steinmetz cp theory and calculation of transient electric phenomena and oscillations

ÆTHERFORCE
THEOEY
AND CALCULATION
OF
TRANSIENT
ELECTRIC PHENOMENA
AND
OSCILLATIONS
BY
CHARLES.
PROTEUS
STEINMETZ
THIRD
EDITION
RTCVISED
AND
ENLARGED
THIRD
IMPRESSION
McGRAW-HILL
BOOK
COMPANY,
ING.
NEW
YORK:
370 SEVENTH
AVENUE
LONDON:
& 8
BOUVEBIE
ST.,

E. C.
4
1920
ÆTHERFORCE
3'/a7
COPYRIGHT,
1920,
BY
THE
MCGRAW-HILL
BOOK
COMPANY,
INC.
COPYRIGHT,
1909,
BY
THE
McGiiAw
PUBLISHING
COMPANY.
FEINTED
IN THE
UNITED
STATES
OF
AMEBICA.
f',
LIBRARY
THE
MAPLE

PRESS
-
YORK PA
ÆTHERFORCE
DEDICATED
TO TUB
MWM.OHY
OK MY
FRIEND
AND
TEACHER
HUDOLtf EKJKEMEYER
ÆTHERFORCE
ÆTHERFORCE
PREFACE TO
THE THIRD
EDITION
SINCE
the
appearance
of
the,
first
edition,
ten
years
ago,
the
study
of

transients has been
greatly
extended
and
the
term
"transient"
has
become
fully
established
in
electrical
literature.
As
the
result
of
the
increasing
importance
of the
subject
and
our
increasing
knowledge,
a
large part
of this book

had
practically
to
be
rewritten,
with
the addition
of
inuch new
material,
espe-
cially
in
Sections
III and
IV.
In
Section
III,
the
chapters
on "Final
Velocity
of
the
Electric
Field"
and on
"High-frequency
Conductors"

have
been
re-
written
and
extended.
As
Section
V,
an
entirely
new section
has been
added,
com-
prising
six new
chapters.
The
effect
of the finite
velocity
of the
electric
field,
that
is,
the
electric
radiation

in
creating
energy components
of inductance
and
of
capacity
and
thereby
effective series
and shunt
resistances
is more
fully
discussed. These
components
may
assume
formid-
able
values
at such
high frequencies
as
are
not
infrequent
in
transmission
circuits,

and
thereby
dominate
the
phenomena.
These
energy components
and
the
equations
of the
unequal
current
distribution
in
the
conductor are
then
applied
to
a fuller
discussion
of
high-frequency
conduction.
In Section
IV,
a
chapter
has

been added
discussing
the relation
of the
common
types
of currents:
direct
current,
alternating
current, etc.,
to the
general equations
of the
electric
circuit.
A discussion
is
also
given
of the
interesting
case
of a
direct
current
with
distributed
leakage,
as

such
gives
phenomena
analogous
to
wave
propagation,
such as
reflection,
etc.,
which are
usually
familiar
only
with
alternating
or
oscillating
currents.
A
new
chapter
is devoted
to
impulse currents,
as
a
class
of
non-periodic

but transient currents
reciprocal
to the
periodic
but
permanent
alternating
currents.
Hitherto
in
theoretical
investigations
of
transients,
the circuit
constants
r
L C and
g
have
been
assumed as constant.
This,
however,
disagrees
with
experience
at
very
high frequencies

ÆTHERFORCE
viii
PREFACE
or
steep
wave
fronts,
thereby
limiting
the
usefulness
of
the
theoretical
investigation,
and
makes
the
calculation
of
many
im-
portant phenomena,
such
as
the
determination
of the
danger
zone

of
steep
wave
fronts,
the
conditions
of circuit
design
limit-
ing
the
danger zone, etc.,
impossible.
The
study
of
these
phenomena
has
been undertaken
and
four
additional
chapters
devoted to the
change
of
circuit
constants
with the

frequency,
the
increase
of
attenuation
constant
resulting
therefrom,
and
the
degeneration,
that
is
rounding
off of
complex
waves,
the
flattening
of wave fronts with
the
time
and
distance
of
travel,
etc.,
added.
The
method

of
symbolic representation
has
been
changed
from
the time
diagram
to the crank
diagram,
in accordance
with the
international
convention,
and
in
conformity
with
the other
books;
numerous
errors
of
the
previous
edition
corrected,
etc.
CHARLES
P. STEINMETZ.

