Theory and Calculation of Electric Circuits — part 8 of 15

SHAPING OF WAVES BY MAGNETIC SATURATION 151 . peaked wave of Fig. 72 contains very pronounced harmonics up to about the 701th, which at 60 cycles of fundamental frequency, gives frequencies up to 42,000, or well within the range of the danger frequencies of high-voltage power transformers, that is, See ieee Te EERGERe STG RRGEE || | RE aaa a Pt TT Tiel TE EE See eee Pt tT a ET REN LEH A tA L | \L : SCOTT | aT TFs TT TN Tt | SSN PT tT tt iA tt tT PTT ETAL ET Ke EA eee NEOPA EE EET PT a Eni a - PT TT eee T_T nt a EHH a — PET TTT ee oe pee | PET TT itt ty tt tt yt Fia. 72. frequencies with which the high-voltage coils of transformers, as circuits of distributed capacity, can resonate. 75. Magnetic saturation, and closed or partly closed magnetic circuits thus are a likely source of wave-shape distortion, resulting in high voltage peaks, and where they are liable to occur, as in

152 ELECTRIC CIRCUITS

current transformers, series transformers at open secondary cir-

cuit, autotransformers or reactors, etc., they may be guarded

against by using a small air-gap in the magnetic circuit, or by | providing the extra insulation required to stand the voltage, and

the secondary circuit, even if of an effective voltage which is not

dangerous to life when a sine wave, should be carefully handled

as the voltage peak may reach values which are dangerous to

life, without the voltmeter—which reads the effective value—

indicating this.

Inversely, such voltage peaks are intentionally provided in some series autotransformers for the operation of individual arcs

of the type, in which slagging and consequent failures to start may . occur, due to a high-resistance slag covering the electrode tips. By designing the autotransformer so as to give a very high voltage peak at open circuit—and providing in the apparatus the insula- tion capable to stand this voltage—reliability of starting is se- cured by puncturing any non-conducting slag on the electrode tips, by the voltage peak.

These high voltage peaks, produced by magnetic saturation, etc., greatly decrease and vanish if considerable current is pro- duced by them. Thus, when the secondary of a closed magnetic circuit series transformer is open, at magnetic saturation, a high

voltage peak appears; with increasing load on the secondary, however, the voltage peak drops and practically disappears

| already at relatively small load. Thus such arrangements are

suitable for producing voltage peaks only when no current is

required, as for disruptive effects, or only very small currents.

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CHAPTER IX WAVE SCREENS. EVEN HARMONICS

76, The elimination of voltage and current distortion, and production of sine waves from any kind of supply wave, that is, the reverse procedure from that discussed in the preceding chapter, is accomplished by what has been called “wave screens.”

Series reactance alone acts to a considerable extent as wave screen, by consuming voltage proportional to the frequency and the current, and thereby reducing the harmonics of voltage in the rest of the circuit the more, the higher their order. .

Let the voltage impressed upon the circuit be denoted sym- bolically by .

e=~eitestestert... = Len (29) where n denotes the order of the harmonic of absolute numerical value én.

If, then, the reactance z (at fundamental frequency) is inserted

into the circuit of resistance, r, the impedance is 21 = Vr?+ 2? for the fundamental frequency, and za = Vr? + 732? for the nth harmonic, (30) and the current thus is pa PSF o_ = z >a [pt 4 nz? (31) or, denoting r. z =¢, (32) it is , poly = + tt TInt +e? THY1 +c? %I/9+ c? 73 Ny ed e e e 33 mepat +++ 6) 153

154 ELECTRIC CIRCUITS if r is small compared with z, c* is negligible compared with 1, 9, 25, etc., and it is a és , &5 , or i-tlate+ete+... | that is, the current, 7, and thus the voltage across the resistance, 7, shows the harmonics of the supply voltage, e, reduced in propor- tion to their order, n. ; ‘

Even if r is large compared with z, and thus c*?>1, finally c? becomes negligible with n*, and the harmonics decrease with their order.

