Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients — part 1 of 7

ELEMENTARY LECTURES ON ELECTRIC DISCHARGES, WAVES AND IMPULSES, | | AND . OTHER TRANSIENTS | BY h CHARLES PROTEUS STEINMETZ, A.M., Px.D. Past President, American Institute of Electrical Engineers SECOND EDITION Revisep AND ENLARGED McGRAW-HILL BOOK COMPANY, Inc. | : 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, BE. C. , 1914 .

PREFACE TO THE SECOND EDITION.

Sincg the issue of the first edition, in 1911, our knowledge of transients has greatly increased, and many of the phenomena, especially those of double energy transients and compound cir- .

_ cuits, have been observed and studied on transmission systems to a considerable extent, and have corroborated the oscillographic records given in the previous edition.

Considerable work has been done on momentary short circuits . of alternators, and the variable component of the self-inductive reactance recognized as a transient reactance resulting from’ the mutual induction of the armature with the field circuit.

Especially in the field of sustained or continual, and of cumu- lative oscillations, a large amount of information has been gathered. The practical importance of these continual and cumulative oscil- lations has been strongly impressed upon operating and designing engineers in recent years, usually in the most disagreeable manner by the destruction of high power, high voltage transformers.. A . chapter on these phenomena has therefore been added in the second edition.

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

February, 1914.

Vv

PREFACE TO THE FIRST EDITION. | In the following I am trying to give a short outline of those : phenomena which have become the most important to the elec- trical engineer, as on their understanding and control depends the further successful advance of electrical engineering. The art has now so far advanced that the phenomena of the steady flow of power are well understood. Generators, motors, transforming devices, transmission and distribution conductors can, with rela- . tively little difficulty, be calculated, and the phenomena occurring ~ in them under normal conditions of operation predetermined and controlled. Usually, however, the limitations of apparatus and lines are found not in the normal condition of operation, the steady flow of power, but in the phenomena occurring under abnormal though by no means unfrequent conditions, in the more or less transient abnormal voltages, currents, frequencies, etc.; and the study of the laws of these transient phenomena, the electric dis- charges, waves, and impulses, thus becomes of paramount impor- tance. In a former work, ‘‘ Theory and Calculation of Transient Electric Phenomena and Oscillations,” I have given a systematic study of these phenomena, as far as our present knowledge per- mits, which by necessity involves to a considerable extent the use _ Of mathematics. As many engineers may not have the time or inclination to a mathematical study, I have endeavored to give in ; the following a descriptive exposition of the physical nature and meaning, the origin and effects, of these phenomena, with the usé’ 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 vu

Vill PREFACE TO THE FIRST EDITION. - with transient phenomena by the inspection of their record on the . photographic film of the oscillograph. I would therefore recom- mend 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 under- a standing of their matheinatical representation, which latter mS obviously is necessary for their numerical calculation and pre- - determination. The book contains a series of lectures on electric discharges, oe waves, and impulses, which was given during the last winter to the graduate classes of Union University as an elementary intro- duction to and “translation from mathematics into English” of the . : “ Theory and Calculation of Transient Electric Phenomena and Oscillations.”” Hereto has been added a chapter on the calculation ‘of capacities and inductances of conductors, since capacity and inductance are the fundamental quantities on which the transients depend. , In the preparation of the work, I have been materially assisted by Mr. C. M. Davis, M.E.E., who kindly corrected and edited the manuscript and illustrations, and to whom I wish to express my thanks. CHARLES PROTEUS STEINMETZ. October, 1911.

CONTENTS. | PAGE

Lecture I.— Narurs AND ORIGIN oF TRANSIENTS..............008 1

1. Electric power and energy. Permanent and transient phenomena. Instance of permanent phenomenon; of transient; of combination of both. Trarisient as intermediary condition between permanents.
2. Inergy storage in electric circuit, by magnetic and dielectric field. Other energy storage. Change of stored energy as origin of tran- sient.
3. Transients existing with all forms of energy: transients of rail- way car; of fan motor; of incandescent lamp. Destructive values. High-speed water-power governing. Fundamental condition of transient. JVlectric transients simpler, their theory further ad-

vanced, of inore direct industrial importance.

