CHAPTER X INSTABILITY OF CIRCUITS : THE ARC A. General 81. During the earlier days of electrical engineering practi- cally all theoretical investigations were limited to circuits in stable or stationary condition, and where phenomena of instability occurred, and made themselves felt as disturbances or troubles in electric circuits, they either remained imunderstood or the theo- retical study was limited to the specific phenomenon, as in the case of lightning, dropping out of step of induction motors, hunt- ing of synchronous machines, etc., or, as in the design of arc lamps and arc-lighting machinery, the opinion prevailed that theoretical calculations are impossible and only design by trying, based on practical experience, feasible. The first class of imstable phenomena, which was systemat- ically investigated, were the transients, and even today it is ques- tionable whether a systematic theoretical classification and in- vestigation of the conditions of instability in electric circuits is yet feasible. Only a preliminary classification and discussion of such phenomena shall be attempted in the following. Three main types of instability in electric systems may be distinguished : I. The transients of readjustment to changed circuit con- ditions. II. Unstable electrical equilibrium, that is, the condition in which the eflfect of a cause increases the cause. III. Permanent instability resulting from a combination of circuit constants which can not coexist. I. Transients 82. Transients are the phenomena by which, at the change of circuit conditions, current, voltage, etc., readjust themselves from the values corresponding to the previous condition to the values corresponding to the new condition of the circuit. For in- u 161 162 ELECTRIC CIRCUITS stance, if a switch is closed, and thereby a load put on the circuit, the ciurent can not instantly increase to the value corresponding to the increased load, but some time elapses, diu-ing which the increase of the stored magnetic energy corresponding to the in- creased current, is brought about. Or, if a motor switch is closed, a period of acceleration intervenes before the flow of current be- comes stationary, etc. The characteristic of transients therefore is, as implied in the term, that they are of limited, usually very short duration, inter- vening between two periods of stationary conditions. Considerable theoretical work has been done, more or less systematically, on transients, and a great mass of information is thus available in the literature. These transients are more ex- tensively treated in "Theory and Calculation of Transient Elec- tric Phenomena and Oscillations," and in " Electric Discharges, Waves and Impulses, '' and therefore will be omitted in the fol- lowing. However, to some extent, the transients of our theoret- ical literature, still are those of the "phantom circuit," that is, a circuit in which the constants r, L, C, g, are assumed as constant. The effect of the variation of constants, as found more or less in actual circuits: the change of L with the current in circuits con- taining iron; the change of C and g with the voltage (corona, etc.) ; the change of r and g with the frequency, etc., has been studied to a limited extent only, and in specific cases. In the application of the theory of transients to actual electric circuits, considerable judgment thus is often necessary to allow and correct for these "secondary" phenomena which are not in- cluded in the theoretical equations. Especially deficient is oiu* knowledge of the conditions under which the attenuation constant of the transient becomes zero or negative, and the transient thereby becomes permanent, or becomes a cumulative surge, and the phenomenon thereby one of unstable equilibrium. II. Unstable Electrical Equilibbixjm 83. If the effect brought about by a cause is such as to oppose or reduce the cause, the effect must limit itself and stability be finally reached. If, however, the effect brought about by a cause increases the cause, the effect continues with increasing intensity, that is, instability results. INSTABILITY OF CIRCUITS 163 This applies not to electrical phenomena alone, but equally to all other phenomena. Instability of an electric circuit may assume three different forms: 1. Instability leading up to stable conditions. For instAnce, in a pyroelectric conductor of the volt-ampere cbaracteristic given in Fig. 78, at the impressed volti^, eg, three different values of current are possible: ii, i% and ig. ii and ii are stable, in unstable. That is, at current, h, passing through the conductor under the constant impressed voltage, e^, a mo- mentary increase of current would give an excess voltage beyond that required by the conductor, thereby increase the current still •■. 7V \ -- ^ =-'' VI l" Wm^ ' i Fia. 78. further, and with increasing rapidity the current would rise, until it becomes stable at the value, ij. Or, a momentary decrease of current, by requiring a higher voltage than available, would further decrease the current, and with increasing rapidity the current would decrease to the stable value, ii. 2, Instability putting the circuit out of service. An instance is the arc on constant-potential supply. With the volt-ampere characteristic of the arc shown as A, in Fig. 79, a current of 4 amp. would require SO volts across the arc terminals. At a constant impressed voltage of 80, the current could not re- main at 4 amp., but the current would either decrease with in- creasing rapidity, until the arc goes out, or the current would in- 164 ELECTRIC CIRCUITS crease with increasing rapidity, up to short-circuit, that is, until the supply source limits the current, 3. Instability leading again to iustability, and thus periodically repeating the phenomena. For instance, if an arc of the volt-ampere characteristic. A, in Fig. 79 is operated in a constant-current circuit of sufficiently high direct voltage