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Digitized by the Internet Archive in 2011 with funding from
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ELECTRIC DISCHAEGES, WAVES AND IMPULSES

& !^ Qraw'MlBook 7ne.
PUBLISHERS OF BOOKS F O R^

Coal Age ^ Electric Railway Journal

Electrical World ^ Engineering News -Record

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Engineering 8 Mining Journal ^ Power

Chemical d> Metallurgical Engineering

Electrical Merchandising

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S7

ELEMENTARY LECTURES

ON

ELECTRIC DISCHARGES, WAYES
AND IMPULSES,
AND
OTHER TRANSIENTS

BY
CHARLES PROTEUS STEINMETZ, A.M., Ph.D.
Past President, American Institute of Electrical Engineers

Second Edition Revised and Enlarged
Third Impression

McGRAW-HILL BOOK COMPANY, Inc. NEW YORK: 370 SEVENTH AVENUE
LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1914

TK
Copyright, 1911, BY THE
McGRAW-HILL BOOK COMPANY

PREFACE TO THE SECOND EDITION.
Since the issue of the first edition, in 1911, our knowledge of
transients has greatly increased, and many of the phenomena, especially those of double energy transients and compound cir-
cuits, have been observed and studied on transmission systems to a considerable extent, and have corroborated the oscillographic
records given in the previous edition.
Considerable work has been done on momentary short circuits of alternators, and the variable component of the self-inductive reactance recognized as a transient reactance resulting from the mutual induction of the armature with the field circuit.
Especially in the field of sustained or continual, and of cumulative oscillations, a large amount of information has been gathered. The practical importance of these continual and cumulative oscillations has been strongly impressed upon operating and designing engineers in recent years, usually in the most disagreeable manner
by the destruction of high power, high voltage transformers. A
chapter on these phenomena has therefore been added in the
second edition.
CHARLES P. STEINMETZ, A.M., Ph.D.
February, 1914-

PREFACE TO THE FIRST EDITION.
In the following I am trying to give a short outline of those
phenomena which have become the most important to the electrical engineer, as on their miderstanding 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 relatively little difficulty, be calculated, and the phenomena occurring in them under normal conditions of 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 phenomena, the electric discharges, 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 permits, 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
vii

Vlll

PREFACE TO THE FIRST EDITION.

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 predetermination.
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 ad-
vanced, 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.

ix

X

CONTENTS.

PAGE 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.

— Lecture III. Single-energy Transients in Continuous-cur-

rent Circuits

19

13. Single-energj^ 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 $, 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 s^^stem. Relation between the transients of the three phases. Starting transient of three-phase magnetic field, and its construction. The
oscillator}^ 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-cicuiit 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.

xi

PAGE
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.

— 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.

XU

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 w^ave length. 30. Relation betw^een inductance, capacity, and frequency of propagation. 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 pow^r 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 w^ave 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 compound circuit.

CONTENTS

xm

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

PAGE

transition point.

— Lecture X. Continual and Cumulative Oscillations

119

43. Continual energy supply to the system as necessary cause, in-

volving frequency transformation. Instance of arcing ground on

transmission hne. Recurrent and continuous continual oscilla-

tions. Their change and intermediate forms. Oscillograms of dif-

ferent types. Singing arc.
44. Mechanism of energy supply to the continual oscillation by negative energy cycle. Hysteresis cycle of transient arc. Mechanism of energy supply and continuous and cumulative hunting of synchronous machines. Conditions of continual and of cumula-

tive oscillations.
45. Frequency of continual oscillation. lation. Cumulative effect on insulation.

Destructiveness of oscilUnlimited energy supply.

Independence of frequency of continual oscillation from that of

exciting cause.
— Lecture XL Inductance and Capacity of Round Parallel Con-

ductors

128

46. Definition of inductance and of capacity. The magnetic and

the dielectric field. The law of superposition of fields, and its use

for calculation.

47. Calculation of inductance of two parallel round conductors. External magnetic flux and internal magnetic flux. 48. Calculation and discussion of the inductance of two parallel conductors at small distances from each other. Approximations

and their practical hmitations.

49. 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. 50. Conductor with ground return, inductance, and capacity.
The image conductor. Limitations of its appHcation. Correction

for penetration of return current in ground.
51. Mutual inductance between circuits. Calculation of equation, and approximation.

52. Mutual capacity between circuits. Symmetrical circuits and asymmetrical circuits. Grounded circuit. 53. The three-phase circuit. Inductance and capacity of two-

wire 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

wath three-phase circuit.

ELEMENTARY LECTURES ON ELECTRIC 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

f

oo,o o
.
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 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

"A ELECTRIC DISCHARGES, WAVES AND IMPULSES.
A condenser C is charged, the current in the Hne 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.
]C
Fig. 2.
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 mechanical power, and supplied to the line A. In the line A 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 phenomenon. While electric power flows over the line A, 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 surrounding 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 sj^stem. If we now close the switch, before the
permanent condition corresponding to the closed switch can occur,
A

GO

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 condition 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.

