Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients — part 4 of 7

60 ELECTRIC DISCHARGES, WAVES AND IMPULSES. : where C = capacity = coefficient of energy storage by the volt- . age, in the dielectric field, and g = conductance = coefficient of power consumption by the voltage, as leakage conductance by the voltage, corona, dielectric hysteresis, etc. Thus the transient of the spontaneous discharge of a condenser would be represented by _2 € = ee ce (4) Similar single-energy transients may occur in other systems, For instance, the transient by which a water jet approaches con- stant velocity when falling under gravitation through a resisting medium would have the duration v where vy = limiting velocity, g = acceleration of gravity, and would be given by t . v=n(1—e r) (6) ’ In a system in which energy can be stored in two different forms, as for instance as magnetic and as dielectric energy in a circuit containing inductance and capacity, in addition to the gradual decrease of stored energy similar to that represented by the single-energy transient, a transfer of energy can occur between its two different forms. Thus, if ¢ = transient current, e = transient voltage (that is, the difference between the respective currents and voltages exist- ing in the circuit as result of the previous circuit condition, and the values which should exist as result of the change of circuit conditions), then the total stored energy is Ti? , Ce? W =a =a? , 273 (7) = W mn +W. While the total energy W decreases by dissipation, W,, may be converted into Wz, or inversely. ° Such an energy transfer may be periodic, that is, magnetic energy may change to dielectric and then back again; or unidirectional, that is, magnetic energy may change to dielectric (or inversely, dielectric to magnetic), but never change back again; but the

LINE OSCILLATIONS. 8 thus is 1 ly = vlog = af’ (32) and, substituting (82) into (81), gives | Qa = a9, (33) or 1 1 ny 34 Fo Vino (4)

This gives a very important relation between inductance Lo and capacity Co per unit length, and the velocity of propagation.

It allows the calculation of the capacity from the inductance,

Co = pL’ (35) and inversely. As in complex overhead structures the capacity . usually is difficult to calculate, while the inductance is easily de- tived, equation (35) is useful in calculating the capacity by means of the inductance.

This equation (35) also allows the calculation of the mutual — capacity, and thereby the static induction between circuits, from the mutual magnetic inductance.

The reverse equation,

1 Ly = PC,’ (36) ; is useful in calculating the inductance of cables from their meas- ured capacity, and the velocity of propagation equation (13).

31. If J, is the length of a line, and its two ends are of different electrical character, as the one open, the other short-circuited, and thereby 7 = 0 at one end, e = 0 at the other end, the oscilla- tion of this line is a quarter-wave or an odd multiple thereof.

The longest wave which may exist in this circuit has the wave length 1p = 41;, and therefore the period to = oolo = 4 ooh, that is, the frequency fo = Tol, , i This is called the fundamental wave

01 of oscillation. In addition thereto, all its odd multiples can exist as higher harmonics, of the respective wave lengths —— 7 and the frequencies (2k — 1)fo, wherek = 1,2,3...

62 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The maximum transient voltage can thus be calculated from the maximum transient current: €o = lo vi = 920, (10) and inversely, _ To = €9 vg = €oYo- (11)

This relation is very important, as frequently in double-energy transients one of the quantities é9 or 2) is given, and it is impor- tant to determine the other.

For instanee, if a line is short-circuited, and the short-circuit current 7 suddenly broken, the maxiinum voltage which can be induced by the dissipation of the stored magnetic energy of the short-circuit current is €9 = Zo.

If one conductor of an ungrounded cable system is grounded, the maximum momentary current which may flow to ground is To = €oYo, Where ey = voltage between cable conductor and ground.

If lightning strikes a line, and the maximum voltage which it

— may produce on the line, as limited by the disruptive strength of the line insulation against momentary voltages, is eo, the maximum discharge current in the line is limited to 7) = eoYo.

If L is high but C low, as in the high-potential winding of a high-voltage transformer (which winding can be considered as a eircuit of distributed capacity, inductance, and resistance), Zo is high and yo low. That is, a high transient voltage can produce only moderate transient currents, but even a small transient cur-

. rent produces high voltages. Thus reactances, and other reactive apparatus, as transformers, stop the passage of large oscillating currents, but do so by the production of high oscillating voltages.

Inversely, if L is low and C high, as in an underground cable, zo is low but yo high, and even moderate oscillating voltages pro- duce large oscillating currents, but even large oscillating currents produce only moderate voltages. Thus underground cables are little liable to the production of high oscillating voltages. This . is fortunate, as the dielectric strength of a cable is necessarily relatively much lower than that of a transmission line, due to the close proximity of the conductors in the former. A cable, therefore, when receiving the moderate or small oscillating cur- rents which may originate in a transformer, gives only very low

