A Text Book of Electrical Machinery
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THE
ALEXANDER GRAY MEMORIAL LIBRARY ELECTRICAL ENGINEERING Tii e ai rx
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Cornell University Library
TK 21S2.R98 A text-book of electrical machinery .v.
3 1924 004 407 536
1
Cornell University Library
The tine
original of
tiiis
book
is in
Cornell University Library.
There are no known copyright
restrictions in
the United States on the use of the
text.
http://www.archive.org/details/cu31924004407536
A TEXT-BOOK OF
Electrical Machinery. VOLUME
I.
ELECTRIC, MAGNETIC, AND ELECTROSTATIC CIRCUITS.
HARRIS
J.
RYAN,
M.E.,
Professor of Electrical Engineering, Leland Stanford Jr. University;
Member
American Institute of Electrical Engineers and American Society of Mechanical Engineers, etc.
of the
the
HENRY
H.
of
NORRIS, M.E.,
Professor of Electrical Engineering, Sib leyCollege, Cornell University; Member of the American Institute of Electrical Engineers and of the Society for the Promotion of Engineering Education, etc.
GEORGE
L.
HOXIE, M.M.E.,
Ph.D.,
Consulting Electrical Engineer;
Member of the American Institute of Electrical Engineers and Member of the American Society of Mechanical Engineers,
Associate
FIRST EDITION. SECOND THOUSAND.
NEW YORK: JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited.
Copyright, igo3,
BY
HARRIS
J.
RYAK.
ROBERT DDUMMOriD, PRJNTER, NEW YORK
PREFACE.
The
student of electrical engineering comes to the technical
and professional part ciples of elementary
and
tions
task
is
to
of his course well grounded in the prin-
and applied mathematics and of
characteristics
leam
physical
in the rela-
phenomena.
to apply this training to the
His
next
working principles
of engineering, both those underlying the design and operation of electrical machinery and those
methods are based.
With
desirable to produce a
upon which general engineering
these facts in
text for the
mind
it
has been found
purpose of communicating to
the student the working principles mentioned above and to pre-
pare him for reading profitably the literature of his profession.
The book has been
designed as a distinctively engineering
not as a work on physics or apphed mathematics.
time
it
text,
At the same
has been found desirable to restate in engineering terms
the elementary laws and principles of those sciences which bear
upon the subject
directly
As a it
in hand.
result of experience in teaching electrical engineering
has been found most satisfactory, both in maintaining the
interest of the student to found the treatment circuit,
nomena
and
from which the follows
illustrated
in economizing his time
upon
and energy,
the laws of the alternating-current
treatment of continuous-current phe-
naturally.
The
application
by means of a few problems.
It
of these
laws
is
has not, however.
PREFACE.
IV
been the purpose
1;o
make
this
a problem book, and the teacher
and student should prepare additional problems
for class
and
In this part of the work, books of the nature of " Elec-
home
use.
trical
Problems," by Hooper and Wells, will be found of service.
Volume
I covers the laws of the electric, magnetic,
and performance
treated in
The
Volume
elec-
such a manner that the analyses of the struc-
trostatic circuits in
tural
and
may be
II,
of electrical machinery,
characteristics
easily followed.
authors express obhgations to
Whi
literature of the profession.
e the
all
contributors to the
method of no one author
has been followed, the aim has been to profit by the work of
and
to provide
an
fessional text, laboratories, lectures,
of electrical-engineering personnel
Volume
material of
I
fifty
all
prepare the pro-
and the unhmited sources
and
literature
when amphfied with
problems, preferably taken from practical cases,
oughly covered in
to
further training from
student to secure most profitably
The
wherewith
in'.roductory text
recitations, the
additional
may be
thor-
two volumes being de-
signed to provide profitable work for approximately one hundred class exercises.
The form
of the material in this
years of experience in
its
New
York,
Sept.
is
the result of several
use as a text for the instruction of classes
in Cornell University. Ithaca,
volume
i,
1903.
COMTEISITS.
PAGB
Chapter
I.
Electricity and Magnetism.
Electricity
and
Electrical
Energy
r
Electromotive Force
z
Magnetism
5
Chapter
II.
Fundamental and Derived Units.
Fundamental Units
12
Derived Units
Power Consumption Problems in the Use
Chapter
III.
15 in Electric Circuits
19.
of Electrical Units
20
Periodic Curves.
Properties of Sine Curves
25 31 42
Combinations of Sine Curves The Fourier's Series for an Alternating Quantity Analysis of a General Periodic Curve
Chapter IV.
45
Complex Quantities. 60
Vectors Alternating Quantities and Vectors
Chapter V.
Laws of the Electric
65
Circuit.
Consumption of e.m.f. in Single Circuits Problems in Single Series Circuits Problems in Simple Multiple Circuits Consumption of e.m.f. in Series-multiple Circuits
Chapter VI.
68 76
80 82
Electric Power.
Function of the Electric Circuit Power with Current and e.m.f. in phase Power with Current and e.m.f. in quadrature Power with Current and e.m.l neither in phase nor in quadrature Average Power with non-sine form e.m.f. and Current
The
Equivalent Sine
92
93 95
96 99 100
Wave V
CONTENTS.
VI
PAGE
Magnetomotive Force and the Laws of the Magnetic
Chapter VII. Circuit.
M.m.f. and the Magnetization Curve Matters Affecting Permeability Reluctance of the Magnetic Circuit Magnetic Hysteresis Ewing's Theory of Magnetism Illustrative Problems in Magnetic Circuits
Chapter
105 112 122
125
132 13S
Rotating Magnetic Fields.
VIII.
Polyphase c.m.f .s, Currents and Fields
142
Components
145 146
of the Rotating Pivot Field
Production of a Rotating Pivot Field Components of the Rotating Cylinder Field
150 162
Production of Rotating Cylinder Fields
The
Chapter IX.
Electrostatic Field.
General Characteristics
The
Electrostatic
173 180
Corona
Dielectric Thickness to avoid
Corona
189
Dielectric Conduction
ipS 196
Problems on the Electrostatic Field
198
Dielectric Hysteresis
Chapter X.
Losses in Electric Circuits.
Sources of Circuit Losses
199 200
Resistance
207 213 216
Inductance Skin Effect in Conductors Eddy Losses in Conductors Eddy Losses in Magnetic Circuits Capacity of Transmission Lines and Cables
Magnetic and Dielectric Hysteresis and Dielectric Conduction
Appendix
-
217 223 230
iiJ
TABLE OF IMPORTANT SYMBOLS AND ABBREVIATIONS.
B
density of magnetic flux or induction.
C
electrostatic capacity.
e.m.f... electromotive force in volts.
E
e.m.f. effective value.
e.m.f. instantaneous value.
e
F
mechanical
f.
frequency in cycles per second.
force.
H
magnetomotive force
/
current in amperes, effective value.
i
current, instantaneous value.
/
in gilberts.
-t/"^^-
L
inductance in henrys.
m.m.f. .magnetomotive force. pL
magnetic permeability.
P
electric
$ Q.
power.
magnetic induction or
.
.
.
.
total
,
.
.
.
quantity of electricity.
r
electric resistance.
(R
magnetic reluctance.
t
time in seconds.
6
angle of phase difference.
W
electric
X
reactance.
z
impedance.
flux.
energy or work.
This table contains only those symbols and abbreviations which are frequently Those which are used locally only are explained when used.
used.
ELECTRICAL MACHINERY. CHAPTER
I.
ELECTRICITY AND MAGNETISM.
and
electrical energy.
1.
Electricity
2.
Electromotive force.
ment 3.
Three methods
an e,m.£
for maintaining
Measure-
of e.m.f.
Magnetism: a. Magnetomotive
force.
Magnetic flux. Water flow, t. Tension of the magnetic field. d. Other hydraulic analogies to magnetism. Magnetic tension and flux density. b.
4.
I.
Electricity and Electrical Energy.
nomena
—
Electrical
are manifestations of molecular action.
unfortunately,
no means available
character of the
for
phe-
There
are,
observing the exact
molecular mechanisms upon which these
phenomena depend.
