lf the two ends of the line 4, have the same electrical charac- teristics, that is, e = 0 at both ends, or 7 = 0, the longest possible wave has the length Jo = 2 d;, and the frequency

1 1 fo= ole” Dads’ or any multiple (odd or even) thereof. If then @ and w again represent the time and the distance angles of the fundamental wave, its harmonics have the respective time and distance angles 2¢, 2a, ; 3 ?, 3 Ww, * 4, 4, ete. . A complex oscillation then has the form a; COS (o Fao 71) + az cos 2 (¢ For 2) +azscos3(@Fw—y)+..., (38)

and the length J; of the line is represented by angle w; = 7, and the

oscillation is called a half-wave oscillation.

The half-wave oscillation thus contains even as well as odd harmonics, and thereby may have a wave shape, in which one half wave differs from the other.

Equations (37) and (88) are of the form of equation (17), but

TRAVELING WAVES. es Te BS ROO Be Sa rn Var © S Bore z MCE! 77) (6 . ay. 5 = Bets . ; wo TA oe! rc Dot ° oe BE me BR de Be ao ns Ss Ow ae Co Bere oF an rr ro) 3 rere; Clr rr ne As 7 weet ; . a Zz n Cte . _ ony Fa Ne 2 _— oe ot vot es ; a a re 2 co mes = Bo an , re xo en I wo . : 3 pe ty ae . ee oO 10 ny — # ee o 3 on . Cel . ne a; Je rae S . ea ee “3 ts chy oy a > a ae a <- ae 23 rn | Cee ete — i ee weemet * vos Sa eo ig ; my a . a a a arr S St a m be . ep 2S 7 a me eT . + ee Sut . a ao 7 rr rl a Vc > hot Ce * } >: ee st So So ee 8 8 ra - E me oa Gee &E A a > ena bo "Se ee eee ee a horn RS ao) IRA Ch ee EE Sg Jann wo 2 Be eeweeess) © of ao ve a ro & 2 ne . po 7 2 we, ; _ i = = . .7 ee. : 8 ~ mo, ‘ ie ie = S NO: o A ey ” +. a rer . Tree fa oo oe ag we EEE ee S elie en) © 5 Sa ea wa Tee : wig Beg 2 mm So re Dn ony le os A Te or = vo gee ES =o ; a on S “oe ve * gk Sab Se ae so 2k a ¢ woe . Py ie 2g phe cane I S — ce, or) Soa et Bg aa 8 ra La oe a > 8 ce, ro g mt ee, ee ener ~ a . ‘= pet ