Theory and Calculation of Transient Electric Phenomena and Oscillations — part 2 of 20

This is frequently expressed pictorially by saying that the lines of magnetic force of the circuit are concentric, the lines . of electrostatic force radial to the conductor.

Where, as is usually the case, the electric circuit consists of several conductors, the electric fields of the conductors super- impose upon each other, and the resultant lines of magnetic

and of electrostatic forces are not concentric and radial respec- tively, except approximately in the immediate neighborhood

of the conductor.

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| THE CONSTANTS OF THE ELECTRIC CIRCUIT 5

Between parallel conductors they are conjugate pencils of circles.

2. Neither the power consumption in the conductor, nor the electromagnetic field, nor the electrostatic field, are pro- portional to the flow of energy through the circuit.

The product, however, of the intensity of the magnetic field, ®, and the intensity of the electrostatic field, V, is proportional to the flow of energy or the power, P, and the power P is there- fore resolved into a product of two components, i and e, which are chosen proportional respectively to the intensity of the magnetic field ® and of the electrostatic field V.

That is, putting

P= ve (1) we have ® = In = the intensity of the electromagnetic field. (2) | Vv = Ce = the intensity of the electrostatic field. (3)

The component 7, called the current, is defined as that factor of the electric power P which is proportional to the magnetic field, and the other component e, called the voltage, is defined as that- factor of the electric power P which is proportional to the electrostatic field. .

Current 7 and voltage e, therefore, are mathematical fictions, factors of the power P, introduced to represent respectively the magnetic and the electrostatic phenomena.

The current 7 is measured by the magnetic action of a circuit, as in the ammeter; the voltage e, by the electrostatic action of a circuit, as in the electrostatic voltmeter, or by producing a current 7 by the voltage e and measuring this current 7 by its magnetic action, in the usual voltmeter.

' The coefficients Z and C, which are the proportionality factors of the magnetic and of the electrostatic component of the electric field, are called the inductance and the capacity of the circuit, respectively.

As electric power P is resolved into the product of current 7 and voltage e, the power loss in the conductor, P,, therefore can also be resolved into a product of current 7 and voltage e, which is consumed in the conductor. That is,

P, im 1).

6 TRANSIENT PHENOMENA It is found that the voltage consumed in the conductor, ¢, is proportional to the factor 2 of the power P, that is, e¢ = nN, (4) where r is the proportionality factor of the voltage consumed by the loss of power in the conductor, or by the power gradient, and is called the resistance of the circuit. Any electric circuit therefore must have three constants,

r, L, and C, where

r = circuit constant representing the power gradient, or the loss of power in the conductor, called resistance.

L = circuit constant representing the intensitv of the electro- magnetic component of the electric fiv.J of the circuit, called inductance.

C = circuit constant representing the intensity of the electro- static component of the electric field of the circuit, called capacity.

3. A change of the magnetic field of the conductor, that is, of the number of lines of magnetic force ¢ surrounding the

conductor, generates an e.m.f.

dp e=F (5) Y in the conductor and thus absorbs a power | ., _ dg P=t= tr (6) or, by equation (2), di , P’ = lh "a ’ (7) and the total energy absorbed by the magnetic field during the rise of current from zero to 7 is Wy = [Prat (8) = L fidi, that is, 2 PL Wu- >: @)

THE CONSTANTS OF THE ELECTRIC CIRCUIT 7 A change of the electrostatic field of the conductor, ¥, absorbs a current proportional to the change of the electro- static field: adv » dv 4 a (10) and absorbs the power ; dv Wf ei! = e——,

P’ =e =e a (11)

or, by equation (3), de v7 —,

P” = Ce ii (12) and the total enérgy absorbed by the electrostatic field during a rise of voltage from 0 to e is ;

Wx = f P’'dt (13) =C fede, that is : Wr= ec. (14) 2 The power consumed in the conductor by its resistance r is

P, = tey (15) and thus, by equation (4),

P, = 04. (16)

That is, when the electric power

P=e (1) exists in a circuit, it is P,= ?r = power lost in the conductor, (16)

2 Wu= 7 =energy stored in the magnetic field of the circuit, (9) Wr= “a = energy stored in the electrostatic field of the cir- cuit, (14)