77. The screening effect of the series reactance is increased by shunting a capacity, C, beyond the inductance, L, that is, across the resistance, r, as shown in Fig. 73. By consuming current

L L LL. . 4 H . Fia. 73. Fia. 74. -

; proportional to frequency and voltage, the condenser shunts the more of the current passing through the reactance, the higher the frequency, and thereby still further reduces the higher harmonics

‘of current in the resistance, r, and thus of voltage across this re- sistance. Its effect is limited, however, by the decreasing voltage distortion at r and thus at the condenser, C.

Thus the screening effect is still further increased by inserting

; a second inductance, L, beyond the condenser, C, in series to the resistance, r, as shown in Fig. 74. By making the second induct- ance equal to the first one, and making the condenser, C, of the same reactance, for the fundamental wave, as each of the two inductances, we get what probably is the most effective wave screen. This 7-connection or resonating circuit will be discussed more fully in Chapter XIV, in its feature of constant-potential constant-current transformation.

Under the condition, that the two inductive reactances and the

WAVE SCREENS. EVEN HARMONICS 155 capacity reactance are equal, the equation of the current in the resistance, r, is (page 291), for the nth harmonic,

pa eo, T= an(n? — 2) — jr(n? — 1) (34) or, absolute, . : . €o 1 = 8 y —___ 35 Te Vatiat — 8) + nt — 1 “ where e=2 (36) . x If c is small, that is, r small compared with z, the current becomes t=-—_% _ (37) wn (n? — 2) or, for higher values of n,

. . t= < , . (38) that is, it decreases with increasing order of harmonic, and pro- portional to the cube of the order n, thus shows an extremely rapid decrease. ;

If cis not negligible, the denominator in (35) is larger, and i, therefore, still smaller. : As illustration may be shown the current, i), and thus the vol- . tage, €, across a resistance, 7, under the very greatly distorted and peaked voltage of Fig. 62: (a) for a series reactance, x, equal tor, that is, c = 1; (b) for the complete wave screen of two inductances and one capacity. Itis - impressed voltage, e@ = 1.27 eo { 11 + 0.9783 + 0.935; + 0.8777 + 0.800) + 0.71311

• • 0.61733 + 0.51715 + 0.41617 + 0.31519 + 0.18921}. (a) Reduction factor of the nth harmonic, 1 1 Vita Vat PT hence, a = aes + 0.442; + 0.258, + 0.1757 + 0.1255 + 0.09141

• 0.06713 + 0.04915 + 0.03417 + 0.0231) + 0.0132:}.

156 ELECTRIC CIRCUITS (b) Reduction factor of the nth harmonic, , 1 hence, ; €2 = 1.27 {1, + 0.047; + 0.008; + 0.003, + 0.001, + 0.0011}. ‘That is, the third harmonic is reduced to less than 5 per cent., the fifth to less than 1 per cent., and the higher ones are practically entirely absent. While in the supply voltage wave, e, the voltage peak (by adding the numerical values of all the harmonics: 1 + 0.978 + 0.935 + . . .) is 7.36 times that of the fundamental wave, it is reduced by series reactance to less than 2.28 times the maximum of the fundamental wave, that is, very greatly reduced, and by the complete wave screen to less than 1.06 times the maximum of the fundamental. That is, in the last case = the voltage is practically a perfect sine Cc S) wave. sf St 78. By “wave screens” the separation of | — pulsating currents into their alternating and | (A) (c) their continuous component, or the separa- i tion of complex alternating currents—and Fie. 75. thus voltages—into their constituent har- monics can be accomplished, and inversely, the combination of alternating and continuous currents or vol- tages into resultant complex alternating or pulsating currents. The simplest arrangement of such a wave screen for separating, or combining alternating and continuous currents into pulsating ones, is the combination, in shunt with each other, of a capacity, C, and an inductance, L, as shown in Fig. 75. If, then, a pulsating voltage, e, is impressed upon the system, the pulsating current, 7, produced by it divides, as the continuous component can not pass through the condenser, C’,, and the alternating component . is barred by the inductance, L, the more completely, the higher this inductance. Thus the current, 71, in the apparatus, A, isa true alternating current, while the current, io, in the apparatus, C, is a slightly pulsating direct current. Inversely, by placing a source of alternating voltage, such as an alternator or the secondary of a transformer, at A, and a source of continuous voltage, such as a storage battery or direct-current

WAVE SCREENS. EVEN HARMONICS 157 generator, at C, in the external circuit a pulsating voltage, e, and pulsating current, 7, result.