Jan.,
1920.
ÆTHERFORCE
PREFACE
TO
THE
FIRST
EDITION
THE
following
work
owes
its
origin
to
a
course
of
instruction
given
during
the
last
few
years
to
the
senior
claas
in

electrical
engineering
at
Union
University
and
represents
the
work of
a
number
of
years.
It
comprises
the
investigation
of
phenomena
which
heretofore
have
rarely
been
dealt
with in
text-books
but
have
now

become of
such
importance
that a
knowledge
of
them
is
essential
for
every
electrical
engineer,
as
they
include
sonic?
of
the
most
important
problems
which
electrical
engineering
will
have
to
solve
in

the
near
future
to
maintain
its
thus
far
unbroken
progress.
A
few of
these
transient
phenomena
were
observed
and
experi-
mentally
investigated
in
the
early
clays
of
electrical
engineering
for
instance,

the
building
up
of
the
voltage
of
direct-current
generators
from
the
remanent
magnetism.
Others,
such
a,s
the
investigation
of
the
rapidity
of
the
response
of
a
compound
generator
or
a

booster
to
a
change
of
load,
have
become
of
impor-
tance
with
the
stricter
requirements
now
made
on
electric
totems
Iransient
phenomena which
were
of
such
abort
duration
and'
small
magnitude

as
to
be
negligible
with
the
small
apparatus
of
former
days
have
become
of
serious, importance
in
the,
hu,
generators and
high
power
systems
of
to-day,
as
the
discharge
of
generator
fields,

the
starting
currents
of
transformers
the
short
circuit
currents
of
alternators,
etc.
Especially
is
this
,
t
tht
ocl
asses
of
phenomena
closely
related
to
"
IK
ÆTHERFORCE
x
PREFACE

and
others,
dealing
with
the
fairly high
frequency
of
sound
waves.
Especially
lightning
and
all
the kindred
high
voltage
and
high frequency
phenomena
in
electric
systems
have
become
of
great
and still
rapidly
increasing

importance,
due
to- the
great
increase
in extent
and
in
power
of
the modern
electric
systems,
to
the
interdependence
of all the electric
power
users
in
a
large territory,
and to
the
destructive
capabilities
resulting
from
such
disturbances.

Where
hundreds of
miles
of
high
and
medium
potential
circuits,
overhead
lines
and
underground
cables,
are
interconnected,
the
phenomena
of
distributed
capacity,
the
effects of
charging
currents
of lines and
cables,
have
become
such as

to
require
careful
study.
Thus
phenomena
which once
were
of
scientific interest
only,
as
the
unequal
current distribu-
tion
in
conductors
carrying alternating currents,
the
finite
velocity
of
propagation
of the
electric
field,
etc.,
now
require

careful
study
by
the electrical
engineer,
who
meets
them
in
the rail
return
of
the
single-phase railway,
in the
effective
impedance
interposed
to the
lightning
discharge
on
which the
safety
of
the entire
system depends,
etc.
The
characteristic

of
all
these
phenomena
is
that
they
are
transient
functions of
the
independent
variable,
time or
distance,
that
is,
decrease
with
increasing
value of
the
independent
variable,
gradually
or in an
oscillatory
manner,
to zero
at

infinity,
while
the functions
representing
the
steady
flow of
electric
energy
are
constants
or
periodic
functions.
While
thus the
phenomena
of
alternating
currents are
repre-
sented
by
the
periodic function,
the
sine
wave
and
its

higher
harmonics or
overtones,
most of
the
transient
phenomena
lead
to a
function which is
the
product
of
exponential
and
trigono-
metric
terms,
and
may
be
called an
oscillating
function,
and
its
overtones
or
higher
harmonics.

A
second
variable,
distance,
also enters into
many
of
these
phenomena;
and
while the
theory
of
alternating-current
appara-
tus
and
phenomena
usually
has to deal
only
with
functions
of
one
independent
variable,
time,
which
variable

is eliminated
by
the
introduction
of
the
complex
quantity,
in
this
volume
we
have
frequently
to
deal with functions
of
time and
of
distance.,
ÆTHERFORCE
PREFACE
xi
We thus
have
to consider
alternating
functions
and transient
functions

of time and
of distance.
The
theory
of
alternating
functions
of
time
is
given
in
"
Theory
and Calculation
of
Alternating
Current
Phenomena." Transient
functions
of
time
are studied
in
the first
section of the
present
work,
and
in the second section

are
given periodic
transient
phenomena,
which
have
become
of
industrial
importance,
for
instance,
in
rectifiers,
for
circuit
control,
etc.
The
third
section
gives
the
theory
of
phenomena
which are
alternating
in time
and

transient
in
distance,
and the
fourth and
last section
gives
phenomena
transient
in
time and
in
distance.
To
some
extent this
volume can
thus be
considered as a
con-
tinuation of
"Theory
and Calculation
of
Alternating
Current
Phenomena."
In
editing
this

work,
I have
been
greatly
assisted
by
Prof. 0.
Ferguson,
of
Union
University,
who has
carefully
revised the
manuscript,
the
equations
and the
numerical
examples
and
checked the
proofs,
so that it
is
hoped
that the
errors
in the
work

are reduced to
a minimum.
Great
credit
is clue
to
the
publishers
and
their
technical
staff
for their valuable
assistance
in
editing
the
manuscript
and
for
the
representative
form of
the
publication
they
have
produced.
CHARLES
P. STEINMETZ.

SCHENECTADY,
December,
1908.
ÆTHERFORCE
PREFACE TO TPIE SECOND
EDITION
DUE to the
relatively
short
time
which has
elapsed
since
the
appearance
of
the
first
edition,
no
material
changes
or
additions
were needed
in
the
preparation
of the second
edition.

The work
has
been
carefully
perused
and
typographical
and
other
errors,
which had
passed
into
the
first
edition,
were
eliminated.
In
this,
thanks
are
due
to those
readers
who
have drawn
my
attention
to errors.

