GIFT OF
ASSOCIATED ELECTRICAL AND MECHANICAL ENGINEERS
MECHANICS DEPARTMENT

ELECTRIC DISCHARGES, WAVES AND IMPULSES

Published by the
Me Grow -Hill Book. Company Ne^vYork.

Successors to tKeBookDepartments of tKe

McGraw Publishing Company

Hill Publishing- Company

Publishers of Books for

Elec trical World
Engineering Record

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Electric Railway Journal Metallurgical and Chemical Engineering*

Coal Age
Power

ELEMENTARY LECTURES
ON
ELECTRIC DISCHARGES, WAVES AND IMPULSES,
AND
OTHER TRANSIENTS
BY
CHARLES PROTEUS STEINMETZ, A.M., PH.D.
\\
Past President, American Institute of Electrical Engineers
McGRAW-HILL BOOK COMPANY
239 WEST 39TH STREET, NEW YOKK
6 BOUVERIE STREET, LONDON, E.G. 1911

rH .
Library
y
COPYRIGHT, 1911, BY THE
McGRAW-HILL BOOK COMPANY
Stanbopc Hfress
F. H. GILSON COMPANY
BOSTON, U.S.A.

PREFACE.

IN the following I am trying to give a short outline of those

phenomena which have become the most important to the elec-

trical engineer, as on their understanding and control depends the

further successful advance of electrical engineering. The art has

now so far advanced that the phenomena of the steady flow of

power are well understood. Generators, motors, transforming

devices, transmission and distribution conductors can, with rela-

tively little difficulty, be.Calculated, and the phenomena occurring

in them under normal (faa^tftmS'bf operation predetermined and

controlled. Usually, however, the limitations of apparatus and

lines are found not in the normal condition of operation, the steady

flow of power, but in the phenomena occurring under abnormal

though by no means unfrequent conditions, in the more or less

transient abnormal voltages, currents, frequencies, etc.; and the

study of the laws of these transient phenomen^4fee electric dis-

charges, waves, and impulses, thus becomes of paramount importance. In a former work, " Theory and Calculation of Transient

Electric Phenomena and Oscillations," I have given a systematic

study of these phenomena, as far as our present knowledge per-

mits, which by necessity involves to a considerable extent the use

of mathematics. As many engineers may not have the time or

inclination to a mathematical study, I have endeavored to give in

the following a descriptive exposition of the physical nature and

meaning, the origin and effects, of these phenomena, with the use

of very little and only the simplest form of mathematics, so as to

afford a general knowledge of these phenomena to those engineers

who have not the time to devote to a more extensive study,

and

also

to

serve

as

an

introduction

to

the

study

of

"
Transient

Phenomena." I have, therefore, in the following developed these

phenomena from the physical conception of energy, its storage and

readjustment, and extensively used as illustrations oscillograms of

such electric discharges, waves, and impulses, taken on industrial

electric circuits of all kinds, as to give the reader a familiarity

749213

vi

PREFACE.

with transient phenomena by the inspection of their record on the photographic film of the oscillograph. I would therefore recommend the reading of the following pages as an introduction to the study of " Transient Phenomena," as the knowledge gained
thereby of the physical nature materially assists in the understanding of their mathematical representation, which latter obviously is necessary for their numerical calculation and pre-
determination.

The book contains a series of lectures on electric discharges, waves, and impulses, which was given during the last winter to the graduate classes of Union University as an elementary introduction to and " translation from mathematics into English" of the " Theory and Calculation of Transient Electric Phenomena and Oscillations." Hereto has been added a chapter on the calculation of capacities and inductances of conductors, since capacity and inductance are the fundamental quantities on which the transients

depend.

In the preparation of the work, I have been materially assisted
by Mr. C. M. Davis, M.E.E., who kindly corrected and edited the manuscript and illustrations, and to whom I wish to express

my thanks.

CHARLES PROTEUS STEINMETZ.

October, 1911.

CONTENTS.

LECTURE I. NATURE AND ORIGIN OF TRANSIENTS

PAGE 1

1 . Electric power and energy. Permanent and transient phenomena. Instance of permanent phenomenon; of transient; of combination of both. Transient as intermediary condition between permanents. 2. Energy storage in electric circuit, by magnetic and dielectric field. Other energy storage. Change of stored energy as origin of tran-
sient.
3. Transients existing with all forms of energy: transients of rail-
way car; of fan motor; of incandescent lamp. Destructive values. High-speed water-power governing. Fundamental condition of
transient. Electric transients simpler, their theory further advanced, of more direct industrial importance.
4. Simplest transients: proportionality of cause and effect. Most
electrical transients of this character. Discussion of simple transient of electric circuit. Exponential function as its expression. Coefficient of its exponent. Other transients: deceleration of ship.
5. Two classes of transients: single-energy and double-energy
transients. Instance of car acceleration; of low-voltage circuit; of pendulum; of condenser discharge through inductive circuit.
Transients of more than two forms of energy. 6. Permanent phenomena usually simpler than transients. Reduction of alternating-current phenomena to permanents by effective values and by symbolic method. Nonperiodic transients.

LECTURE II. THE ELECTRIC FIELD

10

7. Phenomena of electric power flow: power dissipation in con-
ductor; electric field consisting of magnetic field surrounding conductor and electrostatic or dielectric field issuing from conductor. Lines of magnetic force; lines of dielectric force. 8. The magnetic flux, inductance, inductance voltage, and the energy of the magnetic field.
9. The dielectric flux, capacity, capacity current, and the energy of the dielectric field. The conception of quantity of electricity, electrostatic charge and condenser; the conception of quantity of
magnetism. 10. Magnetic circuit and dielectric circuit. Magnetomotive force, magnetizing force, magnetic field intensity, and magnetic density. Permeability. Magnetic materials.
vii

Vlll

CONTENTS.

11. Electromotive force, electrifying force or voltage gradient. Dielectric field intensity and dielectric density. Specific capacity or permittivity. Velocity of propagation. 12. Tabulation of corresponding terms of magnetic and of dielectric field. Tabulation of analogous terms of magnetic, dielectric, and electric circuit.

PAGE

LECTURE III. SINGLE-ENERGY TRANSIENTS IN CONTINUOUS-CUR-

RENT CIRCUITS

19

13. Single-energy transient represents increase or decrease of
energy. Magnetic transients of low- and medium-voltage circuits. Single-energy and double-energy transients of capacity. Discussion of the transients of 4>, i, e, of inductive circuit. Exponential equation. Duration of the transient, time constant. Numerical values of transient of intensity 1 and duration 1. The three forms of the equation of the magnetic transient. Simplification by choosing the starting moment as zero of time. 14. Instance of the magnetic transient of a motor field. Calcula-
tion of its duration.
15. Effect of the insertion of resistance on voltage and duration of the magnetic transient. The opening of inductive circuit. The effect of the opening arc at the switch. 16. The magnetic transient of closing an inductive circuit. General method of separation of transient and of permanent terms during the transition period.

LECTURE IV. SINGLE-ENERGY TRANSIENTS OF ALTERNATING-CUR-

RENT CIRCUITS

30

17. Separation of current into permanent and transient component.
Condition of maximum and of zero transient. The starting of an
alternating current; dependence of the transient on the phase; maxi-
mum and zero value.
18. The starting transient of the balanced three-phase system. Relation between the transients of the three phases. Starting transient of three-phase magnetic field, and its construction. The
oscillatory start of the rotating field. Its independence of the phase
at the moment of start. Maximum value of rotating-field tran-
sient, and its industrial bearing.
19. Momentary short-circuit current of synchronous alternator, and current rush in its field circuit. Relation between voltage,
load, magnetic field flux, armature reaction, self-inductive reactance, and synchronous reactance of alternator. Ratio of momentary to
permanent short-cicurit current. 20. The magnetic field transient at short circuit of alternator. Its effect on the armature currents, and on the field current. Numerical relation between the transients of magnetic flux, armature currents, armature reaction, and field current. The starting transient of the armature currents. The transient full-frequency pulsation of the

CONTENTS.

ix

field current caused by it. Effect of inductance in the exciter field. Calculation and construction of the transient phenomena of a polyphase alternator short circuit. 21. The transients of the single-phase alternator short circuit. The permanent double- frequency pulsation of armature reaction and of field current. The armature transient depending on the phase of the wave. Combination of full-frequency transient and double-frequency permanent pulsation of field current, and the shape of the field current resulting therefrom. Potential difference at field terminal at short circuit, and its industrial bearing.

PAGE

LECTURE V. SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. ... 52
22. Absence of proportionality between current and magnetic flux in ironclad circuit. Numerical calculation by step-by-step method. Approximation of magnetic characteristic by Frohlich's formula, and its rationality. 23. General expression of magnetic flux in ironclad circuit. Its introduction in the differential equation of the transient. Integration, and calculation of a numerical instance. High-current values and steepness of ironclad magnetic transient, and its industrial
bearing.

LECTURE VI. DOUBLE-ENERGY TRANSIENTS

59

24. Single-energy transient, after separation from permanent term, as a steady decrease of energy. Double-energy transient consisting of energy-dissipation factor and energy-transfer factor. The latter periodic or unidirectional. The latter rarely of industrial importance. 25. Pulsation of energy during transient. Relation between maxi-
mum current and maximum voltage. The natural impedance and
the natural admittance of the circuit. Calculation of maximum voltage from maximum current, and inversely. Instances of line
short circuit, ground on cable, lightning stroke. Relative values of transient currents and voltages in different classes of circuits.
26. Trigonometric functions of the periodic factor of the transient. Calculation of the frequency. Initial values of current and voltage.
27. The power-dissipation factor of the transient. Duration of the double-energy transient the harmonic mean of the duration of the magnetic and of the dielectric transient. The dissipation exponent, and its usual approximation. The complete equation of the
double-energy transient. Calculation of numerical instance.

LECTURE VII. LINE OSCILLATIONS

72

28. Review of the characteristics of the double-energy transient: periodic and transient factor; relation between current and voltage; the periodic component and the frequency; the transient component and the duration; the initial values of current and voltage.

X

CONTENTS.

PAGE
Modification for distributed capacity and inductance: the distance phase angle and the velocity of propagation; the time phase angle; the two forms of the equation of the line oscillation. 29. Effective inductance and effective capacity, and the frequency of the line oscillation. The wave length. The oscillating-line section as quarter wave length. 30. Relation between inductance, capacity, and frequency of prop-
agation. Importance of this relation for calculation of line con-
stants.
31. The different frequencies and wave lengths of the quarterwave oscillation; of the half-wave oscillation. 32. The velocity unit of length. Its importance in compound circuits. Period, frequency, time, and distance angles, and the
general expression of the line oscillation.

LECTURE VIII. TRAVELING WAVES

88

33. The power of the stationary oscillation and its correspondence with reactive power of alternating currents. The traveling wave and its correspondence with effective power of alternating currents. Occurrence of traveling waves: the lightning stroke: The traveling wave of the compound circuit. 34. The flow of transient power and its equation. The powerdissipation constant and the power-transfer constant. Increasing and decreasing power flow in the traveling wave. The general
equation of the traveling wave.
35. Positive and negative power-transfer constants. Undamped oscillation and cumulative oscillation. The arc as their source.
The alternating-current transmission-line equation as special case of traveling wave of negative power-transfer constant. 36. Coexistence and combination of traveling waves and stationary oscillations. Difference from effective and reactive alternating
waves. Industrial importance of traveling waves. Their frequencies. Estimation of their effective frequency if very high.
37. The impulse as traveling wave. Its equations. The wave
front.

LECTURE IX. OSCILLATIONS OF THE COMPOUND CIRCUIT

108

38. The stationary oscillation of the compound circuit. The time decrement of the total circuit, and the power-dissipation and power-transfer constants of its section. Power supply from section
of low-energy dissipation to section of high-energy dissipation.
39. Instance of oscillation of a closed compound circuit. The two traveling waves and the resultant transient-power diagram. 40. Comparison of the transient-power diagram with the power diagram of an alternating- current circuit. The cause of power increase in the line. The stationary oscillation of an open com-
pound circuit.

CONTENTS.

xi

41. Voltage and current relation between the sections of a compound oscillating circuit. The voltage and current transformation at the transition points between circuit sections. 42. Change of phase angle at the transition points between sections of a compound oscillating circuit. Partial reflection at the
transition point.

PAGE

LECTURE X. INDUCTANCE AND CAPACITY OF ROUND PARALLEL CON-

DUCTORS

119

43. Definition of inductance and of capacity. The magnetic and the dielectric field. The law of superposition of fields, and its use
for calculation.
44. Calculation of inductance of two parallel round conductors. External magnetic flux and internal magnetic flux. 45. Calculation and discussion of the inductance of two parallel conductors at small distances from each other. Approximations and their practical limitations. 46. Calculation of capacity of parallel conductors by superposition of dielectric fields. Reduction to electromagnetic units by the velocity of light. Relation between inductance, capacity, and
velocity of propagation. 47. Conductor with ground return, inductance, and capacity.
The image conductor. Limitations of its application. Correction
for penetration of return current in ground.
48. Mutual inductance between circuits. Calculation of equation, and approximation. 49. Mutual capacity between circuits. Symmetrical circuits and asymmetrical circuits. Grounded circuit. 50. The three-phase circuit. Inductance and capacity of twowire single-phase circuit, of single-wire circuit with ground return, and of three-wire three-phase circuit. Asymmetrical arrangement of three-phase circuit. Mutual inductance and mutual capacity with three-phase circuit.