5

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.
(6) The theoretical side of electrical engineering is further advanced than the theoretical side of most other sciences, and
especiallj^
(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 current, 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, 12, it would traverse a curve i', which is the same as curve
i, except that all values are changed proportionally, by the ratio

-; that is, i^= iX^-

ii'

^l

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

starting with i^, the transient follows the proportional curve i\

At some time, t, however, the current i has dropped to the value 12,

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 i\ at the moment h; that is, from t
onward, curve i is the same as curve i' from time ti onward. Since

— Fig. 4. Curve of Simple Transient: Decay of Current.

i^ is proportional to i, from any point t onward curve i is proportional to the same curve i from ti onward. At time ti, it is

_ di2 dii

12

dti

dti

ii

But

since -i^ and
dti

4

at

h

are

the

same

as -r and
dt

i

at

time

t, '

it

follows

_ di dii i

dt

dti ii

or.
— = di ct,
dt

^ where

c

=

—

1
^

= dii

constant,

and

the

minus

sign

is

chosen,

as

vi CIL

di . IS negative.
dt

As in Fig. 4:

tan (/)

dii 'di'

aA = ii,

1 dii

tan </>

1

ii dt

aiti

tlt2

:

NATURE AND ORIGIN OF TRANSIENTS.

7

that is, c is the reciprocal of the projection T = tih on the zero hne of the tangent at the starting moment of the transient.

Since

c = ^5

— = —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

= — d log i

-\- C,

i = Ae-''^

^^>

i = Ae~T)

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;

= din
it is -t-

cm,

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

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

— Fig. 5. Deceleration of Ship.

deceleration. Thus also in a low-voltage electric circuit of negli-

gible 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 gravita-

tion. This energy changes to

kinetic mechanical energy until

in the lowest position, a, when

all the potential gravitational

energy has been either con-

verted to kinetic mechanical

energy or dissipated. Then,

during the rise of the weight,

that part of the energy which

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.

9

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 m_omentum of the motor.
6. The study and calculation of the permanent phenomena in electric circuits arc usually far simpler than are the study and calculation of transient phenomena. However, only the phenomena 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 phenomena, 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 function 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, commutation, 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 M.

While power flows through

the conductors A, power is con-

sumed in these conductors by

JV[ conversion into heat, repre-

sented by ^2r. 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 $. 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 ^. 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 proportional to the current, i, with a proportionality factor, L, which
is called the inductance of the circuit.

$ = L^.*

(1)

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

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

Since power is current times voltage

= P

e'i,

(2)

* n^, if the flux <l> interlinks the circuit n fold.

:

12 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
to produce the magnetic field $ of the current i, a voltage 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 $. This voltage e' is called the inductance voltage, or voltage consumed hy self-induction.
Since no power is required to maintain the field, but power is required to produce it, the inductance voltage must be proportional to the rate of increase of the magnetic field

or by (1),

L§ = e'

(4)

di
If i and therefore $ decrease, j- 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 = j Li di;

hence

^=—

(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-, is proportional to the

voltage e, with a proportionality factor, C, which is called the

capacity of the circuit:

^ = 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 ^¥ of the voltage e, a current i^

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

THE ELECTRIC FIELD.

V6

the power p, which suppHes 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 rate of increase of the dielectric field:

y • =

or by (6),

i' = C*

(9)

^ If e and therefore

de
decrease, -^ and therefore i' are negative;

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 = fpdt, w = I Cede;
^ = «>

(10) (11)

is the energy of the dielectric field
^ = 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 electricitj^, 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 b}^ the lines of magnetic force, the termi-
nology 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 inagnetic 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,

F

f=j

(13)

is called the magnetizing force, or w.agnetic gradient, and is expressed in ampere tunis 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 = 4x/ 10~^ lines of magnetic force
per square cm. section of the magnetic circuit. (Here the factor 10"^ results from the ampere being 10~^ of the absolute or cgs.

unit of current.)

.3C- 4 7r/10-^*

(14)

* The factor 4 7r 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^, produced by the magnetizing force of / ampere turns per cm. in emptj' space.
The jnagnetic density, in lines of magnetic force per cm^, produced b}^ the field intensity 5C in any material is

(^ = M^,

(15)

where /x is a constant of the material, a " magnetic conductivity,"
and is called the penneahility . ^t = 1 or very nearly so for most
materials, with the exception of very few, the so-called magnetic materials : iron, cobalt, nickel, and some alloj^s and oxides of these metals and of manganese and chromium.
A If then is the section of the magnetic circuit, the total magnetic

flux is

= ^

A(S>.

(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
dF

In (16), the magnetic flux <l> 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

= j-

(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 4 7r lines of force, and this introduced the factor 4 w into many magnetic quantities. An attempt to drop this factor 4 x has failed, as the magnetic units were already too well
established.
The factor 10-^ 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 ununiform field, as that between two conductors, varies, being higher at the denser, and lower at the less dense, portion of the field, and is

47r

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

(20)

would be the dielectric density, where /c 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

where

W v = SX

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,

K = j^; 4 Try-

(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, except in materials of

appreciable electric conductivity.

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 7r in the magnetic-field intensity 3C, due to the relations between these
quantities.

:

THE ELECTRIC FIELD.

17

TABLE I.

Magnetic Field.

Dielectric Field.

Magnetic flux:
$ = Li 10^ lines of magnetic force.
Inductance voltage:

Dielectric flux:
^ = Ce lines of dielectric force, or
coulombs. Capacity current:

= e' n^-; 10-^ = L-7- volts.

(It

dt

'^Tt=^dt^'^'^'''^''

Magnetic energy:

ID

=

Li^, -^joules.

Dielectric energy:
w = -^joules.

Magnetomotive force:
F = ni ampere turns.

Magnetizing force:

/

=

F
-y

ampere

turns

per

cm.

Electromotive force:
e = volts.
Electrifying force or voltage gradient:
G = volts per cm. J

Magnetic-field intensity:
JC = 47r/10-^ lines of magnetic
force per cm^.
Magnetic density:
CB = mJC lines of magnetic force
per cm^.
Permeability: n
Magnetic flux:
$ = A(^ lines of magnetic force.