to restart the arc when it goes out, and the arc «■ 1.D ^ iin \ C^ -' litn \ V ^^ fn \ in \ \ ■ ^ ■m ~- ^ .0 ■^ >^ ~ rn ^ ^ m ^ 5- ^ y' ^ ^ ^ ^ ^ i: Fig. 79. . ia shunted by a condenser, the condenser nmkes the arc unstable and puts it out; the available supply voltage, however, starts it again, and so periodically the arc starts and extinguishes, aa an "oscillating arc." 84. There are certain circuit elements which tend to produce instability, such as arcs, pyroelectric conductors, condensers, induction and synchronous motors, etc., and their recognition therefore is of great importance to the engineer, in guarding \^ 1 [<^ INSTABILITY OF CIRCUITS 165 s^ainst instability. Whether instability results, and what form it assumes, depends, however, not only on the "exciting element,'' as we may call the cause of the instability, but on all the elements of the circuit. Thus an arc is unstable, form (2), on constant- voltage supply at its terminals; it is stable on constant-current supply. But when shunted by a condenser, it becomes un- stable on constant current, and the instability may be form (2) or form (3), depending on the available voltage. With a resist- ance, r, of volt-ampere characteristic ir shown as S, in Fig. 79, the arc is stable on constant-voltage supply for currents above z'o = 3 amp., unstable below 3 amp., and therefore, with a constant- supply voltage, Co, two current values, ii and ig, exist, of which the former one is stable, the latter one unstable. That is, current, ii, can not persist, but the current either runs up to ii and the arc then gets stable (form 1), or the current decreases and the arc goes out, instabiUty form (2). Thus it is not feasible to separately discuss the different forms of instability, but usually all three may occur, under different circuit conditions. The electric arc is the most frequent and most serious cause of instability of electric circuits, and therefore should first be sus- pected, especially if the instability assumes the form of high- frequency disturbances or abrupt changes of current or voltage, such as is shown for instance in the oscillograms. Figs. 80 and 81. Somewhat similar effects of instability are produced by pyro- electric conductors. Induction motors and synchronous motors may show instability of speed: dropping out of step, etc. III. Permanent instability 86. If the constants of an electric circuit, as resistance, in- ductance, capacity, disruptive strength, voltage, speed, etc., have values, which can not coexist, the circuit is unstable, and remains so as long as these constants remain unchanged. Case (3) of II, imstable equilibrium, to some extent may be considered as belonging in this class. The most interesting class in this group of unstable electric systems are the oscillations resulting sometimes from a change of circuit conditions (switching, change of load, etc.), which con- tinue indefinitely with constant intensity, or which steadily increase in intensity, and may thus be called permanent and 166 ELECTRIC CIRCUITS cumulative surges, hunting, etc. They may be considered as transients in which the attenuation constant is zero or negative. In the transient resulting from a change of circuit conditions, the energy which represents the difference of stored energy of the circuit before and after the change of circuit condition, is dissi- pated by the energy loss in the circuit. As energy losses always occur, the intensity of a true transient thus must always be a maximum at the beginning, and steadily decrease to zero or per- manent condition. An oscillation of constant intensity, or of increasing intensity, thus is possible only by an energy supply to the oscillating system brought about by the oscillation. If this energy supply is equal to the energy dissipation, constancy of the phenomenon results. If the energy supply is greater than the energy dissipation, the oscillation is cumulative, and steadily increases until self-destruction of the system results, or the in- creasing energy loss becomes equal to the energy supply, and a stationary condition of oscillation results. The mechanism of this energy supply to an oscillating system from a source of energy differing in frequency from that of the oscillation is still practi- cally unknown, and very little investigating work has been done to clear up the phenomenon. It is not even generally realized that the phenomenon of a permanent or cumulative line surge involves an energy supply or energy transformation of a fre- quency equal to that of the oscillation. Possibly the oldest and best-known instance of such cumulative oscillation is the hunting of synchronous machines. Cumulative oscillations between electromagnetic and electro- static energy have been observed by their destructive effects in high- voltage electric circuits on transformers and other apparatus, and have been, in a number of instances where their frequency was sufficiently low, recorded by the oscillograph. They obvi- ously are the most dangerous phenomena in high-voltage electric circuits. Relatively little exact knowledge exists of their origin. Usually — if not always — an arc somewhere in the system is instrumental in the energy supply which maintains the oscilla- tion. In some instances, as in wireless telegraphy, they have found industrial application. A systematic theoretical investiga- tion of these cumulative electrical oscillations probably is one of the most important problems before the electrical engineer, today. The general nature of these permanent and cumulative oscilla- tions and their origin by oscillating energy supply from the transi- INSTABILITY OF CIRCUITS 167 ent of a change of circuit condition, is beat illustrated by the in- stance of the hunting of eynchronous machines, and this will, therefore, be investigated somewhat more in detail. B. The Arc as Unstable Conductor 86. The instability of the arc is the result of its dropping volt- ampere characteristic, as discussed in paragraphs 18 to 27 of the V: \ \ \ J ■= .To ARC ON ^CONSTANT VOLTAGE =a SUPPLY \ f s T \ \ \f \ » E im? \ \ ^ V \ \ «^ -^ \ ■N ,-- '^ ^ < \ "s ^ -^ C " ~~- \ ^ ::\ :^ s \ 1 ~^ — . tt \ Q ^ U- ■?