4, Simplest transients: proportionality of cause and effect. Most electrical transients of this character. Discussion of simple tran- sient 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 nore than two forms of energy. 6. Permanent phenomena usually simpler than transients. Re - duction of alternating-current phenomena to permanents by effec- tive values and by symbolic method. Nonperiodic transients.

Lecrurs IJ. — Tue Evecrric FIe€bp..........0..... 0c eee ee eee eects = 10 7. Phenomena of electric power flow: power dissipation in con- ductor; electric field consisting of magnetic field surrounding con- ductor 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 tlie magnetic field. ; 9. The dielectric flux, capacity, capacity current, and the energy of the dielectric field. The conception of quantity of electricity, ; electrostatic charge and condenser; the conception of quantity of magnetism. . 10. Magnetic circuit and dielectric circuit. Magnetomotive force, magnetizing force, magnetic field intensity, and magnetic density. . Permeability. Magnetic materials.

ix

x CONTENTS. PAGE . 11. Electromotive force, electrifying force or voltage gradient. Dielectric field intensity and dielectric density. Specific capacity or permittivity. Velocity of propagation. ‘ 12. Tabulation of corresponding terms of magnetic and of die lectric field. Tabulation of analogous terms of magnetic, dielec- tric, and electric circuit. Lecrurs IJI.—~SiNGLe-ENERGY TRANSIENTS IN CONTINUOUS-CUR- RENT CIRCUITS... 2... eee cece cee nt ent e et neenee 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. Discus- sion of the transients of %, 2, e, of inductive circuit. Exponen- : tial equation. Duration of the transient, time constant. Numer- ical values of transient of intensity 1 and duration 1. The three forms of the equation of the magnetic transient. Simplification by choosing the starting moment as zero of time. 14. Instance of the magnetic transient of a motor field. Calcula- tion of its duration. 15. Effect of the insertion of resistance on voltage and duration of the magnetic transient. The opening of inductive circuit. The effect of the opening are at the switch. 16. The magnetic transient of closing an inductive circuit. General method of separation of transient and of permanent terms during the transition period. Lecture IV.—SinGLe-ENERGY TRANSIENTS OF ALTERNATING-CUR- RENT CIRCUITS... 0... 00 cece cece eect eect ee eetceseecssreee 30 17. Separation of current into permanent and transient component. Condition of maximum and of zero transient. The starting of an alternating current; dependence of the transient on the phase; maxi- mum and zero value. 18. The starting transient of the balanced three-phase system. Relation between the transients of the three phases. Starting transient of three-phase magnetic field, and its construction. The oscillatory start of the rotating field. Its independence of the phase at the moment of start. Maximum value of rotating-field tran- sient, and its industrial bearing. 19. Momentary short-circuit current of synchronous alternator, and current rush in its field circuit. Relation between voltage, load, magnetic field flux, armature reaction, self-inductive reactance, aud 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. Numeri- cal relation between the transients of magnetic flux, armature currents, armature reaction, and field current. The starting transient of the armature currents. The transient full-frequency pulsation of the

. xii 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 sec- tion as quarter wave length. 30. Relation between inductance, capacity, and frequency of prop- : agation. Importance of this relation for calculation of line con- stants. 31. The different frequencies and wave lengths of the quarter- wave oscillation; of the half-wave oscillation. 32. The velocity unit of length. Its importance in compound circuits. Period, frequency, time, and distance angles, and the .

_ general expression of the line oscillation. Lecruxe VIII. — TRaveELING WAVES.......... ccc cece ec eee cece eee §=688 . 33. The power of the stationary oscillation and its correspondence . with reactive power of alternating currents. The traveling wave and its correspondence with effective power of alternating currents. | : Occurrence of traveling waves: the lightning stroke. The traveling wave of the compound circuit. 34. The flow of transient power and its equation. The power- dissipation constant and the power-transfer constant. Increasing , and decreasing power flow in the traveling wave. The general equation of the traveling wave. 35. Positive and negative power-transfer constants. Undamped . oscillation and cumulative oscillation, The are 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 fre- - quencies. Estimation of their effective frequency if very high. ‘ 87. The impulse as traveling wave. Its equations. The wave front. Lecrure IX. — OscrnLaTions oF THE Compounp CrrcuiT.......... 108 38. The stationary oscillation of the compound circuit. The time decrement of the total circuit, and the power-dissipation and power-transfer constants of its section. Power supply from section of low-energy dissipation to section of high-energy dissipation. ; ; 39. Instance of oscillation of a ‘closed compound circuit. The —

two traveling waves and the resultant transient-power diagram. : 40. Comparison of the transient-power diagram with the power . diagram of an alternating-current circuit. The cause of power increase in the line. The stationary oscillation of an open com- pound circuit.