Since
the
appearance
of
the first
edition,
the
industrial
importance
of transients
has
materially
increased,
and con-
siderable
attention
has
thus been devoted to
them
by engineers.
The
term
"transient"
.has
thereby
found an
introduction,
as
noun.,
into

the technical
language,
instead of
the more
cumber-
some
expression
"transient
phenomenon,"
and
the
former
term
is therefore
used to some
extent
in
the
revised edition.
As
appendix
have
been
added tables
of
the
velocity
functions
of
the

electric
field,
sil x and
col
x,
and similar
functions,
together
with
explanation
of
their mathematical
relations,
as
tables
of
these functions
are
necessary
in
calculations
of wave
propagation,
but
are otherwise
difficult
to
get.
These
tables

were
derived
from
tables
of related
functions
published
by
J.
W.
L.
Glaisher,
Philosophical
Transactions
of the
Royal
Society
of
London,
1870,
Vol.
160.
xii
ÆTHERFORCE
CONTENTS
SECTION I.
TRANSIENTS IN
TIME.
CHAPTER
I. THE CONSTANTS OF

THE
ELECTRIC
CIRCUIT.
1. Flow
of electric
energy,
the
electric field
and
its
components.
2. The
electromagnetic
field,
the
electrostatic
field
and the
power
consumption,
and their
relation
to
current and
voltage.
3.
The
electromagnetic
energy,
the

electrostatic
energy,
and
the
power
loss of the
circuit,
and
their
relations to
the
circuit
constants,
inductance,
capacity
and
resistance.
4. Effect
of conductor
shape
and
material on
resistance,
inductance
and
capacity.
5.
The resistance
of materials
:

metals,
electrolytes,
insulators
and
pyroelectrolytes.
6. Inductance
and the
magnetic
characteristics of
materials.
Permeability
and
saturation,
and
its effect on
the
mag-
netic
field of the
circuit.
7.
Capacity
and the
dielectric
constant of
materials. The
disruptive strength
of
materials,
and

its effect on
the
electrostatic field of
the circuit.
8. Power
consumption
in
changing magnetic
and
static
fields:
magnetic
and
dielectric
hysteresis.
Effective
resistance and
shunted conductance.
9.
Magnitude
of
resistance,
inductance and
capacity
in
in-
dustrial
circuits.
Circuits
of

negligible
capacity.
10. Gradual
change
of circuit conditions
in a
circuit of
negli-
gible
capacity.
Effect of
capacity
in
allowing
a
sudden
change
of
circuit
conditions,
causing
a
surge
of
energy
between
magnetic
and static.
CHAPTER II.
INTRODUCTION.

11. The
usual
equations
of
electric
circuit
do
not
apply
to
the
time
immediately
after a
circuit
changes,
but
a
transient
term then
appears.
12.
Example
of the
transient
term
in
closing
or
opening

a con-
tinuous current circuit
:
the
building
up
and the
dying
out of the
direct
current
in
an alternator field,
xiii
PAGE
3
11
12
12
14
16
16
16
ÆTHERFORCE
xiv
CONTENTS
PAGE
13.
Example
of transient

term
pioduced
by
capacity:
the
charge
and
discharge
of
a
condenser, through
an
induc-
tive
circuit.
Conditions
for
oscillations,
and
the
possi-
bility
of excessive
currents
and
voltages.
17
14.
Example
of

the
gradual
and the
oscillatory
approach
of
an
alternating
current
to
its
permanent
value.
20
15.
Conditions
for
appearance
of
transient
terms,
and for
their harmlessness
or
danger.
Effect of
capacity.
21
16.
Relations

of
transient
terms and their character
to
the
stored
energy
of the circuit.
21
17.
Recurrent or
periodic
transient terms
: their
appearance
in
rectification.
_
22
IS.
Oscillating
arcs
and
arcing ground
of transmission
line,
as an
example
of recurrent transient
terms.

22
19.
Cases
in
which
transient
phenomena
are
of
industrial
im-
portance.
23
CHAPTER
III.
INDUCTANCE AND
RESISTANCE
IN
CONTINUOUS-
CURRENT CIRCUITS.
25
20.
Equations
of
continuous-current
circuit,
including
its
transient term.
25

Example
of
a
continuous-current motor
circuit.
27
Excitation
of
a
motor field. Time
required
for
shunt
motor
field
to build
up
or
discharge.
Conditions
of
design
to secure
quick
response
of
field.
27
23.
Discharge

of
shunt motor
field
while
the
motor
is
coming
to
rest.
Numerical
example.
29
24.
Self-excitation
of
direct-current
generator:
the
effect of
the
magnetic
saturation
curve.
Derivation of the
general
equations
of
the
building

up
of the
shunt
generator.
Calculations of
numerical
example.
32
25.
Self
-excitation
of
direct-current
series
machine.
Numeri-
cal
example
of
time
required
by
railway
motor to
build
up
as
generator
or
brake,

