This book was produced in EPUB format by the Internet Archive.

The book pages were scanned and converted to EPUB format automatically. This process relies on optical character recognition, and is somewhat susceptible to errors. The book may not offer the correct reading sequence, and there may be weird characters, non-words, and incorrect guesses at structure. Some page numbers and headers or footers may remain from the scanned page. The process which identifies images might have found stray marks on the page which are not actually images from the book. The hidden page numbering which may be available to your ereader corresponds to the numbered pages in the print edition, but is not an exact match; page numbers will increment at the same rate as the corresponding print edition, but we may have started numbering before the print book's visible page numbers. The Internet Archive is working to improve the scanning process and resulting books, but in the meantime, we hope that this book will be useful to you.

The Internet Archive was founded in 1996 to build an Internet library and to promote universal access to all knowledge. The Archive's purposes include offering permanent access for researchers, historians, scholars, people with disabilities, and the general public to historical collections that exist in digital format. The Internet Archive includes texts, audio, moving images, and software as well as archived web pages, and provides specialized services for information access for the blind and other persons with disabilities.

Created with hocr-to-epub (v.1.0.0)

The text on this page is estimated to be only 25.44% accurate

GIFT OF ASSOCIATED ELECTRICAL AND MECHANICAL ENGINEERS MECHANICS DEPARTMENT

The text on this page is estimated to be only 27.40% accurate

ELECTRIC DISCHARGES, WAVES AND IMPULSES

The text on this page is estimated to be only 22.04% accurate

Published by the Me Grow -Hill Book. Company Ne^vYork. Successors to tKeBookDepartments of tKe McGraw Publishing Company Hill Publishing- Company Publishers of Books for Elec trical World The Engineering and Mining' Journal Engineering Record American Machinist Electric Railway Journal Coal Age Metallurgical and Chemical Engineering* Power

The text on this page is estimated to be only 25.10% accurate

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

The text on this page is estimated to be only 21.90% accurate

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

The text on this page is estimated to be only 29.00% accurate

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 (faa^tftmS'bf operation predetermined and controlled. Usually, however, the limitations of apparatus and lines are found not in the normal condition of operation, the steady flow of power, but in the phenomena occurring under abnormal though by no means unfrequent conditions, in the more or less transient abnormal voltages, currents, frequencies, etc.; and the study of the laws of these transient phenomen^4fee electric 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 749213

The text on this page is estimated to be only 28.89% accurate

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

The text on this page is estimated to be only 28.70% accurate

CONTENTS. PAGE 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. 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. vii

The text on this page is estimated to be only 28.82% accurate

Vlll CONTENTS. PAGE 11. Electromotive force, electrifying force or voltage gradient. Dielectric field intensity and dielectric density. Specific capacity or permittivity. Velocity of propagation. 12. Tabulation of corresponding terms of magnetic and of dielectric field. Tabulation of analogous terms of magnetic, dielectric, and electric circuit. LECTURE III. — SINGLE-ENERGY TRANSIENTS IN CONTINUOUS-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 4>, i, e, of inductive circuit. Exponential equation. Duration of the transient, time constant. Numerical values of transient of intensity 1 and duration 1. The three forms of the equation of the magnetic transient. Simplification by choosing the starting moment as zero of time. 14. Instance of the magnetic transient of a motor field. 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 bet ween 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

The text on this page is estimated to be only 29.08% accurate

CONTENTS. ix PAGE field current caused by it. Effect of inductance in the exciter field. Calculation and construction of the transient phenomena of a polyphase alternator short circuit. 21. The transients of the single-phase alternator short circuit. The permanent double- frequency pulsation of armature reaction and of field current. The armature transient depending on the phase of the wave. Combination of full-frequency transient and double-frequency permanent pulsation of field current, and the shape of the field current resulting therefrom. Potential difference at field terminal at short circuit, and its industrial bearing. LECTURE V. — SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. ... 52 22. Absence of proportionality between current and magnetic flux in ironclad circuit. Numerical calculation by step-by-step method. Approximation of magnetic characteristic by Frohlich's formula, and its rationality. 23. General expression of magnetic flux in ironclad circuit. Its introduction in the differential equation of the transient. Integration, and calculation of a numerical instance. High-current values and steepness of ironclad magnetic transient, and its industrial bearing. LECTURE VI. — DOUBLE-ENERGY TRANSIENTS 59 24. Single-energy transient, after separation from permanent term, as a steady decrease of energy. Double-energy transient consisting of energy-dissipation factor and energy-transfer factor. The latter periodic or unidirectional. The latter rarely of industrial importance. 25. Pulsation of energy during transient. Relation between 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.

The text on this page is estimated to be only 29.26% accurate

X CONTENTS. PAGE Modification for distributed capacity and inductance: the distance phase angle and the velocity of propagation; the time phase angle; the two forms of the equation of the line oscillation. 29. Effective inductance and effective capacity, and the frequency of the line oscillation. The wave length. The oscillating-line section as quarter wave length. 30. Relation between inductance, capacity, and frequency of propagation. Importance of this relation for calculation of line constants. 31. The different frequencies and wave lengths of the quarterwave oscillation; of the half- wave oscillation. 32. The velocity unit of length. Its importance in compound circuits. Period, frequency, time, and distance angles, and the general expression of the line oscillation. LECTURE VIII. — TRAVELING WAVES 88 33. The power of the stationary oscillation and its correspondence with reactive power of alternating currents. The traveling wave and its correspondence with effective power of alternating currents. Occurrence of traveling waves: the lightning stroke: The traveling wave of the compound circuit. 34. The flow of transient power and its equation. The powerdissipation constant and the power-transfer constant. Increasing and decreasing power flow in the traveling wave. The general equation of the traveling wave. 35. Positive and negative power- transfer constants. Undamped oscillation and cumulative oscillation. The arc as their source. The alternating-current transmission-line equation as special case of traveling wave of negative power-transfer constant. 36. Coexistence and combination of traveling waves and stationary oscillations. Difference from effective and reactive alternating waves. Industrial importance of traveling waves. Their frequencies. Estimation of their effective frequency if very high. 37. The impulse as traveling wave. Its equations. The wave front. LECTURE IX. — OSCILLATIONS OF THE COMPOUND CIRCUIT 108 38. The stationary oscillation of the compound circuit. The time decrement of the total circuit, and the power-dissipation and power-transfer constants of its section. Power supply from section of low-energy dissipation to section of high-energy dissipation. 39. Instance of oscillation of a closed compound circuit. The two traveling waves and the resultant transient-power diagram. 40. Comparison of the transient-power diagram with the power diagram of an alternating- current circuit. The cause of power increase in the line. The stationary oscillation of an open compound circuit.

The text on this page is estimated to be only 28.79% accurate

CONTENTS. xi PAGE 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 return current in ground. 48. Mutual inductance between circuits. Calculation of equation, and approximation. 49. Mutual capacity between circuits. Symmetrical circuits and asymmetrical circuits. Grounded circuit. 50. The three-phase circuit. Inductance and capacity of twowire single-phase circuit, of single-wire circuit with ground return, and of three-wire three-phase circuit. Asymmetrical arrangement of three-phase circuit. Mutual inductance and mutual capacity with three-phase circuit.

The text on this page is estimated to be only 28.29% accurate

ELEMENTAEY LECTURES ON ELECTEIC DISCHARGES, WAVES AND IMPULSES, AND OTHER TRANSIENTS. LECTURE I. NATURE AND ORIGIN OF TRANSIENTS. i. Electrical engineering deals with electric energy and its flow, that is, electric power. Two classes of phenomena are met: permanent and transient, phenomena. To illustrate: Let G in 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 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 e1 '; 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 S is closed, and therefore also in the moment of closing the switch, no current flows in the line A. Immediately after the switch S is closed, current begins to flow over line A into the condenser C, charging this condenser up to the voltage given by the generator. When the 1

The text on this page is estimated to be only 28.65% accurate

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

The text on this page is estimated to be only 28.95% accurate

NATURE AND ORIGIN OF TRANSIENTS. 3 in generator, line, and load does not represent the entire phenomenon. While electric power flows over the line A , there is a magnetic field surrounding the line conductors, and an electrostatic field issuing from the line conductors. The magnetic field and the electrostatic or "dielectric " field represent stored energy. Thus, during the permanent conditions of the flow of power through the circuit Fig. 3, there is electric energy stored in the space surrounding the line conductors. There is energy stored also in the generator and in the load ; for instance, the mechanical momentum of the revolving fan in Fig. 1, and the heat energy of the incandescent lamp filaments. The permanent condition of the circuit Fig. 3 thus represents not only flow of power, but also storage of energy. When the switch S is open, and no power flows, no energy is stored in the system. If we now close the switch, before the permanent condition corresponding to the closed switch can occur, Fig. 3. the stored energy has to be supplied from the source of power; that is, for a short time power, in supplying the stored energy, flows not only through the circuit, but also from the circuit into the space surrounding the conductors, etc. This flow of power, which supplies the energy stored in the permanent condition of the circuit, must cease as soon as the stored energy has been supplied, and thus is a transient. Inversely, if we disconnect some of the load L in Fig. 3, and thereby reduce the flow of power, a smaller amount of stored energy would correspond to that lesser flow, and before the conditions of the circuit can become stationary, or permanent (corresponding to the lessened flow of power), some of the stored energy has to be returned to the circuit, or dissipated, by a transient. Thus the transient is the result of the change of the amount of stored energy, required by the change of circuit conditions, and

The text on this page is estimated to be only 29.35% accurate

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

The text on this page is estimated to be only 27.92% accurate

NATURE AND ORIGIN OF TRANSIENTS. O change of the stored energy, of whatever form this energy may be, leads to a transient. Electrical transients have been studied more than transients of other forms of energy because : (a) Electrical transients generally are simpler in nature, and therefore yield more easily to a theoretical and experimental investigation. (b) The theoretical side of electrical engineering is further advanced than the theoretical side of most other sciences, and especially : (c) The destructive or harmful effects of transients in electrical systems are far more common and more serious than with other forms of energy, and the engineers have therefore been driven by necessity to their careful and extensive study. 4. The simplest form of transient occurs where the effect is directly proportional to the cause. This is generally the case in electric circuits, since voltage, current, magnetic flux, etc., are proportional to each other, and the electrical transients therefore are usually of the simplest nature. In those cases, however, where this direct proportionality does not exist, as for instance in inductive circuits containing iron, or in electrostatic fields exceeding the corona voltage, the transients also are far more complex, and very little work has been done, and very little is known, on these more complex electrical transients. Assume that in an electric circuit we have a transient current, as represented by curve i in Fig. 4 ; that is, some change of circuit condition requires a readjustment of the stored energy, which occurs by the flow of transient current i. This current starts at the value ii, and gradually dies down to zero. Assume now that the law of proportionality between cause and effect applies; that is, if the transient current started with a different value, izj it would traverse a curve if, which is the same as curve i, except that all values are changed proportionally, by the ratio ^; that is, i'=iX*iij ii Starting with current ii, the transient follows the curve i; starting with z'2, the transient follows the proportional curve i' . At some time, t, however, the current i has dropped to the value t'2, with which the curve i' started. At this moment t, the conditions in the first case, of current i, are the same as the conditions in

The text on this page is estimated to be only 23.42% accurate

6 ELECTRIC DISCHARGES, WAVES AND IMPULSES. the second case, of current if, at the moment t\; that is, from t onward, curve i is the same as curve i' from time i\ onward. Since t! Fig. 4. — Curve of Simple Transient: Decay of Current. if is proportional to i from any point t onward, curve i' is proportional to the same curve i from t\ onward. Hence, at time t\, it is diz dii ^. i% dti dti ii But since -^ and i2 at t\ are the same as -r and i at time t, it CLL\ dv follows: di dii i or, di where c = — - -r;1 = constant, and the minus sign is chosen, as ii at di . -r is negative. at As in Fig. 4: ~aJi = ii, 1^ dii _ tan _ _1_ . ~ ~ ~ '

The text on this page is estimated to be only 28.24% accurate

NATURE AND ORIGIN OF TRANSIENTS. 7 that is, c is the reciprocal of the projection T = tj* on the zero line of the tangent at the starting moment of the transient. Since di , — = — cdt; that is, the percentual change of current is constant, or in other words, in the same time, the current always decreases by the same fraction of its value, no matter what this value is. Integrated, this equation gives: log i = — ct + C, i = Ae~ct, or> i~#€:':*5 that is, the curve is the exponential. The exponential curve thus is the expression of the simplest form of transient. This explains its common occurrence in electrical and' other transients. Consider, for instance, the decay of radioactive substances : the radiation, which represents the decay, is proportional to the amount of radiating material; it is ~-r- = cm, Cit which leads to the same exponential function. Not all transients, however, are of this simplest form. For instance, the deceleration of a ship does not follow the exponential, but at high velocities the decrease of speed is a greater fraction of the speed than during the same time interval at lower velocities, and the speed-time curves for different initial speeds are not proportional to each other, but are as shown in Fig. 5. The reason is, that the frictional resistance is not proportional to the speed, but to the square of the speed. 5. Two classes of transients may occur: 1. Energy may be stored in one form only, and the only energy change which can occur thus is an increase or a decrease of the stored energy. 2. Energy may be stored in two or more different forms, and the possible energy changes thus are an increase or decrease of the total stored energy, or a change of the stored energy from one form to another. Usually both occur simultaneously. An instance of the first case is the acceleration or deceleration

The text on this page is estimated to be only 28.51% accurate

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 0 10 20 30 40 Seconds 50 60 70 80 90 100 110 120 Fig. 5. — Deceleration of Ship. deceleration. Thus also in a low-voltage electric circuit of negligible capacity, energy can be stored only in the magnetic field, and the transient represents an increase of the stored magnetic energy, during increase of current, or a decrease of the magnetic energy, during a decrease of current. An instance of the second case is the pendulum, Fig. 6 : with the weight at rest in maximum elevation, all the stored energy is potential energy of gravitation. This energy changes to kinetic mechanical energy until in the lowest position, a, when all the potential gravitational energy has been either converted to kinetic mechanical energy or dissipated. Then, during the rise of the weight, that part of the energy which 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 Fig. 6. Double-energy Transient of Pendulum.

The text on this page is estimated to be only 29.30% accurate

NATURE AND ORIGIN OF TRANSIENTS. energy. Thus in electric circuits containing energy stored in the magnetic and in the dielectric field, the change of the amount of stored energy — decrease or increase — frequently occurs by a series of successive changes from magnetic to dielectric and back again from dielectric to magnetic stored energy. This for instance is the case in the charge or discharge of a condenser through an inductive circuit. If energy can be stored in more than two different forms, still more complex phenomena may occur, as for instance in the hunting of synchronous machines at the end of long transmission lines, where energy can be stored as magnetic energy in the line and apparatus, as dielectric energy in the line, and as mechanical energy in the momentum of the motor. 6. The study and calculation of the permanent phenomena in electric circuits are usually far simpler than are the study and calculation of transient phenomena. However, only the phenomena of a continuous-current circuit are really permanent. The alternating-current phenomena are transient, as the e.m.f. continuously and periodically changes, and with it the current, the stored energy, etc. The theory of alternating-current phenomena, as periodic transients, thus has been more difficult than that of continuous-current phenomena, until methods were devised to treat the periodic transients of the alternating-current circuit as permanent phenomena, by the conception of the " effective values," and more completely by the introduction of the general number or complex quantity, which represents the periodic function of time by a constant numerical value. In this feature lies the advantage and the power of the symbolic method of dealing with alternating-current phenomena, — the reduction of a periodic transient to a permanent or constant quantity. For this reason, wherever periodic transients occur, as in rectification, commutation, etc., a considerable advantage is frequently gained by their reduction to permanent phenomena, by the introduction of the symbolic expression of the equivalent sine wave. Hereby most of the periodic transients have been eliminated from consideration, and there remain mainly the nonperiodic transients, as occur at any change of circuit conditions. Since they are the phenomena of the readjustment of stored energy, a study of the energy storage of the electric circuit, that is, of its magnetic and dielectric field, is of first importance.

The text on this page is estimated to be only 28.49% accurate

LECTURE II. THE ELECTRIC FIELD. 7. Let, in Fig. 7, a generator G transmit electric power over line A into a receiving circuit L. While power flows through the conductors A, power is consumed in these conductors by conversion into heat, repre sented by i?r. This, however, Fig. 7. is not all, but in the space surrounding the conductor certain phenomena occur: magnetic and electrostatic forces appear. Fig. 8. — Electric Field of Conductor. The conductor is surrounded by a magnetic field, or a magnetic flux, which is measured by the number of lines of magnetic force . With a single conductor, the lines of magnetic force are concentric circles, as shown in Fig. 8. By the return conductor, the circles 10

The text on this page is estimated to be only 28.32% accurate

THE ELECTRIC FIELD. 11 are crowded together between the conductors, and the magnetic field consists of eccentric circles surrounding the conductors, as shown by the drawn lines in Fig. 9. An electrostatic, or, as more properly called, dielectric field, issues from the conductors, that is, a dielectric flux passes between the conductors, which is measured by the number of lines of dielectric force ty. With a single conductor, the lines of dielectric force are radial straight lines, as shown dotted in Fig. 8. By the return conductor, they are crowded together between the conductors, and form arcs of circles, passing from conductor to return conductor, as shown dotted in Fig. 9. Fig. 9. — Electric Field of Circuit. The magnetic and the dielectric field of the conductors both are included in the term electric field, and are the two components of the electric field of the conductor. 8. The magnetic field or magnetic flux of the circuit,

The text on this page is estimated to be only 26.77% accurate

12 ELECTRIC DISCHARGES, WAVES AND IMPULSES. To produce the magnetic field $ of the current i, a voltage ef must be consumed in the circuit, which with the current i gives the power p, which supplies the stored energy w of the magnetic field . This voltage er 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 increase of the magnetic field: :' ; (3) or by (1), (4) If i and therefore $ decrease, -r and therefore e' are negative; that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power p is w = I p dt, or by (2) and (4), w = I Li di; hence L* (^ w = T (5) is the energy of the magnetic field $ = Li of the circuit. 9. Exactly analogous relations exist in the dielectric field. The dielectric field, or dielectric flux, ty} is proportional to the voltage 6, with a proportionality factor, C, which is called the capacity of the circuit: f = Ce. (6) The dielectric field represents stored energy, w. To produce it, power, p, must, therefore, be supplied by the circuit. Since power is current times voltage, p = i'e. (7) To produce the dielectric field ty of the voltage e, a current ir must be consumed in the circuit, which with the voltage e gives

The text on this page is estimated to be only 27.89% accurate

THE ELECTRIC FIELD. 13 the power p, which supplies the stored energy w of the dielectric field ^. This current i' is called the capacity current, or, wrongly, charging current or condenser current. Since no power is required to maintain the field, but power is required to produce it, the capacity current must be proportional to the increase of the dielectric field: or by (6), i' = C^. (9) de If e and therefore ^ decrease, -j- and therefore if are negative; that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power p is w=j*pdt, (10) or by (7) and (9), w = I Cede; hence rw « = £ (ID is the energy of the dielectric field t = Ce of the circuit. As seen, the capacity current is the exact analogy, with regard to the dielectric field, of the inductance voltage with regard to the magnetic field; the representations in the electric circuit, of the energy storage in the field. The dielectric field of the circuit thus is treated and represented in the same manner, and with the same simplicity and perspicuity, as the magnetic field, by using the same conception of lines of force. Unfortunately, to a large extent in dealing with the dielectric fields the prehistoric conception of the electrostatic charge on the conductor still exists, and by its use destroys the analogy between the two components of the electric field, the magnetic and the

The text on this page is estimated to be only 28.41% accurate

14 ELECTRIC DISCHARGES, WAVES AND IMPULSES. dielectric, and makes the consideration of dielectric fields unnecessarily complicated. There obviously is no more sense in thinking of the capacity current as current which charges the conductor with a quantity of electricity, than there is of speaking of the inductance voltage as charging the conductor with a quantity of magnetism. But while the latter conception, together with the notion of a quantity of magnetism, etc., has vanished since Faraday's representation of the magnetic field by the lines of magnetic force, the terminology of electrostatics of many textbooks still speaks of electric charges on the conductor, and the energy stored by them, without considering that the dielectric energy is not on the surface of the conductor, but in the space outside of the conductor, just as the magnetic energy. 10. All the lines of magnetic force are closed upon themselves, all the lines of dielectric force terminate at conductors, as seen in Fig. 8, and the magnetic field and the dielectric field thus can be considered as a magnetic circuit and a dielectric circuit. To produce a magnetic flux

The text on this page is estimated to be only 27.41% accurate

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 of / ampere turns per cm. in empty space. The magnetic density, in lines of magnetic force per cm2, produced by the field intensity 3C in any material is & = /z3C, (15) where ju is a constant of the material, a " magnetic conductivity," and is called the permeability. ^ = 1 or very nearly so for most materials, with the exception of very few, the so-called magnetic materials: iron, cobalt, nickel, oxygen, and some alloys and oxides of iron, manganese, and chromium. If then A is the section of the magnetic circuit, the total magnetic flux is $ = A®. (16) Obviously, if the magnetic field is not uniform, equations (13) and (16) would be correspondingly modified; / in (13) would be the average magnetizing force, while the actual magnetizing force would vary, being higher at the denser, and lower at the less dense, parts of the magnetic circuit: '-"• In (16), the magnetic flux $ would be derived by integrating the densities (B over the total section of the magnetic circuit. ii. Entirely analogous relations exist in the- dielectric circuit. To produce a dielectric flux ^, an electromotive force e is required, which is measured in volts. The e.m.f. per unit length of the dielectric circuit then is called the electrifying force or the voltage gradient, and is G = f- (18)unit distance with unit force. The unit field intensity, then, was defined as the field intensity at unit distance from unit magnetic mass, and represented by one line (or rather "tube") of magnetic force. The magnetic flux of unit magnetic mass (or "unit magnet pole") hereby became 4w lines of force, and this introduced the factor 4 TT into many magnetic quantities. An attempt to drop this factor 4 TT has failed, as the magnetic units were already too well established. The factor 1Q-1 also appears undesirable, but when the electrical units were introduced the absolute unit appeared as too large a value of current as practical unit, and one-tenth of it was chosen as unit, and called "ampere."