CONTENTS . xiii 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 sec- tions of a compound oscillating circuit. Partial reflection at the transition point. LEcTURE X.— CoNTINUAL AND CUMULATIVE OSCILLATIONS......... 119 43, Continual energy supply to the system as necessary cause, in- volving frequency transformation. Instance of arcing ground on transmission line. Recurrent and continuous continual oscilla- tions. Their change and intermediate forins. Oscillograins of dif- ferent types. Singing arc. 44. Mechanism of energy supply to the continual oscillation by

_negative energy cycle. Hysteresis cycle of transient arc. Mecha- nism of energy supply and continuous and cumulative hunting of synchronous machines. Conditions of continual and of cumula- tive oscillations.

45. Frequency of continual oscillation. Destructiveness of oscil-

. lation. Cumulative effect on insulation. Unlimited energy supply. Independence of frequency of continual oscillation from that of exciting cause. :

Lecture XI. — InpucTaNcE AND Capacity or RounpD PARALLEL Con-

DUCTORS. 0.0. eect eee center et eee ceeeces 128 46. Definition of inductance and of capacity. The magnetic and the dielectric field. The law of superposition of fields, and its use for calculation. 47. Calculation of inductance of two parallel round conductors. External magnetic flux and internal magnetic flux. 48. Calculation and discussion of the inductance of two parallel conductors at small distances from each other. Approximations and their practical limitations.

, 49, Calculation of capacity of parallel conductors by superposition of dielectric fields. Reduction to electromagnetic units by the ; velocity of light. Relation between inductance, capacity, and velocity of propagation. 50. Conductor with ground return, inductance, and capacity. . The image conductor. Limitations of its application. Correction for penetration of return current in ground. 51. Mutual inductance between circuits. Calculation of equation, and approximation. ; ; 52. Mutual capacity between circuits. Symmetrical circuits and . asymmetrical circuits. Grounded circuit. 58. The three-phase circuit. Inductance and capacity of two-

wire single-phase circuit, of single-wire circuit with ground return, and of three-wire three-phase circuit. Asymmetrical arrangement of three-phase circuit. Mutual inductance and mutual capacity with three-phase circuit. .

ELEMENTARY LECTURES ON ELECTRIC DISCHARGES, WAVES AND IMPULSES, | . AND OTHER TRANSIENTS. | LECTURE I. . NATURE AND ORIGIN OF TRANSIENTS.

1. Electrical engineering deals with electric energy and its flow, that is, electric power. Two classes of phenomena are met: permanent and: transient phenomena. To illustrate: Let G in Fig. 1 be a direct-current generator, which over. a circuit A con- nects to a load L, as a number of lamps, ete. In the generator - G, the line A, and the load ZL, acurrent 7 flows, and voltages e A | AF | e( 0) O00 |b , : NZ ee a , ; _ Fig. 1. , exist, which are constant, or permanent, as long as the conditions of the circuit remain the same. If we .connect in some more ; lights, or disconnect some of the load, we get a different current a’, and possibly different voltages e’; but again ¢’ and ¢’ are per- manent, that is, remain the same as long as the circuit remains unchanged. ; Let, however, in Fig. 2, a direct-current generator G be connected | to an electrostatic condenser C’. Before the switch Sis closed, and — therefore also in the moment of closing the switch, no current flows Coe 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

2 ELECTRIC DISCHARGES, WAVES AND IMPULSES. condenser C is charged, the current in the line A and the condenser C is zero again. That is, the permanent condition before closing the switch S, and also some time after the closing of the switch, _is zero current in the line. Immediately after the closing of 7 the switch, however, current flows for a more or less short time. , With the condition of the circuit unchanged: the same generator voltage, the switch S closed on the same circuit, the current nevertheless changes, increasing from zero, at the moment of closing the switch S, to a maximum, and then decreasing again to zero, while the condenser charges from zero voltage to the genera- tor voltage. We then here meet a transient phenomenon, in the , charge of the condenser from a source of continuous voltage. A a As S c(o ) | =e A _ Fig. 2. Commonly, transient and permanent phenomena are super- imposed 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 dis- connected, and the fan motor running at full speed. The question then arises, why the effect of a change in the conditions of an electric circuit does not appear instantaneously, but only after a transition period, requiring a finite, though frequently very short, time. 2. Consider the simplest case: an electric power transmission (Fig. 3). In the generator G electric power is produced from me- chanical power, and supplied to the line A. In the line A some of this power is dissipated, the rest transmitted into the, load L, | where the power is used. The consideration of the electric power