8 TRANSIENT PHENOMENA and the three circuit constants 7, L, C therefore appear as the components of the energy conversion into heat, magnetism, and electric stress, respectively, in the circuit. 4. The circuit constant, resistance r, depends only on the size and material of the conductor, but not on the position of the conductor in space, nor on the material filling the space surrounding the conductor, nor on the shape of the conductor section. | The circuit constants, inductance L and capacity C, almost | entirely depend on the position of the conductor in space, on the material filling the space surrounding the conductor, and on the shape of the conductor section, but do not depend on the material of the conductor, except to that small extent as represented by the electric field inside of the conductor section. 5. The resistance r is proportional to the length and inversely proportional to the section of the conductor, l . r= A ’ (17) where o is a constant of the material, called the resistivity or specific resistance. For different materials, p varies probably over a far greater range than almost any other physical quantity. Given in ohms per centimeter cube,* it is, approximately, “at ordinary tem- peratures : Metals: Cu....... 00000. 1.6 X 107° Allee eee eee eee 2.5 X 107° Fe... eee eee eee eee eee 10 X 107° Hg............... 00. OF & 107° Gray castiron..............up to 100 x 107° High-resistance alloys.......up to 150 x 10° Electrolytes: NO,H.............down to 1.3 at 30-per cent KOH.............down to 1.9 at 25 per cent NaCl..............down to 4.7 at 25 per cent up to Pure river water ............2.2.0-0-+.-.108 and over alcohols, oils, etc., to practically infinity.

• Meaning a conductor of one centimeter length and one square centimeter section. .

THE CONSTANTS OF THE ELECTRIC CIRCUIT 9

So-called “‘insulators”’:

Fiber... 0.0. ec eee eee ee ee eee about 10” Paraffin oil................. eee eee eee about 10% Paraffin........................... about 10" to 10'¢ Mica... 0.2... eee ee ee ee eee about 10" Glass.................0...--24..4.-- about 10" to 10'* Rubber..................2.-.020-+++..-- about 10" Air. eee eee eee eee practically oo

In the wide gap between the highest resistivity of metal alloys, about » = 150 x 107°, and the lowest resistivity of electrolytes, about » = 1, are ,

Carbon: metallic......................about 0.0003 amorphous (dense)..........0.04 and higher . anthracite........................ very high

Silicon and Silicon Alloys:

Cast silicon..........................1 down to 0.04

Ferro silicon................0.04 down to 50 x 107°

The resistivity of arcs and of Geissler tube discharges is of about the same magnitude as electrolytic resistivity.

The res'stivity, p, is usually a function of the temperature, rising slightly with increase of temperature in metallic conduct- ors and decreasing in electrolytic conductors. Only with few materials, as silicon, the temperature variation of ¢ is so enor- mous that p can no longer be considered as even approximately constant for all currents 7 which give a considerable tempera- ture rise in the conductor. Such materials are commonly called pyroelectrolytes.

6. The inductance L is proportional to the section and inversely proportional to the length of the magnetic circuit surrounding the conductor, and so can be represented by

L= a (18) where » is a constant of the material filling the space surround- ing the conductor, which is called the magnetic permeability.

As in general neither section nor length is constant in differ- ent parts of the magnetic circuit surrounding an electric con-

. * | 10 TRANSIENT PHENOMENA ductor, the magnetic circuit has as a rule to be calculated . piecemeal, or by integration over the space occupied by it.

The permeability, 4, is constant and equals unity or very . closely » = 1 for all substances, with the exception of a few materials which are called the magnetic materials, as iron, | cobalt, nickel, etc., in which it is very much higher, reaching sometimes and under certain conditions in iron values as high as » = 6000.

In these magnetic materials the permeability » is not con- stant but varies with the magnetic flux density, or number of lines of magnetic force per unit section, @, decreasing rapidly for high values of @.