Electricity
must,
for
this
reason,
be
studied like heat, light, chemical energy, and other forms of
molecular energy, that
is,
by
the' results of the operation of
in
its
effects.
By
observation of
molecular forces, as manifested
mass motion or in chemical action, some idea of the forces
can be gained.
As
implied in the preceding statement, electricity
of energy.
is
a form
This energy has the same character as has com-
:
ELECTRICAL MACHINERY.
2
mass motion, which
plete
readily transformed into
shown by the
is
[2
fact that
it
any other kind of energy.
may be It may
also be transferred from point to point
by the use of
molecular kinematic connection, and
this ability to transfer
it is
power without mass motion which makes carrier of
energy oyer long distances.
electrical
energy
is
electrical
electrical
is
it'
the only successful
The
transformation of
work, and the rate of
this action
power, just as in the case of mass motion.
Electrical energy
may be
stored in the production of
netism just as mechanical energy velocity of a mass.
It
may
is
energy
charge is
is
be recovered from
its
stored form.
is
when an
taken up by capacity, just as mechanical
by
stored
phenomenon
mag-
stored in accelerating the
Likewise electrical energy becomes potential energy electric
suitable
elasticity in
also reversible.
compressing a spring.
These
This
illustrations point to
the identity of electrical and mechanical energies, and the
important points
in the
(i) the conservation
study of electrical engineering are:
of all energy; (2) the energy character of
electricity.
Electromotive Force.
2. initial
— Electromotive
force*
is
the
cause of the electric current and of electrostatic attrac-
tion ^
This e.m.f.
may
be maintained by one of three methods as
follows a. i>'.
c.
a.
Thermo-electric. Chemico-electric.
Dynamo-electric.
When
the junction of two metals
is
heated, an e.m.f.
is
produced, the value of which depends on the metals and the temperature to which their junction
may
be used to cause a flow of
heated.
This e.m.f.
electric current
by connecting
is
* Usually. written e.m.f.
ELECTRICITY AND MAGNETISM. the unheated terminals and thus part of the applied heat
energy may be
But practically the portion
utilized electrically.
of heat so transformed
is
very small, and because of
of economy the thermal couple
Fig.
I.
is
very
little
used.
this lack
Fig.
—Cox Thermo-electric Generator,
represents a commercial form of the electric' thermo-pile. is
known If
b.
as the
Cox
some
It
generator.
two unlike metals, not
in a bath of
i
in metallic contact, are placed
which attacks one of them more than
liquid
up between the metals, and by suitable connection outside the liquid an electric current may be the other, an e.m.f.
is
set
This chemicail generation of current has
produced.
its
prac-
tical application in the primary battery which has an important
place in small work.
which c.
force
is
in
The is
common third
Fig. 2 shows a form of primary cell
use.
method
for the generation of
an electromotive
the all-important one to the engineer, and
in the application of the principle that a wire
magnetic
field in
it
consists
moved
in
a
such a direction as to cut across the magnetic
b
ELECTRICAL MACHINERY. flux
of the field will have
force the value of
which
Fig. 2
Fig. 3
velocity, trates
produced
will
it
an electromotive
depend on the length of wire,
A Typical
Primary
its
Cell.
Simple Dynamo-electric Machine,
and the strength of the
cut
field
experimentally this method
developing an e.m.f.
in
This principle
for is
by
it.
Fig. 3 illus-
electro-mechanically
used in the construction
ELECTRICITY
3]
of
all
AND MAGNETISM.
dynamo-electric machines which form the main means
for the conversion of
mechanical into electrical energy, and
vice versa.
Measurement of
e.in.f.
difference of potential,
cated and
is'
presence
of an e.m.f.,
or
indi-
amount may be
its
measured by means of an also
is
elec-
This instru-
trostatic voltmeter.
ment, which
—The
known
shown
an electrometer,
is
commercial form
in Fig. 4.
utilizes the facts that
in
there
as
a It is
repulsion or attraction between
two
electrically
charged bodies
and that two bodies may be charged by connection to the terminals of a circuit in which
a difference of potential exists. In
measurement of low
the
electrical pressures * the elec-
trometer plates constituting the
charged bodies are numerous in order that the loss of attrac-
due to the low pressure
tion
may
be made up.
the instrument
is
In this form
known
Fig.
4..
as a
-Multicellular Electrostatic Voltmeter.
multicellular voltmeter. 3. net.
Magnetism. It
is
made
—In Fig.
5,
NS
is
a permanent bar
of hardened crucible steel
mag-
and has been
magnetized through some natural means, such as contact with another bar magnet, or with a piece of loadstone, or
been placed *
The
in a solenoid carrying
an
expression " electric pressure "
is
electric current. often used for e.m.f.
it
has
When
:
ELECTRICAL MACHINERY. bar
this
remote from other magnetic substances and the
is
immediate region about a magnetic field is
£s
field,
it is
examined with a small compass,
such as that illustrated, will be found.
due to magnetism or magnetic
flux,
This,
which emanates,
from
one end of the bar and returns to the other as the
lines
show.
by
physicists
Extensive experimental researches conducted
have led to the following conclusions
in
this;
connection
KiG.
5.
—Magnetic Field surround-
Fig. 6.
a.
—Hydraulic Model of the Mag-
netic Field of a
ing a Bar Magnet.
The magnetism about
the bar
magnet
Bar Magnet.
is
due to a mag-
netomotive force (m.m.f.) that resides in the molecules 6f the bar magnet.
Such m.m.f.
bar, and, therefore, to the
is
proportional to the length of the
number
of molecules
which consti-
tute a single filament of the bar. d.
A
m.m.f. sets up a difference of magnetic pressure
ELECTRICITY AND MAGNETISM.
3]
f
between the two ends of the bar, which causes magnetism or magnetic flux to be established from one end of the bar to the In Fig. 6
other.
is
an
M,
The model
magnet proper
of the
is
located in a vessel of comparatively large size containing
The
water.
M
part
represents
metal tube perforated on
A
an hydraulic model of a
illustration of
permanent bar magnet.
rotating
carrying screw
shaft
the model with
magnet. It is a numerous small holes.
bar
the
sides with
all
propellers,' _/5!5!7")
furnishes
a water-motive force directed from
S
to
N
This force corresponds to the magneto-
within the tube.
motive force that resides in the molecules of the bar magnet.
When
the propellers, ffff, are set in motion, the water will
S and go
enter the tube at
The
out at N.
The
represents the magnetic flux.
lines
flux of this
drawn
represent the direction of the water flux at
water
in the figure
points in the
all
immediate region of the tube, while the space between these lines
is
a measure of the cross-section over which a definite
rate of water flux occurs.
In the same manner, then, as in this model the lines in
show at once the direction of magnetic amount at any point in the region of the magnet. Fig.
c.
5
At
points within
all
is
as follows
tension along
its
:
and
the field of flux there exists
mechanical force related to the magnet. force
flux
The magnetic flux own direction and
The
its
a
nature of this
possesses a mechanical
a mechanical pressure
everywhere at right angles to the direction of the magnetic flux.
The
entire field of flux
netic body from which It
it
is
rigidly attached to the
emanates or by which
has been found by experimental
mag-
established.
means, to be described
proportional to the square of the rate
later, that this tension
is
of magnetic
flux
any
magnetic flux
is
exists along
its
at
it is
point.
The
lateral
pressure
of
so intimately associated with the tension that
own
direction that
it
is
hardly necessary to
S
ELECTRICAL MACHINERY.
distinguish between the two.
to keep in
mind the
necessary to keep in mind
It is
much
their difference, however,
as
[4
necessary in meclianics
it is
between action and reaction.
difference
All engineering problems in magnetism are solved in terms of the tension of the magnetic flux. d.
flux
is
shown
Physical experience has
that magnetic
further
established in a closed circuit just as the water currents
model flow
in the hydraulic
amount of water current passes out at the
N end.
been found to exist
Whatever amount
in closed circuits; i.e.,
An
for the
magnetism about a magnetic body.
of magnetic flux
This amount
emitted from one side or
is
same amount everywhere
is
Of
route from one pole to the other. flux within the
of the tube also
entirely analogous property has
end, called pole, precisely that opposite pole.