If the capacity, C, is so large as to practically short-circuit the alternating voltage, and the inductance, L, so high as to practically open-circuit the alternating voltage, the separation—of combi- nation—is practically complete, and independent of the frequency of the alternating wave.

Wave screens based on resonance for a definite frequency by series connection of capacity and inductance, can be used to sepa- rate the current of this frequency from a complex current or voltage wave, such as those given in Figs. 56 to 63, and thus can be used for separation of complex waves into their Cr LQ Q0 000000 (&:) components, by “harmonic c L analysis.” 3} VO0Q000 (As)

Thus in Fig. 76, if the | 9, Lif GOOOO0 successive capacities and in- (As) ductances are chosen such Cr LQO606 (3) that

° 1 » L000 1 nn Luf QQ (Ai) 8 xfla = FG 1 10 «fly = 75 xfs’ 1 2n fly = 2xfnC,, (39) Fia. 76. where f = frequency of the fundamental wave. . Then, through any of the branch circuits C,, L,, only the nth harmonic, 7,, can pass to an appreciable extent. Such resonant wave screen, however, has the serious disadvan- ~ tage to require very high constancy of f, since the resonance condi- tion between C, and L, depends on the square of f, 1 = 2, G7 4xf?L,.

78. Even harmonics are produced in a closed magnetic circuit by the superposition of a continuous current upon the alternating wave. With an alternating sine wave impressed upon an iron magnetic circuit, saturation, or in general the lack of proportional-

| 158 ELECTRIC CIRCUITS ity between magnetic flux and m.m.f., produces a wave-shape dis-

• tortion, that is, higher harmonics, of voltage with a sine wave of current, of current with a sine wave of impressed voltage. The constant term of a wave, however, is the first even harmonic, and thus, if the impressed wave comprises a fundamental sine and a pie tT TTT PT Ty te . po ee pte TT TT NI a! (oo apn ay HAA tt Pt] VET Net | TT | era J ee Ok a a ZEN Yt , COOSA SS DR AS 7A NA i | | YI A jet NTN | t ft ft yt : AVC LENE NET PCOS SS EIN NIZA ZEERNNSECNET PSO TT NN NC Ber SN FE --ATSG A He See Pt tte eee eT . Fie. 77, constant term, the former gives rise to the odd harmonics, the latter to the even harmonics.

Let, then, on the alternating sine wave of impressed voltage a continuous current by superimposed. The magnetic flux then oscillates sinusoidally, not between equal and opposite values, but between two unequal values, which may be of the same, or of opposite signs. That is, it performs an unsymmetrical magnetic cycle. Neglecting again hysteresis, that is, assuming the rising

WAVE SCREENS. EVEN HARMONICS 159 and the decreasing magnetization curve as coincident—which is permissible as approximation, since the hysteresis contributes little to distortion—and choosing the same magnetization curve as in the preceding, curve I in Fig. 64, we may as an instance con- sider a sinusoidal magnetic pulsation between the limits +15.4 and +19.7, corresponding to a variation of the m.m.f. between H = +10 and H = +100. Fig. 77 then gives, as curve B, the sinusoidally pulsating mag- netic flux density. Taking from curve I, Fig. 64, the values of H corresponding to the values in curve B, Fig. 77, gives curve H. - This, resolved (“‘Engineering Mathematics,” paragraph 92) gives the constant term % = 36, and the alternating current, 1. The latter is unsymmetrical, having one short half-wave of a peak value 64, and one long half-wave of maximum value 26. It thus resolves into the odd harmonics, 1;, alternating between +45, and the even harmonics, mainly the second harmonic, alternating between maximum values +18 and —15. 14, is peaked with flat zero, thus showing a third harmonic, which is separated as 73, and 42 is unsymmetrical, showing further even harmonics, which are separated as 1,4, but are rather small. . ‘ Thus the pulsating exciting current of the sinusoidally varying ‘ unidirectional magnetic flux B = 17.55 + 2.15 cos ¢ is given by H = 36 + 37 cos¢ + 16.5 cos2¢ + 8cos3¢-+2cos4g+.. Instead of superimposing a direct current upon an alternating wave, as by connecting in series an alternator and a direct-current generator or storage battery, two separate coils can be used on the magnetic circuit, one energized by an alternating impressed vol- tage, the other by a direct current. A high inductive reactance would then be connected in the latter circuit, to eliminate the current pulsation which would be caused by the alternating vol- tage induced in this coil. _ ‘Connecting two such magnetic circuits with their direct-current magnetizing coils in series, but in opposition (without the use of a series reactance) eliminates the induced fundamental wave, but leaves the second harmonic in the direct-current circuit, which thus can be separated. Numerous arrangements can then be de- vised by two magnet cores energized by separate alternating-