In such materials the use of the term » is therefore incon- venient, and the inductance, L, is calculated by the relation between -the magnetizing force as given in ampere-turns per unit length of magnetic circuit, or by ‘‘field intensity,” and magnetic induction @. °

The magnetic induction @ in magnetic materials is the sum of the “space induction” 3, corresponding to unit permeability, | plus the “metallic induction” 8’, which latter reaches a finite | limiting value. That is, |

B=-H+B. (19) |

The limiting values, or so-called “saturation values,” of @’ are approximately, in lines of magnetic force per square centi- | meter: |

Tron... ce eee e eee ees 20,000 Cobalt... 0. cece eee ee eee ee 12,000 Nickel... 0.202000. e eee 6,000 Magnetite..:........0......0 0202 e eee eee ee 5,000 Manganese alloys ..................... .up to 4,000

The inductance, L, therefore is a constant of the circuit if

the space surrounding the conductor contains no magnetic material, and is more or less variable with the current, 2, if magnetic material exists in the space surrounding the conductor. In the latter case, with increasing current, 7, the inductance, L, first slightly increases, reaches a maximum, and then decreases, approaching as limiting value the value which it would have in the absence of the magnetic material.

THE CONSTANTS OF THE ELECTRIC CIRCUIT 11 . 7. The capacity, C, is proportional to the section and inversely proportional to the length of the electrostatic field of the con- ductor: A . c=—, (20) | where « is a constant of the material filling the space surround- ‘ ing the conductor, which is called the “dielectric constant,” or the “specific capacity.” Usually the section and the length of the different parts of the electrostatic circuit are different, and the capacity therefore has to be calculated piecemeal, or by integration. The dielectric constant « of different materials varies over a relative narrow range only. It is approximately: . . « = 1 in the vacuum, in air and in other gases, « = 2 in oils, paraffins, fiber, etc., , « = 3 to 4 in rubber and gutta-percha, « = 3 to 5 in glass, mica, etc., reaching values as high as 7 to 8 in organic compounds of heavy ! metals, as lead stearate, and about 12 in sulphur. The dielectric constant, «, is practically constant for all voltages e, up to that voltage at which the electrostatic field intensity, or the electrostatic gradient, that is, the “volts per centimeter,” exceeds a certain value 6, which depends upon the material and which is called the “dielectric strength” or “disruptive strength”’ of the material. At this potential gradient the medium breaks down mechanically, by puncture, and ceases to insulate, but electricity passes and so equalizes the potential gradient. The disruptive strength, 6, given in volts per centimeter is. approximately : Air: 60,000. ; Oils: 250,000 to 1,000,000. Mica: up to 4,000,000. The capacity, C, of a circuit therefore is constant up to the voltage e, at which at some place of the electrostatic field the dielectric strength is exceeded, disruption takes place, and a : part of the surrounding space therefore is made conducting, and by this increase of the effective size of the conductor the capacity C is increased. |

| ° 12 TRANSIENT PHENOMENA

8. Of the amount of energy consumed in creating the electric field of the circuit not all is returned at the disappearance of the electric field, but a part is consumed by conversion into heat in producing or in any other way changing the electric field.

That is, the conversion of electric energy into and from the

‘ electromagnetic and electrostatic stress is not complete, but a loss of energy occurs, especially with the magnetic field in the | so-called magnetic materials, and with the electrostatic field in | unhomogeneous dielectrics. |

The energy loss in the production and reconversion of the | magnetic component of the field can be represented by an | effective resistance 7’ which adds itself to the resistance r, of the conductor and more or less increases it. |

The energy loss in the electrostatic field can be represented | by an effective resistance 7’, shunting across the circuit, and

. consuming an energy current 7”, in addition to the current 7 in the conductor. Usually, instead of an effective resistance 7’, | its reciprocal is used, that is, the energy loss in the electro- | static field represented by a shunted conductance g. |

In its most general form the electric circuit therefore contains | the constants: |

2 |

1. Inductance L, storing the energy, oe, | . . eC |

2. Capacity C, storing the energy, 3” | |

3. Resistance r = r, + 7’, consuming the power, ?r = ?r,+7?r’, |

4. Conductance g, consuming the power, ¢’g, | _ _wherer,is the resistance of the conductor, 7” the effective resist- ance representing the power loss in the magnetic field L, and g represents the power loss in the electrostatic field C.

5. If of the three components of the electric field, the electro- magnetic stress, electrostatic stress, and the power gradient, one | equals zero, a second one must equal zero also. That is, either | all of the three components exist or only one exists.