5 end
enters the
whatever
magnetic body
little
is
re-enters at the in
existence en
the nature of magnetic
known.
Researches,
however, have long since proven that within the magnetic
body there
exists a state of things corresponding to a con-
tinuity of the
magnetic flux whereby a complete
circuit of
such
is
the case with the electric
current in the closed electric circuit.
Because our knowledge
flux
is
always established, just as
body ceases
of the state of things within the magnetic point, that
which completes the
circuit of
at this
magnetic flux within
and through the magnetic body is called induction. In Fig. 7, 4. Magnetic Tension and Flux Density. AAA is a bar of soft wrought iron formed as shown, and
—
the bar magnet of
AAA
as
shown
and
7.— Magnet and ^^^^'^^
^5.
This
is
is
mounted
the figure.
in
circumstances Fig.
NS
in contact
almost
all
with
in front it
much
Under these evidences
of
magnetic flux will be found to have disappeared
due to the
fact
in
the region that
that
the
surrounds
magnetic flux has
:
ELECTRICITY AND MAGNETISM.
4J
disappeared as induction in the soft wrought iron.
It
has
been found that but a very small amount of magnetic pressure is
consumed
maintaining the induction in the iron bar, while
in
the balance of the magnetic pressure generated by the m.m.f. that resides in the steel bar
taneously with
NS.
Thus
magnetic pressure
results
the steel bar
used up point by point, simul-
is
origin, in maintaining the induction
its
that practically no difference of
it is
along NS, which accounts
disappearance of the magnetic total
the
circuit
NSAAA
any other
is
and the end
Fig.
N
practically
the
any
the
that the
cross-section of
same
as that
which
NS
is
mounted on a knife-edge
at
suspended from the stirrup of a scale-beam
is
8.—Apparatus
It is true, too,
exists at
for
cross-section.
In Fig. 8 the magnet S,
flux.
amount of induction which
exists at
through
for Studying the
Tension due to Magnetic Induction.
The mounting ofiV5is with another knife-edge as shown. is formed separating air-gap, ag, so adjusted that a small
N
from
A
.
The
following facts
with this apparatus
may be
observed experimentally
.
ELECTRICAL MACHINERY.
10
The
1,
of induction
existence
NS
in
[+
and
AAA,
and a
corresponding amount of magnetic flux across the air-gaps at
N and S,
will at
once be shown by the tension registered on
the scale-beam.
The tension
2
is
which magnetic flux
An
proportional to the square of the rate at distributed over a given cross-section.
is
indication of the
amount of
from the pole
flux
the face of the soft iron armature or keeper,
AAA,
N into
is
given
by the throw of the galvanometer needle when the turn of wire, T, is drawn away from the position showji in the figure so as to cut the magnetic flux. flux
is
bar
is
now
Fig.
measurement has
Another
Fig. 10.
9.
Change of Air-gap Area.
been made.
just
cross-section and length as the
square in cross-section its
tension of the lines of
used in the place of the one upon which the
Illustrating the Effect of
same
The
weighed on the scale-beam and found to be P.
its
The second bar has the but instead of being
first,
horizontal thickness
is
one-half and
vertical thickness is twice the thickness of the bar that
just
been removed.
Fig. 10,
That
is, it
is
set
and the area of the air-gaps
that found in the preceding case, which
on edge is,
is
as
has
shown
in
therefore, one-half
illustrated in Fig. 9.
.
.
ELECTRICITY yiND MAGNETISM.
4j
The m.m.f. same
of the bar has been adjusted so as to set up the
This
flux as before.
may be shown by
by the galvanometer when the turn of wire,
Upon weighing
flux at the air-gap. find
it
ir
the kick given
T, cuts across the
the tension of the flux
we
Note the significance of these experiments.
to be 2F.
First experiment:
Total flux, ^. Cross-section at air-gap,
Flux
density,
B =
A.
-r.
,
Observed magnetic pull, F.
Second experiment: Total flux, $. Cross-section at air-gap,
Flux
density,
B"
,
=
-r-r-
\A.
= 2B'
Observed magnetic pull, 2F.
Had
the area in this experiment been
A
instead oi^A,.s.nd
had the magnetic density been maintained at B" = 2B' the magnetic pull would evidently have been 4F. Thus we find ,
when
A
remains constant,
Flux density = B' ; the magnetic pull " " " " =: 2B' ; " If
= F. = 4.F.
'
the flux density had been increased in a third experi-
ment to 3^' we should have found the magnetic pull to be gF. Thus we learn experimentally that the contractile tension of the magnetic fluk density.
is
proportional to the square
The numerical
XB'\ where
^
is
a
of the flux
value of this magnetic
constant arbitrarily chosen.
value in the centimeter-gram-second system
is
Its
-^— 07t
pull,'
F,
is
numerical
.
CHAPTER
II.
FUNDAMENTAL AND DERIVED
UNITS.
SYNOPSIS. 5.
Fundamental
The unit of magnetic flux. The unit of current. The unit of electromotive force.
a. i. c.
6.
The
electric circuit.
Through dynamo and simple conductor. Through dynamo, conductor, and condenser. Through dynamo, conductor, and electrolytic
u. b. c.
7.
Derived b. t
.
d. c.
f.
9.
cells,
units.
a.
8.
units.
The The The The The The
unit of resistance.
unit of inductance. unit quantity of electricity.
unit of capacity. unit of power.
unit of energy.
Power consumption in electric circuits. 1 Power consumed by resistance. 2. Power consumed by counter e.m.f. Problems in the use of electrical
5.
—The
Fundamental Units.
tromagnetic
action
is
units.
dynamic character of elec-
much
so
in
accord with
common
mechanical experience that the electromagnetic actions have
been chosen to form the basis absolute electrical units,
second
{c.g.s.) system.
in electrophysics
and
in
—
for the definition of
often called the
This system electrical
is
a system of
centimeter-gram-
universally adopted
engineering.
The
c.g.s.
units are usually of inconvenient magnitude and a system of
practical units
is
necessary.
The
practical units are arbitrary
multiples of the c.g.s. units. _
12
:
.
FUNDAMENTAL AND DERIVED
5] a.
The unit of magnetic
UNITS.
13
flux.
Practical unit, the Maxwell, equal to the c.g.s.
the flux which will produce a tension in I
-H Stt dynes
when
own
its
unit,
is
directio7i
of
distributed uniformly over one sq. cm. of
cross-section.
The name of this unit of magnetic flux The density of magnetic flux is the number
is
the maxwell.
of maxwells per
unit cross-section. b.
The unit of current.
Practical. unit, the
Ampere:
From experimental
one-tenth of the c.g.s. unit.
research
has been found
it
straight conductor carrying an electric current
that
a
a uniform
magnetic flux will be acted upon by a mechanical force
fleld of
tending to move It
in
it
at right angles to the direction of the flux.
has been found that this force
is
proportional to the length
of the conductor in the flux, the current strength, and the flux
These
density.
facts
form the basis
for defining the unit of
current strength as follows
One
unit of current in a wire located in
and at right angles
to
a uniform field of unit flux density will cause a mechanical force af one dyne
to
be applied to each centimeter length of the con-
ductor, at right angles both
This
is
the c.g.s. unit.
to the
flux and
One-tenth of
to the
it
as a convenient unit for practical purposes. practical unit of current
is
has been adopted
The name
of this
the ampere.
The
unit of electromotive force. Practical unit, the Volt: one hundred million c.
conductor
c.g.s. units.
Experimental research has determined that an e.m.f. generated in a conductor moved across a that
is
field
is
of magnetic flux
proportional to the velocity, flux density, and length of
the conductor moving through the facts the value of the unit of e.m.f.
as follows:
flux. is
On
the basis of these
determined by definition
ELECTRICAL MACHINERY.