LINE OSCILLATIONS. 83 usually are more conveniently resolved into the form of equa- tion (19). At extremely high frequencies (2 k —1)f, that is, for very large . . values of k, the successive harmonics are so close together that a very small variation of the line constants causes them to overlap, and as the line constants are not perfectly constant, but may . vary slightly with the voltage, current, etc., it follows that at very high frequencies the line responds to any frequency, has no definite frequency of oscillation, but oscillations can exist of any frequency, provided this frequency is sufficiently high. Thus in long trans- mission lines, resonance phenomena can occur only with moderate . frequencies, but not with frequencies of hundred thousands or millions of cycles. , 32. The line constants 79, go, Lo, Co are given per unit length, as per cm., mile, 1000 feet, ete. The most convenient unit of length, when dealing with tran- sients in cireuits of distributed capacity, is the velocity unit »v. That is, choosing as unit of length the distance of propagation in unit time, or 3X 10” em. in overhead circuits, this gives v = 1, and therefore __ co = VLlr = 1: (39) LoCo = 1, j or Comps le=o That is, the capacity per unit of length, in velocity measure, is inversely proportional to the inductanee. In this velocity unit of length, distances will be represented by x. Using this unit of length, oo disappears from the equations of the transient. This velocity unit of length becomes specially useful if the transient extends over different circuit sections, of different con- stants and therefore different wave lengths, as for instance an overhead line, the underground cable, in which the wave length is . about one-half what it is in the overhead line (« = 4) and coiled windings, as the high-potential winding of a transformer, in which the wave length usually is relatively short. In the velocity measure of length, the wave length becomes the same throughout all these circuit sections, and the investigation is thereby greatly simplified.

64 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . must be zero when the current is a maximum, and inversely; and if the current is represented by the cosine function, the voltage thus is represented by the sine function, that is, e = ey si (6 — 7), (15) where é; = —eosin y = initial value of transient voltage. (16) The frequency f is still unknown, but from the law of propor- tionality it follows that there inust be a frequency, that is, the suc- cessive conversions between the two forms of energy must occur in equal time intervals, for this reason: If magnetic energy converts to dielectric and back again, at some moment the proportion be- tween the two forms of energy must be the same again as at the starting moment, but both reduced in the same proportion by the power dissipation. From this moment on, the same cycle then must repeat with proportional, but proportionately lowered values. MUR TNA ME a ae es id ae vom oR bo aed ted . i { nor Ce, Sn 7k ee Fo ee _ | ye MOO : ed Se . = re Fig. 31.—cp10017.— Oscillogram of Stationary Oscillation of Varying Frequency: Compound Circuit of Step-up Transformer and 28 Miles of 100,000-volt Transinission Line. If, however, the law of proportionality does not exist, the oscil- lation may not be of constant frequency. Thus in Fig. 31 is shown an oscillogram of the voltage oscillation of the compound circuit oe consisting of 28 miles of 100,000-volt transmission line and the 2500-kw. high-potential step-up transformer winding, caused by switching transformer and 28-mile line by low-tension switches off a substation at the end of a 153-mile transmission line, at 88 kv. With decreasing voltage, the magnetic density in the transformer

LINE OSCILLATIONS. 85 beginning of time, that is, for @¢ = 0, and by the values of 7 and e at all times ¢ (or @ respectively) at the ends of the circuit, that is, for w = 0 and w = 5- , For instance, if: (a) The circuit is open at one end w = 0, that is, the current is zero at all times at this end. That is, . 1 = 0 for w = 0; the equations of 7 then must not contain the terms with cos w, cos 2 w, etc., as these would not be zero for w = 0. That is, it must be : Qa, = Q, by = 0, ag = 0, be = Q, (43) a3 = 0, bs = 0, ete. The equation of 2 contains only the terms with sin w, sin 2 w, etc. Since, however, the voltage e is a maximum where the current tis zero, and inversely, at the point where the current is zero, the voltage must be a maximum; that is, the equations of : the voltage must contain only the terms with cos w, cos 2 w, etc. Thus it must be c =0, d’ =0, . Cy’ = Q, d,’ = 0, (44) . 7 ; Ca’ => Q, ds’ = 0, ete. Substituting (48) and (44) into (42) gives t= e"'! fe, cos¢ + d,sin ¢} sin w, " (45) e = « Sa,’ cos¢ + by’ sind} cosw $ and the higher harmonics hereof. (b) If in addition to (a), the open circuit at one end w = 0, the line is short-circuited at the other end w = 5) the voltage e must be zero at this latter end. Cos w, cos3 w, cos5, etc., become zero for w = 5 but cos 2 w, cos 4 w, ete., are not zero for o = 5 and the latter functions thus cannot appear in the expres- sion of e.

66 ELECTRIC DISCHARGES, WAVES AND IMPULSES. as the expression of the frequency of the oscillation, where ¢o=VIiC (20) is a convenient abbreviation of the square root. ‘The transfer of energy between magnetic and dielectric thus — occurs with a definite frequency f = x , and the oscillation thus ‘is a sine wave without distortion, as long as the law of proportion- ‘ality applies. When this fails, the wave may be distorted, as seen on the oscillogram Fig. 31.

The equations of the pertodie part of the transient can now be written down by substituting (13), (19), (14), and. (16) into (12) and (15):

Zt = ip cos (6 — ¥) = i cos y cos d + iy) Sin y sin @ . i ty . = tl, COS - — @, —SIN-» C €y oC and by (11):

• t . € 7 = 1 cos= — yor Sin=, (21) and in the same manner: t .. € ; € = €,cos- + Zgti Sin -) (22) oT o where e, is the initial value of transient voltage, 7, the initial value of- transient current.