Electric systems in which the magnetic component of the field is absent, while the electrostatic component may be consider-

• able, are represented for instance by an electric generator or a battery on open circuit, or by the electrostatic machine. In such systems the disruptive effects due to high voltage, there-

THE CONSTANTS OF THE ELECTRIC CIRCUIT 18

fore, are most pronounced, while the power is negligible, and phenomena of this character are usually called “static.”

Electric systems in which the electrostatic component of the field is absent, while the electromagnetic component is consider- able, are represented for instance by the short-circuited secondary coil of a transformer, in which no potential difference and, there- fore, no electrostatic field exists, since the generated e.m.f. is consumed at the place of generation. Practically negligible also is the electrostatic component in all low-voltage circuits.

The effect of the resistance on the flow of electric energy in industrial applications is restricted to fairly narrow limits: as the resistance of the circuit consumes power and thus lowers the efficiency of the electric transmission, it is uneconomical to permit too high a resistance. As lower resistance requires a larger expenditure of conductor material, it is usually uneco- nomical to lower the resistance of the circuit below that which gives a reasonable efficiency.

As result hereof, practically always the relative resistance, that is, the ratio of the power lost in the resistance to the total ; power, lies between 2 per cent and 20 per cent.

It is different with the inductance L and the capacity C. Of

42

the two forms of stored energy, the magnetic ae and electro- static “ , usually one is so small that it can be neglected com- pared with the other, and the electric circuit with sufficient approximation treated as containing resistance and inductance, or resistance and capacity only.

In the so-called electrostatic machine and its applications, frequently only capacity and resistance come into consideration.

In all lighting and power distribution circuits, direct current or alternating current, as the 110- and 220-volt lighting circuits, the 500-volt railway circuits, the 2000-volt primary distribution circuits, due to the relatively low voltage, the electrostatic energy x is still so very small compared with the electro- magnetic energy, that the capacity C can for most purposes be neglected and the circuit treated as containing resistance and inductance only.

y.

14 - TRANSIENT PHENOMENA

Of approximately equal magnitude is the electromagnetic

2

energy ie and the electrostatic energy ae in the high-potential long-distance transmission circuit, in the telephone circuit, and in the condenser discharge, and so in most of the phenomena resulting from lightning or other disturbances. In these cases all three circuit constants, r, L, and C, are of essential impor- tance.

10. In an electric circuit of negligible inductance L and negligible capacity C, no energy is stored, and a change in the circuit thus can be brought about instantly without any disturb- ance or intermediary transient condition. .

In a circuit containing only resistance and capacity, as a static machine, or only resistance and inductance, as a low or medium voltage power circuit, electric energy is stored essentially in one form only, and a change of the circuit, as an opening of the circuit, thus cannot be brought about instantly, but occurs more or less gradually, as the energy first has to be stored or discharged.

In a circuit containing resistance, inductance, and capacity, and therefore capable of storing energy in two different forms, the mechanical change of circuit conditions, as the opening of a circuit, can be brought about instantly, the internal energy of the circuit adjusting itself to the changed circuit conditions by a transfer of energy between static and magnetic and inversely, that is, after the circuit conditions have been changed, a transient phenomenon, usually of oscillatory nature, occurs in the circuit by the readjustment of the stored energy.

These transient phenomena of the readjustment of stored electric energy with a change of circuit conditions require careful study wherever the amount of stored energy is sufficiently large to cause serious damage. This is analogous to the phenomena of the readjustment of the stored energy of mechanical motion: while it may be harmless to instantly stop a slowly moving light carriage, the instant stoppage, as by collision, of a fast railway train leads to the usual disastrous result. So also, in electric systems of small stored energy, a sudden change of circuit con- ditions may be safe, while in a high-potential power system of very great stored electric energy any change of circuit conditions requiring a sudden change of energy is liable to be destructive.

THE CONSTANTS OF THE ELECTRIC CIRCUIT 15 Where electric energy is stored in one form only, usually little danger exists, since the circuit protects itself against sudden change by the energy adjustment retarding the change, and only where energy is stored electrostatically and magnetically, the mechanical change of the circuit conditions, as the opening of the circuit, can be brought about instantly, and the stored energy then surges between electrostatic and magnetic energy.