Dielectric-field intensity:
K —rt = - r 10^ lines of dielectric 47rz;2
force per cm^, or coulombs per cm^. Dielectric density
D = kK lines of dielectric force
per cm^, or coulombs per cm^. Permittivity or specific capacity: k

Dielectric flux:

^ AD =

lines of dielectric force,

or coulombs.

3X V =

10^0 = velocity of light.

The powers of 10, which appear in some expressions, are reduction factors between the absolute or cgs. units which are used for $, 3C, (B, and the practical electrical units, used for other
constants.
As the magnetic field and the dielectric field also can be considered as the magnetic circuit and the dielectric circuit, some analogy exists between them and the electric circuit, and in Table II the corresponding terms of the magnetic circuit, the dielectric circuit, and the electric circuit are given.

:

:

18 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

TABLE II.

Magnetic Circuit.

Dielectric Circuit.

Electric Circuit.

Magnetic flux (magnetic

current):

= <J>

lines

of

magnetic

force.

Magnetomotive force:
F = ni ampere turns.

Permeance:

Dielectric flux (dielectric current):
^ = lines of dielectric
force.
Electromotive force:
e = volts.

Electric current:
i = electric cur-
rent. Voltage:
e = volts.

Inductance:

71"$

n^

Permittance or capacity: Conductance:

C = — farads.
e

a = - mhos.
e

henry.

Reluctance:

(Elastance)

Resistance:

R = ^.
$

1

e

C ^'

r = - ohms.
I

Magnetic energy:
iy=L-i^2 =-F^^1,0^~^,.joul, es.

Dielectric energy:

Ce^

e^ .

,

^ = 2 ~ 2 J°"^^^-

Electric power:
p = ri^ = ge^ = ei
watts.

Magnetic density:

Dielectric density:

Electric-current densitj':

(B = A-7=MXlinespercm2.

D = A-r = KK lines per cm2.

I = -r = yG am-
perespercm^.

Magnetizing force:

/

=

F y

ampere

turns

per

cm.

Dielectric gradient:
G = volts per cm.
J

Electric gradient:
G = -, volts per cm.

Magnetic-field intensity: Dielectric-field sity:

inten-

JC = Airf.

K = 7^109.

Permeability:
'^ ac Reluctivity:
P^ =^(-B
Specific magnetic energy

Permittivity or specific Conductivity:

capacity:

D

y

=

I
P

mh,o-cm.

(Elastivity ?):
1 _^
K D'
Specific dielectric energy

Resistivity:

p

=

1 -

=

G,
-rOhm-cm.

7/

Specific power:

— ^^5 10~^ joules per cm^
OTT

2 ttv^KD joules per cm^.

Po = pP = G' = GI
watts per cm\

;

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,
currents may be very consider-
able, so that in spite of the very

and the

momentary

discharge
%

small inductance, considerable
magnetic-energy storage may oc-
cur; that is, the system is one storing energy in two forms, and oscillations appear, as in the dis-

t

1

%

A

1
io

C ..1.
c
c

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

— Fig. 10. Magnetic Single-energy
Transient.

a continuous voltage eo 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'

Lio
interlinks with the coil. Assuming

now that the voltage eo 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 $0, 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 $0 up to the moment of

I
*o

A

^^"^^-5
1

B io K-L__

1

1

^0

C />:^^^

^

'

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

time where the short circuit is applied, as indicated by the dotted line ^0. From ^0 on the magnetic flux decreases, as shown by curve

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

latter must follow a curve proportional to $, as shown in Fig. 115.

The impressed voltage is shown in Fig. IIC as a dotted line; ii is

Co up to to, and drops to at ^o- However, since after ^0 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

In the first moment of short circuit, the magnetic flux <J> still
-J-.

has full value $o, and the current i thus also full value U. Hence,
at the first moment of short circuit, the induced e.m.f. e must be equal to eo, that is, the magnetic flux $ must begin to decrease at such rate as to induce full voltage eo, as shown in Fig. IIC

The three curves $, ^, and e are proportional to each other, and as e is proportional to the rate of change of $, $ 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:

= $Qe-c(i-io)
(J)

= I

toe

(1)

eo€-^('-^«).

$, 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 7i times the decrease of the magnetic flux, and the total induced voltage, that is, the area of the induced-voltage curve. Fig. IIC, thus equals n times the total decrease of magnetic flux, that is, equals the initial current io times the inductance L:

Zet = n$olO-« = Lio.

(2)

Whatever, therefore, may be the rate of decrease, or the shape
of the curves of $, i, and e, the total area of the voltage curve must
be the same, and equal to ?i$o = Liq.
If then the current i would continue to decrease at its initial rate, as shown dotted in Fig. IIB (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 eo up to the moment = of time t to -\- T, where the current has fallen to zero, as

22 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

shown dotted in Fig. llC The area of this new voltage curve
would be CqT, 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

Se^ = eoT,

I
= rioT,

(3)

I

and, combining (2) and (3), I'o cancels, and we get the value of T:

T = -.

(4)

r

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 7" = - 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.

23

by the numerical values of the exponential function, = y (T',
given in Table III.
TABLE III.

Exponential Transient of Initial Value 1 and Duration 1.

y = e-^.

e = 2.71828.