38
CHAPTER
IV.
INDUCTANCE
AND
RESISTANCE IN
ALTERNATING-
CURRENT
CIRCUITS.
41
26.
Derivation of
general
equations,
including
transient term. 41
27.
Conditions for
maximum
value,
and
of
disappearance
of
transient
term.
Numerical
examples;
lighting
circuit,

motor
circuit,
transformer
and
reactive
coil. 43
28.
Graphic
representation
of
transient
term.
45
ÆTHERFORCE
CONTENTS
XV
PAGE
CHAPTEE
V.
RESISTANCE,
INDUCTANCE
AND
CAPACITY
IN
SERIES.
CONDENSEB
CHARGE AND DISCHARGE.
47
29. The differential
equations

of
condenser
charge
and dis-
charge.
47
30.
Integration
of these
equations.
48
31.
Final
equations
of condenser
charge
and
discharge,
in
exponential
form.
50
32. Numerical
example.
51
33. The
three cases
of
condenser
charge

and
discharge
:
loga-
rithmic,
critical and
oscillatory.
.
52
34.
The
logarithmic case,
and the effect
of
resistance
in elimi-
nating
excessive
voltages
in
condenser
discharges.
53
35.
Condenser
discharge
in
a non-inductive
circuit.
54

36. Condenser
charge
and
discharge
in
a circuit of
very
small
inductance,
discussion
thereof,
and
numerical
example.
55
37.
Equations
of
the
critical
case
of condenser
charge
and dis-
charge.
Discussion.
56
3S. Numerical
example.
58

39.
Trigonometric
or
oscillatory
case.
Derivation
of the
equations
of the
condenser
oscillation.
Oscillatory
con-
denser
charge
and
discharge.
58
40. Numerical
example.
Cl
41.
Oscillating
waves
of current
and
e.m.f.
produced
by
con-

denser
discharge.
Their
general equations
and
frequen-
cies.
02
42.
High
frequency
oscillations,
and their
equations.
63
43.
The decrement
of the
oscillating
wave.
The effect
of
resist-
ance
on the
damping,
and
the critical resistance.
Numerical
example.

65
CHAPTER VI.
OSCILLATING
CURRENTS.
67
44. Limitation
of
frequency
of
alternating
currents
by genera-
tor
design;
limitation
of
usefulness
of
oscillating
current
by
damping
due to
resistance. 67
45. Discussion
of
sizes
of
inductances
and

capacities,
and their
rating
in
kilovolt-amperes.
68
46.
Condenser
discharge equations,
discussion
and
design.
69
47.
Condenser
discharge
efficiency
and
damping.
71
48.
Independence
of
oscillating
current
frequency
on
size of
condenser
and

inductance.
Limitations
of
frequency
by
mechanical
size
and
power.
Highest
available
frequencies.
72
ÆTHERFORCE
xvi
CONTENTS
PAGE
49. The
oscillating
current
generator,
discussion
of its
design.
74
50.
The
equations
of the
oscillating

current
generator.
76
51.
Discussion
of
equations: frequency,
current,
power,
ratio
of
transformation.
79
52. Calculation
of
numerical
example
of a
generator
having
a
frequency
of hundreds
of
thousands
of
cycles per
second. 82
53.
52

Continued.
86
54.
Example
of
underground
cable
acting
as
oscillating
cur-
rent
generator
of
low
frequency.
87
CHAPTER VII.
RESISTANCE,
INDUCTANCE
AND
CAPACITY
IN
SERIES
IN ALTERNATING
CURRENT CIRCUIT.
SS
55.
Derivation
of the

general equations.
Exponential
form. 88
56. Critical
case.
92
57.
Trigonometric
or
oscillatory
case. 93
58.
Numerical
example.
94
59.
Oscillating
start
of
alternating
current circuit.
96
60. Discussion
of the conditions
of its
occurrence. 98
61.
Examples.
100
62.

Discussion
of
the
application
of
the
equations
to trans-
mission
lines and
high-potential
cable circuits.
102
63.
The
physical
meaning
and
origin
of the
transient term.
103
CHAPTER
VIIL_
LOW-FREQUENCY
SURGES IN HIGH-POTENTIAL
"SYSTEMS.
105
64.
Discussion

of
high potential
oscillations
in
transmission
lines and
underground
cables.
105
65. Derivation
of
the
equations
of current and condenser
potentials
and
their
components.
106
66.
Maximum
and
minimum
values
of oscillation. 109
67.
Opening
the
circuit
of

a transmission
line
under
load.
112
68.
Rupturing
a
short-circuit
of
a transmission
line.
113
69. Numerical
example
of
starting
transmission
line
at
no
load, opening
it
at full
load,
and
opening
short-circuit. 116
70. Numerical
example

of
a
short-circuit oscillation
of under-
ground
cable
system.
119
71.
Conclusions.
120
CHAPTER IX.
DIVIDED
CIRCUIT. 121
72.
General
equations
of a
divided
circuit.
121
73.
Resolution
into
permaneiat
term and transient
term.
124
74.
Equations

of
special
case
of
divided
continuous-current
circuit
without
capacity.
126
ÆTHERFORCE
CONTENTS
xvii
PAGE
75. Numerical
example
of
a
divided circuit
having
a
low-
resistance
inductive,
and a
high-resistance
noninduc-
tive
branch.
129

76.
Discussion
of
the
transient
term
in
divided
circuits,
and
its
industrial
use.
130
77.
Example
of the
effect
of a current
pulsation
in
a circuit
on
a voltmeter
shunting
an
inductive
part
of
the circuit.