In the following, first the phenomena will be considered which result, from the stored energy and its readjustment in circuits storing energy in one form only, which usually is as electro- Magnetic energy, and then the general problem of a circuit . storing energy electromagnetically and electrostatically will be considered.

CHAPTER II. INTRODUCTION. 11. In the investigation of electrical phenomena, currents and potential differences, whether continuous or alternating, are usually treated as stationary phenomena. That is, the assumption is made that after establishing the circuit a sufficient time has elapsed for the currents and potential differences to reach their final or permanent values, that is, become constant, with continuous current, or constant periodic functions of time, with alternating current. In the first moment, however, after establishing the circuit, the currents and potential differences in the circuit have not yet reached their permanent values, . that is, the electrical conditions of the circuit are not yet the | normal or permanent ones, but a certain time elapses while the electrical conditions adjust themselves. 12. For instance, a continuous e.m.f., e,, impressed upon a circuit of resistance r, produces and maintains in the circuit a - | current, i, = 2. | Tr In the moment of closing the circuit of e.m.f. e, on resistance r, | the current in the circuit is zero. Hence, after closing the circuit the current 7 has to rise from zero to its final value 7, If the circuit contained only resistance but no inductance, this would take place instantly, that is, there would be no transition period. Every circuit, however, contains some inductance. The induc- tance L of the circuit means L interlinkages of the circuit with, lines of magnetic force produced by unit current in the circuit, . or tL interlinkages by current 7. That is, in establishing current 2, in the circuit, the magnetic flux 7,L must be produced. A change of the magnetic flux 7Z surrounding a circuit generates in the circuit an e.m.f,, d e=— (iL). dts’ ) 16

INTRODUCTION 17 This opposes the impressed e.m.f. e,, and therefore lowers the . e.m.f. available to produce the current, and thereby the current, which then cannot instantly assume its final value, but rises thereto gradually, and so between the starting of the circuit and the establishment of permanent condition a transition period appears. In the same manner and for the same reasons, if the impressed e.m.f. e, is withdrawn, but the circuit left closed, the current < does not instantly disappear but gradually dies out, as shown in Fig. 1, which gives the rise and the decay of a TTT TTT TET Hekate LO rT TTT rrr Ate mote} LO PTT TIAT TET TT iy TAT TP tee fe heneyd [| Ht t+ +++ tt TY TTT tt Tt tl rT TIA TT Trt i fe PA TT TT PTT yt Tt tty reer IN TT BEY SRN HAE A PT VT TT TTT yt Er rN TTT Pe yt TT TTT yr ty err Pi Tt ree err Pe Ee Te ey ye eer TT TT 0 1 2 8 4 5 0 1 2 3 4 5 Seconds Fig. 1. Rise and decay of continuous current in an inductive circuit. continuous current in an inductive circuit: the exciting current of an alternator field, or a circuit having the constants r = 12 ohms; L = 6 henrys, and e, = 240 volts; the abscissas being seconds of time.

13. If an electrostatic condenser of capacity C is connected to a continuous e.m.f. e,, no current exists, in stationary con- dition, in this direct-current circuit (except that a very small current may leak through the insulation or the dielectric of the condenser), but the condenser is charged to the potential dif- ference e,, or contains the electrostatic charge

Q = Ce,.

In the moment of closing the circuit of e.m.f. e, upon the capacity C, the condenser contains no charge, that is, zero potential difference exists at the condenser terminals. If there were no resistance and no inductance in the circuit in the

18 TRANSIENT PHENOMENA | | moment of closing the circuit, an infinite current would exist | charging the condenser instantly to the potential difference e,. | If r is the resistance of the direct-current circuit containing the condenser, and this circuit contains no inductance, the current . ’ starts at the value i = °°, that is, in the first moment after closing the circuit all the impressed e.m.f. is consumed by the current in the resistance, since no charge and therefore no potential difference exists at the condenser. - With increasing charge of the condenser, and therefore increasing potential difference at the condenser terminals, less and less e.m.f. is available for the resistance, and the current decreases, and ultimately becomes zero, when the condenser is fully charged.