X

y

X

y

1.000

1.0

0.368

0.05

0.951

1.2

0.301

0.1

0.905

1.4

0.247

0.15

0.860

1.6

0.202

0.2

0.819

1.8

0.165

0.25

0.779

2.0

0.135

0.3

0.741

2.5

0.082

0.35

0.705

3.0

0.050

0.4

0.670

3.5

0.030

0.45

0.638

4.0

0.018

0.5

0.607

4.5

0.011

0.6

0.549

5.0

0.007

0.7

0.497

6.0

0.002

0.8

0.449

7.0

0.001

0.9

0.407

8.0

0.000

1.0

0.368

As seen in Lecture I, the coefficient of the exponent of the
T single-energy transient, c, is equal to yp, where is the projection
of the tangent at the starting moment of the transient, as shown in Fig. 11, and by equation (4) was found equal to -. That is,
1

and the equations of the single-energy magnetic transient, (1),
thus may be written in the forms:

t-tn

= = cE> $oe~'^^'~'"^ *o€

= $ -I,.

= = - I'oe- " ^' ^"^

t-tn
T
i'oe

-£<'--k)
ioe

(5)

5

= e eoe""^^'"'"^

t-tp
e^e T

-£"--k)
eoe

Usually, the starting moment of the transient is chosen as the zero of time, ^o = 0, and equations (5) then assume the simpler
form

24 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

= = ^ $oe - ct $oe T = $oe L

t

rt

= = = I

-
ioe-

ct

T
loe

loe 'L

(6)

t

rt

= = = e

e^e-''^

eoe ^

e^e ^.

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

Lf + n = 0,

(7)

n where

L di -7-

is

the

inductance

voltage,

the resistance voltage,

and their sum equals zero, as the coil is short-circuited.
Equation (7) transposed gives

hence

— — di =
I

r
LJ dt,

= — logi

T
J t -\- logC,

Ce-''\

= = = C and, as for t 0: i io, it is:

to; hence

t

=

-£'
iQe

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 dissi-
pates normally 500 watts in the field excitation.
= = That is, if io field-exciting current, n number of field turns per pole, r = resistance, and L = inductance of the field-exciting

circuit, it is

ZoV = 500,

hence

^500

'

^

io'

SINGLE-ENERGY TRANSIENTS.

25

X The magnetic flux is $o = 8 10^, and with 4 7z total turns
the total number of magnetic interlinkages thus is
X = 4 n<E>o 32 n 10^

hence the inductance

— — Lr

=

4n$olO-5
:

=

.32 n, : henrys.

^o

to

The field excitation is

hence

m'o = 6000 ampere turns,
n = 6000
^"0

hence and

X Lr = .32

6000
r^

h^enrys,

^0

^5 7 = ^ =

= 3.84 sec.

r 500

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

— Fig. 12. Magnetic Single-energy Transient.
coil is not short-circuited upon itself, but its circuit closed by a resistance r'. Such would, for instance, be the case in Fig. 12, when opening the switch S.

^6 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 ^o at the moment ^o 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.

*o
A

^\$

io
B
\
/
^0

C

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

The impressed voltage eo is withdrawn at the moment ^o, 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, U, the current is still ^o, and the induced voltage

thus must be

= + eo'

io (r

/),

while the impressed voltage, before to, was

= Co

lor;

hence the induced voltage eo' is greater than the impressed volt-

age Co, 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

+ ^ eo^ r r\

eo

r

SINGLE-ENERGY TRANSIENTS.

27

The duration of the transient now is

r -\- r

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

thereby the induced voltage is higher.
= If r' 00 , that is, no resistance is in shunt to the coil, but the

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

would be :

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.

1 6. 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. ei produces a current ^l in a circuit, and an e.m.f. 62
+ produces in the same circuit a current 2*2, then the e.m.f. ei 62 + produces the current ii ^2, as is obvious.
+ + If now the voltage ei ^2, and thus also the current ii ii, con-
sists of a permanent term, ei and ^l, and a transient term, 62 and ^2,
the transient terms 62, ^"2 follow the same curves, when combined
with the permanent terms ei, ii, 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.

5i« 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.

._

j.

" i
A

[

-^^ ^^^—^^^

''\^-^-^
B

\

^,''

'°

C

i

j

^^^'"^—
!

— 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 eo be impressed upon the coil. At

the moment of time, U^ 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 io, <I>o, eo are reached.

may

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

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

current io, permanent flux fpo, permanent voltage Co, and in addi-

SINGLE-ENERGY TRANSIENTS.

29

— tion thereto a transient current —io, a transient flux $o, and a
transient voltage —eo. 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 2*0, $0, eo. This is shown
in Fig. 14.
A shows the permanent flux ^0, and the transient flux — $0,
which are assumed, up to the time ^o, to give the resultant zero flux. The transient flux dies out by the curve $', in accordance with Fig. 11. $' added to $0 gives the curve ^, which is the transient from zero flux to the permanent flux $o-
In the same manner B shows the construction of the actual
current change i by the addition of the permanent current ^o and the transient current i\ which starts from —io at ^o-
C then shows the voltage relation: eo the permanent voltage, e'
the transient voltage which starts from —eo at ^0, and e the resultant or effective voltage in the coil, derived by adding eo 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 transition 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 ii current existing in the circuit immediately before, and thus at the moment of the change of circuit condition, and 12 = current which should exist at the moment of change in accordance mth the circuit condition after the change, then the actual current
ii can be considered as consisting of a part or component 12, and a
— component ii = 12 iq. The former, 2*2, is permanent, as result-
ing from the established circuit condition. The current component U, 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 12 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
i'l, in Fig. 15 A, the conditions be changed, at the moment t = 0,
so as to produce the current iV The instantaneous value of the
current ii at the moment t = can be considered as consisting
of the instantaneous value of the permanent current (2, shown
— dotted, and the transient io = i\ 22. The latter gradually dies

down,

with

the

duration

T

=

— ,

on

the

usual

exponential

tran-

30

I SINGLE-ENERGY TRANSIENTS.