ELEMENTAEY LECTURES ON ELECTEIC DISCHARGES, WAVES AND IMPULSES, AND OTHER TRANSIENTS.
LECTURE I.
NATURE AND ORIGIN OF TRANSIENTS.
i. Electrical engineering deals with electric energy and its
flow, that is, electric power. Two classes of phenomena are met: permanent and transient, phenomena. To illustrate: Let G in A Fig. 1 be a direct-current generator, which over a circuit con-
nects to a load L, as a number of lamps, etc. In the generator G, the line A, and the load L, a current i flows, and voltages e

Fig. 1.

exist, which are constant, or permanent, as long as the conditions
of the circuit remain the same. If we connect in some more

lights, or disconnect some of the load, we get a different current

i',

and

possibly

different

voltages

1
e ';

but

again

i'

and

e' are

per-

manent, that is, remain the same as long as the circuit remains

unchanged.
Let, however, in Fig. 2, a direct-current generator G be connected to an electrostatic condenser C. Before the switch S is closed, and therefore also in the moment of closing the switch, no current flows in the line A. Immediately after the switch S is closed, current A begins to flow over line into the condenser C, charging this condenser up to the voltage given by the generator. When the

1

DISCHARGES, WAVES AND IMPULSES.
condenser C is charged, the current in the line A and the condenser C is zero again. That is, the permanent condition before closing
the switch S, and also some time after the closing of the switch, is zero current in the line. Immediately after the closing of the switch, however, current flows for a more or less short time. With the condition of the circuit unchanged: the same generator
voltage, the switch S closed on the same circuit, the current nevertheless changes, increasing from zero, at the moment of closing the switch S, to a maximum, and then decreasing again to
zero, while the condenser charges from zero voltage to the genera-
tor voltage. We then here meet a transient phenomenon, in the
charge of the condenser from a source of continuous voltage.
Commonly, transient and permanent phenomena are superimposed upon each other. For instance, if in the circuit Fig. 1 we close the switch S connecting a fan motor F, at the moment of closing the switch S the current in the fan-motor circuit is zero. It rapidly rises to a maximum, the motor starts, its speed increases while the current decreases, until finally speed and current become constant; that is, the permanent condition is reached.
The transient, therefore, appears as intermediate between two permanent conditions: in the above instance, the fan motor disconnected, and the fan motor running at full speed. The question then arises, why the effect of a change in the conditions of an
electric circuit does not appear instantaneously, but only after a transition period, requiring a finite, though frequently very short,
time.
2. Consider the simplest case: an electric power transmission
(Fig. 3). In the generator G electric power is produced from me-
A A chanical power, and supplied to the line . In the line some of
this power is dissipated, the rest transmitted into the load L, where the power is used. The consideration of the electric power

NATURE AND ORIGIN OF TRANSIENTS.

3

in generator, line, and load does not represent the entire phenome-
A non. While electric power flows over the line , there is a magnetic

field surrounding the line conductors, and an electrostatic field

issuing from the line conductors. The magnetic field and the
electrostatic or "dielectric " field represent stored energy. Thus,

during the permanent conditions of the flow of power through the

circuit Fig. 3, there is electric energy stored in the space surround-

ing the line conductors. There is energy stored also in the genera-

tor

and

in

the

load ;

for

instance,

the

mechanical

momentum

of

the

revolving fan in Fig. 1, and the heat energy of the incandescent

lamp filaments. The permanent condition of the circuit Fig. 3

thus represents not only flow of power, but also storage of energy.
When the switch S is open, and no power flows, no energy is

stored in the system. If we now close the switch, before the

permanent condition corresponding to the closed switch can occur,

Fig. 3.
the stored energy has to be supplied from the source of power; that is, for a short time power, in supplying the stored energy, flows not only through the circuit, but also from the circuit into the space surrounding the conductors, etc. This flow of power, which supplies the energy stored in the permanent condition of the circuit, must cease as soon as the stored energy has been supplied, and thus is a transient.
Inversely, if we disconnect some of the load L in Fig. 3, and
thereby reduce the flow of power, a smaller amount of stored energy would correspond to that lesser flow, and before the conditions of the circuit can become stationary, or permanent (corresponding to the lessened flow of power), some of the stored energy has to be returned to the circuit, or dissipated, by a
transient.
Thus the transient is the result of the change of the amount of stored energy, required by the change of circuit conditions, and

4 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
is the phenomenon by which the circuit readjusts itself to the
change of stored energy. It may thus be said that the perma-
nent phenomena are the phenomena of electric power, the transients the phenomena of electric energy.
3. It is obvious, then, that transients are not specifically electrical phenomena, but occur with all forms of energy, under all conditions where energy storage takes place.
Thus, when we start the motors propelling an electric car, a
transient period, of acceleration, appears between the previous permanent condition of standstill and the final permanent con-
dition of constant-speed running; when we shut off the motors, the permanent condition of standstill is not reached instantly,
but a transient condition of deceleration intervenes. When we
open the water gates leading to an empty canal, a transient condition~"of flow and water level intervenes while the canal is
filling, until the permanent condition is reached. Thus in the case of the fan motor in instance Fig. 1, a transient period of speed and mechanical energy appeared while the motor was speeding up
and gathering the mechanical energy of its momentum. When
turning on an incandescent lamp, the filament passes a transient
of gradually rising temperature.
Just as electrical transients may, under certain conditions, rise
to destructive values; so transients of other forms of energy may become destructive, or may require serious consideration, as, for
instance, is the case in governing high-head water powers. The column of water in the supply pipe represents a considerable amount of stored mechanical energy, when flowing at velocity, under load. If, then, full load is suddenly thrown off, it is not
possible to suddenly stop the flow of water, since a rapid stopping would lead to a pressure transient of destructive value, that is, burst the pipe. Hence the use of surge tanks, relief valves, or deflecting nozzle governors. Inversely, if a heavy load comes on suddenly, opening the nozzle wide does not immediately take care of the load, but momentarily drops the water pressure at the nozzle, while gradually the water column acquires velocity, that is, stores energy.
The fundamental condition of the appearance of a transient
thus is such a disposition of the stored energy in the system as differs from that required by the existing conditions of the system; and any change of the condition of a system, which requires a

NATURE AND ORIGIN OF TRANSIENTS.

O

change of the stored energy, of whatever form this energy may be,
leads to a transient. Electrical transients have been studied more than transients of
other forms of energy because : (a) Electrical transients generally are simpler in nature, and
therefore yield more easily to a theoretical and experimental
investigation.
(b) The theoretical side of electrical engineering is further advanced than the theoretical side of most other sciences, and

especially :
(c) The destructive or harmful effects of transients in electrical systems are far more common and more serious than with other forms of energy, and the engineers have therefore been driven by

necessity to their careful and extensive study. 4. The simplest form of transient occurs where the effect is
directly proportional to the cause. This is generally the case in

electric circuits, since voltage, current, magnetic flux, etc., are proportional to each other, and the electrical transients therefore are usually of the simplest nature. In those cases, however,

where this direct proportionality does not exist, as for instance in

inductive circuits containing iron, or in electrostatic fields exceed-
ing the corona voltage, the transients also are far more complex, and very little work has been done, and very little is known, on

these more complex electrical transients. Assume that in an electric circuit we have a transient cur-

rent, as represented by curve i in

Fig.

4 ;

that

is, some

change of

circuit condition requires a readjustment of the stored energy,

which occurs by the flow of transient current i. This current starts at the value ii, and gradually dies down to zero. Assume now that the law of proportionality between cause and effect

applies; that is, if the transient current started with a different

value,

izj

it

would

traverse

a

curve

f
i ,

which is

the same as curve

i, except that all values are changed proportionally, by the ratio

^;j that is, i'=iX*-

ii

ii

Starting with current ii, the transient follows the curve i;

starting with z'2 , the transient follows the proportional curve i' .

At some time, t, however, the current i has dropped to the value t'2, with which the curve i' started. At this moment t, the conditions

in the first case, of current i, are the same as the conditions in

6 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

the

second case,

of

current

f
i

,

at

the

moment

t\;

that

is,

from

t

onward, curve i is the same as curve i' from time i\ onward. Since

t!
Fig. 4. Curve of Simple Transient: Decay of Current.

f
i

is

proportional

to

i

from

any

point

t

onward,

curve

i'

is

propor-

tional to the same curve i from t\ onward. Hence, at time t\, it is

diz dii ^. i% dti dti ii

But

since -^ and CLL\

i2

at

t\

are

the

same

as

-r dv

and

i

at

time

t,

it

follows:

di dii i

or, di

where c =

-

= 1
-r;

constant,

and

the

minus

sign

is

chosen,

as

ii at

di . -r is negative. at

As in Fig. 4:

=

~aJi

ii,

_ ~ 1^ dii ~ tan <f> _~ _1_ . '

NATURE AND ORIGIN OF TRANSIENTS.

7

T = that is, c is the reciprocal of the projection

tj* on the zero line

of the tangent at the starting moment of the transient.

Since

di =

,

cdt;

that is, the percentual change of current is constant, or in other words, in the same time, the current always decreases by the same fraction of its value, no matter what this value is.

Integrated, this equation gives:

+ = log i

ct

C,

i = Ae~ ct

,

or >

i~#:':*5

that is, the curve is the exponential.
The exponential curve thus is the expression of the simplest form of transient. This explains its common occurrence in elec-
trical and' other transients. Consider, for instance, the decay of radioactive substances : the radiation, which represents the decay,
= is proportional to the amount of radiating material; it is ~-r- cm,
Cit
which leads to the same exponential function. Not all transients, however, are of this simplest form. For
instance, the deceleration of a ship does not follow the exponential, but at high velocities the decrease of speed is a greater fraction of
the speed than during the same time interval at lower velocities, and the speed-time curves for different initial speeds are not proportional to each other, but are as shown in Fig. 5. The reason
is, that the frictional resistance is not proportional to the speed, but to the square of the speed.
5. Two classes of transients may occur: 1. Energy may be stored in one form only, and the only energy
change which can occur thus is an increase or a decrease of the
stored energy.
2. Energy may be stored in two or more different forms, and the
possible energy changes thus are an increase or decrease of the
total stored energy, or a change of the stored energy from one form to another. Usually both occur simultaneously.
An instance of the first case is the acceleration or deceleration

8 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
of a train, or a ship, etc. : here energy can be stored only as mechanical momentum, and the transient thus consists of an increase of the stored energy, during acceleration, or of a decrease," during

Seconds 10 20 30 40 50 60 70 80 90 100 110 120
Fig. 5. Deceleration of Ship.

deceleration. Thus also in a low-voltage electric circuit of negligible capacity, energy can be stored only in the magnetic field, and the transient represents an increase of the stored magnetic energy,
during increase of current, or a decrease of the magnetic energy, during a decrease of current.
An instance of the second case is the pendulum, Fig. 6 : with the weight at rest in maximum elevation, all the stored energy is
potential energy of gravitation. This energy changes to kinetic mechanical energy until
in the lowest position, a, when
all the potential gravitational energy has been either converted to kinetic mechanical

energy or dissipated. Then,

during the rise of the weight,

that part of the energy which

Fig. 6.

Double-energy Transient of Pendulum.

is not dissipated again changes to potential gravitational energy, at c, then back again to

kinetic energy, at a; and in this manner the total stored energy

is gradually dissipated, by a series of successive oscillations or

changes between potential gravitational and kinetic mechanical

NATURE AND ORIGIN OF TRANSIENTS.

energy. Thus in electric circuits containing energy stored in the magnetic and in the dielectric field, the change of the amount

of stored energy decrease or increase frequently occurs by a series of successive changes from magnetic to dielectric and back

again from dielectric to magnetic stored energy. This for instance is the case in the charge or discharge of a condenser through an

inductive circuit.

If energy can be stored in more than two different forms, still

more complex phenomena may occur, as for instance in the hunt-

ing of synchronous machines at the end of long transmission lines, where energy can be stored as magnetic energy in the line and

apparatus, as dielectric energy in the line, and as mechanical
energy in the momentum of the motor. 6. The study and calculation of the permanent phenomena in

electric circuits are usually far simpler than are the study and

calculation of transient phenomena. However, only the phe-

nomena of a continuous-current circuit are really permanent.

The alternating-current phenomena are transient, as the e.m.f.

continuously and periodically changes, and with it the current, the stored energy, etc. The theory of alternating-current phe-

nomena, as periodic transients, thus has been more difficult than that of continuous-current phenomena, until methods were devised

to treat the periodic transients of the alternating-current circuit

as

permanent

phenomena,

by

the

conception

of

the

" effective

values," and more completely by the introduction of the general

number or complex quantity, which represents the periodic func-

tion of time by a constant numerical value. In this feature lies

the advantage and the power of the symbolic method of dealing

with alternating-current phenomena, the reduction of a periodic

transient to a permanent or constant quantity. For this reason, wherever periodic transients occur, as in rectification, commuta-

tion, etc., a considerable advantage is frequently gained by their

reduction to permanent phenomena, by the introduction of the

symbolic expression of the equivalent sine wave.

Hereby most of the periodic transients have been eliminated

from consideration, and there remain mainly the nonperiodic

transients, as occur at any change of circuit conditions. Since

they are the phenomena of the readjustment of stored energy, a

study of the energy storage of the electric circuit, that is, of its

magnetic and dielectric field, is of first importance.

LECTURE II.

THE ELECTRIC FIELD.

7. Let, in Fig. 7, a generator G transmit electric power over

A line

into a receiving circuit L.

While power flows through

the conductors A, power is con-

sumed in these conductors by

conversion into heat, repre-

sented by i?r. This, however,

Fig. 7.

is not all, but in the space

surrounding the conductor cer-

tain phenomena occur: magnetic and electrostatic forces appear.

Fig. 8. Electric Field of Conductor.
The conductor is surrounded by a magnetic field, or a magnetic flux, which is measured by the number of lines of magnetic force <J>. With a single conductor, the lines of magnetic force are concentric
circles, as shown in Fig. 8. By the return conductor, the circles
10

THE ELECTRIC FIELD.

11

are crowded together between the conductors, and the magnetic field consists of eccentric circles surrounding the conductors, as shown by the drawn lines in Fig. 9.
An electrostatic, or, as more properly called, dielectric field, issues
from the conductors, that is, a dielectric flux passes between the conductors, which is measured by the number of lines of dielectric force ty. With a single conductor, the lines of dielectric force are
radial straight lines, as shown dotted in Fig. 8. By the return
conductor, they are crowded together between the conductors, and form arcs of circles, passing from conductor to return conductor, as shown dotted in Fig. 9.

Fig. 9. Electric Field of Circuit.

The magnetic and the dielectric field of the conductors both are

included in the term electric field, and are the two components of

the electric field of the conductor.

8. The magnetic field or magnetic flux of the circuit, <, is pro-

portional to the current, i, with a proportionality factor, L, which

is called the inductance of the circuit.

= Li.

(1)

The magnetic field represents stored energy w. To produce it,

power, p, must therefore be supplied by the circuit. Since power

is current times voltage,

= p e'i.

(2)

12 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

To

produce

the

magnetic

field

$

of

the

current

i,

a

voltage

f
e

must be consumed in the circuit, which with the current i gives
the power p, which supplies the stored energy w of the magnetic

field <i>.

This

voltage

r
e

is

called

the

inductance

voltage,

or

voltage

consumed by self-induction.