14
One unit of electromotive force
is
[6-
generated in each centimeter-
length of a conductor moving at a velocity of one centimeter
second through a uniform field of unit flux density angles both
This unit tical is
is is
to
the flux
and
the length
to
and
per
at right
of the conductor.
a convenient definition, but the size of the resulting
inconveniently small for practical purposes.
The
unit that has been adopted, because of its convenient
one hundred million (lO^) times the
determined by
definition.
The name
pracsize,
size of the c.g.s. unit as
of this practical unit
is
the volt.
There are other
electric
and magnetic
units.
They
are
among which must
derived from the above fundamental units,
These derived
be included the unit of time, or the second.
units will be discussed in connection with the control of the electric current 6.
The
and the magnetic Circuit.
Electric
possible unless there
no case
is
electric action
a complete circuit over which a transfer
is
of an electric charge
flux.
—In
may
occur.
This circuit must be com-
plete through the source of e.m.f. as well as
by an external
route. a.
In Fig.
electric
The
circuit
current
circuit of the
is
1 1
the
in
dynamo forms
a part of the conducting
which the actuating
established
by
this e.m.f.
dynamo and conductor
_
1 1.
b.
— Simple Conductor
In
Fig.
12
Circuit.
the circuit
C^I3>
Fig. 12.
may
— Circuit containing Condenser. be closed through the
cumstances a rush of current, as indicated
A,
generated.
through the complete
.dynamo, the conductor, and a condenser.
at
is
as indicated in the figure.
I
Fig.
e.-m.f.
will occur at the instant the circuit
is
Under these cirby the instrument closed.
As soon
as the strain in the dielectric of the condenser will no longer
FUNDAMENTAL AND
7]
DERII^ED UNITS.
15
increase under the pressure of the actuating e.m.f.,
through the condenser
will
Now
cease.
if
the
all
current
conductor
by some such means as illustrated in a momentary current will be shown on the instrument
terminals are reversed Fig-.
A
1
3,
while the process of relieving the dielectric strain in one
direction
and applying
it
in the other
reversal of the condenser terminals
mutator
in Fig.
1
is
going on.
Continued
by revolving the com-
3 will cause a succession of current
impulses
to be established in the circuit.
^ Fig.
c.
in Fig.
through the dynamo, an
14,
1
—
13.— Circuit containing Condenser with Commutator.
Again, as
'I'I'I'I
Fig. 14. Circuit containing Electrolytic Cells.
the circuit
may
electric conductor,
be established
and one or more
In general the cells will transmit the current
electrolytic cells.
with more or less irregularity through processes of internal
molecular changes
merely close an
or
Electrolytic
transfers.
electric circuit as
cells
do not
does a conductor or a con-
denser; they are generally sources of e.m.f. and thus modify the current by subtracting from or adding to the source of e.m.f. 7.
Derived Units.
will
be set up
the
source
capacity,
in
any
e.m.f.,
— Current
Control.
electric circuit
—The
current
and upon the values of the
and inductance of the
that
depends upon the value of
circuit.
resistance,
Capacity and induc-
tance present phenomena like resilience and mass motion in
mechanics. a.
The unit of
resistance.
Practical unit, the It is
in a is
Ohm:
10^ c.g.s. tinits.
found experimentally that
when
a current
is
established
conductor at any constant temperature, electric pressure
consumed
in
direct proportion to the value of the current.
6
:
:
ELECTRICAL M/ICHINERY.
1
This property of a conductor
Based on the above is
fact,
called
is
[7
electric
resistance.
the deiinition of the unit of resistance
as follows
One unit of resistance will consume one unit of pressure per unit of current.
The
practical unit of resistance
sumes one •ohm
is
volt
On
per ampere.
is
called the
ohm.
It
con-
account of the ampere the
ten times and on account of the volt
it
is
a hundred
million (lo^) times, or a total of one thousand million (lo') times, the magnitude of the c.g.s. unit. b.
The unit of inductance.
Practical unit, the Henry: lo^ c.g.s. units.
Every conductor carrying current about
itself
the same
way
into existence
up magnetic
flux
This has been established by experiment.
In
it
is
learned that such flux cannot be brought
by the current without cutting across the con-
ductor about which
it is
the magnetic flux cuts across
changing, an e.m.f
is
flux
given the
As
long as the current
is
always such as
The
name
it
ability to self-generate
The above
inductance.
to
is
The oppose
process of setting up
about a conductor by the current
and the
is
equal to the rate at which
self-generated in the conductor.
of this self-induced e.m.f.
called self-induction, is
is
it.
the corresponding current change.
magnetic
In doing so an e.m.f
established.
generated in the conductor that
direction
sets
carries
is
an e.m.f.
facts constitute the
basis for defining the unit of inductance as follows
A
circuit possesses one unit
of change of current Since this
is
of inductance when a unit rate
in the circuit generates one unit
a derived unit, the
sponding practical unit
is
at
of e.in.f magnitude of the corre-
once determined by reference to
The second always remains as The ampere is one-tenth of, and
the ampere and the volt.
the
practical unit of time.
the
volt one hundred million (lo') times the corresponding absolute
:
FUNDAMENTAL AND DERIVED
7]
i7
UNITS:
This would make the practical unit of inductance one
unit.
thousand million
(lo') times the absolute unit
The name
definition.
determined
of this practical unit of inductance
byis
the henry.
The unit quantity of
c.
electricity.
Practical unit, the Coulomb, one-tenth c.g.s. unit.
When strain
a dielectric
subjected to electric pressure a definite
In the production of this strain a quantity
produced.
is
is
of electricity must be applied by transfer through the circuit.
A unit for quantity of electricity One unit quantity of transferred by one
This unit
sponding
is
it is
is
of time. Its corre-
On
account
one-tenth of the value of the absolute unit.
The unit of
is
called the coulomb.
capacity. ^
c.g.s. unit,
and
the Micro-
10-'^^ c.g.s. unit.
farad,
all
the quantity
to
the ampere-second.
Practical units, the Farad, IQ-
If
equal
in one unit
practical unit quantity of electricity
d.
or
is
derived from the fundamental units.
practical value
of the ampere
The
of current
-unit
therefore, necessary.
is,
electricity
an
electric circuit
be closed through a condenser, some
of the impressed pressure of the electric
taken up by the capacity of the
circuit will
Experiment
dielectric.
reveals in this connection the following fact: constituting the capacity will take up a
be
The
dielectric
quantity of electricity,
or electric charge, in proportion to the electric pressure applied
between electric
its
faces.
As
this applied pressure is
charge accepted by the
changed.
The
dielectric
is
correspondingly
rate of transfer of electric charge
of the electric current by which the change
The
changed, the
unit of capacity
is,
is
is
the value
accomplished.
therefore, defined in terms of the units
of current, pressure, and time, thus A dielectric in an electric circuit has a capacity of unity
l8
ELECTRICAL MACHINERY.
when
the transfer through
it
[7
of unit current requires a unit
rate of change of the applied pressure.
The magnitude
of the
corresponding
practical
unit
capacity, therefore, becomes onq tenth on account of
of tlie
ampere, and one hundred-millionth (lO"*) on account of the
making the
volt,
practical unit
of the absolute unit.
capacity
most
the farad.
is
The name of this practical unit The farad is inconveniently large
purposes,
practical
rated in a unit that
is
one thousand-millionth (io~')
so that condensers are
one millionth of the farad. microfarad.
stitute for the farad is called the
ordinarily
This sub-
The microfarad
one million-thousand-millionth (lO~^') of the
therefore,
is,
of for
absolute unit. e.
The unit of power.
Practical unit, the Watt: id' e.g. s. units. It
has been found by experiment that the power in any
part of an electric circuit e.m.f. at
The
its
is
proportional to the product of the
terminals and the current present.
unit of
power
is
applied zvhen a unit of current
is
established by a unit of pressure.
The corresponding ampere. The name of
practical
unit of
it
is
the volt-
watt.
Being derived
therefore,
one-tenth on
this unit is the
from the ampere and the volt
power
is,
account of the ampere and one hundred million (lo^) times on account of the It
unit. i.e.,
volt,
or ten million
(lo') times the absolute
has a mechanical equivalent of .ooij^oj horse-power;
there are j/fS watts in one horse-power,
f.