The text on this page is estimated to be only 27.38% accurate

16 ELECTRIC DISCHARGES, WAVES AND IMPULSES. This gives the average voltage gradient, while the actual gradient in an ummiform field, as that between two conductors, varies, being higher at the denser, and lower at the less dense, portion of the field, and is then is the dielectric-field intensity, and D = KK (20) would be the dielectric density, where K is a constant of the material, the electrostatic or dielectric conductivity, and is called the specific capacity or permittivity. For empty space, and thus with close approximation for air and other gases, 1 K — ~9» VL where v = 3 X 1010 is the velocity of light. It is customary, however, and convenient, to use the permittivity of empty space as unity: K = 1. This changes the unit of dielectric-field intensity by the factor — , and gives: dielectric-field intensity, dielectric density, = T^-oJ (21) 4 Try2 D = KK, (22) where K = 1 for empty space, and between 2 and 6 for most solids and liquids, rarely increasing beyond 6. The dielectric flux then is ^ = AD. (23) 12. As seen, the dielectric and the magnetic fields are entirely analogous, and the corresponding values are tabulated in the following Table I. * The factor 4 TT appears here in the denominator as the result of the factor 4*- in the magnetic-field intensity 5C, due to the relations between these quantities.

The text on this page is estimated to be only 25.31% accurate

THE ELECTRIC FIELD. TABLE I. 17 Magnetic Field. Dielectric Field. Magnetic flux: 4> = Li 108 lines of magnetic force. Dielectric flux: ^ = Ce lines of dielectric force. Inductance voltage: e'^n-^. 1Q-8 = L -jj volts. at at Capacity current: ., _ d^ _ „ di dt dt Magnetic energy: Li2. . w = -n- joules. Dielectric energy: Ce2 w = -=- joules. Magnetomotive force: F = ni ampere turns. Electromotive force: e = volts. Magnetizing force: F f = -r ampere turns per cm. Electrifying force or voltage gradient: a G = j volts per cm. Magnetic-field intensity: 3C = 47r/10-1 lines of magnetic force per cm2. Dielectric-field intensity: K = - — - lines of dielectric force 4 Try2 per cm2. Magnetic density: (B = M5C lines of magnetic force per cm2. Dielectric density: D = nK lines of dielectric force per cm2. Permeability: /* Permittivity or specific capacity: K Magnetic flux: $ = A($> lines of magnetic force. Dielectric flux: ^ = AD lines of dielectric force. v = 3 X 10 10 = velocity of light. The powers of 10, which appear in some expressions, are reduction factors between the absolute or cgs. units which are used for $, 3C, CB, and the practical electrical units, and 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.

The text on this page is estimated to be only 20.91% accurate

18 ELECTRIC DISCHARGES, WAVES AND IMPULSES. TABLE II. Magnetic Circuit. Dielectric Circuit. Electric Circuit. Magnetic flux (magnetic Dielectric flux (dielectric Electric current: current): current) : T ~~" T OIIIXIS. Magnetic energy: Dielectric energy: Electric power: w=— = — !Q-* joules. Ce2 e^ . , w = -JT- = -jr- joules. p = ri2 = ge2 — ei watts. Magnetic density: Dielectric density: Electric-current density: (B = -j =/z JClinespercm2. A. D = ^ = K/ninespercm2. 7 = -j = yG amA. perespercm2. Magnetizing force: Dielectric gradient: Electric gradient: F / = j ampere turns per G = j volts per cm. G =j volts per cm. cm. Magnetic-field intensity: Dielectric-field intensity: OC = AT/. j^ "" . Permeability: Permittivity or specific Conductivity: capacity: *-|K_D y = ~ mho — cm. Cr Reluctivity: (Elastivity ?): Resistivity: p = & 1 . K* 1 G , p = - = -?ohm — cm. y I Specific magnetic energy: Specific dielectric energy: Specific power: Awf2 /(B1A_8 KGZ GD . , Po = p/2 = G2 = GI W^O ~~* ^ — c\ ^-^ WQ — -. ; — ~~^: — JOUieS joules per cm3. 4irV2 2 ' per cm3. watts per cm3.

The text on this page is estimated to be only 27.89% accurate

LECTURE III. SINGLE-ENERGY TRANSIENTS IN CONTINUOUSCURRENT CIRCUITS. 13. The simplest electrical transients are those in circuits in which energy can be stored in one form only, as in this case the change of stored energy can consist only of an increase or decrease ; but no surge or oscillation between several forms of energy can exist. Such circuits are most of the low- and medium-voltage circuits, — 220 volts, 600 volts, and 2200 volts. In them the capacity is small, due to the limited extent of the circuit, resulting from the low voltage, and at the low voltage the dielectric energy thus is negligible, that is, the circuit stores appreciable energy only by the magnetic field. A circuit of considerable capacity, but negligible inductance, if of high resistance, would also give one form of energy storage only, in the dielectric field. The usual high-voltage capacity circuit, as that of an electrostatic machine, while of very small inductance, also is of very small resistance, and the momentary discharge currents may be very considerable, so that in spite of the very small inductance, considerable __ magnetic-energy storage may occur; that is, the system is one eo storing energy in two forms, and ^ oscillations appear, as in the dis- ' ~ charge of the Leyden jar. Fig 10._Magnetie Single.energy Let, as represented in Fig. 10, Transient, a continuous voltage e0 be impressed upon a wire coil of resistance r and inductance L (but negligible capacity). A current iQ = — flows through the coil and a magnetic field $0 10~8 = - - interlinks with the coil. Assuming now that the voltage e0 is suddenly withdrawn, without changing 19

The text on this page is estimated to be only 27.48% accurate

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 o, by the dissipation of its stored energy in the resistance of the coil circuit as i~r. Plotting, therefore, the magnetic flux of the coil as function of the time, in Fig. 11 A, the flux is constant and denoted by $0 up to the moment of Fig. 11. — Characteristics of Magnetic Single-energy Transient. time where the short circuit is applied, as indicated by the dotted line t0. From t0 on the magnetic flux decreases, as shown by curve

The text on this page is estimated to be only 27.43% accurate

SINGLE-ENERGY TRANSIENTS. 21 This is the e.m.f. induced by the decrease of magnetic flux 0, and the current i thus also full value iQ. Hence, at the first moment of short circuit, the induced e.m.f. e must be equal to eQ, that is, the magnetic flux $ must begin to decrease at such rate as to induce full voltage e0, as shown in Fig. 11C. The three curves ,

The text on this page is estimated to be only 27.93% accurate

22 ELECTRIC DISCHARGES, WAVES AND IMPULSES. shown dotted in Fig. 11C. The area of this new voltage curve would be e0T, and since it is the same as that of the curve e, as seen above, it follows that the area of the voltage curve e is = ri.r, and, combining (2) and (3), i0 cancels, and we get the value of T: : • .:' :V : T-\- >•'••; • (4) That is, the initial decrease of current, and therefore of magnetic flux and of induced voltage, is such that if the decrease continued at the same rate, the current, flux, and voltage would become zero after the time T = — • r The total induced voltage, that is, voltage times time, and therefore also the total current and magnetic flux during the transient, are such that, when maintained at their initial value, they would last for the time T = -=• • Since the curves of current and voltage theoretically never become zero, to get an estimate of the duration of the transient we may determine the time in which the transient decreases to half, or to one-tenth, etc., of its initial value. It is preferable, however, to estimate the duration of the transient by the time T, which it would last if maintained at its initial value. That is, the duration of a transient is considered as the time T = - • r This time T has frequently been called the " time constant " of the circuit. The higher the inductance L, the longer the transient lasts, obviously, since the stored energy which the transient dissipates is proportional to L. The higher the resistance r, the shorter is the duration of the transient, since in the higher resistance the stored energy is more rapidly dissipated. Using the time constant T = - as unit of length for the abscissa, and the initial value as unit of the ordinates, all exponential transients have the same shape, and can thereby be constructed

The text on this page is estimated to be only 24.47% accurate

SINGLE-ENERGY TRANSIENTS. by the numerical values of the exponential function, y = e given in Table III. TABLE III. Exponential Transient of Initial Value 1 and Duration 1. y = e~x. e = 2.71828. X y X y 0 1.000 1.0 0.368 0.05 0.951 1.2 0.301 0.1 0.905 1.4 0.247 0.15 0.860 1.6 0.202 0.2 0.819 1.8 0.165 0.25 0.779 2.0 0.135 0.3 0.741 2.5 0.082 0.35 0.705 3.0 0.050 0.4 0.670 3.5 0.030 0.45 0.638 4.0 0.018 0.5 0.607 4.5 0.011 0.6 0.549 5.0 0.007 0.7 0.497 6.0 0.002 0.8 0.449 7.0 0.001 0.9 0.407 8.0 0.000 1.0 0.368 As seen in Lecture I, the coefficient of the exponent of the single-energy transient, c, is equal to ^, where T is the projection of the tangent at the starting moment of the transient, as shown in Fig. 11, and by equation (4) was found equal to -. That is, r r and the equations of the single-energy magnetic transient, (1), thus may be written in the forms: I = lot~ c (t ~ 'o) = IQ€ e = e0e~c('~'o) = e0e - 7 # - to = ^Qe L , - y (t - *0) ^r /?„£ L (5) Usually, the starting moment of the transient is chosen as the zero of time, Zo = 0, and equations (5) then assume the simpler form:

The text on this page is estimated to be only 27.19% accurate

24 ELECTRIC DISCHARGES, WAVES AND IMPULSES, (6) The same equations may be derived directly by the integration of the differential equation: where L -=- is the inductance voltage, ri the resistance voltage. and their sum equals zero, as the coil is short-circuited. Equation (7) transposed gives hence logi =i = Ce~~L\ and, as for t = 0: i = to, it is: C = i0; hence 14. Usually single-energy transients last an appreciable time, and thereby become of engineering importance only in highly inductive circuits, as motor fields, magnets, etc. To get an idea on the duration of such magnetic transients, consider a motor field: A 4-polar motor has 8 ml. (megalines) of magnetic flux per pole, produced by 6000 ampere turns m.m.f. per pole, and dissipates normally 500 watts in the field excitation. That is, if IQ = field-exciting current, n = number of field turns per pole, r = resistance, and L = inductance of the field-exciting circuit, it is iQ2r = 500, hence 500

The text on this page is estimated to be only 26.05% accurate

SINGLE-ENERGY TRANSIENTS. 25 The magnetic flux is $0 = 8 X 106, and with 4 n total turns the total number of magnetic interlinkages thus is 4 n$0 = 32 n X 106, hence the inductance L0~8 .32 n T L = ^o , henrys. The field excitation is ra'o = 6000 ampere turns, 6000 hence hence and n = , .32 X 6000 , L = - —r henrys, *

The text on this page is estimated to be only 27.23% accurate

26 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The transients of magnetic flux, current, and voltage are shown as A, B, and C in Fig. 13. The magnetic flux and therewith the current decrease from the initial values $o and i0 at the moment to of opening the switch S, on curves which must be steeper than those in Fig. 11, since the current passes through a greater resistance, r + r', and thereby dissipates the stored magnetic energy at a greater rate. Fig. 13. — Characteristics of Magnetic Single-energy Transient. The impressed voltage eQ is withdrawn at the moment 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, to, the current is still iQ, and the induced voltage thus must be eo = io (r + r'), while the impressed voltage, before to, was eQ = ior; hence the induced voltage eo' is greater than the impressed voltage eo, in the same ratio as the resistance of the discharge circuit r + r' is greater than the resistance of the coil r through which the impressed voltage sends the current e0

The text on this page is estimated to be only 27.42% accurate

SINGLE-ENERGY TRANSIENTS. 27 The duration of the transient now is T L T = 7+-r" that is, shorter in the same proportion as the resistance, and thereby the induced voltage is higher. If rf = 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 : e0f = co ; that is, an infinite voltage would be induced. That is, the insulation of the coil would be punctured and the circuit closed in this manner. The more rapid, thus, the opening of an inductive circuit, the higher is the induced voltage, and the greater the danger of 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. e\ produces a current i\ in a circuit, and an e.m.f. ez produces in the same circuit a current i2, then the e.m.f. e\ + ez produces the current i\ -\- 1%, as is obvious. If now the voltage e\ + ez, and thus also the current ii + iZj consists of a permanent term, e\ and ii, and a transient term, e2 and iz, the transient terms ez, iz follow the same curves, when combined with the permanent terms e\, i\, as they would when alone in the circuit (the case above discussed). Thus, the preceding 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.

The text on this page is estimated to be only 27.53% accurate

28 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The same reasoning also applies to the transient resulting from several forms of energy storage (provided that the law of proportionality of i, e, $, etc., applies), and makes it possible, in investigating the phenomena during the transition period of energy readjustment, to separate the permanent and the transient term, and discuss them separately. B Fig. 14. — Single-energy Starting Transient of Magnetic Circuit. For instance, in the coil shown in Fig. 10, let the short circuit A be opened, that is, the voltage eQ be impressed upon the coil. At the moment of time, tQ, when this is done, current i, magnetic flux 0, e0 are reached. We may then, as discussed above, separate the transient from the permanent term, and consider that at the time U the coil has a permanent current i0, permanent flux 0, permanent voltage e0, and in addi 

The text on this page is estimated to be only 25.66% accurate

SINGLE-ENERGY TRANSIENTS. 29 tion thereto a transient current —i0j a transient flux — 0, e0. This is shown in Fig. 14. A shows the permanent flux 0, which are assumed, up to the time tQ, to give the resultant zero flux. The transient flux dies out by the curve , which is the transient from zero flux to the permanent flux 3>0. In the same manner B shows the construction of the actual current change i by the addition of the permanent current iQ and the transient current i', which starts from —iQ at to. C then shows the voltage relation: eQ the permanent voltage, e' the transient voltage which starts from — e0 at t0, and e the resultant or effective voltage in the coil, derived by adding eQ and e'.

The text on this page is estimated to be only 27.92% accurate

LECTURE IV. SINGLE-ENERGY TRANSIENTS IN ALTERNATINGCURRENT CIRCUITS. 17. Whenever the conditions of an electric circuit are changed in such a manner as to require a change of stored energy, a transition period appears, during which the stored energy adjusts itself from the condition existing before the change to the condition after the change. The currents in the circuit during the transition period can be considered as consisting of the superposition of the permanent current, corresponding to the conditions after the change, and a transient current, which connects the current value before the change with that brought about by the change. That is, if i\ = current existing in the circuit immediately before, and thus at the moment of the change of circuit condition, and i% = current which should exist at the moment of change in accordance with the circuit condition after the change, then the actual current ii can be considered as consisting of a part or component iz, and a component ii — iz — IQ. The former, iz, is permanent, as resulting from the established circuit condition. The current component IQ, however, is not produced by any power supply, but is a remnant of the previous circuit condition, that is, a transient, and , therefore gradually decreases in the manner as discussed in paragraph 13, that is, with a duration T = — The permanent current i2 may be continuous, or alternating, or may be a changing current, as a transient of long duration, etc. The same reasoning applies to the voltage, magnetic flux, etc. Thus, let, in an alternating-current circuit traversed by current t'i, in Fig. 15 A, the conditions be changed, at the moment t = 0, so as to produce the current i2. The instantaneous value of the current ii at the moment t = 0 can be considered as consisting of the instantaneous value of the permanent current i2, shown dotted, and the transient io = i\ — i*. The latter gradually dies down, with the duration T — — , on the usual exponential tran30

The text on this page is estimated to be only 27.72% accurate

SINGLE-ENERGY TRANSIENTS. 31 sient, shown dotted in Fig. 15. Adding the transient current iQ to the permanent current i2 gives the total current during the transition period, which is shown in drawn line in Fig. 15. As seen, the transient is due to the difference between the instantaneous value of the current i\ which exists, and that of the current i2 which should exist at the moment of change, and Fig. 15. — Single-energy Transient of Alternating-current Circuit. thus is the larger, the greater the difference between the two currents, the previous and the after current. It thus disappears if the change occurs at the moment when the two currents ii and 12 are equal, as shown in Fig. 15B, and is a maximum, if the change occurs at the moment when the two currents i\ and iz have the greatest difference, that is, at a point one-quarter period or 90 degrees distant from the intersection of i\ and 12, as shown in Fig. 15C.