B. Power dissipation. ;

27. In Fig. 32 are plotted as A the periodic component of the oscillating current 7, and-.as B the voltage e, as C the stored mag- —

. 2 2 netic energy = , and as D the stored dielectric energy ce :

As seen, the stored magnetic energy pulsates, with double frequency, 2f, between zero and a maximum, equal to the total stored energy. The average value of the stored magnetic energy thus is one-half of the total stored energy, and the dissipation of magnetic energy thus occurs at half the rate at which it would occur if all the energy were inagnetic energy; that is, the transient resulting from the power dissipation of the magnetic energy lasts twice as long as it would if all the stored energy were magnetic, or in other words, if the transient were a single (magnetic) energy

. LINE OSCILLATIONS. 87 In these equations (50), d and a’ are the maximum values of’ current and of voltage respectively, of the different harmonic waves. Between the maximum values of current, 7», and of volt- age, é, of a stationary oscillation exists, however, the relation €o0 _ L io = a/E, where Z is the natural impedance or surge impedance. That is a’ = dzo, (51) and substituting (51) into (50) gives t=e" Sd sn d sn wtadssins¢@sndwt+d;sind5¢ sin 5 w € = zye*t fd cos ¢ cos w+ d3 cos 3d cos 38 w+d; cos5 ¢ cosd w +... f, (d) If then the distribution of voltage e along the circuit is given at the moment of start of the transient, for instance, the voltage is constant and equals e, throughout the entire circuit at the starting moment ¢@ = 0 of the transient, this gives the relation, by substituting in (52), . é, = me" J$d,cosw+d;cos8w+td;cosiwt... }, (58) for all values of w. . Herefrom then calculate the values of di, ds, d;, etc., in the manner as discussed in ‘‘ Engineering Mathematics,’”’ Chapter ITI.

68 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ; as only half the energy is dielectric, the dissipation is half as rapid, that is, the dielectric transient has the duration 2 Te =27) = 26, (24) and therefore adds the factor : k=ae 7? to the equations (21) and (22). While these equations (21) and (22) constitute the periodic part of the phenomenon, the part which represents the dissipa- tion of power is given by the factor t t 1 1 hk =e Tie Ting '(T tT), (25) The duration of the double-energy transient, 7, thus is given by 1 1,1 . T~ Tt Tr’ (26) 2\T. ' To) and this is the harmonic mean of the duration of the single-energy magnetic and the single-energy dielectric transient. It is, by substituting for J) and Ty’, 1 l/r g\ _ ae @7) where wu is the abbreviation for the reciprocal of the duration of the double-energy transient. Usually, the dissipation exponent of the double-energy transient _1/fr .g u=5 (; + #) | is given as a ; 2L° | This is correct only if g = 0, that is, the conductance, which rep- resents the power dissipation resultant from the voltage (by leak- age, dielectric induction and dielectric hysteresis, corona, etc.), is negligible. Such is the case in most power circuits and trans- mission lines, except at the highest voltages, where corona appears. It is not always the case in underground cables, high-potential .

TRAVELING WAVES. 89 physical meaning a wave has, in which current and voltage are in phase with each other:

t = %pe7*! cos (¢ Fw Y), (4) e = ee“! cos (¢ Fw — 7). In this case the flow of power is p = e&, = €yioe 2%! cos? (6 F w — ¥), = fe [1 + cos 2 (6 Fw — y)I, (5) and the average flow of power is Po = ave Pp, = S ent, (6) Such a wave thus consists of a combination of a steady flow of power along the circuit, 70, and a pulsation or surge, 1, of the same nature as that of the standing wave (2): p—i = eo cos 2 (¢ Fw — 7). (7) Such a flow of power along the circuit is called a traveling wave. It occurs very frequently. For instance, it may be caused if by a lightning stroke, etc., a quantity of dielectric energy is impressed 1\A 7| \B ° v 4 C 4 D 6 ee Fig. 39. — Starting of Impulse, or Traveling Wave. upon a part of the circuit, as shown by curve A in Fig. 39, or if by a local short circuit a quantity of magnetic energy is impressed upon a part of the circuit. This energy then gradually distributes over . the circuit, as indicated by the curves B, C, etc., of Fig. 39, that is, moves along the circuit, and the dissipation of the stored energy thus occurs by a flow of power along the circuit.

70 ELECTRIC DISCHARGES, WAVES AND IMPULSES. , : T, = 28 = 0.001 sec. = 1 millisecond, T; = 7 = ().0005 sec. = 0.5 millisecond, T= wy = 0.000333 sec. = 0.33 millisecond; TT | eit tit i ty te sen a \4
2090-4 —100)-- - }
Pale ariteieane HAA FARHAN Vv TY pA LNG fin KA LN Ze NZ Miv/aan ee Coors SEES a ef Pr CREE COC E EEE ef AT LT [tush [PLP || . “TX NOLL EEE two VE Net Te | Tt ACT NT Se AGEL S=seF]-£ |_| ne KIA TE} ttt tt 7 i= eR Pte TET tt EE EE EEE | | Fig. 33. | hence, substituted in equation (28), ~ = e140 cos 0.2¢ — 80 sin 0.2 é}, ? e = e~*4{2000 cos 0.2 ¢ + 3500 sin 0.2 ¢}, j where the time ¢ is given in milliseconds.

ES. 91 , TRAVELING WAVES. : ag | . i - . mt ; e : 7 3 . moe 7. ae me , _ 2 TE . Dot : a , ee, . og

• i he bE 7 a a - 2g ao 4 ee 32 a ‘en = > . . Q ok nn tn . . Poy, eo . ces 2 8 lon y ee’ é — eo a Sa . . . , " . . sae . 2 “9
• ., . ° . ce we . 7 3 7 rs ae BO as “oe oy on EN ; —_ & ; _ es . Ez . : co. ' A a : — we o . Y. i ee Ba . on ae Coen -? . : = & z eer ae ae | Wo . S 4 re _ AS s ; Tom eet - SS & Tl SoS . Set ~ &

LECTURE VII. LINE OSCILLATIONS. 28. In a circuit containing inductance and capacity, the tran- sient consists of a periodic component, by which the stored energy . Li? . . Ce? ; surges between magnetic > and dielectric ce and a transient: component, by which the total stored energy decreases. Considering only the periodic component, the maximum value of magnetic energy must equal the maximum value of dielectric ; energy, Liz Ce?

