Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients — part 3 of 7

SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 565 netic circuit, and the saturation value of the flux in the iron. , * That is, for 7 = 0, = 1T,; and fori =0,¢’ = ; If r = resistance, the duration of the component of the transient resulting from the air flux would be —8 T, = 2 = rel0 (5) r r and the duration of the transient which would result from the initial inductance of the iron flux would be In, naild-8 hr ar ae (6) The differential equation of the transient is: induced voltage plus resistance drop equal zero; that is, ne 10-8 + ri = 0. Substituting (3) and differentiating gives nal0-§ di, ag di . (+ hyp di + ne 10 at + r= 0, and, substituting (5) and (6), (7, di, so. | )aa be tT tts % hence, separating the variables, Tdi Todt _ Tatbhyt T = (7) The first term is integrated by resolving into partial fractions: it 2it) __ > i1+6i)? «+ L+ot (1 +62)?’ and the integration of differential equation (7) then gives t . T; _ Py log pp; + Telogi + pay, t+ = 0. (8) If then, for the time ¢ = éo, the current is 7 = %, these values substituted in (8) give the integration constant C: Zo . Ti _ Tilogyapy, t+ Teles t+ page th + C = 0, (9)

38 ELECTRIC DISCHARGES, WAVES AND IMPULSES. with the armature current, that is, is instantaneous. The arma- ture reaction, however, is the m.m.f. of the armature current in its . reaction on the m.m.f. of the field-exciting current. That is, that part %2 = 29 — 2 of the synchronous reactance which corresponds to the armature reaction is not a true reactance at all, consumes no voltage, but represents the consumption of field ampere turns by the m.m.f. of the armature current, and the corresponding change of field flux. Since, however, the field flux represents stored magnetic energy, it cannot change instantly, and the arma- ture reaction thus does not appear instantaneously with the arma- ture current, but shows a transient which is determined essentially by the constants of the field circuit, that is, is the counterpart of the field transient of the machine.

If then an alternator is short-circuited, in the first moment only the true self-inductive part x, of the synchronous reactance exists, and the armature current thus is 4, = 2, where é is the induced em.f., that is, the voltage corresponding to the magnetic-field.

| excitation flux existing before the short circuit. Gradually the armature reaction lowers the field flux, in the manner as repre- sented by the synchronous reactance 2, and the short-circuit cur- rent decreases to the value 7) = = :

The ratio of the momentary short-circuit current to the perma-

. nent short-circuit current thus is, approximately, the ratio = = m that is, synchronous reactance to self-inductive reactance, or armature reaction plus armature self-induction, to armature self-induction. In machines of relatively low self-induction and high armature reaction, the momentary short-circuit cur- rent thus may be many times the permanent short-circuit current. ;

The field flux remaining at short circuit is that giving the volt- age consumed by the armature self-induction, while the decrease of field flux between open circuit and short circuit corresponds to the armature reaction. The ratio of the open-circuit field flux to the short-circuit field flux thus is the ratio of armature reaction plus self-induction, to the self-induction; or of the synchronous reactance to the self-inductive reactance: es .

LECTURE VI. DOUBLE-ENERGY TRANSIENTS.

24. Ina circuit in which energy can be stored in one form only, the change in the stored energy which can take place as the result of a change of the circuit conditions is an increase or decrease. The transient can be separated frétm tlie permanent condition, and then always is the representation of a gradual decrease of energy. Even if the stored energy after the change of circuit conditions is greater than before, and during the transition period an increase of energy occurs, the representation still is by a decrease of the transient. This transient then is the difference between the energy storage in the permanent condition and the energy storage during the transition period.

If the law of proportionality between current, voltage, magnetic flux, etc., applies, the single-energy transient is a simple exponential function:

t y = yor 7, (1) where Yo = initial-value of the transient, and T, = duration of the transient, that is, the time which the transient voltage, current, etc., would last if maintained at its initial value.