The text on this page is estimated to be only 27.41% accurate

32 ELECTRIC DISCHARGES, WAVES AND IMPULSES. If the current ii is zero, we get the starting of the alternating current in an inductive circuit, as shown in Figs. 16, A, B, C. The starting transient is zero, if the circuit is closed at the moment when the permanent current would be zero (Fig. 16B), and is a maximum when closing the circuit at the maximum point of the permanent-current wave (Fig. 16C). The permanent current and the transient components are shown dotted in Fig. 16, and the resultant or actual current in drawn lines. B Fig. 16. — Single-energy Starting Transient of Alternating-current Circuit. 1 8. Applying the preceding to the starting of a balanced three-phase system, we see, in Fig. 17 A, that in general the three transients t'i°, i20, and 4° of the three three-phase currents ii, iz, is are different, and thus also the shape of the three resultant currents during the transition period. Starting at the moment of zero current of one phase, ii, Fig. 175, there is no transient for this current, while the transients of the other two currents, iz and i3, are equal and opposite, and near their maximum value. Starting, in Fig. 17C, at the maximum value of one current ia, we have the maximum value of transient for this current i'3°, while the transients of the two other currents, i\ and ii, are equal, have

The text on this page is estimated to be only 26.77% accurate

SINGLE-ENERGY TRANSIENTS. 33 half the value of 13°, and are opposite in direction thereto. In any case, the three transients must be distributed on both sides of the zero line. This is obvious: if ii, i2', and is' are the instantaneous values of the permanent three-phase currents, in Fig. 17, the initial values of their transients are: —i\, — iz, —isFig. 17. — Single-energy Starting Transient of Three-phase Circuit. Since the sum of the three three-phase currents at every moment is zero, the sum of the initial values of the three transient currents also is zero. Since the three transient curves ii°, i'2°, iz° are proportional to each other fas exponential curves of the same duration T = — ], and the sum of their initial values is zero, it follows

The text on this page is estimated to be only 29.14% accurate

34 ELECTRIC DISCHARGES, WAVES AND IMPULSES. that the sum of their instantaneous values must be zero at any moment, and therefore the sum of the instantaneous values of the resultant currents (shown in drawn line) must be zero at any moment, not only during the permanent condition, but also during the transition period existing before the permanent 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

The text on this page is estimated to be only 27.01% accurate

SINGLE-ENERGY TRANSIENTS. 35 the permanent and the transient components of the three currents, as is done in the preceding. Let ii, i2) is be the instantaneous values of the permanent currents at the moment of closing the circuit, t = 0. Combined, these would give the resultant field (Mo in Fig. 18. The three transient currents in this moment are i'i° =—ii, i^_== — i2, 13° =—i^', and combined these give a resultant field OB0, equal and opposite to OA0 in Fig. 18. The permanent field rotates synchronously on the concentric circle a; the transient field OB remains constant in the direction OB0, since all three transient components of current decrease in proportion to each other. It decreases, however, with the decrease of the transient current, that is, shrinks together on the line BQ0. The resultant or actual field thus is the combination of the permanent fields, shown as OAi OA2, . . . , and the transient fields, shown as OBi, OBZ, etc., and derived thereby by the parallelogram law, as shown in Fig. 18, as OC\, OC2, etc. In this diagram, Bid, B2C2) etc., are equal to OAi, OA2, etc., that is, to the radius of the permanent circle a. That is, while the rotating field in permanent condition is represented by the concentric circle a, the resultant field during the transient or starting period is represented by a succession of arcs of circles c, the centers of which move from BQ in the moment of start, on the line BQ0 toward 0, and can be constructed hereby by drawing from the successive points B0, BI} B2) which correspond to successive moments of time 0, tij t2 ... , radii BiCi, B2C2, etc., under the angles, that is, in the direction corresponding to the time 0, ^ t2, etc. This is done in Fig. 19, and thereby the transient of the rotating field is constructed. Fig. 19. — Starting Transient of Rotating Field: Polar Form.

The text on this page is estimated to be only 29.19% accurate

36 ELECTRIC DISCHARGES, WAVES AND IMPULSES. _From this polar diagram of the rotating field, in Fig. 19, values OC can now be taken, corresponding to successive moments of time, and plotted in rectangular coordinates, as done in Fig. 20. As seen, the rotating field builds up from zero at the moment of closing the circuit, and reaches the final value by a series of 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 Transient of Rotating Field: Rectangular Form. We have here an oscillatory transient, produced in a system with only one form of stored energy (magnetic energy), by the combination of several simple exponential transients. However, it must be considered that, while energy can be stored in one form only, as magnetic energy, it can be stored in three electric circuits, and a transfer of stored magnetic energy between the three electric circuits, and therewith a surge, thus can occur. It is interesting to note that the rot at ing-field transient is independent of the point of the wave at which the circuit is closed. That is, while the individual transients of the three three-phase currents vary in shape with the point of the wave at which they start, as shown in Fig. 17, their polyphase resultant always has the same oscillating approach to a uniform rotating field, of duration T — — • r 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 double value, and if then it reaches far above magnetic saturation, excessive momentary currents may appear, similar as in starting transformers of high magnetic density. In polyphase rotary

The text on this page is estimated to be only 28.89% accurate

SINGLE-ENERGY TRANSIENTS. 37 apparatus, however, these momentary starting currents usually are far more limited than in transformers, by the higher stray field (self-inductive reactance), etc., of the apparatus, resulting from the air gap in the magnetic circuit. 19. As instance of the use of the single-energy transient in engineering calculations may be considered the investigation of the momentary short-circuit phenomena of synchronous alternators. In alternators, especially high-speed high-power mar chines as turboalternators, the momentary short-circuit current may be many times greater than the final or permanent shortcircuit current, and this excess current usually decreases very 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 and of double frequency, and in the beginning the armature currents also show unsymmetrical shapes. Some oscillograms of three-phase, quarter-phase, and single-phase short circuits of turboalternators are shown in Figs. 25 to 28. By considering the transients of energy storage, these rather complex-appearing phenomena can be easily understood, and predetermined from the constants of the machine with reasonable exactness. In an alternator, the voltage under load is affected by armature reaction and armature self-induction. Under permanent condition, both usually act" in the same way, reducing the voltage at noninductive and still much more at inductive load, and increasing it at antiinductive load; and both are usually combined in one quantity, the synchronous reactance XQ. In the transients resulting from circuit changes, as short circuits, the self-inductive armature reactance and the magnetic armature reaction act very differently:* the former is instantaneous in its effect, while the latter requires time. The self-inductive armature reactance Xi consumes a voltage x\i by the magnetic flux surrounding the armature conductors, which results from the m.m.f . of the armature current, and therefore requires a component of the magnetic-field flux for its production. As the magnetic flux and the current which produces it must be simultaneous (the former being an integral part of the phenomenon of current flow, as seen in Lecture II), it thus follows that the armature reactance appears together * So also in their effect on synchronous operation, in hunting, etc.

The text on this page is estimated to be only 28.32% accurate

38 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 xz = XQ — Xi of the synchronous reactance which corresponds to the armature reaction is not a true reactance at all, consumes no voltage, but represents the consumption of field ampere turns by the m.m.f. of the armature current and the corresponding change of field flux. Since, however, the field flux represents stored magnetic energy, it cannot change instantly, and the armature reaction thus does not appear instantaneously with the armature current, but shows a transient which is determined essentially by the constants of the field circuit, that is, is the counterpart of the field transient of the machine. If then an alternator is short-circuited, in the first moment only the true self -inductive part Xi of the synchronous reactance exists, and the armature current thus is i\ = — , where 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 x0, and the short-circuit current decreases to the value i'0 = — • XQ The ratio of the momentary short-circuit current to the permanent short-circuit current thus is, approximately, the ratio — = — > IQ Xi that is, synchronous reactance to self-inductive reactance, or armature reaction plus armature self-induction, to armature self-induction. In machines of relatively low self-induction and high armature reaction, the momentary short-circuit 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: — •

The text on this page is estimated to be only 26.98% accurate

SINGLE-ENERGY TRANSIENTS. 39 Thus it is: momentary short-circuit current _ open-circuit field flux * _ permanent short-circuit current ~" short-circuit field flux armature reaction plus self-induction _ synchronous reactance _ XQ self-induction self-inductive reactance ~~ x\ 20. Let $1 = field flux of a three-phase alternator (or, in general, polyphase alternator) at open circuit, and this alternator be shortcircuited at the time t = 0. The field flux then gradually dies down, by the dissipation of its energy in the field circuit, to the short-circuit field flux 3>0, as indicated by the curve $ in Fig. 21A. If m = ratio armature reaction plus self-induction _ XQ armature self-induction ~ x\ it is $1 = m$0, and the initial value of the field flux consists of the permanent part 0, and the transient part 0, or m times, and are represented in Fig. 21 B. The resultant m.m.f. of the armature currents, or the armature reaction, is proportional to the currents, and thus follows the same field transient, as shown by F in Fig. 2 1C. The field-exciting current is i0 at open circuit as well as in the permanent condition of short circuit. In the permanent condition of short circuit, the field current iQ combines with the armature reaction F0, which is demagnetizing, to a resultant m.m.f., which produces the short-circuit flux 3>0. During the transition period the field flux $ is higher than 3>0, and the resultant m.m.f. must therefore be higher in the same proportion. Since it is the difference between the field current and the armature reaction F, and the latter is proportional to 3>, the field current thus must also be * If the machine were open-circuited before the short circuit, otherwise the field flux existing before the short circuit. It herefrom follows that the momentary short-circuit current essentially depends on the field flux, and thereby the voltage of the machine, before the short circuit, but is practically independent of the load on the machine before the short circuit and the field excitation corresponding to this load.

The text on this page is estimated to be only 26.18% accurate

40 ELECTRIC DISCHARGES, WAVES AND IMPULSES. proportional to $>. Thus, as it is i = iQ at and the armature currents. B Fig. 21. — Construction of Momentary Short Circuit Characteristic 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 $1 against the demagnetizing armature reaction. In other words, the field flux $ decreases at such a rate as to induce in the field circuit the e.m.f. required to raise the field current in the proportion m, from iQ to i0f, and maintain it at the values corresponding to the transient i, Fig. 2 ID. As seen, the transients 3>; z'i, i'2, iz] F; i are proportional to each other, and are a field transient. If the field, excited by current iQ

The text on this page is estimated to be only 27.12% accurate

SINGLE-ENERGY TRANSIENTS. 41 at impressed voltage e0, were short-circuited upon itself, in the first moment the current in the field would still be iQ, and therefore-the voltage e0 would have to be induced by the decrease of magnetic flux ; and the duration of the field transient, as discussed in Lecture III, would be TQ = — ro The field current in Fig. 2 ID, of the alternator short-circuit transient, starts with the value ij = mi0, and if eQ is the e.m.f. supplied in the field-exciting circuit from a source of constant voltage supply, as the exciter, to produce the current i0f, the voltage Co' = meo must be acting in the field-exciting circuit; that is, in addition to the constant exciter voltage e0, a voltage (m — I)e0 must be induced in the field circuit by the transient of the magnetic flux. As a transient of duration — induces the voltage e0, TO to induce the voltage (m — I)e0 the duration of the transient must be -1 o — / -i \ ) (m- 1) TO where L0 = inductance, r0 = total resistance of field-exciting circuit (inclusive of external resistance). The short-circuit transient of an alternator thus usually is of shorter duration than the short-circuit transient of its field, the more so, the greater m, that is, the larger the ratio of momentary to permanent short-circuit current. In Fig. 21 the decrease of the transient is shown greatly exaggerated compared with the frequency of the armature currents, and Fig. 22 shows the curves more nearly in their actual proportions. The preceding would represent the short-circuit phenomena, if there were no armature transient. However, the armature circuit contains inductance also, that is, stores magnetic energy, and thereby gives rise to a transient, of duration T = — , where L = inductance, r = resistance of armature circuit. The armature transient usually is very much shorter in duration than the field transient. The armature currents thus do not instantly assume their symmetrical alternating values, but if in Fig. 215, iV, iz, is are the instantaneous values of the armature currents in the moment of start, t — 0, three transients are superposed upon these, and

The text on this page is estimated to be only 27.71% accurate

42 ELECTRIC DISCHARGES, WAVES AND IMPULSES. start with the values —ii, —iz, —is'. The resultant armature currents are derived by the addition of these armature transients upon the permanent armature currents, in the manner as discussed in paragraph 18, except that in the present case even the permanent armature currents ii, i2, is are slow transients. In Fig. 22B are shown the three armature short-circuit currents, in their actual shape as resultant from the armature transient and the field transient. The field transient (or rather its beginning) is shown as Fig, 22 A. Fig. 22B gives the three armature Fig. 22. — Momentary Short Circuit Characteristic of Three-phase Alternator. currents for the case where the circuit is closed at the moment when t'i should be maximum ; ii then shows the maximum transient, and iz and ^3 transients in opposite direction, of half amplitude. These armature transients rapidly disappear, and the three currents become symmetrical, and gradually decrease with the field 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

The text on this page is estimated to be only 28.51% accurate

SINGLE-ENERGY TRANSIENTS. 43 in Fig. 21(7 by F. During the initial part of the short circuit, however, while the armature transient is appreciable and the armature currents thus unsymmetrical, as seen in Fig. 225, their resultant polyphase m.m.f. also shows a transient, the transient of the rotating magnetic field discussed in paragraph 18. That is, it approaches the curve F of Fig. 21 C by a series of oscillations, as indicated in Fig. 21E. Since the resultant m.m.f. of the machine, which produces the flux, is the difference of the field excitation, Fig. 21 D and the armature reaction, then if the armature reaction shows an initial oscillation, in Fig. 21 E, the field-exciting current must give the same oscillation, since its m.m.f. minus the armature reaction gives the resultant field excitation corresponding to flux $>. The starting transient of the polyphase armature reaction thus appears in the field current, as shown in Fig. 22(7, as an oscillation of full machine frequency. As the mutual induction between armature and field circuit is not perfect, the transient pulsation of armature reaction appears with reduced amplitude in the field current, and this reduction is the greater, the poorer the mutual inductance, that is, the more distant the field winding is from the armature winding. In Fig. 22(7 a damping of 20 per cent is assumed, which corresponds to fairly good mutual inductance between field and armature, as met in turboalternators. If the field-exciting circuit contains inductance outside of the alternator field, as is always the case to a slight extent, the pulsations of the field current, Fig. 22(7, are slightly reduced and delayed in phase; and with considerable inductance intentionally inserted into the field circuit, the effect of this inductance would require consideration. From the constants of the alternator, the momentary shortcircuit characteristics can now be constructed. Assuming that the duration of the field transient is sec., (m —I)r0 the duration of the armature transient is T = ~ = .1 sec. And assuming that the armature reaction is 5 times the armature

The text on this page is estimated to be only 27.38% accurate

44 ELECTRIC DISCHARGES, WAVES AND IMPULSES. self-induction, that is, the synchronous reactance is 6 times the selfinductive reactance, — = m = 6. The frequency is 25 cycles. Xi If i is the initial or open-circuit flux of the machine, the short3>i 1 circuit flux is 0 = — = ~ $1, and the field transient $ is a tranm o sient of duration 1 sec., connecting $1 and

The text on this page is estimated to be only 29.15% accurate

SINGLE-ENERGY TRANSIENTS. 45 on the point of the wave at which the phenomenon begins, but not so in their resultant effect. 21. The conditions with a single-phase short circuit are different, since the single-phase armature reaction is pulsating, varying between zero and double its average value, with double the machine frequency. The slow field transient and its effects are the same as shown in Fig. 21, A to D. However, the pulsating armature reaction produces a corresponding pulsation in the field circuit. This pulsation is of double Fig. 23. — Symmetrical Momentary Single-phase Short Circuit of Alternator. frequency, and is not transient, but equally exists in the final shortcircuit current. Furthermore, the armature transient is not constant in its reaction on the field, but varies with the point of the wave at which the short circuit starts. Assume that the short circuit starts at that point of the wave where the permanent (or rather slowly transient) armature current should be zero: then no armature transient exists, and the armature current is symmetrical from the beginning, and shows the slow transient of the field, as shown in Fig. 23, where A

The text on this page is estimated to be only 28.99% accurate

46 ELECTRIC DISCHARGES, WAVES AND IMPULSES. is the field transient (the same as in Fig. 22 A) 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 Fig. 24. — Asymmetrical Momentary Single-phase Short Circuit of Alternator. current, and thus maximum of armature reaction, and a minimum for every zero value of armature current, and thus armature reaction. Such single-phase short-circuit transients have occasionally been recorded by the oscillograph, as shown in Fig. 27. Usually, however, the circuit is closed at a point of the wave where the permanent armature current would not be zero, and an armature transient appears, with an initial value equal, but opposite to, the initial value of the permanent armature current. This is shown in Fig. 24 for the case of closing the circuit at the moment where the

The text on this page is estimated to be only 29.05% accurate

SINGLE-ENERGY TRANSIENTS. 47 armature current should be a maximum, and its transient thus a maximum. The field transient

The text on this page is estimated to be only 27.50% accurate

48 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

The text on this page is estimated to be only 27.53% accurate

SINGLE-ENERGY TRANSIENTS. 49 frequency, and as the result an increase of voltage and a distortion of the quadrature phase occurs, as shown in the oscillogram Fig. 25. Various momentary short-circuit phenomena are illustrated by the oscillograms Figs. 26 to 28. Figs. 26A and 265 show the momentary three-phase short circuit of a 4-polar 25-cycle 1500-kw. steam turbine alternator. The Fig. 26 A. — CD9399. — Symmetrical. Fig. 2QB. — CD9397. — Asymmetrical. Momentary Three-phase Short Circuit of 1500-Kw. 2300- Volt Three-phase Alternator (ATB-4-1500-1800) . Oscillograms of Armature Current and Field Current. lower curve gives the transient of the field-exciting current, the upper curve that of one of the armature currents, — in Fig. 26A that current which should be near zero, in Fig. 26B that which should be near its maximum value at the moment where the short circuit starts. Fig. 27 shows the single-phase short circuit of a pair of machines in which the short circuit occurred at the moment in which the armature short-circuit current should be zero; the armature cur 

The text on this page is estimated to be only 27.14% accurate

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

The text on this page is estimated to be only 29.32% accurate

SINGLE-ENERGY TRANSIENTS. 51 symmetrical, and the double-frequency pulsation of the field current shows during the first few cycles the alternate high and low peaks resulting from the superposition of the full-frequency transient pulsation of the rotating magnetic field of armature reaction. Interesting in this oscillogram is the irregular initial decrease of the armature current and the sudden change of its wave shape, which is the result of the transient of the current transformer, through which the armature current was recorded. On the true armaturecurrent transient superposes the starting transient of the current transformer. Fig. 25 shows a single-phase short circuit of a quarter-phase alternator; the upper wave is the voltage of the phase which is not short-circuited, and shows the increase and distortion resulting from the double-frequency pulsation of the armature reaction.

The text on this page is estimated to be only 28.68% accurate

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 field, etc. ; and the proportionality between current or voltage and their respective fields, the magnetic and the dielectric, thus ceases, or, as it may be expressed, the inductance L is not constant, but varies with the current, or the capacity is not constant, but varies with the voltage. The most important case is that of the ironclad magnetic circuit, as it exists in one of the most important electrical apparatus, the alternating-current transformer. If the iron magnetic circuit contains an air gap of sufficient length, the magnetizing force consumed in the iron, below magnetic saturation, is small compared with that consumed in the air gap, and the magnetic flux, therefore, is proportional to the current up to the values where magnetic saturation begins. Below saturation values of current, the transient thus is the simple exponential discussed before. If the magnetic circuit is closed entirely by iron, the magnetic flux is not proportional to the current, and the inductance thus not constant, but varies over the entire range of currents, following the permeability curve of the iron. Furthermore, the transient due to a decrease of the stored magnetic energy differs in shape and in value from that due to an increase of magnetic energy, since the rising and decreasing magnetization curves differ, as shown by the hysteresis cycle. Since no satisfactory mathematical expression has yet been found for the cyclic curve of hysteresis, a mathematical calculation is not feasible,, but the transient has to be calculated by an '^''"r '*_/ ? :,": \ : 52

The text on this page is estimated to be only 27.29% accurate

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: which is usually represented in the form given by Kennelly: T/> p = - = a + crOC; (2) that is, the reluctivity is a linear function of the field intensity. It gives a fair approximation for higher magnetic densities. This formula is based on the fairly rational assumption that the permeability of the iron is proportional to its remaining magnetizability. That is, the magnetic-flux density (B consists of a component 3C, the field intensity, which is the flux density in space, and a component (B' = (B — 3C, which is the additional flux density carried by the iron. (B' is frequently called the " metallic-flux density." With increasing 3C, (B' reaches a finite limiting value, which in iron is about &x' = 20,000 lines per cm2. * At any density (B', the remaining magnetizability then is (B^' — (B', and, assuming the (metallic) permeability as proportional hereto, gives and, substituting gives a, = cftco'rc^ * See "On the Law of Hysteresis," Part II, A.I.E.E. Transactions, 1892, page 621.