2 3" Af) where 7) = maximum value of transient current, ¢ = maximum value of transient voltage.

This gives the relation between ¢ and %, oo /L_ a1 in VO™ Oy’ @) where 2» is called the natural impedance’or surge impedance, 7 © the natural or surge adinittance of the circuit. . As the maximum of current must coincide with the zero of voltage, ancl inversely, if the one is represented by the cosine function, the other is the sine function; hence the periodic com- ponents of the transient are 1, = %) cos (p — ¥) t (3) €1 = eosin ( — y) J’ where ¢ = 2zft, (4) and f= 6) 2a VLC is the frequency of oscillation. The dissipative or “ transient ” component is hk = «~*, (6) 72

TRAVELING WAVES. 93 , the rate e~“‘, corresponding to the dissipation of the stored energy by e~“‘, as indicated by A’ in Fig. 42; while in the case (6) the power flow decreases faster, in case.(c) slower, than corresponds to the energy dissipation, and is illustrated by B’ and C’ in Fig. 42. (a) If the flow of power is constant in the direction of propa- gation, the equation would be <= toe"! cos (6 — w — ¥), € = ee"! cos (6 —w — Y), (9) po = ett In this case there must be a continuous power supply at the one end, and power abstraction at the other end, of the circuit or circuit section in which the flow of power is constant. This could occur approximately only in special cases, as in a circuit section of medium rate of power dissipation, u, connected between a section of low- and a section of high-power dissipation. For instance, if as illustrated in Fig. 43 we have a transmission line Line a = Line Fig. 43. -- Compound Circuit. connecting the step-up transformer with a load on the-step-down end, and the step-up transformer is disconnected from the gener- ating system, leaving the system of step-up transformer, line, and load to die down together in a stationary oscillation of a compound circuit, the rate of power dissipation in the transformer then is much lower, and that in the load may be greater, than the average rate of power dissipation of the system, and the trans- former will supply power to the rest of the oscillating system, the __ load receive power. If then the rate of power dissipation of the line « should happen to be exactly the average, uo, of the entire system, power would flow from the transformer over the line into the load, but in the line the flow of power would be uniform, as the line neither receives energy from nor gives off energy to the rest of the system, but its stored energy corresponds to its rate of power dissipation.

74. ELECTRIC DISCHARGES, WAVES AND IMPULSES. are the current and voltage at the point A, this oscillation will appear at a point B, at distance 2 from A, at a moment of time later than at A by the time of propagation ¢, from A to B, if the oscillation is traveling from A to B; that is, in the equation (11), instead of é the time (f — é:) enters. Or, if the oscillation travels from B to A, it is earlier at B by the _ time é,; that is, instead of the time ¢, the value (¢ + ¢:) enters the equation (11). In general, the oscillation at A will appear at B, and the oscillation at B will appear at A, after the time 4; that is, both expressions of (11), with (@ — ¢,) and with (¢ + 4), will . oceur. The general form of the line oscillation thus is given by substi- tuting (¢ = 4) instead of ¢ into the equations (11), where ¢, is the time of propagation over the distance I. If v = velocity of propagation of the electric field, which in air, as with a transmission line, is approximately v= 3 X 10%, (12) and in a medium of permeability » and permittivity (specific capacity) « is ; ; xX 10! aint | . 13 wa (13) and we denote 1 a= ?? (14) then i = al; (15) and if we denote 2 aft; = w = 2zfal, (16) we get, substituting ¢ =F ¢, for t and ¢ = w for ¢ into the equation (11), the equations of the line oscillation: i = ce"! cos (6 F w — ¥) € = zoce“t sin (6 = w — 7) (17) In these equations, . : o = 2nft is the time angle, and (18) wo = 27fal is the space angle, andc = c+“ is the maximum value of current, zc the maximum value of voltage at the point 1.

TRAVELING WAVES. 95 the average power then is Do = avg el, = eoty eT 2 (um st g-2sh Colo eT 2 ul et2s(t—d) _ (12) ; 2 2 Both fornis of the expressions of 7, e, and po of equations (11) and (12) are of use. The first form shows that the wave de- creases slower with the time ¢, but decreases with the distance X. The second form shows that the distance » enters the equation only in the form ¢ — \ and ¢ — w respectively, and that thus for a constant value of t — A the decrement is e?“, that is, in the direction of propagation the energy dies out by the power dissi- pation constant w. Equations (10) to (12) apply to the case, when the direction of propagation, that is, of wave travel, is toward increasing i. For a wave traveling in opposite direction, the sign of \ and thus of w is reversed. (c) If the flow of power increases along the line, more power leaves every line clement than enters it; that is, the line clement is drained of its stored energy by the passage of the wave, and thus the transient dies down with the time at a greater rate than corre- sponds to the power dissipation by r and g. That is, not all the stored energy of the line elements supplies the power which is being dissipated in the line element, but a part of the energy leaves the line element in increasing the power which flows along the line. The rate of dissipation thus is increased, and instead of u, (wu +s) enters the equation. That is, the exponential time decrement is et se (13) but inversely, along the line \ the power flow increases, that is, the intensity of the wave increases, by the same factor e+®, or rather, the wave decreases along the line at a slower rate than corresponds to the power dissipation. . The equations then become: T= lye UT)! ET COS (H— w— y) = pew" 8 “—®) COS (P-—wW—Y), l (4) €= eye (UtS)t eT COS (— w—Y) = eoe~“E~8"—®) COS (P—w—y), J and the average power is Do = SO eB ut a et2a = Pent en? 8(E A), (15)