The duration 7) is the ratio of the energy-storage coefficient to the power-dissipation coefficient. Thus, if energy is stored by the current 7, as magnetic field,

L To = Yr (2) where L = inductance = coefficient of energy storage by the cur- . rent, r = resistance = coefficient of power dissipation by the current.

If the energy is stored by the voltage e, as dielectric field, the duration of the transient would be C

‘= — Ty =o (3) 59

. 40 ELECTRIC DISCHARGES, WAVES AND IMPULSES, proportional to’. Thus, as it isi = 7) at o, during the transition © period it is? = 2 to. Hence, the field-exciting current traverses 0 . the same transient, from an initial value 79’ to the normal value 7, as the field flux & and the armature currents. ®, a? & << A eee ‘, Ty | B CX SSeS 1 a EES 1 / be Le F ’ Cc B : | Xt ol om eeeEg . i, Je . oO H Tm ae . Fig. 21. — Construction of Momentary Short Circuit Characteristic of Poly- phase Alternator. _ Thus, at the moment of short circuit a sudden rise of field current must occur, to maintain the field flux at the initial value ®, against the demagnetizing armature reaction. In other words, the field flux @ decreases at such a rate as to induce in the field . circuit the e.m.f. required to raise the field current in the propor- tion m, from 7% to 79’, and maintain it at the values corresponding to the transient 2, Fig. 21D. As seen, the transients ®; 7, 72, 73; /; 7 are proportional to each other, and are a field transient. If the field, excited by current 7

| | DOUBLE-ENERGY TRANSIENTS. 61 energy is dissipated before this. This latter case occurs when the . . dissipation of energy is very rapid, the resistance (or conductance) high, and therefore gives transients, which rarely are of industrial importance, as they are of short duration and of low power. It therefore is sufficient to consider the oscillating double-energy transient, that is, the case in which the energy changes periodically between its two forms, during its gradual dissipation. This may be done by considering separately the periodic trans- fer, or pulsation of the energy between its two forms, and the gradual dissipation of energy. A. Pulsation of energy. 25. The magnetic energy is a maximum at the moment when the dielectric energy is zero, and when all the energy, therefore, is magnetic; and the magnetic energy is then Lio? 2” where 7) = maxinium value of transient current. The dielectric energy is a maximum at the moment when the magnetic energy is zero, and all the energy therefore dielectric, and is then Ceo? 2” where ey) = maximum value of transient voltage. As it is the same stored energy which alternately appears as magnetic and as dielectric energy, it obviously is Lip? — Ceo? a 8) This gives a relation between the maximum value of transient current and the maximum value of transient voltage: €o vz — = =~ 9 to C : ( ) vz therefore is of the nature of an impedance 2p, and is called the natural impedance, or the surge impedance, of the circuit; and _ its reciprocal, yg = Yo, is the natural admittance, or the surge admittance, of the circuit.

42 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ; start with the values —7’, —7’, —73’. The resultant armature currents are derived by the addition of these armature transients upon the permanent armature currents, in the manner as dis- cussed in paragraph 18, except that in the present case even the permanent armature currents 7%, 72, 73 are slow transients. In Fig. 22B are shown the three armature short-circuit currents, in their actual shape as resultant from the armature transient and the field transient. The field transient (or rather its begin- ning) is shown as Fig, 22A, Fig, 22B gives the three armature t=0 nt 2 3 zr 5 6 Sees, , tb | | A | ° ; , B i TARA soho (NA COCOA iNig C Y\ . io iy Fig. 22. — Momentary Short Circuit Characteristic of Three-phase Alternator. currents for the case where the circuit is closed at the moment when 2, should be maximum; 7; then shows the maximum transient, and 72 and 73; transients in opposite direction, of half amplitude. These armature transients rapidly disappear, and the three currents become syminetrical, and gradually decrease with the field tran- sient to the final value indicated in the figure. The resultant m.m.f. of three three-phase currents, or the arma- ture reaction, is constant if the currents are constant, and as the currents clecrease with the field transient, the resultant armature reaction decreases in the same proportion as the field, as is shown

_ DOUBLE-ENERGY TRANSIENTS. | 63 oscillating voltages, that is, acts as a short circuit for the trans-_ former oscillation, and therefore protects the latter. Inversely, if the large oscillating current of a cable enters a reactive device, as a current transformer, it produces enormous voltages therein.