4 ELECTRIC DISCHARGES, WAVES AND IMPULSES. — _ is the phenomenon by which the circuit readjusts itself to the ' change of stored energy. It may thus be said that the perma- nent phenomena are the phenomena of electric power, the tran- sients the phenomena of electric energy.

3. It is obvious, then, that transients are not specifically electri- . cal phenomena, but occur with all forms of energy, under all condi- tions where energy storage takes place.

Thus, when we start the motors propelling an electric car, a

transient period, of acceleration, appears between the previous permanent condition of standstill and the final permanent con- dition of constant-speed running; when we shut off the motors, the permanent condition of standstill is not reached instantly, but a transient condition of deceleration intervenes. When we open the water gates leading to an empty canal, a transient condition” of flow and water level intervenes while the canal is filling, until the permanent condition is reached. Thus in the case of the fan motor in instance Fig. 1, a transient period of speed and mechanical energy appeared while the motor was speeding up and gathering the mechanical energy of its momentum. When turning on an incandescent lamp, the filament passes a transient of gradually rising temperature.

Just as electrical transients may, under certain conditions, rise to destructive values; so transients of other forms of energy may become destructive, or may require serious consideration, as, for

instance, is the case in governing high-head water powers. The column of water in the supply pipe represents a considerable amount of stored mechanical energy, when flowing at velocity, under load. If, then, full load is suddenly thrown off, it is not possible to suddenly stop the flow of water, since a rapid stopping . . would lead to a pressure transient of destructive value, that. is, burst the pipe. Hence the use of surge tanks, relief valves, or deflecting nozzle governors. Inversely, if a heavy load comes on suddenly, opening the nozzle wide does not immediately take care of the load, but momentarily drops the water pressure at the

~ nozzle, while gradually the water column acquires velocity, that

is, stores energy.

The fundamental condition of the appearance of a transient thus is such a disposition of the stored energy in the system as differs from that required by the existing conditions of the system;

• and any change of the condition of a system, which requires a

NATURE AND ORIGIN OF TRANSIENTS. 5 change of the stored energy, of whatever form this energy may be, . leads to a transient. oe . Electrical transients have been studied more than transients of other forms of energy because:

(a) Electrical transients generally are simpler in nature, and therefore yield more easily to a theoretical and experimental investigation. .

(b) The theoretical side of electrical engineering is further advanced than the theoretical side of most other sciences, and especially:

(c) The destructive or harmful effects of transients in electrical systems are far more common and more serious than with other forms of energy, and the engineers have therefore been driven by necessity to their careful and extensive study. . _

4. The simplest form of transient occurs where the effect is directly proportional to the cause. This is generally the case in electric circuits, since voltage, current, magnetic flux, etc., are proportional to each other, and the electrical transients therefore are usually of the simplest nature. In those cases, however, where this direct proportionality does not exist, as for instance in

. inductive circuits containing iron, or in electrostatic fields exceed- ing the corona voltage, the transients also are far more complex, and very little work has been done, and very little is known, on these more complex electrical transients. : ' Assume that in an electric circuit we have a transient cur- rent, as represented by curve 7 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 7. This current starts at the value 7,, 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, %2, it would traverse a curve 2’, which is the same as curve i, except that all values are changed proportionally, by the ratio . 4. that is, = ixX2. Y at

Starting with current 7, the transient follows the curve 7; . starting with 7%, the transient follows the proportional curve 7’.

At some time, ¢, however, the current 7 has dropped to the value 2,, with which the curve 7’ started. At this moment ¢, the conditions . in the first case, of current 7, are the same as the conditions in .