: 160 ELECTRIC CIRCUITS current exciting coils and saturated by one common direct-current exciting coil, surrounding both cores, or their common return, etc. - 80. The preceding may illustrate some of the numerous wave- shape distortions which are met in electrical engineering, their characteristics, origin, effects, use and danger. Numerous other wave distortions, such as those produced by ares, by unidirec- tional conductors, by dielectric effects such as corona, by Y con- nection of transformers for reactors, by electrolytic polarization, by pulsating resistance or reactance, etc., are discussed in other chapters or may be studied in a similar manner. |

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 ununderstood 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 unstable 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 effect 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-

11 , 161

162 ELECTRIC CIRCUITS . stance, if a switch is closed, and thereby a load put on the circuit,

' the current can not instantly increase to the value corresponding to the increased load, but some time elapses, during which the increase of the stored magnetic energy corresponding to the in- creased current, is broughtabout. 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 our 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. UnstastE EvectRicaL EquImLiBpRIUM

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 characteristic given in Fig. 78, at the impressed -voltage, é, three different values of current are possible: 7;, 73 and 73. 7, and ts are stable, 72 unstable. That is, at current, #3, passing through the conductor under the constant impressed voltage, é&, a mo- mentary increase of current would give an excess voltage beyond that required by the conductor, thereby increase the current still

Pet TE ETT EE ET EE TNE TN | Ee Ee tT EE TT ENE PL eo TT er PT TENA fe eee TT Tee t Te et eet ty FEE tte tee ty EE TT TL PEt ty yt tee et Pi tte tT tei tT tT ETT PET TEE TTT ET tet tt Fia. 78. further, and with increasing rapidity the current would rise, until it becomes stable at the value, 73. 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, 4.

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 80 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 instability, 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

|= eA} | Tt ttt eT TE TT PUAN i SRR 1a NSS RRR ae LAN a a 0 ee ee ee ee ° DSS A ee 7 | LIN eer tt tt Tt ty of | IN, ET EET tT Ty Ey ttt | a SEE | COC SSB | eff te : ol | | ttt tt tert | ef fitter | tt tt -CCCCC ERECT eet tt tilt tt ttt tt tt tt aAiie tT bi de bt tede | Fia. 79. is shunted by a condenser, the condenser makes the arc unstable and puts it out; the available supply voltage, however, starts it again, and so periodically the arc starts and extinguishes, as 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

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_ INSTABILITY OF: CIRCUITS 165 against 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 B, in Fig. 79, the arc is stable on constant-voltage supply for currents above 1 = 3 amp., unstable below 3 amp., and therefore, with a constant- supply voltage, éo, two current values, 7, and 72, exist, of which the former one is stable, the latter one unstable. That is, current, 74q, can not persist, but the current either runs up to 7; and the arc then gets stable (form 1), or the current decreases and the arc

; goes out, instability 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

85. 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, unstable 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 best illustrated by the in- stance of the hunting of synchronous 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 wh TVET TT TE tT TT Tt TT TN | tT ttt tt Et tt Tt AA tof Stoner Vac ae e