Thus, cable oscillations are more liable to be destructive to the reactive apparatus, transformers, etc., connected with the cable, than to the cable itself.

A transmission line is intermediate in the values of zo and yo

between the cable and the reactive apparatus, thus acting like a reactive apparatus to the former, like a cable toward the latter. Thus, the transformer is protected by the transmission line in oscillations originating in the transformer, but endangered by the transmission line in oscillations originating in the transmission line. _

The simple consideration of the relative values of z) = vi in the different parts of an electric system thus gives considerable information on the relative danger and protective action of the parts on each other, and shows the reason why some elements, as current transformers, are far more liable to destruction than others;

but also shows that disruptive effects of transient voltages, observed in one apparatus, may not and very frequently do not originate in the damaged apparatus, but originate in another part of the system, in which they were relatively harmless, and become dangerous only when entering the former apparatus.

26. If there is a periodic transfer between magnetic and dielec- tric energy, the transient current 7 and the transient voltage e . successively increase, decrease, and become zero.

The current thus may be represented by

t = ly vos (@ — Y), (12) where 7) is the maximum value of current, discussed above, and ; @ = 27 ft, (13) where f = the frequency of this transfer (which is still undeter- mined), and y the phase angle at the starting moment of the transient; that is,

a, = 19 cOS y = initial transient current. (14)

As the current 7 is a maximum at the moment when the magnetic energy is a maximum and the dielectric energy zero, the voltage e

44 ELECTRIC DISCHARGES, WAVES AND IMPULSES. self-induction, that is, the synchronous reactance is 6 times the self- oe inductive reactance, = = m= 6. The frequency is 25 cycles. I

If ; is the initial or open-circuit flux of the machine, the short- circuit flux is dy) = me = 5h and the field transient @ is a tran- sient of duration 1 sec., connecting ®; and %o, Fig. 22A, repre- sented by the expression

t @ = By + (h; _- Poe To,

The permanent armature currents 7, %2, 73 then are currents

starting with the values m =, and decreasing to the final short- 0 . circuit current =, on the field transient of duration 7. To these 0

currents are added the armature transients, of duration 7’, which start with initial values equal but opposite in sign to the initial values of the permanent (or rather slowly transient) armature currents, as discussed in paragraph 18, and thereby give the asym- inetrical resultant currents, Fig. 22B. _

The field current 7 gives the same slow transient as the flux %, starting with 7%’ = m7, and tapering to the final value 7. Upon this is superimposed the initial full-frequency pulsation of the armature reaction. The transient of the rotating field, of duration.

T = .1 sec., is constructed as in paragraph 18, and for its instan- taneous values the percentage deviation of the resultant field from its permanent value is calculated. Assuming 20 per cent damping in the reaction on the field excitation, the instantaneous values of .the slow field transient (that is, of the current (i — 70), since 7) is the permanent component) then are increased or de- creased by 80 per cent of the percentage variation of the ‘transient field of armature reaction from uniformity, and thereby the field curve, Fig. 22C, is derived. Here the correction for the external field inductance is to be applied, if considerable.

Since the transient of the armature reaction does not depend on the point of the wave where the short circuit occurs, it follows that the phenomena at the short circuit of a polyphase alternator . are always the same, that is, independent of the point of the wave at which the short circuit occurs, with the exception of the initial wave shape of the armature currents, which individually depend