>- ^ k^ / _ , / ^ / F -^ — — i_ ^ 5 2. 3. G L i. G EL G chapter on "Electric Conductors." As shown there, the arc is always unstable on constant voltage impressed upon it. Series 168 ELECTRIC CIRCUITS resistance or reactance produces stability for currents above a certain critical value of current, io. Such curves, giving the vol- tage consumed by the arc and its series resistance as function of the current, thus may be termed stability curves of the arc. Their minimum values, that is, the stability limits corresponding to the different resistances, give the stability characteristic of the arc. The equations of the arc, and of its stability curves and stability characteristic, are given in paragraphs 22 and 23 of the chapter on ^' Electric Conductors. " Let, in Fig. 82, A present the volt-ampere characteristic of an arc, given approximately by the equation c(l + d) e = a + .^ where (1) e' = 4= (2) is the stream voltage, that is, voltage consumed by the arc stream. Fig. 82 is drawn with the constants, a = 35, c = 51, I = 1.8, d = 0.8, hence, 6 = 35 + i^- Assuming this arc is operated from a circuit of constant-voltage supply, E = 150 volts, through a resistance, vo The voltage consumed by the resistance, ro, then is 62 = rot, (3) and the voltage available for the arc thus ei = E - roi (4) Lines B, C and D of Fig. 82 give ei, for the values of resistance, ro = 20 ohms (B) = 10 ohms (C) = 13 ohms (D). INSTABILITY OF CIRCUITS 169 As seen, line B does not intersect the volt-ampere characteris- tic, A, of the arc, that is, with 20 ohms resistance in series, this I = 2.5 cm. arc can not be operated from E = 150 volt supply. Line C intersects A at a and 6, i = 6.1 and 1.9 amp. respect- ively. At a, z = 6.1 amp., the arc is stable; At 6, i = 1.9 amp., the arc is unstable; for the reasons discussed before: an increase of current decreases the voltage consumed by the circuit, e + ^2, and thus still further increases the current, and inversely. Thus the arc either goes out, or the current rims up to i = 6.1 amp., where the arc gets stable. Line D is drawn tangent to A, and the contact point, c, thus gives the minimum current, i = 3.05 amp., of operation of the arc on E = 150 volts, that is, the value of current or of series resist- ance, at which the arc ceases to be stable: a point of the stability characteristic, S, of the arc. This stability characteristic is determined by the condition where Co = e + roi (6) this gives and r = - ^ (7) 2 ? V7 2 i , 1.5 6 = a + 1-5 ei (8) as the equation of the stability characteristic of the arc on a con- stant-voltage circuit. 87. In general, the condition of stability of a circuit operated on constant-volt&ge supply, is f.>0 (9) where e is the voltage consumed by the current, i, in the circuit. The ratio of the change of voltage, de, as fraction of the total voltage, e, brought about by a change of current, di, as fraction of 170 ELECTRIC CIRCUITS the total current, i, thus may be called the stability coefficient of the circuit, de dd t de dt (10) In a circuit of constant resistance, r, it is e de __ hence, d = 1, that is, the stability coefficient of a circuit of constant resistance, r, is unity. In general, if the efifective resistance, r, is not constant, but varies with the current, z, it is e = n, de , ,dr df + 'di' hence, the stability coefficient dr 6 = 1+^ r (11) thus in a circuit, in which the resistance increases with the current, the stability coefficient is greater than 1. Such is that of a con- ductor with positive temperature coefficient of resistance, in which the temperature rise due to the increase of current increases the resistance. A conductor with negative temperature coeffici- ent of resistance gives a stability coefficient less than 1, but as long as 5 is still positive, that is, the decrease of resistance slower than the increase of current, the circuit is stable. d>0 (12) INSTABILITY OF CIRCUITS 171 is the condition of stability of a circuit on constant-voltage supply, and 5 < (13) is the condition of instability, and 5 = (14) thus gives the stabiUty characteristic of the circuit. In the arc, , b the stability coefficient is, by (10), _ b 6^ . «v 2 e\/7 2e e' that is, equals half the stream voltage, « > divided by the arc voltage, e. Or, substituting for e in (15), and rearranging, 1 5 = - 1 (16) 2(1 + 0.2625 VT) in Fig. 82. Fori = 0, itis 5= —0.5; i = 00, it is 5 = 0. The stability coefficient of the arc having the volt-ampere characteristic, A, in Fig. 82 is shown as F in Fig, 82. 88. On constant-voltage supply, E = 150 volts, the arc having the characteristic. A, Fig. 82, can not be operated at less than 3.05 amperes. At i = 3.05 is its stability limit, that is, the stability coefficient of arc plus series resistance, ro, required to give 150 volts, changes from negative for lower currents, to positive for higher currents. The stability coefficient of such arcs, operated on constant- voltage supply through various amounts of series resistance, ro, then would be given by dep ■ di do = — , • where 172 ELECTRIC CIRCUITS and the resistance tq chosen so as to give ei = 150 volts, from (17) follows, do = > Co and, substituting from (17), 6 tro = eo — a — eo = a + —7= + roi (17) Vi gives , 1.5 6 a + 8„ = 1 ^ (1«> or, 5o = 1 - - (19) eo where eo is the supply voltage, eo the voltage given by the stability characteristic, S, 8oy the stability characteristic of the arc, A,on E — 150 volt constant-potential supply, is given as curve, G, in Fig. 82. As seen, it passes from negative — ^instability — ^to positive — stability — at the point, k, corresponding to c and h on the other curves. 