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Fie. 82, 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, %. 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 e=a + ol + 8) : Vi (1) b at where b is the stream voltage, that is, voltage consumed by the arc stream. Fig. 82 is drawn with the constants, a = 35, c= 51, t= 18, 6 = 0.8, hence, 133 e= 35 + Vi Assuming this arc is operated from a circuit of constant-voltage supply, E = 150 volts, through a resistance, ro The voltage consumed by the resistance, 70, then is ae = Tot, (3) and the voltage available for the arc thus e, = E — rot (4) Lines B, C and D of Fig. 82 give e:, for the values of resistance, 7 = 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 1 = 2.5 cm. arc can not be operated from # = 150 volt supply. Line C intersects A at a and 6, 7 = 6.1 and 1.9 amp. respect- ively. At a,¢ = 6.1 amp., the arc is stable; At b, 7 = 1.9 amp., the arc is unstable; for the reasons discussed before: an increase of current decreases the voltage consumed by the circuit, e + é2, and thus still further increases the current, and inversely. Thus the arc either goes out, or the current runs up to7 = 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, 7 = 3.05 amp., of operation of the are 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 are. This stability characteristic is determined by the condition . deo a = 0, (5) where eo =e+t+re (6) matters =a+ Vi + rot, this gives ; . 1 oii 24 @ and ; 1.5 @&=a+ Vi | (8) . =ea + 1.5 ey 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-voltage supply, is ie . aE? - (9) . where e is the voltage consumed by the current, 7, 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

7 : f | 170 ELECTRIC CIRCUITS . the total current, z, thus may be called the stability coefficient of the circuit, . @ . ; $= di t de . (10) ) aa “3 a In a circuit of constant resistance, r, it is Cur ; t um Ay . de _ . aS hence, . 6=1, that is, the stability coefficient of a circuit of constant resistance, r, is unity. ;

• In general, if the effective resistance, r, is not constant, but varies with the current, 7, it is : e=M, de dr. ‘ at =r-+t a’ hence, the stability coefficient ; dr s=14+% (11) i 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 6 is still positive, that is, the décrease of resistance slower than the increase of current, the circuit is stable. 6>0 (12)

INSTABILITY OF CIRCUITS 171 is the condition of stability of a circuit on constant-voltage supply, and

5<0 (18) is the condition of instability, and 7 6=0 (14) thus gives the stability characteristic of the circuit. In the are, b e=at Vi? the stability coefficient is, by (10), b é . o= ~Fi - Be (15) ‘ a that is, equals half the stream voltage, 5 divided by the are voltage, e. ‘

Or, substituting for e in (15), and rearranging,

1 § = — —_____

a; (16)

2(1 + 5vi)

__ 1 2(1 + 0.2625 V/1) in Fig. 82. For: = 0, itis = —0.5; t= 0, itisé = 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, H = 150 volts, the are having the characteristic, A, Fig. 82, can not be operated at less than 3.05 amperes. Ati = 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, To, then would be given by

deo a: 50 = a t where

172 ELECTRIC CIRCUITS &=at a2 + rot (17) 0 Vi 0 and the resistance ro chosen so as to give , é: = 150 volts, from (17) follows, , b . » = avi EO ooanv—? €0 and, substituting from (17), . iro = @—-a- a gives 1.56 ot (18) do = 1 — ———— € . or, b=1-2 (19) Co where ép is the supply voltage, e> the voltage given by the stability characteristic, S. éo, 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, &, 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,r:. Thus in Fig. 83, let J = 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 are be shunted by resistance, 7:. If e = voltage consumed by the arc, the current shunted by the resistance, 71, is , . e€ u= rh (20) and the current available for the arc thus is t=I-i (21) . =-jJ—4 "1 or e= nit - 4). (22)