DOUBLE-ENERGY TRANSIENTS. 65 decreases, and as at lower magnetic densities the permeability of the iron is higher, with the decrease of voltage the permeability of the iron and thereby the inductance of the electric circuit inter- linked with it increases, and, resulting from this increased magnetic energy storage coefficient L, there follows a slower period of oscil- , lation, that is, a decrease of frequency, as seen on the oscillogram, from 55 cycles to 20 cycles per second. If the energy transfer is not a simple sine wave, it can be repre- sented by a series of sine waves, and in this case the above equa- tions (12) and (15) would still apply, but the calculation of the frequency f would give a number of values which represent the different component sine waves. The dielectric field of a condenser, or its “charge,” is capacity times voltage: Ce. It is, however, the product of the current flowing into the condenser, and the time during which this current flows into it, that is, it equals 7 ¢. Applying the law Ce = it (17) to the oscillating energy transfer: the voltage at the condenser , changes during a half-cycle from —eo to +éo, and the condenser charge thus is 2 €oC; the current has a maximum value 7, thus an average value 2, ~ , . v . and as it flows into the condenser during one-half cycle of the frequency f, that is, during the time ree itis - . | | 42.1 : 2 €oC = z to af’ : . ' which is the expression of the condenser equation (17) applied to the oscillating energy transfer. Transposed, this equation gives f ~ 2 TeoC’ (18) . and substituting equation (10) into (18), and canceling with 7%), gives 1 1 = = 19 f anViG Ine (19)

46 ELECTRIC DISCHARGES, WAVES AND IMPULSES. , is the field transient & (the same as in Fig. 224A) and B the arma-

ture current, decreasing from an initial value, which is m times

the final value, on the field transient.

Assume then that the mutual induction between field and armature is such that 60 per cent of the pulsation of armature reaction appears in the field current. Forty per cent damping for the double-frequency reaction would about correspond to the 20) per cent damping assumed for the transient full-frequency pulsa- tion of the polyphase machine. The transient field current thus pulsates by 60 per cent around the slow field transient, as shown by Fig. 23C; passing a maximum for every maximum of armature

t=0 al .2 23 4 Seconds a d . *, } A 4 MANOA AANA. J a, VUUUUUUY YY Cc . TVPG, | i, Fig. 24. — Asymmetrical Momentary Single-phase Short Circuit of Alternator. current, and thus maximum of armature reaction, and a minimum for every zero value of armature current, and thus armature reac- tion.

Such single-phase short-circuit transients have occasionally been recorded by the oscillograph, as shown in Fig. 27. Usually, how- - ever, the circuit is closed at a point of the wave where the perma- nent armature current would not be zero, and an armature transient appears, with an initial value equal, but opposite to, the initial value of the permanent armature current. This is shown in Fig. 24 for the case of closing the circuit at the moment where the.

DOUBLE-ENERGY TRANSIENTS. 67 transient. In the latter case, the duration of the transient would be L To = 7? and with only half the energy magnetic, the duration thus is | twice as long, or 1, =2T) = ae, (23) and hereby the factor t hae ft multiplies with the values of current and voltage (21) and (22). . “ i i . . {| (A | H H { H { } | € Bi} } ae ! . A Li’ | : am 2 ff LC} i “. 1 \ ! . \ { Ce’ \y wy N . D i Fig. 32. — Relation of Magnetic and Dielectric Energy of Transient. The same applies to the dielectric energy. If all the energy were dielectric, it would be dissipated by a transient of the dura- tion: ; To = ¢; g

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DOUBLE-ENERGY TRANSIENTS. 09 . transformers, etc., and is not the case in telegraph or telephone lines, etc. It is very nearly the case if the capacity is due to elec- trostatic condensers, but not if the capacity is that of electrolytic condensers, aluminum cells, ete. Combining now the power-dissipation equation (25) as factor with the equations of periodic energy transfer, (21) and (22), gives the complete equations of the double-energy transient of the circuit containing inductance and capacity: , ee: £2 )™ | t= e— “5 2) COS— — Yoei SIN - 0? a a} ; ; (28) e=e—tt ) €, COS ~ ++ 2921 SIN=e? g o where _ ‘Lo. 29 = Vz = yo (29) vaist4gl, 21L'C oc = VIC, (30) and 2; and e are the initial values of the transient current and volt- age respectively. As instance are constructed, in Fig. 33, the transients of current and of voltage of a circuit having the constants: Inductance, L = 1.25 mh = 1.25 X 10-3 henrys; Capacity, C = 2af = 2 X 10-§ farads; _ Resistance, r = 2.5 ohms; Conductance, g = 0.008 mho, in the case, that . The initial transient current, 7% = 140 amperes; The initial transient voltage, e, = 2000 volts. It is, by, the preceding equations: , o = VIC =5 X 1075, : f= =— = 3180 cycles per second, 2040 Zo = vi = 25 ohms, Yo = vs = 0.04 mho, » a o-aaf