Since no power is required to maintain the field, but power is

required to produce it, the inductance voltage must be propor-

tional to the increase of the magnetic field:

:'

;

(3)

or by (1),

(4)

If i and therefore $ decrease, -r and therefore e' are negative;
that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power p is

w = I p dt,

or by (2) and (4),

w = I Li di;

hence

T w = L*

(^
(5)

is the energy of the magnetic field

$ = Li

of the circuit.

9. Exactly analogous relations exist in the dielectric field.
The dielectric field, or dielectric flux, ty} is proportional to the voltage 6, with a proportionality factor, C, which is called the

capacity of the circuit:

f = Ce.

(6)

The dielectric field represents stored energy, w. To produce it,

power, p, must, therefore, be supplied by the circuit. Since power

is current times voltage,

= p

i'e.

(7)

To

produce

the

dielectric

field

ty

of

the

voltage

e,

a

current

r
i

must be consumed in the circuit, which with the voltage e gives

THE ELECTRIC FIELD.

13

the power p, which supplies the stored energy w of the dielectric
field ^. This current i' is called the capacity current, or, wrongly, charging current or condenser current.
Since no power is required to maintain the field, but power is required to produce it, the capacity current must be proportional
to the increase of the dielectric field:

or by (6),

i' = C^.

(9)

de

^ If e and therefore decrease,

and

therefore

f
i

are

negative;

-j-

that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power p is

or by (7) and (9),
hence

w=j*pdt, w = I Cede;
= rw

(10)
(ID

is the energy of the dielectric field
t = Ce
of the circuit.
As seen, the capacity current is the exact analogy, with regard to the dielectric field, of the inductance voltage with regard to the magnetic field; the representations in the electric circuit, of the energy storage in the field.
The dielectric field of the circuit thus is treated and represented in the same manner, and with the same simplicity and perspicuity, as the magnetic field, by using the same conception of lines of
force.
Unfortunately, to a large extent in dealing with the dielectric fields the prehistoric conception of the electrostatic charge on the
conductor still exists, and by its use destroys the analogy between the two components of the electric field, the magnetic and the

14 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

dielectric, and makes the consideration of dielectric fields un-

necessarily complicated.
There obviously is no more sense in thinking of the capacity current as current which charges the conductor with a quantity of electricity, than there is of speaking of the inductance voltage as charging the conductor with a quantity of magnetism. But
while the latter conception, together with the notion of a quantity of magnetism, etc., has vanished since Faraday's representation
of the magnetic field by the lines of magnetic force, the terminology of electrostatics of many textbooks still speaks of electric charges on the conductor, and the energy stored by them, without considering that the dielectric energy is not on the surface of the conductor, but in the space outside of the conductor, just as the

magnetic energy. 10. All the lines of magnetic force are closed upon themselves,
all the lines of dielectric force terminate at conductors, as seen in Fig. 8, and the magnetic field and the dielectric field thus can be considered as a magnetic circuit and a dielectric circuit.
To produce a magnetic flux <, a magnetomotive force F is required.
Since the magnetic field is due to the current, and is proportional to the current, or, in a coiled circuit, to the current times the num-
ber of turns, magnetomotive force is expressed in current turns or

ampere turns.

F = ni.

(12)

If F is the m.m.f., I the length of the magnetic circuit, energized

by F,

,

/=7

(13)

is called the magnetizing force, and is expressed in ampere turns per cm. (or industrially sometimes in ampere turns per inch).
In empty space, and therefore also, with very close approximation, in all nonmagnetic material, / ampere turns per cm. length
of magnetic circuit produce 3C = 4 TT/ 10" 1 lines of magnetic force
per square cm. section of the magnetic circuit. (Here the factor 10" 1 results from the ampere being 10" 1 of the absolute or cgs.
unit of current.)
(14)
* The factor 4 * is a survival of the original definition of the magnetic field intensity from the conception of the magnetic mass, since unit magnetic mass was defined as that quantity of magnetism which acts on an equal quantity at

THE ELECTRIC FIELD.

15

is called the magnetic-field intensity. It is the magnetic density,

that is,

the

number of lines of magnetic

force per

cm 2 ,

produced

by the magnetizing force of / ampere turns per cm. in empty space.

The magnetic

density,

in lines

of

magnetic

force

per

cm 2 ,

pro-

duced by the field intensity 3C in any material is

&= /z3C,

(15)

where

ju

is

a

constant

of

the

material,

a

"
magnetic

conductivity,"

and is called the permeability. ^ = 1 or very nearly so for most

materials, with the exception of very few, the so-called magnetic

materials: iron, cobalt, nickel, oxygen, and some alloys and oxides

of iron, manganese, and chromium.
A If then is the section of the magnetic circuit, the total magnetic

flux is

$ = A.

(16)

Obviously, if the magnetic field is not uniform, equations (13) and (16) would be correspondingly modified; / in (13) would be the average magnetizing force, while the actual magnetizing force would vary, being higher at the denser, and lower at the less dense, parts of the magnetic circuit:
'-"

In (16), the magnetic flux $ would be derived by integrating the

densities (B over the total section of the magnetic circuit.

ii. Entirely analogous relations exist in the- dielectric circuit.
To produce a dielectric flux ^, an electromotive force e is required, which is measured in volts. The e.m.f. per unit length of the

dielectric circuit then is called the electrifying force or the voltage

gradient, and is

G=
f-

(18)-

unit distance with unit force. The unit field intensity, then, was defined as the field intensity at unit distance from unit magnetic mass, and represented by one line (or rather "tube") of magnetic force. The magnetic flux of unit magnetic mass (or "unit magnet pole") hereby became 4w lines of force, and this introduced the factor 4 TT into many magnetic quantities. An attempt to drop this factor 4 TT has failed, as the magnetic units were already too well
established.
The factor 1Q- 1 also appears undesirable, but when the electrical units were introduced the absolute unit appeared as too large a value of current as practical unit, and one-tenth of it was chosen as unit, and called "ampere."

16 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
This gives the average voltage gradient, while the actual gradient
in an ummiform field, as that between two conductors, varies, being higher at the denser, and lower at the less dense, portion of the field, and is

then is the dielectric-field intensity, and
D = KK

(20)

would be the dielectric density, where K is a constant of the material, the electrostatic or dielectric conductivity, and is called the spe-

cific capacity or permittivity.

For empty space, and thus with close approximation for air and

other gases,

1

K

V~9L

where

X v = 3

10 10

is the velocity of light.

It is customary, however, and convenient, to use the permit-

tivity of empty space as unity:

= K

1.

This changes the unit of

dielectric-field intensity by the factor , and gives: dielectric-field

intensity,

=
T4^T-ryo2J

(21)

dielectric density,

D = KK,

(22)

where K = 1 for empty space, and between 2 and 6 for most solids
and liquids, rarely increasing beyond 6. The dielectric flux then is

^ = AD.

(23)

12. As seen, the dielectric and the magnetic fields are entirely analogous, and the corresponding values are tabulated in the following Table I.
* The factor 4 TT appears here in the denominator as the result of the factor 4*- in the magnetic-field intensity 5C, due to the relations between these
quantities.

THE ELECTRIC FIELD.

17

TABLE I.

Magnetic Field.

18 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
TABLE II.
Magnetic Circuit.

LECTURE III.

SINGLE-ENERGY TRANSIENTS IN CONTINUOUSCURRENT CIRCUITS.

13. The simplest electrical transients are those in circuits in

which energy can be stored in one form only, as in this case the

change

of

stored

energy

can

consist

only

of

an

increase

or

decrease ;

but no surge or oscillation between several forms of energy can

exist. Such circuits are most of the low- and medium-voltage

circuits, 220 volts, 600 volts, and 2200 volts. In them the capac-

ity is small, due to the limited extent of the circuit, resulting from

the low voltage, and at the low voltage the dielectric energy thus

is negligible, that is, the circuit stores appreciable energy only by

the magnetic field.
A circuit of considerable capacity, but negligible inductance, if
of high resistance, would also give one form of energy storage only, in the dielectric field. The usual high-voltage capacity circuit, as

that of an electrostatic machine, while of very small inductance,

also is of very small resistance, and the momentary discharge

currents may be very consider-

able, so that in spite of the very

small inductance, considerable

__

magnetic-energy storage may oc-

cur; that is, the system is one

e o

storing energy in two forms, and

^

oscillations appear, as in the dis- '

~

charge of the Leyden jar.
Let, as represented in Fig. 10,

Fig 10 ._ Magnetie Single.energy
Transient,

a continuous voltage e be im-
pressed upon a wire coil of resistance r and inductance L (but

A negligible capacity).

current iQ = flows through the coil and

a magnetic field $0 10~ 8 = - - interlinks with the coil. Assuming
now that the voltage e is suddenly withdrawn, without changing
19

20 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
the constants of the coil circuit, as for instance by shortcircuiting the terminals of the coil, as indicated at A, with no voltage impressed upon the coil, and thus no power supplied to it, current i and magnetic flux < of the coil must finally be zero. However, since the magnetic flux represents stored energy, it cannot instantly vanish, but the magnetic flux must gradually decrease from its initial value 3>o, by the dissipation of its stored
energy in the resistance of the coil circuit as i~r. Plotting, therefore, the magnetic flux of the coil as function of the time, in Fig.
11 A, the flux is constant and denoted by $ up to the moment of

Fig. 11. Characteristics of Magnetic Single-energy Transient.

time where the short circuit is applied, as indicated by the dotted

line t . From t on the magnetic flux decreases, as shown by curve

<. Since the magnetic flux is proportional to the current, the

latter must follow a curve proportional to <, as shown in Fig. IIB. The impressed voltage is shown in Fig. 1 1C as a dotted line; it is

CQ up to t , and drops to at t . However, since after t a current i flows, an e.m.f. must exist in the circuit, proportional to the

current.

= e

ri.

SINGLE-ENERGY TRANSIENTS.

21

This is the e.m.f. induced by the decrease of magnetic flux <, and
is therefore proportional to the rate of decrease of <, that is, to
d<&
. In the first moment of short circuit, the magnetic flux $ still
-j-
has full value 3> , and the current i thus also full value i Q . Hence, at the first moment of short circuit, the induced e.m.f. e must be
equal to eQ, that is, the magnetic flux $ must begin to decrease at such rate as to induce full voltage e , as shown in Fig. 11C.
The three curves <, i, and e are proportional to each other, and as e is proportional to the rate of change of 3>, < must be proportional to its own rate of change, and thus also i and e. That is, the transients of magnetic flux, current, and voltage follow the
law of proportionality, hence are simple exponential functions, as seen in Lecture I:

(1)

<, i, and e decrease most rapidly at first, and then slower and slower, but can theoretically never become zero, though practically they become negligible in a finite time.
The voltage e is induced by the rate of change of the magnetism, and equals the decrease of the number of lines of magnetic force, divided by the time during which this decrease occurs, multiplied by the number of turns n of the coil. The induced voltage e times the time during which it is induced thus equals n times the decrease of the magnetic flux, and the total induced voltage,
that is, the area of the induced-voltage curve, Fig. 11C, thus
equals n times the total decrease of magnetic flux, that is, equals the initial current i times the inductance L:

w Zet =

= 10- 8 Li Q .

(2)

Whatever, therefore, may be the rate of decrease, or the shape
of the curves of $, i, and e, the total area of the voltage curve must
be the same, and equal to w = Li .

If then the current i would continue to decrease at its initial

rate, as shown dotted in Fig. 115 (as could be caused, for instance,

by a gradual increase of the resistance of the coil circuit), the

induced voltage would retain its initial value e up to the moment

+ = of time t tQ

T, where the current has fallen to zero, as

22 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
shown dotted in Fig. 11C. The area of this new voltage curve would be e T, and since it is the same as that of the curve e, as
seen above, it follows that the area of the voltage curve e is

= ri.r,

and, combining (2) and (3), i cancels, and we get the value of T:

:

V:

.:'

:

T-\-

>';

(4)

That is, the initial decrease of current, and therefore of magnetic flux and of induced voltage, is such that if the decrease continued at the same rate, the current, flux, and voltage would
become zero after the time T =
r
The total induced voltage, that is, voltage times time, and therefore also the total current and magnetic flux during the transient, are such that, when maintained at their initial value,
they would last for the time T = -=
Since the curves of current and voltage theoretically never become zero, to get an estimate of the duration of the transient
we may determine the time in which the transient decreases to
half, or to one-tenth, etc., of its initial value. It is preferable,
however, to estimate the duration of the transient by the time T, which it would last if maintained at its initial value. That is,
the duration of a transient is considered as the time T = -
r
This time T has frequently been called the " time constant "
of the circuit.
The higher the inductance L, the longer the transient lasts, obviously, since the stored energy which the transient dissipates
is proportional to L.
The higher the resistance r, the shorter is the duration of the transient, since in the higher resistance the stored energy is more
rapidly dissipated.
Using the time constant T = - as unit of length for the abscissa,

and the initial value as unit of the ordinates, all exponential transients have the same shape, and can thereby be constructed

SINGLE-ENERGY TRANSIENTS.

by the numerical values given in Table III.

of the exponential
TABLE III.

function,

y=e

Exponential Transient of Initial Value 1 and Duration 1.

= y

e~ x .

e = 2.71828.

X

24 ELECTRIC DISCHARGES, WAVES AND IMPULSES,

(6)
The same equations may be derived directly by the integration
of the differential equation:

where L -=- is the inductance voltage, ri the resistance voltage. and their sum equals zero, as the coil is short-circuited.
Equation (7) transposed gives

hence

=-
logi

i = Ce~~L \

= = = and, as for t

0: i

C to, it is:

i hence ;

14. Usually single-energy transients last an appreciable time, and thereby become of engineering importance only in highly inductive circuits, as motor fields, magnets, etc.
To get an idea on the duration of such magnetic transients,
consider a motor field:
A 4-polar motor has 8 ml. (megalines) of magnetic flux per
pole, produced by 6000 ampere turns m.m.f. per pole, and dissipates normally 500 watts in the field excitation.
That is, if IQ = field-exciting current, n = number of field turns per pole, r = resistance, and L = inductance of the field-exciting

circuit, it is

= i Q 2r

500,

hence 500

SINGLE-ENERGY TRANSIENTS.

25

X The magnetic flux is $ = 8

106 and with 4 n total turns ,

the total number of magnetic interlinkages thus is

4 n$ = 32 n X 106 ,

hence the inductance

LT =

L0~ 8
^o

.32 n ,
henrys.

