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ELEMENTARY LECTURES
ON
ELECTRIC DISCHARGES, WAVES AND IlIPULSES,
AND
OTHER '~ritANSIENTS
BY
CHARLES PROTEUS STEINMETZ, A.M., PH.D.
Pa:st P1·eside1it, American institute of Electrical Engineer3
SECOND ECDITION
R&vt&Eu AND E:in.AROED
McGltAW-HILL BOOK COMPANY, INc.
239 WEST 39TH STREET, NEW YORK
6 BOUVERIE STREET, LONDON, E. C.
1914

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 cumu-

lative oscillations, a large amount of information has been gathered.

The practical importance of these continual and cumulative oscil-

lations has been strongly impressed upon operating and designing

engineers in recen~ 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.

V

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 understanding and control depends the further successful advance of electrical engineering. The art has now so far advanced that the phenomena of the steady flow of power are well understood. Generators, motors, transforming devices, transmission and distribution conductors can, with 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 hav.e 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 Phenor{iena." 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 7'0 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 an<l 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 an<l Calculation of Transient Electric Phenomena and Oscillations." Hereto has been ad<led a chapter on the calculation ·or capacities an<l 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. J\L Davis, lvI.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.

PA.GB

LECTURE I. - NATURE AND ORIGIN OF TRANSIENTS.................

1

1. Electric power and energy. Permanent and transient phenomena. Instance of permanent phenomenon; of transient; of combination of both. Trmisient 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 transient. 3. Transients existing with all forms of energy: transients of railway car; of fan motor; of incandescent lamp. Destructive values. High-speed water-power governing. Fundamental condition of transient. Electric transients simpler, their theory further advanced, of more direct industrial importance. 4. Silllplcst 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 clnsses 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 conduct.or; 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 i:etensity, and magnetic density.. Permeability. Magnetic materials.
ix

X

CONTENTS.

PA.GI:
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. - S1NGLE·ENERGY TRANSIENTS IN CONTINUOUS-CUR•

RENT C1ncurrs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • •

19

13. Single-energy transient represents increase or decrease of energy. Magnetic transients of low- and medium-voltage circuits. Single-energy and double-energy transients of capacity. Discussion of the transients of 'P, 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 1:;quation of the magnetic transient. Simplification by choosing the starting moment as zero of time.

14. Instance of the magnetic transient of a motor field. Calculation 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. 10. 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
Ii. 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; maximum and zero value. 18. The starting transient of the balanced three-phase system. Relation between the transients of the three phases. Starting transient of three-phase magnetic field, and its construction. The oscillatory start of the rotating field. Its independence of the phase at the moment of start. Maximum value of rotating-field transient, and its industrial bearing. 19. Momentary short-circuit current of synchronous alternator, and current rush in its fiel<l circuit. Relation between voltage. load, magnetic fiel<l flux, nrmature reaction, self-inductive reactance, and synchronous reactance of alternator. Ratio of momentary to permanent short-cicurit current. 20. The magnetic field transient at short circuit of alternator. I ts 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

..

Xll

CONTENTS.

PA.GE
Modification for distributed capacity and inductance: the distance phase angle and the velocity of propagation; the time phase angle; the two forms of the equation of the line oscillation. 29. Effective inductance and effective capacity, and the frequency of the line oscillation. The wave length. The oscillating-line section as quarter wave length. 30. Relation between inductance, capacity, aud frequency of propagation. Importance of this relation for calculation of line constants. 31. The different frequencies antl 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.

LECTUHE VIII. - TRAVELING WAVES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
33. The power of the stationary oscillation and its correspondence with reactive power of alternating currents. The traveling wave and its correspondence with effective power of alternating currents. Occurrence of traveling waves: the lightning stroke. The traveling wave of the compound circuit. 34. The flow of transient power and its equation. The powerdissipation constant and the power-transfer constant. Increasing and decreasing power flow in the traveling wave. The general equation of the traveling wave. 35. Positive and negative power-transfer constants. Undamped oscillation and cumulative oscillation. The arc as their source. The alternating-current transmission-line equation as special case of traveling wave of negative power-transfer constant. 36. Coexistence and combination of traveling waves and stationary oscillations. Difference from effective and reactive alternating waves. Industrial importance of traveling waves. Their frequencies. Estimation of their effective frequency if very high. 37. The impulse as travelin~ wave. Its equations. The wave front.

LECTUHE 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

xiii

PAGE
41. Voltage and current relation between the sections of a compound oscillating circuit. The voltage and current transformation :1t the transition points between circuit sections. 42. Change of phase angle at the transition points between sections of a compound oscillating circuit. Partial reflection at the
transition point.

LECTURE X. -CONTINUAL AND Cu11rnLATIVE OscILLATIONS..... .. . . 119

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

volving frequency transformation. Instance of nrcing ground on

transmission line. 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. Mecha-

nism of energy supply and continuous and cumulative hunting of

synchronous machines. Conditions of continual and of cumula-

tive oscillations.

45. Frequency of continual oscillation. Destructiveness of oscil-

lation. Cumulative effect on insulation. Unlimited energy supply.

Independence of frequency of continual oscillation from that of

exciting cause.

•

LECTURE XI. -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 limitations.

49. Calculation of capacity of pnrallel conductors by superposition

of dielectric fields. Reduction to electromagnetic units by the

velocity of light. Relation between inductance, capacity, arid

velocity of propagation.

50. Conductor_ with ground return, inductance, and capacity.

The image conductor. Limitations of its application. 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

with three-phase circuit.

ELE~IENTARY LECTURES ON ELECTRIC DISCHARGES, '\YAVES AND IMPULSES, AND OTHER TRANSIENTS.
LECTURE I.
NATURE AND ORIGIN OF TRANSIENTS.
1. 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 Fig. 1 be a direct-current generator, which over a circuit A connects 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 •
A
G
A
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 permanent, 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 Sis 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 begins to fl.ow over line A into the condenser C, charging' this condenser· u1>'to the voltage given by· the generator. When the
1

2 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
condenser C is charged, the current in the line A and the condenser C is zero again. That is, the permanent condition before closing the switch S, and also some time after the closing of the switch, is zero current in the line. Immediately after the closing of the switch, however, current flows for a more or less short time. ·with the condition of the circuit unchanged: the same generator voltage, the switch S closed on the same circuit, the current nevertheless changes, increasing from zero, at the moment of closing the switch S, to a maximum, and then decreasing again to zero, ·while the condenser charges from zero voltage to the generator voltage. \Ve then here meet a transient phenomenon, in the charge of the condenser from a source of continuous voltage.
A
G
A
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 arid 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 ma. chanical 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 powe~ is used. The consideration of the electric power

4 ELECTRIC DISCHARGES, lVAVES AND IMPULSES.
is the phenomenon by which the circuit readjusts itself to the change of stored energy. It may thus be said that the permanent phen9mena 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 standst.ill 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 wbile 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, whicli 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.

(b) The theoretical side of electrical engineering is further

advanced than the theoretical side of most other sciences, and

especially:

(c) The destructive or harmful effects of transients in electrical

systems are far more common and more serious than with other

forms of energy, and the engineers have therefore been driven by

necessity to their careful and extensive study.

4. The simplest form of transient occurs where the effect is

directly proportional to the cause. This is generally the case in

electric circuits, since voltage, current, magnetic flux, etc., are

proportional to each other, and the electrical transients therefore

are usually of the simplest nature. In those cases, however,

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

inductive circuits containing iron, or in electrostatic fields exceed-

ing the corona voltage, the transients also are far more complex,

and very little work has been done, and very little is known, on

these more complex electrical transients.

Assume that in an electric circuit we have ~ transient cur-

rent, as represented by curve i in Fig. 4; that is, some change of

circuit condition requires a readjustment of the stored energy,

which occurs by the flow of transient current i. This current

starts at the value i1, 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, ¼, it would traverse a curve i', which is the same as curve

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

-¼:- ; that I.S, t., = •t• X -¼:-•

t1

t1

Starting with current i1, the transient follows the curve i;

starting with i2, the transient follows the proportional curve i'.

At some time, t, however, the current i has dropped to the value ¼,

with which the curve i' started. At this moment t, the <;onditions

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 t1; that is, from ( onward, curve i is the same as curve i' from time ti onward. Since

0

t1

t

t 1

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

i' is proportional to i, from a.ny point t onward curve i is proportional to the same curve i from ti onward. At time t1, it is

di2 = di1 X ~-
dt1 dt1 i1
But sm• ce ddit-ii and 1•-2 at t1 are the same as ddti and i• at t·1me t, 1•t

follows:
or,

d-i
dt

=dd-~t'1.1-ii1,

di

.

dt = - ci,

where c = -

1
ii

ddi1t

=

constant,

and

t he minus sign is chosen,

as

ddti

.
1s

negat·1ve.

As in Fig. 4:

tan</;,=d-i-1,
dt

a1t1 = i1,

c =1-d-i1- =tan-<J= , -1,.

i1 dt

a1t1

ti~

NATURE AND ORIGIN OF TRANSIENTS.

7

that is, c is the reciprocal of the projection T = t1½. on the zero line of the tangent ·at the starting moment of the transient.

Since

1
C = T' d--:i- = -cdt;
1,

that is, the percentual change of current is constant, or in other words, in the same time, the current always decreases by the saine fraction of its value, no matter what this value is.
Integrated, this equation gives:
log i = - ct + C,
i = Acct,

or,

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 electrical and other transients. Consider, for instance, the decay of radioactive substances: the radiation, which represents the decay,
1•s proporti•ona1to t11e amount of rad1' at·m~ mater1•a1; 1•t 1•s ddmt = 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 en, •gy, or a change of the stored energy from one form to another. Usually both occur simultaneously. An instance of the first case is the acce! 0 rntion 6r deceleration

8

ELECTRIC DISCHARGES, WAVES AND IMPULSES.

of a train, or a ship, etc.: here energy can be stored only as mechanical momentum, and the transient thus consists of an increase of the stored energy, during acceleration, or of a decrease; during

Seconds
10 20 30 40 60 00 70 80 90 100 11() 120-
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,

-+----t=------------~-t-- . during the rise of the weight,

a
Fig. 6. - Double-energy Transient of Pendulum.

that part of the energy .which 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

· 12 ELECTRIC DISCHARGES, WA YES AND IMPULSES.

to produce the magnetic field qi 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 qi_ This voltage e' is called the inductance voltage, or voltage
consumed by self-induction. Since no power is required to maintain the field, but power is
required to produce it, the inductance voltage must be proportional to the rate of increase of the magnetic field:

, e

=

d-dqt i ,

(3)

or by (1),

e' = L di.
dt

(4)

If i and therefore cf, decrease, : a~d therefore e' are negative;
that is, p becom'es negative, an<l power is returned into the circuit. The energy supplied by the power p is

or by (2) and (4),

w = Jpdt,

hence

w =L2-i.2

(5)

is the energy of the magnetic field

qi = Li

of the circuit.

9. Exactly analogous relations exist in the dielectric field.

The dielectric field, or dielectric ftia, '11', is proportional to the

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

capacity of the circuit:

'V = 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 '11' of the voltage e, a current i'

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

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 'l!. 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, 4', is proportional to the current, i, with a proportionality factor, L, which is called the inductance of the circuit.

cp = Li.*

(1)

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

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

Since power is current times voltage:

= p

eI • i,

(2)

* n<i>, if the flux1<i> interlinks the circuit n fold. I

THE ELECTRIC FIELD.

13

the power p, which supplies the stored energy w of the dielectric field'¥. This current i' is called the:capacity curren< 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:

.,
i

=dd-'Yt,

(8)

or by (6),

i' = Cde.
dt

(9)

If e and therefore 'I' 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

w = JP df,

(10)

or by (7) and (9),

w = J Cede;

hence

w =C2e-2

(11)

is the energy of the dielectric field

'I'= 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 ,vith the same simplicity and perspicuity,

as the magnetic field, by using the same conception of lines of

force.