The text on this page is estimated to be only 26.16% accurate

54 ELECTRIC DISCHARGES, WAVES AND IMPULSES. or, substituting 1_ 1 *** / t*« ,—fc / (/ • gives equation (1). For OC = 0 in equation (1), ^ = - ; for 3C = oo » = - ; that is, uv a: cr in equation (1), - = initial permeability, - = saturation value of Oi (7 magnetic density. If the magnetic circuit contains an air gap, the reluctance of the iron part is given by equation (2), that of the air part is constant, and the total reluctance thus is p = ft + ffK, where 3 = a plus the reluctance of the air gap. Equation (1), therefore, remains applicable, except that the value of a is increased. In addition to the metallic flux given by equation (1), a greater or smaller part of the flux always passes through the air or through space in general, and then has constant permeance, that is, is given by 23. In general, the flux in an ironclad magnetic circuit can, therefore, be represented as function of the current by an expression of the form where , . = & is that part of the flux which passes through 1 -f- ut the iron and whatever air space may be in series with the iron, and a is the part of the flux passing through nonmagnetic material. Denoting now L2 = nc 10-8, i where n = number of turns of the electric circuit, which is interlinked with the magnetic circuit, L2 is the inductance of the air part of the magnetic circuit, LI the (virtual) initial inductance, that is, inductance at very small currents, of the iron part of the mag 

The text on this page is estimated to be only 22.57% accurate

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 55 netic circuit, and =- the saturation value of the flux in the iron. 72,CJ>' d That is, for i = 0, — r- = Z/i ; and for i = oo , . and the integration of differential equation (7) then gives If then, for the time t = tQ, the current is i = i0, these values substituted in (8) give the integration constant C: T1log- + !T2logio + T- + ^o + C = 0, (9)

The text on this page is estimated to be only 25.84% accurate

56 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and, subtracting (8) from (9), gives 1 + 6i 5 ' (10) This equation is so complex in i that it is not possible to calculate from the different values of t the corresponding values of i; but inversely, for different values of i the corresponding values of t can be calculated, and the corresponding values of i and t, derived in this manner, can be plotted as a curve, which gives the single-energy transient of the ironclad magnetic circuit. Tra sient o Ironclad Inductive Circuit : t=2.92i + t-.6i j l+.6i (dotted: t = 1.0851g i— .50?) 2 3 4 5 Fig. 29. 6 seconds Such is done in Fig. 29, for the values of the constants a = 4 X 105, c = 4 X 104, b = .6, n = 300.

The text on this page is estimated to be only 27.50% accurate

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 57

The text on this page is estimated to be only 28.07% accurate

58 ELECTRIC DISCHARGES, WAVES AND IMPULSES. This gives T = 4 Assuming i0 = 10 amperes for t0 = 0, gives from (10) the equation : 4 T = 2.92 - 1 9.21 log10 ^ + . 921 .6 i Herein, the logarithms have been reduced to the base 10 by division with logwe = .4343. For comparison is shown, in dotted line, in Fig. 29, the transient of a circuit containing no iron, and of such constants as to give about the same duration: t = 1.0S5log™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 is slower in the range of medium magnetic densities. Thus, in ironclad transients very high-current values of short duration may occur, and such transients, as those of the starting current of alternating-current transformers, may therefore be of serious importance by their excessive current values. An oscillogram of the voltage and current waves in an 11,000-kw. high-voltage 60-cycle three-phase transformer, when switching onto the generating station near the most unfavorable point of the wave, is reproduced in Fig. 30. As seen, an excessive current rush persists for a number of cycles, causing a distortion of the voltage wave, and the current waves remain unsymmetrical for many cycles.

The text on this page is estimated to be only 27.98% accurate

LECTURE VI. DOUBLE-ENERGY TRANSIENTS. 24. In a circuit in which energy can be stored in one form only, the change in the stored energy which can take place as the result of a change of the circuit conditions is an increase or decrease. The transient can be separated from the permanent condition, and then always is the representation of a gradual decrease of energy. Even if the stored energy after the change of circuit conditions is greater than before, and during the transition period an increase of energy occurs, the representation still is by a decrease of the transient. This transient then is the difference between the energy storage in the permanent condition and the energy storage during the transition period. If the law of proportionality between current, voltage, magnetic flux, etc., applies, the single-energy transient is a simple exponential function : j_ y = i/oe T°, (1) where ?/o = initial value of the transient, and TO = duration of the transient, that is, the time which the transient voltage, current, etc., would last if maintained at its initial value. The duration 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, T0 = £, (2) where L = inductance = coefficient of energy storage by the current, r = resistance = coefficient of power dissipation by the current. If the energy is stored by the voltage e, as dielectric field, the duration of the transient would be TJ = -, (3) s/ 59

The text on this page is estimated to be only 28.13% accurate

60 ELECTRIC DISCHARGES, WAVES AND IMPULSES. where C = capacity = coefficient of energy storage by the voltage, in the dielectric field, and g = conductance = coefficient of power consumption by the voltage, as leakage conductance by the voltage, corona, dielectric hysteresis, etc. Thus the transient of the spontaneous discharge of a condenser would be represented by e = e0e~£ct. (4) Similar single-energy transients may occur in other systems. For instance, the transient by which a water jet approaches constant velocity when falling under gravitation through a resisting medium would have the duration T = -°, (5) where VQ = limiting velocity, g = acceleration of gravity, and would be given by v = v0(l-6~r}. (6) In a system in which energy can be stored in two different forms, as for instance as magnetic and as dielectric energy in a circuit containing inductance and capacity, in addition to the gradual decrease of stored energy similar to that represented by the single-energy transient, a transfer of energy can occur between its two different forms. Thus, if i = transient current, e = transient voltage (that is, the difference between the respective currents and voltages 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 w Li* Ce* ) 'T+-2-'£ (7) = Wm +Wd. > While the total energy W decreases by dissipation, Wm may be converted into Wd, or inversely. Such an energy transfer may be periodic, that is, magnetic energy may change to dielectric and then back again; or unidirectional, that is, magnetic energy may change to dielectric (or inversely, dielectric to magnetic), but never change back again; but the

The text on this page is estimated to be only 27.42% accurate

DOUBLE-ENERGY TRANSIENTS. 61 energy is dissipated before this. This latter case occurs when the dissipation of energy is very rapid, the resistance (or conductance) high, and therefore gives transients, which rarely are of industrial importance, as they are of short duration and of low power. It therefore is sufficient to consider the oscillating double-energy transient, that is, the case in which the energy changes periodically between its two forms, during its gradual dissipation. This may be done by considering separately the periodic transfer, or pulsation of the energy between its two forms, and the gradual dissipation of energy. A . Pulsation of energy. 25. The magnetic energy is a maximum at the moment when the dielectric energy is zero, and when all the energy, therefore, is magnetic ; and the magnetic energy is then where t'0 = maximum transient current. The dielectric energy is a maximum at the moment when the magnetic energy is zero, and all the energy therefore dielectric, and is then Ce02 2 ' where e0 = maximum transient voltage. As it is the same stored energy which alternately appears as magnetic and as dielectric energy, it obviously is W _ Ceo2 ~2~ ~2" This gives a relation between the maximum transient current and the maximum transient voltage: v/: -^ therefore is of the nature of an impedance z0, and is called the natural impedance, or the surge impedance, of the circuit ; and fc its reciprocal, V/y = yo, is the natural admittance, or the surge T J j admittance, of the circuit.

The text on this page is estimated to be only 27.61% accurate

62 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The maximum transient voltage can thus be calculated from the maximum transient current: #0 = 'Z'O V/ 7> = i&Qj (10) and inversely, /C io = eo y j = e02/o. (11) This relation is very important, as frequently in double-energy transients one of the quantities e$ or i0 is given, and it is important to determine the other. For instance, if a line is short-circuited, and the short-circuit current IQ suddenly broken, the maximum voltage which can be induced by the dissipation of the stored magnetic energy of the short-circuit current is e0 = igZo. If one conductor of an ungrounded cable system is grounded, the maximum momentary current which may flow to ground is io = eo2/o, where e0 = 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 i0 = e<>yo. If L is high but C low, as in the high-potential winding of a high-voltage transformer (which winding can be considered as a circuit of distributed capacity, inductance, and resistance), z0 is high and T/O 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 oscillating voltages. Inversely, if L is low and C high, as in an underground cable, ZQ is low but 2/0 high, and even moderate oscillating voltages 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, as 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

The text on this page is estimated to be only 28.48% accurate

DOUBLE-ENERGY TRANSIENTS. 63 oscillating voltages, that is, acts as a short circuit for the transformer oscillation, and therefore protects the latter. Inversely, if the large oscillating current of a cable enters a reactive device, as a current transformer, it produces enormous voltages therein. Thus, cable oscillations are more liable to be destructive to the reactive apparatus, transformers, etc., connected with the cable, than to the cable itself. A transmission line is intermediate in the values of z0 and yQ between the cable and the reactive apparatus, thus acting like a reactive apparatus to the former, like a cable toward the latter. Thus, the transformer is protected by the transmission line in oscillations originating in the transformer, but endangered by the transmission line in oscillations originating in the transmission line. The simple consideration of the relative values of ZQ = V ^ in the different parts of an electric system thus gives considerable information on the relative danger and protective action of the parts on each other, and shows the reason why some elements, as current transformers, are far more liable to destruction than others; but also shows that disruptive effects of transient voltages, observed in one apparatus, may not and very frequently do not originate in the damaged apparatus, but originate in another part of the system, in which they were relatively harmless, and become dangerous only when entering the former apparatus. 26. If there is a periodic transfer between magnetic and dielectric energy, the transient current i and the transient voltage e successively increase, decrease, and become zero. The current thus may be represented by i = locosfa -7), (12) where iQ is the maximum value of current, discussed above, and = 27Tft, (13) where / = the frequency of this transfer (which is still undetermined), and 7 the phase angle at the starting moment of the transient; that is, ii = IQ cos 7 = initial transient current. (14) As the current i is a maximum at the moment when the magnetic energy is a maximum and the dielectric energy zero, the voltage e

The text on this page is estimated to be only 28.53% accurate

64 ELECTRIC DISCHARGES, WAVES AND IMPULSES. must be zero when the current is a maximum, and inversely; and if the current is represented by the cosine function, the voltage thus is represented by the sine function, that is, e = e0 sin (0 - 7), (15) where ei = — e0 sin 7 = initial value of transient voltage. (16) The frequency / is still unknown, but from the law of proportionality it follows that there must be a frequency, that is, the successive conversions between the two forms of energy must occur in equal time intervals, for this reason: If magnetic energy converts to dielectric and back again, at some moment the proportion between the two forms of energy must be the same again as at the starting moment, but both reduced in the same proportion by the power dissipation. From this moment on, the same cycle then must repeat with proportional, but proportionately lowered values. Fig. 31. — CD10017. — Oscillogram of Stationary Oscillation of Varying Frequency: Compound Circuit of Step-up Transformer and 28 Miles of 100,000-volt Transmission Line. If, however, the law of proportionality does not exist, the 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, caused by switching transformer and 28-mile line by low-tension switches off a substation at the end of a 153-mile transmission line, at 88 kv. With decreasing voltage, the magnetic density in the transformer

The text on this page is estimated to be only 28.01% accurate

DOUBLE-ENERGY TRANSIENTS. 65 decreases, and as at lower magnetic densities the permeability of the iron is higher, with the decrease of voltage the permeability of the iron and thereby the inductance of the electric circuit interlinked with it increases, and, resulting from this increased magnetic energy storage coefficient L, there follows a slower period of oscillation, 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 represented by a series of sine waves, and in this case the above equations (12) and (15) would still apply, but the calculation of the frequency / would give a number of values which represent the different component sine waves. The dielectric field of a condenser, or its " charge," is capacity times voltage: Ce. It is, however, the product of the current flowing into the condenser, and the time during which this current flows into it, that is, it equals i t. Applying the law Ce = it (17) to the oscillating energy transfer: the voltage at the condenser changes during a half-cycle from — eQ to -fe0, and the condenser charge thus is 2e0C; 2 the current has a maximum value i'0, thus an average value -i0, IT and as it flows into the condenser during one-half cycle of the frequency /, that is, during the time =-}, it is 2eQC = -io o7» 7T 2J which is the expression of the condenser equation (17) applied to the oscillating energy transfer. Transposed, this equation gives and substituting equation (10) into (18), and canceling with i0, gives

The text on this page is estimated to be only 26.23% accurate

66 ELECTRIC DISCHARGES, WAVES AND IMPULSES. as the expression of the frequency of the oscillation, where a = VLC (20) is a convenient abbreviation of the square root. The transfer of energy between magnetic and dielectric thus occurs with a definite frequency / = ~ — - , and the oscillation thus Z TTCT is a sine wave without distortion, as long as the law of proportionality applies. When this fails, the wave may be distorted, as seen on the oscillogram Fig. 31. The equations of the periodic part of the transient can now be written down by substituting (13), (19), (14), and (16) into (12) and (15) : i = io cos (0 — 7) = io cos 7 cos + i0 sin 7 sin and by (11): t IQ . t — i\ cos e\ — sin - , (7 €Q

The text on this page is estimated to be only 26.80% accurate

DOUBLE-ENERGY TRANSIENTS. 67 transient. In the latter case, the duration of the transient would be and with only half the energy magnetic, the duration thus is twice as long, or 2 T 7\=2T0 = ^=, (23) and hereby the factor multiplies with the values of current and voltage (21) and (22). /C Fig. 32. — Relation of Magnetic and Dielectric Energy of Transient. The same applies to the dielectric energy. If all the energy were dielectric, it would be dissipated by a transient of the duration: rp f k • IV--;

The text on this page is estimated to be only 26.54% accurate

68 ELECTRIC DISCHARGES, WAVES AND IMPULSES. as only half the energy is dielectric, the dissipation is half as rapid, that is, the dielectric transient has the duration T2 = 2 T0' = — , (24) y and therefore adds the factor 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 _JL _JL t(± + ±\ hk = € T^ T,= € \T^TJt (25) The duration of the double-energy transient, T, thus is given by I..! !_ T IV 2V 1/1 I (26) and this is the harmonic mean of the duration of the single-energy magnetic and the single-energy dielectric transient. It is, by substituting for T0 and TV, where u is the abbreviation for the reciprocal of the duration of the double-energy transient. Usually, the dissipation exponent of the double-energy transient is given as r 2L' This is correct only if g = 0, that is, the conductance, which represents 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

The text on this page is estimated to be only 24.77% accurate

DOUBLE-ENERGY TRANSIENTS. 69 transformers, etc., and is not the case in telegraph or telephone lines, etc. It is very nearly the case if the capacity is due to electrostatic condensers, but not if the capacity is that of electrolytic condensers, aluminum cells, etc. Combining now the power-dissipation equation (25) as factor with the equations of periodic energy transfer, (21) and (22), gives the complete equations of the double-energy transient of the circuit containing inductance and capacity: = € e = e t . t cos yQei sin - > > (7 flCCS - + z0ii sin / ' o- ° ) (28) where (29) a = VLC, (30) and ii and e\ are the initial values of the transient current and voltage respectively. As instance are constructed, in Fig. 33, the transients of current and of voltage of a circuit having the constants : L = 1.25 mh = 1.25 X 10~3 henrys; C = 2 mf = 2 X lO"6 "farads; r = 2.5 ohms; g = 0.008 mho, Inductance, Capacity, Resistance, Conductance, in the case, that The initial transient current, ii = 140 amperes; The initial transient voltage, e\ = 2000 volts. It is, by the preceding equations: a = Vie = 5 x io-5, / = - — = 3180 cycles per second, Z TTff ZQ = y ~ = 25 ohms, /C 2/o = y T = 0.04 mho,

The text on this page is estimated to be only 22.19% accurate

TO ELECTRIC DISCHARGES, WAVES AND IMPULSES. Tl = — = 0.001 sec. = 1 millisecond, 2C = 0.0005 sec. = 0.5 millisecond, 1 0.000333 sec. = 0.33 millisecond; 3000 X15& / i \ -1,0 £1 ,i\i \ \ \ Milliseconds \ Fig. 33. hence, substituted in equation (28), i = €-3

The text on this page is estimated to be only 23.51% accurate

DOUBLE-ENERGY TRANSIENTS. 71 Fig. 33 A gives the periodic components of current and voltage: if = 140 cos 0.2 1 - 80 sin 0.2 1, er = 2000 cos 0.2 1 + 3500 sin 0.2 1. Fig. 335 gives The magnetic-energy transient, h = e~', The dielectric-energy transient, k = e~2

The text on this page is estimated to be only 24.18% accurate

LECTURE VII. LINE OSCILLATIONS. 28. In a circuit containing inductance and capacity, the transient consists of a periodic component, by which the stored energy 7" /j'2 f^ r/>2 surges between magnetic -^- and dielectric — , and a transient £i A component, by which the total stored energy decreases. Considering only the periodic component, the maximum magnetic energy must equal the maximum dielectric energy, Lio2 _ Ceo2 "2" ~2~' where i0 = maximum transient current, e0 = maximum transient voltage. This gives the relation between eQ and io, e0 V/L_ 1 i-0 = \C-ZQ-yQ' where ZQ 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 ii = IQ cos (> — 7) ei = e0 sin (0 — 7) l where # = 2»ft (4) and ' = 27^ (5) is the frequency of oscillation. The transient component is hk = e-*, (6) 72

The text on this page is estimated to be only 25.55% accurate

LINE OSCILLATIONS. 73 where e = — €Q sin 7 hence the total expression of transient current and voltage is i = loe-^cos (0 - 7) 6 = eoe-^sinfa - 7) 7, e0, and i.Q follow from the initial values ef and i' of the transient, at £ = Oor 0 = 0: hence The preceding equations of the double-energy transient apply to the circuit in which capacity and inductance are massed, as, for instance, the discharge or charge of a condenser through an inductive circuit. Obviously, no material difference can exist, whether the capacity and the inductance are separately massed, or whether they are intermixed, a piece of inductance and piece of capacity alternating, or uniformly distributed, as in the transmission line, cable, etc. Thus, the same equations apply to any point of the transmission line. A B Fig. 34. However, if (8) are the equations of current and voltage at a point A of a line, shown diagrammatically in Fig. 34, at any other point B, at distance I from the point A, the same equations will apply, but the phase angle 7, and the maximum values eQ and IQ, may be different. Thus, if i = c

The text on this page is estimated to be only 24.82% accurate

74 ELECTRIC DISCHARGES, WAVES AND IMPULSES. are the current and voltage at the point A, this oscillation will appear at a point B, at distance I from A, at a moment of time later than at A by the time of propagation ti 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 — t\) enters. Or, if the oscillation travels from B to A, it is earlier at B by the time ti' that is, instead of the time t, the value (t + ti) 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 t\; that is, both expressions of (11), with (t — t\) and with (t + ti), will occur. The general form of the line oscillation thus is given by substituting (t T ti) instead of t into the equations (11), where t\ is the time of propagation over the distance I. If v = velocity of propagation of the electric field, which in air, as with a transmission line, is approximately v = 3 X 1010, (12) and in a medium of permeability /z and permittivity (specific capacity) K is 3 X 1010 ( . v =5 - T=^> (13) VfUJ and we denote ;•; • .v •'.,. a-j, ffifil (14) then ti = al; (15) and if we denote co = 27rM (16) we get, substituting t =F t\ for Z and 0 =F co for $ into the equation (11), the equations of the line oscillation: i = ce~ut cos (0 T co - 7) ) ,17, 6 = Z0ce-u( sin (> =F co — 7) ) In these equations, 0 = 2 7T/Z ^ is the time angle, and (18) co = 2 7r/aZ ) is the space angle, and c is the maximum value of current, ZQC the maximum value of voltage at the point I.