The field excitation is

hence

ra'o = 6000 ampere turns, n = 6000

hence and

X L, = .-32

6000 ,

r

henrys,

*<r

L

- 1920

3 ' 8OA4 sec '

That is, the stored magnetic energy could maintain full field
excitation for nearly 4 seconds. It is interesting to note that the duration of the field discharge
does not depend on the voltage, current, or size of the machine, but merely on, first, the magnetic flux and m.m.f., which determine the stored magnetic energy, and, second, on the excitation power, which determines the rate of energy dissipation.
15. Assume now that in the moment where the transient be-
gins the resistance of the coil in Fig. 10 is increased, that is, the

I

Fig. 12. Magnetic Single-energy Transient.

'
coil is not short-circuited upon itself, but its circuit closed by a

resistance

1
r '.

Such would, for instance, be the case in Fig. 12,

when opening the switch S.

26 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
The transients of magnetic flux, current, and voltage are shown
as A, B, and C in Fig. 13.
The magnetic flux and therewith the current decrease from the initial values $o and i at the moment to of opening the switch S, on curves which must be steeper than those in Fig. 11, since the
+ current passes through a greater resistance, r r', and thereby
dissipates the stored magnetic energy at a greater rate.

Fig. 13. Characteristics of Magnetic Single-energy Transient.

The impressed voltage eQ is withdrawn at the moment t , and a

voltage thus induced from this moment onward, of such value as
+ to produce the current i through the resistance r r'. In the

first moment, to, the current is still i Q, and the induced voltage

thus must be

= + eo

io (r

r'),

while the impressed voltage, before to, was

= e Q

ior;

hence the induced voltage eo' is greater than the impressed voltage eo, in the same ratio as the resistance of the discharge circuit
+ r r' is greater than the resistance of the coil r through which the
impressed voltage sends the current

e

SINGLE-ENERGY TRANSIENTS.

27

The duration of the transient now is

T

=

L
7+-r "

that is, shorter in the same proportion as the resistance, and

thereby the induced voltage is higher.

= f
If r

oo
,

that

is,

no

resistance

is

in

shunt

to

the

coil,

but

the

circuit is simply opened, if the opening were instantaneous, it

= would

be :

f
e

co ;

that

is,

an infinite

voltage

would be

induced.

That is, the insulation of the coil would be punctured and the

circuit closed in this manner.

The more rapid, thus, the opening of an inductive circuit, the

higher is the induced voltage, and the greater the danger of break-

down. Hence it is not safe to have too rapid circuit-opening

devices on inductive circuits.

To some extent the circuit protects itself by an arc following the

blades of the circuit-opening switch, and thereby retarding the cir-

cuit opening. The more rapid the mechanical opening of the

switch, the higher the induced voltage, and further, therefore, the

arc follows the switch blades and maintains the circuit.

16. Similar transients as discussed above occur when closing a

circuit upon an impressed voltage, or changing the voltage, or the

A current, or the resistance or inductance of the circuit.

discus-

sion of the infinite variety of possible combinations obviously

would be impossible. However, they can all be reduced to the

same simple case discussed above, by considering that several

currents, voltages, magnetic fluxes, etc., in the same circuit add

algebraically, without interfering with each other (assuming, as

done here, that magnetic saturation is not approached).
If an e.m.f. e\ produces a current i\ in a circuit, and an e.m.f. ez
+ produces in the same circuit a current i2 , then the e.m.f. e\ ez

produces the current i\ -\- 1%, as is obvious.
+ + If now the voltage e\ ez, and thus also the current ii iZj con-
sists of a permanent term, e\ and ii, and a transient term, e2 and iz , the transient terms ez, iz follow the same curves, when combined with the permanent terms e\, i\, as they would when alone in the

circuit (the case above discussed). Thus, the preceding discus-

sion applies to all magnetic transients, by separating the transient

from the permanent term, investigating it separately, and then

adding it to the permanent term.

28 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
The same reasoning also applies to the transient resulting from
several forms of energy storage (provided that the law of proportionality of i, e, $, etc., applies), and makes it possible, in investigating the phenomena during the transition period of energy readjustment, to separate the permanent and the transient term, and discuss them separately.
B

Fig. 14. Single-energy Starting Transient of Magnetic Circuit.

A For instance, in the coil shown in Fig. 10, let the short circuit

be opened, that is, the voltage eQ be impressed upon the coil. At

the moment of time, tQ, when this is done, current i, magnetic

flux <, and voltage e on the coil are zero. In final condition, after

We the transient has passed, the values i , 3> , e are reached.

may

then, as discussed above, separate the transient from the perma-

nent term, and consider that at the time U the coil has a permanent

current i , permanent flux <i> , permanent voltage e , and in addi-

SINGLE-ENERGY TRANSIENTS.

29

tion thereto a transient current

i 0j a transient flux

< and a ,

transient voltage e Q . These transients are the same as in Fig. 11

(only with reversed direction). Thus the same curves result, and

to them are added the permanent values i , <J> , e . This is shown

in Fig. 14.

A shows the permanent flux <

and the transient flux
,

<J> ,

which are assumed, up to the time tQ, to give the resultant zero

flux. The transient flux dies out by the curve <', in accordance

& with Fig. 11.

added to < gives the curve 3>, which is the tran-

sient from zero flux to the permanent flux 3> .

In the same manner B shows the construction of the actual

current change i by the addition of the permanent current iQ and

the transient current i', which starts from i Q at to.
C then shows the voltage relation: e Q the permanent voltage, e'

the transient voltage which starts from

e at t and e the re,

sultant or effective voltage in the coil, derived by adding e Q and e'.

LECTURE IV.

SINGLE-ENERGY TRANSIENTS IN ALTERNATINGCURRENT CIRCUITS.

17. Whenever the conditions of an electric circuit are changed

in such a manner as to require a change of stored energy, a transi-

tion period appears, during which the stored energy adjusts itself

from the condition existing before the change to the condition

after the change. The currents in the circuit during the transition

period can be considered as consisting of the superposition of

the permanent current, corresponding to the conditions after the

change, and a transient current, which connects the current value

before the change with that brought about by the change. That
= is, if i\ current existing in the circuit immediately before, and thus at the moment of the change of circuit condition, and i% =

current which should exist at the moment of change in accordance

with the circuit condition after the change, then the actual current

ii can be considered as consisting of a part or component iz, and a component ii iz IQ. The former, iz , is permanent, as resulting from the established circuit condition. The current compo-

nent IQ, however, is not produced by any power supply, but is a

remnant of the previous circuit condition, that is, a transient, and

therefore
,

gradually

decreases

in

the

manner

as

discussed

in

para-

graph 13, that is, with a duration T = -

The permanent current i2 may be continuous, or alternating, or
may be a changing current, as a transient of long duration, etc.
The same reasoning applies to the voltage, magnetic flux, etc.
Thus, let, in an alternating-current circuit traversed by current
t'i, in Fig. 15 A, the conditions be changed, at the moment t = 0,
so as to produce the current i2 . The instantaneous value of the
current ii at the moment t = can be considered as consisting
of the instantaneous value of the permanent current i2, shown
dotted, and the transient io = i\ i*. The latter gradually dies

down, with the duration T

, on the usual exponential tran-

30

SINGLE-ENERGY TRANSIENTS.

31

sient, shown dotted in Fig. 15. Adding the transient current iQ to the permanent current i2 gives the total current during the transition period, which is shown in drawn line in Fig. 15.
As seen, the transient is due to the difference between the instantaneous value of the current i\ which exists, and that of the current i2 which should exist at the moment of change, and

Fig. 15. Single-energy Transient of Alternating-current Circuit.
thus is the larger, the greater the difference between the two currents, the previous and the after current. It thus disappears if the change occurs at the moment when the two currents ii and 12 are equal, as shown in Fig. 15B, and is a maximum, if the change occurs at the moment when the two currents i\ and iz have the greatest difference, that is, at a point one-quarter period or 90 degrees distant from the intersection of i\ and 12, as shown
in Fig. 15C.

32 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
If the current ii is zero, we get the starting of the alternating current in an inductive circuit, as shown in Figs. 16, A, B, C. The starting transient is zero, if the circuit is closed at the moment when the permanent current would be zero (Fig. 16B), and is a maximum when closing the circuit at the maximum point of the permanent-current wave (Fig. 16C). The permanent current and the transient components are shown dotted in Fig. 16, and the resultant or actual current in drawn lines.
B

Fig. 16. Single-energy Starting Transient of Alternating-current Circuit.

1 8. Applying the preceding to the starting of a balanced

three-phase system, we see, in Fig. 17A, that in general the three

4 transients t'i, i2 , and

of the three three-phase currents ii, iz , is

are different, and thus also the shape of the three resultant

currents during the transition period. Starting at the moment

of zero current of one phase, ii, Fig. 175, there is no transient for

this current, while the transients of the other two currents, iz

and i3, are equal and opposite, and near their maximum value. Starting, in Fig. 17C, at the maximum value of one current ia,

we have the maximum value of transient for this current i'3 , while

the transients of the two other currents, i\ and ii, are equal, have

SINGLE-ENERGY TRANSIENTS.

33

half the value of 13, and are opposite in direction thereto. In

any case, the three transients must be distributed on both sides

of the zero line.

This

is

obvious:

if

ii,

i2 ', and

'
is

are

the

instan-

taneous values of the permanent three-phase currents, in Fig.

17, the initial values of their transients are: i\, iz,

is-

Fig. 17. Single-energy Starting Transient of Three-phase Circuit.

Since the sum of the three three-phase currents at every moment is zero, the sum of the initial values of the three transient currents
also is zero. Since the three transient curves ii, i'2 , iz are pro-

portional to each other fas exponential curves of the same dura-

tion T =

and the sum of their initial values is zero, it follows

],

34 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
that the sum of their instantaneous values must be zero at any moment, and therefore the sum of the instantaneous values of the resultant currents (shown in drawn line) must be zero at any moment, not only during the permanent condition, but also during the transition period existing before the permanent condi-
tion is reached. It is interesting to apply this to the resultant magnetic field
produced by three equal three-phase magnetizing coils placed under equal angles, that is, to the starting of the three-phase rotating magnetic field, or in general any polyphase rotating magnetic field.
Fig. 18. Construction of Starting Transient of Rotating Field.
As is well known, three equal magnetizing coils, placed under equal angles and excited by three-phase currents, produce a resultant magnetic field which is constant in intensity, but revolves synchronously in space, and thus can be represented by a concen-
tric circle a, Fig. 18.
This, however, applies only to the permanent condition. In
the moment of start, all the three currents are zero, and their resultant magnetic field thus also zero, as shown above. Since
the magnetic field represents stored energy and thus cannot be produced instantly, a transient must appear in the building up of the rotating field. This can be studied by considering separately

SINGLE-ENERGY TRANSIENTS.

35

the permanent and the transient components of the three currents,

as is done in the preceding. Let ii, i2) is be the instantaneous

values of the permanent currents at the moment of closing the

= circuit, t

0.

Combined, these would give the resultant field

(Mo in Fig. 18. The three transient currents in this moment

=ii, are i'i

i^_== i2 , 13 =i^', and combined these give a

OB OA resultant field

, equal and opposite to

in Fig. 18.

The

permanent field rotates synchronously on the concentric circle a;
the transient field OB remains constant in the direction OB ,

since all three transient components of current decrease in propor-

tion to each other. It decreases, however, with the decrease of
B the transient current, that is, shrinks together on the line Q0.

The resultant or actual field thus is the combination of the per-

OA manent fields, shown as OAi

2,

.

.

.

and the transient
,

fields,

shown as OBi, OBZ , etc., and derived thereby by the parallelo-

gram law, as shown in Fig. 18, as OC\, OC2, etc. In this diagram,

B OA Bid, C2 2) etc., are equal to OAi,

2 , etc., that is, to the radius

of the permanent circle a. That is, while the rotating field in

permanent condition is represented by the concentric circle a,

the resultant field during the transient or starting period is repre-

sented by a succession of arcs of circles c, the centers of which
move from BQ in the moment of start, on the line B Q toward 0,

and can be constructed hereby by drawing from the successive

points

B ,

BI }

B 2)

which

correspond

to

successive

moments

of

B C time 0, tij t2 ... , radii BiCi, 2 2, etc., under the angles, that is,

^ in the direction corresponding to the time 0,

t2 , etc. This is

done in Fig. 19, and thereby the transient of the rotating field

is constructed.

Fig. 19. Starting Transient of Rotating Field: Polar Form.

36 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

_From this polar diagram of the rotating field, in Fig. 19, values OC can now be taken, corresponding to successive moments of

time, and plotted in rectangular coordinates, as done in Fig. 20.

As seen, the rotating field builds up from zero at the moment of

closing the circuit, and reaches the final value by a series of oscil-

lations ;

that

is,

it

first

reaches

beyond

the

permanent

value, then

drops below it, rises again beyond it, etc.

Fig. 20.

3

4 cycles

Starting Transient of Rotating Field: Rectangular Form.

We have here an oscillatory transient, produced in a system
with only one form of stored energy (magnetic energy), by the combination of several simple exponential transients. However, it must be considered that, while energy can be stored in one form only, as magnetic energy, it can be stored in three electric circuits, and a transfer of stored magnetic energy between the three electric circuits, and therewith a surge, thus can
occur.
It is interesting to note that the rot at ing-field transient is
independent of the point of the wave at which the circuit is closed. That is, while the individual transients of the three three-phase currents vary in shape with the point of the wave at which they start, as shown in Fig. 17, their polyphase resultant always has the same oscillating approach to a uniform rotating
T field, of duration r The maximum value, which the magnetic field during the transi-
tion period can reach, is limited to less than double the final value, as is obvious from the construction of the 'field, Fig. 19. It is evident herefrom, however, that in apparatus containing rotating fields, as induction motors, polyphase synchronous machines, etc.,
the resultant field may under transient conditions reach nearly
double value, and if then it reaches far above magnetic saturation,
excessive momentary currents may appear, similar as in starting
transformers of high magnetic density. In polyphase rotary

SINGLE-ENERGY TRANSIENTS.