-

•

Unfortunately, to a large extent in dealing with the dielectric

fields the prehistoric conception of the electrostatic charge on the

conductor still exists, and by its use destroys the analogy between

the two components of the electric field, the magnetic and the

14 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

dielectric, and makes the consideration of dielectric fields un-

necessarily complicated.

There obviously is no more sense in thinking of the capacity

current as current which charges the conductor with a quantity

of electricity, than there is of speaking of the inductance voltage

as charging the conductor with a quantity of magnetism. But

while the latter conception, together with the notion of a quantity

of magnetism, etc., has vanished sinqe Faraday's representation

of the magnetic field by 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 conduct<;>rs, as seen in

Fig. 8, and the magnetic field and the dielectric field thus can be

considered as a magnetic circuit and a dielectric circuit.

To produce a magnetic flux <I>, a magnetomotive force Fis 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 Fis the m.m.f., l the length of the magnetic circuit, energized

byF,

f = FT

(13)

is called the magnetizing force, or magnetic gradient, and is ex-

pressed in ampere turns per cm. (or industrially sometimes in

ampere turns per inch).

In empty space, and therefore also, with very close approxi-

mation, in all nonmagnetic material, f ampere turns per cm. length

of magnetic circuit produce JC= 4 r f 10-t lines of magnetic force

per square cm. section of the magnetic circuit. (Here the factor

10-1 results from the ampere being 10--1 of the absolute or cgs.

unit of current.)

JC= 4rf 10--l *

• (14)

* The factor 4 1r 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 us 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 cm2, produced by the magnetizing force off ampere turns p~r cm. in empty space.
The magnetic density, in lines of magnetic force per cm2, produced by the field intensity :JC in any material is

(15)

whereµ is a constant of the material, a "magnetic conductivity,"

and is called the penneability. µ = 1 or very nearly so for most

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

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

these metals and of manganese and chromium.

If then A is the section of the magnetic circuit, the total ma~netic

flux is

cp = A<B.

(16)

Obviously, if the magnetic field is not uniform, equations (13)

n.nd (16) would be correspondingly modified; f in (13) would be

the average magnetizing force, while the actual magn€tizing force

would vary, being higher at the denser, and lower at the less dense,·

parts of the magnetic circuit:

f = adF1·

(17)

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

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

u. Entirely analogous relations exist in the dielectric circuit.

To produce a dielectric flux'¥, an electromotive force e is required, .

which is measured ~n volts'. The e.m.f. per unit length of the

dielectric .circuit then is called the electrifying force or the voltage

gradient, and is

•

G=er

(18)

unit.distance wit~1 unit force. The unit field intensity, then, was defined as
the field intensity :l.t 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 ,r lines of force, and this introduced the factor 4 ,r into many magnetic quantities. An attempt
to drop this factor 4 ,r has failed, as the magnetic units were already too well
established. The factor 10-1 also appears undesirable, but when the electrical units
were introduced the absolute unit appeared as too large a value of current as practical unit, and one-tenth of it was chosen M unit, and called "ampere.''

16 ELECTRIC DISCHARGES, JVAVES 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

G= de. dl

l(19)

K =4G-'Ir *

then is the dielectric-Jkld intensity, and

D = 1<.K

(20)

would be the dielectric density, where K. is a constant of the material,

the electrostatic or dielectric conductivity, ·and is called the spe-

cific capacity or permittivity.·

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

other gases,

K. = -1, v2

where

= V 3 X 1010

is the velocity of light.

It is customary, however, and convenient, to use the permit-
tivity of empty space as unity: K. = I. This changes the unit of

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

K

G .
= -'l4 rV z'

(21)

dielectric density,

D = K.K,

(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

'l' = 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 ,r appears here in the denominator as the result of the fa.ctor 4 r in the magnetic-field intensity JC, due to the relations between these
quantities.

THE ELECTRIC FIELD.

17

TABLE I.

Magnetic Field.

Dielectric Field.

Magnetic flux:
<I> = Li 108 lines of magnetic force.
Inductance voltage:
e'= n dd<tf! 10-s = L ddti volts.

Dielectric flux:
'1t = Ce lines of dielectric force, or

coulombs. Capacity current:

.,
i

=

dd,tf;

=

Cddet amperes.

Magnetic energy:

w

=

L•·2
2

J•ou1es.

Magnetomotive force:
F = ni ampere turns.

Magnetizing force:
f = yF ampere turns per cm.

Dielectric energy:

w

=

Ce
2

2

J• oules.

Electromotive force:
e = volts.

Elect-rifying force or voltage gra-

dient:

• •

G = Ie volts per cm.

Magnetic-field intensity:
JC = 41rf 10-1 lines of magnetic
force per cm2•
Magnetic density: CS= µJC lines of magnetic force per cm2•
Permeability: µ. Magnetic flux:
,p = ACS lines ·of magnetic force.

Dielectric-field intensity: ·
K = _!]_ 109 lines of dielectric 4111,.2 force per cm2, or coulombs per cm2•
Dielectric density:
D = KK lines of dielectric force
per cm2, or coulombs per cm2• Permittivity or specific capacity: K
Dielectric flux:
'1t = AD lines of dielectric force,
or coulombs.

v = 3 X 1010 = 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 4>, JC, <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.

SINGLE-ENERGY TRANSIENTS.

25

The magnetic flux is cl>0 = 8 X 106, and with 4 n total turns
the total number of magnetic interlinkages thus is
4 ncl>0 = 32 n X 106,

hence the inductance

L = 4 ncl>0. 10-s = -.32.-n henrys.

to

to

The field excitation is

hence hence and

ni0 = 6000 ampere turns,

n =6-0.0-0
io

L

_
-

.32

X 6000
t.o2

h enrys,

T

=

L r

=

1920 500

=

3.84

sec.

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

i:::::ii:::::iiri.i-:::.n r lt:ltl:l:tttll L
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.

LECTURE_ II.

THE ELECTRIC FIELD.

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

line A into a receiving circuit lv[.

While power flows through

A

~ the conductors A, power is con-

sumed in these conductors by

lVI conversion in,to beat, repre-

A

sented by i'lr. 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 'q:i. With a single conductor, the lines of magnetic force are concentric circles, as shown in Fig. 8. By the return conductor, the circles
10

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, GOO volts, an<l 2200 volts. In them the capac-

• ity is small, <lue to the limited extent of the circuit, resulting from

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

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

the magnetic field.

A circuit of considerable capacity, but negligible inductance, if

of high resistance, would also give one form of energy storage only,

in the dielectric field. The usual high-voltage capacity circuit, as

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

also is of very small resistance, and the momentary discharge

currents may be very consider-

able, so that in spite of the very

small inductance, considerable

magnetic-energy storage may occur; that is, the system is one storing energy in two forms, and oscillations appear, as in the dis-

T

e I

A

JO

I

J,

lo

(

)

(

>

r
L

(

)

charge of the Leyden jar.

Fig. 10.-Mngnetie Single-energy

Let, as represented in Fig. 10,

Transient..

a continuous voltage eo be im-

pressed upon a wire coil of resista.nce r and inductance L (but

negligible capacity). A current io = ero flows through the coil and

a magnetic field 'Po 10-s = Lnio. 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 <i> 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 <i>o, by the dissipation of its stored energy in the resistance o( the coil circuit as i 2r. Plotting, therefore, the magnetic flux of the coil as function of the time, in Fig.
llA, the flux i::o~stant anlenoted by <JS, up to the moment of

A

if.,

io :B
' ,I --e;

C

---T---

0

I
t

t

0

Fig. 11. - Charn.cterist.ics of Magnetic Single-energy Transient.

time where the short cir~uit is applied, as indicated by the dotted

line t0• From to on the magnetic flux decreases, as shown by-curve

<i>. Since the magnetic flux is pr~portional to the current, the

latter must follow a curve proportional to <i>, as shown in Fig. 11B.

The impressed voltage is shown in Fig. 11C as a dotted line; it is

e0 up to t 0, and drops to Oat to. However, ~ince after t0 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 <P, and is therefore proportional to the rate of decrease of <P, that is, to
~~. In the first moment of short circuit, the magnetic flux <P still

has full value <1>0, and the current i thus also full value i 0• Hence,

at the first moment of short circuit, the induced e.m.f. e must be

equal to e0, that is, the magnetic flux <P must begin to decrease at

such rate as to induce full voltage eo, as shown in Fig. UC.

The three curves <P, i, an<l e are proportional to each other, and

as e is proportional to the rate of change of <P, <P must be propor-

tional to its own rate of change; and thus also i and e. That is,

the transients of magnetic flux, current, an<l voltage follow the

l law of proportionality, hence are simple exponential functions, as
seen in Lecture I:
= 4> cf>oE - c (t - to),

= i ioe-c<t-lo),

(1)

= e eoe - c <t- lo).

<P, 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 rnte 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 <luring which it is induced thus equals n times the decrease of the magnetic flux, and the total induced voltage, • that is, the area of the induced-voltage curve, Fig. UC, thus equals n times the total decrease of magnetic flux, that is, equals the initial current io times the inductance L:

Iet = n<Polo-s = Lio,

(2)

Whatever, therefore, may be the rate of decrease, or the shape
of the curves of <I>, i, and e, the total area of the voltage curve must
be the same, and equal to n<Po = Lio,
If then the current i would continue to decrease at its initial
rate, as shown dotted in Fig. UB (as could be caused, for instance,
by a gradual increase of the resistance of the coil circuit), the
inducea. voltage would retain its initial value eo up to the moment
of time t = to + T, where the current has fallen to zero, as

SINGLE-ENERGY TRANSIENTS.

27

The duration of the transient now is

T

= r

-+L-r,'

that is, shorter in the same proportion as the resistance, and thereby the induced voltage is higher.
If r' = oo, that is, no resistance is in shunt to the coil, but the
circuit is simply opened, if the opening were instantaneous, it
would be: eo' = oo; 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, an<l the greater the danger of breakdown. 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 circuit opening. The more rapid the mechanical opening of the switch, the higher· the induced voltage, and further, therefore, the arc follows the switch blades and maintains the circuit.
16. Similar transients as discussed above occur when closing a circuit upon an impressed voltage, or changing the voltage, or the current, or the resistance or inductance of the circuit. A discussion 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. e1 produces a current i1 in a circuit, and an e.m.f. e2
produces in the same circuit a current i2, then the e.m.f. e1 + e2 produces the current i1 + i2, as is obvious.
If now the voltage e1 + ~, and thus also the current i1 + i2, con-
sists of a permanent term, e1 and i1, and a transient term, e2 and ¼,
the transient terms e2, i2 follow the same curves, when combined with the permanent terms e1, ii, as they would when alone in the circuit (the case above discussed). Thus, the preceding discussion applies to all magnetic transients, by separating the transient from the permanent term, investigating it separately, and then adding it to the permanent term.

18 • ELECTRIC DISCHARGES, WAVES AND IMPULSES.

TABLE II.

Magnetic Circuit.

Dielectric Circuit.

Electric Circuit.

lVfognetic fiux (magnetic current):
<f> = lines of magnetic
force.
l\fognetomotive force:
F = ni nrnpere turns.
Permeance:

Dielectric flux (dielectric

current):

it = lines of dielectric

force.

•

Electromotive force:
e = volts.

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

ill=_!_· .41rll
Inductance: L= n;:10-s= n:10-s

Permittance or capacity: Conductance:

C = !e farads.

g = :e mhos.

henry. Reluctance:
R= !·

Magnetic energy:

w

=

L
2

i2

=

F,f>
2

1

0-s

joules.

l\fagnetic density:

CB= ,Af> =µJCliuespcrcm 2•

Magnetizing force:
f = TFampere turns per

(Ela.stance):

c 1

=

e
ii.

Dielectric energy:

w

=

Cc 2
2

=

2c'1t.Joules.