31

sient, shown dotted in Fig. 15. Adding the transient current io to the permanent current 22 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 ii which exists, and that of the current 2*2 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 2*2 are equal, as shown in Fig. 155, and is a maximum, if the change occurs at the moment when the two currents ii and 12 have the greatest difference, as shown in Fig. 15C, that is, at a
point one-quarter period or 90 degrees distant from the intersection of ii and 12.

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. 165), 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.
— 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 transients and 2*1°, {2°, 13° of the three three-phase currents ii, 22, i^ 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, 2*2
and 23, are equal and opposite, and near their maximum value. Starting, in Fig. 17 C, at the maximum value of one current 23, we have the maximum value of transient for this current 2*3^ while
the transients of the two other currents, 21 and 2*2, are equal, have

SINGLE-ENERGY TRANSIENTS.

33

half the value of iz^, 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, iV, and is are the instantaneous values of the permanent three-phase currents, in Fig.
— 17, the initial values of their transients are: —ii, zV, —4'.

.1

—

'>?jQc>r>:

.1

X

''

i,

'

— 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^, 12^, 4° 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

84 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, iV, 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

OAo in Fig. 18. The three transient currents in this moment

are ii^ =—ii, = 12^ —ii', iz^ =—iz, and combined these give a

resultant field OBo, equal and opposite to OAq in Fig. 18. The

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

since all three transient components of current decrease in propor-

tion to each other. It decreases, however, with the decrease of

the transient current, that is, shrinks together on the line BqO.

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

manent fields, shown as OAi

OA2,

•

.

.

and the transient
,

fields,

shown as OBi, OB2, etc., and derived thereby by the parallelo-

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

BiCi, B2C2, etc., are equal to OAi, OA2, 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 BqO toward 0, and can be constructed hereby by drawing from the successive points Bo, Bi, B2, which correspond to successive moments of

time 0, h, t2 . . . , radii BiCi, B2C2, etc., under the angles, that is, in the direction corresponding to the time 0, ^1, 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.

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 oscillations ; that is, it first reaches beyond the permanent value, then drops below it, rises again beyond it, etc.

4 cycles
— Fig, 20. Starting Transient of Rotating Field: Rectangular Form.

"^e 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 rotating-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 sjaichronous 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 machines 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 fairly
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 shov/n 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 Xii 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 armature 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 X2

a;o

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 arma-

ture reaction thus does not appear instantaneously with the arma-

ture 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 ^l = , where eo 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 Xq, and the short-circuit cur-

— = rent decreases to the value ?"o

•

The ratio of the momentary short-circuit current to the perma-

nent

short-circuit

current thus is, approximately, the ratio

-^

=

— y

to 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: •
Xi

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 Xi

= 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 $o, as indicated by the curve $ in Fig. 21A.

m = If

ratio

armature reaction plus self-induction _ Xq

armature self-induction

Xi'

= it is $1 m$o, and the initial value of the field flux consists of the permanent part $o, and the transient part ^' = $i — $o = (m— 1)
$0- This is a rather slow transient, frequently of a duration of a second or more.
The armature currents ii, 12, iz are proportional to the held 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 $0, or times, and are represented in Fig. 2\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. 21C
The field-exciting current is 2*0 at open circuit as well as in the permanent condition of short circuit. In the permanent condition of short circuit, the field current 2*0 combines with the armature reaction Fq, which is demagnetizing, to a resultant m.m.f., which produces the short-circuit flux $o- During the transition period the held flux $ is higher than $0, and the resultant m.m.f. must therefore be higher in the same proportion. Since it is the difference between the field current and the armature reaction F, and the latter is proportional to <!>, 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 ?; io at $0, during the transition

= — period it is i

Iq. Hence, the field-exciting current traverses

$0

the same transient, from an initial value iq to the normal value U,

as the field flux ^ and the armature currents.

'^l

^ ^'^^ A ^'
*0

%

To ^

— 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 proportion m, from Iq to Iq , and maintain it at the values corresponding to the transient ?*, Fig. 2 ID.
As seen, the transients ^I^; I'l, 2*2, H', 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 eo, were short-circuited upon itself, in the first moment the current in the field would still be ^o, and therefore the voltage eo 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 To =

•

To

The field current in Fig. 21 D, of the alternator short-circuit
= transient, starts with the value io^ rnio, and if eo 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 iq', the
= voltage eo' 7?ieo must be acting in the field-exciting circuit; that — is, in addition to the constant exciter voltage eo, a voltage (?7^ 1) eo

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

— netic flux. As a transient of duration

induces the voltage eo,

to induce the voltage (m — l)eo the duration of the transient must

be

T—

-^0

— (m 1) To

where Lq = inductance, Tq = 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 exaggerated 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 circuit 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, ii, iV, 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, iV, —is. The resultant armature
currents are derived by the addition of these armature transients upon the permanent armature currents, in the manner as discussed in paragraph 18, except that in the present case even the permanent armature currents ii, 4, 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 ii should be maximum ; ^l then shows the maximum transient, and
12 and iz transients in opposite direction, of half amplitude. These armature transients rapidly disappear, and the three currents become symmetrical, and gradually decrease with the field transient to the final value indicated in the figure.
The resultant m.m.f. of three three-phase currents, or the armature 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 C 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. 22B, their resultant polyphase m.m.f. also shows a transient, the transient of the rotating magnetic field discussed in paragraph 18. That is,
it approaches the curve F of Fig. 21 C by a series of oscillations,
as indicated in Fig. 21E. Since the resultant m.m.f. of the machine, which produces the
flux, is the difference of the field excitation. Fig. 2 ID 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 j&eld current, as shown in Fig. 22C, 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. 22C, 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

— = {7n 1) ro

1 sec,

the duration of the armature transient is

=- T

= A sec.

r

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-
— m inductive reactance, = = 6. The frequency is 25 cycles.

