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 bvanishing. 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 SHUNTING ARC E aa

• 1 fe |
  I 1 I : : e 7 N . L a A . a. ty rey . ; —__ . rt: Fia. 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, A, Fig. 86, when traversed by a current t=], If, then, in this circuit the current changes by I, to t=I+4 8, the voltage consumed by the circuit changes by 5 E, to e=E+5E, 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, 7, and voltage, e, in the circuit, A, are constant, no current passes through the condenser,C’. If, however,thevoltage _ . of A changes, a current, 7;, passes through the condenser, given by the equation

1=C de, (26) dt

If, then, the supply current, J, suddenly changes by éJ, from

. I to I + I, and the circuit, A, is a dead resistance, r, without the condenser, C,, the voltage of A would just as suddenly change, from Eto H+ 5E. By (26) this would, however, give an infinite current, 71, in the condenser. However, the current in the con- denser can not exceed 6J, as with

ty = ér . 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 ¢ in the circuit, A, starting from the previous value, E, at the starting moment, the time, é = 0.

. Thus, if = current, e = voltage of circuit, A, at time, ¢, after the increase of the supply current, I, by 4J, it is current in condenser, ;

ty = Cc de. dt current in circuit, A, t=I+sl-i, . (27) thus, voltage of circuit, A, of resistance, r, e=rt =r + rol — ry (28) substituting (26) into (28), gives ” e= r+ al) —rc%, _ at or, ° de 1 rd + oly) ae ~ ro (29) integrated by

180 ELECTRIC CIRCUITS . _+t . . rc e=rI —réI (1 " ) | (30) =E-sE(1-e ") since e = E for ¢ = 0 is the terminal condition which determines the integration constant.

With a sudden change of the supply current, J, by 6Z, as shown by the dotted lines, J, in Fig. 86, the voltage, e, and current, 7, in the circuit, A, and the current, 7, in the condenser, C, thus change by the exponential transients shown in Fig. 86 as e, i and 1.

92. Suppose now, however, that the circuit, A, has a dropping volt-ampere characteristic, is an arc.

A sudden decrease of the supply current, I by 4J, to I — 8I, would by the arc characteristic, e = a + 7 cause an increase of the voltage of circuit, A, from EF to EH + 6H. Such a sudden increase ef 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 shunt tothe circuit, A, the voltage, A, can not vary instantly, but at a decrease of the supply current, J, by 4I, 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 6/, the condenser, C, thus must feed the current, 51, 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 I, by 5I, 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 ashort-circuit in the first moment, passing the current without voltage drop, and the voltage thus has to be taken up by the shunt resistance, r:, 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 arising volt-ampere characteristic, as shown in Fig. 87, the readjustment occurring at a sudden change of the supply current, J, is not exponential, as in Fig. 86, but oscillatory, as in Fig. 87. Asin the circuit, A, assum- ing it consists of a resistance, 7, 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 I é . on ° é Cc 1) > A . "7 . i; i L . Ns, ; 0 Kia. 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, J, 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

. t

i=1(1-£) (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, J, the condenser thus must absorb

. the decrease of current in A, that is, the condenser current is ‘, t 1, = I-- (32) to -

With decrease of current, 7, 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, 74, thus the condenser voltage tends to rise over the cir- cuit voltage of A, and thus checks the decrease of the voltage and thus of the current 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, éo. .

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, é1.