Dielectric density:

D =~=,cf( 1ines per cmi.

Dielectric gradient:
G = le volts per cm.

Resistance:
r =~ohms.
t
Electric power:. '
p = ri2 = gc2 = ei •
watts.
Electric-current density:
I= Ai = -yGam-
perespercm2•. Electric gradient:
G = le volts per cm.

cm.

Magnetic-field intensity: Dielectric-field sity:

JC= .41rf.

K=4-1Gr1v2 09•

in ten-

. Permeability:
µ = (B.
JC

Permittivity or specific Conductivity:

capo.city:

,c= KD .

-y = GI mho-cm.

Reluctivity:
p = L. (B

(Elastivity ?):

; l

=

K D.

Resistivity:

p

1 =;=

1Gohm-cm.

Specific mngnetic energy: Specific dielectric energy: Specific power:

,,:,

= = tl)o

.4 ,rµf.:. = f<9., 10-s =
2 •2

= Wo

/((JI.
4- 1d

09

GD -21

0

9

=

cl[$
S1r

10-7

J. oulespercm 3•

2 1rrPKD joules per cm3•

= = = Po pP. (JI. GI
watts per cm3.

SINGLE-ENERGY· TRANSIENTS.

31

sient, shown clotted in Fig. 15. Adding the transient current io to the permanent curre~t ·i2 gives the total current during the transition period; which is shown in drawn line in Fig. 15.
As seen, the transient is clue to the difference between the instantaneous value of the current i1 which exists, and that of the current i2 which should exist at the moment of change, and

Fig. 15. - Single-energy Transient of Alternating-current Circuit.
thus is the larger, the greater the difference between the two currents, the previous and the after current. It thus disappears if the change occurs at the moment when the two currents 1·1 and i2 are equal, as shown in Fig. 15B, and is a maximum, if the change occurs at the moment when the two currents i 1 and iz 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 i1 and i2,

SINGLE-ENERGY. TRANSIENTS.

33

half the value of is0, 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 it', 't2', and is' are the instan-
taneous values of the permanent three-phase currents, in Fig.
17, the initin.I values of their tra.nsients are: -it', - i2', -is'.

A

il
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 i1°, i,i.°, i3° are proportional to each other (as exponential curves of the same dura-
tion T = Lr ), and the sum of their initial values is .zero, it follows

22 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

shown dotted in Fig. 11G. The area of this new voltage curve would be eoT, 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

'Zet = e~T, } = rioT,

(3)

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

T = -Lr·

(4)

That is, the initial decrease of current, and· therefore of mag• netic 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 = !r:. •
Since the curves of current and voltage theoretically never become zero, to get an estimate of the duration of the transient we may determine the time in which the transient decreases to half, or to one.:.tenth, etc., of its initial value. It is preferable, however, to estimate the duration of the transient by the time T, which it would last if maintained at its initial value. That is,

the duration of a transient is considered as the time T = !r:. •

This time T has frequently been called the " time constant "

of the circuit.

The higher the inductance L, the longer the transient lasts,

obviously, since the stored energy which the transient dissipates

is proportional to L.

The higher the resistance r, the shorter is the duration of the

transient, since in the higher resistance the stored energy is more

rapidly dissipated.

.

Using the time constant T = !r:. 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.

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 shortcircuit current, an<l this excess current usually decreases fairly slowly, lasting for many cycles. At the same time, a big current rush occurs in the field. This excess field current shows curious pulsations, of single an<l of double frequency, and in the beginning the armature currents also show unsymmetrical shapes. Sorrie oscillograms of three-phase, quarter-phase, and single-phase short circuits of turboalternators are shown in Figs. 25 to 28.
By considering the transients of energy storage, these rather complex-appearing phenomena can be easily understood, and predetermined from the constants of the machine with reasonable exactness.
In an alternator, the voltage under load is affected by armature reaction and armature self-induction. Under permanent condition, both usually act: in the same way, reducing the voltage at noninductiveand still much more at inductive load, and increasing it at antiinductive load; and both are usually combined in one quantity, the synchronous reactance Xo. In the transients resulting from circuit changes, as short circuits, the self-inductive armature reactonce and t.he magnetic armature reaction act very differently:* the former is instantaneous in its effect, while the latter requires time. The self-inductive armature reactance x1 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 synchronollB operation, in hunting, etc.

24 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

t

rt

<I> = 4>oE- ct = 4>oE-T = 4>oE-1,,

t

rt

i = ioE- ct = ioE- T = ioE- L,

(6)

t

rt

e = eoE-ct = eoE- T == eoE- L.

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

+ . L ddti

0
ri:::::a'

(7)

where L :: is the inductance voltage, ri the resistance voltage:

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

di i = - Lr dt,

hence

L logi = - t + logC,

_!:. t
i = CE L'
and, as for t = 0: i = io, it is: C = io; hence

14. Usually single-energy transients last an appreciable time,

and thereby become of engineering importance, only in highly

inductive circuits, as motor fields, magnets, etc.

To get an idea on the duration of such magnetic transients,

consider a motor field: A 4-polar motor haso: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

io2r = 500,

hence

r--5-io020·

SINGLE-ENERGY TRANSIENTS.

89

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 xo

self-induction

= self-inductivereactance = X1 •

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 = O. The field flux then gradually dies
<lawn, by the dissipation of its energy in the field circuit, to the short-circuit field flux cI>o, as indicated by the curve <I> in Fig. 21A.
If m = ratio
armature reaction plus self-induction armature self-induction

it is <I>1 = m¢o, and the initial value of the field flux consists of the
permanent part cI>o, and the transient part <I>' = <I>1 -<1>0 = (m-1)
tl>0• This is a rather slow trn.nsient, frequently of a duration of a second or more.
The armature currents ii, i~, ia are proportional to the field flux <Ji which produces them, and thus gradually decrease, from initial values, which are as many times higher than the final values as <I>1 is higher than <I>0, or m times, and are represented in Fig. 21B.
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. 21 C.
The field-exciting current is i0 at open circuit as well as in the permanent condition of short circuit. In the permanent condition of short circuit, the field current i 0 combines with the armature reaction F0, which is demagnetizing, to a resultant m.m.f., which produces the short-circuit flux ¢0. During the transition period the field flux <I> is higher than tl>o, 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 <I>, 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.

26 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

The transients of magnetic flux, current, and voltage are shown as A, B, and C in Fig. 13.
The magnetic flux and therewith the current decrease from the initial values <I>o and io at the moment t0 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.

<I>o I

I

I

I

A

I

I

I

1o I

B

I I

I

I

I
e'_I

__

_

_

_

_

_ I

0

I

I

I

-~--

e I

I

C

'

tI o

t

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

The impressed voltage e0 is withdrawn at the moment t0, 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, t0, the current is still io, and the induced voltage

thus must be

eo' = io (r + r'),

while the impressed voltage, before to, was
eo = ior;

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

age e0, in the same ratio as the resistance of the discharge circuit
r + r' is greater than the resistance of the coil r thri::1ugh which the

impressed voltage sends the current

eo'
e0

=r-+r-r'·

SINGLE-ENERGY TRANSIENTS.

43

in Fig. 21C 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. 21C 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. 21D and the

armature reaction, then if the armature reaction shows an initial os-

cillation, in Fig. 21E, 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 cf?. The starting

transient of the polyphase armature reaction thus appears in the

field 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 wind-

ing. In Fig. 22C 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 pul-

sations of the field current, Fig. 22C, are slightly reduced and

delayed in phase; and with considerable inductance intentionally

inserted into the field circuit, the e:ffect of this inductance would

require consideration.

From the constants of the alternator, the momentary short-

circuit characteristics can now be constructed.

Assuming that the duration of the field transient is

.To

=

( m

Lo - 1) To

= 1 sec.,

the duration of the armature transient is
T = -Lr = .I sec.

.. And assuming that the armature reaction is 5 times the armature

28 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
The same reasoning also applies to the transient resulting from several forms of energy storage (provided that the law of proportionality ()f i, e, <P, 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.

A

--·~-------- I

1

_,,,.,--

---------------•-

_-_<_I_' 0_ .......i.. ,,,,,,""

I

I

I

---•- --+I ·----------- • -----·...-----.--- --------------- ----

1 I

• 0 • I

B

' ' I

I

I

C

Fig. 14. - Single-energy Starting Transient of Magnetic Circuit.
For instance, in the coil shown in Fig. 10, let the short circuit A be opened, that is, the voltage eo be impressed upon the coil. At the moment of time, to, when this is <lone, current i, magnetic flux 4>, and voltage e on the coil are zero. In final condition, after the transient has passed, the values io, 4>o, eo are reached. We may then, as discussed above, separate the transient from the permanent term, and consider that at the time to the coil has a permanent current io, permanent flux cl>o, permanent voltage e0, and in adcli-

SINGLE-ENERGY 'l'RANSIENTS.

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, vary-

ing 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

Fig. 21, A to D.

However, the pulsating armature reaction produces a corre-

sponding pulsation in the field circuit. This pulsation is of double

.1

.2

.4 Seconds

A

B

C
lo
Fig. 23. -Symmetricn.l Moment0,ry Single-phase Short Circuit of Alternator.
frequency, and is not transient, but equally exists in the final shortcircuit current.
Furthermore, the annature 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
shows the slow transient of the field, as shown in Fig. 23, where A .

LECTURE IV.
SINGLE-ENERGY TRANSIENTS IN ALTERNATINGCURRENT CIRCUITS.
I7. Whenever the conditions of an electric circuit are change~ 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 i1 = current existing in the circuit immediately before, and
thus at the moment of the change of circuit condition, and i2 =
current which should exist at the moment of change in accordance with the circuit condition after the change, then the actual current i1 can be considered as consisting of a part or component i2, and a
component i1 - i2 = io, The former, i2, is permanent, as result-
ing from the established circuit condition. The current component io, 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 = .rf. • = G,L.
The permanent current i2 may be continuous, or alternating, or may be a changing current, as a transient of long duration, etc.
The same reasoning applies to the voltage, magnetic flux, etc. Thus, let, in an alternating-current circuit traversed by current
i1, in Fig. 15A, the conditions be changed, at the moment t = 0,
so as to produce the current i2• The instantaneous value of the
current i1 at the moment t = 0 can be considered as consistin.g of the instantaneous value of the permanent current ¼, shown dotted, and the transient io = i1 - i2, The latter gradually dies
down, with the duration T = Lr , on the usual exponential tran-
30

49 frequenry, and nf'< tho r0sult; an incn•ase of volt.age a111l u disfort.ion of t.lie quadrature phase occur:-,, ns shown in the uscillogram Fig. 2G.
Various morneutary short-circuit phenomena are illustrated by the oseillogrnms Figs. 26 to 28.
Figs. 2011 und 2GB show the momentary three-phase short cir~uit of n 4-polnr 2G-cycle 1500-kw. steam t,urhine alternator. The
FiJ,!:. 20. I. - cD!l:1\l!l. - Rymmetrical.
Fig. 2fiR. - rn030i. - Asymmetrical. l\Iomeutnry Thn•<'-pha"'" 1--:hort Cimiit, or 1300-Kw. 2300-Volt -11uce-phnse Alt erna1.nr (A Tn--1- HiOO-- I SOil). 0!4dllograrni:; of Arnmture Current am l Fidu Current.
lower cun·<i gi n•s 1hP trnnsiPnt, of Uw lidd-1•xeit.ing current, the upper em--v<' that of one of the armature cm-r<'nts, - in Fig. 2GA :hut curn•nt '\Yhi<'h shoukl h1i near zero, in Fig. 20B that ,vhich should be 111•ar its rnaximum value at the moment ,vhere the short circuit starts.
Fig. 27 sho,Ys the single-phase short circuit of a pair of machines in which the short, <'ir('uit 1H·currP(l n.t the momPnt in which the armature short-1·ir1·uil, ,~mT<·ut, ~hould Jip Z<'ro; t.hc armature cur-

32 ELECTRIC DISCHARGES, WAVES AND IMPULSES. If the current i1 is zero, we get the starting of the alternating
current in an inductive circuit, as shown in Figs. 16, A, B, C. The starting transient is zero, if the circuit is closed at the moment when the permanent current would be zero (Fig. 16B), and is a maximum when closing the circuit at the maximum point of the permanent-current wave (Fig. 16C). The permanent current and the transient components are shown dotted in Fig. 16, and the resultant or actual current in drawn lines.
A
B
C
Fig. 16. - Single-energy Starting Transient of Alternating-current Circuit.
18. Applying the preceding to the starting of a balanced three-phase system, we see, in Fig. 17A, that in general the three
° transients i 1°, i2°, and i3 of the three three-phase currents i 1, i 2, i3
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, i1, Fig. 17B, there is no transient for this current, while the transients of the other two currents, i2 and i3, are equal and opposite, and near their maximum value. Starting, in Fig. 17C, at the maximum value of one current i3,
we have the maximum value of transient for this current i3°, while
the transients of the two other currents., i1 and i2, are equal, have

SINGLE-ENERGY TRANSIENTS.