The text on this page is estimated to be only 28.73% accurate

LINE OSCILLATIONS. 75 Resolving the trigonometric expressions of equation (17) into functions of single angles, we get as equations of current and of voltage products of the transient e~ut, and of a combination of the trigonometric expressions: cos 0 cos co, sin 0 cos co, cos 0 sin co, sin 0 sin co. Line oscillations thus can be expressed in two different forms, either as functions of the sum and difference of time angle 0 and distance angle co: (0 ± co), as in (17); or as products of functions of 0 and functions of co, as in (19). The latter expression usually is more convenient to introduce the terminal conditions in stationary waves, as oscillations and surges; the former is often more convenient to show the relation to traveling waves. In Figs. 35 and 36 are shown oscillograms of such line oscillations. Fig. 35 gives the oscillation produced by switching 28 miles of 100-kv. line by high-tension switches onto a 2500-kw. step-up transformer in a substation at the end of a 153-mile threephase line; Fig. 36 the oscillation of the same system caused by switching on the low-tension side of the step-up transformer. 29. As seen, the phase of current i and voltage e changes progressively along the line Z, so that at some distance 1Q current and voltage are 360 degrees displaced from their values at the starting point, that is, are again in the same phase. This distance Z0 is called the wave length, and is the distance which the electric field travels during one period to = j of the frequency of oscillation. As current and voltage vary in phase progressively along the line, the effect of inductance and of capacity, as represented by the inductance voltage and capacity current, varies progressively, and the resultant effect of inductance and capacity, that is, the effective inductance and the effective capacity of the circuit, thus are not the sum of the inductances and capacities of all the line elements, but the resultant of the inductances and capacities of all the line elements combined in all phases. That is, the effective inductance and capacity are derived by multiplying the total 2 inductance and total capacity by avg/cos/, that is, by - •

The text on this page is estimated to be only 27.83% accurate

T6 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

The text on this page is estimated to be only 29.00% accurate

LINE OSCILLATIONS. 77

The text on this page is estimated to be only 24.96% accurate

78 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Instead of L and C, thus enter into the equation of the doubleO T Q H energy oscillation of the line the values - - and — . 7T 7T In the same manner, instead of the total resistance r and the 2 T 2 Q total conductance g, the values - — and - - appear. 7T 7T The values of z0, y0, u, 0, and co are not changed hereby. The frequency /, however, changes from the value corresponding to the circuit of massed capacity, / = - . , to the value 2 IT VLC f = 4 Vic * Thus the frequency of oscillation of a transmission line is where (7 = VLC. (21) If h is the length of the line, or of that piece of the line over which the oscillation extends, and we denote by LO, Co, TO, go (22) the inductance, capacity, resistance, and conductance per unit length of line, then -i / „ ~ \ (23) that is, the rate of decrease of the transient is independent of the length of the line, and merely depends on the line constants per unit length. It then is o- = Z*ro, (24) where — -\/T C* fOf^\ (TO — * J-JQ\s 0 \^"/ is a constant of the line construction, but independent of the length of the line. The frequency then is /.-rrr- (26)

The text on this page is estimated to be only 27.77% accurate

LINE OSCILLATIONS. 79 The frequency / depends upon the length Zi of the section of line in which the oscillation occurs. That is, the oscillations occurring in a transmission line or other circuit of distributed capacity have no definite frequency, but any frequency may occur, depending on the length of the circuit section which oscillates (provided that this circuit section is short compared with the entire length of the circuit, that is, the frequency high compared with the frequency which the oscillation would have if the entire line oscillates as a whole). If Zi is the oscillating line section, the wave length of this oscillation is four times the length Z0 = 4 ZL (27) This can be seen as follows: At any point I of the oscillating line section Zi, the effective power Po = avg ei = 0 (28) is always zero, since voltage and current are 90 degrees apart. The instantaneous power p = ei, (29) however, is not zero, but alternately equal amounts of energy flow first one way, then the other way. Across the ends of the oscillating section, however, no energy can flow, otherwise the oscillation would not be limited to this* section. Thus at the two ends of the section;~the instantaneous power, and thus either e or i, must continuously be zero. Three cases thus are possible: 1. e = 0 at both ends of Zx; 2. i = 0 at both ends of Zi; 3. e = 0 at one end, i = 0 at the other end of Zi. In the third case, i = 0 at one end, e = 0 at the other end of the line section Zi, the potential and current distribution in the line section Zi must be as shown in Fig. 37, A, B, C, etc. That is, Zi must be a quarter-wave or an odd multiple thereof. If Zi is a three-quarters wave, in Fig. 375, at the two points C and D the power is also zero, that is, Zi consists of three separate and independent oscillating sections, each of the length ^ ; that is, the o

The text on this page is estimated to be only 26.32% accurate

80 ELECTRIC DISCHARGES, WAVES AND IMPULSES. unit of oscillation is -5, or also a quarter-wave. o The same is the 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 li is a half-wave, but the middle, C, of li is a node or point of zero power, and the oscillating unit again is a quarter-wave. In the same way, in Fig. 385, the section /i consists of 4 quarter- wave units, etc. Fig. 37. Fig. 38. The same applies to case 1, and it thus follows that the wave length 10 is four times the length of the oscillation l\. 30. Substituting /0 = 4 li into (26) gives as the frequency of oscillation / = ^r • (30) However, if / = frequency, and v = - , velocity of propagation, the wave length 1Q is the distance traveled during one period: ^o = -* = period, (31)

The text on this page is estimated to be only 26.68% accurate

LINE OSCILLATIONS. 81 thus is Zo = trfo = ^., (32) and, substituting (32) into (31), gives a = (70, (33) or (34) This gives a very important relation between inductance LO and capacity Co per unit length, and the velocity of propagation. It allows the calculation of the capacity from the inductance, C0 = ^ , (35) and inversely. As in complex overhead structures the capacity usually is difficult to calculate, while the inductance is easily derived, equation (35) is useful in calculating the capacity by means of the inductance. This equation (35) also allows the calculation of the mutual capacity, and thereby the static induction between circuits, from the mutual magnetic inductance. The reverse equation, - (36) is useful in calculating the inductance of cables from their measured capacity, and the velocity of propagation equation (13). 31. If li is the length of a line, and its two ends are of different electrical character, as the one open, the other short-circuited, and thereby i = 0 at one end, e = 0 at the other end, the oscillation 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 Z0 = 4 Zi, and therefore the period tQ = cr0/o = 4 o-0/i, that is, the frequency /0 = -A —r . This is called the fundamental wave 4 ooti of oscillation. In addition thereto, all its odd multiples can exist as higher harmonics, of the respective wave lengths ^ ^ °_ and the frequencies (2 k — 1)/0, where k = 1, 2, 3 . . .

The text on this page is estimated to be only 25.98% accurate

82 ELECTRIC DISCHARGES, WAVES AND IMPULSES. If then 0 denotes the time angle and co the distance angle of the fundamental wave, that is, 0 = 2 TT represents a complete cycle and co = 2ir a complete wave length of the fundamental wave, the time and distance angles of the higher harmonics are 30, 3 co, 50, 5 co, 70, 7 co, etc. A complex oscillation, comprising waves of all possible frequencies, thus would have the form «i cos (0 =F co — 71) + a3 cos 3 (0 =F co — 73) + a5 cos 5 (0 T co - 75) + . . . , (37) and the length h of the line then is represented by the angle co = ~, and the oscillation called a quarter-wave oscillation. If the two ends of the line h have the same electrical characteristics, that is, e = 0 at both ends, or i = 0, the longest possible wave has the length 10 = 2 l\, and the frequency r 1 1 J° ~ T ~" o T ' " or any multiple (odd or even) thereof. If then 0 and co again represent the time and the distance angles of the fundamental wave, its harmonics have the respective time and distance angles 20, 2 co, 30, 3 co, 40, 4 co, etc. A complex oscillation then has the form a\ cos (0 =F co — 71) + «2 cos 2 (0 T co — 72) + a3 cos 3 (0 =F co - 73) + . . . , (38) and the length l\ of the line is represented by angle coi = TT, and the oscillation is called a half -wave oscillation. The half-wave oscillation thus contains even as well as odd harmonics, and thereby may have a wave shape, in which one half wave differs from the other. Equations (37) and (38) are of the form of equation (17), but

The text on this page is estimated to be only 28.11% accurate

LINE OSCILLATIONS. 83 usually are more conveniently resolved into the form oi equation (19). At extremely high frequencies (2 k —I)/, that is, for very large values of k, the successive harmonics are so close together that a very small variation of the line constants causes them to overlap, and as the line constants are not perfectly constant, but may vary slightly with the voltage, current, etc., it follows that at very high frequencies the line responds to any frequency, has no definite frequency of oscillation, but oscillations can exist of any frequency, provided this frequency is sufficiently high. Thus in transmission lines, resonance phenomena can occur only with moderate frequencies, but not with frequencies of hundred thousands or millions of cycles. 32. The line constants r0, go, L0, C0 are given per unit length, as per cm., mile, 1000 feet, etc. The most convenient unit of length, when dealing with transients in circuits of distributed capacity, is the velocity unit v. That is, choosing as unit of length the distance of propagation in unit time, or 3 X 1010 cm. in overhead circuits, this gives v = 1, and therefore "T 1 or GO — -j- ; LIQ — -ftj1 j -L/o That is, the capacity per unit of length, in velocity measure, is inversely proportional to the inductance. In this velocity unit of length, distances will be represented by X. Using this unit of length, <70 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 and therefore different wave lengths, as for instance an overhead line, the underground cable, in which the wave length is about one-half what it is in the overhead line (K = 4) and coiled windings, as the high-potential winding of a transformer, in which the wave length usually is relatively short. In the velocity measure of length, the wave length becomes the same throughout all these circuit sections, and the investigation is thereby greatly simplified.

The text on this page is estimated to be only 21.18% accurate

84 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Substituting O-Q = 1 in equations (30) and (31) gives ^o = Ao> O ^ 27rX CO = 2 7T/X = — — ; AO (40) and the natural impedance of the line then becomes, in velocity measure, 4 / LQ T 1 1 ^O /A1\ z° = V r = L° = T = ?T = T (41) ^o ^o 2/o ^o where e0 = maximum voltage, i0 = maximum current. That is, the natural impedance is the inductance, and the natural admittance is the capacity, per velocity unit of length, and is the main characteristic constant of the line. The equations of the current and voltage of the line oscillation then consist, by (19), of trigonometric terms cos 0 cos co, sin 0 cos cu, cos 0 sin co, sin sin co, multiplied with the transient, e~ut, and would thus, in the most general case, be given by an expression of the form i = e~ "* I ai cos > cos co + 61 sin 0 cos co + Ci cos 0 sin co -f-disin>sinco|, e _ €- ut | fll/ cos ^ cos w _|_ ^ sm ^ cos w _j_ Cl/ cos ^ sm w + di sin 0 sin co j , and its higher harmonics, that is, terms, with 20, 2 co, 30, 3 co, 40, 4 co, etc. In these equations (42), the coefficients a, 6, c, d, a', 6', c', d' are determined by the terminal conditions of the problem, that is, by the values of i and e at all points of the circuit co, at the

The text on this page is estimated to be only 25.28% accurate

LINE OSCILLATIONS. 85 beginning of time, that is, for respectively) at the ends of the circuit, that is, for co = 0 and co = •=• ft For instance, if: (a) The circuit is open at one end co = 0, that is, the current is zero at all times at this end. That is, i = 0 for co = 0; the equations of i then must not contain the terms with cos co, cos 2 co, etc., as these would not be zero for co = 0. That is, it must be Ol == 0, 61 = 0, ) a2 = 0, 62 = 0, (43) a3 = 0, 63 = 0, etc. ) The equation of i contains only the terms with sin co, sin 2 co, 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 the voltage must contain only the terms with cos co, cos 2 co, etc. Thus it must be ci' = 0, d/ = 0, ) C2' = 0, cV = 0, (44) c8' = 0, d8' = 0, etc.J Substituting (43) and (44) into (42) gives i = c~ui \d cos > + di sin

The text on this page is estimated to be only 25.59% accurate

86 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 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, ^ = 0, a/ = 0, 62' = 0, ) c4 = 0, d, = 0, a,' = 0, 64' = 0, (46) C6 = 0, d6 = 0, a6' = 0, 66' = 0, etc. ) The complete expression of the stationary oscillation in a circuit open at the end co = 0 and short-circuited at the end co = ^ thus would be i = e~ut I (ci cos 0 + di sin 0) sin co + (c3 cos 3 0 + d3 sin 3 0) sin3co + . . . j, e = e~ut I (a/ cos 0 + bi sin 0) cos co + (a/ cos 3 0 + 63' sin 3 0) cos 3 w +...}. (c) Assuming now as instance that, in such a stationary oscillation as given by equation (47), the current in the circuit is zero at the starting moment of the transient for 0 = 0. Then the equation of the current can contain no terms with cos 0, as these would not be zero for 0 = 0. That is, it must be c3 = 0, [ (48) c5 = 0, etc. ) At the moment, however, when the current is zero, the voltage of the stationary oscillation must be a maximum. As i = 0 for 0 = 0, at this moment the voltage e must be a maximum, that is, the voltage wave can contain no terms with sin 0, sin 3 0, etc. This means V = 0, ) 63' = 0, (49) 65' = 0, etc. ) Substituting (48) and (49) into equation (47) gives sin 0 sin co + d3 sin 3 0 sin 3 co + c?5 sin 5 0 sin 5 co (50) e = «-"' \ai cos cos w+O) cos 3 0 cos 3 w+o5' cos 5 cos 5 w + . . . 1.

The text on this page is estimated to be only 24.77% accurate

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, i0, and of voltage, eo, of a stationary oscillation exists, however, the relation where z0 is the natural impedance or surge impedance. That is a'=dz0) (51) and substituting (51) into (50) gives i — e~ut \ di sin sin co + d$ sin 3 sin 3 co + d$ sin 5 sin 5 co e = z0 €~ut I di cos 0 cos co + d3 cos 3

The text on this page is estimated to be only 27.14% accurate

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 i = iQe~ut cos (0 T co - 7), ) e = e0e~ut sin (> T co — 7), ) where 0 is the time angle, co the distance angle, u the exponential decrement, or the "power-dissipation constant," and i0 and eQ the maximunl current and voltage respectively. The power flow at any point of the circuit, that is, at any distance angle co, and at any time t, that is, time angle T co — 7) sin (0 =F co — 7), = ^|V2«=Fco-7), (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, p0, 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

The text on this page is estimated to be only 27.08% accurate

TRAVELING WAVES. 89 physical meaning a wave has, in which current and voltage are in phase with each other: i = loe~ut COS (0 =F co — 7), e = eQe~ut cos (

The text on this page is estimated to be only 29.00% accurate

90 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Such a flow of power must occur in a circuit containing sections of different dissipation constants u. For instance, if a circuit consists of an unloaded transformer and a transmission line, as indicated in Fig. 40, that is, at no load on the step-down trans^> Line Transformer Line Fig. 40. former, the high-tension switches are opened at the generator end of the transmission line. The energy stored magnetically and dielectrically in line and transformer then dissipates by a transient, as shown in the oscillogram Fig. 41. This gives the oscillation of a circuit consisting of 28 miles of line and 2500-kw. 100-kv. step-up and step-down transformers, and is produced by disconnecting this circuit by low-tension switches. In the transformer, the duration of the transient would be very great, possibly several seconds, as the stored magnetic energy (L) is very large, the dissipation of power (r and g) relatively small; in the line, the transient is of fairly short duration, as r (and g) are considerable. Left to themselves, the line oscillations thus would die out much more rapidly, by the dissipation of their stored energy, than the transformer oscillations. Since line and transformer are connected together, both must die down simultaneously by the same transient. It then follows that power must flow during the transient from the transformer into the line, so as to have both die down together, in spite of the more rapid energy dissipation in the line. Thus a transient in a compound circuit, that is, a circuit comprising sections of different constants, must be a traveling wave, that is, must be accompanied by power transfer between the sections of the circuit.* A traveling wave, equation (4), would correspond to the case of effective power in a permanent alternating-current circuit, while the stationary wave of the uniform circuit corresponds to the case of reactive power. Since one of*the most important applications of the traveling wave is the investigation of the compound circuit, it is desirable * In oscillogram Fig. 41, the current wave is shown reversed with regard to the voltage wave for greater clearness.

The text on this page is estimated to be only 28.67% accurate

TRAVELING WAVES. 91

The text on this page is estimated to be only 27.23% accurate

92 ELECTRIC DISCHARGES, WAVES AND IMPULSES. to introduce, when dealing with traveling waves, the velocity unit as unit of length, that is, measure the length with the distance of propagation during unit time (3 X 1010 cm. with a straight conductor in air) as unit of length. This allows the use of the same distance unit through all sections of the circuit, and expresses the wave length X0 and the period T0 by the same numerical values, X0 = TQ = -, and makes the time angle 0 and the distance angle co directly comparable: 0 = 2vft = 27T— , AO CO = 2 7T — = 2 7T/X. A (8) 34. If power flows along the circuit, three cases may occur: (a) The flow of power is uniform, that is, the power remains constant in the direction of propagation, as indicated by A in Fig. 42. B B c' A' B' Fig. 42. — Energy Transfer by Traveling Wave. (b) The flow of power is decreasing in the direction of propagation, as illustrated by B in Fig. 42. (c) The flow of power is increasing in the direction of propagation, as illustrated by C in Fig. 42. Obviously, in all three cases the flow of power decreases, due to the energy dissipation by r and g, that is, by the decrement e~ut. Thus, in case (a) the flow of power along the circuit decreases at

The text on this page is estimated to be only 28.12% accurate

TRAVELING WAVES. 93 the rate e~ut, corresponding to the dissipation of the stored energy by e-"', as indicated by A ' in Fig. 42; while in the case (6) the power flow decreases faster, in case (c) slower, than corresponds to the energy dissipation, and 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 = io — 7), e = e^~ut cos (0 - co - 7), (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 Transformer LoadCT Line ^-) Fig. 43. — Compound Circuit. connecting the step-up transformer with a load on the step-down end, and the step-up transformer is disconnected 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 should happen to be exactly the average, w0, 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.

The text on this page is estimated to be only 28.17% accurate

94 ELECTRIC DISCHARGES, WAVES AND IMPULSES. (b) If the flow of power decreases along the line, every line element receives more power at one end than it gives off at the other end. That is, energy is supplied to the line elements by the flow of power, and the stored energy of the line element thus decreases at a slower rate than corresponds to its power dissipation by r and g. Or, in other words, a part of the power dissipated in the line element is supplied by the flow of power along the line, and only a part supplied by the stored energy. Since the current and voltage would decrease by the term e~w

The text on this page is estimated to be only 27.51% accurate

TRAVELING WAVES. 95 the average power then is Po = avg e, -s)t e-2s\ — L° €-2ute+2s(t-\) ^ 2 Both forms of the expressions of i, e, and po of equations (11) and (12) are of use. The first form shows that the wave decreases 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 4> — co respectively, and that thus for a constant value of t — \ the decrement is e~2ut} that is, in the direction of propagation the energy dies out by the power dissipation constant u. Equations (10) to (12) apply to the case, when the direction of propagation, that is, of wave travel, is toward increasing X. For a wave traveling in opposite direction, the sign of X and thus of co is reversed. (c) If the flow of power increases along the line, more power leaves every line element than enters it; that is, the line element 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 corresponds 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 exponential time decrement is e~ — a — * and the average power is

The text on this page is estimated to be only 26.60% accurate

96 ELECTRIC DISCHARGES, WAVES AND IMPULSES. that is, the power decreases with the time at a greater, but with the distance at a slower, rate than corresponds to the power dissipation. For a wave moving in opposite direction, again the sign of X and thus of co would be reversed. 35. In the equations (10) to (15), the power-transfer constant s is assumed as positive. In general, it is more convenient to assume that s may be positive or negative; positive for an increasing, negative for a decreasing, flow of power. The equations (13) to (15) then apply also to the case (6) of decreasing power flow, but in the latter case s is negative. They also apply to the case (a) for s = 0. The equation of current, voltage, and power of a traveling wave then can be combined in one expression: i = '^ _ _ ^ Q = ^£~VW-r*Vc=csAr»rka i ^3^,., — /v } ==/?„£— at £—s u-r A; r»r\c f .^^ /.i — o/ 1 I V§W where the upper sign applies to a wave traveling in the direction toward rising values of X, the lower sign to a wave traveling in opposite direction, toward decreasing X. Usually, waves of both directions of travel exist simultaneously (and in proportions depending on the terminal conditions of the oscillating system, as the values of i and e at its ends, etc.). s = 0 corresponds to a traveling wave of constant power flow (case (a)). s > 0 corresponds to a traveling wave of increasing power flow, that is, a wave which drains the circuit over which it travels of some of its stored energy, and thereby increases the time rate of dying out (case (c)). s < 0 corresponds to a traveling wave of decreasing power flow, that is, a wave which supplies energy to the circuit over which it travels, and thereby decreases the time rate of dying out of the transient. If s is negative, for a transient wave, it always must be since, if — s > u, u -\- s would be negative, and e~(u + s}t would increase with the time; that is, the intensity of the transient would