37

apparatus, however, these momentary starting currents usually are far more limited than in transformers, by the higher stray field (self-inductive reactance), etc., of the apparatus, resulting from the air gap in the magnetic circuit.
19. As instance of the use of the single-energy transient in
engineering calculations may be considered the investigation of
the momentary short-circuit phenomena of synchronous alternators. In alternators, especially high-speed high-power mar chines as turboalternators, the momentary short-circuit current
may be many times greater than the final or permanent short-
circuit current, and this excess current usually decreases very
slowly, lasting for many cycles. At the same time, a big cur-
rent rush occurs in the field. This excess field current shows
curious pulsations, of single and of double frequency, and in the beginning the armature currents also show unsymmetrical shapes. Some oscillograms of three-phase, quarter-phase, and single-phase short circuits of turboalternators are shown in Figs.
25 to 28.
By considering the transients of energy storage, these rather
complex-appearing phenomena can be easily understood, and predetermined from the constants of the machine with reasonable
exactness.
In an alternator, the voltage under load is affected by armature reaction and armature self-induction. Under permanent condition, both usually act" in the same way, reducing the voltage at noninductive and still much more at inductive load, and increasing it at antiinductive load; and both are usually combined in one quantity, the synchronous reactance XQ. In the transients resulting from circuit changes, as short circuits, the self-inductive armature reactance and the magnetic armature reaction act very
differently:* the former is instantaneous in its effect, while the
latter requires time. The self-inductive armature reactance Xi consumes a voltage x\i by the magnetic flux surrounding the armature conductors, which results from the m.m.f. of the armature current, and therefore requires a component of the magnetic-field flux for its production. As the magnetic flux and the current which produces it must be simultaneous (the former being an integral part of the phenomenon of current flow, as seen in Lecture
II), it thus follows that the armature reactance appears together * So also in their effect on synchronous operation, in hunting, etc.

38 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

with the armature current, that is, is instantaneous. The arma-
ture reaction, however, is the m.m.f. of the armature current in its reaction on the m.m.f. of the field-exciting current. That is, that
part xz = XQ Xi of the synchronous reactance which corresponds
to the armature reaction is not a true reactance at all, consumes
no voltage, but represents the consumption of field ampere turns by the m.m.f. of the armature current and the corresponding change of field flux. Since, however, the field flux represents stored magnetic energy, it cannot change instantly, and the armature reaction thus does not appear instantaneously with the armature current, but shows a transient which is determined essentially by the constants of the field circuit, that is, is the counterpart of the field transient of the machine.
If then an alternator is short-circuited, in the first moment only
the true self-inductive part Xi of the synchronous reactance exists,

and the armature current thus is i\ =

where e is the induced
,

Xi

e.m.f., that is, the voltage corresponding to the magnetic-field excitation flux existing before the short circuit. Gradually the
armature reaction lowers the field flux, in the manner as represented by the synchronous reactance x , and the short-circuit cur-

= rent decreases to the value i'
XQ
The ratio of the momentary short-circuit current to the perma-

= nent short-circuit current thus is, approximately, the ratio

>

IQ Xi

that is, synchronous reactance to self-inductive reactance, or armature reaction plus armature self-induction, to armature self-induction. In machines of relatively low self-induction
and high armature reaction, the momentary short-circuit cur-
rent thus may be many times the permanent short-circuit

current.

The field flux remaining at short circuit is that giving the voltage consumed by the armature self-induction, while the decrease of field flux between open circuit and short circuit corresponds to the armature reaction. The ratio of the open-circuit field flux to
the short-circuit field flux thus is the ratio of armature reaction

plus self-induction, to the self-induction; or of the synchronous

reactance to the self-inductive reactance:

SINGLE-ENERGY TRANSIENTS.

39

Thus it is:

momentary short-circuit current ~_" open-circuit field flux * _

permanent short-circuit current

short-circuit field flux

armature reaction plus self-induction _ synchronous reactance _~~ XQ

self-induction

self-inductive reactance x\

= 20. Let $1 field flux of a three-phase alternator (or, in general,

polyphase alternator) at open circuit, and this alternator be short-
= circuited at the time t 0. The field flux then gradually dies

down, by the dissipation of its energy in the field circuit, to the

short-circuit field flux 3> , as indicated by the curve $ in Fig. 21A.

m = If

ratio

armature reaction plus self-induction _~ XQ

armature self-induction

x\

= m$ it is $1

, and the initial value of the field flux consists of the

(ml) permanent part <i> , and the transient part <' = $1 < =

$0. This is a rather slow transient, frequently of a duration of a

second or more.

^ The armature currents i1} iz are proportional to the field flux
$ which produces them, and thus gradually decrease, from initial values, which are as many times higher than the final values as $1
m is higher than 3> , or times, and are represented in Fig. 21 B.
The resultant m.m.f. of the armature currents, or the armature
reaction, is proportional to the currents, and thus follows the same
field transient, as shown by F in Fig. 2 1C.
The field-exciting current is i at open circuit as well as in the
permanent condition of short circuit. In the permanent condition
of short circuit, the field current iQ combines with the armature
reaction F , which is demagnetizing, to a resultant m.m.f., which
produces the short-circuit flux 3> . During the transition period
the field flux $ is higher than 3> , and the resultant m.m.f. must
therefore be higher in the same proportion. Since it is the dif-
ference between the field current and the armature reaction F, and the latter is proportional to 3>, the field current thus must also be

* If the machine were open-circuited before the short circuit, otherwise the field flux existing before the short circuit. It herefrom follows that the momentary short-circuit current essentially depends on the field flux, and thereby the voltage of the machine, before the short circuit, but is practically independent of the load on the machine before the short circuit and the field excitation corresponding to this load.

40 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

= proportional to $>. Thus, as it is i i Q at < , during the transition

= period it is i

<
iQ . Hence, the field-exciting current traverses

<P O

the same transient, from an initial value iY to the normal value i' ,

as the field flux 3> and the armature currents.

B

Fig. 21.

Construction of Momentary Short Circuit Characteristic of Poly-
phase Alternator.

Thus, at the moment of short circuit a sudden rise of field current must occur, to maintain the field flux at the initial value
$1 against the demagnetizing armature reaction. In other words,
the field flux $ decreases at such a rate as to induce in the field

circuit the e.m.f. required to raise the field current in the propor-

tion

m,

from

iQ

to

i

f ,

and

maintain

it

at

the

values

corresponding

to the transient i, Fig. 2 ID.

As seen, the transients 3>; z'i, i'2 , iz] F; i are proportional to each

other, and are a field transient. If the field, excited by current iQ

SINGLE-ENERGY TRANSIENTS.

41

at

impressed

voltage

e ,

were

short-circuited

upon

itself,

in

the

first moment the current in the field would still be i Q , and there-

fore -the voltage e would have to be induced by the decrease of

magnetic

flux ;

and

the

duration

of

the

field

transient,

as discussed

in Lecture III, would be TQ =

-

ro

The field current in Fig. 2 ID, of the alternator short-circuit

= transient, starts with the value ij

mi ,

and

if

eQ

is

the

e.m.f.

supplied in the field-exciting circuit from a source of constant

voltage

supply,

as

the

exciter,

to

produce

the

current

f
i ,

the

voltage Co' = meo must be acting in the field-exciting circuit; that

is, in addition to the constant exciter voltage e , a voltage (m I)e

must be induced in the field circuit by the transient of the mag-

netic flux. As a transient of duration

induces the voltage e ,

TO

to induce the voltage (m I)e the duration of the transient must

be

-1 o

(m- /

\ -i

)

1) TO

L = = where

inductance, r

total resistance of field-exciting cir-

cuit (inclusive of external resistance).

The short-circuit transient of an alternator thus usually is of

shorter duration than the short-circuit transient of its field, the

more so, the greater m, that is, the larger the ratio of momentary

to permanent short-circuit current.

In Fig. 21 the decrease of the transient is shown greatly exagger-

ated compared with the frequency of the armature currents, and

Fig. 22 shows the curves more nearly in their actual proportions. The preceding would represent the short-circuit phenomena, if

there were no armature transient. However, the armature cir-

cuit contains inductance also, that is, stores magnetic energy, and

thereby gives rise to a transient, of duration T =

where L =
,

= inductance, r resistance of armature circuit. The armature
transient usually is very much shorter in duration than the field
transient.

The armature currents thus do not instantly assume their
symmetrical alternating values, but if in Fig. 215, iV, iz, is are
the instantaneous values of the armature currents in the moment

of start, t 0, three transients are superposed upon these, and

42 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
start with the values ii, iz, is'. The resultant armature currents are derived by the addition of these armature transients upon the permanent armature currents, in the manner as dis-
cussed in paragraph 18, except that in the present case even the permanent armature currents ii, i2 , is are slow transients.
In Fig. 22B are shown the three armature short-circuit currents,
in their actual shape as resultant from the armature transient and the field transient. The field transient (or rather its beginning) is shown as Fig, 22A. Fig. 22B gives the three armature

Fig. 22.

Momentary Short Circuit Characteristic of Three-phase
Alternator.

currents for the case where the circuit is closed at the moment when

t'i

should

be

maximum ;

ii then shows the maximum transient,

and

iz and ^3 transients in opposite direction, of half amplitude. These

armature transients rapidly disappear, and the three currents

become symmetrical, and gradually decrease with the field tran-

sient to the final value indicated in the figure.

The resultant m.m.f. of three three-phase currents, or the arma-

ture reaction, is constant if the currents are constant, and as the

currents decrease with the field transient, the resultant armature

reaction decreases in the same proportion as the field, as is shown

SINGLE-ENERGY TRANSIENTS.

43

in Fig. 21(7 by F. During the initial part of the short circuit, however, while the armature transient is appreciable and the
armature currents thus unsymmetrical, as seen in Fig. 225, their resultant polyphase m.m.f. also shows a transient, the transient of the rotating magnetic field discussed in paragraph 18. That is,
F it approaches the curve C of Fig. 21 by a series of oscillations,
as indicated in Fig. 21E. Since the resultant m.m.f. of the machine, which produces the
D flux, is the difference of the field excitation, Fig. 21 and the
armature reaction, then if the armature reaction shows an initial oscillation, in Fig. 21 E, the field-exciting current must give the same oscillation, since its m.m.f. minus the armature reaction gives the resultant field excitation corresponding to flux $>. The starting transient of the polyphase armature reaction thus appears in the field current, as shown in Fig. 22(7, as an oscillation of full machine frequency. As the mutual induction between armature and field circuit is not perfect, the transient pulsation of armature reaction appears with reduced amplitude in the field current, and this reduction is the greater, the poorer the mutual inductance, that is, the more distant the field winding is from the armature winding. In Fig. 22(7 a damping of 20 per cent is assumed, which corresponds to fairly good mutual inductance between field and armature, as met in turboalternators.
If the field-exciting circuit contains inductance outside of the alternator field, as is always the case to a slight extent, the pulsations of the field current, Fig. 22(7, are slightly reduced and delayed in phase; and with considerable inductance intentionally inserted into the field circuit, the effect of this inductance would
require consideration.
From the constants of the alternator, the momentary shortcircuit characteristics can now be constructed.
Assuming that the duration of the field transient is

(m I)r

sec.,

the duration of the armature transient is

T = ~ = .1 sec.

And assuming that the armature reaction is 5 times the armature

44 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

self-induction, that is, the synchronous reactance is 6 times the self-

inductive reactance,

m =

= 6. The frequency is 25 cycles.

Xi

If <f>i is the initial or open-circuit flux of the machine, the short-

= m = circuit flux is <J>

3>i

1

~
o

$1,

and

the

field

transient

$

is

a

tran-

sient of duration 1 sec., connecting $1 and < , Fig. 22A, represented by the expression

The permanent armature currents ii, i%, is then are currents

m starting with the values

, and decreasing to the final short-

XQ

circuit current

on the field transient of duration T . To these

,

XQ

currents are added the armature transients, of duration T, which

start with initial values equal but opposite in sign to the initial

values of the permanent (or rather slowly transient) armature

currents, as discussed in paragraph 18, and thereby give the asym-

metrical resultant currents, Fig. 225.
The field current i gives the same slow transient as the flux <,
= starting with i f miQ, and tapering to the final value i . Upon

this is superimposed the initial full-frequency pulsation of the

armature reaction. The transient of the rotating field, of duration
T = .1 sec., is constructed as in paragraph 18, and for its instan-

taneous values the percentage deviation of the resultant field

from its permanent value .is calculated. Assuming 20 per cent

damping in the reaction on the field excitation, the instantaneous

values of the slow field transient (that is, of the current (i i' ),

since i is the permanent component) then are increased or de-

creased by 80 per cent of the percentage variation of the transient

field of armature reaction from uniformity, and thereby the field

curve, Fig. 22C, is derived. Here the correction for the external

field inductance is to be applied, if considerable.

Since the transient of the armature reaction does not depend

on the point of the wave where the short circuit occurs, it follows

that the phenomena at the short circuit of a polyphase alternator

are always the same, that is, independent of the point of the wave

at which the short circuit occurs, with the exception of the initial

wave shape of the armature currents, which individually depend

SINGLE-ENERGY TRANSIENTS.

45

on the point of the wave at which the phenomenon begins, but not
so in their resultant effect.
21. The conditions with a single-phase short circuit are differ-
ent, since the single-phase armature reaction is pulsating, varying between zero and double its average value, with double the machine frequency.
The slow field transient and its effects are the same as shown in
A Fig. 21, to D.
However, the pulsating armature reaction produces a corresponding pulsation in the field circuit. This pulsation is of double

Fig. 23. Symmetrical Momentary Single-phase Short Circuit of Alternator.
frequency, and is not transient, but equally exists in the final short-
circuit current.
Furthermore, the armature transient is not constant in its
reaction on the field, but varies with the point of the wave at which the short circuit starts.
Assume that the short circuit starts at that point of the wave where the permanent (or rather slowly transient) armature current should be zero: then no armature transient exists, and the armature current is symmetrical from the beginning, and
A shows the slow transient of the field, as shown in Fig. 23, where

46 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

B is the field transient <i> (the same as in Fig. 22 A) and the arma-

m ture current, decreasing from an initial value, which is

times

the final value, on the field transient.

Assume then that the mutual induction between field and

armature is such that 60 per cent of the pulsation of armature

reaction appears in the field current. Forty per cent damping for the double-frequency reaction would about correspond to the 20

per cent damping assumed for the transient full-frequency pulsation of the polyphase machine. The transient field current thus

pulsates by 60 per cent around the slow field transient, as shown
by Fig. 23C; passing a maximum for every maximum of armature

Fig. 24. Asymmetrical Momentary Single-phase Short Circuit of Alternator.
current, and thus maximum of armature reaction, and a minimum
for every zero value of armature current, and thus armature reac-
tion.
Such single-phase short-circuit transients have occasionally been recorded by the oscillograph, as shown in Fig. 27. Usually, however, the circuit is closed at a point of the wave where the permanent armature current would not be zero, and an armature transient appears, with an initial value equal, but opposite to, the initial value of the permanent armature current. This is shown in
Fig. 24 for the case of closing the circuit at the moment where the

SINGLE-ENERGY TRANSIENTS.

