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., eOJ impressed upon a circuit of resistance r, produces and maintains in the circuit a current, In the moment of closing the circuit of e.m.f. e0 on resistance r, the current in the circuit is zero. Hence, after closing the circuit the current i has to rise from zero to its final value i0. 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 iL interlinkages by current i. That is, in establishing current i0 in the circuit, the magnetic flux iQL must be produced. A change of the magnetic flux iL surrounding a circuit generates in the circuit an e.m.f., d e - * 16 INTRODUCTION 17 This opposes the impressed e.m.f. e0, 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 In the same manner and for the same reasons, if the impressed e.m.f. eQ is withdrawn, but the circuit left closed, the current i does not instantly disappear but gradually dies out, as shown in Fig. 1, which gives the rise and the decay of a ^ ,__— • IS 1 / \ 5 -2- KM ?/i 1 / 1 1 \ I, = • 01 en r.v- 1° \ I a a \ 1 < \ / V / \ / \ s ' — , -*^. -0- ( 1 I I , 5 ! i j ; L I 1 ! I 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 eQ = 240 volts; the abscissas being seconds of time. 13. If an electrostatic condenser of capacity C is connected to a continuous e.m.f. e0, 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 eo; or contains the electrostatic charge Q = to0. In the moment of closing the circuit of e.m.f. e0 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 e0. If r is the resistance of the direct-current circuit containing the condenser, and this circuit contains no inductance, the current Q 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 inductance 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 iL in the inductance produces an e.m.f. Ldi/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 iL 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 i, and the potential difference at the con- denser terminals, as e, with a continuous impressed e.m.f. e0, for the circuit constants r = 250 ohms; L = 100 mh.; C = 10 mf., and e0 = 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. e0, current still exists. This current cannot instantly stop, since the decrease of current and there- with the decrease of its magnetic flux iL generates an e.m.f., 1000 4 — 800 Fig. 2. Charging a condenser through a circuit having resistance and inductance. Constant potential. Logarithmic charge: high resistance. 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. e0, 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. e0, 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 circuit, by a continuous impressed e.m.f. e0. The current is represented by i, 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 eQ = 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 f s N 6 1200 / A \ \ ^ x^ * H 4 7? 8004— f 1 \ \ ^\? - — 4 oSOOJ-4 / \ ^ x* !^° «• =1000 vo oh ml ts o , Ann 1 1 L4= 100 , a lJyJ7 \ . V \ c = 10 mf. \6 =, Dt gr^es AT S k r^ •— ^ 8 u It U\ ' "2A tO 8 !7 4 0 45 0 5( < &J ^s ^2 0 / \ 1 -4 . 1 — \ s t / Fig. 3. Charging a condenser through a circuit having resistance and inductance. Constant potential. Oscillating charge: low resistance. 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.,e0 sufficiently to cause disruptive effects. 14. If an alternating e.m.f., e = E cos 0, is impressed upon a circuit of such constants that the current lags 45°, that is, the current is i = I cos (d - 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 i. This approach of the current from the initial value, in the INTRODUCTION 21 present case zero, to the final value of the curve i, can either be gradual, as shown by the curve il of Fig. 4, or by a series of oscillations of gradually decreasing amplitude, as shown by curve i2 of Fig. 4. 15. The general solution of an electric current problem there- fore includes besides the permanent term, constant or periodic, l /i >c- Gradual or Logarithm o start of current: Oscillatory or 1 S arjthui e start rigonometrio s I**™** [rtartV 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 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, tne 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 inductance 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 arcs). 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 >uch 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. (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 niultigap lightning arresters; the transmission of 24 TRANSIENT PHENOMENA current impulses in telephony; the distribution of alternating current in a conductor, as the 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.