.
131
78.
Capacity
shunting
a
part
of
the
continuous-current circuit.
Derivation
of
equations.
133
79.
Calculations
of
numerical
example.
136
80.
Discussions
of
the
elimination
of
current
pulsations
by
shunted

capacity.
137
81.
Example
of
elimination
of
pulsation
from non-inductive
circuit, by
shunted
capacity
and scries inductance. 139
CHAPTER X. MUTUAL
INDUCTANCE. 141
82.
The
differential
equations
of
mutually
inductive
cir-
cuits.
141
83. Their discussion.
143
84.
Circuits
containing

resistance,
inductance
and
mutual
inductance,
but
no
capacity.
144
85.
Integration
of
their
differential
equations,
and their
dis-
cussion.
146
86. Case
of constant
impressed
e.m.fs. 147
87.
The
building up
(or
down)
of an
over-compounded

direct-
current
generator,
at
sudden
changes
of
load.
.
149
88.
87
Continued.
152
89. 87
Continued.
154
90.
Excitation
of series
booster,
with
solid
and
laminated
field
poles.
Calculation
of
eddy

currents
in
solid
field
iron.
155
91.
The
response
of a
series
booster
to
sudden
change
of
load.
158
92.
Mutual
inductance
in circuits
containing
self-inductance
and
capacity.
Integration
of the
differential
equations.

161
93.
Example
: the
equations
of the Ruhmkorff coil or
induc-
torium.
'
164
94. 93 Continued.
166
CHAPTER
XL
GENERAL
SYSTEM
OF
CIRCUITS.
168
95.
Circuits
containing
resistance
and
inductance
only.
168
96.
Application
to an

example.
171
ÆTHERFORCE
xviii
CONTENTS
PAGE
97. Circuit
containing
resistance,
self
and
mutual inductance
and
capacity.
174
98.
Discussion of the
general
solution
of the
problem.
177
CHAPTER XII.
MAGNETIC SATURATION AND
HYSTERESIS IN
MAG-
NETIC CIRCUITS.
179-
99.
The

transient term in a
circuit
of constant inductance.
179
100.
Variation
of inductance
by
magnetic
saturation
causing
excessive transient currents.
ISO
101.
Magnetic cycle causing
indeterminate values
of
transient
currents.
181
102. Effect of
frequency
on
transient terms
to be
expected
in
transformers.
181
103. Effect of

magnetic
stray
field
or
leakage
on
transient
starting
current of
transformer.
182
104.
Effect of
the
resistance,
equations,
and method
of con-
struction
of
transient current
of
transformer when
starting.
185
105.
Construction of
numerical
examples,
by

table.
188
106.
Approximate
calculation
of
starting
current
of
transformer.
190
107.
Approximate
calcxilation
of transformer
transient
from
Froehlich's
formula.
192
108. Continued
and
discussion
194
CHAPTER
XIJJ.
TRANSIENT
TERM
OF
THE ROTATING

FIELD.
197
109.
Equation
of the resultant
of a
sytem
of
polyphase
m.m.i's.,
in
any
direction,
its
permanent
and
its transient
term.
Maximum value
of
permanent
term.
Nu-
merical
example.
.
197
110. Direction
of maximum
intensity

of
transient
term.
Velocity
of
its rotation.
Oscillating
character of
it.
Intensity
of
maximum value.
Numerical
example.
200
111.
Discussion.
Independence
of
transient
term
on
phase
angle
at start.
203
CHAPTER
XIV.
SHORT-CIRCUIT
CURRENTS

OF ALTERNATORS.
205
112.
Relation
of
permanent
short-circuit
current to armature
reaction and
self-inductance.
Value of
permanent
short-circuit
current.
205
ÆTHERFORCE
CONTENTS
xix
PAGE
113.
Relation
of
momentary
short-circuit current
to
arma-
ture reaction
and
self-inductance. Value
of

momen-
tary
short-circuit
current.
200
114.
Transient
term
of
revolving
field
of armature
reaction.
Pulsating
armature reaction
of
-single-phase
alternator.
207
115.
Polyphase
alternator. Calculation
of
field current
during
short-circuit.
Equivalent
reactance of armature
reac-
tion.

Self-inductance
in field
circuit. 210
116.
Equations
of armature short-circuit
current
and short-
circuit armature reaction.
213
117. Numerical
example.
214
118.
Single-phase
alternator. Calculation of
pulsating
field
current at short-circuit.
215
119.
Equations
of armature
short-circuit current
and
short-
circuit armature reaction.
216
120.
Numerical

example.
218
121.
Discussion.
Transient
reactance.
218
SECTION II.
PERIODIC
TRANSIENTS.
CHAPTER
I. INTRODUCTION.
223
1.
General character
of
periodically
recurring
transient
phenomena
in
time,
223
2.
Periodic transient
phenomena
with
single
cycle.
224

3.
Multi-cycle
periodic
transient
phenomena.
224
4.
Industrial
importance
of
periodic
transient
phenomena:
circuit
control,
high frequency
generation,
rectification.
226
5.
Types
of rectifiers.
Arc
machines.
227
CHAPTER
II.
CIRCUIT CONTROL
BY
PERIODIC

TRANSIENT
PHENOM-
ENA.
229
6.
Tirrill
Regulator. 229
7.
Equations.
230
8.
Amplitude
of
pulsation.
232
CHAPTER
III. MECHANICAL
RECTIFICATION.
235
9.
Phenomena
during
reversal,
and
types
of mechanical
rec-
tifiers.
235
10.