47

armature current should be a maximum, and its transient thus a maximum. The field transient < is the same as before. The
armature current shows the initial asymmetry resulting from the armature transient, and superimposed on the slow field transient.
On the field current, which, due to the single-phase armature reaction, shows a permanent double-frequency pulsation, is now
superimposed the transient full-frequency pulsation resultant from the transient armature reaction, as discussed in paragraph 20. Every second peak of the permanent double-frequency pulsation then coincides with a peak of the transient full-frequency pulsation, and is thereby increased, while the intermediate peak of the
double-frequency pulsation coincides with a minimum of the fullfrequency pulsation, and is thereby reduced. The result is that
successive waves of the double-frequency pulsation of the field current are unequal in amplitude, and high and low peaks alternate. The difference between successive double-frequency waves
is a maximum in the beginning, and gradually decreases, due to
the decrease of the transient full-frequency pulsation, and finally the double-frequency pulsation becomes symmetrical, as shown in Fig. 24C.
In the particular instance of Fig. 24, the double-frequency and the full-frequency peaks coincide, and the minima of the fieldcurrent curve thus are symmetrical. If the circuit were closed at another point of the wave, the double-frequency minima would become unequal, and the maxima more nearly equal, as is easily
seen.
While the field-exciting current is pulsating in a manner determined by the full-frequency transient and double-frequency permanent armature reaction, the potential difference across the
field winding may pulsate less, if little or no external resistance or inductance is present, or may pulsate so as to be nearly alternating and many times higher than the exciter voltage, if consid-
erable external resistance or inductance is present; and therefore
it is not characteristic of the phenomenon, but may become impor-
tant by its disruptive effects, if reaching very high values of voltage. With a single-phase short circuit on a polyphase machine, the
double-frequency pulsation of the field resulting from the singlephase armature reaction induces in the machine phase, which is in quadrature to the short-circuited phase, an e.m.f. which contains the frequencies /(2 1), that is, full frequency and triple

48 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

SINGLE-ENERGY TRANSIENTS.

49

frequency, and as the result an increase of voltage and a distortion of the quadrature phase occurs, as shown in the oscillogram
Fig. 25.
Various momentary short-circuit phenomena are illustrated by
the oscillograms Figs. 26 to 28.
Figs. 26A and 265 show the momentary three-phase short circuit of a 4-polar 25-cycle 1500-kw. steam turbine alternator. The

Fig. 26 A. CD9399. Symmetrical.
Fig. 2QB. CD9397. Asymmetrical. Momentary Three-phase Short Circuit of 1500-Kw. 2300-Volt Three-phase Alternator (ATB-4-1500-1800) . Oscillograms of Armature Current and Field Current.
lower curve gives the transient of the field-exciting current, the
upper curve that of one of the armature currents, in Fig. 26A that current which should be near zero, in Fig. 26B that which should be near its maximum value at the moment where the short
circuit starts.
Fig. 27 shows the single-phase short circuit of a pair of machines
in which the short circuit occurred at the moment in which the
armature short-circuit current should be zero; the armature cur-

50 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
rent wave, therefore, is symmetrical, and the field current shows only the double-frequency pulsation. Only a few half-waves were recorded before the circuit breaker opened the short circuit.
Fig. 27. CD5128. Symmetrical. Momentary Single-phase Short Circuit of Alternator. Oscillogram of Armature Current, Armature Voltage, and Field Current.
Fig. 28. CD6565. Asymmetrical. Momentary Single-phase Short Circuit of 5000-Kw. 11, 000-Volt Three-phase Alternator (ATB-6-5000-500) . Oscillogram of Armature Current and Field Current.
Fig. 28 shows the single-phase short circuit of a 6-polar 5000-kw. 11,000-volt steam turbine alternator, which occurred at a point of the wave where the armature current should be not far from its
maximum. The transient armature current, therefore, starts un-

SINGLE-ENERGY TRANSIENTS.

51

symmetrical, and the double-frequency pulsation of the field current shows during the first few cycles the alternate high and low peaks resulting from the superposition of the full-frequency transient pulsation of the rotating magnetic field of armature reaction.
Interesting in this oscillogram is the irregular initial decrease of the
armature current and the sudden change of its wave shape, which is the result of the transient of the current transformer, through which the armature current was recorded. On the true armature-
current transient superposes the starting transient of the current transformer.
Fig. 25 shows a single-phase short circuit of a quarter-phase alternator; the upper wave is the voltage of the phase which is not short-circuited, and shows the increase and distortion resulting from the double-frequency pulsation of the armature reaction.

LECTURE V.
SINGLE-ENERGY TRANSIENT OF IRONCLAD
CIRCUIT.

22. Usually in electric circuits, current, voltage, the magnetic

field and the dielectric field are proportional to each other, and the

transient thus is a simple exponential, if resulting from one form of

stored energy, as discussed in the preceding lectures. This, how-

ever, is no longer the case if the magnetic field contains iron or

other magnetic materials, or if the dielectric field reaches densities

beyond the dielectric

strength

of

the

carrier

of

the

field,

etc. ;

and

the proportionality between current or voltage and their respective

fields, the magnetic and the dielectric, thus ceases, or, as it may be expressed, the inductance L is not constant, but varies with the

current, or the capacity is not constant, but varies with the voltage.
The most important case is that of the ironclad magnetic cir-

cuit, as it exists in one of the most important electrical apparatus,

the alternating-current transformer. If the iron magnetic circuit

contains an air gap of sufficient length, the magnetizing force con-

sumed in the iron, below magnetic saturation, is small compared

with that consumed in the air gap, and the magnetic flux, therefore,

is proportional to the current up to the values where magnetic

saturation begins. Below saturation values of current, the tran-

sient thus is the simple exponential discussed before.

If the magnetic circuit is closed entirely by iron, the magnetic

flux is not proportional to the current, and the inductance thus not

constant, but varies over the entire range of currents, following

the permeability curve of the iron. Furthermore, the transient

due to a decrease of the stored magnetic energy differs in shape

and in value from that due to an increase of magnetic energy, since

the rising and decreasing magnetization curves differ, as shown by

the hysteresis cycle.

Since no satisfactory mathematical expression has yet been

found for the cyclic curve of hysteresis, a mathematical calcula-

tion is not feasible,, but the transient has to be calculated by an

'^''"r

'*_/ ? :,": \ :

52

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 53
approximate step-by-step method, as illustrated for the starting transient of an alternating-current transformer in "Transient Elec-
tric Phenomena and Oscillations," Section I, Chapter XII. Such methods are very cumbersome and applicable only to numerical
instances.
An approximate calculation, giving an idea of the shape of the transient of the ironclad magnetic circuit, can be made by neglect-
ing the difference between the rising and decreasing magnetic characteristic, and using the approximation of the magnetic characteristic given by Frohlich's formula:

which is usually represented in the form given by Kennelly:

T/>

+ - p =

=a

crOC;

(2)

that is, the reluctivity is a linear function of the field intensity.

It gives a fair approximation for higher magnetic densities.

This formula is based on the fairly rational assumption that the

permeability of the iron is proportional to its remaining magnetiza-

bility. That is, the magnetic-flux density (B consists of a compo-

nent 3C, the field intensity, which is the flux density in space, and

a component (B' = (B 3C, which is the additional flux density

carried by the iron.

(B'

is

frequently

called

the

" metallic-flux

density." With increasing 3C, (B' reaches a finite limiting value,

which in iron is about

& = '
x

20,000 lines per cm 2 . *

At any density (B', the remaining magnetizability then is (B^' (B', and, assuming the (metallic) permeability as proportional
hereto, gives

and, substituting

gives

a = , cftco'rc^

* See "On the Law of Hysteresis," Part II, A.I.E.E. Transactions, 1892,
page 621.

54 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

or, substituting

1_

1

***

/

t* ,fc /

(/

gives equation (1).

^ = = For OC

in equation (1), uv

= - for 3C oo ;
a:

=-
;

that is,

cr

= = in equation (1), - initial permeability, - saturation value of

Oi

(7

magnetic density. If the magnetic circuit contains an air gap, the reluctance of the
iron part is given by equation (2), that of the air part is constant, and the total reluctance thus is

= + p

ft

ffK ,

where 3 = a plus the reluctance of the air gap. Equation (1),
therefore, remains applicable, except that the value of a is in-
creased.
In addition to the metallic flux given by equation (1), a greater or smaller part of the flux always passes through the air or through space in general, and then has constant permeance, that is, is given by

23. In general, the flux in an ironclad magnetic circuit can, therefore, be represented as function of the current by an expression of the form

where

& =
,.

is that part of the flux which passes through

1 -f- ut

the iron and whatever air space may be in series with the iron, and a is the part of the flux passing through nonmagnetic

material.

Denoting now

L = 2

nc

10- 8 ,

i

where n = number of turns of the electric circuit, which is interlinked with the magnetic circuit, L2 is the inductance of the air
part of the magnetic circuit, LI the (virtual) initial inductance, that is, inductance at very small currents, of the iron part of the mag-

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 55

netic circuit, and =- the saturation value of the flux in the iron.

= = = = 72,CJ>'

d

That is, for i

0, r-

Z/i ; and for i

oo <' ,

T.

i

If r = resistance, the duration of the component of the transient

resulting from the air flux would be

T~ _ L2 nc 10~ 8
*V-7"

and the duration of the transient which would result from the initial inductance of the iron flux would be

The differential equation of the transient is: induced voltage

plus

resistance

drop

equal

zero ;

that

is,

Substituting (3) and differentiating gives
'
+ + na 10~ 8 di . ncl.,0_rSa di . (i+Wdi dt
and, substituting (5) and (6),

+ t(l

2
bi)

Z

'

d*

5

hence, separating the variables,

+ + Tidi

Tidi dt = Q

The first term is integrated by resolving into partial fractions

1

"1

+ i(l

2
6i)

i

6
+ 1 6i

6

+ 2>

(1

6i) .

and the integration of differential equation (7) then gives

= = If then, for the time t

t Q, the current is i

i these values ,

substituted in (8) give the integration constant C:

+ + T1 log-

!T2 logio

+ + T-

^o

C = 0,

(9)

56 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
and, subtracting (8) from (9), gives
+ ' (10)
1 6i
5
This equation is so complex in i that it is not possible to calculate from the different values of t the corresponding values of i; but inversely, for different values of i the corresponding values of t can be calculated, and the corresponding values of i and t, derived in this manner, can be plotted as a curve, which gives the single-energy transient of the ironclad magnetic circuit.

Tra sient o

Ironclad Inductive Circuit :

t=2.92-

(dotted: t = 1.0851g i

+ i l+t.-.66ii j
.50?)

2

3

4

5

6 seconds

Fig. 29.

Such is done in Fig. 29, for the values of the constants

a = 4 X 105

,

c = 4 X 104 ,

= b

.6,

n = 300.

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 57

58 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

This gives

T=4

Assuming i = 10 amperes for t = 0, gives from (10) the equa-

tion :

- T = 2.92

9.21

10
log

1

^ + . 921

4
.6 i

Herein, the logarithms have been reduced to the base 10 by

= division

with

w
log e

.4343.

For comparison is shown, in dotted line, in Fig. 29, the transient

of a circuit containing no iron, and of such constants as to give

about the same duration:

t = 1.0S5logi- .507.

As seen, in the ironclad transient the current curve is very much steeper in the range of high currents, where magnetic saturation is reached, but the current is slower in the range of medium
magnetic densities. Thus, in ironclad transients very high-current values of short
duration may occur, and such transients, as those of the starting current of alternating-current transformers, may therefore be of
serious importance by their excessive current values.
An oscillogram of the voltage and current waves in an 11,000-kw.
high-voltage 60-cycle three-phase transformer, when switching onto the generating station near the most unfavorable point of the wave, is reproduced in Fig. 30. As seen, an excessive current rush persists for a number of cycles, causing a distortion of the voltage wave, and the current waves remain unsymmetrical for many
cycles.

LECTURE VI.

DOUBLE-ENERGY TRANSIENTS.

24. In a circuit in which energy can be stored in one form only,

the change in the stored energy which can take place as the result

of a change of the circuit conditions is an increase or decrease.

The transient can be separated from the permanent condition, and

then always is the representation of a gradual decrease of energy.

Even if the stored energy after the change of circuit conditions is

greater than before, and during the transition period an increase

of energy occurs, the representation still is by a decrease of the

transient. This transient then is the difference between the energy

storage in the permanent condition and the energy storage during

the transition period.

If the law of proportionality between current, voltage, magnetic

flux, etc., applies, the single-energy transient is a simple exponential

function :

j_

=T

y

i/oe

,

(1)

where

= ?/o initial value of the transient, and

TO = duration of the transient,

that is, the time which the transient voltage, current, etc., would last if maintained at its initial value.
The duration T is the ratio of the energy-storage coefficient
to the power-dissipation coefficient. Thus, if energy is stored by the current i, as magnetic field,

T= ,

(2)

where L = inductance = coefficient of energy storage by the current, r = resistance = coefficient of power dissipation by the current.
If the energy is stored by the voltage e, as dielectric field, the
duration of the transient would be

TJ = -,

(3)

s/

59

60 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

where C = capacity = coefficient of energy storage by the volt-

= age, in the dielectric field, and g = conductance

coefficient of

power consumption by the voltage, as leakage conductance by

the voltage, corona, dielectric hysteresis, etc.

Thus the transient of the spontaneous discharge of a condenser

would be represented by

= e

e

e~

ct
.

(4)

Similar single-energy transients may occur in other systems.
For instance, the transient by which a water jet approaches constant velocity when falling under gravitation through a resisting medium would have the duration

T = -,

(5)

where VQ = limiting velocity, g = acceleration of gravity, and would
be given by

v = v (l-6~r}.

(6)

In a system in which energy can be stored in two different

forms, as for instance as magnetic and as dielectric energy in a

circuit containing inductance and capacity, in addition to the

gradual decrease of stored energy similar to that represented by

the single-energy transient, a transfer of energy can occur between

its two different forms.