— ( a] spay ' al ALG - “he * re (0 ELECTRIC DISCHARGES, WAVES AND IMPULSES. oe o A ", 7 - " cee) 2 vee tom, * ae ar . ves a gh hPa eye pa ee de eee eke ee te at lea fea Re om nS EECA SE SC OLDIE apy” So a Oe OR EC CH B pt UN ete ke Bae Be ae 7 eee Eee ene es a en ee Q ° Re tee Bo ae ae oe TN 5 iw Bote . et ee a — ene SE a eT he et eo mg Ro gen et I Tn OE et ge ges q ee . se ot RA er ae 5 RS SE aT Nats wet Stewart ls weby * Boma cacy get By PE ee UE engi SOE mae URIS Sta fe er he Ba a i pal Fats a wo he tet OR alo - ECC aR SST EE RI RS SOO SS roe Bete Bei a eT Oe etic Eh, EA Ss a se Se ta OTE ee Bl ae OO GR + ES 2 OR at A AD ah aati Be tee teeta a eee er a wan wv y M te “iy er ear, aA Set ee EA 2 Sey ME ee RE te ea - <I PR ee Tyaty pa Sage fecha Sa Ta oe weet ee Ce aoe. Str. eset Ta ey o Sia ss PR A OS, RR A Z Le yen St a reat ea ee > ca oe I SR St Ae Re Sa = ee Ee a Pe a ae eee eye iy AST Ran ae G4 Mc sar ee ths Che ie an Me Se, OBE As 2 BR ee a F rr Se Fe ee, tis A om) cre ea SP eget PP Sete Tae Ler Min IB ne ee 8d er eT Tee ape VER So Pen Pi 32 4 OE See Prt p Boe ne Eee et Feb: ah eee stay Wok, ee a ny “ pe a, woe ROSS came aie ee ie De [>] 3 EE tu OM The te LY Saeed SANE EEE SP ET ae > Mee a Eee ga Ss geet at . & Re ee I ee es s Se en I at ee ty ° OT ee NEAL Fe Se err weg gi OS OU Lie el fo 2 gg mitt inten eae ag. BYRD, So = Te Be ee cage OPE DS Ce Sa ene Te RAE ORE RET] A a a OR SS OE y ee OR Bea UN te Mite aca Ge tate f et MAP I Ma Ee eee , Bree gated we a ge a ey ey ot ere Meyer ea Dae ae RP Oc te oy PRL IN PTR A ese ° nN a “ Oe Ath ee . SS Rte tg ae, gee ek. wt ee Me te He D tae moe: TOR ITE Mee T ee eh le rn at ged a ES Pe ofa m . eet th, a oo * we tLe pepe Min tees et hee to Sane Pr eer ae wi ee Pua ate un eit! Bes ere ee Nea ita SE i) “Ee SWE SL Ae ee a a ee LE Cote as ug Tee Le dete Oe T= wp Bete ae og RST UTE aR So aT oe Senate EES es Py Pt = me Sreha TM hE ga E g ae Soap ee ES ee ra pe a PR Oa . * Seer de Maia, ee eR oP Na ae et ae garnet a eS wath geet ”n Ser ete ek a yt TOES eg PEL ee tg ee PR Oo ri ae a ts Oe eT int oS Sy pet era GO cee MES Gil HE ee STU LEOI TS ET a bee peCNnSe cee = Rees eR Sie Ste ete a gS eg Oo ee! Dee ot a we a ns BT Sir eke et a ae Ts ODE Bo ee SPs. ro Be ea sin cece” gt Sets aes oes TE NEE A a a PE aie? ro) UO reg ee ge te . operat. : pee AE gg Sey See ua gt is aes cate ete i TS bt Na dee The oi ae ayes NCD Se CL St De ee ate Tet OURS ops ONE. Ee en ore Se pe ENE EES. ¢ Sgn EY Ae SRLS Or er ee wate alee ore tte ERE. ° their ER a Se Da ee a Po ge So Spent et come by RE. om aera Toe a Se bk ete ® Sa rs pe ie ree — agit eee we bus Pas ASS? —_ apthac n Stat ts SY Ip oe ct OE meme OP a era eb a: 3 PI Ea Be Der be SM se Sn pene PT Re RD Ue ES = Seta Cg Bank a ETE bt Sa Mbeki: Be ied OW SSS ee BA eee RE Spy Gy MO eames ae PEM Cae BD ed PN I ee Oo soy OUR ones eh ek Um a ae yee ee SE ae Pte a ee PES SEIS geet See 2 TD SEE, quae MEE TEE aS aE TEE ep . aA og eb te gether 8 Go a re ™ teeny ee SN de fe er a aes meen renee Se eee . See ten sg OS i=} CL Se SAU aie Be eet we RE eR Lee cd = yee ae ES a re fo care pheasew rere es ty ee, Sf foie tf ey > DD ete a eo ce ge Etc ee Te Ee ta ADD A a UN Fey pm SUN ee — an a ee ea a OR CR PS CO 3 por a te ES peemee amen FE a aaa DD sr US a ee ns - a rs tae ee . ner _ em ae a eL Chew t TOW Qe hss tee Tt Eg emmmmeammrnsen: EET Steerer ee a TT peng es disk pipe tenets Mad toe US gee eee ne dete Se eit a) . ee ED ee he TPE le eames Sy les Sei yy SS Tree eee ED Get Qh ged De OS a lee tate ae PT ae - phage: Nae 6 ae fo Nae ts g ere " mT am ad Os oe bes ere. eS Cag arya : ene To wR ae TRE cE nl te Sey ke ee ae nee lal eee BE ste sep oe eee Tee OE So pemete ke oet tee BL ly STS UP Nah aiden Selo ete ee ata, Me UP tase 3 ie ate Et ee eae SE inte Simm ce ge Tet ns . at TE IRS prem SR ao snes Sab Ba olism meetepetce Atty ieee ot eee cee ee BB Beta ee ea TE loa en Te el nt aS) CEE en ee gig SE Be Gg ge ob Aare = ee I ee SR oe tT ra TAT EE a 2 pe yer end. wa ty 2. ‘ es See eS et eek OE eT aa oo ra on Stile eyarte “a. . ro Mat a So: Ba re ao sat . ae nt ee . aaa 3 RB : I MO SA pS io) ree Te mg Nae ar seem PO Net re LIM nS poe Y. ena a a er TTS OO . t; these, Parr Re MY ee See ee Te fa de Oty Ble ¢ ne - an os mrt” FoF ae =, tee an : “ = . oa ar : pas ve rane “oie © opis ee. rt i Sal See wen TE S . ca . wh te . rer. * ¥ ott te we emer were whee mot POR ata ta Te Te eT vas paren i=] ote . et rn cree) ott fhe oman Ip me Det Peat ONT e : hae ee A rr Beever wel etn § a oversee . we . . a ‘ue ar aaa met eh eee Nae se coe) rer . ee A ey slo or Gliae era ants ee “ a we fe Bat ey “ue . See QT eke ariel o ; . ‘ & es