Single-phase
constant-current
rectification:
compounding
of alternators
by
rectification.
237
11.
Example
and numerical calculations.
239
12.
Single-phase constant-potential
rectification:
equations.
242
ÆTHERFORCE
XX
CONTENTS
PAGE
13.
Special
case,
calculation
of
numerical
example.
245
14.

Quarter-phase
rectification.
:
Brush
arc
machine.
Equations.
248
15. Calculation
of
example.
252
CHAPTER
IV.
ARC
RECTIFICATION.
255
16. The
rectifying
character
of
the
arc.
255
17.
Mercury
arc rectifier.
Constant-potential
and
constant-

current
type.
25(3
18.
Mode
of
operation
of
mercury
arc
rectifier:
Angle
of
over-lap.
258
19.
Constant-current
rectifier:
Arrangement
of
apparatus.
261
20.
Theory
and
calculation: Differential
equations.
262
21.
Integral

equations.
264
22.
Terminal conditions and
final
equations.
266
23.
Calculation of
numerical
example.
268
24.
Performance curves and
oscillograms.
Transient
term. 269
25.
Equivalent
sine waves:
their
derivation.
273
26.
25 Continued.
275
27.
Equations
of the
equivalent

sine
waves
of
the
mercury
arc
rectifier. Numerical
example.
277
SECTION
^5)
TRANSIENTS IN SPACE.
CHAPTER
I.
INTRODUCTION.
283
1.
Transient
phenomena
in
space,
as
periodic
functions of
time
and transient functions of
distance,
represented
by
transient

functions of
complex
variables.
283
2.
Industrial
importance
of
transient
phenomena
in.
space.
284
CHAPTER
II. LONG DISTANCE TRANSMISSION
LINE.
285
3. Relation
of
wave
length
of
impressed
frequency
to
natural
frequency
of
line,
and limits

of
approximate
line
cal-
culations.
285
4.
Electrical and
magnetic phenomena
in
transmission line.
287
5.
The
four
constants
of the
transmission
line
:
r,
L,
g,
C.
288
6. The
problem
of the
transmission
line.

289
7. The
differential
equations
of the
transmission
line,
and
their
integral
equations.
289
8. Different
forms of the transmission
line
equations.
293
9.
Equations
with, current
and
voltage given
at one
end
of
the
line.
295
10,
Equations

with
generator
voltage,
and
load
on
receiving
circuit
given,
297
ÆTHERFORCE
CONTENTS xxi
PAQP
11.
Example
of
60,000-volt
200-mile line.
298'
12.
Comparison
of
result
with
different
approximate
calcula-
tions.
300
13.

Wave
length
and
phase
angle.
301
14.
Zero
phase
angle
and
45-degree
phase
angle.
Cable
of
negligible
inductance.
302
15.
Examples
of
non-inductive,
lagging
and
leading
load,
and
discussion
of flow of

energy. 303
16.
Special
case
:
Open
circuit at
end of line.
305
17.
Special
case:
Line
grounded
at end.
310
18.
Special
case :
Infinitely
long
conductor.
311
19.
Special
case:
Generator
feeding
into
closed circuit.

312
20.
Special
case: Line
of
quarter-wave
length,
of
negligible
resistance.
312
21. Line of
quarter-wave
length,
containing
resistance
r
and
conductance
g.
31,5
22.
Constant-potential
constant-current transformation
by
line of
quarter-wave
length.
316
23.

Example
of
excessive
voltage
produced
in
high-potential
transformer coil
as
quarter-
wave
circuit.
31g
24.
Effect
of
quarter-wave
phenomena
on
regulation
of
long
transmission
lines;
quarter-wave
transmission.
319
25.
Limitations
of

quarter-wave
transmission.
320
26.
Example
of
quarter-wave
transmission
of
60,000
kw. at 60
cycles,
over
700
miles.
321
CHAPTEE III. THE NATURAL PERIOD OF
THE
TRANSMISSION
LINE.
326
27.
The oscillation of the transmission
line
as
condenser.
326
28. The conditions
of
free

oscillation.
327
29.
Circuit
open
at one
end, grounded
at
other end.
328
30.
Quarter-wave
oscillation of transmission
line.
330
31.
Frequencies
of
line
discharges,
and
complex
discharge
wave.
333
32.
Example
of
discharge
of

line of
constant
voltage
and zero
current.
335
33.
Example
of
short-circuit
oscillation of
line.
337
34. Circuit
grounded
at both ends :
Half-wave
oscillation.
339
35. The even harmonics of the half-wave
oscillation.
340
36. Circuit
open
at
both
ends.
341
37. Circuit closed
upon

itself:
Full-wave
oscillation.
342
38.
Wave
shape
and
frequency
of
oscillation.
344
39.
Time
decrement of
oscillation,
and
energy
transfer
be-
tween
sections
of
complex
oscillating
circuit.
345
ÆTHERFORCE
xxii
CONTENTS