= Thus, if i = transient current, e

transient voltage (that is,

the difference between the respective currents and voltages exist-

ing in the circuit as result of the previous circuit condition, and

the values which should exist as result of the change of circuit

conditions), then the total stored energy is

w + Li* Ce* ) 'T -2-'

(7)

W +W = m

d. >

W W While the total energy

decreases by dissipation, m may be

converted into Wd, or inversely.

Such an energy transfer may be periodic, that is, magnetic energy

may change to dielectric and then back again; or unidirectional,

that is, magnetic energy may change to dielectric (or inversely,

dielectric to magnetic), but never change back again; but the

DOUBLE-ENERGY TRANSIENTS.

61

energy is dissipated before this. This latter case occurs when the
dissipation of energy is very rapid, the resistance (or conductance) high, and therefore gives transients, which rarely are of industrial importance, as they are of short duration and of low power. It therefore is sufficient to consider the oscillating double-energy transient, that is, the case in which the energy changes periodically between its two forms, during its gradual dissipation.
This may be done by considering separately the periodic trans-
fer, or pulsation of the energy between its two forms, and the
gradual dissipation of energy.
A . Pulsation of energy.
25. The magnetic energy is a maximum at the moment when the dielectric energy is zero, and when all the energy, therefore, is
magnetic ; and the magnetic energy is then

where t' = maximum transient current. The dielectric energy is a maximum at the moment when the
magnetic energy is zero, and all the energy therefore dielectric, and is then
Ce 2
'
2
where e = maximum transient voltage.
As it is the same stored energy which alternately appears as magnetic and as dielectric energy, it obviously is
W _ Ceo2
~2~ ~2"
This gives a relation between the maximum transient current and the maximum transient voltage:

v/:-^ therefore is of the nature of an impedance z , and is called

the

natural

impedance,

or

the

surge

impedance,

of

the

circuit ;

and

= fc
its reciprocal, VT /Jyj

yo, is the natural admittance, or the surge

admittance, of the circuit.

62 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
The maximum transient voltage can thus be calculated from the maximum transient current:

and inversely,

= = #0

V/ 'Z'O

7>

i&Qj

/C

= j = y io eo

e 2/o.

(10) (11)

This relation is very important, as frequently in double-energy transients one of the quantities e$ or i is given, and it is important to determine the other.

For instance, if a line is short-circuited, and the short-circuit
current IQ suddenly broken, the maximum voltage which can be

induced by the dissipation of the stored magnetic energy of the

= short-circuit current is e

igZo.

If one conductor of an ungrounded cable system is grounded,
the maximum momentary current which may flow to ground is = io eo2/o, where e = voltage between cable conductor and ground.

If lightning strikes a line, and the maximum voltage which it

may produce on the line, as limited by the disruptive strength of

the line insulation against momentary voltages, is e , the maximum

= discharge current in the line is limited to i

e<>yo.

If L is high but C low, as in the high-potential winding of a

high-voltage transformer (which winding can be considered as a

circuit of distributed capacity, inductance, and resistance), z is

high and T/O low. That is, a high transient voltage can produce

only moderate transient currents, but even a small transient cur-

rent produces high voltages. Thus reactances, and other reactive

apparatus, as transformers, stop the passage of large oscillating

currents, but do so by the production of high oscillating voltages.
Inversely, if L is low and C high, as in an underground cable,

ZQ is low but 2/0 high, and even moderate oscillating voltages pro-

duce large oscillating currents, but even large oscillating currents

produce only moderate voltages. Thus underground cables are

little liable to the production of high oscillating voltages. This

is fortunate, as the dielectric strength of a cable is necessarily

relatively much lower than that of a transmission line, due to

A the close proximity of the conductors in the former.

cable,

therefore, when receiving the moderate or small oscillating cur-

rents which may originate in a transformer, gives only very low

DOUBLE-ENERGY TRANSIENTS.

63

oscillating voltages, that is, acts as a short circuit for the transformer oscillation, and therefore protects the latter. Inversely, if the large oscillating current of a cable enters a reactive device, as a current transformer, it produces enormous voltages therein.
Thus, cable oscillations are more liable to be destructive to the reactive apparatus, transformers, etc., connected with the cable, than to the cable itself.
A transmission line is intermediate in the values of z and yQ
between the cable and the reactive apparatus, thus acting like a reactive apparatus to the former, like a cable toward the latter. Thus, the transformer is protected by the transmission line in oscillations originating in the transformer, but endangered by the
transmission line in oscillations originating in the transmission
line.

V The simple consideration of the relative values of ZQ =

^ in

the different parts of an electric system thus gives considerable

information on the relative danger and protective action of the

parts on each other, and shows the reason why some elements, as

current transformers, are far more liable to destruction than others;

but also shows that disruptive effects of transient voltages,

observed in one apparatus, may not and very frequently do not

originate in the damaged apparatus, but originate in another

part of the system, in which they were relatively harmless, and become dangerous only when entering the former apparatus.

26. If there is a periodic transfer between magnetic and dielec-

tric energy, the transient current i and the transient voltage e

successively increase, decrease, and become zero.
The current thus may be represented by

i = locosfa -7),

(12)

where iQ is the maximum value of current, discussed above, and

=

<t>

27Tft,

(13)

where / = the frequency of this transfer (which is still undeter-

mined), and 7 the phase angle at the starting moment of the

transient; that is,

= = ii IQ cos 7 initial transient current.

(14)

As the current i is a maximum at the moment when the magnetic energy is a maximum and the dielectric energy zero, the voltage e

64 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

must be zero when the current is a maximum, and inversely; and

if the current is represented by the cosine function, the voltage

thus is represented by the sine function, that is,

where

= - e e sin (0 7),

(15)

= = ei

e sin 7 initial value of transient voltage.

(16)

The frequency / is still unknown, but from the law of proportionality it follows that there must be a frequency, that is, the successive conversions between the two forms of energy must occur in
equal time intervals, for this reason: If magnetic energy converts
to dielectric and back again, at some moment the proportion between the two forms of energy must be the same again as at the starting moment, but both reduced in the same proportion by the power dissipation. From this moment on, the same cycle then must
repeat with proportional, but proportionately lowered values.

Fig. 31. CD10017. Oscillogram of Stationary Oscillation of Varying
Frequency: Compound Circuit of Step-up Transformer and 28 Miles of
100,000-volt Transmission Line.
If, however, the law of proportionality does not exist, the oscil-
lation may not be of constant frequency. Thus in Fig. 31 is shown
an oscillogram of the voltage oscillation of the compound circuit consisting of 28 miles of 100,000-volt transmission line and the 2500-kw. high-potential step-up transformer winding, caused by switching transformer and 28-mile line by low-tension switches off a substation at the end of a 153-mile transmission line, at 88 kv. With decreasing voltage, the magnetic density in the transformer

DOUBLE-ENERGY TRANSIENTS.

65

decreases, and as at lower magnetic densities the permeability of

the iron is higher, with the decrease of voltage the permeability of

the iron and thereby the inductance of the electric circuit inter-

linked with it increases, and, resulting from this increased magnetic

energy storage coefficient L, there follows a slower period of oscil-

lation, that is, a decrease of frequency, as seen on the oscillogram,

from 55 cycles to 20 cycles per second.

If the energy transfer is not a simple sine wave, it can be repre-

sented by a series of sine waves, and in this case the above equa-

tions (12) and (15) would still apply, but the calculation of the

frequency / would give a number of values which represent the

different component sine waves.

The

dielectric

field

of

a

condenser,

or

its

" charge,"

is

capacity

times voltage: Ce. It is, however, the product of the current

flowing into the condenser, and the time during which this current

flows into it, that is, it equals i t.

Applying the law

Ce = it

(17)

to the oscillating energy transfer: the voltage at the condenser

changes during a half-cycle from eQ to -fe , and the condenser

charge thus is

2e C;

2

the current has a maximum value i' , thus an average value

-i ,

IT

and as it flows into the condenser during one-half cycle of the

frequency /, that is, during the time =-}, it is

2eQC = -io o7
7T 2J
which is the expression of the condenser equation (17) applied to the oscillating energy transfer.
Transposed, this equation gives

and

substituting

equation

(10)

into

(18),

and

canceling

with

i ,

gives

66 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

as the expression of the frequency of the oscillation, where
a = VLC

(20)

is a convenient abbreviation of the square root.
The transfer of energy between magnetic and dielectric thus

= occurs with a definite frequency /

~ - and the oscillation thus
Z, TTCT

is a sine wave without distortion, as long as the law of proportion-
ality applies. When this fails, the wave may be distorted, as seen
on the oscillogram Fig. 31.
The equations of the periodic part of the transient can now be written down by substituting (13), (19), (14), and (16) into (12)
and (15) :

= i io cos (0

= + 7)

io cos 7 cos <j>

i sin 7 sin

and by (11):

t
i\ cos
(7

IQ . t
e\ sin - ,

Q

<J

= i i\ cos
(T

1/001 sin - ,
ff

(21)

and in the same manner:

= + e

e\ cos -

z

ii

sin

-
,

(7

a

(22)

where e\ is the initial value of transient voltage, ii the initial value

of transient current.

B. Power dissipation.
A 27. In Fig. 32 are plotted as the periodic component of the B oscillating current i, and as the voltage e, as C the stored mag-

^ D Li 2

netic energy

z

and as
,

Ce 2 the stored dielectric energy z

As seen, the stored magnetic energy pulsates, with double frequency, 2/, between zero and a maximum, equal to the total stored energy. The average value of the stored magnetic energy thus is one-half of the total stored energy, and the dissipation of magnetic energy thus occurs at half the rate at which it would occur if all the energy were magnetic energy; that is, the transient resulting from the power dissipation of the magnetic energy lasts twice as long as it would if all the stored energy were magnetic, or in other words, if the transient were a single (magnetic) energy

DOUBLE-ENERGY TRANSIENTS.

67

transient. In the latter case, the duration of the transient would be

and with only half the energy magnetic, the duration thus is

twice as long, or

7\ = 2T

=2T ^=,

(23)

and hereby the factor

multiplies with the values of current and voltage (21) and (22).

/C
Fig. 32. Relation of Magnetic and Dielectric Energy of Transient.
The same applies to the dielectric energy. If all the energy were dielectric, it would be dissipated by a transient of the dura-
tion:
IrpVf --k;

68 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

as only half the energy is dielectric, the dissipation is half as rapid, that is, the dielectric transient has the duration

T2 = 2 T ' =

,

y

and therefore adds the factor

(24)

to the equations (21) and (22). While these equations (21) and (22) constitute the periodic
part of the phenomenon, the part which represents the dissipation of power is given by the factor

hk =

_JL _JL
T^ T,=

t(\T^+TJ\ t

(25)

The duration of the double-energy transient, T, thus is given by

I..! !_
T IV 2V

1/1

I

(26)

and this is the harmonic mean of the duration of the single-energy
magnetic and the single-energy dielectric transient.
T It is, by substituting for and TV,

where u is the abbreviation for the reciprocal of the duration of
the double-energy transient. Usually, the dissipation exponent of the double-energy transient

is given as

r
2L'

= This is correct only if g 0, that is, the conductance, which rep-
resents the power dissipation resultant from the voltage (by leakage, dielectric induction and dielectric hysteresis, corona, etc.), is negligible. Such is the case in most power circuits and transmission lines, except at the highest voltages, where corona appears. It is not always the case in underground cables, high-potential

DOUBLE-ENERGY TRANSIENTS.

69

transformers, etc., and is not the case in telegraph or telephone lines, etc. It is very nearly the case if the capacity is due to electrostatic condensers, but not if the capacity is that of electrolytic condensers, aluminum cells, etc.
Combining now the power-dissipation equation (25) as factor
with the equations of periodic energy transfer, (21) and (22), gives the complete equations of the double-energy transient of the circuit containing inductance and capacity:

where

= e=e

t
cos
(7

y Qei

.
sin

-t

>

>

fl-

+ CCS -

z ii sin / '

o-

)

(28)

(29)

a = VLC,

(30)

and ii and e\ are the initial values of the transient current and volt-

age respectively.

As instance are constructed, in Fig. 33, the transients of current

and of voltage of a circuit having the constants :

Inductance, Capacity, Resistance, Conductance,

L = 1.25 mh = 1.25 X 10~ 3 henrys;
C = 2 mf = 2 X lO" 6 "farads; r = 2.5 ohms; g = 0.008 mho,

in the case, that
The initial transient current, The initial transient voltage,

ii = 140 amperes; = e\ 2000 volts.

It is, by the preceding equations:
a = Vie = 5 x io- 5 ,

= = /

Z TTff

3180 cycles per second,

y ZQ = ~ = 25 ohms,

2/o

=

/C
yT

=

0.04

mho,

TO ELECTRIC DISCHARGES, WAVES AND IMPULSES.

Tl =

= = 0.001 sec.

1 millisecond,

2C = 0.0005 sec. = 0.5 millisecond,

1

0.000333 sec. = 0.33 millisecond;

\

\

3000 X15&

/i \
\
-1,0

1
,i\i
\

Milliseconds

Fig. 33.
hence, substituted in equation (28),
i = - 3 <S140cos0.2-80sin0.2Zj,
= + e e- 3< S2000 cos 0.2 1 3500 sin 0.2 },
where the time t is given in milliseconds.

DOUBLE-ENERGY TRANSIENTS.

71

A Fig. 33 gives the periodic components of current and voltage:

= - f
i

140 cos 0.2 1

80 sin 0.2

1,

= + r
e

2000 cos 0.2 1

3500 sin 0.2 1.

Fig. 335 gives

The magnetic-energy transient, The dielectric-energy transient, And the resultant transient,

= h e~',

= k

e~ 2<

,

hk

e~ 3< .

And Fig. 33(7 gives the transient current, i = hki', and the tran-
sient voltage, e like'.

LECTURE VII.

LINE OSCILLATIONS.

28. In a circuit containing inductance and capacity, the tran-

sient consists of a periodic component, by which the stored energy

7" /j'2

f^r />2

surges between magnetic -^- and dielectric
i

and a transient
A,

component, by which the total stored energy decreases.
Considering only the periodic component, the maximum magnetic energy must equal the maximum dielectric energy,

_ Lio 2 Ceo2
"2" ~2~'

where i = maximum transient current, e = maximum transient
voltage.
This gives the relation between eQ and io,

\C- -y /L_ e = V

1

ZQ

'

i-

Q

where ZQ is called the natural impedance or surge impedance, y the natural or surge admittance of the circuit.
As the maximum of current must coincide with the zero of

voltage, and inversely, if the one is represented by the cosine

function, the other is the sine function; hence the periodic com-

ponents of the transient are

= ii

IQ cos (</>

7)

= ei e sin (0

l
7)

where

# = 2ft

(4)

and

' = 27^

(5)

is the frequency of oscillation.