51

symmetrical, and the double-frequency .p1;isution 6f-i,h~ field cur-

rent shows during the first few cycles the :.1.lte!'U.1?,te high-iind low

peaks resulting from the full-frequency transieti:t,. pulsation of

the rotating magnetic field of armature reaction. • '!'he irregular

initial decrease of the armature current and the sud1.lcn: change

of its wave shnpe are due to the transient of the current +.ran3-

former, 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 dist~rtion resulting

from the double-frequency pulsation of the ·armature reaction.

While the synchronous reactance xo can be predetermined with

fair accuracy, the self-inductive X1 is not such a definite quantity.

It includes a transient component. The armature magnetic cir-

cuit 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 tran-

sients. This effect is materially affected by the amount of resist-

ance 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 e1iergy

transfer between the phases, during the armature transient.

The instantaneous power of the momentary short-circuit

current, and '\Yith 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 prop~rtioual to the square of· the

short-circuit current.

•

84 ELECTRIC DISCHARGES, JVAVES AND 11vJPULSES. 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 condition 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 concentric 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 TRANSIENT OF IRONCLAD CIRCUIT. 53

approximate step-by-step method, as illustrated for the starting transient of an alternating-current transformer in "Transient Electric 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 neglecting the difference between the rising and decreasing magnetic characteristic, and using the approximation of the magnetic characteristic given by Frohlich's formula:

O?, -- a+JCuJC'

(1)

which is usually represented in the form given by Kennelly:

p

=

JC
O?,

=

a

+

uJC;

(2)

that is, the reluctivit.y is a li1war 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 O?i consists of a component JC, the field intensity, which is the flux density in space, and
= a component O?,' O?, - JC, which is the additional flux density
carried by the iron. O?,' is frequently called the "metallic-flux density." \Vith increasing JC, <B' reaches a finite limiting value, which in iron is about

<B00 ' = 20,000 lines per cm2• *

At any density <B', the remaining magnetizability then is

IB..,'-aV, and, assuming the (metallic) permeability as proportional

hereto, gives

µ = c(<B,,/ - <B'),

and, substituting

gives

<B' -

1c<B+ao

'JC' cJC''

• See "On the Law of Hysteresis,'' Pa.rt II, A.I.E.E. Transactions, 1892,

page 621.

•

86 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
From this polar diagram of the rotating field, in Fig. 19, values OC can now be taken, corresponding to successive moments of time, and plotted in rectangular coordinates, as done in Fig. 20~ As seen, the rotating field builds up from zero at the moment of closing the circuit, and reaches the final value by a series of oscillations; that is, it first reaches beyond the permanent value, then drops below it, rises .again beyond it, etc.

3

4 cycles

Fig. 20. - Starting Tmusicnt of Rotating Fiekl: Rectangular Form.

\Ve 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 he 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 wit.h 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
field, of duration T = Lr •
The maximum value, which the magnetic field during the transition period can reach, is limited to less than double the final value, as is obvious from the construction of the field, Fig. 19. It is evident herefrom, however, that in apparatus containing rotating fields, as induction motors, polyphase synchronous machines, etc., the resultant field may under transient conditions reach nearly <louLle value, an<l 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 TRANSIENT OF IRONCLAD CIRCUIT. 55
i netic circuit, and the saturation value of the flux in the iron.

That is, for i = 0, nc~' = Li; ~nd for i = oo, <I>' = ~b • i If r = resistance, the duration of the component of the transient
resulting from the air flux would be

T2

= L-rt

=

nc 10-s - -r - ,

(5)

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

Ti= L1 = na10-s_

r

r

(6)

The differential equation of the transient is: induced voltage plus resistance drop equal zero; that is,

n c: 10-s + ri = 0.

Substituting (3) and differentiating gives

(·n1·-a+-1-·0b--i·)s-2 -ddti + nc 10-s -ddti + ri. = 0'

an<l, substituting (5) and (G),

j ! ~ (l

1
bi) 2

+

T2

:: + i = 0;

hence, separating the variables,

i(lT+1dibi) 2

+

T2di
-i-.

+

dt

=

O.

(7)

The first term is integrated by resolving into partial fractions:

1

1 b

b

i (1 + bi) 2 = i - I + bi - + (1 bi) 2'

an<l the integration of differential equation (7) then gives

Tilog 1 ~ bi + T2 log i + 1 ~bi + t + C = 0.

(8)

If then, for the time t = to, the current is i = i 0, these values
substituted in (8) give the integ:cation co~stant C:

T1

log

1

+iobio

+

T
2

log

.
io

+

1

+Tibio

+

to

+

C

=

0,

(9)

88 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
with the armature current, that is, is instantaneous. The armature 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 Z2 = x0 - xi of the synchronous reactance which corresponds
to the armature reaction is not a true reactance at all, consumes no voltage, but represents the consumption of field ampere turns by the m.m.f. of the armature current, and the corresponding change of field flux. Since, however, the field flux represents stored magnetic energy, it cannot change instantly, and the armature reaction thus does not appear instantaneously with the armature current, but shows a transient which is determined essentially by the constants of the field circuit, that is, is the counterpart of the field transient of the machine.
If then an alternator is short-circuited, in the first moment only the true self-inductive part X1 of the synchronous reactance exists,
and the armature current thus is i1 = eo, where e0 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 xo, and the short-circuit cur-
rent decreases to the value -i0 = eo • Xo The ratio of the momentary short-circuit current to the perma-
uent sl1ort-c1•rcm·t current tlms 1•s, approx1•mate1y, tl1e rati•o i-:-1- = X-o , 'lo X1
that is, synchronous reactance to self-inductive rcactance, 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 current 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: Xo • X1

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 frdm 'tlte 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 befo_re, and during the transition period an increase

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

transient. This t,ransicnt then is the difference between the energy

storage in the permanent condition and the energy storage during

the transition period.

If the law of proportionality between current, voltage, magnetic

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

function:

t

= Y YoE - To,

(1)

where

Yo = 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 T0 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=Lr-,

(2)

where L = inductance = coefficient of energy storage by the current, r = resistance= coef?cient 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

To'= C, g

(3)

59

40 ELECTRIC DISCHARGES, W.tl l'ES AND IMPULSES.
! proportional to cf>. Thus, as it is i = i0 at cJ>0, during the transition
period it is i = i0• Hence, the field-exciting current traverses 0
the same transient, from an initial value io' to the normal value i 0, as the field flux <I> and the armature currents.
B - ··---~ ---...... --- - ----- - ---- --- -- -
C
0
Fig. 21. - Construction of Momentary Short Circuit Chamcteristic of Polyphase 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 4>1 against the demagnetizing armature reaction. In other words, the field flux <I> 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 i0 to io', and maintain it at the values corresponding to the transient i, Fig. 21D.
As seen, the transients <I>; ii, i2, i3 ; Fi i are proportional to each other, and are a field transient. If the field, excited by current io
.·

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 cousider the oscillating <louble-energy transient., that is, the case in which the energy changes periodically between its two forms, <luring its gradual dissipation.
This may be done by considering separately the periodic transfer, or pulsation of the energy between its two forms, and the gra<lual dissipation of energy.
A. Pulsation of energy. 25. The magnetic energy is a maximum at the moment when the dielectric energy is zero, and when all the energy, therefore, is magnetic; an<l the magnetic energy is then
Lio2
T'
where io = maximum value of t-ransient 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

where e0 = maximum value of transient voltage.
As it is the same stored energy which alternately appears as magnetic an<l as dielectric energy, it obviously is

Lio2 Ceo2
T =T

(8)

This gives a relation between the maximum value of transient current an<l the maximum value of transient voltage:.

eo
io

=

V• Ifc:..

(g)

v§ therefore is of the nature of an impedance Zo, and is called

the natural impedance, or the surge impedance, of the circuit; and

Vi .its reciprocal,

= Yo, is the natural admittance, or the surge

admittance, of the circuit.

42 ELECTRIC DISCHARGES, WAVES AND IMPULSES.
start with the values -.i1', - i2', - i3'. 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 i1, i2, ia 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

t= 9

.1

.2

.a

. 4

.5

.6 Secs.

4>1

A

i1
B

C
1
lo
Fig. 22. - 11omentary Short Circuit Characteristic of Three-phase Alternator.
currents for the case where the circuit is closed at the moment when i1 should be maximum; i1 then shows the maximum transient, and i2 and ia 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

DOUBLE-ENERGY TRANSIEN'J.'S.

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 cabl~ enters a reactive device,

as a current transformer, it produces enormous voltages therein.

Thus, cable oscillations arc 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 z0 and Yo between the cable and the reactive apparatus, thus acting like a

reactive apparatus to the former, like a cable toward the latter.

Thus, the transformer is protected by the transmission line in

oscillations originating in the transformer, but endangered by the

transmission line in oscillations originating in the transmission

line.

v§ The simple consideration of the re;~tive values of Zo =

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 ancl the transient voltage e

successively increase, decrease, and become zero.

The current thus may be represented by

i = io cos(¢, - -y),

(12)

where i0 is the maximum value of current, discussed above, and

= ¢, 2 1r ft,

(13)

where f = the frequency of this transfer (which is still undeter-
mined), and -y the phase angle at the starting moment of the +,ransient; that is,

i1 = io cos -y = initial transient current.

(14)

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

'44 ELECTRIC DISCIIARGES, WAVES AND IlvfP.ULSES.
self-induction, that is, the synchronous reactance is 6 times the self-
inductive reactance, Xo = m = 6. The frequency is 25 cycles. Xi If cI>1 is the initial or open-circuit flux of the machine, the short-
! circuit flux is cI>o = cI>i = cI>1, an<l the field transient cI> is a tran1n u
sient of duration 1 sec., connecting cI>1 and <I>o, Fig. 22A, represented by the expression
t
c.I> = cl>o + (cI>1 - <I>o)E- To.
The permanent armature currents ii, i2, i3 then are currents
starting with the values 1n eo , and decreasing to the final short-
Xo
circuit current eo, 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 <liscussecl in paragraph 18, and thereby give the asymmetrical resultant currents, Fig. 22B.
The field current i gives the same slow transient as the flux cl>,
starting with io' = 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 (i - i0), f;ince io 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, anc.l thereby the field curve, Fig. 22C, is derived. Here the correction for the external field inductance is to be applied, if considerable.
Since the transient of the armature reaction does not depend on the point of the wave where the short circuit occurs, it follows that the phenomena at the short circuit of a polyphase alternator are always the same, that is, independent of the point of the wave at which the short circuit occurs, with the exception of the initial wave ~hape of the armature currents, which individually depend

. DOUBLE-ENERGY TRANSIENTS.