If, now, at a decrease of the arc current, 7, the arc voltage, éo, rises faster than the condenser voltage, e:, the increase of eg overe,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 are 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, éo increases slower than e;, 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.. ©

at co ; hence, by (32), ; 1

. é t :

dt ~ iC (33)

from the volt-ampere characteristic of the arc,

• e@nat—. (34) Vi

follows,

; INSTABILITY OF CIRCUITS 183 the voltage rise at the arc terminals, § de b di ~ = —- — 35 | dd a @°) and, by (31), a __t. dt to’ : ‘ hence, substituted into (34), ; de lor . dt 2tyivé (36) ’ The condition of stability is, that the voltage rise at the con- . denser, (33), is greater than that at the arc, (36), thus, . | a > bl toC 2 tive or, ; , 2uVi . 0 >1 (37) or, substituting for ¢ from equation (31), gives , 2 tivi (I — 4) . ee (88)

• as the condition of stability, and ; 2 boiSt (I - t) _ Fa 1 (39) thus is the stability limit. . 94. Integrating (33) and substituting the terminal condition: ’ t= 0;e = E£, gives a= B+2 (40) 2 tC as the equation of the voltage at the condenser terminals. Substitute (31) into (34) gives - b = —- 41 €0 a + J 1 i ( ) VI- 5g as the equation of the are voltage. For, a = 35, 6 = 200, . . T= 3, hence,

. ! ! | | 184 ELECTRIC CIRCUITS -

= 151, :

and to = 10-‘ sec., : and, for the three values of capacity, Cc = 10-8 (e1) 0.75 X 10-* (és) , 0.5 X 10-* (és) PT TTT ETT tT tet tT TY | | et tT ETT ET TET TE EE . eee, | peo || [CAPACITY sHuNTING arc] | [7] || vol | | TT TT TT Ty Tr | | | | | tT | Le le o| | | | | | | Le ee PLLC | Leer TT rrr Pitt tT tT tT tet te EE TT TT pret tT tT TT TE ET tT SRR ~ See ttt | | | Peter tt pt tt tt abe y eet Pt | pterrt tT | TTT | | EN Lee tT tt tT TT TT bk Fia. 88. the curves of the arc voltage, éo, and of the condenser voltage, ¢1, é2, és, are shown on Fig. 88, together with the values of 7 and 4. As seen, ¢: is below é over the entire range. That is, 1 mf. makes the arc unstable over the entire range. 0.5 mf., es, gives instability up to about ¢ = 0.25 X 10‘ sec., then stability results. With 0.75 mf., es, there is a narrow range of stability, between

INSTABILITY OF CIRCUITS 185 4% and 734 X 10-‘ sec., before and after this instability exists.

From equation (37), the condition of stability, it follows that for small values of ¢, 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

€o = 41, b vl a+ Vi [- —+F =F + 2hC’ \ h _+ to or, expanded into a series, b t er ats litset. a } -E+550 b cancelling HE = a + VI and rearranging, gives . ¢ h=7F Vi (42) thus, at the time, , - WC, 1 IVI the condition changes from unstable to stable. , As ¢, must be smaller than f, the total time of change, it follows: bC or, . C< IVI (44) are expressions of the (approximate) stability limit of an arc with condenser shunt. . ° As seen from (44),

the larger ty is, that is, the slower the arc changes, the larger is the permissible shunted capacity, and inversely.

As an instance, let

. b = 200, I= 3, and

186 ELECTRIC CIRCUITS . (a) b= 107°, which is probably the approximate magnitude in the carbon arc. This gives C < 26 mf. Let: (b) & = 10°, : which is probably the approximate magnitude in the mercury arc. This gives . C < 0.26 mf.

95. Consider the case of a circuit, A, Fig. 87, supplied by a constant current, J, but shunted by a capacity, C, inductance, L, and resistance, 7, in series.

RESONATING CIRCUIT SHUNTING ARC i —_ e - I 6o ;

4 PP 7 é y a NN ere: eo

. °

__ 6-69 = I-¢ Fia. 89.

As long as the current in the circuit, A—whether resistance or arc—is steady, no current passes the condenser circuit, and the current and voltage in A thus are constant, 7 = I, e = é.

Suppose now a pulsation of the current, 7, should be produced in circuit, A, as shown as 7 in Fig. 89. Then, with constant-sup- ply current, J, an alternating current,

1 = I - 1, would traverse the condenser circuit, C, since the continuous com- ponent of current can not traverse the condenser, C.