The text on this page is estimated to be only 28.25% accurate

TRAVELING WAVES. 97 increase with the time, which in general is not possible, as the transient must decrease with the time, by the power dissipation in r and 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 are 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. 45 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 (60-cycle system) is given in Fig. 46. It is caused by switching off 28 miles of line by high-tension switches, at 88 kilovolts. As seen, the oscillation rapidly increases in intensity, 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 = i0e±sX cos (0 =F co — 7), e = e0e±sX cos (0 T co - 7), v

The text on this page is estimated to be only 27.57% accurate

98 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 8

The text on this page is estimated to be only 28.67% accurate

TRAVELING WAVES. 99

The text on this page is estimated to be only 26.62% accurate

100 ELECTRIC DISCHARGES, WAVES AND IMPULSES. that is, are not transient, but permanent or alternating currents and voltages. Writing the two waves in (18) separately gives cos (0 - co - 70 - i'0'e-sX e = e0e+sx cos (0 - co and these are the equations of the alternating-current transmission line, and reduce, by the substitution of the complex quantity for the function of the time angle , to the standard form given in "Transient Phenomena," Section III. 36. Obviously, traveling waves and standing waves may occur simultaneously in the same circuit, and usually do so, just as in alternating-current circuits effective and reactive waves occur simultaneously. In an alternating-current circuit, that is, in permanent condition, the wave of effective power (current in phase with the voltage) and 'the wave of reactive power (current in quadrature with the voltage) are combined into a single wave, in which the current is displaced from the voltage by more than 0 but less than 90 degrees. This cannot be done with transient waves. The transient wave of effective power, that is, the traveling wave, i = iQ€- ut €- s (t ±\) cos (^ =p w _ T)? e = eQ€~ ut e~ s (i ±X) cos (0 =F co — 7), cannot be combined with the transient wave of reactive power, that is, the stationary wave, i = io'e-ut cos (0 T co - 7'), e = e0'e-ut sin (

The text on this page is estimated to be only 28.93% accurate

TRAVELING WAVES. 101 When traveling waves and stationary waves occur simultaneously, very often the traveling wave precedes the stationary wave. The phenomenon may start with a traveling wave or impulse, and this, by reflection at the ends of the circuit, and combination of the reflected waves and the main waves, gradually changes to a stationary wave. In this case, the traveling wave has the same 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 in a transmission line by the repeated rei, Fig. 47. — CD11168. — Reproduction of an Oscillogram of Stationary Line Oscillation by Reflection of Impulse from Ends of Line. flection from the ends of the line of the single impulse caused by short circuiting the energized line at one end. 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-frequency oscillations intervening between

The text on this page is estimated to be only 28.28% accurate

102 ELECTRIC DISCHARGES, WAVES AND IMPULSES. the alternating waves before the beginning of the transient and the low-frequency stationary oscillation of the complete circuit. Such oscillograms are given in Figs. 48 to 49. Fig. 48A gives the oscillation of the compound circuit consisting of 28 miles of line and the high-tension winding of the 2500-kw. step-up transformer, caused by switching off, by low-tension switches, from a substation at the end of a 153-mile three-phase transmission line, at 88 kilovolts. Fig. 4SA. — CD10002. — Oscillogram of High-frequency Oscillation Preceding Low-frequency Oscillation of Compound Circuit of 28 Miles of 100,000-volt Line and Step-up Transformer; Low-tension Switching. Fig. 48# gives the oscillation of the compound circuit consisting of 154 miles of three-phase line and 10,000-kw. step-down transformer, when switching this line, by high-tension switches, off the end of another 154 miles of three-phase line, at 107 kilovolts. The voltage at the end of the supply line is given as ei, at the beginning of the oscillating circuit as e2. Fig. 49 shows the oscillations and traveling waves appearing in a compound circuit consisting of 154 miles of three-phase line and 10,000-kw. step-down transformer, by switching it on and off the generating system, by high-tension switches, at 89 kilovolts. Frequently traveling waves are of such high frequency — reaching into the millions of cycles — that the oscillograph does not record them, and their existence and approximate magnitude are determined by inserting a very small inductance into the

The text on this page is estimated to be only 28.00% accurate

TRAVELING WAVES. 103

The text on this page is estimated to be only 26.76% accurate

104 ELECTRIC DISCHARGES, WAVES AND IMPULSES. circuit and measuring the voltage across the inductance by spark gap. These traveling waves of very high frequency are extremely local, often extending over a few hundred feet only. An approximate estimate of the effective frequency of these very high frequency local traveling waves can often be made from their striking distance_across a small inductance, by means of the relation -^ = V/ 7^ = z0, discussed in Lecture VI. lo * Co For instance, in the 100,000- volt transmission line of Fig. 48A, the closing of the high-tension oil switch produces a high-frequency oscillation which at a point near its origin, that is, near the switch, jumps a spark gap of 3.3 cm. length, corresponding to ei = 35,000 volts, across the terminals of a small inductance consisting of 34 turns of 1.3 cm. copper rod, of 15 cm. mean diameter and 80 cm. length. The inductance of this coil is calculated as approximately 13 microhenrys. The line constants are, L = 0.323 henry, C = 2.2 X 10~6 farad; hence z0 = y 5 = Vo.1465 X 103 = 383 ohms. The sudden change of voltage at the line terminals, produced i on nno by closing the switch, is - -~— = 57,700 volts effective, or a V_3 maximum of e0 = 57,700 X V2 = 81,500 volts, and thus gives a maximum transient current in the impulse, of i0 = — = 212 amperes. iQ = 212 amperes maximum, traversing the inductance of 13 microhenrys, thus give the voltage, recorded by the spark gap, of e\ = 35,000. If then / = frequency of impulse, it is e\ = 2-jrfLiQ. Or' '=2^' ; .' . Y 35,000 27rX 13 X 10-6 X212 = 2,000,000 cycles. 37. A common form of traveling wave is the discharge of a local accumulation of stored energy, as produced for instance by a direct or induced lightning stroke, or by the local disturbance caused by a change of circuit conditions, as by switching, the blowing of fuses, etc.

The text on this page is estimated to be only 27.41% accurate

TRAVELING WAVES. 105 Such simple traveling waves frequently are called "impulses." When 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, Fig. 50. — Traveling Wave. one, which decreases very rapidly, e~(u + s}i} and thus preponderates in the beginning of the phenomenon, and one, which decreases slowly, e-(u~s)t. Hence it may be expressed in the form: a2e-2^-s)^e-2sX, (20) where the value of the power-transfer constant s determines the " steepness of wave front." Figs. 51 to 53 show oscillograms of the propagation of such an impulse over an (artificial) transmission line of 130 miles,* of the constants : r = 93.6 ohms, L = 0.3944 henrys, C = 1.135 microfarads,— thus of surge impedance ZQ = y ~ = 590 ohms. The impulse is produced by a transformer charge, f Its duration, as measured from the oscillograms, is TQ = 0.0036 second. In Fig. 51, the end of the transmission line was connected to a noninductive resistance equal to the surge impedance, so as to * For description of the line see "Design, Construction, and Test of an Artificial Transmission Line," by J. H. Cunningham, Proceedings A.I.E.E., January, 1911. t In the manner as described in "Disruptive Strength of Air and Oil with Transient Voltages," by J. L. R. Hayden and C. P. Steinmetz, Transactions A.I.E.E., 1910, page 1125. The magnetic energy of the transformer is, however, larger, about 4 joules, and the transformer contained an air gap, to give constant inductance.

The text on this page is estimated to be only 24.56% accurate

106 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Fig. 51. — CD11145. — Reproduction of Oscillogram of Propagation of Impulse Over Transmission Line; no Reflection. Voltage, Fig. 52. — CD 11 152. — Reproduction of Oscillogram of Propagation of Impulse Over Transmission Line; Reflection from Open End of Line. Voltage.

The text on this page is estimated to be only 28.30% accurate

TRAVELING WAVES. 107 give no reflection. The upper curve shows the voltage of the impulse at the beginning, the middle curve in the middle, and the lower curve at the end of the line. Fig. 52 gives the same three voltages, with the line open at the end. This oscillogram shows the repeated reflections of the voltage impulse from the ends of the line, — the open end and the transformer inductance at the beginning. It also shows the increase of voltage by reflection. Fig. 53. — CD11153. — Reproduction of Oscillogram of Propagation of Impulse Over Transmission Line; Reflection from Open End of Line. Current. Fig. 53 gives the current impulses at the beginning and the middle of the line, corresponding to the voltage impulses in Fig. 52. This oscillogram shows the reversals of current by reflection, and the formation of a stationary oscillation by the successive reflections of the traveling wave from the ends of the line.

The text on this page is estimated to be only 26.92% accurate

LECTURE IX. OSCILLATIONS OF THE COMPOUND CIRCUIT. 38. The most interesting and most important application of the traveling wave is that of the stationary oscillation of a compound circuit, as industrial circuits are never uniform, but consist of sections of different characteristics, as the generating system, transformer, line, load, etc. Oscillograms of such circuits have been shown in the previous lecture. If we have a circuit consisting of sections 1, 2, 3 . . . , of the respective lengths (in velocity measure) Xi, X2, X3 . . . , this entire circuit, when left to itself, gradually dissipates its stored energy by a transient. As function of the time, this transient must decrease at the same rate u0 throughout the entire circuit. Thus the time decrement of all the sections must be 6-**. Every section, however, has a power-dissipation constant, u\t Uz, u3 . . . , which represents the rate at which the stored energy of the section would be dissipated by the losses of power in the section, €-"»', €-«*', €-"*' . . . But since as part of the whole circuit each section must die down at the same rate e~Uot, in addition to its power-dissipation decrement e~Ul*, e~"2' . . . , each section must still have a second time decrement, €-(«*-*J*, e-(u0-u,)t t t t This latter does not represent power dissipation, and thus represents power transfer. That is, 51 = U0 — Ui, 52 = UQ — Uz, (1) It thus follows that in a compound circuit, if u0 is the average exponential time decrement of the complete circuit, or the average 108

The text on this page is estimated to be only 28.31% accurate

OSCILLATIONS OF THE COMPOUND CIRCUIT. 109 power-dissipation constant of the circuit, and u that of any section, this section must have a second exponential time decrement, S = UQ — U, (2) which represents power transfer from the section to other sections, or, if s is negative, power received from other sections. The oscillation of every individual section thus is a traveling wave, with a power-transfer constant s. As UQ is the average dissipation constant, that is, an average of the power-dissipation constants u of all the sections, and s = UQ — u the power-transfer constant, some of the s must be positive, some negative. In any section in which the power-dissipation constant u is less than the average UQ of the entire circuit, the power-transfer constant s is positive ; that is, the wave, passing over this section, increases in intensity, builds up, or in other words, gathers energy, which it carries away from this section into other sections. In any section in which the power- dissipation constant u is greater than the average UQ of the entire circuit, the power-transfer constant s is negative; that is, the wave, passing over this section, decreases in intensity and thus in energy, or in other words, leaves some of its energy in this section, that is, supplies energy to the section, which energy it brought from the other sections. By the power-transfer constant s, sections of low energy dissipation supply power to sections of high energy dissipation. 39. Let for instance in Fig. 43 be represented a circuit consisting of step-up transformer, transmission line, and load. (The load, consisting of step-down transformer and its secondary circuit, may for convenience be considered as one circuit section.) Assume now that the circuit is disconnected from the power supply by low-tension switches, at A. This leaves transformer, line, and load as a compound oscillating circuit, consisting of four sections: the high-tension coil of the step-up transformer, the two lines, and the load. Let then Xi = length of line, X2 = length of transformer circuit, and Xs = length of load circuit, in velocity measure.* If then * If Zi = length of circuit section in any measure, and L0 = inductance, Co = capacity per unit of length Zi, then the length of the circuit in velocity measure is Xi = o-oZi, where

The text on this page is estimated to be only 26.36% accurate

110 ELECTRIC DISCHARGES, WAVES AND IMPULSES. HI = 900 = power- dissipation constant of the line, u* = 100 = power-dissipation constant of transformer, and uz = 1600 = powerdissipation constant of the load, and the respective lengths of the circuit sections are Xi = 1.5 X 10-3; X2 = 1 X 10~3; \3 = 0.5 X 10~3, it is: Line. Transformer. Line. Load. Sum. Length: X = 1.5X10-3 1X10~3 1.5X10~3 .5X10~3 4.5X1Q-3 Power-dissipation constant: u = 900 100 900 1600 «X = 1.35 .1 1.35 .8 3.6 hence, Wo= average, u = — — = 800, and: ZA Power-transfer constant: S = MO-M= -100 +700 -100 -800 The transformer thus dissipates power at the rate u2 = 100, while it sends out power into the other sections at the rate of s2 = 700, or seven times as much as it dissipates. That is, it supplies seven-eighths of its stored energy to other sections. The load dissipates power at the rate Uz = 1600, and receives power at the rate —s = 800; that is, half of the power which it dissipates is supplied from the other sections, in this case the transformer. The transmission line dissipates power at the rate HI = 900, that is, only a little faster than the average power dissipation of the entire circuit, u0 = 800; and the line thus receives power only at the rate —s= 100, that is, receives only one-ninth of its power from the transformer; the other eight-ninths come from its stored energy. The traveling wave passing along the circuit section thus increases or decreases in its power at the rate e+2*x; that is, in the line: p = pie~200X, the energy of the wave decreases slowly; in the transformer: p = 7?2C+1400X, the energy of the wave increases rapidly; length li = n, and the length in velocity measure, X = aQn = n VLC. Or, if L = inductance, C = capacity of the entire transformer, its length in velocity measure is X = v LC. Thus, the reduction to velocity measure of distance is very simple.

The text on this page is estimated to be only 28.77% accurate

OSCILLATIONS OF THE COMPOUND CIRCUIT, 111 in the load: p — p3e~l600X, the energy of the wave decreases rapidly. Here the coefficients of pi, p2, PS must be such that the wave at the beginning of one section has the same value as at the end of the preceding section. In general, two traveling waves run around the circuit in opposite direction. Each of the two waves reaches its maximum intensity in this circuit at the point where it leaves the transformer and enters the line, since in the transformer it increases, while in the line it again decreases, in intensity. Fig. 54. — Energy Distribution in Compound Oscillation of Closed Circuit; High Line Loss. Assuming that the maximum value of the one wave is 6, that of the opposite wave 4 megawatts, the power values of the two waves then are plotted in the upper part of Fig. 54, and their difference, that is, the resultant flow of power, in the lower part of Fig. 54. As seen from the latter, there are two power nodes, or points over which no power flows, one in the transformer and one in the load, and the power flows from the transformer over the line into the load; the transformer acts as generator of the power, and of this

The text on this page is estimated to be only 28.08% accurate

112 ELECTRIC DISCHARGES, WAVES AND IMPULSES. power a fraction is consumed in the line, the rest supplied to the load. 40. The diagram of this transient power transfer of the system thus is very similar to that of the permanent power transmission by alternating currents: a source of power, a partial consumption in the line, and the rest of the power consumed in the load. However, this transient power-transfer diagram does not represent the entire power which is being consumed in the circuit, as power is also supplied from the stored energy of the circuit; and the case may thus arise — which cannot exist in a permanent power transmission — that the power dissipation of the line is less than corresponds to its stored energy, and the line also supplies power to the load, that is, acts as generator, and in this case the power would not be a maximum at the transformer terminals, but would still further increase in the line, reaching its maximum at the load terminals. This obviously is possible only with transient power, where the line has a store of energy from which it can draw in supplying power. In permanent condition the line could not add to the power, but must consume, that is, the permanent power-transmission diagram must always be like Fig. 54. Not so, as seen, with the transient of the stationary oscillation. Assume, for instance, that we reduce the power dissipation in the line by doubling the conductor section, that is, reducing the resistance to one-half. As L thereby also slightly decreases, C increases, and g possibly changes, the change brought about in the constant u = =lj ~^7>) *s no^ necessarily a reduction to half, but depends upon the dimensions of the line. Assuming therefore, that the power-dissipation constant of the line is by the doubling of the line section reduced from u\ = 900 to HI = 500, this gives the constants: Line. Transformer. Line. Load. Sum. X= 1.5X10-3 1X10-3 1.5X10-3 .5X10-3 4.5X1Q-3 u= 500 100 500 1600 wX= .75 .1 .75 .8 2.4 SwX hence, MO = average, u = — - = 533, and: 2/A s= +33 +433 +33 -1067 one 

The text on this page is estimated to be only 29.13% accurate

OSCILLATIONS OF THE COMPOUND CIRCUIT. 113 That is, the power-transfer constant of the line has become positive, si = 33, and the line now assists the transformer in supplying power to the load. Assuming again the values of the two traveling waves, where they leave the transformer (which now are not the maximum values, since the waves still further increase in intensity in passing over the lines), as 6 and 4 megawatts respectively, the power diagram of the two waves, and the power diagram of their resultant, are given in Fig. 55. Fig. 55. — Energy Distribution in Compound Oscillation of Closed Circuit; Low Line Loss. In a closed circuit, as here discussed, the relative intensity of the two component waves of opposite direction is not definite, but depends on the circuit condition at the starting moment of the transient. In an oscillation of an open compound circuit, the relative intensities of the two component waves are fixed by the condition that at the open ends of the circuit the power transfer must be zero. As illustration may be considered a circuit comprising the highpotential coil of the step-up transformer, and the two lines, which are assumed as open at the step-down end, as illustrated diagrammatically in Fig. 56.

The text on this page is estimated to be only 27.71% accurate

114 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Choosing the same lengths and the same power-dissipation constants as in the previous illustrations, this gives: Line. Transformer. Line. Sum. 1.5X10-3 1X10-3 1.5X10-3 4X10-3 900 100 900 1.35 .1 1.35 2.8 SwX x= u\ = hence, w0 = average, u = ^^ = 700, and: s= -200 +600 -200 Line Transformer Line Fig. 56. The diagram of the power of the two waves of opposite directions, and of the resultant power, is shown in Fig. 57, assuming 6 megawatts as the maximum power of each wave, which is reached at the point where it leaves the transformer. Transmission Line Transformer Transmission Line U =900 U =100 U=900 Fig. 57. — Energy Distribution in Compound Oscillation of Open Circuit. In this case the two waves must be of the same intensity, so as to give 0 as resultant at the open ends of the line. A power node then appears in the center of the transformer. 41. A stationary oscillation of a compound circuit consists of two traveling waves, traversing the circuit in opposite direction, and transferring power between the circuit section in such a manner

The text on this page is estimated to be only 26.69% accurate

OSCILLATIONS OF THE COMPOUND CIRCUIT. 115 as to give the same rate of energy dissipation in all circuit sections. As the result of this power transfer, the stored energy of the system must be uniformly distributed throughout the entire circuit, and if it is not so in the beginning of the transient, local traveling waves redistribute the energy throughout the oscillating circuit, as stated before. Such local oscillations are usually of very high frequency, but sometimes come within the range of the oscillograph, as in Fig. 47. During the oscillation of the complex circuit, every circuit element d\ (in velocity measure), or every wave length or equal part of the wave length, therefore contains the same amount of stored energy. That is, if e0 = maximum voltage, i0 = maximum current, and X0 = wave length, the average energy ° ° Q must be constant throughout the entire circuit. Since, however, in velocity measure, Xo is constant and equal to the period TO throughout all the sections of the circuit, the product of maximum voltage and of maximum current, e0^o, thus must be constant throughout the entire circuit. The same applies to an ordinary traveling wave or impulse. Since it is the same energy which moves along the circuit at a constant rate, the energy contents for equal sections of the circuit must be the same except for the factor e~ 2 "*, by which the energy decreases with the time, and thus with the distance traversed during this time. Maximum voltage e0 and maximum current i'o, however, are related to each other by the condition_ e° i /^° fo\ — = ZQ = y -FT , (3) and as the relation of L0 and <70 is different in the different sections, and that very much so, ZQ, and with it the ratio of maximum voltage to maximum current, differ for the different sections of the circuit. If then ei and ii are maximum voltage and maximum current respectively of one section, and z\ = y -^ is the "natural impedance " of this section, and ez, 12, and z2 — V/TT are the correspondV 02 ing values for another section, it is • _ • / ' A\

The text on this page is estimated to be only 27.12% accurate

116 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and since ^ — ei — iz ' ii substituting e2 = i'2z2,l = 'z I ^ into (4) gives or and z2 or fz» /ON That is, in the same oscillating circuit, the maximum voltages 60 in the different sections are proportional to, and the maximum currents i0 inversely proportional to, the square root of the natural impedances z0 of the sections, that is, to the fourth root of the ratios of inductance to capacity -^ • to At every transition point between successive sections traversed by a traveling wave, as those of an oscillating system, a transformation of voltage and of current occurs, by a transformation ratio which is the square root of the ratio of the natural impedances, ZQ = V TT > of the two respective sections. * Co When passing from a section of high capacity and low inductance, that is, low impedance z0, to a section of low capacity and high inductance, that is, high impedance z0, as when passing from a transmission line into a transformer, or from a cable into a transmission line, the voltage thus is transformed up, and the current transformed down, and inversely, with a wave passing in opposite direction. A low-voltage high-current wave in a transmission line thus becomes a high-voltage low-current wave in a transformer, and inversely, and thus, while it may be harmless in the line, may become destructive in the transformer, etc.