65

decreases, an<l 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, an<l in this case the above equa-

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

frequency f 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, an<l the time <luring which this current

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

Applying the luw

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

2 eoC;

the current has a maximum value io, thus an average value ~ io, • 7r

and •as it flows into the -con<lense'r during one-half cycle of the

frequency f, that is, during tl~e time 2\, .it is

2

eoC

=

~
1r

io

.2.!_f ,

·

which is the expression of the condenser equation (17) applied to the oscillating energy transfer.
Transposed, this equation gives

to
f = 21reoC'

(18)

and substituting equation (10) into (18), and canceling with i0,

(19)

46 ELECTRIC DISCHARGES, WAVES AND L"r!PULSES.
is the fiel<l transient cJ> (the same as in Fig. 22A) and B the armature current, decreasing from an initial value, which is m. times the final value, on the field transient.
Assume then that the mutual induction between field and armature is such that 60 per cent of the pulsation of armature reaction appears in the field current. Forty per cent damping for the double-frequency reaction would about correspond to the 20 per cent damping assumed for the transient full-frequency pulsation of the polyphase machine. The transient field current thus pulsates by 60 per cent around the slow field transient, as shown by Fig. 23C; passing a maximum for every maximum of armature

.1

.2

.4 Seconds

A

B

C

Fig. 24. -Asymmetrical Momentary Single-phnse Short Circuit of Alternator.
current, and thus maximum of armature reaction, and a minimum for every zero value of armature current, and thus armature reaction.
Such single-phase short-circuit tmnsients 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 annature 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.

DOUBLE-ENERGY TRANSIENTS.•

67

transient. In the lat'ver case, the duration of the transient would be
To=L-r ,
and with only half the energy magnetic, the duration thus is twice as long, or
(23)

and hereby the factor

t
h = E-T1

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

/1
, I
/ I
/Al

n
,,
f I
,,' II
.,'Ic :I

Fig. 32. - lleln.tion of l\·fognetic and Dielectric Energy of Transient.
The same applies to the dielectric energy. If all the energy were dielectric, it would be dissipated by a transient of the duration:
T 0' ::::; -C,.
g

l+00

t,:;
t-<
t::;i
('") -..:,
~.....
('")
-'C
tr.
("':,
t::.,:.:.
~ (;'.}
t::;i
. 9J
;..;..:..

v~.:

::i.

<

.b...

~

"ti

Fig. 25. -co9762.- Momentary Single-phase Short Circuit of 1875-Kw. 2300-Volt Two-phase Alternator (AQD-4-1875lll-1800). Oscillograrn of Armature Voltage of Open Phase, Armature Current of Short-circuited

I

Phase, and Field Current.

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 pcrio<lic energy transfer, (21) and (22), gives the complete equations of the double-energy transien,t of the circuit containing in<luctance and capacity:

l i• = E- ut)·i1 cost; - yoe1 sm• -t;;. l~ , )"

! i ' ~ ~ + e = c ut e1 cos zoi1 sin

(28)

where

v~ Zo =

= :o' I

(29)

u=!~f+~i,

u = VU',

(30)

and i1 and e1 are the initial values of the transient current an<l voltage respectively.
As instance are constructed, in Fig. 33, the transients of current and of voltage of a circuit liaving the constants:

Inductance, Capacity, Resistance, Conductance,

L = 1.25 mh = 1.25 X 10-3 henrys;
C = 2Jftf = 2 X 10-6 farads;
r = 2.5 ohms;
g = 0.008 mho,

in the case, that

The initial transient current, i1 = 140 amperes;
The initial transient voltage, e1 = 2000 volts.

It is, by.,the preceding equations:

u = v1LC = 5 X 10-s,

f

=

-1 2 1f'(I

=

3180

cycles

per

second,

v'€ zo =

= 25 ohms,

Vi Yo = = 0.04 mho,

50 Ef,ECTJUC lJJSC:IJARGES, W..1FES ANJJ IMPULSES. rPut w:tn•, tl1Prdor0., is synrnwtrical, and the field current shows only tlw do11ble-freqtH•11cy pulsation. Only a few half-waves were reeonled before U1c circuit. lir<•n.kPr opened the short circuit.
Fig. 27. - c1>ii I'..!S. -- HyrnmPl.ri1·:d. i\lo11wnlary Single-phase Short Circuit of /\ It Prnat.or. ( lP.cillog;ra111 of An11al 111•p ( '.11rrc111'., Armature Voltage, :llld FiPld ( '111TP11I. (C'ircnil hrC'akPr opens.)
Fi~. 2X. - c1>1ii;1>ii. -- ,\,-ym111Plri(·:t I. i\lomcutary 8ingle-phusc 8hort Circuit of ii000-1\.w. 11,0110-Vnlt ThrPe-phase ..\11.ernator (A'rB-6-5000-500). Uscillogrnm of Armature Current uml Ficlll Current.
Fig. 28 shows the single-phase short circuit of a 6-polar 5000-kw. 11,000-Yolt steam turbine alternator, which occurred at a point of th<' wm·P. wlwrc the armat.11n~ current should be not far from its rnaxmm111. The t.nmsiPnt armat,urc currC'nt, therefore, st.arts un-

DOUBLE-ENERGY 1'RANSIE1V'l'S.

71

Fig. 33A gives the periodic components of current and voltage:
i' = 140 cos 0.2 t - SO sin 0.2 t,
e' = 2000 cos 0.2 t + 3500 sin 0.2 t.
Fig. 33B gives
= The magnetic-energy transient, h i:- 1,
The dielectric-energy transient, k = E- 2!, ·
And the resultant transient, hk = E-a:.
And Fig. 33C gives the transient current, i = hki', and the transient voltage, e = hke'.

LECTURE V.
SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT.
22. Usually in electric circuits, current, voltage, the magnetic field and the dielectric field are proportional to each other, and the transient thus is a simple exponential, if resulting from one form of stored energy, as discussed in the preceding lectures. This, 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 fiel<l, et'c.; and the proportionality between current or voltage and their respective fields, the magnetic and the dielectric, thus ceases, or, as· it may be expressed, the inductance L is not constant, but varies with the current, or the capacity is not constant, but varies with the voltage.
The most important case is that of the ironclad magnetic 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 foun<l for the cyclic curve of hysteresis, a mathematical calculation is not feasible, but the transient has to be calculated by an
52

LINE OSCILLA71IONS.

73

where

it = ½(f + t) ;

(7)

hence the total expression of transient current and voltage is

·i = i 0c ut cos (q;, - -y);
e = eoc"t:::;in (<f., - -y) ~ •

(8)

"Y, e0, and io follo,v from the initial values e' and ·i' of the transient,

at t = 0 or <f., = 0:

i' = iu cos 'Y l .
c' =-cosm• -y (J '

(0)

hence

tan 'Y = - e-;-'- -io = -yo e-;-' •

(10)

i' eo

i'

The preceding equations of the double-energy transient apply

to the circuit in which capacity and inductance are massed, as, for

instance, the di:::;chargc or charge of a condenser through an in-

ductive circuit.

•

Obviously, no material difference can exist, whether the capacity

and the inductance arc 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 trans~ission

line.

I

I

I

I

1-------------1 ·---------~

l

:

I

I

A

I
B

Fig. 34.

However, if (8) are the equations of current and voltage at a

point A of a line, shown diagrammatically in Fig. 34, at any other

point B, _at distance l from the point A, the same equations will

apply, but the phase angle 'Y, an<l the maximum values e0 and i 0, may be different.

Thus- it

= i CoE-ut COS (cp '- '}') }
e = z0C0E-ut sin (cf> - 'Y)

(11)

54 ELECTRIC DISCHARGES, WAVES .tlND IMPULSES.

or, substituting gives equation (1).

cB1 '

= a,

1
(J?, ,

=

u'

C 00

00

For JC = 0 in equation (1), JCO?, = ! ; for JC = oo, O?, = ! ; that is,

a

u

m• equat·1011 (1), a-1 = n• u•tm• l permeab1'l1' ty, -u1 = saturati•on value of

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 = fJ + a-JC,

where (3 = a plus the reluctance of the air gap. Equation (1),
therefore, remains applicable, except that the value of a is increased.
In addition to the metallic flux given by eguation (1), a greater or smaller part of the flux always passes through the air or through space in general, an<l then has constant permeance, that is, is given by
O?, = cJC.

23. In general, the flux in an ironclad magnetic circuit can, therefore, be represented as function of the currePt by an expression of the form
(3)

where 1 : bi = w' is that part of the flux which passes through

the iron and whatever air space may be in series with the iron,

an<l ci is the part of the flux passing through nonmagnetic

material.

Denoting now

L1 = na 10-s, l

~ = nc 10-s, ~

(4)

where n = number of turns of the electric circuit, which is inter-
linked with the magnetic c,ircuit, L2 is the inductance of the air part of the magnetic circuit, L1 the (virtual) initial inductance, that is, inducta1ll'c at very small currents, of the iron part of the mag-

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-ue, and of a combination of the trigonometric expressions:

cos q:, cos w, }

sin q:, cos w, cos q:, sin w,

(19)

sin q:, sin w.

Line oscillations thus can be expressed in two different forms, either as functions of the sum and difference of time angle q:, and
distance angle w: (q:, ± w), as in (li); or as products of functions of q:, and functions of w, 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. 3G 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 l, so that at some distance lo current and voltage are 360 degrees displaced from their values at the starting point, that is, are again in the same phase. This distance lo is called the wave length, and is the distance which the electric field
travels during one period to = ] 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 ~he[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 an~ capacity are derived by multiplying the total
inductance and total capacity by a,yg/cos/, that is, by ~ • 1r

56 ELEC1'JUC JJISCIIAR<lES, WAYES AND IA/PULSES.

and, subtracting (8) from (9), gives

t

_

to

_ -

T1log i1:0((l1++bbioi))

+

T2

log

io i

+

~
T1 (1

+1 bio _

l

+ 1

( bi~

•

(10)

This equation is so complex in i that it is not possible to calculate from the different values oft 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.

1: 10

9

8

Trnusient o
7

lro 11:l1ul 11lt1nctlv1 Circuit:

1+\ t=2.92-{9.21 lg

i + .921 lg i + l :.6 i }- 6

\
\

( dotted: t =1.085 lg i-.507)
6
4

\

3
' ' ' ' ', \

' i\
-~ --,._ ·-
-~ ..::..-........._

2 l

t=-1

2

a

4

5

6 seconds

Fig. 29.

Such is <lone in Fig. 29, for the values of the constants:
r = .3,
a=4X105, c=4X104,
b = .6,
n = 300.

.,
.:.:.-..,
►
~
C
,V....
.:.;. tt-
~
~
~
C
>tr.
Fig. 36. - en10002. -Oscillogmm of Sturting Oscillation of 28 Mile:1 of 100,000-\·olt Transmission Line: Lowtension Switching.
-1 -1

58 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

This gives

T1 = 4,
T2 = .4.

.Assuming io = 10 amperes for t0 = 0, gives from (10) the equa-

tioo:

•

j ~ T = 2.92 - 9.21 log10 l +i.6i + .921 log10 i + l :.6i •

Herein, the logarithms have been reduced to the base 10 by
division with log10E = .4343.
For comparison is shown, in <lotted line, in Fig. 29, the transient of a circuit containing no iron, and of such constants as to give about the same duration:
t = 1.085 foglO i - .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-k,y. 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.

LINE OSCILL,11'IONS.