INSTABILITY OF CIRCUITS 187

Due to the pulsation of current, 7 in A, the voltage, e, of cir- cuit, A, would pulsate also. These voltage pulsations are in the same direction as the current pulsation, if A is a resistance, in opposite direction, if A is an arc; in either case, however, they are in phase with the current pulsation, and the alternating vol- tage on the condenser,

€1 = & — @, thus is in phase with the alternating current, 7,, that is, capacity, -C, and inductance, L, neutralize. ,

Thus, the only pulsation of current and voltage, which could occur in a circuit, A, shunted by capacity and inductance, is that of the resonance frequency of capacity and inductance.

Suppose the circuit, A,is a dead resistance. The voltage pulsa- tion produced by a current pulsation, 7, in this circuit then would be in the same direction as 7, that is, would be as shown in dotted line by e’ in Fig. 89. In the condenser circuit, C, the alternat- ing component of voltage thus would be

e’, =e’ — , thus would be in opposition.to the alternating current, 7), as shown in Fig. 89 in dotted line. That is, it would require a supply of power to maintain such pulsation.

Thus, with a dead resistance as circuit, A, or in general with A as a circuit of rising volt-ampere characteristic, the maintenance

of a resonance pulsation of current and voltage between-A and C, at constant current, J, requires a supply of alternating-current power in the condenser circuit, and without such power supply the pulsation could not exist, hence, if started, would rapidly die out, as oscillation, as shown in Fig. 87.

96. Suppose, however, A isan arc. A current pulsation,7, then gives a voltage pulsation in opposite direction, as shown by e in Fig. 89, and the alternating current, 7; = J — 7, and the alter- nating voltage, ¢1 = ¢ — é, in the condenser circuit, thus would | be in phase with each other, as shown by 1; and e; in Fig. 89.

• ‘That is, they would reptesent power generation, or rather trans- formation of power from the constant direct-current supply, J, into the alternating-current resonating condenser circuit, C.

Thus, such a local pulsation of the are current, 7, and corre- sponding alternating current, 7,, in the condenser circuit, if once started, would maintain itself without external power supply,

188 ELECTRIC CIRCUITS

and would even be able to supply the power represented by vol- tage, €1, with current, ¢;, into an external circuit, as the resistance, r, shown in Fig. 87, or through a transformer into a wireless send- ing circuit, ete.

Thus, due to the dropping arc characteristic, an arc shunted by capacity and inductance, on a constant-current supply, be- comes a generator of alternating-current power, of the frequency set by the resonance of C and L. .

If the resistance, r, or in general, the load on the oscillating cir- cuit, C, is greater than r; = = that is, if a higher voltage would be required to send the current, 7;, through the resistance, r, than the voltage, €1, generated by the oscillating arc, A, the pulsations die

out as oscillations.

If r is less than e the pulsations increase in amplitude, that is, current, 7, and voltage, ¢1, increase, until either, by the internal

. reaction in the arc, the ratio, i drops to equality with the effective resistance of the load, r, and stability of oscillation is reached, or, if = never falls to equality with r—for instance, if r = 0, the oscillations increase up to the destruction of the circuit: the extinction of the arc. .

If, in the latter case, the voltage back of the supply current, J, is sufficiently high to restart the arc, A, the phenomena repeats, and we have a series of successive arc oscillations, each rising until it puts the arc out, and then the arc restarts.

We thus have here the mechanism which produces a cumulative oscillation, that is, a transient, which does not die out, but in-

: creases in amplitude, until the increasing energy losses limit its further increase, or until it destroys the circuit, and in the latter case, it may become recurrent.

It is very important to realize in electrical engineering, that

• any electric circuit with dropping volt-ampere characteristic is capable of transforming power into a cumulative oscillation, and thereby is able under favorable conditions to produce cumulative oscillations, such as hunting, etc.