i “9 sey y ’ r > : cr 50 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . rent wave, therefore, is symmetrical, and the field current shows only the double-frequeney pulsation. Only a few half-waves were recorded before the circuit breaker opened the short circuit. poe a SMR gdiN Cob ga ee : Ae {Nis eel ns. N\ NORE Splec bY ss [legmest se Se Pe Posh Foo, ne cee Os fae wie tet a Be : Bet ep Bee Cane gh. Pepes von fo: \ pase: orn Wn eae Nar een V nV Bee oe Mee Ma AVIRA Mig WO] peeesiaes Ghar oh uN NENA en SE . nad ee OT ae _ me RE a eit ESS ote oot Fig. 27. — epdl2s. -—Syminetrienl Momentary Single-phase Short Circuit of Alternator. Osciflogram of Armature Current, Armature Voltage, and Field Current. (Circuit: breaker opens.)

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of 5000-Kw. J1,000-Volt. Three-phase Alternator (atB-6-5000-500). Oscillogram of Armature Current and Field Current.

Fig. 28 shows the single-phase short circuit of a 6-polar 5000-kw. 11,000-volt steam turbine alternator, which oceurred at a point of the wave where the armature current should be not far from its maximum. The transient armature current, therefore, starts un-

DOUBLE-ENERGY TRANSIENTS. 71 ; Fig. 33.4 gives the periodic components of current and voltage: av = 140 cos 0.2% — 80 sin 0.22, e’ = 2000 cos 0.2 t 4+- 3500 sin 0.2 t. Fig, 33B gives The magnetic-energy transient, h = «‘, The dielectric-energy transient, k = «?',- And the resultant transient, Ak = 7%, And Fig. 33C gives the transient current, 7 = hki’, and the tran- sient voltage, e = hke’.

LECTURE V. SINGLE-ENERGY TRANSIENT OF IRONCLAD . CIRCUIT.

22. Usually in electric circuits, current, voltage, the magnetic field and the dielectric field are proportional to cach other, and the transient thus is a simple exponential, if resulting from one form of stored energy, as discussed in the preceding lectures. This, how- ever, is no longer the case if the magnetic field contains iron or other magnetic materials, or if the dielectric field reaches densities beyond the dielectric strength of the carrier of the field, etc.; and

the proportionality between current or voltage and their respective

fields, the magnetic and the dielectric, thus ceases, or, as it may be expressed, the inductance L is not constant, but varies with the current, or the capacity is not constant, but varies with the voltage.

The most important case is that of the ironclad magnetic cir- cuit, as it exists in one of the most important electrical apparatus, the alternating-current transformer. If the iron magnetic circuit contains an air gap of sufficient length, the magnetizing force con- sumed in the iron, below magnetic saturation, is small compared with that consumed in the air gap, and the magnetic flux, therefore, is proportional to the current up to the values where magnetic saturation begins. Below saturation values of current, the tran- sient thus is the simple exponential discussed before.

If the magnetic circuit is closed entirely by iron, the magnetic flux is not proportional to the current, and the inductance thus not constant, but varies over the entire range of currents, following the permeability curve of the iron. Furthermore, the transient . due to a decrease of the stored magnetic energy differs in shape and in value from that due to an increase of magnetic energy, since . the rising and decreasing magnetization curves differ, as shown by the hysteresis cycle.