INSTABILITY OF CIRCUITS 173 Curves B, C and D of Fig. 83 show the values of equation (22) f . . ° 7, = 32 ohms: line B = 48 ohms: line C = 40.8 ohms: line D. PT TTT ETT PET yer ry eS | CONSTANT CURRENT Sa LV ETT det seat TT TT TT pp Af Nee eee |} RY |W A- HH ae NEN tH} RAPE | | : ANNE NESE eA Site wy AN | | | est | ae PORE mT NNN ETE EEPNEXSU CSS pt ONE ee ot | tT SSS — mL ETN C hee alee ae to oe oo os do UNG? ols co ds to | | Fig. 83. Line B does not intersect the arc characteristic, A, that is, with a resistance as low as r; = 32, no arc can be maintained on the 5-amp. constant-current circuit. Line C intersects A at two points: (a) t = 2.55 amp., e = 118 volts, stable condition; (b) 7 = 0.55 amp., e = 214 volts, unstable condition. _ Line D is drawn tangent to A, touches at c:71 = 1.4 amp.,

174 ELECTRIC CIRCUITS e = 148 volts, the limit of stability. At J = 5 amp., the point h, ate = 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 abscisse, ¢, tangents on A, and transferring their contact points, c, b, to the abscisse, from which the tangent is drawn, gives the points h, g, of the constant-current stability characteristic of the arc, that is, the curve of arc voltages in a constant-current circuit, J, when . by shunting the arc with a resistance, 11, consuming current, 41, the stability limit of the are with current 7 = I—7; is reached. P then gives the curve of the arc currents, 7, corresponding to '* the are voltage, e, of curve Q, for the different values of the con- . stant-circuit current, I. The equations of Q and P are derived as follows: _ The stability limit, point c, corresponding to circuit current, I, as given by de _ . a”! where e = arc voltage, and 7 = arc current. Or,

1 ivi (

• It is, however, ; =a + Vi . : and e T-i Fy = 171. From these three equations follows, by eliminating r; and i or e, Q, _ 623 e — a) | tea es) P, p= ib +2 avi). (95) 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 as series resistance makes it unstable in a constant-voltage supply circuit.

90. Due to the dropping volt-ampere characteristic, two arcs “ean not be operated in parallel, unless at least one of them has a sufficiently high resistance in series. PET ey Tee TE EE ET TT] HERES cae Ff P| tt pe | orancs [x | | | | ti | reel sey | TTT te - i tT | te det Ee LM TE ot fe Met ft Pr Tt | pet — ARCCCECCCBe Ce PNT NE er TT TT de INE T Seer TAL TT TT Pek TOE EEE] et NEE er P| INS er eer i tT pt TIN peer tt tt — COECEBEEFCrCErPeres mee || fs} eee} e}-ee t= PTT} ET ete eT yy Ty yl tt ttt tT eet TTT ET UL |b vo ts 2b ds a0 86 co als so os co os co | | Fie. 84. Let, as shown in Fig. 84, two arcs be connected in parallel into the circuit of a constant current I = 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 ¢ = current in the first arc, thus 7’ = I — 7 = 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 A’.

As the two parallel arcs must have the same voltage, the oper- ating point is the point, a, of the intersection of A and A’ 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 are, 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: b, g and h.

Of these, point } 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 h, however, are unstable.

At b, stable condition, the characteristics, B and B’, are rising; at a, unstable 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

1.56 é a+ Vi

Fig. 85 shows the case of two parallel arcs, which are not equal

and do not have equal resistances, 7, in series, one being a longarc,

INSTABILITY OF CIRCUITS 177 having no resistance in series, the other a short arc with a resist- ance r = 40 ohms in series. .

. The volt-ampere characteristic of the long are is given by A, that of the short arc by B, and that of the short arc plus resistance, r, by C.

A and C intersect at three points,a,bandc. Of these, only the point a is stable, as any change of current from this point limits ah Pes renin

Pt | Tt Pee TT] pr ares | | mi A | | tel get | | et tT et Jt NOV ae eee bol NAL | fot tT LT TT Et pel Nt ol INN TEE el} | INSEE EE EE ET mol {| | TOSSEE TT TT mE SSR ! IRE EERESN ASE errs | ef ttt ttt tt ye tt tt a ae eee ot tT TET TET TT TTT ot | | | {| pete TE TT . | alo de alo ale slo als clo als ole os co os to |_| Fie. 85. itself; b 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. 12