Where the arc oscillations limit themselves, and the alternating current and voltage in the condenser circuit thus reach a constant value, the arc often is called a “singing arc,” due to the musical note given by the alternating wave. Where the arc oscillations

SPA | i 21 | . | |

|

|

| | 7 oe E | 2 | 3 : _ : ' 8

= ee eee

INSTABILITY OF CIRCUITS 189 | rise cumulatively to interruption, and the arc then restarts by | the supply voltage and repeats the same phenomenon, it may be called a “rasping arc,” by the harsh noise produced -by the | interrupted cumulative oscillation. |

| PETE EPET TTT YET | | | farcasogcutator | |7| | | | | | | | 44 4 | el | PTT PPA EE | ot | ET ETA ET | Pf, LE || a oA | et tT Pt Tt le Set tT it tt ET | eo! | | | | | ti tt tt tt : wi\ |} ttt ttt ttt _ mf AI} wi IW | ttt ttt tt ty o| | AEE PETE Tt tt BN eee SRE SRR COCCCER RSE el | | | yy yt PE of | TT EET TTT Tt eXCOCOPEE AAT pe, PSST] Beton CLERC Cer eee 5 10 15 20 25 20 36 do 45 Ho 85 @o 65 to ws | 7 7, » , : ‘ Fia. 94. pee ee Figs. 90 and 91 give oscillograms of singing arcs; Figs.92 and _. ° _ 93, of rasping arcs, 90 and 92 in circuits with massed constants, - ; 91 and 93 in transmission lines. NO 97. As‘an illustration, let curve, A,in Fig. 94 represent the volt- ampere characteristic of an arc, and assume that this arc is operat- ing steadily at J amp., consuming é volts. .

190 ELECTRIC CIRCUITS . Suppose this arc is shunted by capacity, C, inductance, L, and resistance, r, as shown in Fig. 87. . For a small pulsation of the arc current around its average value 7 = I, the corresponding voltage pulsation is given by ; | de_ _ _ ob dt Livi Or, in general, for any pulsation of current, 1, by 6, between v’ and 7”, around the mean value I, the corresponding voltage pulsation 5e, between e’ and e”, 1s given by the volt-ampere characteristic of the arc, A, as be oe — Fy er aan , é, = de thus is the voltage, made available for the condenser circuit, by the arc pulsation, and in phase with the current, t, = — 67 in the condenser circuit, and : bee’ — , R= ~ 3 — v— dd thus is the permissible effective resistance in the condenser circuit, that is, the maximum value of resistance, through which the pulsating arc can maintain its alternating power supply: with a larger resistance, the oscillations die out; with a smaller , resistance, they increase. From the are characteristic, A, thus can be derived a curve of effective resistances, R, as the values of as for pulsations between 7 + 82 andi — 61, and such a curve is shown as RF in Fig. 94. We may say, that the arc, when shunted by an oscillating circuit, has an effective negative resistance, be i and thereby: generates alternating power, from the consumed _ . direct-current power, and is able to supply alternating power through an effective resistance of the oscillating circuit, of be R=- a

INSTABILITY OF CIRCUITS 191

The arc characteristic in Fig. 94 is drawn with the equation

200 e= 35 + —= Vi and for . i=I1=3 amp. as mean value, the values of the effective resistance, R, increase from . ‘ R=-— de. = 18.5 ohms di for very small oscillations, to ’ R = 20.3 ohms for oscillations of 1 amp., between 1 = 2 andi = 4, to R = 27.5 ohms , for oscillations of 2 amp., between? = 1 andi = 5, etc.

Thus, if with this oscillating arc, Figs. 87 and 94, a load resist- ance r < 18.5 ohms is used, oscillation starts immediately, and cumulatively increases.

If the resistance, 7, is greater than 18.5 ohms, for instance, is _

r = 22.5 ohms, then no oscillation starts spontaneously, but the arc runs steady, and no appreciable current passes through the condenser circuit. But if once the current in the arc is brought below 1.5 amp., or above 4.5 amp., the oscillation begins and cumulatively increases, since for oscillations of an amplitude greater than between 1.5 and 4.5 amp., the effective resistance, R, is greater than 22.5 ohms.