If $1 is the initial or open-circuit flux of the machine, the short-

^1 1

= — = circuit flux is $o

;^ ^i, and the field transient <l> is a tran-

mo

sient of duration 1 sec, connecting $i and $o, Fig. 22A, repre-

sented by the expression

_±

= + — «J>

$0

($1 ^o)e ^o.

The permanent armature currents ii, i^, is then are currents
m starting with the values —, and decreasing to the final short-
Xo
— circuit current , on the field transient of duration To. To these
Xo
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 asymmetrical resultant currents. Fig. 225.
The field current i gives the same slow transient as the flux <J>,
= starting with ^o' mio, and tapering to the final value io. 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 (^ z'o),
since I'o is the permanent component) then are increased or decreased by 80 per cent of the percentage variation of the transient field of armature reaction from uniformity, and thereby the field curve. Fig. 22(7, 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

i:6 ELECTRIC DISCHARGES, WAVES AND IMPULSES. is the field transient $ (the same as in Fig. 22 A) and B 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 pulsa-
tion of the polyphase machine. The transient field current thus pulsates by 60 per cent around the slow field transient, as shown b}^ Fig. 23 C; 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-frequenc}^ 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 con-
tains the frequencies /(2 ±1), that is, full frequency and triple

48 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
o ;±5
ao
51
CO

O bJD .a a
bO
go
oS
U

^
I I
It ^^

faC

^

A

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 2QB 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. 265. CD9397. Asymmetrical.
Momentary Three-phase Short Circuit of 1500-Kw. 2300-Volt Three-phase Alternator (atb-4-1500-1800). Oscillograms of Armatm-e 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. 26
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, TrAT'^>S AND IMPULSES.
rent wave, therefore, is s}^nmetrical, 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. (Circuit breaker opens.)
— — Fig. 28. cd656o. 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 oOOO-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 full-frequency transient pulsation of the rotating magnetic field of armature reaction. The irregular initial decrease of the armature current and the sudden change of its wave shape are due to the transient of the current transformer, through which the armature current was recorded.
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.
While the synchronous reactance Xq can be predetermined with fair accuracy, the self-inductive Xi is not such a definite quantity. It includes a transient component. The armature magnetic circuit is in mutual inductive relation with the field-exciting circuit. At constant alternating current in the armature, the resultant of the armature m.m.f's. and e.m.f's. is constant with regard to the field, and the mutual inductance thus does not come into play. During a transient, however, the armature conditions change, and the self-inductance of the exciting circuit is partly transformed into the armature circuit by the ratio of field turns to armature turns, giving rise to a transient effective component of armature self-induction, which depends on the relative rate of
change of the armature and the field, and thereby is a maximum
in the beginning, and gradually decreases to zero in stationary
conditions. This tends to lower the maximum values of the field
transients and to increase the duration of the armature transients. This effect is materially affected by the amount of resistance and reactance in the exciting circuit outside of the field
winding.
There also exists a mutual inductance between the armature circuits of the three-phase machine, which results in an energy transfer between the phases, during the armature transient.
The instantaneous power of the momentary short-circuit current, and with it the forces acting on driving shaft and prime
mover, are proportional to the short-circuit current, being short-
circuit current times magnetic field flux. The forces exerted be-
— tween the armature conductors which tend to tear and strip — the end windings, etc. are proportional to the square of the
short-circuit current.

LECTURE V.
SINGLE-ENERGY TRA.NSIENT 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, however, 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 w^ith the
current, or the capacity is not constant, but varies with the voltage.
The most important case is that of the ironclad magnetic circuit, 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 consumed 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 transient 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 calculation is not feasible, but the transient has to be calculated by an
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:

a -f- crJv which is usually represented in the form given by Kennelly:

p = ^ = « + <rX;

(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 magnetizability. That is, the magnetic-flux density (B consists of a component X, the field intensity, which is the flux density in space, and
— a component (B' = (B 5C, which is the additional flux density
carried by the iron. (B' is frequently called the " metallic-flux density." With increasing CfC, (B' reaches a finite limiting value, which in iron is about
= (S!,J 20,000 lines per cm^. *

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

hereto, gives

M = c((B^'-(BO,

and, substituting

M

rrp/'

gives

^, ^ CCEJOC'

+ 1

cOC''

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

54 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

or, substituting

= a, -:=r—i

(T,

gives equation (1).

For X =

/O

1

-j

- = = = in equation (1),

- ; for 5C

oo , (B

- ; that is,

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

(X

(J

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

/3

<7ac,

where Q = 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

= (B

c5e.

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

^. * =

+ a,

(3)

where

=
,.

^' is

that part

of

the

flux

which

passes

through

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

material.

Denoting now

Z/i = na 10-^ )

U = nc 10-s, )

^^

where n = number of turns of the electric circuit, which is inter-
linked 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 j- the saturation value of the flux in the iron.

—^ = That is, for i 0,

= Li\ and for 2 = oo , $' = -.

^

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

resulting from the air flux would be

r

r

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,
n^l0-8 + n = 0.

Substituting (3) and differentiating gives

+ + (1 +6^)2 dt

'

nc 10~^ -TT
dt

= ^^ 0,

and, substituting (5) and (6),

hence, separating the variables.

T.di +7W.-^,^^o.

^(1 -^hiY ' i

(7)

The first term is integrated by resolving into partial fractions:

1

^1

h

h__

i{i^uy i + l 6^ ii + uy'

and the integration of differential equation (7) then gives

= = If then, for the time t U, the current is i Iq, these values
substituted in (8) give the integration constant C:

56 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
and, subtracting (8) from (9), gives
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.