If the circuit also contains inductarice L, then the current cannot rise instantly but only gradually: in the moment after closing the circuit the potential difference at’ the condenser is still zero, and rises at such a rate that the increase of magnetic flux iZ in the inductance produces an e.m.f. Ld/dt, which consumes the impressed e.m.f. Gradually the potential differ- ence at the condenser increases with its increasing charge, and the current and thereby the e.m.f. consumed by the resistance increases, and so less e.m.f. being available for consumption by the inductance, the current increases more slowly, until ulti- mately it ceases to rise, has reached a maximum, the inductance consumes no e.m.f., but all the impressed e.m.f. is consumed by . the current in the resistance and by the potential difference at the condenser. The potential difference at the condenser con- tinues to rise with its increasing charge; hence less e.m.f. is available for the resistance, that is, the current decreases again, and ultimately becomes zero, when the condenser is fully charged, During the decrease of current the decreasing mag- netic flux 7Z in the inductance produces an e.m.f., which assists . the impressed e.m.f., and so retards somewhat the decrease of

, current.

Fig. 2 shows the charging current of a condenser through an inductive circuit, as 7, and the potential difference at the con- denser terminals, as e, with a continuous impressed e.m.f. @,, for the circuit constants r = 250 ohms; L = 100 mh.; C = 10 mf., and e, = 1000 volts.

If the resistance is very small, the current immediately after

INTRODUCTION 19 closing the circuit rises very rapidly, quickly charges the con- denser, but at the moment where the condenser is fully charged to the impressed e.m.f. e,, current still exists. This current cannot instantly stop, since the decrease of current and there- with the decrease of its magnetic flux 7Z generates an e.m.f.,

1000 ] Faral T T “ aap =—annnl L— womke | | | SCC AAT CSS Tom tome. FT] Se-So0 |_| | |i | lap fe) eT CL A < ol pt fate ~ | + ge tte So Seeeee o—olLl iit | oe) ae a a ee bal Pt 4 8 kee6e@wenunueese 8 6B Fig. 2. Charging a condenser through a circuit having resistance and inductance. Constant potential. Logarithmic charge. which maintains the current, or retards its decrease. Hence electricity still continues to flow into the condenser for some time after it is fully charged, and when the current ultimately stops, the condenser is overcharged, that is, the potential dif- ference at the condenser terminals is higher than the impressed e.m.f. e,, and as result the condenser has partly to discharge again, that is, electricity begins to flow in the opposite direction, or out of the condenser. In the same manner this reverse current, due to the inductance of the circuit, overreaches and discharges the condenser farther than down to the impressed e.m.f. e,, so that after the discharge current stops again a charg- ing current — now less than the initial charging current — starts, and so by a series of oscillations, overcharges and under- charges, the condenser gradually charges itself, and ultimately the current dies out.

Fig. 3 shows the oscillating charge of a condenser through an inductive cireuit, by a continuous impressed e.m.f. e,. The current is represented by 7, the potential difference at the con- denser terminals by e, with the time as abscissas. The con- stants of the circuit are: r = 40 ohms; L = 100 mh.; C = 10 mf., and e, = 1000 volts.

In such a continuous-current circuit, containing resistance, inductance, and capacity in series to each other, the current at the moment of closing the circuit as well as the final current

20 TRANSIENT PHENOMENA is zero, but a current exists immediately after closing the ‘ circuit, as a transient phenomenon; a temporary current, " steadily increasing and then decreasing again to zero, or con- ; sisting of a number of alternations of successively decreasing ‘ amplitude: an oscillating current. : If the circuit contains no resistance and inductance, the cur- rent into the condenser would theoretically be infinite. That 1600 Teavaanesestiaiie = a : “LX a ce . 2— 400 +—— | | || tnd 100/mh. fn ae fama NE Pee Eee A a ocoee eee HHH és WW Pro Fig. 3. Charging a condenser through a circuit having resistance and inductance, Constant potential. Oscillating charge. ; is, with low resistance and low inductance, the charging current of a condenser may be enormous, and therefore, although only transient, requires very serious consideration and investigation. If the resistance is very low and the inductance appreciable, the overcharge of the condenser may raise its voltage above the impressed e.m.f., e, sufficiently to cause disruptive effects. 14. If an alternating e.m.f., e = Ecos6, is impressed upon a circuit of such constants that the current lags 45°, that is, the current is t = I cos (@ — 45°), and the circuit is closed at the moment 6 = 45°, at this moment the current should be at its maximum value. It is, however, zero, and since in a circuit containing inductance (that is, in practically any circuit) the current cannot change instantly, it follows that in this case the current gradually rises from zero as initial value to the permanent value of the sine wave 1. This approach of the current from the initial value, in the t