6 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ; the second case, of current 7’, at the moment 4; that is, from ¢ oe onward, curve? is the same as curve?’ from time ¢, onward. Since ii a, : , tas Xi ; ' ' :
i 1 . t J XN 3 ot, t t, Fig. 4. — Curve of Simple Transient: Decay of Current. 7’ is proportional to 7, from any point ¢t onward curve 7 is propor- . tional to the same curve 7 from 4; onward. At time 4, it is diz diy _, 22 diy ~ dt; iy" | _ Ale . dt . . . But since ai, and % at ¢, are the same as Pr and 7 at time ¢, it 1 follows: . di _ di; a dt dt; 1’ or, dy | dé °° 1 diy . . where ¢ = — 7 aT constant, and the minus sign is chosen, as 1 ai is negative ‘ at ** neBAnive: As in Fig. 4: diy tan o¢=—- dt’ ait = iy 1d, tang 1, ¢=z— +> a FR Oo FES) . % dt Aly hile

NATURE AND ORIGIN OF TRANSIENTS. T that is, c is the reciprocal of the projection T’ = tf on the zero line of the tangent ‘at the starting moment of the transient. Since c= 7 ti =—cdt; 4 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: . logi=—ca +C, _ , t= Ae*, or, i= Ae F; that is, the curve is the exponential. . The exponential curve thus is the expression of the simplest form of transient. This explains its common occurrence in elec- trical and other transients. Consider, for instance, the decay of ~ radioactive substances: the radiation, which represents the decay, is proportional to the amount of radiating material; it is om = cm, ‘which leads to the same exponential function. Not all transients, however, are of this simplest form. For instance, the deceleration of a ship does not follow the exponential, | but at high velocities the decrease of speed is a greater fraction of . the speed than during the same time interval at lower velocities, and the speed-time curves for different initial speeds are not pro- . portional 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. s. Two classes of transients may occur: 1, Energy may be stored in one form only, and the only energy change which can occur thus is an increase or a decrease of the stored energy. 2. Energy may be stored in two or more different forms, and the possible energy changes thus are an increase or decrease of the total stored en. zy, or a change of the stored energy from one form to another. Usually both occur simultaneously. ; An instance of the first case is the acceleration or deceleration

8 ELECTRIC DISCHARGES, WAVES AND IMPULSES. of a train, or aship, etc.: here energy can be stored only as mechan- ical momentum, and the transient thus consists of an increase of the stored energy, during acceleration, or of a decrease, during Si . ! Ss H | i 3 H Seconds ‘0 10 20 30 40 50 60 70 80 9 100 110 120. Fig. 5. — Deceleration of Ship. deceleration. Thus also in a low-voltage electric circuit of negli- gible capacity, energy can be stored only in the magnetic field, and the transient represents an increase of the stored magnetic energy, during increase of current, or a decrease of the magnetic energy, during a decrease of current. An instance of the second case is the pendulum, Fig. 6: with the ; weight at rest in maximum elevation, all the stored energy is potential energy of gravita- tion. This energy changes to kinetic mechanical energy until in the lowest position, a, when

• all the potential gravitational energy has been either con- verted to kinetic mechanical ‘ CY) 4 energy or dissipated. Then, 3 (AN aE _during the rise of the weight, that part of the energy .which Fig. 6. — Double-energy Transient 1s not dissipated again changes , of Pendulum. to potential’ gravitational en- ergy, at c, then back again to kinetic energy, at a; and in this manner the total stored energy y is gradually dissipated, by a series of successive oscillations or . . changes between potential gravitational and kinetic mechanical

-12 ELECTRIC DISCHARGES, WAVES AND IMPULSES. to produce the magnetic field @ of the current 7, a voltage e’ . roust be consumed in the circuit, which with the current 7 gives the power p, which supplies the stored energy w of the magnetic field 6. This voltage e’ is called the inductance voltage, or voltage consumed by self-induction. Since no power is required to maintain the field, but power is required to produce it, the inductance voltage must be propor- tional to the rate of increase of the magnetic field: d® ra, ° di (8) or by (1), di ; re 7 e, . . ea ba (4) If z and therefore © decrease, a and therefore e’ are negative; that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power 7p is w= f p dt, or by (2) and (4), w= f Li di; hence ‘2 is the energy of the magnetic field o=Li ) of the circuit. ; . . 9. Exactly analogous relations exist in the dielectric field. , | The dielectric field, or dielectric flux, ¥, is proportional to the _ - yoltage e, with a proportionality factor, C, which is called the capacity of the circutt: . VW = 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= te, (7) to produce the dielectric field W of the voltage e, a current 7’ must be consumed in the circuit, which with the voltage e gives