Since no satisfactory mathematical expression has yet been found for the cyclic curve of hysteresis, a mathematical calcula- tion is not feasible, but the transient has to be calculated by an

52

LINE OSCILLATIONS. 73 , where w=5(F +4): (7) . hence the total expression of transient current and voltage is t = te“ cos (@ — ¥) } _ ul (° (8) . e = eye“ sin (6 — y) § +, €o, and 7 follow from the initial values e’ and 7’ of the transient, att = Oorg = 0: t’ = ty cosy co ees (9) ce’ =—esiny J hence et e’ tany=— ae = You (10) The preceding equations of the double-energy transient apply to the circuit in which capacity and inductance are massed, as, for instance, the discharge or charge of a condenser through an in- ductive circuit. Obviously, no material difference can exist, whether the capacity and the inductance are separately massed, or whether they are intermixed, a piece of inductanee and piece of capacity alternating, or uniformly distributed, as in the transmission line, cable, ete. Thus, the same equations apply to any point of the transmission line. a . | ‘ 1 { [+----------}----------~9} | sans EESREInD SanmERRREemmmnemmeemmnemmmpenomerernmemeenenmecne™ nee eateereeete et | A B Fig. 34. However, if (8) are the equations of current and voltage at a point A of a line, shown diagrammatically in Fig. 34, at any other point B,.at distance J from the point A, the same equations will apply, but the phase angle y, and the maximum values é9 and 2g, may be different. Thus. i t= ce“ cos ('— 0€ nut ot (? 7) (11) € = ze“ sin (6 — 7)

54. ELECTRIC DISCHARGES, WAVES AND IMPULSES. or, substituting 1 1 cB.’ = @, a? =d, gives equation (1). . . 8 .

For 3C = 0 in equation (1), 7 *; for K=0, B= : that is,

in equation (1), 27 initial permeability, : = saturation value of

‘magnetic density.

If the magnetic circuit contains an air gap, the reluctance of the iron part is given by equation (2), that of the air part is constant, and the total reluctance thus is

p=B+ ok, where 8 = a@ plus the reluctance of the air gap. Equation (1), therefore, remains applicable, except that the value of q@ is in- creased.

In addition to the metallic flux given by equation (1), a greater

. or smaller part of the flux always passes through the air or through space in general, and then has constant permeance, that is, is given by

@ = cK,

23. In general, the flux in an ironclad magnetic circuit can, therefore, be represented as function of the currert by an expression of the form

P= a + ct (3) 1+ bi 7 where ro = @/ is that part of the flux which passes through the iron and whatever air space may be in series with the iron, and cz is the part of the flux passing through nonmagnetic _material. Denoting now DL; = na 10-8 meares, Qo LT, = ne 1078, where n = number of turns of the electric circuit, which is inter- . linked with the magnetic circuit, LZ, is the inductance of the air part of the magnetic circuit, Z, the (virtual) initial inductance, that is, inductance at very small currents, of the iron part of the mag-

LINE OSCILLATIONS. (6) . Resolving the trigonometric expressions of equation (17) into functions of single angles, we get as equations of current and of voltage products of the transient «~“‘, and of a combination of the trigonometric expressions: COS > COS w, sin ¢ COS w, cos ¢ SiN w, (19) sin @ Sin w. } Line oscillations thus can be expressed in two different forms, either as functions of the suin and difference of time angle ¢ and distance angle w: (@ + w), as in (17); or as products of functions of @ and functions of w, as in (19). The latter expression usually is more convenient to introduce the terminal conditions in station- ary waves, as oscillations and surges; the former is often more ’ convenient to show the relation to traveling waves. in Figs. 35 and 36 are shown oscillograms of such line oscilla- tions. Fig. 35 gives the oscillation produced by switching 28. miles of 100-kv. line by high-tension switches onto a 2500-kw. step-up transformer in a substation at the erid of a 153-mile three- phase line; Fig. 36 the oscillation of the same system caused by switching on the low-tension side of the step-up transformer. 29. As seen, the phase of current 7 and voltage e changes pro- gressively along the line 7, so that at some distance J) current and voltage are 360 degrees displaced from their values at the starting point, that is, are again in the same phase. This distance Ip is , called the wave length, and is the distance which the electric field travels during one period f = of the frequency of oscillation. ° As current and voltage vary in phase progressively along the line, the effect of inductance and of capacity, as represented by the {inductance voltage and capacity current, varies progressively, and the resultant effect of inductance and capacity, that is, the effective inductance and the effective capacity of the circuit, thus are not the sum of the inductances and capacities of all the line elements, but the resultant of the inductances and capacities of all the line elements combined in all phases. That is, the effective inductance and capacity are derived by multiplying the total inductance and total capacity by avg/cos/, that is, by .

