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 (