89. On a constant-current supply, an arc is inherently stable. Instability, however, may result by shunting the arc by a resist- ance, Vi. Thus in Fig. 83, let / = 5 amp. be the constant supply current. The volt-ampere characteristic of the arc is given by A, and shows that on this 5-amp. circuit, the arc consumes 94 volts, point d. Let now the arc be shunted by resistance, Vi. K e = voltage consumed by the arc, the current shunted by the rec^istance, ri, is ii = ~ (20) and the current available for the arc thus is i = I - ix (21) ri or e = rid - il (22) INSTABILITY OF CIRCUITS 173 Curves B, C and D of Fig. 83 ehow the values of equation (22) or Ti = 32 ohms: line 5 = 48 ohms: line C = 40.8 ohms: line D. «: I ARC ON CONSTANT CURRENT SUPPLY 1 I \ ^ ^1 \ i ^ ^ \ I - \ B * I l \ S »? s \ \ V ^ > ^ \ ~~ A \ [ s \ a -J __ s \ -^ A " \ D\ S^ ;^ 'P' =" -^ ^ \^ ^ \ \; ^ . — ■ ^ \ A ES- -* fc £j B_i £_i p. a !.* v,^ B E a 6 S T [) Fia. 83. Line B does not intersect the arc characteristic, A, that is, with a resistance as low as ri = 32, no arc can be maintained on the S-amp. constant-current circuit. Line C intersects A at two points : (a) i = 2.55 amp., e = 118 volts, stable condition; (&) i = 0.55 amp., e = 214 volts, unstable condition. Ifine D is drawn tangent to A, touches at c: i = 1.4 amp., 174 ELECTRIC CIRCUITS e = 148 volts, the limit of stability. At / = 5 amp., the point fe, at e = 148 volts, thus gives the voltage consumed by an arc when by shunting it with a resistance the stability limit is reached. Drawing then from the different points of the abscissae, t, tangents on A, and transferring their contact points, c, 6, to the abscissae, from which the tangent is drawn, gives the points h, gf, of the constant-current stability characteristic of the arc, that is, the curve of arc voltages in a constant-current circuit, /, when by shunting the arc with a resistance, ri, consuming current, ii, the stability limit of the arc with current i = /— ii is reached. P then gives the curve of the arc currents, i, corresponding to the arc voltage, e, of curve Q, for the different values of the con- stant-circuit current, /. The equations of Q and P are derived as follows: The stability limit, point c, corresponding to circuit current, /, as given by de where e = arc voltage, and i = arc current. Or, It is, however, and ri = — ^- (23) . b e = a + 6 -. = ri. From these three equations follows, by eliminating ri and i or e, I = '-^^^ (24) (e — a)' P, I = i(3A+Wl). (25) These curves are of lesser interest than the constant-voltage stability curve of the arc, S in Fig. 82. It is interesting to note, that the resistance, ri (23), which makes an arc unstable as shunting resistance in a constant- current circuit, has the same value as the resistance, ro, (7), which INSTABILITY OF CIRCUITS 175 OS Beries reaistatice makes it unstable in a coDstant-voItage supply circuit. 90. Due to the dropping volt-ampere characteristic, two arcs can not be operated in parallel, unless at least one of them has a sufficiently high resistance in series. I - \ 6 A PARALLEL OPERATION OP ARCS a 1 r-i ^ ,r 7 1 1 i • • ' I h /- 1 / ™, N > s y \ ^ ^ 1 r» ^ K '' /I «r — // -a N y, ' C — ■ — - ■^ ~^ — < i— — V if» V \ ^ / ™ A' > < i«l — — — -^ — i_ ^ 6 t 1 5 a « I 6 i 4 5 S t G 6 . I a Let, as shown in Fig. 84, two arcs be connected in parallel into the circxiit of a constant current J = 6 amp. Assume at first both arcs of the same length and same electrode material, that is, the same volt-ampere characteristic. 176 ELECTRIC CIRCUITS Let i = current in the first arc, thus i' = I — i = current in the second arc. The volt-ampere characteristic of the first arc, then, is given by A in Fig. 84, that of the second arc by -4'. As the two parallel arcs must have the same voltage, the oper- ating point is the point, a, of the intersection of A and -4' in Fig. 84. The arcs thus would divide the current, each operating at 3 amp. However, the operation is unstable: if the first arc should take a little more current, its voltage decreases, on curve A, that of the second arc increases, on A', due to the decrease of its current, and the first arc thus takes still more current, thus robs the second arc, the latter goes out and only one arc continues. Thus two arcs in parallel are unstable, and one of them goes out, only one persists. Suppose now a resistance of r = 30 ohms is connected in series with each of the two arcs, as shown in Fig. 84. The volt-ampere characteristics of arc plus resistance, r, then, are given by curves B and B\ These intersect in three points: 6, g and h. Of these, point b is stable: an increase of the current in one of the arcs, and corresponding decrease in the other, increases the voltage consumed by the circuit of the former, decreases that con- sumed by the circuit of the latter, and thus checks itself. The points g and ft, however, are unstable. At 6, stable condition, the characteristics, B and jB', are rising; at a, imstable condition, the characteristics, A and A', are drop- ping, and the stability limit is at that value of resistance, r, at which the circuit characteristics plus resistance, are horizontal, the point c, where the characteristics, C and C, touch each other. c is the stability limit of C or C, thus a point of the stability characteristic of either arc, or given by the equation , 1.5 6 e = a -\ 7=. V « Fig. 85 shows the case of two parallel arcs, which are not equal and do not have equal resistances, r, in series, one being a long arc, INSTABILITY OF CIRCUITS 177 having do resistance in series, the other a short arc with a resist- ance r = 40 ohms in series. The volt-ampere characteristic of the long arc is given by A, that of the short arc by B, and that of the short arc plus resistance, T,hyC. A and C intersect at three points, a, b and c. Of these, only the point a is stable, as any change of current from this point limits I - PARALLEL OPERATION 1 6 A OF RC \ a «1 I \ 2 "• ^ ^n \ ^ ' ' l. > K \ \ S I ■™ \ .