The text on this page is estimated to be only 27.39% accurate

OSCILLATIONS OF THE COMPOUND CIRCUIT. 117 42. At the transition point between two successive sections, the current and voltage respectively must be the same in the two sections. Since the maximum values of current and voltage respectively are different in the two sections, the phase angles of the waves of the two sections must be different at the transition point; that is, a change of phase angle occurs at the transition point. This is illustrated in Fig. 58. Let z0 = 200 in the first section (transmission line), ZQ = 800 in the second section (transformer). /800 The transformation ratio between the sections then is V onn=2; » ^Uu that is, the maximum voltage of the second section is twice, and the maximum current half, that of the first section, and the waves of current and of voltage in the two sections thus may be as illustrated for the voltage in Fig. 58, by e\e^. Fig. 58. — Effect of Transition Point on Traveling Wave. If then ef and if are the values of voltage and current respectively at the transition point between two sections 1 and 2, and e\ and ii the maximum voltage and maximum current respectively of the first, e% and iz of the second, section, the voltage phase and current phase at the transition point are, respectively: For the wave of the first section: _ = cos 71 and -r = cos 5i. For the wave of the second section: e' i' — — cos 72 and — = cos §2. (9)

The text on this page is estimated to be only 26.40% accurate

118 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Dividing the two pairs of equations of (9) gives cos 72 _ 61 cos 71 62 = ii = Jz* f iz V ?! cos <5i hence, multiplied, cos 72 cos 82 cos 71 cos di or (11) cos 72 _ cos di COS 7i COS £2 or cos 71 cos 5i = cos 72 cos 52; that is, the ratio of the cosines of the current phases at the transition point is the reciprocal of the ratio of the cosines of the voltage phases at this point. Since at the transition point between two sections the voltage and current change, from ei, ii to 62, is, by the transformation ratio — , this change can also be represented as a partial reflection. That is, the current i\ can be considered as consisting of a component z'2, which passes over the transition point, is " transmitted " current, and a component i\ = i\ — iz, which is " reflected " current, etc. The greater then the change of circuit constants at the transition point, the greater is the difference between the currents and voltages of the two sections; that is, the more of current and voltage are reflected, the less transmitted, and if the change of constants is very great, as when entering from a transmission line a reactance of very low capacity, almost all the current is reflected, and very little passes into and through the reactance, but a high voltage is produced in the reactance. v/

The text on this page is estimated to be only 28.62% accurate

LECTURE X. INDUCTANCE AND CAPACITY OF ROUND PARALLEL CONDUCTORS. A. Inductance and capacity. 43. As inductance and capacity are the two circuit constants which represent the energy storage, and which therefore are of fundamental importance in the study of transients, their calculation is discussed in the following. The inductance is the ratio of the interlinkages of the magnetic flux to the current, £ = ?- (i) i/ where = magnetic flux or number of lines of magnetic force, and n the number of times which each line of magnetic force interlinks with the current i. The capacity is the ratio of the dielectric flux to the voltage, where \f/ is the dielectric flux, or number of lines of dielectric force, and e the voltage which produces it. With a single round conductor without return conductor (as wireless antennae) or with the return conductor at infinite distance, the lines of magnetic force are concentric circles, shown by drawn lines in Fig. 8, page 10, and the lines of dielectric force are straight lines radiating from the conductor, shown dotted in Fig. 8. Due to the return conductor, in a two-wire circuit, the lines of magnetic and dielectric force are crowded together between the conductors, and the former become eccentric circles, the latter circles intersecting in two points (the foci) inside of the conductors, as shown in Fig. 9, page 11. With more than one return conductor, and with phase displacement between the return currents, as in a three-phase three-wire circuit, the path of the 119

The text on this page is estimated to be only 29.01% accurate

'iJBLtiGTRIC DISCHARGES, WAVES AND IMPULSES. lines of force is still more complicated, and varies during the cyclic change of current. The calculation of such more complex magnetic and dielectric fields becomes simple, however, by the method of superposition of fields. As long as the magnetic and the dielectric flux are proportional respectively to the current and the voltage, — which is the case with the former in nonmagnetic materials, with the latter for all densities below the dielectric strength of the material,— the resultant field of any number of conductors at any point in space is the combination of the component fields of the individual conductors. Fig. 59. — Magnetic Field of Circuit. Thus the field of conductor A and return conductor B is the combination of the field of A, of the shape Fig. 8, and the field of B, of the same shape, but in opposite direction, as shown for the magnetic fields in Fig. 59. All the lines of magnetic force of the resultant magnetic field must pass between the two conductors, since a line of magnetic force, which surrounds both conductors, would have no m.m.f., and thus could not exist. That is, the lines of magnetic force of A beyond B, and those of B beyond A, shown dotted in Fig. 59, neutralize each other and thereby vanish; thus, in determining the resultant magnetic flux of conductor and return conductor (whether the latter is a single conductor, or divided into two con 

The text on this page is estimated to be only 28.98% accurate

ROUND PARALLEL CONDUCTORS. 121 ductors out of phase with each other, as in a three-phase circuit), only the lines of magnetic force within the space from conductor to return conductor need to be considered. Thus, the resultant magnetic flux of a circuit consisting of conductor A and return conductor B, at distance s from each other, consists of the lines of magnetic force surrounding A up to the distance s, and the lines of magnetic force surrounding B up to the distance s. The former is attributed to the inductance of conductor A, the latter to the inductance of conductor B. If both conductors have the same size, they give equal inductances; if of unequal size, the smaller conductor has the higher inductance. In the same manner in a three-phase circuit, the inductance of each of the three conductors is that corresponding to the lines of magnetic force surrounding the respective conductor, up to the distance of the return conductor. B. Calculation of inductance. 44. If r is the radius of the conductor, s the distance of the return conductor, in Fig. 60, the magnetic flux consists of that external to the conductor, from r to s, and that internal to the conductor, from 0 to r. Fig. 60. — Inductance Calculation of Circuit. At distance x from the conductor center, the length of the mag netic circuit is 2 irx, and if F = m.m.f. of the conductor, the magnetizing force is and the field intensity hence the magnetic density (B 2F x (4) (5)

The text on this page is estimated to be only 25.26% accurate

122 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and the magnetic flux in the zone dx thus is d^=^fdx, I (6) and the magnetic flux interlinked with the conductor thus is X hence the total magnetic flux between the distances x\ and z2 is rx*2 thus the inductance X 1. External magnetic flux, xi = r; xz = s; jP = i, as this flux surrounds the total current; and n = 1, as each line of magnetic force surrounds the conductor once, ju = 1 in air, thus: ?-""-:- 2. Internal magnetic flux. Assuming uniform current ^density throughout the conductor section, it is Cx\2 -J , as each line of magnetic force surrounds only a part of the conductor and the total inductance of the conductor thus is C 9 // i L = LI + L2 = 2 j log- +T( per cm. length of conductor, (11) or, if the conductor consists of nonmagnetic material, ju = 1 : (12)

The text on this page is estimated to be only 27.23% accurate

ROUND PARALLEL CONDUCTORS. 123 This is in absolute units, and, reduced to henry s, = 109 absolute units : = 2 j log ? + 1 1 10-9 h per cm. (13) (14) In these equations the logarithm is the natural logarithm, which is most conveniently derived by dividing the common or 10 logarithm by 0.4343.* C. Discussion of inductance. 45. In equations (11) to (14) s is the distance between the conductors. If s is large compared with r, it is immaterial whether as s is considered the distance between the conductor centers, or between the insides, or outsides, etc.; and, in calculating the inductance of transmission-line conductors, this is the case, and it therefore is immaterial which distance is chosen as s; and usually, in speaking of the "distance between the line conductors," no attention is paid to the meaning of s. Fig. 61. — Inductance Calculation of Cable. However, if s is of the same magnitude as r, as with the conductors of cables, the meaning of s has to be specified. Let then in Fig. 61 r = radius of conductors, and s = distance between conductor centers. Assuming uniform current density in the conductors, the flux distribution of conductor A then is as indicated diagrammatically in Fig. 61. * 0.4343 = log10*,

The text on this page is estimated to be only 27.52% accurate

124 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The flux then consists of three parts: 3>i, between the conductors, giving the inductance and shown shaded in Fig. 61. $2, inside of conductor A, giving the inductance $3, the flux external to A, which passes through conductor B and thereby incloses the conductor A and part of the conductor F J5, and thus has a m.m.f. less than i, that is, gives - < 1. % That is, a line of magnetic force at distance s — r2 and $3 are zero, and the inductance is . (15)

The text on this page is estimated to be only 27.79% accurate

ROUND PARALLEL CONDUCTORS. 125 That is, in other words, with small conductors and moderate currents, the total inductance in Fig. 61 is so small compared with the inductances in the other parts of the electric circuit that no very great accuracy of its calculation is required; with large conductors and large currents, however, the unequal current distribution and resultant increase of resistance become so considerable, with round conductors, as to make their use uneconomical, and leads to the use of flat conductors. With flat conductors, however, conductivity and frequency enter into the value of inductance as determining factors. The exact determination of the inductance of round parallel conductors at short distances from each other thus is only of theoretical, but rarely of practical, importance. An approximate estimate of the inductance L3 is given by considering two extreme cases: (a) The return conductor is of the shape Fig. 62, that is, from s — r to s + f the m.m.f. varies uniformly. B Fig. 62. Fig. 63. Inductance Calculation of Cable. (6) The return conductor is of the shape Fig. 63, that is, the m.m.f. of the return conductor increases uniformly from s — r to s, and then decreases again from s to s + r. (a) For s — r < x < s + r, it is -f r — x 2r 2r 2r hence by (8), /»s+rg_|_r fa r*+*dx J8_r r x Ja_r r s — r by the approximation log (1 ± x) = (16) (17) (18)

The text on this page is estimated to be only 23.56% accurate

126 ELECTRIC DISCHARGES, WAVES AND IMPULSES. it is , s + r . s + r , s — r , L . r\ , /., r\ rtr log— = log — - log — = log (l + -) - log(l - -) = 2-g, hence r (6) For s — r < x < s, it is f-l-sl^^r^h (20) and for s < x < s + r, it is :' 'hence, and integrated this gives fc-aiog^ + ^log'-f'- ii^log^-3, (23) o — if o / o — / and by the approximation (18) this reduces to L,-^, (24) O that is, the same value as (19); and as the actual case, Fig. 60, should lie between Figs. 61 and 62, the common approximation of the latter two cases should be a close approximation of case 4. That is, for conductors close together it is L = L! + L2 + L3 (25) However, - can be considered as the approximation of — log s ( 1 -- )= log - , and substituting this in (25) gives, by com\ s/ s — r _ o _ y o o bining log -- h log - = log - : T S T T (26)

The text on this page is estimated to be only 28.31% accurate

ROUND PARALLEL CONDUCTORS. 127 where s = distance between conductor centers, as the closest approximation in the case where the distance between the conductors is small. This is the same expression as (13). In view of the secondary phenomena unavoidable in the conductors, equation (26) appears sufficiently accurate for all practical purposes, except when taking into consideration the secondary phenomena, as unequal current distribution, etc., in which case the frequency, conductivity, etc., are required. D. Calculation of capacity. 46. The lines of dielectric force of the conductor A are straight radial lines, shown dotted in Fig. 64, and the dielectric equipotential lines are concentric circles, shown drawn in Fig. 64. Fig. 64. — Electric Field of Conductor. If e = voltage between conductor A and return conductor B, and s the distance between the conductors, the potential difference between the equipotential line at the surface of A, and the equipotential line which traverses B, must be e. If e = potential difference or voltage, and I = distance, over which this potential difference acts, G = - = potential gradient, or electrifying force, (27)

The text on this page is estimated to be only 23.88% accurate

128 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and K = - — 2 = - — ^ = dielectric field intensity, (28) 4 Trf 4 irV L where v2 is the reduction factor from the electrostatic to the electromagnetic system of units, and v = 3 X 1010 cm. sec. = velocity of light; (29) the dielectric density then is where K = specific capacity of medium, = 1 in air. The dielectric flux then is where A = section of dielectric flux. Or inversely: -IS?* : || (32) If then ^ = dielectric flux, in Fig. 60, at a distance x from the conductor A, in a zone of thickness dx, and section 2 TTZ, the voltage is, by (32), , de and the voltage consumed between distances x\ and x2 thus is /»*2 2v2^ Xz ei2 = / de = —^-Llog-, (34) hence the capacity of this space : C2 r K /'Q^^ i — —5 — * (po) The capacity of the conductor A against the return conductor B then is the capacity of the space from the distance Zi = r to the distance x^ = s, hence is, by (35), C = — per cm. (36) 2t;2log 

The text on this page is estimated to be only 26.63% accurate

ROUND PARALLEL CONDUCTORS. 129 in absolute units, hence, reduced to farads, C= *1Q9 /per cm., (37) 2z;2logand in air, for K = 1 : 1H9 (38) Immediately it follows: the external inductance was, by (9), Li = 2 log- 10~9 h per cm., and multiplying this with (38) gives or CL> = £' (39) that is, the capacity equals the reciprocal of the external inductance LI times the velocity square of light. The external inductance LI would be the inductance of a conductor which had perfect conductivity, or zero losses of power. It is VLC = velocity of propagation of the electric field, and this velocity is less than the velocity of light, due to the retardation by the power dissipation in the conductor, and becomes equal to the velocity of light v if there is no power dissipation, and, in the latter case, L would be equal to LI, the external inductance. The equation (39) is the most convenient to calculate capacities in complex systems of circuits from the inductances, or inversely, to determine the inductance of cables from the measured capacity, etc. More complete, this equation is CLt = ^, (40) where K = specific capacity or permittivity, /* = permeability of the medium.

The text on this page is estimated to be only 29.01% accurate

130 ELECTRIC DISCHARGES, WAVES AND IMPULSES. E. Conductor with ground return. 47. As seen in the preceding, in the electric field of conductor A and return conductor B, at distance s from each other, Fig. 9, the lines of magnetic force from conductor A to the center line CC' are equal in number and in magnetic energy to the lines of magnetic force which surround the conductor in Fig. 59, in concentric circles up to the distance s, and give the inductance L of conductor A. The lines of dielectric force which radiate from conductor A up to the center line CC', Fig. 9, are equal in number and in dielectric energy to the lines of dielectric force which issue as straight lines from the conductor, Fig. 8, up to the distance s, and represent the capacity C of the conductor A. The center line CC' is a dielectric equipotential line, and a line of magnetic force, and therefore, if it were replaced by a conducting plane of perfect conductivity, this would exert no effect on the magnetic or the dielectric field between the conductors A and B. If then, in the electric field between overhead conductor and ground, we consider the ground as a plane of perfect conductivity, we get the same electric field as between conductor A and central plane CC' in Fig. 9. That is, the equations of inductance and capacity of a^conductor with return conductor at distance s can be immediately applied to the inductance and capacity of a conductor with ground return, by using as distance s twice the distance of the conductor from the ground return. That is, the inductance and capacity of a conductor with ground return are the same as the inductance and capacity of the conductor against its image conductor, that is, against a conductor at the same distance below the ground as the conductor is above ground. As the distance s between conductor and image conductor in the case of ground return is very much larger — usually 10 and more times — than the distance between conductor and overhead return conductor, the inductance of a conductor with ground return is much larger, and the capacity smaller, than that of the same conductor with overhead return. In the former case, however, this inductance and capacity are those of the entire circuit, since the ground return, as conducting plane, has no inductance and capacity; while in the case of overhead return, the inductance of the entire circuit of conductor and return conductor is twice, the capacity half, that of a single conductor, and therefore the total inductance of a circuit of two overhead conductors is greater,

The text on this page is estimated to be only 28.39% accurate

ROUND PARALLEL CONDUCTORS. 131 the capacity less, than that of a single conductor with ground return. The conception of the image conductor is based on that of the ground as a conducting plane of perfect conductivity, and assumes that the return is by a current sheet at the ground surface. As regards the capacity, this is probably almost always the case, as even dry sandy soil or rock has sufficient conductivity to carry, distributed over its wide surface, a current equal to the capacity current of the overhead conductor. With the magnetic field, and thus with the inductance, this is not always the case, but the conductivity of the soil may be much below that required to conduct the return current as a surface current sheet. If the return current penetrates to a considerable depth into the ground, it may be represented approximately as a current sheet at some distance below the ground, and the "image conductor " then is not the image of the overhead conductor below ground, but much lower; that is, the distance s in the equation of the inductance is more, and often much more, than twice the distance of the overhead conductor above ground. However, even if the ground is of relatively low conductivity, and the return current thus has to penetrate to a considerable distance into the ground, the inductance of the overhead conductor usually is not very much increased, as it varies only little with the distance s. For instance, if the overhead conductor is J inch diameter and 25 feet above ground, then, assuming perfect conductivity of the ground surface, the inductance would be and r = i"; s = 2 X 25' = 600", hence - = 2400, L = 2 ] log - + 10~9 = 16.066 X 10~9 h. T Z \ If, however, the ground were of such high resistance that the current would have to penetrate to a depth of over a hundred feet, and the mean depth of the ground current were at 50 feet, this would give s = 2 X 75' = 1800", hence - = 7200, and L = 18.264 X 10-9 h, or only 13.7 per cent higher. In this case, however, the ground sec 

The text on this page is estimated to be only 27.08% accurate

132 ELECTRIC DISCHARGES, WAVES AND IMPULSES. tion available for the return current, assuming its effective width as 800 feet, would be 80,000 square feet, or 60 million times greater than the section of the overhead conductor. Thus only with very high resistance soil, as very dry sandy soil, or rock, can a considerable increase of the inductance of the overhead conductor be expected over that calculated by the assumption of the ground as perfect conductor. F. Mutual induction between circuits. 48. The mutual inductance between two circuits is the ratio of the current in one circuit into the magnetic flux produced by this current and interlinked with the second circuit. That is, j _ $2 _ $1 Lim — ~ -- -r i ll li where $2 is the magnetic flux interlinked with the second circuit, which is produced by current i\ in the first circuit. . In the same manner as the self-inductance L, the mutual inductance Lm between two circuits is calculated; while the (external) self-inductance cor° B responds to the magnetic flux between the distances r and s, the mutual inductance of a conductor k a A upon a circuit ab corresponds to the magnetic flux 0 ° produced by the conductor A and passing between Fig- 65' the distances Aa and Ab, Fig. 65. Thus the mutual inductance between a circuit AB and a circuit ab is mutual inductance of A upon ab, Jiutual inductance of B upon ab, hence mutual inductance between circuits AB and ab, Lm = Lm" — Lm , where A a, Ab, Ba, Bb are the distances between the respective conductors, as shown in Fig. 66.