THE ELECTRIC PIELD. 11 are crowded together between the conductors, and the magnetic field consists of eccentric circles surrounding the conductors, as shown by the drawn lines in Fig. 9. An electrostatic, or, as more properly called, dielectric field, issues from the conductors, that is, a dielectric flux passes between the conductors, which is measured by the number of lines of dielectric force ¥. With a single conductor, the lines of dielectric force are radial straight lines, as shown dotted in Fig. 8. By the return conductor, they are crowded together between the conductors, and form ares of circles, passing from conductor to return conduc- tor, as shown dotted in Fig. 9. { / ve Cc ‘ . \ — ¥

• ; 6 \ een tonns \ : a ? Guts One na + <a. | . CA < cane wy ; AY ’ ue o t@ ~, - ~{-}- Ce ©): . . 7 Sd / 1 wf ote AK r \ \ ag foot NV MAS. NT ! \ \ ~._| ee J }
  } \ _ ‘ Cc : a“ / Vig. 9. — Electrie Field of Circuit. The magnetic and the dielectric field of the conductors both are included in the term electric field, and are the two components of
• the electric field of the conductor.

8. The magnetic field or magnetic flux of the circuit, &, is pro- portional to the current, 2, with a proportionality factor, LZ, which is called the inductance of the circuit. ; © = Li.* (1) . The magnetic field represents stored energy w. To produce it, power, p, must therefore be supplied by the circuit. Since power is current times voltage: ; p=e', (2) .

• n&, if the flux & interlinks the circuit n fold.

THE ELECTRIC FIELD. | 13 , the power p, which supplies the stored energy w of the dielectric field ¥. This current 7’ is called theycapacity current; or, wrongly, charging current or condenser current. . Since no power is required to maintain the field, but power is required to produce it, the capacity current must be proportional to the rate of increase of the dielectric field: rot, 8) or by (6), . de to _—e v=C Zi (9) If e and therefore WV decrease, oe and therefore 2’ are negative; — : that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power p is . w= f p dt, (10) or by (7) and (9), . w= f Ce de; hence Ce? v= 2 (11) is the energy of the dielectric field , VW = Ce mo of the circuit. As seen, the capacity current is the exact analogy, with regard to the dielectric field, of the inductance voltage with regard to the magnetic field; the representations in the electric circuit, of the energy storage in the field. . The dielectric field of the circuit thus is treated and represented in the same manner, and with the same simplicity and perspicuity, as the magnetic field, by using the same conception of lines of . force. Unfortunately, to a large extent in dealing with the dielectric fields the prehistoric conception of the electrostatic charge on the conductor still exists, and by its use destroys the analogy between the two components of the electric field, the magnetic and the

14 ELECTRIC DISCHARGES, WAVES AND IMPULSES. dielectric, and makes the consideration of dielectric fields un-_ necessarily complicated. There obviously is no more sense in thinking of the capacity current as current which charges the conductor with a quantity of electricity, than there is of speaking of the inductance voltage . as charging the conductor with a quantity of magnetism. But while the latter conception, together with the notion of a quantity of magnetism, etc., has vanished since Faraday’s representation of the magnetic field by the lines of magnetic force, the termi- nology of electrostatics of many textbooks still speaks of electric charges on the conductor, and the energy stored by them, without considering that the dielectric energy is not on the surface of,;the conductor, but in the space outside of the conductor, just as the magnetic energy. ‘ro. 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 circurzt and a dielectric circuit. To produce a magnetic flux &, a magnetomotive force F is required. Since the magnetic field is due to the current, and is proportional to the current, or, in a coiled circuit, to the current times the num- ber of turns, magnetomotive force is expressed in current turns or ampere turns. F=ni. (12) If F is the m.m.f., | the length of the magnetic circuit, energized by F, , ;

J f=F | 3) is called the magnetizing force, or magnetic gradient, and is ex- pressed in ampere turns per cm. (or industrially sometimes in ampere turns per inch).