. . TRAVELING WAVES. 97 increase with the time, which in general is not possible; as the transient must decrease with the time, by the power dissipation ; “in r and g. Standing waves and traveling waves, in which the coefficient in the exponent of the time exponential is positive, that is, the wave increases with the time, may, however, occur in electric cir- cuits in which the wave is supplied with energy from some outside source, as by a generating system flexibly connected (electrically) through an are. Such waves then are “cumulative oscillations.” They may cither increase in intensity indefinitely, that is, up to destruction of the circuit insulation, or limit themselves by the power dissipation increasing with the increasing intensity of the oscillation, until it becomes equal to the power supply. Such oscillations, which frequently ure most destructive ones, are met in electric systems as “arcing grounds,’ “grounded phase,” etc. - They are frequently called “undamped oscillations,” and as such find a use in wireless telegraphy and telephony. Thus far, the only source of cumulative oscillation seems to be an energy supply over an are, especially an unstable arc. In the self-limiting cumu- : lative oscillation, the so-called damped oscillation, the transient becomes a permanent phenomenon. Our theoretical knowledge of the cumulative oscillations thus far is rather limited, however. An oscillogram of a “grounded phase” on a 154-mile three- phase line, at 82 kilovolts, is given in Figs. 44 and 45. Fig. 44 shows current and voltage at the moment of formation of the ground; Fig. 45 the same one minute later, when the ground was fully developed. ’ An oscillogram of a cumulative oscillation in a 2500-kw. 100,000- volt power transformer (60-cycle system) is given in Fig. 46. It is caused by switching off 28 iniles of line by high-tension switches, at 88 kilovolts. As seen, the oscillation rapidly increases in in- tensity, until it stops by the are extinguishing, or by the destruc- _ tion of the transformer. Of special interest is the limiting case, —s=uU; in this case, u-+s = 0, and the exponential function of time vanishes, and current and voltage become 1 = Ipe=* cos ( F w — Ys 2 € = ere=" cos (f F w — 7), J (18)

78 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . Instead of L and C, thus enter into the equation of the double- ¥% energy oscillation of the line the values 22 and 2. ’ In the same manner, instead of the total resistance 7 and the * total conductance g, the values cat and =f appear. The values of 2, yo, u, ¢, and w are not changed hereby. The frequency f, however, changes from the value correspond- . _ : 1 ing to the circuit of massed capacity, f = ———==, to the value yf 22% VLC 1 I= TVie Thus the frequency of oscillation of a transmission line is . 1 1 ; = — = 20 Se Te’ 29) where _ o = VLC. (21) If 1, is the length of the line, or of that piece of the line over which the oscillation extends, and we denote by Lo, Co, To, Jo (22) the inductance, capacity, resistance, and conductance per unit length of line, then i | _1(T 2); that is, the rate of decrease of the transient is independent of the ° length of the line, and merely depends on the line constants per , unit length. It then is = hoo, (24) where do = VLCo (25) is a constant of the line construction, but independent of the length of the line. The frequency then is 1 =. 2 f 4 hoo ( 6) % ERRGR IN STEMMETS ANAC ESS | FUR MSDS, Z BEcom és ° £4. a & 2 Le a ae