The unit of energy.
Practical unit, .the Joule : id' e.g. s. units.
Power
is
the rate of transformation of energy from one
form to another.
The amount
of energy thus transformed is
the product of the power and the time.
:
.
:
FUNDAMENT/I L AND DERIVED UNITS.
8]
T/ie Unit
of energy
is
I9>
transformed by unit power
in unit
time.
The
corresponding
The name
(lo'' c.g.s.').
Power Consumption
8.
consumption of power
in
unit
practical
of this unit in
is
Electric
an electric
watt-second
the
is
the joule. Circuits.
circuit is
— By
formation of electrical energy into some other form. transformation
may
Power Consumed by consumed by the resistance
Resistance.
By
lishing the current.
setting
the
E
is
power that
is-
IE, where /
is-
the pressure used in estab-
definition the value of the resistance
number of the
up the current
—The
of a conductor
the current in the conductor and
ohms means
This
occur in two ways
1.
in
the
meant the trans-
volts used per
in the conductor.
ampere
It follows, then,
in
that
E^ Ir. The power consumed by the
resistance of the wire
W= 1E= This power
changed
to
is
lost
from
is,
therefore,
Pr.
the wire as heat.
heat in a conductor due
The
electric
to its resistance is
power
propor-
tional to the square of the current.
The
relation
E=
Ir, as
above determined,
may be
transr
posed so as to stand
^=§' when
it
becomes Ohm's law, which
In a closed
electric circuit the
(O states
current equals the ratio of the
electromotive force to the resistance of such circuit. 2
Power Consumed by Counter-Electromotive
When circuit or
electrical is
energy
is
Force.
stored in or about an electric
transformed into mechanical or chemical energy.
ELECTRICAL M/ICHINERY.
20
an e.m.f.
produced
is
motive
in the circuit with
Such an
to the current.
e.m.f.
[9
a direction opposed
called a counter-electro-
is
force.
The rate at which the forjned
is
equal
is
stored or trans-
product of the counter-electromotive force
to the
and the current. The energy of a
energy
electrical
may be transformed into heat by and magnetic naolecular action in and about the
dielectric
circuit
Such processes are substantially the same as the
circuit.
dissipation of heat in resistance. 9.
Problems in the Use of Electrical Units.
—Prob.
the space between two poles,
B, Fig. 15,
//
is
i.
A
If
and
a magnetic field and the
area of each pole
is
10 sq.
find
in.,
the pull between the poles in pounds
when
the density of magnetism in the
10,000 maxwells per
field is
One pound Prob.
445,000 dynes.
is
5/6. 9 lbs.
In the magnetic circuit
2.
shown
in Fig.
is
a density of magnetism of 10,000 maxwells per
{B
=
10.000).
sq. ins.:
What
should
To produce
(«)
be, the
area
a pull of 100
lbs..'
cm.
sq.
Ans. 16 there
cm
sq.
of each pole in ((5)
To produce
a pull of 50 lbs..' (a)
Prob.
If the
3.
exposes 10
sq. in. surface to
(a)
50
(b)
lbs..'
(a) (b)
The
wire
in
A7is.
Fig. 16
each pole and weighs 10 sq.
100
cm.
lbs..'
"
lbs.,
needed to support a (c) 200 lbs..' is
2280 maxwells per " {a) What average current flows for the time is 7 microfarads, {b)
How much
pressure.?
energy
is
stored in the condenser at 1000 volts '
(«)
.84 ampere.
(^) 3-5 joules.
Ans. "
ELECTRIC/tL MACHINERY.
24
Prob. II.
changed from
The 3
current in the circuit to
is
tance
is
The
Fig. 24.
current in a circuit similar to that of Fig.
What
resistance of the circuit rising
?
(6)
When
Ans.
^-o
from 10 to 20 amperes 3 henrys.
is
volt
^wsm^
Fig. 23.
Prob. 12.
24
If .01
.000285 henry.
o
rises
in Fig.
the value of the inductance in
?
-E
24
shown
10 amperes in .2 second.
opposes the change, what
henrys
[9
in
The
two seconds.
induc-
average volts are consumed is
if
the
ohms {a) while current become uniform ?
three
current has
,
(a)
60
volts.
(
quency, and are generated by the radius vectors It is
required to find the form of th« curve
sum.
Construct the parallelogram
The
ordinate to the sine curve a
OA The
sin
xOA —
ordinate to the sine curve b
OB The ordinate
sin
xOB —
where
A
and B.
which
is
their
OASB. is
at
any
instant,
x^^
A-^x^. is,
at the
same
instant,
B^x^.
to the curve of sums, SjjTj
s,
fre-
s, is,
at the
same
instant,
= AjX^-{- B^x^ S^A^
=
B'^x^,
AS sin A^AS = OB
sin
xOB = B^x^=
xOS = OA
sin
xOA
S^A^,
and
OS Therefore
through
x^.
sin
S^x^^
is
+ ^5 sin A^S
the projection of
OS
on the ordinate
,
"]
PERIODIC CURVES. Since this
OASB,
it
true for
is
35
any position of the parallelogram
follows that
The sum of two sine curves having' the same frequency is the which is generated by the diagonal of the parallelo-
sine curve
gram formed on the radius vectors of the component curves. c. Sum of Sine Curves in Quadrature. The equation
—
may
y"
= B cos
X
y"
= B sin
[x
be written
when
it is
seen to be a sine curve that
y ^^ A The sum
is
90° ahead of the curve
sin X.
of the curves jj/'
is
+ J)
=^
sin
X
y"
and
therefore a sine curve which
is
= B cos x
generated by a radius vector
forming the diagonal of the rectangle produced by drawing
making the angle
x,
and B, making the angle x
-\
,
A
with
Fig. 33.
the zero position.
(See Fig. 33.)
The
must be
S=
VA^
+ ^.
value of this diagonal
ELECTRICAL MACHINERY.
36
The magnitude of
A
and B, while
makes with A, Thus
A
is
of the its
sum
phase position, or the angle a which
governed by the
= tan a.
or
=
a
a may therefore vary between the and A and B are positive, S
"
A A
negative and
B
positive,
=
dependent on the magnitude'
is
zero,
With
[X
5
ratio
tan
between
—i
limits o°
A
it
and B.
-—r.
and 90°.
If
x be
lies in
the
lies in
the second quadrant.
first
quadrant.
PERIODIC CURVES.
ii]
sum
express a sine curve as the
37
of the corresponding sine
and
cosine curves, thus:
J
=A
sm X
-\-
y
=
sin(;ir
+ "d
B
x
cos
(/)
instead of
nT
d.
S
•
Product of Sine Curves of the curves
product of sine
is
Same
Frequency.
meant the curves
— By
of products of
instantaneous values. I
.
When
Let
the sine curves are in phase.
to find the product of the sine curves b
and
b' ,
it
be required
Fig. 35.
Fig. 35.
Their equations are
= ^ sin X, y = A' sin X, j>/
and the product
of their ordinates at
yy'
Expanding
this, it
y^
any
instant
= y^ = AA' Ax? X
is
(8)
becomes
A A' = ^— (I — 2
cos 2x) ,
(9>
ELECTRICAL MACHINERY.
38
Transposing the axis of this curve from of
,
its
AA'
AA'
(^2jr
, .
above the axis of
distance
at a
.
+ -j, ....
a sine curve of twice the frequency of ^ or
AA' axis
^cos2x,
J-=
^2= - -y-sm is
XX to X'X' a distance
equation becomes
^2=^1
which
[11
l>
i>',
and
average value of
may
This
curve referred to tne axis
this
^'.
be written
—A= —A'= .
\/2
,
when
The average value of is
2.
When
AA' is
.
b'.
product of two sine values in
tlie
maximum
This
is
one-half
values.
the sine curves are in
quadrature.
Let
required to find the product of the sine values b and 36, which are
The
seen to be the prod-
the product of their effective values.
the product of their
its
j/2
uct of the effective values of b and
phase
it is
XX
(11)
with
2
,
(10)
b' ,
it
be
Fig.
in quadrature.