79

The frequency f llepen<ls upon the length l1 of the section of line in which the oscillation occurs. That is, the oscillations occurring in a transmission line or other circuit of distributed capacity have no definite frequency, but any frequency may occur, depending on the length of the circuit section which oscillates (provided that this circuit section is short compared with the entire length of the circuit, that is, the frequency high compared with the frequency which the oscillation would have if the entire line oscillates as a whole).
If l1 is t.he oscillating line :c;ection, the wave length of this oscillation is four times the length

(27)

This can be seen as follows:

At any point l of the oscillating line section l1, the effective

power

]Jo= avg ei = 0

(28)

is always zNo, since voltage and current arc 90 degrees apart. The instantaneous power

p = ei,

(29)

however, is not zero, hut alternately equal amounts of energy flow first one way, then the other way.
Across the ends of the oscillating section, however, :µo 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 = 0 at both ends of l1;
2. i = 0 at both ends of 11;
3. e = 0 at one end, i = 0 at the other end of Z1•

In the third case, i = 0 at one end, e = 0 at the other end of
the line section 11, the potential and current distribution in the line section Z1 must be as shown in Fig. 37, A, B, C, etc. That is, Z1 must be a quarter-wave or an odd multiple thereof.
If l1 is a three-quarters wave, in Fig. 37B, at the two points C and
t; D the power is also zero, that is, Z1 consists of three separate and
independent oscillating sections, each of the length that is, the

60 ELECTRIC DJSCIIARGES, 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

(4)

Similar single-energy transients may occur in other systems. For imito.nee, the transient by which a water jct approaches constant velocity when falling un<ler gravitation through a resisting medium would have the duration

T = V-go ,

(5)

where v0 = limiting velocity, g = acceleration of gravity, and would
be given by

(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 <lifferent 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

TV = Li2 + Ce2' ~

2 2

(7)

= lVm +wd.

'Vhile the total energy lV decreases by dissipation, Wm may be converted into lVd, 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

LINE OSCILLATIONS.

81

thus i~

lo= vto = a-1f,

(32)

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

(33)

or

v =1- = - -1 - ·

(34)

<To -VLoCo

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

(35)

and inversely. As in complex overhead structures the capacity

qsually is difficult to calculate, while the inductance is easily de-

rived, equation (35) is useful in calculating the capacity by means

of the inductance.

This equation (35) also allows the calculation of the mutual

capacity, and thereby the static induction between circuits, from

the mutual magnetic inductance.

The reverse equation,

.

1

Lo= V2cO'

(36)

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

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

31. If l1 is the length of a line, an<l its two ends are of different

electrical character, as the one open, the other short-circuited,
and thereby i = 0 at one end, e = 0 at the other. end, the oscilla-

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

The longest wave which may exist in this circuit has the wave
length lo = 4 Z1, and therefore the period lo = u0Zo = 4 u0l1, that

is, the frequency Jo

=

-4

1 l
<To 1

.

This is called the Junda1nental wave

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

as higher harmonics, of the respective wave lengths 2 kl:_ 1 and
the frequencies (2 k - l)fo, where k = ~' 2, 3 . . .

62 ELECTRIC DISC/JARGES, WAVES AND 11\JPULSES.

The maximum transient voltage can thus be calculated from

the maximum transient current:

/ eo

=

.
lo

\

-LC =

.
toZo,

(10)

and inversely,

(11)

This relation is very important, as frequently in double-energy transients one of the quantities eo or io is given, and it is important to determine the other.
For instancP, 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 = i'.ozo.
If one conductor of an ungrounded cable system is grounded, the maximum momentary current which may flow to ground is io = enYo, 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 e0, the maximum
discharge current in the line is limited to io = 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), z0 is high and Yo low. That is, a high transient voltage can produce only moderate transient currents, but even a small transient current 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 osciilating voltages.
Inversely, if L is low and C high, as in an underground cable, zo is low but y0 high, and even moderate oscillating voltages produce 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, a.<, the dielectric strength of a cable is necessarily relatively much lower than that of a transmission line, due to the close proximity of the conductors in the former. A cable, therefore, when receiving the moderate or small oscillating currents which may originate in a transformer, gives. only very low

LINE OSCILLATIONS.

83

usually are more conveniently resolved into the form oi equa-

tion (19).

At extremely high frequencies (2 k - l)f, that is, for very large

values of k, the successive harmonics are so close together that a

very small variation of the line constants causes them to overlap,

and as the line constants are not perfectly constant, but may

vary slightly with the voltage, current, etc., it follows that at very

high frequencies the line responds to any frequency, has no definite

frequency of oscillation, but oscillations can exist of any frequency,

provided this frequency is sufficiently high. Thus in long trans-

mission lines, resonance phenomena can occur only with moderate

frequencies, but not with frequencies of hundred thousands or

millions of cycles.

32. The line constants r0, Yo, Lo, Co are given per unit length,

as per cm., mile, 1000 feet, etc.

The most convenient unit of length, when dealing with tran-

sient!:, in cirC"uit.~ of distrilmtP<l capacity, is the velocity unit v.

That, i:-i, choo:-1ing as unit of lt>ugt.h the tfo;tancc of propagation
in unit tinw, or :3 X 1010 cm. in uv1•rhea<l circuits, this gives v = 1,

and therefurf'

uo = VLoCo = 1, f

LoCo = 1,

J

(39)

or

CO

=

1 Lo

;

Lo

=

1 Co.

That is, the capacity per unit of length, in velocity measure, ii.; inversely proportional to the imluctancc. In this velocity unit of length, distances will be represented by X.
Using this unit of length, uo disappears from the equations of the transient.
This velocity unit of length becomes specially useful if the transient extends over different circuit sections, of different constants und therefore different wave lengths, as for instance an overhea<l line, the underground cable, in which the wave length is
about one-half what it is in the overhead line (K = 4) and coiled
windings, as the high-potential winding of a transformer, in which the wave length usually is relatively short. In the velocity measure of length, the wave length becomes the same throughout all these circuit sections, and the investigation is thereby greatly simplified.

64 ELECTRIC DISCHARGES, WAVES AND I.ilJPULSES.

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,

e = eo sin (¢ - -y),

(15)

where

e1 = -e0 sin 'Y = initial value of transient voltage. (16)

The frequency f 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 oscillation 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, ca.used by s,vitching 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

LINE OSCILLATIONS.

85

beginning of time, that is, for cf, = 0, and by the values of i and e at all times t (or cf, respectively) at the ends of the circuit, that is,
for w= o and w = ,:r,f
For instance, if:
(a) The circuit is open at one end w = 0, that is, the current
is zero at all times at this end. That is,
,: = 0 for w = O;

the equations of i then must not contain the terms with cos w,

l ros 2 w, etc., as these would not be zero for w = 0.
must be
ai = 0, bi = O, lll! = 0, b2 = O,

That is, it (43)

aa = O, ba = 0, etc.

The equation of i contains only the terms with sin w, sin 2 w,

etc. Since, however, the voltage e is a maximum where the

current i is zero, and inversely, at the point where the current is

zero, the voltage must be a maximum; that is, the equations of

l t.l1e voltage must contain only the terms with cos w, cos 2 w, etc.

Thus it must be

ci' = 0, c/ = 0,

di'
d/

= =

O, 0,

(44)

ca' = 0, da' = 0, etc.

Substituting (43) and (44) into (42) gives

i = c" 1 )c1 coscf,+d1sincf,l sinw, !
e = c"1 1ai' cos cf, + bi' sin cf,} cos w ,

(45)

and the higher harmonics hereof.
(b) If in addition to (a), the open circuit at one end w = 0, the line is short-circuited at the other end w = ; , the voltage e
must be zero at this latter end. Cos w, cos 3 w, cos 5 w, etc.,
become zero for w = ,2r , but cos 2 w, cos 4 w, etc·., are not zero for w= ; , and the latter functions thus cannot appear in the expres-
sion of e.

66 ELECTRIC DISCIIARGES, WAVES AND lklPULSES.

as the expression of the frequency of the oscillation, where

(20)

is a convenient abbreviation of the square root. ·The transfer of ener;gy between magnetic and dielectric thus

occurs with a definite f1:equency f = _!_ 2 7f"(1" , and the oscillation thus
is a sine wave without distortion, as long as the law of proportionality applies. ·when this fails, the wave may be <listorted, as seen on the oscillogram Fig. 31.
The Pqtmt.iuni-1 of tlw wriodfr part of the tranrrient can now he writtt'n down by substituting (13), (19), (14), and (16) into (12) and (15):

z'. = iu cos (¢ - 'Y) = io cos 'Y cos ¢ + 1'.o sin 'Y sin ¢

= . t 'lo . t !1 COS a-- - e1 e- u Slll a-- ,

and by (11):

= .
i

.
11

cost-

-

. t
yue1s1n-,

(1"

(j

(21)

and in the same 11m1mer:

e = ei cos -t + . . Zot1 sm t- ,

(22)

(1"

(1"

where e1 is the initial value of transient voltage, i1 the initial value

of· transient current.

B. Power dissipation.

27. In Fig. 32 are plotted as A the periodic component of the

oscillating current i, and as B the voltage e, as C the stored mag-

• L·2

C 2

netic energy-·~ , and as D the stored dielectri<;! energy 2e •

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

LINE OSCILLATIONS.

87

In these equations (50), d and a' are the maximum values of·

current and of voltage respectively, of the different harmonic

·waves. Between the maximum values of current, io, and of volt-

age, eo, of a stationary oscillation exists, however, the relation

/L eo
Tr;=

z~\

c'

,vhere zo is the natural impedance or surge impedance. That is

a' = dzo,

(51 )

uml Ruh::::t.it.uting (;51) into (50) gives

a = i d t:-,,1 ) 1 sin cp sin w + d3 sin cp sin 3 w + d,, sin 5 cp sin 5 wl

e =

z0

cu 1

Jd1 coscpcos

+ •••L
w+d3 cos3cpcos3

w+d,,cos5cpcos5

wj

(r,):2)

+ ... l-

(d) If then the distribution of voltage e along the circuit is given at the moment of start of the transient, for instance, the voltage is constant and equals e1 throughout the entire circuit at the
starting momei1t </, = 0 of the transient, this gives the relation,
by substituting in (52),
e1 = Zo E-"t l d1 cos w + d:; cos 3 w + d" cos 5 w + . . . J, (53)
for all values of w. Herefrom then calculate the values of d1, d3, d5, etc., in the
manner as discussed in" Engineering Mathematics," Chapter III.

68 ELEC'l'RJC DJSCIJARGES, WAVES AND /Ml'ULSES.
as only half the energy is dielectric, the dissipation is half as rapid, that is, the dielectric transient has the duration
(24) and therefore adds the factor
t

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

t

t

E - T1 E - T:

E -

t

(I Ti

+

1)
Ti .

(25)

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

= ;,l ·+ ;,2'

(26)

=

~
2

(1-
To

+

T_o!,),

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',

~ = ~(f + ~)= u,

(27)

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

is given as

u = ~(f +~)
r 2L·

This is correct only if g = O, 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

TRAVELING TVAVES.

89

physical meaning a wave has, in which current and voltage are in phase with each other:

= i ioE-ut COS (cf, =F w - '1'), (
e = eocut cos (q, =F w - ,,). !

(4)

In this case the flow of power is

p = ei,
= eoioc 2ut cos2 (<J, =F w - -y),

+ = -eo2i-o-c 2"' [l cos 2 (<J, =F w - 'Y) ],

(5)

an<l the average flow of power is

Po = avg p,

= eo2to • E-2ut

(6)

Such a wave thus consists of a combination of a steady flow of power along the circuit, 7Jo, and a pulsation or surge, pi, of the same nature as that of the stan<ling wave (2):

JJ1 = 2eo1:o c .2ut cos 2 (</, =F w - '1') .

(7)

Such a flow of power along the circuit is called a traveling wave. It occurs very frequently. For instance, it may be caused if by a lightning stroke, etc., a quantity of Jielectric energy is impressed

Fig. 39. - Starting of Impulse, or Tr:i.veling Wave.
upon a part of the circuit, as shown by curve A in Fig. 39, or if by a local short circuit a quantity of magnetic energy is impressed upon a part of the circuit. This energy then gradually distributes over the circuit, as indicated by the curves B, C, etc., of Fig. 39, that is, moves along the circuit, and the dissipation of the stored energy thus occurs by a flow of power along the circuit.