^ s ;:* ^ «» <% "■ :^ ^ ~3 / ~~ — / m y — — i. ^ ' D 1 G £ z 5 i 3 6 t t G 6 « 6 7 g Fig. 85. itself; ft and c, however, are unstable. Thus, at the latter points, the arcs can not run, but the current changes until either one arc has gone out and one only persists, or both run at point a. However, the angle under which the two curves, A and C, inter- sect at a is so small, that even at a the two arcs are not very stable. 178 ELECTRIC CIRCUITS Furthermore, a small change in either of the two curves, A or C, results in the two points of intersection a and 6 vanishing. Thus, if r is reduced from 40 ohms to 35 ohms, the curve C changes to C, shown dotted in Fig. 85, and as the latter does not intersect A except at the unstable point c, parallel operation is not possible. That is, two such arcs can be operated in parallel only over a limited range of conditions, and even then the parallel operation is not very stable. The preceding may illustrate the effect of resistance on the stability of operation of arcs. Similarly, other conditions can be investigated, as the stability CAPACITY SHUmiNG ARC e 1 Fig. 86. condition of arcs with resistance in series and in shunt, on constant, voltage supply, etc. 91. Let e = E be the voltage consumed by a circuit, -4, Fig. 86, when traversed by a current i = /. If, then, in this circuit the current changes by 81, to z = / + 57, the voltage consumed bythe circuit changes by 5 ^, to e = E ± dE, and the change of voltage is of the same sign as that of the current producing it, if A is a resistance or other circuit in which the INSTABILITY OF CIRCUITS 179 voltage rises with the current, or is of opposite sign, if the circuit, A, has a dropping volt-ampere characteristic, as an arc. Suppose now the circuit, A, is shunted by a condenser, C. As long as current, t, and voltage, e, in the circuit, A, are constant, no current passes through the condenser, C. If, however, the voltage of -4 changes, a current, zi, passes through the condenser, given by the equation n = C f- (26) If, then, the supply current, 7, suddenly changes by 81, from 7 to / + 81, and the circuit. A, is a dead resistance, r, without, the condenser, C, the voltage of A would just as suddenly change, from EtoE + 8E, By (26) this would, however, give an infinite current, ii, in the condenser. However, the current in the con- denser can not exceed 81, as with ii = 81 at the moment of supply current change, the total excess current would in the first moment flow through the condenser, and the circuit. A, thus in this moment not change in current or voltage. A finite current in the condenser, C, requires a finite rate of change of e in the circuit. A, starting from the previous value, E, at the starting moment, the time, t = 0. Thus, if i = current, e = voltage of circuit, A, at time, t, after the increase of the supply current, I, by 81, it is current in condenser, ^ de current in circuit. A, i = I + 81 - ii (27) thus, voltage of circuit. A, of resistance, r, e = ri = rl + r8l - rii (28) substituting (26) into (28), gives e = r(I + 81) - rC ^. or, TT-r^N = 4^^ (29) r{I + 81) — e rC ^ integrated by 180 ELECTRIC CIRCUITS rCy e = rl - rdl {1 - € ''') t_ = E - 8E (1 -€ '^) (30) since e = -B for < = is the terminal condition which determines the integration constant. With a sudden change of the supply current, 7, by 67, as shown by the dotted lines, 7, in Fig. 86, the voltage, e, and current, i, in the circuit. A, and the current, ii, in the condenser, C, thus change by the exponential transients shown in Fig. 86 as e, i and f i. 92. Suppose now, however, that the circuit, A, has a dropping volt-ampere characteristic, is an arc. A sudden decrease of the supply current, 7 by 57, to 7 — 57^ would by the arc characteristic, e = a + —y^., cause an increase of the voltage of circuit. A, from E to E + 8E. Such a sudden increase of E would send an infinite current through C, that is, all the supply current would momentarily go through the con- denser, C, none through the arc, A, and the latter would thus go out, and that, no matter how small the condenser capacity, C. Thus, with the condenser in shimt to the circuit. A, the voltage. A, can not vary instantly, but at a decrease of the supply current, 7, by 57, the voltage of A at the first moment must remain the same, E, and the current in A thus must remain also, and as the supply current has decreased by 67, the condenser, C, thus must feed the current, 57, back into the arc, A. This, however, requires a de- creasing voltage rating of A, at decreasing supply current, and this is not the case with an arc. Inversely, a sudden increase of 7, by 57, decreases the voltage of A, thus causes the condenser, C, to discharge into A, still further decreases its voltage, and the condenser momentarily short-cir- cuits through the arc, A ; but as soon as it has discharged and the arc voltage again rises with the decreasing current, the condenser, C, robs the arc. A, and puts it out. Thus, even a small condenser in shunt to an arc makes it un- stable and puts it out. If a resistance, ro, is inserted in series to the arc in the circuit. A, stability results if the resistance is sufficient to give a rising volt- ampere characteristic, as discussed previously. Resistance in series to the condenser, C, also produces stability, if sufficiently large: with a sudden change of voltage in the arc INSTABILITY OF CIRCUITS 181 circuit, A, the condenser acts as a short-circuit in the first moment, passing the current without voltage drop, and the voltage thus has to be taken up by the shunt resistance, ri, giving the same con- dition of stability as with an arc in a constant-current circuit, shunted by a resistance, paragraph 89. If, in addition to the capacity, C, an inductance, L, and some re- sistance, r, are shunted across the circuit. A, of a rising volt-ampere characteristic, as shown in Fig. 87, the readjustment occurring at a sudden change of the supply current, 7, is not exponential, as in Fig. 86, but oscillatory, as in Fig. 87. As in the circuit. A, assum- ing it consists of a resistance, r, current and voltage vary simultaneously or in phase, current and voltage in the condenser branch circuit also must be in phase with each other, that is, the Fig. 87. frequency of the oscillation in Fig. 87 is that at which capacity, C, and inductance, L, balance, or is the resonance frequency. If circuit. A, in Fig. 87 is an arc circuit, and the resistance, r, in the shunt circuit small, instability again results, in the same man- ner as discussed before. 