INTRODUCTION 21 | | / present case zero, to the final value of the curve 7, can either | be gradual, as shown by the curve 7, of Fig. 4, or by a series . of oscillations of gradually decreasing amplitude, as shown by curve 2, of Fig. 4. 15. The general solution of an electric current problem there- fore includes besides the permanent term, constant or periodic, | tt tt fs ies coe | OCC CCA eer re, | PT Tt ta ay TT TT . Le PREANTT T T TTT T TET T T (27 SRS ee ete IN PRT TTT TT TT A . | ver | fist BEN TTT TT TT eT A) A ft NA L/h ! a ttTtTT TERING TTT ZETA | PT TT TTT PIN NET TA TY TT a LT TT TIT Ti N&O YT TT Pt PT TT TN Sede tt SRR | Fig. 4. Starting of an alternating-current circuit having inductance. | ° a transient term, which disappears after a time depending upon | the circuit conditions, from an extremely small fraction of a . second to a number of seconds. . These transient terms appear in closing the circuit, opening . the circuit, or in any other way changing the circuit conditions, . as by a change of load, a change of impedance, etc. In general, in a circuit containing resistance and inductance only, but no capacity, the transient terms of current and volt- - | age are not sufficiently large and of long duration to cause a harmful nor even appreciable effects, and it is mainly in circuits containing capacity that excessive values of current and poten- ! tial difference may be reached by the transient term, and there- . with serious results occur. The investigation of transient terms ; therefore is largely an investigation of the effects of electro- static capacity. 16. No transient terms result from the resistance, but only . those circuit constants which represent storage of energy, mag- . netically by the inductance L, electrostatically by the capacity C, give rise to transient phenomena, and the more the resist- | | |

22 ; ‘TRANSIENT PHENOMENA | ance predominates, the less is therefore the severity and dura- tion of the transient term. . When closing a circuit containing inductance or capacity or both, the energy stored in the inductance and the capacity has first to be supplied by the impressed e.m.f. before the circuit conditions can become stationary. That is, in the first | moment after closing an electric circuit, or in general changing the circuit conditions, the impressed e.m.f., or rather the source producing the impressed e.m.f., has, in addition to the power consumed in maintaining the circuit, to supply the power which stores energy in inductance and capacity, and so a transient term appears immediately after any change of circuit condi- tion. If the circuit contains only one energy-storing constant, as either mductance or capacity, the transient term, which connects the initial with the stationary condition of the circuit, necessarily can be a steady logarithmic term only, or a gradual approach. An oscillation can occur only with the existence of | two energy-storing constants, as capacity.and inductance, which . permit a surge of energy from the one to the other, and there- with an overreaching. 17. Transient terms may occur periodically and in rapid suc- cession, as when rectifying an alternating current by synchro- nously reversing the connections of the alternating impressed e.m.f. with the receiver circuit (as can be done mechanically or without moving apparatus by undirectional conductors, as ares). At every half wave the circuit reversal starts a tran- sient term, and usually this transient term has not yet disap- peared, frequently not even greatly decreased, when the next ’ reversal again starts a transient term. These transient terms may predominate to such an extent that the current essentially consists of a series of successive transient terms. 18. If a condenser is charged through an inductance, and the condenser shunted by a spark gap set for a lower voltage than the impressed, then the spark gap discharges as soon as the condenser charge has reached a certain value, and so starts a transient term; the condenser charges again, and discharges, and so by the successive charges and discharges of the condenser a series of transient terms is produced, recurring at a frequency depending upon the circuit constants and upon the ratio of the disruptive voltage of the spark gap to the impressed e.m.f.