The transient component is

hk = e-*,

(6)

72

LINE OSCILLATIONS.

73

where

hence the total expression of transient current and voltage is
i = loe-^cos (0 - 7) 6 = eoe-^sinfa - 7)

7,

e

,

and

i. Q follow from

the

initial

values

f
e

and

i'

of

the

transient,

at = Oor = 0:

hence

e=

Q sin 7

The preceding equations of the double-energy transient apply to the circuit in which capacity and inductance are massed, as, for instance, the discharge or charge of a condenser through an in-
ductive circuit.
Obviously, no material difference can exist, whether the capacity and the inductance are separately massed, or whether they are intermixed, a piece of inductance and piece of capacity alternating, or uniformly distributed, as in the transmission line, cable, etc.
Thus, the same equations apply to any point of the transmission
line.

A

B

Fig. 34.

However, if (8) are the equations of current and voltage at a
A point of a line, shown diagrammatically in Fig. 34, at any other
point B, at distance I from the point A, the same equations will
apply, but the phase angle 7, and the maximum values eQ and IQ,
may be different.

Thus, if

= - i

c<r ui cos (0

7) )

sin(0 -7) )

nx 1

74 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

are the current and voltage at the point A, this oscillation will

appear at a point B, at distance I from A, at a moment of time
A A later than at by the time of propagation ti from to B, if the A oscillation is traveling from to B- that is, in the equation (11),

instead of t the time (t t\) enters.

B B Or, if the oscillation travels from to A, it is earlier at by the

+ time ti' that is, instead of the time t, the value (t ti) enters the

A equation (11). In general, the oscillation at will appear at B,

B and the oscillation at will appear at A, after the time t\; that

is, both expressions of (11), with (t

+ t\) and with (t

ti), will

occur.

The general form of the line oscillation thus is given by substi-
T tuting (t ti) instead of t into the equations (11), where t\ is the
time of propagation over the distance I.
= If v velocity of propagation of the electric field, which in air,

as with a transmission line, is approximately

X v = 3

10 10

,

(12)

and in a medium of permeability /z and permittivity (specific

capacity) K is

-X 3

1010

v =5 T=^>

VfUJ

.
((13)

and we denote

;;
then

.v '.,.

and if we denote

a-j,

=

ti

al;

ffifil (14)
(15)

co = 27rM

(16)

we get, substituting t =F t\ for Z and =F co for $ into the equation

(11), the equations of the line oscillation:

T - = i

ce~ ut cos (0

co

7)

)

= 6

Z ce- u( sin (</> =F co

7) )

,

,

17

In these equations, is the time angle, and

= 2 7T/Z ^

= co

2 7r/aZ )

(18)

is the space angle, and c is the maximum value of current, ZQC the maximum value of voltage at the point I.

LINE OSCILLATIONS.

75

Resolving the trigonometric expressions of equation (17) into

functions of single angles, we get as equations of current and of

voltage

products of the

transient

e~ ut ,

and of

a

combination of the

trigonometric expressions:

cos sin cos sin

cos co,
cos co, sin co, sin co.

Line oscillations thus can be expressed in two different forms, either as functions of the sum and difference of time angle and distance angle co: (0 co), as in (17); or as products of functions of and functions of co, as in (19). The latter expression usually is more convenient to introduce the terminal conditions in stationary waves, as oscillations and surges; the former is often more convenient to show the relation to traveling waves.
In Figs. 35 and 36 are shown oscillograms of such line oscillations. Fig. 35 gives the oscillation produced by switching 28 miles of 100-kv. line by high-tension switches onto a 2500-kw. step-up transformer in a substation at the end of a 153-mile threephase line; Fig. 36 the oscillation of the same system caused by switching on the low-tension side of the step-up transformer.
29. As seen, the phase of current i and voltage e changes progressively along the line Z, so that at some distance 1Q current and voltage are 360 degrees displaced from their values at the starting point, that is, are again in the same phase. This distance Z is called the wave length, and is the distance which the electric field
= travels during one period to j of the frequency of oscillation.
As current and voltage vary in phase progressively along the line, the effect of inductance and of capacity, as represented by the inductance voltage and capacity current, varies progressively, and the resultant effect of inductance and capacity, that is, the effective inductance and the effective capacity of the circuit, thus are not the sum of the inductances and capacities of all the line elements, but the resultant of the inductances and capacities of all the line elements combined in all phases. That is, the effective inductance and capacity are derived by multiplying the total inductance and total capacity by avg/cos/, that is, by -2

T6 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

LINE OSCILLATIONS.

77

78 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

Instead of L and C, thus enter into the equation of the double-

OT

QH

energy oscillation of the line the values - - and

.

7T

7T

In the same manner, instead of the total resistance r and the

total

conductance

g,

the values

2T - and

-2 -Q

appear.

7T

7T

The values of z , y , u, 0, and co are not changed hereby. The frequency /, however, changes from the value correspond-

VLC ing to the circuit of massed capacity, / = -

.

2 IT

to the value
,

f = 4 Vic *

Thus the frequency of oscillation of a transmission line is

where

(7 = VLC.

(21)

If h is the length of the line, or of that piece of the line over which the oscillation extends, and we denote by

LO, Co, TO, go

(22)

the inductance, capacity, resistance, and conductance per unit

length of line, then

-i /

~\

(23)

that is, the rate of decrease of the transient is independent of the

length of the line, and merely depends on the line constants per

unit length.

It then is

=

o-

Z*ro,

(24)

where

(TO

-*\/JT-JQ\Cs*

fOf^\
\^"/

is a constant of the line construction, but independent of the length of the line.
The frequency then is

/.-rrr-

(26)

LINE OSCILLATIONS.

79

The frequency / depends upon the length Zi of the section of line in which the oscillation occurs. That is, the oscillations occurring in a transmission line or other circuit of distributed capacity have
no definite frequency, but any frequency may occur, depending on
the length of the circuit section which oscillates (provided that this circuit section is short compared with the entire length of the circuit, that is, the frequency high compared with the frequency which the oscillation would have if the entire line oscillates as a

whole).
If Zi is the oscillating line section, the wave length of this oscilla-

tion is four times the length

= Z

4 ZL

(27)

This can be seen as follows:

At any point I of the oscillating line section Zi, the effective

power

Po = avg ei =

(28)

is always zero, since voltage and current are 90 degrees apart. The instantaneous power

= p

ei,

(29)

however, is not zero, but alternately equal amounts of energy flow first one way, then the other way.
Across the ends of the oscillating section, however, no energy can flow, otherwise the oscillation would not be limited to this* section. Thus at the two ends of the section;~the instantaneous power, and thus either e or i, must continuously be zero.
Three cases thus are possible:

= 1. e = 2. i = 3. e

at both ends of Zx ;
at both ends of Zi;
at one end, i = at the other end of Zi.

In the third case, i = at one end, e = at the other end of
the line section Zi, the potential and current distribution in the line section Zi must be as shown in Fig. 37, A, B, C, etc. That is, Zi must be a quarter-wave or an odd multiple thereof.
If Zi is a three-quarters wave, in Fig. 375, at the two points C and
D the power is also zero, that is, Zi consists of three separate and

independent oscillating sections, each of the length ^o ; that is, the

80 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
unit of oscillation is -o5, or also a quarter-wave. The same is the
case in Fig. 37C, etc.
In the case 2, i = at both ends of the line, the current and
voltage distribution are as sketched in Fig. 38, A, B, C, etc. That is, in A, the section li is a half-wave, but the middle, C,
of li is a node or point of zero power, and the oscillating unit again is a quarter-wave. In the same way, in Fig. 385, the section /i consists of 4 quarter-wave units, etc.

Fig. 37.

Fig. 38.

The same applies to case 1, and it thus follows that the wave
length 1 is four times the length of the oscillation l\.
30. Substituting / = 4 li into (26) gives as the frequency of
oscillation

/ = ^r

(30)

However,

if /

=

frequency,

and

v

=

-
,

velocity

of

propagation,

the wave length 1Q is the distance traveled during one period:

= = ^o -* period,

(31)

LINE OSCILLATIONS.

thus is

==

Zo

trfo

^.,

and, substituting (32) into (31), gives

= a

(7

,

or

81
(32) (33) (34)

This gives a very important relation between inductance LO and capacity Co per unit length, and the velocity of propagation.
It allows the calculation of the capacity from the inductance,

^ C =

,

(35)

and inversely. As in complex overhead structures the capacity

usually is difficult to calculate, while the inductance is easily de-
rived, equation (35) is useful in calculating the capacity by means

of the inductance.

This equation (35) also allows the calculation of the mutual capacity, and thereby the static induction between circuits, from the mutual magnetic inductance.

The reverse equation,

(36)

is useful in calculating the inductance of cables from their meas-

ured capacity, and the velocity of propagation equation (13).

31. If li is the length of a line, and its two ends are of different

electrical character, as the one open, the other short-circuited,
and thereby i = at one end, e = at the other end, the oscilla-

tion of this line is a quarter-wave or an odd multiple thereof.

The longest wave which may exist in this circuit has the wave

= = = length Z

4 Zi,

and

therefore

the

period

tQ

cr /o

4 o- /i, that

= is, the frequency /

4A

r.
ooti

This is called the fundamental wave

of oscillation. In addition thereto, all its odd multiples can exist

as higher harmonics, of the respective wave lengths ^ ^ _

the frequencies (2 k

= 1)/ , where k

3 1, 2,

...

and

82 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

If then denotes the time angle and co the distance angle of the
fundamental wave, that is, = 2 TT represents a complete cycle and co = 2ir a complete wave length of the fundamental wave,
the time and distance angles of the higher harmonics are

30, 3 co,

50, 5 co,

70, 7 co, etc.

A complex oscillation, comprising waves of all possible fre-
quencies, thus would have the form

i cos (0 =F co

+ 71)

a3 cos 3 (0 =F co

73)

+ T - + a5 cos 5 (0

co

75)

.

.

.

,

(37)

and the length h of the line then is represented by the angle
= co ~, and the oscillation called a quarter-wave oscillation.

If the two ends of the line h have the same electrical charac-

= = teristics, that is, e

at both ends, or i 0, the longest possible

wave has the length 1 = 2 l\, and the frequency

Jr ~

1

1

T ~" o T'
"

or any multiple (odd or even) thereof. If then and co again represent the time and the distance
angles of the fundamental wave, its harmonics have the respective time and distance angles

20, 2 co, 30, 3 co, 40, 4 co, etc.

A complex oscillation then has the form

a\ cos (0 =F co

+ T 71)

2 cos 2 (0

co

72)

+ - + a3 cos 3 (0 =F co 73)

.

.

.

,

(38)

= and the length l\ of the line is represented by angle coi TT, and the
oscillation is called a half-wave oscillation.
The half-wave oscillation thus contains even as well as odd

harmonics, and thereby may have a wave shape, in which one
half wave differs from the other.

Equations (37) and (38) are of the form of equation (17), but

LINE OSCILLATIONS.

83

usually are more conveniently resolved into the form oi equa-
tion (19).
At extremely high frequencies (2 k I)/, that is, for very large values of k, the successive harmonics are so close together that a very small variation of the line constants causes them to overlap,
and as the line constants are not perfectly constant, but may
vary slightly with the voltage, current, etc., it follows that at very high frequencies the line responds to any frequency, has no definite frequency of oscillation, but oscillations can exist of any frequency, provided this frequency is sufficiently high. Thus in transmission
lines, resonance phenomena can occur only with moderate frequencies, but not with frequencies of hundred thousands or millions of
cycles.
32. L C The line constants r , go, , are given per unit length,
as per cm., mile, 1000 feet, etc.
The most convenient unit of length, when dealing with tran-
sients in circuits of distributed capacity, is the velocity unit v.
That is, choosing as unit of length the distance of propagation
X = in unit time, or 3 10 10 cm. in overhead circuits, this gives v 1,
and therefore
"-

or

GO

1

T

-jj- ; LIQ

1
-ftj-

-L/o

That is, the capacity per unit of length, in velocity measure, is inversely proportional to the inductance. In this velocity unit of length, distances will be represented by X.
Using this unit of length, <7 disappears from the equations of the transient.

This velocity unit of length becomes specially useful if the

transient extends over different circuit sections, of different con-

stants and therefore different wave lengths, as for instance an

overhead line, the underground cable, in which the wave length is

about one-half what it is in the overhead line

=
(K 4)

and

coiled

windings, as the high-potential winding of a transformer, in which

the wave length usually is relatively short. In the velocity

measure of length, the wave length becomes the same throughout

all these circuit sections, and the investigation is thereby greatly

simplified.

84 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

= Substituting O-Q 1 in equations (30) and (31) gives

=

^o

Ao>

(40)

= ^ = CO

O2 7T/X

27rX
;

AO

and the natural impedance of the line then becomes, in velocity

measure,

V r T z

= 4 / LQ = LT
^o

= 1 = 1 = ^O

^o

?T
2/o

T^o

/A1\ (41)

where e = maximum voltage, i = maximum current.
That is, the natural impedance is the inductance, and the natural admittance is the capacity, per velocity unit of length, and is the main characteristic constant of the line.

The equations of the current and voltage of the line oscillation then consist, by (19), of trigonometric terms

cos cos co, sin cos cu, cos sin co, sin <f> sin co,

multiplied

with

the

transient,

e~ ut ,

and

would

thus,

in

the

most

general case, be given by an expression of the form

= + + i

e~ "* I ai cos </> cos co

61 sin cos co

Ci cos sin co

_
e

-f-disin</>sinco|,

^ - ut

^ |

/ fll

cos

cos

w

_|_

sm

^

cos

w

_j_

/
Cl

cos

^

sm

w

+ di sin

sin co j,

and its higher harmonics, that is, terms, with

20,

2 co,

30, 3 co,

40,

4 co,

etc.

In these equations (42), the coefficients a, 6, c, d, a', 6', c', d'
are determined by the terminal conditions of the problem, that is, by the values of i and e at all points of the circuit co, at the