93. Another way of looking at the phenomena resulting from a condenser, C, shunting a circuit, A, is: Suppose in Fig. 86 at constant-supply current, 7, the current in the circuit. A, should begin to decrease, for some reason or another. Assuming as simplest case, a uniform decrease of current. The current in the circuit, A, then can be represented by ('-y (31) where to is the time which would be required for a uniform de- crease down to nothing. 182 ELECTRIC CIRCUITS At constant-supply current, /, the condenser thus must absorb the decrease of current in A, that is, the condenser current is ti = / f • (32) to With decrease of current, z, if A is a circuit with rising character- istic, for instance, an ohmic resistance, the voltage of A decreases. The voltage at the condenser increases by the increasing charging current, ii, thus the condenser voltage tends to rise over the cir- cuit voltage of Ay and thus checks the decrease of the voltage and thus of the ciu-rent in A. Thus, the conditions are stable. Suppose, however, A is an arc. A decrease of the current in A then causes an increase of the voltage consumed by A, the arc voltage, eo. The same decrease of the current in A, by deflecting the current into the condenser, causes an increase of the voltage consumed by C, the condenser voltage, d. If, now, at a decrease of the arc current, i, the arc voltage, 60, rises faster than the condenser voltage, 61, the increase of eo over ei de- flects still more current from A into C, that is, the arc current decreases and the condenser current increases at increasing rate, until the arc current has decreased to zero, that is, the arc has been put out. In this case, the condenser thus produces in- stability of the arc. If, however, eo increases slower than ei, that is, the condenser voltage increases faster than the arc voltage, the condenser, C, shifts current over into the arc circuit. A, that is, the decrease of current in the arc circuit checks itself, and the condition becomes stable. The voltage rise at the condenser is given by de 1 . hence, by (32), dfit IT (33) from the volt-ampere characteristic of the arc. dt C''' de II dt hC follows, e = a + -^ (34) INSTABILITY OF CIRCUITS 183 the voltage rise at the arc terminals, de b di dt 2i\r% ^^ and, by (31), di dt I to' hence, substituted into (34), de 16/ dt 2 1 (37) bC as the condition of stability, and 2 toiVi (I - i) > 1 (38) = 1 (39) bC thus is the stability limit. 94. Integrating (33) and substituting the terminal condition: t = 0; 6 = B, gives «^ = ^ + S (4«) as the equation of the voltage at the condenser terminals. Substitute (31) into (34) gives as the equation of the arc voltage. For, a = 35, b = 200, / = 3, hence. 184 ELECTRIC CIRCUITS and E - 151, 1, - 10-" sec. and, for the three values of capacity. C - 10-' 0.75 X 10-' 0.5 X 10-' (e.) (e.) / .: / / / CAPACITY SHUNTING ARC y / '/ / >« / , / «! ^ '^•, ^ ' y ^ ^ m ^ ^ ^ ■«l 4 ^ ^ g ^ == * -J ...... ■~-i .■ , 1 rn """- .-, ^_^ ^ — ~~, ^ ^ XIO S~ Fig. 88. the curves of the arc voltage, eo, and of the condenser voltage, ei, ej, g%, are shown on Fig. 88, together with the values of i and i\. As seen, ci is below eo over the entire range. That is, 1 mt. makes the arc unstable over the entire range. 0.5 mf., cj, gives instability up to about ( = 0.25 X 10"* sec, then stability results. With 0.75 mf., Ca, there is a narrow range of stability, between INSTABILITY OF CIRCUITS 185 4% and 7)4 X 10~* sec, before and after this instability exists. From equation (37), the condition of stability, it follows that for small values of t, that is, small current fluctuations, the con- ditions are always unstable. That is, no matter how small a condenser is, it always has an effect in increasing the current fluctuations in the arc, the more so, the higher the capacity, until conditions become entirely unstable. From equations (40) and (41) follows as the stability limit eo = ei, y/l r t 2 \ ifn \ ,E m IID / ^ V \ \ 5^ ^ L "^ ' \y ^^ \ r 7 A -- ;;-''-- .^ Q / D ' — ^ I- -> I D 1 S 2 a £ S & } B B 4 D Fio. 95. pends on the resistance of the arc stream and the potenti^ drop of the terminals, is different, the variation of voltage, for the Bame variation of current, is less, and the effective negative arc resist- ance thereby is lowered, or may entirely vanish. Fig. 95 shows a number of such transient arc characteristics, INSTABILITY OF CIRCUITS 193 • estimated from oscillographic tests of alternating arcs, and their corresponding eflFective resistances, R. They are: (A) Carbon. (B) Hard carbon. (C) Acheson graphite. (D) Titanium carbid. (E) Hard carbon, stationary characteristic. (F) Titanium carbid, stationary characteristic. As seen from the curves oi R in the upper part of Fig. 95, the efiFective resistances, U, which represent the alternating power generated by the oscillating arc, are much lower with the transi- ent arc characteristic, than would be with the permanent arc characteristic in Fig. 94. Curve Dy titanium carbide, gives under these conditions an unstable or "rasping" arc. That is, with a resistance in the con- denser circuit of less than R = 3.8 ohms, the oscillation starts spontaneously and cumulatively increases to the extinction of the arc; with a resistance of more than 3.8 ohms, the oscillation does not spontaneously start, but if once started with an amplitude which