INTRODUCTION 23 Such a phenomenon for instance occurs when on a high- potential alternating-current system a weak spot appears in the cable insulation and permits a spark discharge to pass to the ground, that is, in shunt to the condenser formed by the cable conductor and the cable armor or ground. 19. In most cases the transient phenomena occurring in electric circuits immediately after a change of circuit conditions are of no importance, due to their short duration. They require serious consideration, however, — (a) In those cases where they reach excessive values. Thus in connecting a large transformer to an alternator the large initial value of current may do damage. In short-circuiting a large alternator, while the permanent or stationary short-circuit current is not excessive and represents little power, the very much larger momentary short-circuit current may be beyond the capacity of automatic circuit-opening devices and cause damage by its high power. In high-potential transmissions ~ the potential differences produced by these transient terms may reach values so high above the normal voltage as to cause disruptive effects. | (6) Lightning, high-potential surges, etc., are in their nature . essentially transient phenomena, usually of oscillating character. a (c) The periodical production of transient terms of oscillating character is one of the foremost means of generating electric cur- rents of very high frequency as used in wireless telegraphy, etc. . (d) In alternating-current rectifying apparatus, by which the direction of current in a part of the circuit is reversed every half wave, and the current so made unidirectional, the stationary condition of the current in the alternating part of the circuit is usually never reached, and the transient term is frequently of primary importance. . (e) In telegraphy the current in the receiving apparatus essentially depends on the transient terms, and in long-distance cable telegraphy the stationary condition of current is never approached, and the speed of telegraphy depends on the duration of the transient terms. (f) Phenomena of the same character, but with space instead of time as independent variable, are the distribution of voltage and current in a long-distance transmission line; the phenomena occurring in multigap lightning arresters; the transmission of

24 TRANSIENT PHENOMENA current impulses in telephony; the distribution of alternating current in a conductor, asthe rail return of a single-phase railway; the distribution of alternating magnetic flux in solid magnetic material, etc.

Some of the simpler forms of transient terms are investigated and discussed in the following pages.

CHAPTER III. INDUCTANCE AND RESISTANCE IN CONTINUOUS- CURRENT CIRCUITS.

20. In continuous-current circuits the inductance does not enter the equations of stationary condition, but, if e, = impressed e.m,.f., r = resistance, L = inductance, the permanent value of current is 1, = 2 .

Therefore less care is taken in direct-current circuits to reduce the inductance than in alternating-current circuits, where the inductance usually causes a drop of voltage, and direct-current circuits as a rule have higher inductance, especially if the circuit is used for producing magnetic flux, as in solenoids, electro- magnets, machine-fields.

Any change of the condition of a continuous-current circuit, as a change of e.m.f., of resistance, etc., which leads to a change of current from one value 7, to another value 7,, results in the appearance of a transient term connecting the current values ?, and 7,, and into the equation of the transient term enters the inductance.

Count the time ¢ from the moment when the change in the continuous-current circuit starts, and denote the impressed em.f. by e,, the resistance by r, and the inductance by L.

i= ° = current in permanent or stationary condition after the change of circuit condition.

Denoting by 7, the current in circuit before the change, and therefore at the moment ¢ = 0, by 7 the current during the change, the e.m.f. consumed by resistance r is

ar, and the e.m.f. consumed by inductance L is di , L hi where 7 = current in the cireuit.

| 26 TRANSIENT PHENOMENA Hence, €é, =ir+ Lo, . (1) or, substituting e, = 7,r, and transposing, r di —--d=——.- 2 L 1% ( ) This equation is integrated by : — Ft = log (i — 1,) — loge, where — log is the integration constant, or, . i-—it= ce u, However, for ¢ = 0,2 = 2,. Substituting this, gives . %— 1, =¢, te hence, t=ayAt+ (%—U4)e 7 ,— (3) the equation of current in the circuit. The counter e.m.f. of self-inductance is di... Ht ge LG aries @) hence a maximum for t = 0, thus: e, =r (t, — 1,). (5) The e.m.f. of self-inductance e, is proportional to the change of current (i, — 7,), and to the resistance r of the circuit after the change, hence would be » for r = », or when opening the circuit. That is, an inductive circuit cannot be opened instantly, _ but the arc following the break maintains the circuit for some time, and the voltage generated in opening an inductive circuit is the higher the quicker the break. Hence in a highly inductive circuit, as an electromagnet or a machine field, the insulation may be punctured by excessive generated e.m.f. when quickly ; opening the circuit. _ As example, some typical circuits may be considered.