56 ELECTRIC DISCHARGES, WAVES AND IMPULSES. , and, subtracting (8) from (9), gives t— to = T, log —————~ + T; log = + T,3-——~- —-——;.¢. (10 = Poet thin + eT t Maa ray OO . This equation is so complex in 7 that it is not possible to cal- culate from the different values of ¢ the corresponding values of 7; but inversely, for different values of 7 the corresponding values of t can be calculated, and the corresponding values of 7 and 4, derived in this manner, can be plotted as a curve, which gives the single-energy transient of the ironclad magnetic circuit. BRR: PTT Try FEEEECCH 8 Ft tt nade | 7 ia lronclad Iiductive Circuit: t=2.92—{9.21 Ig iT + 92ligi+ wa \ 6 , | (dotted: t=1.085ig i—.507) 5 ee Ae ee Vi ttt tt Job pt tf | PS i PE ee t=1 2 3 4 5 6 seconds Fig. 29. Such is done in Fig. 29, for the values of the constants: . r=.3, . a=4xX 105, c=4X 104, b= 6, n = 300.

. ; dnd na 7 tet ryt rar LINE OSCILLATIONS. ri 7 - oo ’ a aero re wn seeing gion Re Se PIN ae 9g CUES ai 2 thet ae en ee oe Ta eb eh ra EBT et iL hat te ie PUT ean net ve ine ams CE ey ee as Jetted red TS Se) penny Sn ae Le o EOS Pa Ae on ee a ee — Hite. ty hse os eae £6 aaa UE Te ERR aE ae ar One i er a eed de aha | SC a we Nt Ted EE SIO AE Fe ee Es? SDE et = we et ee bo S cae a a eee : rrr wo OE gett Ba ro) Wa en Tate ; Toate Sep ee Rn NE OU nea . te a mee eb oe _ wn Ra a yi Sob OR ae Bite i ery Tee = le See Ge ; =n CORE NET eae lene Se mmr Ee reg Oe Le seen PRE 0p Nay Sige ese cig Bs Beak, ee. A a Se TO ae eS = ae er ar a i ean OE 2 Speed Ta SB eet 8 “eS Re UR oe. at Pte hte Ge Ee i, ° es aie! OR ee ERE | = nT baled * sacri Se tae eg pe ES Sita pe oC Ne ik ree a PS EE le i nh ae eee Sr eee ag Bee tbe OTe ea aye ae a

