 1,566 and 1,568.

~Imbricated Winding.~--This is a species of spiral coil winding in
which the end connections are built up one above the other, either in a
radial, or in a horizontal direction.

The winding is used especially on the armatures of turbine alternators
and dynamos.

~Spiral Winding.~--This is a winding in which "spiral" coils, as shown
in fig. 1,569, are used. The spiral form of coil is very extensively
used for armature windings of alternators.

[Illustration: FIG. 1,569.--Diagram showing a spiral coil. This type
of coil is one in which each successive turn lies entirely within
the previous turn, starting with the outermost turn of the coil. The
successive turns of a spiral coil are thus not of the same size, and
are not overlapping as in a "lap" coil.]

~Mummified Winding.~--The word _mummified_ as applied to a winding is
used to express the treatment the coils of the winding receive in the
making; that is, when a winding, after being covered with tape or other
absorbent material, is saturated in an insulating compound and baked
until the whole is solidified, it is said to be mummified.

~Shuttle Winding.~--This type of winding consists of a single coil
having a large number of turns, wound in two slots spaced 180° apart.
It was originally used on Siemens' armature and is now used on
magnetos, as shown in figs. 1,459 to 1,461.

[Illustration: FIG. 1,570.--Frame and armature winding of Westinghouse
pedestal bearing alternator. Armature frames are of cast iron and
ventilated. Interior transverse ribs strengthen the frame and support
the core laminations. The armature core is built up of annealed and
japanned punched laminations. Armature slots are open. Armature coils
are form wound, impregnated, and interchangeable; they are held in
place with fiber wedges. Ventilating spaces are provided at intervals
in the armature core and also between all coil ends.]

~Creeping Winding.~--Another species of winding, known as a creeping
winding is applicable to particular cases.

    If three adjacent coils, each having a pitch of 120 electrical
    degrees, be set side by side, they will occupy the same breadth
    as 4 poles, and, by repetition, will serve for any machine
    having a multiple of 4 poles, but cannot be used for machines
    with 6, 10 or 14 poles. Fig. 1,571 shows this example.

[Illustration: FIGS. 1,571 and 1,572.--Diagram of creeping windings.
Fig. 1,571, three coils subtending four poles; fig. 1,572, nine coils
subtending eight poles.]

    In the same way 9 coils, each of 160 electrical degrees, will
    occupy the same angular breadth as 8 poles.

    Further, 9 coils of 200 electrical degrees will occupy the same
    angular breadth as 10 poles.

    Now of these 9 coils, any three contiguous ones are nearly in
    phase, if wound alternately clockwise and counter-clockwise.

    For the 8 pole machine, the phase difference between adjacent
    coils is 20 degrees.

    For the 10 pole machine, the phase difference is also 20
    degrees.

    The cosine of 20 degrees is .9397, consequently, if 3 adjacent
    coils be united in series, their joint pressure will be 2.897
    multiplied by that of the middle one of the three.

    The 9 coils may therefore be joined up in three groups of 3
    adjacent coils, for the three phases.

    By repetition, the same grouping will suit for any machine
    having a multiple of 8 or of 10 poles. These two cases are
    illustrated in figs. 1,572 and 1,573. In the figures, the coils
    are represented as occupying two slots each, but they might be
    further distributed.

[Illustration: FIG. 1,573.--Developed diagram of creeping winding: nine
coils subtending ten poles.]

[Illustration: FIG. 1,574.--Triumph pedestal and brush rigging for
large revolving field alternators. Carbon brushes are used, carried
in box type brush holders. The stand or pedestal here shown is the
kind used with the engine and fly wheel types of alternator. The brush
studs are mounted on the stand in such a manner that the brushes are
easily accessible. The latter carry only the low voltage direct current
necessary for exciting the field.]

~Turbine Alternator Winding.~--For the reason that steam turbines
run at so much higher speed than steam engines, the construction of
armatures and windings for alternators intended to be direct connected
to turbines must be quite different from those driven by steam engines.
Accordingly, in order that the frequency be not too high, turbine
driven alternators must have very few poles--usually two or four, but
rarely six.

[Illustration: FIGS. 1,575 and 1,576.--Westinghouse turbine alternator
armature construction. Fig. 1,575. View showing dovetail grooves in
armature casting; fig. 1,576, laminæ assembled in dovetail grooves of
armature casting.]

The following table will show the relation between the revolutions and
frequencies for the numbers of poles just designated.

  ~TABLE OF FREQUENCY AND REVOLUTIONS~

   --------------------------------------
  |           |                          |
  |           |        REVOLUTIONS       |
  | Frequency +--------------------------+
  |           |        |        |        |
  |           | 2 pole | 4 pole | 6 pole |
  +-----------+--------+--------+--------|
  |           |        |        |        |
  |    25     | 1,500  |   750  |   500  |
  |    60     | 3,600  | 1,800  | 1,200  |
  |   100     | 6,000  | 3,000  | 2,000  |
  +-----------+--------+--------+--------|

[Illustration: FIG. 1,577.--Armature of Westinghouse turbine alternator
with end bells removed showing method of bracing the coil ends.]