70 ELECTRIC DISCllARGES, WAVES .riND IltfPULSES.
'l.\ = 2 L = 0.001 sec. = 1 millisecond, 7•
T2 = 2 C = 0.0005 sec. = 0.5 millisecond,
!J
T = 1 1 1 = 0.000333 sec. = 0.33 millisecond;
T1 + 'l.'2

e 1•

I I

~coo I

I
SDOO , I

\

- ' e, I
I
, _ I
i_:

ttt r\-~t ~ \- f ~ / I

l

I .- \

~

' ~ - t-

,-

-/

11· A

I \ \ \ I I
/ I

I

\

jI \

\ '1 I I I
I
j

l

\ 'I I

\

" !

I
!

\\ I

\~ )

'

- - I
lL I

\/

\ J

-1:ol~

-- --- ,__ +-:81
d.-- ---- ----:6,

~ ........
" ~

- --· --i-- I

I

-- -~!

h_:r-- ,__

-

.2

I
1 I

.01I

... ·-·,___~-11
r--..... :--. 1--__ _-k.
1{k-r--

I
_...,. __
i---:---l-' ff

e i HOOD

-

1

5

I
0

~

/

~

.1

\ 1000 ~no

I\•

-lOIIO t-50:

1 ,

0 cl

e
I\

.2

• .8 J .4

)1'm1s.-condl!

., .--. . /r---.. / J '\._ ' I\.

.5

.6

-I / ,, ........ .....

\ I I

I I

\

I

' !'- <_ i....-

l /V

C"

- I \ I
I

K

I

- I

I

1

l

Fig. 33.

hence, substituted in equation (28),
i = e- 3,p40 cos 0.2 t - 80 sin 0.2 tL (
e = e- 31 !2000 cos 0.2 t + 3500 sin 0.2 t I,)
where the time t is given in milliseconds.

~
:i,...
~._..
~ c.)
~
~
~
Fig. 41. -en10045. -Oscillogram of Compound Circuit of 154 Miles 100,000-volt Transmission Line and High-tension Coils of 10,000-kw. Step-up Tram,formcrs, Switchiag off by Low-tension Switches. High-tension Current and Low-tension Voltage.
~
J-l

LECTURE VIL

LINE OSCILLATIONS.

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

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

surge's

between

mag1wti. c

Li2
2

and

di•electri•c

Ce
2

2

,

and

a

transi. ent·

component, by which the total stored energy decreases.

Considering only the pcrirnlic component, the maximum value

of magnetic energy must equal the maximwn v~lue of dielectric

energy,

where i0 = maximum value of transient current, e0 = maximum
value of transient voltage.

This gives the relation between e0 and i0,

V ;: = ~ = Zo = :o'

(2)

where z0 is called the natural impedance· or surge impedance, y0 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

i1 = io cos (</> - 'Y) l,
= e1 eo sin (</> - 'Y) ~ '

(3)

where

</> = 2 1rft,

(4)

and

1
J= 21rvLC

(5)

is the frequency of oscillation.

The dissipative or " transient " component is

= hk E-u1,

(6)

72

TRAVELING lVAVES.

98

the rate e-u,, corresponding to the dissipation of the stored energy by e-" 1, as indicated by A' in Fig. 42; while in the case (b) the power flow decreases faster, in case. (c) slower, than corresponds to the energy dissipation, an<l is illustrated by B' and C' in Fig. 42.
(a) If the flow of power is constant in the direction of propagation, the equation would be

= i iof:-ut COS(¢ - W - 'Y),

= 6 eoE-ul COS (cp - W - ')') 1

(9)

In this case there must be a continuous power supply at the one end, and power abstraction at the other end, of the circuit or circuit section in which the flow of power is constant. This could occur approximately only in special cases, as in. a circuit section of medium rate of power dissipation, u, connected between a section of low- and a section of high-power dissipation. For instance, if as illustrated in Fig. 43 we have a transmission line
Line
Load
Line
Fig. 43. - Compound Circuit.
connecting the step-up transformer with a load on the -step-down end, and the step-up transformer is disconnecte<l from the generating system, leaving the system of step-up transformer, line, and load to die down together in a stationary oscillation of a compound circuit, the rate of power dissipation in the transformer then is much lower, and that in the load may be greater, than the average rate of power dissipation of the system, and the transformer will supply power to the rest of the oscillating system, the load receive power. If then the rate of power dissipation of the line u shoul<l happen to he exactly the average, ·u0, of the entire system, power would flow from the transformer over the line into the load, but in the line the flow of power would be uniform, as the line neither receives energy from nor gives off energy to the rest of the system, but its stored energy corresponds to its rate of power dissipation.

74 ELECTRIC lJISCJJARGES, WAVES AND IivlPULSES.

are the current and voltage at the point A, this oscillation will
appear at a point B, at distance l from A, at a moment of time later than at A by the time of propagation l1 from A to B, if the
oscillation is traveling from A to B; that is, in the equation (11),
instead of t the time (t - ti) enters. Or, if the oscillation travels from B to A, it is earlier at B by the
time t1; that is, instead of the time t, the value (t + t1) enters the
equation (11). In general, the oscillation at A will appear at B, and the oscillation at B will appear at A, after the time t1; that
is, both expressions of (11), with (t - t1) and with (t + t1), will
occur. The general form of the line oscillation thus is given by substi-
tuting (t =t= t1) instead of t into the equations (11), where t1 is the time of propagation over the distance l.
If v = velocity of propagation of the electric field, which in air,
as with a transmission line, is approximately

V = 3 X 1010,

(12)

nml in a medium of permeability µ. and permittivity (specific

capacity) K is

v= 3 X 1010

(13)

-y' µ.K

and w,e denote

a=-1,

(14)

V

then

t1 = al;

(15)

and if we denote

2 1rft1 = w = 2 1l"fal,

(16)

we get, substituting t =t= ti for t and <f, =t= w for<!> into the equation (11), the equations of the line oscillation:

i = cc"t cos (<I> =t= w - -y) } = e ZoCE-ut sin (<f, =t= W - -y) •

(17)

In these equations,

l q:, = 2 1rJt

is the time angle, and

(18)

w = 21r"jal )

is the space angle, and c = eoe=ut1 is the maximum value of current,
z0c the maximum value of voltage at the point l.

TRAVELING WAVES.

the average power then is

Po = avg ei,

= = ~~~c2(u-$)tc2s>-

e~:OE-2u.tE+2s(t->-).

95
(12)

Both forms of the expressions of l:, e, and J)o of equations (11)

and (12) are of use. The first form shows that the wave de-

creases slower with the time t, but decreases with the distance X.

The second form shows that the distance X enters the equation

only in the form t - X and </;, - w respectively, and that thus for

a const-ant value of t - X the decrement is c 2ut, that is, in the

direction of propagation the energy dies out by the power dissi-

pation constant u.

Equations (10) to (12) apply to the case, when the direction

of propagation, that is, of wave travel, is toward increasing >..

Fur a wave traveling in opposite direction, the sign of X and thus

of w is reversed.

(e) If the now of power increases along tho line, more power

leaves every line eleuH'nt than enters it; that is, the line clement

is drained of its stored energy by the passage of the wave, and thus

the transient dies down with the time at a greater rate than corre-

spomls to the power dissipation.by r and g. That is, not all the

stored energy of the line elements supplies the power which is

being dissipated in the line element, but a part of the energy

leaves the line element in increasing the power which flows along

the line. The rate of dissipation thus is increased, and instead
of u, (u + s) enters the equation. That is, the e1q)onential time

decrement is

E-(u+ s)t

(13)

'

but inversely, along the line X the power flow increases, that is,

the intensity of the wave increases, by the same factor E+s>-, or

rather, the wave decreases along the line at a slower rate than

corresponds to the power dissipation.

The equations then become:
i i=ioE-cu+s)tE+.•>-cos (</;,-w--y)=-iocutcs(t->-)cos (</;,-w--y),
e= eoc<u+s>t E+s>- cos(</;,- w--y) = €0E- 11'cs(t- >-> cos (</;,-w--y ), 1 (1 4)

and the average power is

.

.

= E-2ut ]Jo= e~o E-2 (u+s)I E+2s>. e~o

E-2s(t-;>,)

1

(15)

iG r Ji)[,lff7TNIC /)/,'WI/ AR<:ES, 1r.,1 HS .1lN I) IM f>UL8ES.

TRAVELJ;'VG WAVES.

97

increase with the time, which in general is not possible; as the

traPsient must decrease with the time, by the power dissipation

in rand g. Standing waves and traveling waves, in which the coefficient
in the exponent of the time exponential is positive, that is, the wave increases with the time, may, however, occur in electric circuits in which the wave is supplied with energy from some outside source, as by a generating system flexibly connected (electrically) through an arc. Such waves then are "cumulative oscillations." They may either increase in intensity indefinitely, that is, up to destruction of the circuit insulation, or limit themselves by the power dissipation increasing with the increasing intensity of the oscillation, until it becomes equal to the power supply. Such oscillations, which frequently u.re most destructive ones, are met in electric systems as "arcing grounds," "grounded phase," etc. They are frequently called "undamped oscillations," and as such find a use in wireless telegraphy and telephony. Thus far, the only source of cumulative oscillation seems to be an energy supply over an arc, especially an unstable arc. In the self-limiting cumulative oscillation, the so-called damped oscillation, the transient becomes a permanent phenomenon. Our theoretical knowledge of the cumulative oscillations thus far is rather limited, however. •

An oscillogram of a "grounded phase " on a 154-mile threephase line, at 82 kilovolts, is given in Figs. 44 and 45. Fig. 44 shows current and voltage at the moment of formation of the ground; Fig. 4.5 the same one minute later, when the ground was fully developed.

An oscillogram of a cumulative oscillation in a 2500-kw. 100,000volt power transformer (GO-cycle system) is given in Fig. 46. It is caused by slvitching off 28 miles of line by high-tension switches, at 88 kilovolts. As seen, the oscillation rapidly increases in in-

tensity, until it stops by the arc extinguishing, or by the destruction of the transformer.
Of special interest is the limiting case,

- s = u;

in this case, u + s = 0, and the exponential function of time
vanishes, and current and voltage become

i = ioe:!:a>. cos(¢ =F w - 1'), (

e = Coe=">. cos (¢ =F w - -y), ~

(18)

18 ELECTRIC DISCHARGES, WA YES AND IMPULSES.

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

energy oscillation of the line the values 2 L and 2 C.. • "If

7r

7r

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

total conductance g, the values -2r and -2g appear. "

7r

7r

The values of zo, y0, u, ¢, and w are not changed hereby.

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

ing to the circuit of massed capacity, f = ~ , to the value
21r LC

! -- 4--V1-LC·

Thus the frequency of oscillation of a transmission line is

1

1

f = 4 VW = 4 u'

(20)

where

(21)

If l1 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, (Jo

(22)

the inductance, capacity, resistance, and conductance per unit

length of line, then

u = ½(~: + i:);

(23)

that is, the rate of decrease of the transient is independent of the length of the line, and merely depends on the line constants per unit length.
It then is (24)
where
(25)
is a constant of the line construction, but independent of the length of the line.
The frequency then is
(26)

~
.~...
~ l:;1'j
.t.-.-.
~
~
.....
:~:.
~ ~
0
Fig. 46.-cnlO0Ol.-Oscillogrum of Cumulative OsciUation in High-potential Coil of 2500-kw. Step-up Transformer· Caused by Disconnecting 28 Miles of 100,000-volt Transmission Line; High-temlion Switching.
<:o
(Cl

80 •ELECTRIC JJISCIIARGES, WAVES ,1ND /1v!PULSES.
um•t of osc1·11at1•on 1•s 311 , or a1so a quarter-wave. The same 1•s th?
case in Fig. 37C, etc. In the case 2, i = 0 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 l1 is a half-wave, but the middle, C,
of l1 is a node or point of zero power, and the oscillating unit again is a quarter-wave. In the same way, in Fig. 38B, the section l1 consists of 4 quarter-wave units, etc.

rI' ---e---..__._
I
Al I I

I

I

I

... _

I'I

, ....._ , II

IA
I

I
I
I ' I
',I
Bl I I

I
' 1 / I _,..-

I I
I
' I
' ', l

C,',...