Trai isient

Iro iclad Inductive Circuit

t~2 9 ' '

JQ

-m^i'1^^'; ,

4

)

r^^'^ i+.6i '

' i+.6i i

(dotted: t = 1.085 Ig i-.507)

'\^ \
\

-C:^-^
— ""^^-^^ii: ^^=^-^ -.

4

5

Fig. 29.

seconds

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

= r

.3,

a = 4 X 10^

X = c

4

104,

= h

.(),

n = 300.

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 57 O
o s
-1-3
M
s

58 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

This gives

T2 = A.

Assuming io = 10 amperes for ^o = 0, gives from (10) the equa-
tion:

T = - +,\ + + 2.92

{ 9.21 log'^ ,
1

.

.921 log'^ i '

.6?; '

="

i '

^

^

.6?;^

Herein, the logarithms have been reduced to the base 10 by
= division with log^^e .4343.
For comparison is shown, in dotted hne, in Fig. 29, the transient of a circuit containing no iron , and of such constants as to give about the same duration:
t = im^log'H- .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 change 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., apphes, the single-energy transient is a simple exponential

function

_ j_

= y

2/oe ^°,

(1)

where

= 2/0 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 To 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.

To = -,

(2)

r

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

TV - -,

(3)

g

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 = eoe~'^\

(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>

9

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

v = Vo[l-e~'^).

(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 existing 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

2^2^

(7)

W Wm While the total energy

decreases by dissipation,

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. Pulsatio7i 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 Iq = maximum value of 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

cv
'
2

where eo = maximum value of transient voltage.
As it is the same stored energy which alternately appears as magnetic and as dielectric energy, it obviously is

This gives a relation between the maximum value of transient current and the maximum value of transient voltage:

'"

U \Vlc^-

(9)

v/-^ therefore is of the nature of an impedance z^^ and is called
the natural impedance, or the surge im'pedance, of the circuit ; and

\ its reciprocal,

Ic
j

=

Vq,

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,

= y g = eo

^0

^'o^Jo,

fc

= y = ^0

eo Y

eo2/o.

(10) (11)

This relation is very important, as frequently in double-energy

transients one of the quantities eo or io is given, and it is impor-

tant to determine the other.

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

induced by the dissipation of the stored magnetic energy of the
= short-circuit current is eo IoZq.

If one conductor of an ungrounded cable system is grounded,
the maximum momentary current which may flow to ground is iQ = e^yo, where eo = 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 Iq eoyo.
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), Zq is

high and 2/0 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 yo 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 trans-
former 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 Zq and t/o
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.

__

The simple = consideration of the relative values of ^o y 7^ 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 dielectric energy, the transient current i and the transient voltage e successively increase, decrease, and become zero.
The current thus may be represented by

= — i

iocos ((f)

y),

(12)

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

= <l>

2 7rft,

(13)

where / = the frequency of this transfer (which is still undetermined), and y the phase angle at the starting moment of the
transient; that is,

= = ^l ^o cos 7 initial transient current.

(14)

As the current iissi 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

eo sin (0

7),

= — ei

60 sin 7 = initial value of transient voltage.

(15) (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 —eo to +eo, and the condenser charge thus is
2eoC; 2
the current has a maximum value io, thus an average value -io,
IT
and as it flows into the condenser during one-half cycle of the

^, frequency /, that is, during the time

it is

2eoC = ~iQ ^—.,
TV ZJ
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 io,
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

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

^o cos (0

= t)

io cos y cos 4>

io sin y sin

and by (11):

t
^l cos a

— sm to . t

ei

-

eo

(T

= i ii cos
(7

2/0^1 sin-,
(T

(21)

and in the same manner:

+ ei cos -

Zoii sin -,

(7

(7

(22)

where ei 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'^

.

.

Ce^

netic energy

, and as the stored dielectric energy

•

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.
be

In the latter case, the duration of the transient would
To--,

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

twice as longj or

T,=2T, = —,

(23)

r

and hereby the factor

= h

T^ e

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

— Fig. 32. Relat on of Magnetic and Dielectric Energy of Transient.

The same appl ies to the dielectric energy. If all the energy

were dielectric, it would be dissipated by a transient of the dura-

tion:

= rp f
i

^. —f

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 To' =

,

9

and therefore adds the factor

_t_

= /c

e

^'2

(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 = e T^e T,^^ '\t,^tJ^

(25)

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

^ 2 \To To'

(26)

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

T = l{i+iy'^'

(27)

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

-K2+8
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

(a \ .

t

= I e~^^ < Zi cos

sm.

t
-

2/0^1

a

= A + e

sm i
e-^^<ei cos -

,
Zoii

-

i
- I'

M a

L1

(28)

(29)

(7 - VlC,

(30)

and ii and e\ are the initial values of the transient current and voltage 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~^ henrys; C = 2mf = 2 X 10'^ farads; r = 2.5 ohms;
g = 0.008 mho,

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

ii = 140 amperes; = ei 2000 volts.

It is, by the preceding equations:
a = VLC - 5 X 10-5,
— / = - = 3180 cycles per second, Z TTCr

2;o = y ^ = 25 ohms.

yo = \ J = 0.04 mho,

TO ELECTRIC DISCHARGES, WAVES AND IMPULSES.