• ile EEE en ee ee S SAE doe Be —grereenene eer | | PS AP es eee a eT Nie ee aH he as) mo pecan 8e, e E e to Ratestae TS Toa Ses a ne Oe ° : Pr SULA OY : Da tai a a et ve Ey it get rcs nnn Aa a " ai ae kee es rs Pe TS ate BA” a ePaper ale Pe si a fede S -“ ots bag Tae Pee . 3 wo ee tert ost iret eae 2 : SNE at eh a Patten ote E er er eee ae CRE Tee ye eT See Si. “4 } wie EE Beet Ee os : WR SIT) ase py wn . MEM howe bags + : Sine EE OR TL OY ee sedan oot see ~~, -° 5 Sins a ae = OSS Bet ual ae TEN. eh! ‘ss ebb Bape ee ee e i) at Xo ee 3 ae et we o ee Oy ib Pate. 2 : Paid Sra te dng ET on Site Re ra = DETAR. Te te ~ eT ear re = we AE ates BAP Pe nn BT Blas a wi Si oe ce : ——— ms vl svi = = oh Sig svete, - re we te oOo 3 ee eS + te a a a A = OR ag ae et ete deel oe tat 2H Maree Make) CO 2 a INST OYE rem a ee Qo os aes an ae coe ~ a ene tae “ * ca aes rs ws . TM EEE Sy eT - ; we emarrcnes oS Ree Nl ate = pet Ba tay ee - : foe PE A eS) SS a ee er RE ee he ee pa cata os wn . ee TRY ter, 3 tes rg he UE See . Meo ame ST! + ve HEISE yee Se eee oS RE Ey EP mo DR ae BESET 3 er Eso are | bat Og Ree Ce me AT) ae on che wo BT pie oO os oan . . . a ‘ xv : vos . — er . = oS ami ye Ae, = Poon bah os DT ee eee <2) Set SD, sae ON perenne Scamomaomm f y o) VAL a a we Te te . . DOE Ee ame OE ea So woe Be Neb ey so oem Te, ee . vegan ere Ta TOSS ioe pgm re a we TN aes rr Pre te Ses ‘ . . wo oo mnt It . la * e ape EL ag ttle tae oe AMD er . Whine “eaves elim ay. . . cho o Tee weg Ee nape tend? os an a FT i "2
• oo. pe ey Ts T St Dae on TI en A . mk PU gO BBR Bee Po weal tre dy oo

58 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . ,

This gives ,

T; = 4, . Ts = 4.

Assuming 7) = 10 amperes for to = 0, gives from (10) the equa-

tion: , T = 2.92 — $9.21 log +. 921 logins + 4. r 1+.67 1+ 67)

Herein, the logarithms have been reduced to the base 10 by division with loge = .4343.

For comparison is shown, in dotted line, in Fig. 29, the transient of a circuit containing no iron, and of such constants as to give about the same duration: ;

t = 1.085 log!©t — .507.

As seen, in the ironclad transient the current curve is very much steeper in the range of high currents, where magnetic sat- uration is reached, but the current change is slower in the range of medium magnetic densities. °

Thus, in ironelad transients very high-current values of short duration may occur, and such transients, as those of the starting

current of alternating-current transformers, may therefore be of .

serious iniportance by their excessive current values.

An oscillogram of the voltage and current waves in an 11,000-kw. high-voltage 60-cycle three-phase transformer, when switching onto the generating station near the most unfavorable point of the wave, is reproduced in Fig. 30. As seen, an excessive current rush persists for a number of cycles, causing a distortion of the volt- . age wave, and the current waves remain unsymmetrical for many

cycles, 7

LINE OSCILLATIONS. 79 . The frequency f depends upon the length J; of the section of line in which the oscillation occurs. That is, the oscillations occurring ; , ° in a transmission line or other circuit of distributed capacity have no definite frequency, but any frequency may occur, depending on the length of the circuit section which oscillates (provided that this circuit section is short compared with the entire length of the circuit, that is, the frequency high compared with the frequency which the oscillation would have if the entire line oscillates as a whole). If l, is the oscillating line section, the wave length of this oscilla- tion is four times the length lo = 4h. (27)

• This can be seen as follows: , At any point | of the oscillating line section 4, the effective power po = avg et = 0 (28) is always zero, since voltage and current are 90 degrees apart. oy The instantaneous power p= et, (29) : . however, is not zero, but alternately equal amounts of energy flow | first one way, then the other way. ' Across the ends of the oscillating section, however, no energy can flow, otherwise the oscillation would not be limited to this section. Thus at the two ends of the section, the instantaneous power, and thus either e or 27, must continuously be zero. Three cases thus are possible:

1. e = 0 at both ends of ,;
2. • = 0 at both ends of 4;
3. e = 0 at one end, 7 = 0 at the other end of J. ; In the third case,z = 0 at one end, e = 0 at the other end of the line section 4, the potential and current distribution in the line section J; must be as shown in Fig. 37, A, B, C, etc. That is, l; must be a quarter-wave or an odd multiple thereof. If [, is a three-quarters wave, in Fig. 37B, at the two points C and D the power is also zero, that is, 1; consists of three separate and independent oscillating sections, each of the length : ; that is, the