Their equations are
y
y = A' sin yy'
This
is
=
{A
sin
\x
^^
-|
A
sin X,
j
A' cos
x){A' cos x)
x,
= AA'
sin 2x.
.
(12)
a sine curve of double frequency, the axis of which
coincides with the axis of b and for a
=
complete period
is
zero,
b'.
Its
average value taken
,
"J
PERIODIC CURVES.
39
The average value of the product of two quadrature
sine curves
in
is zero.
Fig. 36.
3
.
When
rature.
curves b and
may be the
phase and not
the sine curves are not in
Let
it
Fig. 37.
quadrature,
in
quad-
b' ,
One
of these sine curves, say
divided into components, one of which
other
in
be required to find the product of the sine
with
is
in phase,
b'
and
These components are
b.
generated by the radius vectors
OB^ OB,^
where
d
is
= OB cos — OB sin 6,
e,
the angle of phase difference between
The product
of
A
and
B is
the
each of the components of B. (11)
and
(12).
maximum
A
The average
sum
of the products of
and B.
A
with
These products are given by
quadrature product
value of the component of
B
is
OB^
A
= OB cos
6.
which
is
The
is
zero.
in
phase with
40
ELECTRIG/IL MJCHINERY.
["
ll]
PERIODIC CURVES.
The
average product of the sine values
OA
OB
V2
V2
COS
A
41
and
therefore
d.
The average value of the product of two product of their
B is
sine curves is the
effective values, times the cosine
of their angle
of phase difference. e. The Rate of Change Curve of Sine Values. The rate at which the ordinates to a sine curve change their values is
—
variable,
and
if
plotted in rectangular co-ordinates produces a
Fig. 38.
sine curve in quadrature with the one from which
(See Fig. 38.)
The
first
jf/
curve
=
is
« sin X,
or plotted with time as abscissae .
y where
T is
a sm
it is
27r -=,^,
the time of one complete period.
Then df :=
-i^fra
T
cos -i^tdt,
T
it is
derived.
42
and the
[i*
ELECTRICAL MACHINERY.
;
change
rate of
^ = —asm[-+-^t) 'The rate of change curve
is
therefore
advance of the original curve, and
a
sine curve go° in
values are -~r times as
its
The integral of a
Conversely:
great.
(I3>
sine curve is
sine curve po° behind the original curve,
and
its
a second
—T
values are
times as great.
f The
The Addition of Sine Curves of Different Frequencies.
—
curve resulting from the addition of sine curves of different
frequencies
non-sine in form.
is
It is
always irregular
and negative portions taken with reference to the
tive
may
may
or
The
;
X axis
not be alike.
investigation
of this fact led
to
always be found
sum
by
discovery
the
Fourier that periodic sine curves of different frequencies
that their
posi-
may
phase positions, and frequencies such
in value,
produce a periodic alternating curve of any
will
desired form.
Fourier determined a systematic method for obtaining the
which when added would reproduce any given periodic curve however irregular periodic sine curves of different frequencies
it
might be.
series
is
The Fourier's Series
12.
Curve.
Such a
—
an
If
irregular
known for
curve
as a Fourier's Series.
Any be
Periodic Alternating
plotted
may be
rectangular
in
co-ordinates and the length of a complete period
is
made
equal
by the addition of sine waves properly selected as to amplitude and phase position, one of which has the same frequency, or wave length, as the to 2n, or 360°,
it
irregular wave,
times,
.
.
.
reconstructed
and others having twice, three times,
n times that frequency, and having
of 1/2,-1/3, i/4'
•
•
•
I
four
wave lengths
/^ that of the irregular wave.
PERIODIC CURyES.
12]
43
The component sine curves which combine to produce the total curve are known as the first, second, third, etc. harmonics. The first harmonic, or fundamental, has the fi-e.
quency of the
total curve
and so on.
firequency,
that adopted in music
of the fundamental in convention
The
=
A'
A'"
A^, A^,
ylj,
twice that
differs slightly
from
harmonic.
first
The change
used for convenience only.
made up
of harmonic sine
is
sin {x
+
.
where a tone having twice the frequency called the
general equation. for a curve
components jf
is
,
is
the second harmonic
;
This convention
.
-{- o-j) -|-
sin (3;r
.
sine waves, and
.
.
A"
+ «3) +
A^&re
.
.
.
a^y
-j-
+ A" sin
(nx
+ «„).
(14)
the amplitudes of the component
a^, a^,
a.^,
(2x
sin
.
.
.
a„ are constant angles which
comany multiple of 360°.
specify the position of the radius vectors generating the
ponent curves when x
These
is
shown
relations are
zero or
is
in Fig. 39.
In this figure the irregular curve
5
is
made up
of three
harmonic sine components, having frequencies of once, •and
times
three
A',
T'ectors
that
of the
irregular
curve.
The
bered
I,
2,
A",
3 to correspond to their velocities.
figure these radius vectors are
position in ;r
=
jf J
=
which x 45".
radius
and A'" which generate the component
curves rotate with different angular velocities, and are
and
twice,
is
The
and third harmonic
drawn
in
heavy
numIn the
lines for the
and
in dotted lines for the position
equations
of the fundamental, second,
zero,
sine
components of 5 are
y = A' sin {x + 30°), y = A" sin (2x + 60°), y" = A'" sm + 120°), (3;f
44
'ELECTRICAL MACHINERY.
[iz
.
:
PERIODIC CURVES.
13]
and the equation of the curve
J
=
A' sin{x-\-
30°)
45
5 is
+ A" sin
{2x
+ 60°)
4-^'"sin (3^:+ Note that the
this
120°).
(15)
curve does not have Hke loops above and below
X axis. From
the
statements already
made
it
follows
that
by
properly selecting the values A', A", A'", etc., and the constant angles of phase displacement; a^, a^, a^, etc., the above
may
equation
represent any finite, continuous, and periodic
curve whatever, no matter
how
irregular
it
may
be.
The
13. Analysis of a General Periodic Curve tion of
an irregular periodic curve
possible,
and usually
it
will
into
its
components
be found that the
first
is
separa-
always
five
ponents give a sufficiently close approximation to the curve to answer
may
all
practical purposes.
The
be made graphically or analytically.
method 1
is
comtotal
actual analysis
In either case the
based on the following laws
The average of the product of any harmonic with any when taken for a complete cycle of the
other harnuinic is zero
irregular curve.
The average of the product of two sine curves in phase, the same frequency, is half the product of their having and 2.
amplitudes.
The average of the product of two sine curves in quadrature and having the same frequency is zero. The discussion of the first law is beyond the scope of the 3
present text.*
The meaning
of the law
graphically as in Figs. 40 and 41.
may
In Fig.
be illustrated
40 the
b has twice the frequency of the fundamental curve
sine curve a.
Their
product taken over a complete period of a gives a curve
having loops of equal areas above and below the * See " Fourier's Series and Spherical Harmonics"
X axis.
— Byerly.
The
46 10.
ELECTRICAL MACHINERY.
in
PERIODIC CURVES.
13]
average ordinate of the product 41 the product of
in Fig.
and
47
therefore zero.
is
having twice the frequency of
b,
having three times the frequency of
c,
Similarly
a,
a,
gives a product
curve which has the average ordinate zero.
38,
For the proof of the second law see Section 11, and for the third law see Section 1 1, d, 2, page
d,
i
,
page
39.
The product of one quantity by another, is identical with the sum of the products of that quantity and each of the comThus,
ponents of the other.
x{a
= xa
-\- b)
It follows, therefore, that the
xb
-\-
(16)
average ordinate, taken over a
complete period, of the product of an irregular curve and a sine curve having n times
its
ordinate of the product of
of the
with
nth.
is
equal to the average
curve and the component
harmonic of the irregular curve which
The average
it.
frequency,
this sine
ordinate, taken
in
phase
between the same
limits,
is
of the product of this sine curve and any other harmonic com-
ponent of the components,
The
J
= A'
+ A" sin {2x + a^ + ^3) + + A" sin {nx +
sin {x -\- Wj)
A'"
sin (3;r
.