In either case, however, as soon as an oscillation starts, it cumu- latively increases, since the effective resistance, R, steadily in- creases with increase of the amplitude of oscillation. That is, stability of oscillation, or a “singing arc” can not be reached, but : an oscillation, once started, proceeds to the extinction of the arc, and only a “‘rasping arc” could be produced.

98. However, the arc characteristic, A, of Fig. 94 is the sta- tionary characteristic, that is, the volt-ampere relation at constant , current, 7, and voltage, e.

If current, 7, and thus voltage, e, rapidly fluctuate, the arc char- acteristic, A, changes, and more or less flattens out. That is, for

a.

192 ELECTRIC CIRCUITS any value of the current, 7, the volume of the arc stream and the temperature of the arc terminals, still partly correspond to pre- vious values of current, thus are lower for rising, higher for decreasing current, and as the result, the arc voltage, e, which de- atest PtP | elite. | ttt | marae S| | TANT Tt td jt | | | Leet Pt tT tT tT EA TEE ET | | | tat bb dois eo els | | | Pit tT tT et | Ne] Eel PL ATL PON TAN EL tel ANSE OREAPE Ao NAR pt PY Net TT TNT tab iel —ORERKCECETP RE NZS BV.Gr aE Ree ([OS8ele Ee ibs SSR PE Pt ty tt tT tT TT tT tl PEPER eEHEEEEEE PT TT TT Pt tt dee TE EET TT [5 uo ds oo ds ap te oo | | | LI | Fia. 95. pends on the resistance of the are stream and the potential drop of the terminals, is different, the variation of voltage, for the same 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 effective 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 of R in the upper part of Fig. 95, the effective resistances, R, 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 D, 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

_ 8pontaneously 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 2 from curve, D, 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 effective resistance, R, 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 7, = 0,. rises to R = 2.6 ohms at 7; = 0.75, and then drops to zero at 7; = 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, R, steadily drops with increasing oscillating current, 7,, down to zero at 7; = 2.4amp. Thus, with @ resistance in the condenser circuit, of 10 ohms, the oscillations . would have an amplitude of 7, = 0.9 amp.; with 8 ohms resistance an amplitude of 2.1 amp., etc. ;

. 13 |

194 ELECTRIC CIRCUITS

From these curves of R, Fig. 95, the regulation curves of the alternating-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 8, 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, e, consists of the arc terminal drop, a, and the arc stream voltage, e1:

; e=a+e4. ; The stream voltage, é:, is the voltage consumed in the effective resistance of the arc stream; but as the arc stream is produced by the current, the volume of the arc stream and its resistance thus depends on the current, 7, in the arc, that is, the stream vol- tage is | - eg a

a Vi . and the resistance of the arc stream thus nate a i iv Thus, if, . a= 35 — b = 200, for t = 2amp., it is r, = 70.7 ohms, é, = 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 1, the volume of the arc stream

INSTABILITY OF CIRCUITS 195 and thus its resistance is still that corresponding to the previous current, 2’1. :

If thus, at the moment where the current in the arc has become

7, = 2 amp., the arc stream still has the volume and thus the resistance ; corresponding to the previous current, . a’, = 3 amp., , this resistance is 200 r’) = == = 38.50h 1=3 V3 38.5 ohms, and the stream voltage, at the current a; = 2amp., but with the stream resistance, r’1, corresponding to the previous current, t’, = 3 amp., thus is e1 = rity . = 77 volts, instead of e, = 141.4 volts, as it would be under stationary conditions.

That is, the stream voltage and thus the total are voltage at rapidly decreasing current is lower, at rapidly increasing current higher than at stationary current.

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.