' - 4 7 99 TRAVELING WAVES. ne ae ET ce ee pps wee age teas 18 Le ee Mar Sa se AE vee os ae Be Mg Loe. 2 ASK Peg ge . eo ae ee STN Bo EE comely PRE PRS A 3] Cr Se ee et . gt ees nein EEF Gok" an aren we Pe ane a a rai: ce ee EE & foc eet eae St ee Re Bamberg i el on 3 peek rita On + a Lobe eT Dashes. eee eee = . flo wt er pe ae aS Ste ie, ° soe tl base : Me og tae SE Ge haa) car - oer ” g latency iets Senn Eh. i] ne we ep poe a we mmm EE aS Veg OE =| ew. Cot se Pekan eae te Sapte eth encase need ee | Lea a da SD ep eR = IEEE eee ae Ln chejenacneat ae wwe Ta a the = SRL tector Et) Sat aaa a PS CE RS Ce poe MR a TE Es 2 me roe ee A wa SN a8 a ee A meee ESR ane ee eo ae ee sco ne ae ha at Tyce TER ee SO a I bs} er Ta a pa came em TN eel te aphed obo ns actementgnmanen EME 4 2 7 a ahaa) omen p sete "Nae wee tapeeetngieit spat tebe Pore, :. os ‘ oe Dut tassel en a) Bate cmd entnowe eps ct ~ E-4 on wo ean -_ we we ete a ee pte bee! cee er ee . mE EM SE id) RD eT SD Seer scene eae Se pt 8 tempat Regt ge cate set geeg. TTT Byes ro > 2. en, eet MEE eS oom pe Ons So cguet eo Cap akin ee ate = 5 ata nr Son ee a has a STA Sg er Pte eS emma wee! ‘ Rone ei hen en oe . fee fT Nepal east eS caneeaets oF Ltn ets a ] eT St eel ED ——— ee yg ate atey OE ot t 2 “oe on La a’ ale . . . a al i SC — ee TEE ee ~~, : a Oo = PN SP Lap eee ete a + : EEE ls scat ae a So 8 Spence an rete RET tr Ae “ ope Repeats One a ard : le a AEE T CP aCe SRT Ee ls ep vets tae BF a En SR TAC ro Eo! Whe PY ww | “ OD awe NBR os ra ee et, ES Rss te Pin ey ET EE a S ag - fe acorns ease owt TTS” Regt Tadeo tay Py jen wage ENT ck PES, Lt mene De ane, Rian Doan a tig ol a Saran, YS eg Ny ea ahah oT aS aot es pemmne By a ae 8 oo Oe ORE rare Ug me le eats eT re a = a re ee) ae - me oR ee I SS ae tee — oo 2 ns ’. Sr Dette ta epee Hae PRE i Bl ete in, ET ee etge nae tin omlee aoe ne ce mreatnpmetng sont oo aeh sect tse ade HF — aa eae - ap ee EE OU Ta SET oe GEER eens FT — Loe mee, res onc an SRNR CaARCiLO ONE | eg a tet cee rx TS ee Fe TAS ea a 5 PE TE gD MEE Nees os aan, Vale re A 2 = ce ne 2 aC “OT ce MPs TY ~ Rn ale as samme ee ROT as Se ae eR Ee he © 2 2 SOs a nS - — eae aT Oe SRO OE RERL Lee TSS a OE eet gas o peg iat act nd He NE a & Rat yg eal ET npr ite Sy cae gant ogame Eee SE tg “x Se ee Ee ES Se cna SE IE eee et ie = s Le dap esate ee a Trini LA eee a coal an! eT OO Nema 2 ONS ELS att BF tne ON TE ge B LP EES oe Ramen Oe a TE ane rn eR r=] a ee ee ae eT Ly aE Se ORS om" a ait aes Tr heehee! ww yee, iv} shoul / gee ever a Pye Sh PIR IR time Nor fe et S Aa ae -='O Veal neh 2 Seen rg MT Oe i peep BES = > SORE Se ee EEE . ree ee mente eg ER Te ge Oo - a Coe eerie, z. is eet awiy! seat eg Ge feo Lota typos. oe anne, Laem La a SD eh aera om ot poet tye vee Oe SHE A a a . ee EI ore ee a = 86 te meme wd TNE OT . ! See nT ee te v= “a i ry eee nes ek ee - 5 fae! Vas ne a anaeenr ons fans) a sierra -_ ate OTE: Fo eh D. a S aE ae nee te BO be et ee ete Dg jo) hp ite AE . yprerr see et Ta te tre _ fF so name — wee ee © “= : Ea sree me te RE > iin. Tee memes eT 7 ee ILE tee ‘S a . teal! : : : ee lt te te OR EUR oO 7 ar . ee a ME wr es 3 — . he Teal eared Be eee = ont : es , a age did yee Oe gis os buat: eS pe UL SI ON a) ' ream ot. ” aan ge TF a Rt 5 ey rr ae . we Pe Birr s stg dey wo a ne AR ge ee > 2 pores ———=—_ SEES SS OA Fa an SO vege me gh OT we ae a — ae) fe Oe . : ey eee Te re eM tie Uf eon eR ath ge go, i ee eT Trae, 8 =< + Wipe gg wees ee wt : = re a Ly SEARS 5 ro . are aS ee ag Ta Bed he gt gh zs a fe Peat get acho wee PS ete ee eee ke . ee Ta tl ead Ct nt Ta PALE tre es ee aye &) aS a Ta or Sr a Bi epee ft RS SLA Sh ey S i=) oe Re Eee onl oy EB Tc Eire CRS eee = 9 tT SES gg pee 2a ab ee a OT m= 2 iit ete cee, foe ae ee I gr Be oS 2 WAR ta ee a ye mp ne TE ees OE a Brae a om OCP Witan gv, Th bes ar = Te Te eee ; eed Te RE Sa OG 2 we Ee a Tata PREIS, (o) fan} wr FERPA, ST es TT ee eT. ee ee CM ET Dolce ot PQ Oar ee Toy, Pea en et tas, EES By alee gee teins oe Se LG SES te ee 1 CES TR tre | DS pet pee ee mare Lee es are We ta ae na] ee Se Ee SUM on TE eh a I ee : fy oe SAS we ge St ee al TE. at Re ae — pe ete Se Sui ee ean an ge] ey eee a es ° ces, SS SO Re en en ae Bieter ee EN eT NES 2S We yg ERD Lael OF a a a — sete SD te Bed 22 Se . . aan 7 i me et te : 6 OO enh fete SS ~ fy as we ng Pt ate eA te pemhaataen Te. = ae ee ne PO Si ens ee ee Sol tr TL ke" Btls sya ut ree ee So. ee ne Pa Dy ens - ge OE ey Oe pat ar pee Tete ae phe Mage Mone oU eh a Yt Zz on asin tr Stent tes Soe Pee. Bite TS ee Twas SOP ep Ba ee ey See - Me ET ee Taha et BRUT toe ee See Te . ° . = ee ae 5 eR OT a a at epee as oaae Mime te Man ea ~~ he eM seo US Tote cme TER la e “ re ee pr i ee Soa ead ete * Soon oer at *” a eee Peer wag re 4 eR PL egy US Ca le Map eg, Se a wet ata CUE a btn Mae NOTE te, rears oe Sas oe: eM anes SPE ts i Se Ty