=
A^sin X
-\-
A^
it
be required to
sin 'SV
phase angle, a^
From
,
find
.
.
any
of this equation,
+ A^ sin
=
ix
-\-
.
.
-\-
A„
-(-
^3 cos 2^-\-
.
.
sin nx,
(17)
+ B„ cos nx. coefficient,
when
A,^
say A^, and the
the curve itself
Section 11, pages 35 and 2^,
(A'"f
ar„
11, c, p. 37, eq. (7), as follows:
+ ^j cos x-\- £j cos 2x Let
the other harmonic
all
zero.
is
be rewritten by Section
f
of
general equation
+ may
sum
total curve, or the
+ B,\
is
given.
[i3
ELECTRICAL MACHINERY.
48
and
"3 Multiply graphically a complete wave of the total curve by the curve
=
jj/3'
The average
sin 2,x.
E,
of this product,
ordinate
average ordinate of the product of the curve j^
=
A^
sin
latter
since the product of
ix,
each of the other components of
y
is
is
zero.
y^
equal to the factor
= sin
by the 2,x
with
But the average
ordinate of the product of j/3'
=
sin
IX
and j/3
is
=
^3
sin 2>x
+.,,
and
may
this
be placed equal to the average ordinate of the
graphic multiplication, or
A, Similarly, as total curve
F
is
the average ordinate of the product of the
and the curve ^3"
it
follows that B^
this
2E.
=.
=
=
The
2F.
cos ^x,
and cosine components of It was shown
sine
harmonic, A'", have thus been determined.
above that the amount and phase position of therefore, immediately
The
analysis of a general curve
fied
by an inspection
the
first,
third,
fifth,
between components
this
harmonic
are,
known.
of
its
form.
etc.,
will
may If
be often much simpli-
only odd harmonics,
i.e.,
are present, the phase relations
be similar at the end of each half
.
I3j
PERIODIC CURyES.
wave-length of the fundamental.
have
will
49
(See Fig. 45.)
X
above and below the
like loops
Such a curve
axis, and, con-
versely,
Curves having tain only the
This
is
like loops
above and below the
con-
odd hartnonics
the case with nearly all curves which represent the
behavior of electrical machinery.
In this case
ordinate of the graphical multiplications half
X axis
may
the average
be taken over a
-wave instead of over a complete wave, since the product
curve repeats
itself
every half period of the fundamental, where
only odd harmonics are present.
Where
the even harmonics are present,
i.e.,
fourth, sixth, etc., the phase positions of the
the second,
components are
similar only at the close of each complete period of the general
The
periodic curve.
result
is
that all such general curves must
be unsymmetrical about their axes tions of
each period must
differ in
;
the plus and minus por-
form.
Curves having unlike loops above and below the
X axis in
general contain both odd and even harmonics.
The
following problem
cation of the above
alternating curve
is
given to illustrate fully the appli-
method of
may
analysis
by which any
be resolved into
its
periodic
harmonic com-
ponents.
Problem.
It is
required to determine the harmonic
.
com-
ponents of the irregular curve drawn with the heavy line
in
Fig. 42.
This curve
is
symmetrical about
no even harmonics.
This fact
is
its
axis and should contain
tested in the diagram of Fig.
42, in the case of the second harmonic.
The
unit sine curve of double
diagram constitutes the analyzer second harmonic. original curve
It
by the
is
ratio
drawn 40
:
i.
frequency drawn in this
for the sine
to a
The
component
of the
larger scale than original curve
is
the
multi-
5°
ELECTRICAL MACHINERY.
in
^3]
PERIODIC CURVES.
plied
by
this analyzer, the result
down
laid
An
in the
5'
being the product curve as
diagram.
inspection of this product curve shows that
value must be zero. tive areas for a
From symmetry
sum must be
average value of the product curve due evidently to the
is
average
complete period that the curve encloses with
the axis must be equal, their
This result
its
the positive and nega-
zero,
and the
rnust, therefore,
be zero.
fact that the relative
phase
positions of the second harmonic with reference to the plus
minus portions of the original curve are such as to give
and
alter-
nately plus and minus signs to corresponding product values.
The
phase positions of the original curve and the
relative
second harmonic sine analyzer are such as
to give
product
values throughout a complete cycle symmetrically located with reference to the axis.
For every plus value there
is
a corre-
sponding negative value causing the average value of the product curve to be zero.
A trial of
a cosine second harmonic analyzer will produce
the same result, and for the same reasons.
Further
trial
of any even harmonic analyzer will produce
an average zero product It
for similar reasons.
follows from this test, therefore, that this symmetrical
irregular curve contains
product of
it
no even harmonics, otherwise the
and some of the even harmonic analyzers would
give positive average values.
The
curve being quite irregular and symmetrical will be
found to be rich
in the
odd harmonics.
In Figs. 43 and 44
are given diagrams showing the application of the
sine,
and
cosine third harmonic analyzers to determine the value and
phase position of the third harmonic of the original curve.
In
Fig. 43 the unit sine analyzer for the third harmonic analyzer is
drawn and the product curve determined
as shown.
The
52
ELECTRICAL MACHINERY.
[13
^3]
PERIODIC CURVES.
gggggggg 3
n hj
n
e
53
:
eiECTRICAL MACHINERY.
54 positive
and negative areas of the product curve are then
The
determined by means of a planimeter. divided
[13
by the length taken
final
to actual scale of a
net area
is
complete
period, producing thereby the average value of the product.
This value
is
drawn
the numerical work
in Fig. 43, is
and so labeled.
as follows
Positive area
5140
Negative area
454^
Net
positive area
594
Length of one period
360
Average product
594 —7-
The maximum
In doing this
:=
-\- 1. 65
value of the sine component of the third
harmonic of the original curve
^3
= 2(+
is
therefore
i-6s)
= + 3-30.
44 gives graphically a corresponding analysis to determine the cosine cohiponent of the third harmonic of the The numerical work for this is as follows: original curve. Fig.
Positive area
-
PERIODIC CURVES.
13]
The
total third
harmonic component
= VAi+Bi, A'" = ^3-3^ — 6.82^
55 therefore
is
A"'
^'"
=
7.56,
-64°
'
3-3
6'.
This third harmonic component
and phase position
.-6.82 *
+ tan is
drawn
to scale in
amount
in Fig. 45.
The fundamental and the fifth harmonics have been determined by this method. The only difference in detail met with consists in the use of sine
and cosine unit analyzers having
periodicities corresponding to those of the first
monics which they are
and
fifth
har-
The numerical values
to determine.
and phase positions of these harmonics are given
in the follow-
ing equation:
J
=
66.8 sin
X
-\-
4.4 cos
x
— *
The accompanying
-\-
3.3 sin
7.24 sin
^x
—
5;if -j-
6.82 cos 2^
1-99 cos 5x.
(18)
figure will assist the reader to see that the ratio of the
-X—
-Y cosine to the sine components gives the tangent of the angle of phase difference, not the cotangent as might at first be supposed.
and
:
ELECTRICAL MACHINERY.
56
The
equation
in the following
f or, it
=
(66.8^
may
conveniently and obviously be written
manner:
+ 3.33 - 7.24J +y(4.4i - 6.823 +
by combining the quadrature components
may
1-995).*
of the harmonics,
be written
> = 66.9 sin (x + 3° 70
[13
54')
+ 7.56 sin (sx - 64° + 7-5 sin (5;p+ 164°
6')
36').
PERIODIC CURl^ES.
13]
harmonics.
Evidently
nounced minor
the original curve had
irregularities there
between
difference
if
57
would be more of a residual
and the summation curve
it
more proFig. 45.
in
Such residual could then be expressed by the seventh, etc.,
By means
of the
method
illustrated in the
any periodic curve may be broken up
into
Conversely, any periodic curve
ponents. the
ninth,
harmonics.
sum
of
its
may
above problem
harmonic combe expressed as
harmonic components.
its
All problems that arise in the treatment of general periodic or general alternating quantities
ment of
may
be solved by the treat-
harmonic components by the methods and laws
their
given heretofore for the treatment of simple alternating or sine-
(
wave problems.* Problerns numbered 47 and 48 illustrate this method of solving problems that arise in dealing with alternating quantities. Effective
Value of a Non-sine Alternating Curve.
effective value of an alternating curve
mean
square of
nating curve
y=
its
values.