Assuming 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, J, is derived. In this figure is shown as S and T' the effective resistance corresponding to the stationary characteristic, S, respectively the transient characteristic, 7’.

|

196 ELECTRIC CIRCUITS As seen, the stationary characteristic, S, gives an are oscillation which is cumulative and self-destructive, that is, the effective resistance, R, rises indefinitely with increasing amplitude of pulsation. The transient characteristic, however, gives an effect- ive resistance, R, which with increasing amplitude of pulsation

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first increases, but then decreases again, down to zero, so that the cumulative oscillations produced by this arc are self-limiting, 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,

i | 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 crosses. As seen, they fairly well coincide with the transient volt-ampere characteristic, D, of the titanium arc, at least for.the larger currents.

VE TET ET TT EEE TTT AREER HHH ACEC CETTE ee PANEL ET Paes A SE fetes RGR PAE PT IN TT tT Tey SERENE VERGE LF \ SNE APPIN TT ETT yy PIN TT INe EE ET PI IN TEN EET TT PT tTINTTNIEET RET pt | Ne EIN PT POOP ERSEEPSE Er SEER EEES=42n~28 eee t ere tT tt Pw Fia, 97.

In the electric arc we thus have an electric 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 ¢ the voltage impressed between these two conductors. Let C repre- sent the ground.

The capacity of the conductors, A and B, against ground, then, may be represented diagrammatically by two condensers, C; and C:, and the voltages from the lines to ground bye: and és. In gen-

eral, the two line capacities are equal, C, = C2, and the two volt- : ages to ground thus equal also, e, = e: = e, with a single-phase;

= aa with a three-phase line. Assume now that a ground, P, is brought near one of the lines, A, to within the striking distance of the voltage, e. A discharge then occurs over the conductor, P. Such may occur by the punc- 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 e; = e (assuming single- 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 e¢, now comprises the condenser, C2, in series tothe multiple circuit of con-

. denser, C,, and arc, P, and the condenser, C;, rapidly discharges, voltage, ¢:, decreases, and the voltage, és, increases. With a de- crease of voltage, ¢1, the discharge current, 7, also decreases, and the voltage consumed by the discharge arc, e’, increases until the two voltages, e, and e’, cross, as shown in the curve diagram of Fig. 97. At this moment the current, 7, in the arc vanishes, the arc ceases, and the shunt of the condenser, Ci, formed by the dis- charge over Pthus ceases. The voltage, e:, then rises, e, decreases and the two voltages tend toward equality, e, = e2 = e. Before this point is reached, however, the voltage, e:, has passed the dis- ruptive strength of the discharge gap, P, the discharge by the arc over P again starts, and the cycle thus repeats indefinitely.

In Fig. 97 are diagrammatically sketched voltage, ¢1, of con- denser, C1, the voltage, e’, consumed by the discharge arc over P, and the current, 7, of this arc, under the assumption that r is suffi- ciently high to make the discharge non-oscillatory. If 7 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. .

| =| ba. | | Bi " - | oe rE Fs ( ie eee et 48ep |7 /\tt4ag A s | | | | | 2 |

| | | | | | } : | | £ | | | P : | | ; | =.

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, C,, after it has been discharged by the arc over P, the voltage, e:, 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 e, provided that it is still below the voltage ‘ momentarily reached by the oscillatory charge of the line conden- ser, P 1s :

This combination of two transmission line conductors and the ground conductor, P, approaching near line, A, toa distance giving © ; ‘ a striking voltage above e, but below the momentary charging °

• voltage, of C,, then constitutes a circuit which has two permanent conditions, one of stability and one of instability. If the voltage is gradually applied, e: = e2: = e, 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 are 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.