PAGE
CHAPTER
IV.
DISTRIBUTED
CAPACITY
OF
HIGH-POTENTIAL TRANS-
FORMER.
348
40.
The transformer
coil as
circuit
of
distributed
capacity,
and
the character
of
its
capacity.
348
41. The
differential
equations
of
the transformer
coil,
and
their

integral
equations)
terminal
conditions
and final
approximate equations.
350
42.
Low attenuation
constant
and'
corresponding
liability
of
cumulative
oscillations.
353
CHAPTER
V. DISTRIBUTED
SERIES CAPACITY.
354
43. Potential distribution
in
multigap
circuit.
354
44. Probable
relation
of the
multigap

circuit to the
lightning
flash
in
the
clouds.
356
45. The differential
equations
of the
multigap circuit,
and
their
integral equations.
356
46.
Terminal
conditions,
and final
equations.
358
47.
Numerical
example.
359
CHAPTER VI.
ALTERNATING
MAGNETIC
FLUX
DISTRIBUTION.

361
48.
Magnetic
screening by secondary
currents in
alternating
fields.
361
49. The
differential
equations
of
alternating magnetic
flux
in
a lamina.
362
50. Their
integral
equations.
363
51. Terminal
conditions,
and the
final
equations.
364
52.
Equations
for

very
thick
laminae.
365
53. Wave
length,
attenuation,
depth
of
penetration.
366
54.
Numerical
example,
with
frequencies
of
60,
1000
and
10,000
cycles
per
second.
368
55.
Depth
of
penetration
of

alternating
magnetic
flux in
different
metals.
369
56.
Wave
length,
attenuation,
and
velocity
of
penetration.
371
57.
Apparent
permeability,
as function
of
frequency,
and
damping.
372
58.
Numerical
example
and
discussion.
373

CHAPTER
VII.
DISTRIBUTION
OF
ALTERNATING-CURRENT
DENSITY
IN
CONDUCTOR.
375
59.
Cause
and
effect
of
unequal
current
distribution. In-
dustrial
importance.
375
60.
Subdivision
and
stranding.
Flat
conductor
and
large
conductor.
377

ÆTHERFORCE
CONTENTS
xxiii
PACK
61.
The
differential
equations
of
alternating-current
distri-
bution
in a flat conductor.
380
62.
Their
integral
equations.
381
63.
Mean
value
of
current,
and effective resistance.
382
64. Effective
resistance
and
resistance ratio. 383

65.
Equations
for
large
conductors.
384
66.
Effective resistance
and
depth
of
penetration.
386
67.
Depth
of
penetration,
or
conducting
layer,
for different
materials
and
different
frequencies,
and
maximum
economical
conductor
diameter.

391
CHAPTER VIII. VELOCITY OF
PROPAGATION
OF ELECTRIC FIELD. 394
68. Conditions
when the finite
velocity
of
the
electric
field
is of
industrial
importance.
394
69.
Lag
of
magnetic
and
dielectric field
leading
to
energy
com-
ponents
of
inductance
voltage
and

capacity
current and
thereby
to
effective
resistances. 395
70. Conditions under which this effect
of
the
finite
velocity
is
considerable
and therefore of
importance.
396
A
.
Inductance
of
a
Length
lo
of
an
Infinitely Long
Conductor
without
Return Conductor,
71.

Magnetic flux,
radiation
impedance,
reactance and
resistance.
72.
The
sil and col
functions.
73.
Mutually
inductive
impedance
and
mutual
inductance.
Self-inductive
radiation
impedance,
resistance
and
react-
ance.
Self-inductance and
power.
402
B. Inductance
of
a
Length

la
of
an
Infinitely
Long
Conductor
with
Return Conductor
at
Distance
I'.
74. Self-inductive
radiation
impedance,
resistance and
self-
inductance.
404
75.
Discussion.
Effect of
frequency
and
of
distance
of
return
conductor.
405
76. Instance.

Quarter-wave
and half-wave distance
of
return
conductor.
407
ÆTHERFORCE
xxiv
CONTENTS
PAGE
C.
Capacity
of
a
Length
lo
of
an
Infinitely
Long
Conductor.
77.
Calculation
of
dielectric field.
Effective
capacity.
40S
78. Dielectric radiation
impedance.

Relation to
magnetic
radiation
impedance.
410
79. Conductor without return conductor
and with
return con-
ductor. Dielectric radiation
impedance,
effective re-
sistance,
reactance
and
capacity.
Attenuation constant.
411
D. Mutual Inductance
of
Two Conductors
of
Finite
Length
at
Con-
siderable
Distance
from
Each
Other.

80.
Change
of
magnetic
field
with
distance
of finite
and infinite
conductor,
with
and without return
conductor.
414
81.
Magnetic
flux
of conductor of finite
length,
sill
and
coll
functions. 415
82. Mutual
impedance
and
mutual
inductance. Instance.
410
E.