T,= 2L
r

= 0.001 sec. 1 millisecond,

— T2 = 2(7 = 0.0005 sec. = 0.5 millisecond,

g

T=

1

= 0.000333 sec. = 0.33 millisecond;

Fig. 33.
hence, substituted in equation (28),
{ = e-3^U40cos0.2^ -80 sin 0.2^ J
+ = e e- ^'S 2000 cos 0.2 ^ 3500 sin '.2il,\
where the time t is given in milliseconds.

DOUBLE-ENERGY TRANSIENTS.

71

Fig. 33A gives the periodic components of current and voltage:
= - i' 140 cos 0.2 t 80 sin 0.2 1,
+ = e' 2000 cos 0.2 i 3500 sin 0.2 1.

Fig, 335 gives

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

h = €~\ = k e~^-, hk = e~^K

And Fig. 33 C gives the transient current, i = hki', and the tran= sient voltage, e hke'.

)

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

— — surges between magnetic

and dielectric

, and a transient

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

'^'^e^gy-

Li„^ C"^e^,0

(1)

where Iq = maximum value of transient current, 60 = maximum
value of transient voltage.
This gives the relation between Bq and Iq,

Jl ^^ =

,^ = 1,

(2)

where Zq is called the natural impedance or surge impedance, 2/0 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 components of the transient are

= — ii

Iq cos (0

7)

— ei = eo sm (0 7)

where

= 2 Tft,

(4)

and

is the frequency of oscillation.
The dissipative or " transient " component is
M = €-"', (6) 72

LINE OSCILLATIONS.

T6

where

U ^ u

2

C;

(7)

hence the total expression of transient current and voltage is

= — ^

^oe~ "^ cos (0

7)

= — e

eoe~ ^^ sin (0

7)

(8)

7, eo, and io follow from the initial values e' and i' of the transient,

= 2bt t

or = 0: (t>

= ^

^o cos 7

e' = —eo sin 7

(9)

hence

tan 7

(10)

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.
j J

- ..1-

_

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 eo and Iq, may be different.

Thus, if

I=

c^e -"* cos ((f)
ut

-t) )
-t) !

(11)

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 ^, at a moment of time
A A later than at by the time of propagation U from to B, if the A oscillation is traveling from to B; that is, in the equation (11), — instead of t the time {t ti) enters.
5 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 ^i) 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 ti; that
— + is, both expressions of (11), with {t ^i) and with {t ^i), will

occur.

The general form of the line oscillation thus is given by substi-

tuting {t =F ^i) instead of t into the equations (11), where ^i is the

time of propagation over the distance I.
= li V velocity of propagation of the electric field, which in air,

as with a transmission line, is approximately

V = 3X W,

(12)

and in a medium of permeability fx and permittivity (specific

capacity) k is

v= y=-y

(13)

and we denote

a = -,

(14)

then
and if we denote

= h

at',

= = 2 tt/^i

CO

2 Trfal,

(15) (16)

T we get, substituting t k for t and =F co for (/> into the equation

(11), the equations of the line oscillation:

= - i

ce-"' COS ((/) =F CO

7)

)

.

= — e

2;oCe-"' sin (0 =F co

7) j

In these equations, is the time angle, and

= 4>

27rft

^
>-

= CO

2 irfal )

(18)

= is the space angle, and c C(,e=^"'i is the maximum value of current, ZfjC 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~"', and of a combination of the
trigonometric expressions

cos cf) cos 0)^ sin cos CO, cos cf) sin CO, sin cf) sin co.

(19)

Line oscillations thus can be expressed in two different forms, either as functions of the sum and difference of time angle cf) and distance angle co: (0 =t co), as in (17); or as products of functions of cf) 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 I, so that at some distance Iq current and voltage are 360 degrees displaced from their values at the starting point, that is, are again in the same phase. This distance Zo 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 multipljdng the total
2
inductance and total capacity by avg/cos/, that is, by - •

76 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
00 ^
Is o3
02
f=4

LINE OSCILLATIONS.

77

00 .
.2 ^
03
•3 O
O fl
bC -^^

^

78 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

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

— — energy oscillation of the line the values

and

IT

IT

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

— total conductance g, the values

and —^ appear.

TT

IT

The values of Zq, yo, u, 4>, and co are not changed hereby.

The frequency /, however, changes from the value correspond-

7= ing to the circuit of massed capacity, / = vLC 2 7r

, to the value

4:VLC

Thus the frequency of oscillation of a transmission line is
— / = 7= = T-^

(20)

where

a = 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,ro,go

(22)

the inductance, capacity, resistance, and conductance per unit length of line, then

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

= (T

ll<T0,

(24)

where

o-o = VLoCo

(25)

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

^ = 4170-

(2«)

:

:

LINE OSCILLATIONS.

79

The frequency / depends upon the length ^i of the section of hne 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 li is the oscillating line section, the wave length of this oscilla-
tion is four times the length

= Zo

4/i.

(27)

This can be seen as follows

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

power

Pq = 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. 2 = 3. e

at both ends of k;
at both ends of h;
at one end, ^ = at the other end of k.

In the third case, ^ = at one end, e = at the other end of
the line section k, the potential and current distribution in the line section h must be as shown in Fig. 37, A, B, C, etc. That is, h must be a quarter-wave or an odd multiple thereof.
If h is a three-quarters wave, in Fig. S7B, at the two points C and D the power is also zero, that is, li consists of three separate and

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

:
80 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
unit of oscillation is -^^ 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 k 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. SSB, the section h 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 lo is four times the length of the oscillation k.
30. Substituting U = 4:li into (26) gives as the frequency of

oscillation

A / =

•

Iqcfq

(30)

However,

if /

=

frequency,

and

v

=

1
-,

velocity of propagation,

the wave length U is the distance traveled during one period

= = ^0

-> period,

(31)