The
is
—The
the square root of the
equation of the general alter-
is
X cos X
A^svc\.
Ar
j5j
-\-
2x B^ cos 2x
A^
sin
A^s'm
'ix
-f
.
.
.
+ B^ cos
T,x
-{-
.
.
.
2,^
+
.
.
.
.
.
-\-
-j-
(19)
Squaring,
f — A^ sin^ X + A^ sin^ 2x + A^ sin^ B^ cos^ X + B^ cos2 2x + B^ cos^ -j-
ix -^
-f
.
(20)
terms of the form
{A^
sin
mx){A„
(A„
sin
nx)(B„ cos nx)
sin nx) -{-,..-{•
{B^ cos mx)(B„ cos nx) * From
this it is seen that
methods employed
are necessary for the convenient solution of
all
have, therefore, an entirely general application.
+ -\-
.
.
+
.
.
.
.
waves Such methods
for the treatment of sine
non-sine waves.
.
.
ELECTRICAL MACHINERY.
58
All of these product terms have a
See Section
From
1
3
,
page 45
law
,
Section 10,/,
it
sin^
= J(^^« + ^/+^/+ effective value
Effective
.
.
.
;i:)
= ^^«
(21)
+ B,'+B,' + B,^+
.
.
.).
(22)
.
.
.).
(23)
therefore,
is,
j/
= -^VA,^+Ai+Ai+. Since ^A^ and iB^ are the
ing curves, the result square of any harmonic
^J = where
value of zero.
follows that
(Mean ^)
The
mean
i
i.
(Mean ^«
From which
[13
H„
is
.
.B,^
mean
may
+ Bi+B,' +
squares of their correspond-
be further reduced.
The mean
is
KA„^
+ B^),
(see Sec.
1 1, — ^, = ar -\-jbr -\-jax \-j^x = ar -}-j'{br ax) — bx. -\-
This product
may
Its absolute value
— bx)
-fj{br
-\-
ax)
(30)
— bx)^-{-{br + axY.
we may Va'
this result
.
.
.
reduce this value to the form
+ blx
Vr^
+ x'.
....
is
and
6,
respectively,
be the angles which
a +jb, r +jx,
and
(32)
equal to the absolute value of their
product. 6",
(31)
follows that the product of the absolute
it
values of two vectors
6',
(29)
therefore.
is,
V{ar After expanding
,
be written
{ar
Let
be
Thus, {a -{-jUf
From
—
divided like ordinary
only necessary to bear in mind the meaning
It is
of the factor 7'.
(3-72).
and Division of Complex
Multiplication
Complex
(i+y6).
(4-73) (9+76) (- 6 -78)
(«)....
their product,
(ar
— bx)
-\-j\br
+ ax),
ELECTRICAL MACHINERY.
64
make
Then
with the horizontal. cos
[14
=
0'
+
(^2
r
=
cos 6"
^3)4'
b sin d'
+ b^f
{a'
ar
=
cos
(«'
cos Substituting,
(6"
we
cos
+
(9")
X
=
sin 6"
+
= cos
^)^
^'
{e'
+ e") =
+ ^ = + («^
this it follows that the
angles.
is
(^
+ ^)*'
- sin
cos e"
^'
From
X
^'
sin 0".
have, after reducing,
wherefore
two vectors
— bx
(33)
*
(9".
phase angle of the product of
sum of
equal to the
Phase angle
ar-bx X (^* + ^)*'
b^Y-
phase
their individual
the angle which the vector makes.
is
with the line of reference, usually-
taken as the horizontal.
The
quantity resulting
from
the multiplication or division of
complex
down
quantities
graphically.
47, the vectors
be
to
is
Fig. 47
these
2^
=
Expressed vectors
are
1
The numerical
V72-f
are-
+y2 and 3 +73. Their product 2 +76 +72 1-6=15 +727.
The
Multiplication of Vectors.
laid
Thus, in Fig.
oA and oB
multiplied.
analytically,
7
may be
values of
7.27,
1/3^
oA
+
32
,
or
oB, and
=4.24,
oR
absolute are
COMPLEX QUANTITIES.
15]
65
and ViS^
+ 27^ =
30.8
=
7.27
X
4-24.
The absolute value of the product of two or more vectors is the product of the absolute values of the vectors. The angle which this product makes with the angles
As
made by
initial line is the
sum of the
the several vectors.
the operation of division
plication, a similar
law
may be
the reverse of that of multi-
is
stated for the division of
com-
plex quantities.
The absolute value of the by another,
is
quotient,
The angle which
initial line is the difference
by the two vectors with the initial
As vector
a corollary is
reversed.
vector is divided
obtained by dividing the absolute value of the
dividend by that of the divisor.
makes with the
when one
it
may
its
in Fig. 48,
Fig. 48.
of the angles made
line.
be stated that the reciprocal of a
the reciprocal of
Thus,
this quotient
absolute value with
OA'
is
the reciprocal of
its
angle
OA.
—Division of Vectors.
15. Alternating Quantities
Quantities Represented by
and Vectors.
Vectors.
—The
—
a.
Alternating
actual value of an
66
ELECTRICAL MACHINERY.
alternating quantity if
[i5
changing from moment to moment, and
is
and negative alternations are equal the average any whole number of alternations is zero. The
the positive
value for
average
effect
produced by such a quantity
An
necessarily zero.
instance
electric current is present in
rent
is
given.
The
is
is
cur-
to heat the con-
this is the
when
same
the alternating
electric current or e.m.f.
effective
measured
The average
be shown later that
It will
quantity which was defined on page 29 quantity
however,
A new definition of effective value may
ductor continuously.
now be
not,
found where an alternating
a conductor.
zero, but the effect of the current
is
is
in
value of an alternating quantity
terms of an average
effect
which
is its
value
may produce
it
continuously.
Such a quantity, being an average value, may be expressed
by a
positive numeral or represented
by a
straight line.
It is
not in the strictest sense a directed quantity or vector.
It is
nevertheless
a very great convenience
arbitrarily assign to
many
in
cases
an effective value a direction which shall
indicate the phase position of the alternating quantity
The
alternating quantity
is
quantity,
the effective value of the
and a direction representing is
some other same frequency, which is chosen is
its
a relative item, and
the phase position of
This standard
itself.
then represented graphically by a
vector having a length equal to
phase position
to
is
phase position.
The
specified in terms of
alternating quantity of the as a standard of reference.
always represented by a vector
in the zero
position. b. tities.
Operations on Vectors Representing Alternating Quan-
—Alternating
quantities
may be added
or subtracted
adding or subtracting their respective vectors. such an operation
is
The
by
result of
represented by the resulting vector.
Independent alternating quantities
may
not be multiplied
complex quantities.
is]
or divided
ing their
67
by performing these operations on vectors representeffective values. Such an operation is meaningless
and the vector resulting from
it
does not represent the result
of the multiplication or division of the alternating quantities.
A vector
representing an effective value
may be
multiplied
by a vector representing a constant quantity having magnitude and direction, i.e., by a simple vector. The result
or divided
of such an operation
is
a vector which represents the result of
a similar operation on the quantities themselves. It is often
very convenient to determine the result of opera-
tions performed
on alternating quantities by performing the
same operations on vectors representing their effective values. This method is used for convenience only, and its limitations, as given above, must always be kept in mind.
Illustrations
of the statements of this section will be given in the course of
the discussions of alternating currents.
CHAPTER
V,
LAWS OF THE ELECTRIC
16.
17. 18. 19.
^
Consumption of e.m.f. in single circuits. a. E.m.f. consumed in resistance. b. E.m.f. consumed in inductance. c. E.m.f. consumed in capacity. d. E.m.f. consumed in (o), (b\ and (
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