# Elementary lectures on electric discharges, waves and impulses, and other transients

ELEMENTARY LECTURES

ON

ELECTRIC DISCHARGES, WAVES
AND IMPULSES,

AND

OTHER TRANSIENTS

BY

CHARLES PROTEUS STEINMETZ, A.M., PH.D.

Past President, American Institute of Electrical Engineers

McGRAW-HILL BOOK COMPANY

239 WEST 39TH STREET, NEW YORK

6 BOUVERIE STREET, LONDON, E.C.

1911

by the
McGRAW-HILL BOOK COMPANY

Stanbope Press
F. H. GILSON COMPANY
BOSTON, U.S.A.

PREFACE.

In the following I am trying to give a short outline of those phenomena which have become the most important to the 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 have given a systematic study of these phenomena, as far as our present knowledge permits, which by necessity involves to a considerable extent the use of mathematics. As many engineers may not have the time or inclination to a mathematical study, I have endeavored to give in the following a descriptive exposition of the physical nature and meaning, the origin and effects, of these phenomena, with the use of very little and only the simplest form of mathematics, so as to afford a general knowledge of these phenomena to those engineers who have not the time to devote to a more extensive study, and also to serve as an introduction to the study of “Transient Phenomena.” I have, therefore, in the following developed these phenomena from the physical conception of energy, its storage and readjustment, and extensively used as illustrations oscillograms of such electric discharges, waves, and impulses, taken on industrial electric circuits of all kinds, as to give the reader a familiarity with transient phenomena by the inspection of their record on the photographic film of the oscillograph. I would therefore recommend the reading of the following pages as an introduction to the study of “Transient Phenomena,” as the knowledge gained thereby of the physical nature materially assists in the understanding of their mathematical representation, which latter obviously is necessary for their numerical calculation and predetermination.

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

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

CHARLES PROTEUS STEINMETZ.

October, 1911.

CONTENTS.