brings the value of R from curve, 2), above that of the resist- ance in the condenser circuit, cumulative oscillation occurs. With the carbon arc. A, no oscillations can occur under any condition, the efiFective resistance,/?, is negative, and the arc char- acteristic rising. With the hard carbon arc, B, an oscillation starts with a resist- ance less than 2.4 ohms, cumulatively increases, but its amplitude finally limits itself, to 1.45 amp. if the resistance in the oscillating circuit is zero, to 1.05 amp. with 2 ohms resistance, etc., as seen from the curve, B, in the upper part of Fig. 95. Even with more than 2.4 ohms resistance, up to 2.6 ohms resistance, an oscillation can exist, if once started, as the curve of R, starting from R = 2.4 ohms at ii — 0, rises to U = 2.6 ohms at ii = 0.75, and then drops to zero at ii = 1.45 ohms, and beyond this becomes negative. The curve, C, of Acheson graphite, starts with a resistance R = 10.8 ohms, but the resistance, /?, steadily drops with increasing oscillating current, ii, down to zero at ii = 2.4 amp. Thus, with a resistance in the condenser circuit, of 10 ohms, the oscillations would have an amplitude of ii = 0.9 amp. ; with 8 ohms resistance an amplitude of 2.1 amp., etc. -La 194 ELECTRIC CIRCUITS From these curves of R^ Fig. 95, the regulation curves of the altemating-current generation could now be constructed. It is interesting to note, that in many of these transient arc characteristics, Fig. 95, the voltage does not indefinitely rise with decreasing current, but reaches a maximum and then decreases again, in B and C, and the oscillation resistance, that is, the re- sistance through which an alternating current can be maintained by the oscillating arc, thus decreases with increasing amplitude of the oscillation. Thus, if the resistance in the oscillating con- denser circuit is less than the permissible maximum, an oscilla- tion starts, cumulatively increases, but finally limits itself in amplitude. The decrease of the arc voltage with decreasing current, for low values of current in a rapidly fluctuating arc, is due to the time lag of the arc voltage behind the current. 99. The arc voltage, 6, consists of the arc terminal drop, a, and the arc stream voltage, ei: e = a + ei. The stream voltage, ei, is the voltage consumed in the effective resistance of the arc stream; but as the arc stream is produced by the current, the voliune of the arc stream and its resistance thus depends on the current, f , in the arc, that is, the stream vol- tage is b ei = —p. and the resistance of the arc stream thus ^ __ «i _ - h \/% Thus, if, a = 35 h = 200, for i = 2 amp., it is n = 70.7 ohms, ei = 141.4 volts, e = 176.4 volts. But, if the arc current rapidly varies, for instance decreases, then, when the current in the arc is t'l, the volume of the arc stream INSTABILITY OF CIRCUITS 195 and thus its resistance is still that corresponding to the previous current, i\. If thus, at the moment where the current in the arc has become ii = 2 amp., the arc stream still has the volume and thus the resistance corresponding to the previous current, i'l = 3 amp., this resistance is 200 „^ , ^ r 1 = /- = 38.5 ohms, and the stream voltage, at the current ii = 2 amp. , but with the stream resistance, r\, corresponding to the previous current, i\ = 3 amp., thus is e\ = r'lt'i = 77 volts, instead of ei = 141.4 volts, as it would be under stationary conditions. That is, the stream voltage and thus the total arc voltage at rapidly decreasing current is lower, at rapidly increasing current higher than at stationary cmrent. With a periodically pulsating current, it follows herefrom, that at the extreme values of current— maximum and minimum— the voltage has not yet reached the extreme values corresponding to these currents, that is, the amplitude of voltage pulsation is reduced. This means the transient volt-ampere characteristic of the arc is flattened out, compared with the permanent charac- teristic, and caused to bend downward at low currents, as shown by C and B in Fig. 95. Assimiing a sinusoidal pulsation of the current in the arc and assuming the arc stream resistance to lag behind the current by a suitable distance, we then get, from the stationary volt-ampere characteristic of the arc, the transient characteristics. Thus in Fig. 96, from the stationary arc characteristic, S, the transient arc characteristic, T, is derived. In this figure is shown as S and T the effective resistance corresponding to the stationary characteristic, /S, respectively the transient characteristic, T. 196 ELECTRIC CIRCUITS As seen, the stationary characteristic, S, gives an arc oscillation which is cumulative and self-destructive, that is, the effective re^tance, R, rises indefinitely with increasing amplitude of pulsation. The transient characteristic, however, gives an effects ive resistance, R, which with increasing amplitude of pulsation \ 1 1 1 t 1 o- \ OSCILLATION RESISTA NCE m \^ OF ARC inn \ \ ™ T \ \ N\ \ S X \ V' ^ 111. > nn « •r S / « T 7. N -i E-1 .*. . ■i, . 1 1 G S 2 6 3. ,L. 4 B £ ..... t I Fig. 06. first increases, but then decreases again, down to zero, so that the cumulative oscillations produced by this arc are self-limitii^, increase in amplitude only up to the value, where the effective resistance, R, has fallen to the value corresponding to the load on the oscillating circuit. INSTABILITY OF CIRCUITS 197 As further illustration, from the stationary volt-ampere char- acteristic of the titanium arc, shown as F in Fig. 95, values of the transient characteristic have been calculated and are shown in Fig. 95 by crosaes. As seen, they fairly well coincide with the transient volt-ampere characteristic, D, of the titanium are, at least for the larger currents. \ t 4 it ^ r ":::_ t" ~" \ _^—--4ii 3 X A V V ^ \ \ \ ^. ^s^ \>. ^^ ^ ^^ \^ IC ^4 ^^ / ^ >L " ^^ ^^- Vu In the electric arc we thus have an clcctrii; circuit with dropping volt-ampere characteristic. Such a circuit is unstable under various conditions which may occur in industrial circuits, and thereby may be, and frequently is, the source of instability of electric circuits, and of cumulative oscillations appearing in such circuits. 