Capacity
of
a
Sphere
in
Space.
83. Derivation
of
equations.
418
CHAPTEB
IX.
HIGH-FREQUENCY
CONDUCTORS. 420
84.
Effect of
the
frequency
on
the constants of the
conductor.
420
85.
Types
of
high-frequency
conduction in
transmission
lines.
421

86.
Equations
of
unequal
current
distribution
in
conductor. 423
87.
Equations
of
radiation
resistance and
reactance. 425
88.
High-frequency
constants of
conductor with and without
return
conductor.
427
89.
Instance.
428
90.
Discussion of effective
resistance
and
frequency.
430

91.
Discussion
of
reactance and
frequency.
433
92.
Discussion
of
size,
shape
and
material
of
conductor,
and
frequency.
434
93.
Discussion of
size, shape
and
material
on circuit
constants. 435
94.
Instances, equations
and
tables.
430

95.
Discussion
of
tables.
437
96.
Continued.
442
97.
Conductor
without
return
conductor.
444
ÆTHERFORCE
CONTENTS xxv
SECTION
IV.
TRANSIENTS IN TIME
AND
SPACE.
PAGE
CHAPTER I.
GENERAL
EQUATIONS.
449
1.
The
constants of the
electric

circuit,
and
their
constancy.
449
2.
The
differential
equations
of
the
general
circuit,
and
their
general
integral
equations.
451
3.
Terminal conditions.
Velocity
of
propagation.
454
4.
The
group
of terms in
the

general
integral
equations
and
the relations between its
constants.
455
5.
Elimination
of the
complex exponent
in the
group
equa-
tions.
458
6. Final form
of
the
general
equations
of
the electric
circuit.
461
CHAPTER II.
DISCUSSION
OF SPECIAL
CASES. 464
7.

Surge
impedance
or natural
impedance.
Constants
A,
a,
1>
and I.
464
8.
l>
=
0:
permanents.
Direct-current circuit
with
distributed
leakage.
465
9.
Leaky
conductor
of infinite
length. Open
conductor.
Closed
conductor.
405
10.

Leaky
conductor closed
by
resistance.
Reflection
of
voltage
and
current.
467
11. a 0:
(a)
Inductive
discharge
of
closed
circuit,
(b)
Non-
inductive condenser
discharge.
469
12. Z
=
0:
general
equations
of circuit with massed constants.
470
13. I

=
Q,
6=0: direct
currents.
1=0,
b
=
real:
impulse
currents.
471
14.
Continued :
direct-current circuit with
starting
transient.
472
15.
I
=
0,
6
=
imaginary: alternating
currents.
473
16. I
=
0,
&

=
general: oscillating
currents.
474
17.
&
=
real:
impulse
currents. Two
types
of
impulse
currents. 475
18.
b
=
real,
a
=
real;
non-periodic
impulse
currents.
476
19.
&
=
real,
a

=
imaginary: impulse
currents
periodic
in
space.
477
20.
6
=
imaginary:
alternating
currents. General
equations.
478
21.
Continued. Reduction
to
general
symbolic expression.
479
CHAPTER
III.
IMPULSE
CURRENTS.
481
22.
Their
relation
to

the
alternating
currents
as coordinate
special
cases of
the
general
equation.
481
23. Periodic and
non-periodic
impulses.
483
ÆTHERFORCE
xxvi
CONTENTS
PAGE
A.
Non-periodic
Impulses.
24.
Equations.
484
25.
Simplification
of
equations; hyperbolic
form.
485

26.
The two
component
impulses.
Time
displacement,
lead
and
lag;
distortionless circuit.
486
27.
Special
case.
4S7
28.
Energy
transfer
constant, energy dissipation
constant,
wave front
constant.
487
29. Different form
of
equation
of
impulse.
488
30. Resolution

into
product
of time
impulse
and
space
impulse.
Hyperbolic
form.
489
31.
Third form of
equation
of
impulse.
Hyperbolic
form.
490
B,
Periodic
Impulses.
32.
Equations.
491
33.
Simplification
of
equations;
trigonometric
form.

492
34. The two
component
impulses.
Energy
dissipation
constant,
enery
transfer
constant,
attentuation
constants.
Phase
difference.
Time
displacement.
493
35.
Phase relations in
space
and
time.
Special
cases.
495
36.
Integration
constants,
Fourier series.
495

CHAPTEH IV.
DISCUSSION OF
GENERAL
EQUATIONS. 497
37.
The
two
component
waves
and
their
reflected
waves.
Attenuation in time and
in
space.
*
497
38.
Period,
wave
length,
time
and
distance
attenuation
constants.
499
39.
Simplification

of
equations
at
high
frequency,
and
the
velocity
unit of
distance.
500
40.
Decrement of
traveling
wave.
502
41.
Physical
meaning
of
the
two
component
waves.
503
42.
Stationary
or
standing
wave.

Trigonometric
and
logarith-
mic
waves.
504
43.
Propagation
constant
of wave.
506
CHAPTER
V.
STANDING WAVES.
509
44.
Oscillatory,
critical
and
gradual
standing
wave.
509
45. The
wave
length
which
divides the
gradual
from the

oscillatory
wave.
513
ÆTHERFORCE