In empty space, and therefore also, with very close approxi- mation, in all nonmagnetic material, f ampere turns per cm. length of magnetic circuit produce K = 4z/f 10-' lines of magnetic force per square cm. section of the magnetic circuit. (Here the factor 10-! results from the ampere being 10~' of the absolute or cgs. . unit of current.) ,

. H = 47f10''* (14)

• The factor 4 z is a survival of the original definition of the magnetic field

intensity from the conception of the magnetic mass, since unit magnetic mass

was defined as that quantity of magnetism which acts on an equal quantity at

THE ELECTRIC FIELD. «i216 is called the magnetic-field intensity. It is the magnetic density, that is, the number of lines of magnetic force per cm?, produced by the magnetizing force of f ampere turns per cm. in empty space.

• The magnetic density, in lines of magnetic force per cm?, pro- duced by the field intensity 3 in any material is B = pI, (15) where » 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, and some alloys and oxides of - these metals and of manganese and chromium. cS .

If then A is the section of the magnetic circuit, the total magnetic

flux is © = AGB, (16)

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

dF | f= (17) In (16), the magnetic flux 6 would be derived by integrating the densities @ over the total section of the magnetic circuit.

11. Entirely analogous relations exist in the dielectric circuit.

To produce a dielectric flux V, 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=5 (18) unit distance with unit force. The unit field intensity, then, was defined as the field intensity at unit distance from unit magnetic mass, and represented by one line (or rather ‘“‘tube’”’) of magnetic force. The magnetic flux of unit magnetic mass (or ‘‘unit magnet pole’’) hereby became 4 7 lines of force, and - this introduced the factor 4 into many magnetic quantities. An attempt to drop this factor 4 r has failed, as the magnetic units were already too well established.

The factor 10—-' also appears undesirable, but when the electrical units were introduced the absolute unit appeared as too large a value of current as practical unit, and one-tenth of it was chosen as unit, and called ‘‘ampere.” ;

; 16 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . This gives the average voltage gradient, while the actual gradient in an ununiform field, as that between two conductors, varies, being higher at the denser, and lower at the less dense, portion of . the field, and is de ' | G = i’ 1 (19) G * K= ti; then is the dielectric-field intensity, and D=xkK (20) would be the dielectric density, where « is a constant of the material, the electrostatic or dielectric conductivity, and is called the spe- cific capacity or permittivity.- ‘For empty space, and thus with close approximation for air and other gases, , | c=, where - v= 3X 101° is the velocity of light.

; It is customary, however, and convenient, to use the permit- tivity of empty space as unity: « = 1. This changes the unit of dielectric-field intensity by the factor, and gives: dielectric-field intensity,

K=—&,; : (21) 4 xv? dielectric density, ; D = kK, ; (22) where x = 1 for empty space, and between 2 and 6 for most solids , and liquids, rarely increasing beyond 6,. except in materials of appreciable electric conductivity. The dielectric flux then is W = 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 7 appears here in the denominator as the result of the factor 4a” in the magnetic-field intensity 3¢, due to the relations between these quantities. ;

. THE ELECTRIC FIELD. 17 . TABLE I. Magnetic Field. | Dielectric Field. Magnetic flux: Dielectric flux: & = Li 108 lines of magnetic force. WY = Ce lines of dielectric force, or _ coulombs. Inductance voltage: Capacity current: , di ay _ de = na, 10- = Ls, volts. v= 7 = C= amperes. Magnetic energy: Dielectric energy: p= ] w = joul = = joules. = “7 joules. Magnetomotive force: Electromotive force: . F = ni ampere turns. ' e = volts. . Magnetizing force: Electrifying force or voltage gra- PF dient: _ f =~ ampere turns per cm. e l G= I volts per cm. Magnetic-field intensity: Dielectric-field intensity: - _ a : Fe eres masnetic | = —_ 10° lines of dielectric force per cm’, or coulombs . per cm?. Magnetic density: Dielectric density: |. @=y3e lines of magnetic force D = «K lines of dielectric force per cm?, per em*, or coulombs per em? Permeability: u Permittivity or specific capacity: x Magnetic flux: . Dielectric flux: & = AQ lines-of magnetic force. wv = AD lines of dielectric foree, ; or coulombs. v=3 X 10" = velocity of light. The powers of 10, which appear in some expressions, are reduc- tion factors between the absolute or cgs. units which are used for ®, H, @, and the practical electrical units, used for other constants.

• As the magnetic field and the dielectric field also can be con- sidered 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. a,