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1. Electric power and energy. Permanent and transient phenomena. Instance of permanent phenomenon; of transient; of combination of both. Transient as intermediary condition between permanents.
2. Energy storage in electric circuit, by magnetic and dielectric field. Other energy storage. Change of stored energy as origin of 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. Simplest transients: proportionality of cause and effect. Most electrical transients of this character. Discussion of simple transient of electric circuit. Exponential function as its expression. Coefficient of its exponent. Other transients: deceleration of ship.
5. Two classes of transients: single-energy and double-energy transients. Instance of car acceleration; of low-voltage circuit; of pendulum; of condenser discharge through inductive circuit. Transients of more than two forms of energy.
6. Permanent phenomena usually simpler than transients. Reduction of alternating-current phenomena to permanents by effective values and by symbolic method. Nonperiodic transients.
 Lecture II. — The Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7. Phenomena of electric power flow: power dissipation in conductor; electric field consisting of magnetic field surrounding conductor and electrostatic or dielectric field issuing from conductor. Lines of magnetic force; lines of dielectric force.
8. The magnetic flux, inductance, inductance voltage, and the energy of the magnetic field.
9. The dielectric flux, capacity, capacity current, and the energy of the dielectric field. The conception of quantity of electricity, electrostatic charge and condenser; the conception of quantity of magnetism.
10. Magnetic circuit and dielectric circuit. Magnetomotive force, magnetizing force, magnetic field intensity, and magnetic density. Permeability. Magnetic materials.
11. Electromotive force, electrifying force or voltage gradient. Dielectric field intensity and dielectric density. Specific capacity or permittivity. Velocity of propagation.
12. Tabulation of corresponding terms of magnetic and of dielectric field. Tabulation of analogous terms of magnetic, dielectric, and electric circuit.
 Lecture III. — Single-Energy Transients in Continuous-current Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
13. Single-energy transient represents increase or decrease of energy. Magnetic transients of low- and medium-voltage circuits. Single-energy and double-energy transients of capacity. Discussion of the transients of $\Phi$ , $i$ , $e$ , of inductive circuit. Exponential equation. Duration of the transient, time constant. Numerical values of transient of intensity 1 and duration 1. The three forms of the equation of the magnetic transient. Simplification by choosing the starting moment as zero of time.
14. Instance of the magnetic transient of a motor field. 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.
16. The magnetic transient of closing an inductive circuit. General method of separation of transient and of permanent terms during the transition period.
 Lecture IV. — Single-Energy Transients of Alternating-current Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
17. Separation of current into permanent and transient component. Condition of maximum and of zero transient. The starting of an alternating current; dependence of the transient on the phase; 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 field circuit. Relation between voltage, load, magnetic field flux, armature reaction, self-inductive reactance, and synchronous reactance of alternator. Ratio of momentary to permanent short-cicurit current.
20. The magnetic field transient at short circuit of alternator. Its effect on the armature currents, and on the field current. Numerical relation between the transients of magnetic flux, armature currents, armature reaction, and field current. The starting transient of the armature currents. The transient full-frequency pulsation of the
field current caused by it. Effect of inductance in the exciter field. Calculation and construction of the transient phenomena of a polyphase alternator short circuit.
21. The transients of the single-phase alternator short circuit. The permanent double- frequency pulsation of armature reaction and of field current. The armature transient depending on the phase of the wave. Combination of full-frequency transient and double-frequency permanent pulsation of field current, and the shape of the field current resulting therefrom. Potential difference at field terminal at short circuit, and its industrial bearing.
 Lecture V. — Single-Energy Transients of Ironclad Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
22. Absence of proportionality between current and magnetic flux in ironclad circuit. Numerical calculation by step-by-step method. Approximation of magnetic characteristic by Fröhlich’s formula, and its rationality.
23. General expression of magnetic flux in ironclad circuit. Its introduction in the differential equation of the transient. Integration, and calculation of a numerical instance. High-current values and steepness of ironclad magnetic transient, and its industrial bearing.
 Lecture VI. — Double-Energy Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
24. Single-energy transient, after separation from permanent term, as a steady decrease of energy. Double-energy transient consisting of energy-dissipation factor and energy-transfer factor. The latter periodic or unidirectional. The latter rarely of industrial importance.
25. Pulsation of energy during transient. Relation between maximum current and maximum voltage. The natural impedance and the natural admittance of the circuit. Calculation of maximum voltage from maximum current, and inversely. Instances of line short circuit, ground on cable, lightning stroke. Relative values of transient currents and voltages in different classes of circuits.
26. Trigonometric functions of the periodic factor of the transient. Calculation of the frequency. Initial values of current and voltage.
27. The power-dissipation factor of the transient. Duration of the double-energy transient the harmonic mean of the duration of the magnetic and of the dielectric transient. The dissipation exponent, and its usual approximation. The complete equation of the double-energy transient. Calculation of numerical instance.
 Lecture VII. — Line Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
28. Review of the characteristics of the double-energy transient: periodic and transient factor; relation between current and voltage; the periodic component and the frequency; the transient component and the duration; the initial values of current and voltage.
Modification for distributed capacity and inductance: the distance phase angle and the velocity of propagation; the time phase angle; the two forms of the equation of the line oscillation.
29. Effective inductance and effective capacity, and the frequency of the line oscillation. The wave length. The oscillating-line section as quarter wave length.
30. Relation between inductance, capacity, and frequency of propagation. Importance of this relation for calculation of line constants.
31. The different frequencies and wave lengths of the quarter-wave oscillation; of the half- wave oscillation.
32. The velocity unit of length. Its importance in compound circuits. Period, frequency, time, and distance angles, and the general expression of the line oscillation.
 Lecture VIII. — Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
33. The power of the stationary oscillation and its correspondence with reactive power of alternating currents. The traveling wave and its correspondence with effective power of alternating currents. Occurrence of traveling waves: the lightning stroke: The traveling wave of the compound circuit.
34. The flow of transient power and its equation. The power-dissipation constant and the power-transfer constant. Increasing and decreasing power flow in the traveling wave. The general equation of the traveling wave.
35. Positive and negative power- transfer constants. Undamped oscillation and cumulative oscillation. The arc as their source. The alternating-current transmission-line equation as special case of traveling wave of negative power-transfer constant.
36. Coexistence and combination of traveling waves and stationary oscillations. Difference from effective and reactive alternating waves. Industrial importance of traveling waves. Their frequencies. Estimation of their effective frequency if very high.
37. The impulse as traveling wave. Its equations. The wave front.
 Lecture IX. — Oscillations of the Compound Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
38. The stationary oscillation of the compound circuit. The time decrement of the total circuit, and the power-dissipation and power-transfer constants of its section. Power supply from section of low-energy dissipation to section of high-energy dissipation.
38. 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.
41. Voltage and current relation between the sections of a compound oscillating circuit. The voltage and current transformation at the transition points between circuit sections.
42. Change of phase angle at the transition points between sections of a compound oscillating circuit. Partial reflection at the transition point.
 Lecture X. — Inductance and Capacity of Round Parallel Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
43. Definition of inductance and of capacity. The magnetic and the dielectric field. The law of superposition of fields, and its use for calculation.
44. Calculation of inductance of two parallel round conductors. External magnetic flux and internal magnetic flux.
45. Calculation and discussion of the inductance of two parallel conductors at small distances from each other. Approximations and their practical limitations.
46. Calculation of capacity of parallel conductors by superposition of dielectric fields. Reduction to electromagnetic units by the velocity of light. Relation between inductance, capacity, and velocity of propagation.
47. Conductor with ground return, inductance, and capacity. The image conductor. Limitations of its application. Correction for penetration of eturn current in ground.
48. Mutual inductance between circuits. Calculation of equation, and approximation.
49. Mutual capacity between circuits. Symmetrical circuits and asymmetrical circuits. Grounded circuit.
50. The three-phase circuit. Inductance and capacity of 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.