80 -BLECTRIC DISCHARGES, WAVES AND IMPULSES. unit of oscillation is a, or also a quarter-wave. The same is the ease in Fig. 37C, ete.

In the case 2, 7 = O at both ends of the line, the current and voltage distribution are as sketched in Fig. 38, A, B, C, ete.

That is, in A, the section J; is a half-wave, but the middle, C, of l, is a node or point of zero power, and the oscillating unit again is a quarter-wave. In the same way, in Fig. 38B, the section J; consists of 4 quarter-wave units, etc.

. 1 —_ i H ' i Poe fo he Al os | Al i we ! H i ! ~~) I jaa BB) A Cis, Bi i | i { SAR { se H ate tt ™ aT OK | F~ Ly a4 ! { | “AL I . ae | | 17 “J Bi 1 ~ 17 i BR $ ri BI TaN EL“ Bi i ! tS } | hr ‘ ! 1 s+? 1 i i ! ' } 1 ; I a \ ! ! I ’ ; cr | a | /* | cr . | , ran ' 1 1 EI / F! nN I | El’ FI H A CKD PB ORAS A SRS “LL , S4- “4 1 b i | Fig. 37. Fig. 38.

The same applies to case 1, and it thus follows that the wave length lo is four times the length of the oscillation J.

30. Substituting I) = 41, into (26) gives as the frequency of oscillation

1 f= ino (30) |

However, if f = frequency, and v = :, velocity of propagation,

the wave length ly is the distance traveled during one period: ly = ; = period, , (31)

TRAVELING WAVES. 101 When traveling waves and stationary waves occur simultane- ; ously, very often the traveling wave precedes the stationary wave. The phenomenon may start with a traveling wave er impulse, and this, by reflection at the ends of the circuit, and -combination

• of the reflected waves and the main waves, gradually changes-to a . stationary wave. In this case, the traveling wave has the sainc frequency as the stationary wave resulting from it. In Fig. 47‘is shown the reproduction of an oscillograin of the formation of a stationary oscillation in a transmission line by the repeated re- jo Fig. 47. — cp11168. — Reproduction of an Oscillogram of Stationary Line Oscillation by Reflection of Impulse from Ends of Line. (The lowest curve gives a 60-cycle current as time measure.) flection from the ends of the line of the single impulse caused by short circuiting the energized line at oneend. In the beginning of astationary oscillation of a compound circuit, that is, a circuit com- prising sections of different constants, traveling waves frequently occur, by which the energy stored magnetically or dielectrically in . the different circuit sections adjusts itself to the proportion cor- responding to the stationary oscillation of the complete circuit. Such traveling waves then are local, and therefore of much higher frequency than the final oscillation of the complete circuit, and thus die out at a faster rate. Occasionally they are shown by the oscillograph as high-frequency oscillations intervening between

82 ELECTRIC DISCHARGES, WAVES AND IMPULSES. a . If then ¢ denotes the time angle and w the distance angle of the fundainental wave, that is, ¢ = 27 represents a complete cycle and w = 27 a complete wave length of the fundamental wave, the time and distance angles of the higher harmonies are 39, 34, . 5, Sa, 7%, 7 w, ete. A complex oscillation, comprising waves of all possible fre- quencies, thus would have the form COS (@ F w — 1) + a3 eos 3 (6 F w — 43) +a, cos5 (@Fw—ys)+..., (37) and the length i; of the line then is represented by the angle o= 5 and the oscillation called a quarter-wave oscillation.