[Illustration: FIG. 1,578.--Stationary armature of Westinghouse turbine
alternator with part of the winding in place. Because of the small
number of coils in a turbine alternator as compared with a slow speed
machine of the same kva. rating, each coil carries a great amount of
power on large load, particularly at times of short circuits or grounds
on the external circuit. The "throw" of the coils is large, leaving a
considerable part of the winding in the end turns unsupported by the
armature core. For these reasons great stresses, which are dangerous,
if effective means be not adopted to withstand them, may exist between
the coils. The inductors are of such cross section that they can be
made rigid and insulated satisfactorily. The end turns are given a fan
like form as shown, affording ventilation and effective bracing as
shown in fig. 1,577. Cord lashings are, except in the smallest frames,
used only for holding in the small spacing blocks between the coils.
They are not depended on to support the coils. Malleable iron braces,
hard maple blocks, and brass or steel bolts with brass washers are used
to withstand the mechanical stresses imposed on the armature coils by
external short circuits.]

From the table, it is evident that a large number of poles is not
permissible, considering the high speed at which the turbine must be
run.

[Illustration: FIG. 1,579.--Two pole radial slot field. Radial slot
fields are used on very small and very large alternators. The field
diameters are so small that the end turns of the winding can be
effectively bound into place, such binding being necessary with a
radial slot machine. The shaft and disc are a one piece forging of
steel.]

~Ques. How is the high voltage obtained with so few poles?~

Ans. There must be either numerous inductors per slot or numerous slots
per pole.

[Illustration: FIGS. 1,580 to 1,582.--Westinghouse two pole parallel
slot field with ends removed showing construction. The parallel slot
design of field construction is used in Westinghouse machines up to
10,000 kva. capacity. In fig. 1,581, the large holes at the end near
the circumference of the cylinder are for the accommodation of the
bolts that hold the bronze end discs and stub shafts. In winding,
the cylinder is mounted on a horizontal turntable that rotates in a
horizontal plane. The copper strap field coil winding is wound turn
by turn under pressure and strip insulation is wound in between. When
completed the turns are held rigidly in position with heavy brass
wedges. An end disc made of bronze holds the stub shaft and is bolted
to each end of the steel center. When the leads are attached to the
collector rings the field is complete.]

~Ques. What form of armature is generally used?~

Ans. A stationary armature.

~Ques. What difficulty is experienced with revolving armatures?~

Ans. The centrifugal force being considerable on account of the
high speed, requires specially strong construction to resist it,
consequently closed or nearly closed slots must be used.

~Ques. How is the design of the rotor modified so as to reduce the
centrifugal force?~

Ans. It is made long and of small diameter.

    Some examples of revolving fields are shown in figs. 1,579 to
    1,584. Figs. 1,577 and 1,578 show some construction details of
    a stationary armature of turbine alternator.

[Illustration: FIG. 1,583.--Westinghouse two pole parallel slot turbine
alternator field.]

[Illustration: FIG. 1,584.--Westinghouse four pole parallel slot
turbine alternator field.]

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~ELECTRICAL GUIDE, NO. 1~

    Containing the principles of Elementary Electricity, Magnetism,
    Induction, Experiments, Dynamos, Electric Machinery.

~ELECTRICAL GUIDE, NO. 2~

    The construction of Dynamos, Motors, Armatures, Armature
    Windings, Installing of Dynamos.

~ELECTRICAL GUIDE, NO. 3~

    Electrical Instruments, Testing, Practical Management of
    Dynamos and Motors.

~ELECTRICAL GUIDE, NO. 4~

    Distribution Systems, Wiring, Wiring Diagrams, Sign Flashers,
    Storage Batteries.

~ELECTRICAL GUIDE, NO. 5~

    Principles of Alternating Currents and Alternators.

~ELECTRICAL GUIDE, NO. 6~

    Alternating Current Motors, Transformers, Converters,
    Rectifiers.

~ELECTRICAL GUIDE, NO. 7~

    Alternating Current Systems, Circuit Breakers, Measuring
    Instruments.

~ELECTRICAL GUIDE, NO. 8~

    Alternating Current Switch Boards, Wiring, Power Stations,
    Installation and Operation.

~ELECTRICAL GUIDE, NO. 9~

    Telephone, Telegraph, Wireless, Bells, Lighting, Railways.

~ELECTRICAL GUIDE, NO. 10~

    Modern Practical Applications of Electricity and Ready
    Reference Index of the 10 Numbers.

~Theo. Audel & Co., Publishers. 72 FIFTH AVENUE, NEW YORK~





    TRANSCRIBER'S NOTES


    Silently corrected simple spelling, grammar, and typographical
    errors.

    Retained anachronistic and non-standard spellings as printed.

    Enclosed italics markup in _underscores_.

    Enclosed bold markup in ~tildes~.





End of Project Gutenberg's Hawkins Electrical Guide v. 5 (of 10), by Hawkins