Bl

I

',.._

I

--, ,,

I

.._..,

I

I '-, I
'--+ I

Bl

!

I ' ' iI

I

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 l1.

30. Substituting lo = 4 l1 into (26) gives as the frequency of

oscillation

f =lo-1110·

(30)

However,

if f

= frequency,

and v =

! ,
a

velocity

of

propagation,

the wave length lo is the distance traveled during one period:

to_= J1 = per1•0d,

(31)

TRAVELING WAVES.

101

When traveling waves and stationary waves occur simultaneously, very often the traveling wave prec1::1fos the stationary wave.
The phenomenon may start with a traveling wave !)r impulse, and this, by reflection at the ends of the circuit, and ·COflir.>ination of the reflected waves aml the main waves, gradually chai>gt;s.-to a stationary wave. In this case, the traveling wave has the sai~o frequency as the stationary wave resulting from it. In Fig. 47'is shown the reproduction of an oscillogram of the formation of a stationary oscillation iu a transmission line by the repeated re-

'<......._:.: ____;;:..>

:::::::,.., <::..:...:..::

Fig. 47. -col 1168. - Reproduction of :i.n Oscillogmm or Stationary Line Oscillation by Reflection of Impulse from Ends or Line. (The lowest curve
gives a 6()..cycle current as time mea::;ure.)

flection from the ends of the line of the single impulse caused by short circuiting the energized line at one en<l. In the beginning of a stationary oscillation of a compound circuit, that is, a circuit comprising sections of different constants, traveling waves frequently occur, by which the energy stored magnetically or dielectrically in the different circuit sections adjusts itself to the proportion corresponding to the stationary oscillation of the complete circuit. Such traveling waves then are local, and therefore of much higher

frequency than the final oscillation of the complete circuit, and thus die out at a faster rate. Occasionally they are shown by the oscillograph as high-f;requency oscillations intervening between

82 ELECTRIC DISCIIARGES, -WAVES AND IMPULSES.

If then <I> denotes the time angle and w the distance angle of the
fundamental wave, that is, <I> = 2 1r represents a complete cycle
and w = 2 1r a complete wave length of the fundamental wave,
the time and distance angles of the higher harmonics are

;3 <I>, 3 w, 5 <I>, fi w, 7 <I>, 7 w, etc.

A complex oscilln.tion, compr1smg waves of all possible frequencies, thus would have the form

+ a1 cos (</> =i= w - 1'1) 0.3 cos 3 (<I> =F w - 'Ya)

+ a,, cos i'5 (</> =F w - + -y5) . . . ,

(37)

and the length Z1 of the line then is represented by the angle
w = ; , arnl the oscillation called a qnarter-wave oscalation.

If the t,vo ends of the line Z1 have the same electrical charac-
t<'rh;;t.ics, that is, e = 0 at both ends, or i = 0, the longest possible wave has the length lo = 2 li, and the frequency

/o = -l =--1-,
crolo 2 crol1

or n.ny multiple (odd or even) thereof. If then <t> and w again represent the time and the distance
angles of the fumlmnentul wave, its harmonics have the respective time and distance angles

2 <I>, 2 w,
3 <I>, 3 w,
4 <I>, 4 w, etc.

A complex oscillation then has the form

+ Cl1 COS (<J> =t= W - 1'1) a'l COS 2 (<J> =F w - 'Y2)

+ a3 cos 3 (ip =F w - 'Ya) + . . . ,

(38)

and the length l1 of the line is represented by angle w1 = 1r, and the
oscillation is called a half-wave oscillation. The half-wave oscillation thus contains even as well as odd
harmonics, and thereby may have a wave shape, in which one half wave differs from the other.
Equations (37) and (38) are of the form of equation (17), but

Fig. 48B. - co10049. - Oscillogram of High-frequency Oscillation Preceding Low-frequency OsciHation of Compound Circuit Caused ;;

by Switching UH miles of 100,000 Volts Transmission Linc and Step-down Transformer off another 154 1\-Iiles of 100,000 Volts i.:..

Linc; High-tension Switching.

~

.t.-.-.
<:
Cw

~
~
l;:j
~

.....,. Fig. 49. -cn10036. -Oscillogmm of Oscillation of Compound Circuit Consisting of 154 Miles of 100,000 Volts Line and Step-up e0o
Transformer; Connecting and Disconnecting by Low-tension Switches. High-tension Current and Low-tension Voltage.

84 ELEC7'RJC DISCHARGES, WAVES AND IMPULSES.
Substituting u0 = 1 in equations (30) an<l (31) gives

(40)

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

measure,

Vt: J = Zo

~ = Lo = O = : 0 =

(41)

where e0 = maximum voltage, io = ma.ximum current.
That is, the natural impedance is the inductance, and the natural admittance is the capacity, per velocity unit of length, anti is the main characteristic constant of the line.
The equations of the current and voltage of the line oscillation then consist, by (19), of trigonometric terms

cos <p cos w, sin¢ cos w, cos¢ sin w, sin¢ sin w,

multiplied with the transient, e- ut, and would thus, in the most general case, be given by an expression of the form

i = cut ia1 cos cp cos w + b1 s~n cp c?s w + c1 cos cp sin w )

+ d1 sm ¢ sm wL

•

e = cw iai' cos¢ cos w + b1' sin¢ cos w + ci' cos ¢ sin w

42
(

)

+ di' sin ¢ sin w j,

and its higher harmonics, that is, terms, with
2 ¢, 2 w, 3 ¢, 3 w, 4 ¢, 4 w, etc.

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

'l'RA VF:J.,f N(J W,1 VES.

105

Such simple traveling waves frequently are called "inipulses." \Vhen such an impulse passes along the line, at any point of the line, the wave energy is zero up to the moment where the wave front of the impulse arrives. The energy then rises, more or less rapidly, depending on the steepness of the wave front, reaches a maximum, and then decreases again, about as shown in Fig. 50. The impulse thus is the combination of two waves,

Fiµ;. rm. - Traveling \\"an!.

one, which df'crcases very rapidly, e-<"+••> 1 , and thus prepon<leratcs in the beginning of the phenomenon, and one, which <lecreases slowly, e -(u - s> 1• Hence it may be expressed in the form:

(20)

where the value of the power-transfer constant s determines the

" st,et'})llPS:'l of wave front."
Figs. Gl to rs:3 :,;;how oscillograms of the propagation of such an

impul:-;e ov<'r au (artifitial) 1.r:m:-;111i8sio11 li1w of 130 miles,* of the

constants:

r = 93.6 ohms,
L = 0.3944 henrys, C = 1.135 microfarads,

\/5 thus of surge impedance zo = = 500 ohms.

The impulse is produced by a transformer charge.t
Its duration, as measuretl from the oscillograms, is T0 = 0.0030
second.
In Fig. 51, the end of the transmission line was connected to a
nonin<luctive resistance equal to the surge impedance, so as to

* For <lescription of the line see "Design, Construction, :rn<l Test of an Artificial Transmission Line," by J. 11. Cunningham, Procee<lings A.I.E.E., January, 1911.
t In the manner us <lescribc<l in "Disruptive Strength of Air an<l Oil with
Transient Voltages," by J. L. R. Hayden and C. P. Steinmetz, Trnnsactions A.I.E.E., 1010, page 1125. The magnetic energy o{ the transformer is, however, larger, about 4 joules, and the transformer contained au air gap, to give constant in<luctancc.

86 ELECTRIC DISCHARGES, lVAVES AND 11\J,PULSES.

That is, the voltage e can contain no even harmonics. If,
however, the voltage contains no even harmonics, the current produced by this voltage also can contain no even harmonics. That is, it must be

C2 = 0,
C,1 = 0,
= c6 0,

d2 = 0, d4 = O,
d6 = 0,

a2' = 0,
a/= 0,
al= o,

! b2' = O,
b/ = 0,
ba' = 0, etc.

(46)

The complete expression of the stationary oscillation in a circuit
i open at the end w = 0 and short-circuited at the end w = thus

l would be
i: = c"' i (c1 cos <I>+ d1 sin</>) sin w + (ca cos 3 <I>+ da sin 3 </>)

e

=

c

111

sin 3 w + . . . I,
I(ai'cos<1>+b1'sin<1>)cosw+(aa'cos3<t>+ba'sin3<1>)1

(4-)
1

cos 3 w + . . . 5.

•

(c) ARsuming now as instance that, in such a stationary oscilla-

tion as giv('ll by equation (47), the current in the circuit is zern
at the Rtartiug moment of the transient for </> = 0. Then the

Pquatiou of the current can contain no terms with cos</>, as these

would not be zero for <I> = 0.

That is, it must be

Ct= 0,

Cs= 0,

(48)

Cr,= 0,

At the moment, however, when the current is zero, the voltag~
of the stationary oscillation must be a maximum. As i = 0 for

<I> = 0, at this moment the voltage e must be a maximum, that

is, t,lie volt.age wave cnn contain no terms with sin</>, sin 3 <I>, etc.

This means

b1' = 0, )

ba' = 0, l

(49)

bs' = 0, etc.)

l Substituting (48) and (49) into equation (47) gives
i = c•t1 {d1 sin <I> sin w + da sin 3 <I> sin 3 w + ds sin 5 <f, sin 5 w

l + ... !,

('"O)

e = E-ut { a1' cos <I> cos w+aa' cos 3 <I> cos 3 w+as' cos 5 <f, cos 5 w a

+ ... i·

'l'R,lVELING WAVES.

107

give no reflection. The upper curve shows the voltage of the impulse ut the beginning, the mi<ldle curve in the middle, an<l the lower curve at the end of the line.
Fig. 52 gives the same three voltages, with the line open at the en<l. This osci!Iogram shows the repeated reflections of the voltage impulse from the ends of the line, -the open en<l an<l the transformer in<luctunce at the beginning. It also shows the increase of vol tagc by reflectiou.

~
Fig. 53. _; col1153. - Reproduction of Oscillogram of Propagation of Impulse Over Transmission Line; Reflection from Open End of Linc. Current.
Fig. 53 gives the current impulses at the beginning and the middle of the line, corresponding to the voltage impulses in Fig. 52, together with the primary current of the transformer, i0• This oscillogmm shows the reversals of current by reflection, and the formution of a stationary oscillation by the successive reflections of the traveling wave from the en<ls of the line.
• "".

LECTURE VIII.

TRAVELING WAVES.

33. In a. stationary oscillation of a circuit having uniformly distributed capacity and inductance, that is, the transient of a circuit storing energy in the dielectric and magnetic field, current and voltage are given by the expression

l i = ioc"t cos <<1> =i= w - ,,), f e = eoE-ut sin (</> =i= w - i'),

(1)

where <J, is the time angle, w the distance angle, u the exponential decrement, or the "power-dissipation constant," and io and e0 the maximum current and voltage respectively.
The power flow at any point of the circuit, that is, at any distance angle w, and at any time t, that is, time angle <J,, then is

p == ei, = eoioE- 2"t cos (<I> =F w - -y) sin(</> =F w - -y),

= ;e iOE- 2"1 Sin 2 (</, =F W - i'),

(2)

and the average power flow is

Po= avg p,

(3)

= 0.

Hence, in a stationary .oscillation, or standing wave. of a uniform circuit, the average flow of power, Po, is zero, and no power flows along the circuit, but there is a surge of power, of double frequency. That is, power flows first one way, during one-quarter cycle, and then in the opposite direction, during the next quartercycle, etc.
Such a transient wave thus is analogous to the permanent wave· of reactive power.
As in a stationary wave, current and voltage are in quadrature with each other, the question then arises, whether, and what
88