CHAPTER XI INSTABILITY OF CIRCUITS: INDUCTION AND SYN- CHRONOUS MOTORS C. Instability of Induction Motors 102. Instability of electric circuits may result from causes which are not electrical: thus, mechanical relations between the torque given by a motor and the torque required by its load, may lead to instability. Let ' . D = torque given by a motor at speed, S, and D’ = torque required by the load at speed, S. The motor, then, could theoretically operate, that is, run at . constant speed, at that speed, S, where D =D!’ (1) However, at this speed and load, the operation may be stable, that is, the motor continue to run indefinitely at constant speed, or the condition may be unstable, that is, the speed change with increasing rapidity, until stability is reached at some other speed, . or the motor comes to a standstill, or it destroys itself. In general, the motor torque, D, and the load torque, D’, change with the speed, S. If, then, dD' _ dD aS > dS @) the conditions are stable, that is, any change of speed, S, changes the motor torque less than the load torque, and inversely, and thus checks itself. If, however, , dD’ _ dD as <8 (3) the operation is unstable, as a change of speed, S, changes the motor torque, D, more than the load torque, D’, and thereby fur- ther increases the change of speed, etc. dD’ dD aS dS (@) 201

202 ELECTRIC CIRCUITS : thus is the expression of the stability limit.

For instance, assuming a load requiring a constant torque at all

_ speeds. The load torque thus is given by a horizontal line . D’ = const. (5).

in Fig. 101.

Let then the speed-torque curve of the motor be represented by the curve, D, in Fig.101. D approximately represents the torque curve of a series motor. At the constant-load torque, D’, the motor runs at the speed, S = 0.6, point a of Fig. 101, and the speed is stable, as any tendency to change of speed, checks itself. If a2 REE EEE RE SENG EEE Pt tT INoT TT TT TT ey ET tT yt Td, PLT TN TTEETE ETT Tet ty tl ERR ENA Re SERRE RNE SERENE PTT TTT PTT NEE Pert t pi Tt ttt tT TENET Pt tT tt fot TT T wWeE TTT EET Pt tT ett TT TT TT ANE TE TTT TT PT tT ET tT eT foe TT PEA TTT, Pt ttt tt TT TT TT TP FREE SREaer ER SORES SREToe SREP ee Sees

Fia. 101. ,

the load torque decreases to D’o, the speed rises to S = 0.865, point do; if the load torque increases to D’;, the speed drops to S = 0.29, point a:, but the conditions are always stable, until _ finally with increasing load torque, D’, and decreasing speed, standstill is reached at point as. ;

Let now the speed-torque curve of a motor be represented by D in Fig. 102: the curve of a squirrel-cage induction motor with moderately high resistance secondary. The horizontal line, D’, corresponding to a load torque of D’ = 10, intersects D at two points, a and b.

INSTABILITY OF CIRCUITS 203

At a, S = 0.905, the speed is stable. At b, however, S = 0.35, the conditions are unstable, and the motor thus can not run at b, but either—if the speed should drop or the load rise ever so little ~—the motor begins to slow down, thereby, on curve, D, its torque falls below that of the load, D’, thus it slows down still more, and 80, with increasing rapidity the motor comes to a standstill. Or, if the motor speed should be a little higher, or the load momen-

• "tarily a little lower, the motor speed rises, until stability is reached at point a. . =} ==} teg HBR Hy |} PT TT TTT Ty yt feet tT ENT TT ~L>T ETT? TTT Pett tt tt ALT Tt Ta! PT tt epee tet fof | TT TT Yel I ptt tier tt tT ET Et TT ATT Bp eae et tt ttt tt tT EE ET TT TA TT PT Tt ttt yt tte ee te ET YY LL, PE Tt tee Tt tt fod fT Tt TT fel T, oe fae} tt ttt Et ; | PLT TT eet yee eT TET A 4 PL iT Ete ty ee ee et ET eT Ty Ta Pld tate td tse Pe tt Tk TT hol . Fie. 102,

With increasing load torque, D’, the speed gradually drops, | from S = 0.905 at D’ = 10, point a, down to point c, at S = | 0.75, D’ = 14.3; from there, however, the speed suddenly drops to standstill, that is, it is not possible to operate the motor at speeds less than S = 0.75, at constant load-torque, and the .