The text on this page is estimated to be only 23.75% accurate

ROUND PARALLEL CONDUCTORS. 133 If one or both circuits have ground return, they are replaced by the circuit of the overhead conductor and its image conductor below ground, as discussed before. If the distance D between the circuits AB and ab is great compared to the distance S between the conductors of circuit A B, and the distance s between the conductors of circuit ab, and 0 = angle which the plane of circuit AB makes with the distance D, ty the corresponding angle of shown in Fig. 66, it is circuit a&, as approximately Fig. 66. Aa = D -f- £ cos 0 + - cos Ab = D + — cos 0 — - cos A A Ba = D — — cos 0 -{- ~ cos 2i 2i Bb = D — - cos 0 — ^ cos (42) hence m = 21ogn , D+ 2 log D2 - I- cos 0 - ~ cos D2- (7:COS0 -fxCOS = 2 log 2 COS 0 — jz COS - log 1 x io~s /?, hence by ( T __ rt 18) PC s s \2 AS s )2 D2

The text on this page is estimated to be only 26.34% accurate

134 ELECTRIC DISCHARGES, WAVES AND IMPULSES. thus 2 **!()-.*. (43) For 0 = 90 degrees or ty = 90 degrees, Lm is a minimum,, and the approximation (43) vanishes. G. Mutual capacity between circuits. 49. The mutual capacity between two circuits is the ratio of the voltage between the conductors of one circuit into the dielectric flux produced by this voltage between the conductors of the other circuit. That is where ^2 is the dielectric flux produced between the conductors of the second circuit by the voltage e\ between the conductors of the first circuit. If e = voltage between conductors A and B, the dielectric flux of conductor A is, by (36), t = Ce = - , (44) where R is the radius of these conductors and S their distance from each other. This dielectric flux produces, by (32), between the distances Aa and A b, the potential difference Aa g' and the dielectric flux of conductor B produces the potential difference 2v2-fr, Ba. /A0* e = - — l°g^r> (4w K no hence the total potential difference between a and b is 2v2iK AbBa. substituting (44) into (47), e Ab Ba

The text on this page is estimated to be only 22.00% accurate

ROUND PARALLEL CONDUCTORS. 135 and the dielectric flux produced by the potential difference e" — ef between the conductors a and b is . K€ , Ab Ba 2 v2 log- log ^ hence the mutual capacity K 2 v2 log - log — or, by approximation (18), as in (43), Cm= «&«*»*«»* 1(y ,. (49) This value applies only if conductors A and B have the same voltage against ground, in opposite direction, as is the case if their neutral is grounded. If the voltages are different, e\ and e2, where e\ + e2 = 2 e, as for instance one conductor grounded: ei = 0, 62 = e, (50) the dielectric fluxes of the two conductors are different, and that of A is: crt/r; that of B is: c2^, where = f2. 2 e ' and d + c2 = 2, the equations (45) to (49) assume the forms; Aa K AO 2 v2^ , Ba (52) (53) // / — ( Y1" i i j s'< -, ./I rt / e" - e' = - -^ j c2 log BT - ci log^r [ , (54) ir r*\r\ /\ r\ \ % -

The text on this page is estimated to be only 20.79% accurate

136 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Cm = -, j C2 log^ - c, log^ | lO-9/ « -i- *i • » I Bo Ab \ 2 v2 log - log -5 (55) y £42): Ba Aa ~5l -cilogT7 COS 0 — S COS + SCOS' + ir^-)COS 0 — S COS 2D and this gives : fe — Ci) s cos D + + Cfc) /Ss COS 0 (56) hence c.=x . S COS ^ , / N fe-cO—^+fe + d) 2D, and for 61 = 0, and thus c\ = 0, c2 = 2: COS + cos 9 n ^ LJ io9/, (57) (58) hence very much larger than (49). However, equation (58) applies only, if the ground is at a distance very large compared with Z), as it does not consider the ground as the static return of the conductor B. H. The three-phase circuit. 50. The equations of the inductance and the capacity of a conductor (26) 109/ (37)

The text on this page is estimated to be only 28.38% accurate

ROUND PARALLEL CONDUCTORS. 137 apply equally to the two-wire single-phase circuit, the single wire circuit with ground return, or the three-phase circuit. In the expression of the energy per conductor: Li' (59) and of the inductance voltage e' and capacity current i', per conductor: ' = (60) i is the current in the conductor, thus in a three-phase system the Y or star current, and e is the voltage per conductor, that is, the voltage from conductor to ground, which is one-half the voltage between the conductors of a single-phase two-wire circuit, —T=- the voltage between the conductors of a three-phase circuit (that is, it is the Y or star voltage), and is the voltage of the circuit in a conductor to ground, s is the distance between the conductors, and is twice the distance from conductor to ground in a single conductor with ground return.* If the conductors of a three-phase system are arranged in a triangle, s is the same for all three conductors; otherwise the different conductors have different values of s, and A B c the same conductor may have two different values of ° ° o s, for its two return conductors or phases. For instance, in the common arrangement of the o^ three-phase conductors above each other, or beside each other, as shown in Fig. 67, if s is the distance between middle conductor and outside conductors, the OQ distance between the two outside conductors is 2 s. Fig. 67. The inductance of the middle conductor then is: (61) The inductance of each of the outside conductors is, with respect to the middle conductor: * See discussion in paragraph 47.

The text on this page is estimated to be only 28.50% accurate

138 ELECTRIC DISCHARGES, WAVES AND IMPULSES. (62) With respect to the other outside conductor: L = 2Jlogy + ^jlO-U. (63) The inductance (62) applies to the component of current, which returns over the middle conductor, the inductance (63), which is larger, to the component of current which returns over the other outside conductor. These two currents are 60 degrees displaced in phase from each other. The inductance voltages, which are 90 degrees ahead of the current, thus also are 60 degrees displaced from each other. As they are unequal, their resultant is not 90 degrees ahead of the resultant current, but more in the one, less in the other outside conductor. The inductance voltage of the two outside conductors thus contains an energy component, which is positive in the one, negative in the other outside conductor. That is, a power transfer by mutual inductance occurs between the outside conductors of the three-phase circuit arranged as in Fig. 67. The investigation of this phenomenon is given by C. M. Davis in the Electrical Review and Western Electrician for April 1, 1911. If the line conductors are transposed sufficiently often to average their inductances, the inductances of all three conductors, and also their capacities, become equal, and can be calculated by using the average of the three distances s, s, 2 s between the conductors, 4 s that is, - s, or more accurately, by using the average of the log - > o r s 2s log - and log -5- , that is: r o • 3 In the same manner, with any other configuration of the line conductors, in case of transposition the inductance and capacity Q can be calculated by using the average value of the log - between the three conductors. The calculation of the mutual inductance and mutual capacity between the three-phase circuit and a two-wire circuit is made

The text on this page is estimated to be only 24.30% accurate

ROUND PARALLEL CONDUCTORS. 139 in the same manner as in equation (41), except that three terms appear, and the phases of the three currents have to be considered. Q^ Thus, if A, B, C are the three three-phase conductors, and a and b the conductors of the second circuit, as shown in Fig. 68, and if ii, iz, i3 are °C OB the three currents, with their respective phase angles 71, 72, 73, and i the average current, b a denoting: o Fig. 68. 1\ 12 ^3 conductor A gives: conductor B: conductor C: Lm'" = 2 c3 cos (0 - 240° - 73) log^?>* hence, Lm = 2 ) ci cos 03 - 71) log 4r + C2 cos 08 - 120° - 72) log |?, Lm" = 2 c2 cos 08 - 120°- 72) log!?, no 4- c3 cos (0 - 240°- 73) log ^ | 10-9 /i, and in analogous manner the capacity Cm is derived. In these expressions, the trigonometric functions represent a rotation of the inductance combined with a pulsation.

The text on this page is estimated to be only 29.02% accurate

INDEX. PAGE Acceleration as mechanical transient 4 single-energy transient 8 Admittance, natural or surge, of circuit 61, 84 Alternating current in line as undamped oscillation 97 phenomena as transients 9 reduction to permanents 9 Alternators, momentary short-circuit currents 37 construction 40 calculation 44 Arcing grounds 97 Armature transient of alternator short circuit 41 Attenuation of transient, see Duration. Cable inductance, calculation 81, 123 surge 62 Capacity 18 calculation 127 of circuit, definition 12 current 13 definition 119 effective, of line transient 75 equation 129 and inductance calculation of three-phase circuit 136 CAPACITY AND INDUCTANCE OF ROUND PARALLEL CONDUCTORS 119 Capacity, mutual, calculation .T 134, 138 specific 16, 17, 18 Charge, electric, of conductor 14 Charging current 13 Circuit, dielectric 14, 17, 18 of distributed capacity and inductance, also see Line. electric 17, 18 magnetic 14, 17, 18 Closed compound-circuit transient Ill, 112 Combination of effective and reactive power 100 transient 100 of standing and traveling waves 100 COMPOUND CIRCUIT OSCILLATION 108 Compound circuit, power flow 90 velocity unit of length 92 oscillation of closed circuit Ill, 112 of open circuit 0 114 141

The text on this page is estimated to be only 28.94% accurate

142 INDEX. PAGE Condenser current 13 Conductance 18 effective, of line transient 78 Conductivity, electric 18 Cumulative oscillation 97 Current, electric 18 in field at alternator short circuit 40 transient pulsation 43 permanent pulsation 45 transient, maximum 61 Danger from single-energy magnetic transient 27 Decay of single-energy transient 21 Deceleration as mechanical transient 4 Decrease of transient energy 59 Decrement of distance and of time 94 exponential 88 Decrease of power flow in traveling wave 92 Density, dielectric 16, 17, 18 electric current 18 magnetic 15, 17, 18 Dielectric field 11 as stored energy 3 forces 10 flux 15, 17, 18 gradient 18 transient, duration 59 Dielectrics 15, 17, 18 Disruptive effects of transient voltage 63 Dissipation constant of circuit 94 compound circuit 109 double-energy transient 68 line 78 dielectric energy in double energy-transient 67 exponent of double-energy transient 68 of magnetic energy in double-energy transient 67 Distortion of quadrature phase in single-phase alternator short circuit . . 47 Distance decrement 94 Distributed capacity and inductance 73 Double-energy transient 7 equation 69 DOUBLE-ENERGY TRANSIENTS 59 Double frequency pulsation of field current at single-phase alternator short circuit 45 Duration of double-energy transient 68 single-energy transient 22, 27 transient 59 alternator short-circuit current 41

The text on this page is estimated to be only 28.91% accurate

INDEX. 143 PAGE Effective values, reducing A.C. phenomena to permanents 9 Elastance 18 Elastivity 18 Electrifying force 15, 17 Electromotive force 15, 17, 18 Electrostatic, see Dielectric. Energy, dielectric 18 of dielectric field 13 dielectric and magnetic, of transient 67 magnetic 18 of magnetic field 12 storage as cause of transients 3 transfer in double-energy transient 60 by traveling wave 92 of traveling wave in compound circuit 110 Equations of double-energy transient 69 line oscillation 74, 75} 84 simple transient 6 single-energy magnetic transient 21, 24 Excessive momentary short circuit of alternator 37 Exponential decrement 88 magnetic single-energy transient 21 transient 7 numerical values. ..'..- 23 Field current at alternator short circuit, rise 40 transient pulsation 43 permanent pulsation 45 FIELD, ELECTRIC 10 rotating, transient 34 superposition 120 transient, of alternator 38 construction 40 calculation 44 Flux, dielectric 11, 15 magnetic 10, 14 Frequencies of line oscillations 79 Frequency of double-energy transient, calculation 66 oscillation of line transient 78 Frohlich's formula of magnetic-flux density 53 Fundamental wave of oscillation . . 81 Gradient, electric 15, 17, 18 Grounded phase 97 Grounding surge of circuit 62 Ground return of conductor, inductance and capacity 130

The text on this page is estimated to be only 29.06% accurate

144 INDEX. PAGE Half- wave oscillation 82 Hunting of synchronous machines as double-energy transient 9 Hydraulic transient of water power 4 Image conductor of grounded overhead line 130 Impedance, natural or surge, of circuit 61, 84 Impulse propagation over line and reflection 105 as traveling wave 105 Increase of power flow in traveling wave 92 Independence of rotating-field transient from phase at start 36 Inductance of cable 123 calculation 123 and capacity calculation of three-phase circuit 136 INDUCTANCE AND CAPACITY OF ROUND PARALLEL CONDUCTORS 119 Inductance of circuit, definition 11 definition 119 effective, of line transient 75 equation 123, 126, 131 mutual, calculation 132, 138 voltage 12, 18 Intensity, dielectric 16, 17, 18 magnetic 15, 17, 18 IRONCLAD CIRCUIT, SINGLE-ENERGY TRANSIENT 52 Ironclad circuit transient, oscillogram 57 Kennelly's formula of magnetic reluctivity 53 Length of circuit in velocity measure, calculation 109 Lightning surge of circuit 62 as traveling wave 89 Line as generator of transient power 112 LINE OSCILLATION 72 general form 74 also see Transmission line. Magnetic field 10 as stored energy 3 flux 14, 17, 18 forces 10 single-energy transient 19, 25 construction 20, 25 duration 59 Magnetics 14, 17, 18 Magnetizing force 14, 17, 18 Magnetomotive force 14, 17, 18 Massed capacity and inductance 73 Maximum transient current 61 voltage 61

The text on this page is estimated to be only 29.02% accurate

INDEX. 145 PAGE Maximum value of rotating-field transient 36 Measurement of very high frequency traveling wave 104 Mechanical energy transient 4 Momentary short-circuit current of alternators 37 construction 40 calculation 44 Motor field, magnetic transient 24 Mutual capacity, calculation 134, 138 of lines, calculation 81 inductance and capacity with three-phase circuit 138 calculation 132, 138 Natural admittance and impedance of circuit 61, 84 Nonperiodic transient 9 Nonproportional electric transient 52 surge of transformer 64 Open-circuit compound oscillation 113 Oscillating currents. 62 voltages 62 Oscillation frequency of line transient • 78 Oscillation of open compound circuit 113 stationary, see Stationary oscillations and standing waves. Oscillations, cumulative 97 of closed compound circuit Ill, 112 OSCILLATIONS, LINE 72 OSCILLATIONS OF THE COMPOUND CIRCUIT 108 Oscillatory transient of rotating field 36 Oscillograms of arcing ground on transmission line 98 cumulative transformer oscillation 99 decay of compound circuit 91 formation of stationary oscillation by reflection of traveling wave 101 high-frequency waves preceding low-frequency oscillation of compound circuit 102, 103 impulses in line and their reflection 106 single-phase short circuit of alternators 50 single-phase short circuit of quarter-phase alternator 48 three-phase short circuit of alternators 49 starting current of transformer 57 starting oscillation of transmission line 76, 77 varying frequency transient of transformer 64 Pendulum as double-energy transient 8 Periodic component of double-energy transient, equation 66 energy transfer in transient 60 and transient component of transient 72 transients, reduction to permanents 9

The text on this page is estimated to be only 28.90% accurate

146 INDEX. PAGE Period and wave length in velocity units 92 Permanent phenomena, nature 1 Permeability 15, 17, 18 Permeance 18 Permittance 18 Permittivity 16, 17, 18 Phase angle, change at transition point 117 of oscillation, progressive change in line 75 Phenomena, transient, see Transients. Polyphase alternator short circuit 44 oscillograms 48, 49 Power diagram of open compound-circuit transient 114 closed compound-circuit transient Ill, 113 dissipation constant 88 of section of compound circuit 108 double-energy transient 66 electric 18 flow in compound circuit 90 of 'line transient 88, 89 of line oscillation 79 transfer constant of circuit 94 sectiDn of compound circuit 108 in compound-circuit oscillation 90 of traveling wave 95 Progressive change of phase of line oscillation 75 Propagation of transient in line 74 velocity of electric field 74 field 129 Proportionality in simple transient 4 Pulsation, permanent, of field current in single-phase alternator short circuit 45 of transient energy 61 transient, of magnetomotive force and field current at polyphase alternator short circuit 41 Quadrature relation of stationary wave 88 Quantity of electricity 14 Quarter-wave oscillation of line 81, 82 Reactance of alternator, synchronous and self-inductive 37 Reaction, armature, of alternator 37 Reactive power wave 88 Reflected wave at transition point 118 Reflection at transition point 118 Relation between capacity and inductance of line 81 standing and traveling waves 101 Reluctance 18 Reluctivity 18

The text on this page is estimated to be only 28.90% accurate

INDEX. 147 PAGE Resistance 18 effective, of line transient 78 Resistivity 18 Resolution into transient and permanent 30 Rise of field current at alternator short circuit 40 Rotating field, transient 34 Self-induction, e.m.f. of 12 Separation of transient and permanent 27, 30 Ships; deceleration as transient 7 Short-circuit current of alternator, momentary 37 construction 40 calculation 44 surge of circuit 62 Simple transient 4 equation 6 Single-phase alternator short circuit 45 oscillogram 50 short circuit of alternators 45 oscillograms 48, 50 Single-energy transient . 7 SINGLE-ENERGY TRANSIENT OF IRON-CLAD CIRCUITS. 52 SINGLE-ENERGY TRANSIENTS, CONTINUOUS CURRENT. 19 SINGLE-ENERGY TRANSIENTS IN A.C. CIRCUITS 30 Specific capacity 16, 17, 18 Standing waves 97 originating from traveling waves 101 see Stationary oscillation and Oscillation, stationary. Start of standing wave by traveling wave 101 Starting current of transformer, oscillogram ^ 57 oscillation of line 86 transmission line, oscillogram 76, 77 of rotating field 34 transient of A.C. circuit 32 magnetic circuit 28 three-phase circuit 32 Static induction of line, calculation 81 Stationary oscillation of open line 86 see Line oscillation and Standing wave. Steepness of ironclad transient 58 wave front and power-transfer constants of impulse 105 Step-by-step method of calculating transient of ironclad circuit 53 Storage of energy as cause of transients 3 Superposition of fields 120 Surge admittance and impedance of circuit 61, 84 Symbolic method reducing A.C. phenomena to permanents 9 Symmetrical pulsation of field current at single-phase alternator short circuit . . 45

The text on this page is estimated to be only 28.73% accurate

148 INDEX. PAGE Terminal conditions of line oscillation 84 Three-phase alternator short circuit 44 oscillograms 49 Three-phase circuit, inductance and capacity calculations 136 mutual inductance and capacity 138 current transient 32 .-. magnetic-field transient 34 Time constant 22 decrement 94 of compound circuit 108 Transfer of energy in double-energy transient 60 by traveling wave 92 in compound circuit oscillation 90 Transformation ratio at transition point of compound circuit 117 Transformer as generator of transient energy Ill, 113 Transformer surge 62 Transient current in A.C. circuit 31 double-energy 7 power transfer in compound-circuit 112 single-energy 7 of rotating field 34 separation from permanent 27, 30 short-circuit current of alternator 37 construction 40 calculation 44 and periodic components of oscillation 72 Transients, double-energy 59 caused by energy storage 3 fundamental condition of appearance 4 general with all forms of energy 4 as intermediate between permanents 2 nature 1 single-energy A.C. circuit 30 continuous current 19 Transition point, change of phase angle 117 reflection 118 voltage and current transformation 117 Transmission-line surge 63 transient 73 also see Line. Transmitted wave at transition point 118 Transposition of line conductors 138 TRAVELING WAVES 88 Traveling wave, equation 95 as impulses 105 preceding stationary oscillation 101 of very high frequency 104

The text on this page is estimated to be only 29.30% accurate

INDEX. 149 PAGE Turboalternators, momentary short-circuit current 37 construction 40 calculations 44 Undamped oscillations 97 Unidirectional energy transfer in transient 60 Uniform power flow in traveling wave 92 Unsymmetrical pulsation of field current at single-phase alternator short circuit 46 Varying frequency oscillation of transformer 64 Velocity constant of line 78, 83 relation between line capacity and inductance 81 measure, calculation of circuit length 109 of propagation of electric field 74, 129 transient of ship 7 unit of length of line 83 of length in compound circuit 92 Very high frequency traveling wave 104 Voltage 18 gradient . 15, 17 relation of section of compound oscillating circuit 17 rise of quadrature phase in single-phase alternator short circuit 47 at transition point of compound oscillation 117 of single-energy magnetic transient 27 transient, maximum 61 Wave front of impulse 105 length of line oscillation 79, 80 and period in velocity unit 92 standing, see Standing wave. WAVES, TRAVELING 88

The text on this page is estimated to be only 21.55% accurate

TOO i 33417 749213 753 Engineering Library UNIVERSITY OF CALIFORNIA LIBRARY