198 ELECTRIC CIRCUITS 100. For instance, let, in Fig. 97, A and B be two conductors of an ungrounded high-potential transmission line, and 2 e the voltage impressed between these two conductors. Let C repre- sent the ground. The capacity of the conductors, A and B, against groundy theiii may be represented diagrammatically by two condensers, Ci and C2y and the voltages from the lines to ground by ej and es. In g^i- eral, the two line capacities are equal, Ci = d, and the two volt- ages to ground thus equal also, 6i == 62 = e, with a single-phase; = —7^ with a three-phase line. Assume now that a ground, P, is brought near one of the IjneBi A, to within the striking distance of the voltage, e. A dischaigB then occurs over the conductor, P. Such may occur by the puiu>* ture of a line insulator as not infrequently the case. Let r «■ re- sistance of discharge path, P. While without this discharge path, the voltage between A and C would be ei = e (assuming sini^ phase circuit) with a grounded conductor, P, approaching line A within striking distance of voltage, e, a discharge occurs over P forming an arc, and the circuit of the impressed voltage, 2 s, now comprises the condenser, C2, in series to the multiple circuit of con- denser, Ci, and arc, P, and the condenser, Ci, rapidly discharges^ voltage, eij decreases, and the voltage, 62, increases. With a de- crease of voltage, ei, the discharge current, i, also decreaseSi and the voltage consumed by the discharge arc, e', increases until the two voltages, ei and e', cross, as shown in the curve diagram of Fig. 97. At this moment the current, i, in the arc vanishes, the arc ceases, and the shunt of the condenser, Ci, formed by the dis- charge over P thus ceases. The voltage, ei, then rises, €% decreases and the two voltages tend toward equality, ei = ej = e. Before this point is reached, however, the voltage, 61, has passed the dis- ruptive strength of the discharge gap, P, the discharge by the an over P again starts, and the cycle thus repeats indefinitdy. In Fig. 97 are diagrammatically sketched voltage, Ci, of con- denser, Ci, the voltage, «', consumed by the discharge arc overP, and the current, i, of this arc, under the assumption that r is sufli^ ciently high to make the discharge non-oscillatory. If r is small, each of these successive discharges is an oscillation. Such an unstable circuit gives a continuous series of successive discharges, which are single impulses, as in Fig. 97, or more com- monly are oscillations. INSTABILITY OF CIRCUITS 199 If the line conductors, A and B, in Fig. 97 have appreciable in- ductance, as is the case with transmission lines, in the charge of the condenser, Ci, after it has been discharged by the arc over P, the voltage, ei, would rise beyond e, approaching 2 e, and the dis- charge would thus start over P, even if the disruptive strength of this gap is higher than 6, provided that it is still below the voltage momentarily reached by the oscillatory charge of the line conden- ser, Pi. This combination of two transmission line conductors and the ground conductor, P, approaching near line, A, to a distance giving a striking voltage above e, but below the momentary charging voltage, of Ci, then constitutes a circuit which has two permanent conditions, one of stability and one of instability. If the voltage is gradually applied, 6i = 62 = 6, the condition is stable, as no discharge occurs over P. If, however, by some means, as a mo- mentarily overvoltage, a discharge is once produced over the spark-gap, P, the unstable condition of the circuit persists in the form of successive and recurrent discharges. 101. Usually, the resistance, r, of the discharge path is, or after a number of recurrent discharges, becomes sufficiently low to make the discharge oscillatory, and a series of recurrent oscilla- tions then result, a so-called "arcing ground." Oscillograms of such an arcing grounds on a 30-mile 30-kv. transmission line are shown in Figs. 98, 99 and 100. If, however, the resistance of the discharge path is very low, a sustained or cumulative oscillation results, as discussed in the pre- ceding, that is, the arcing ground becomes a stationary oscillation of constant-resonance frequency, increasing cumulatively in cur- rent and voltage amplitude until limited by increasing losses or by destruction of apparatus. In transmission lines, usually the resistance is too high to pro- duce a cumulative oscillation; in underground cables, usually the inductance is too low and thus no cumulative oscillation results, except perhaps sometimes in single-conductor cables, etc. In the high-potential windings of large high-voltage power trans- formers, however, as circuits of distributed capacity, inductance and resistance, the resistance commonly is below the value through which a cumulative oscillation can be produced and maintained, and in high-potential transformers, destruction by high voltages resulting from the cumulative oscillation of some arc in the 200 ELECTRIC CIRCUITS system, and building up to high stationary waves, have frequently been observed. The "arcing ground" as recurrent single impulses, the "arcing ground oscillation'' as more or less rapidly damped recurrent oscillations in transmission lines — of frequencies from a few hun- dred to a few thousand cycles — ^and the "stationary oscillations" causing destruction in high-potential transformer windings, at frequencies of 10,000 to 100,000 cycles, thus are the same phenom- ena of the dropping arc characteristic, causing permanent in- stability of the electric circuit, and differ from each other merely by the relative amount of resistance in the discharge path.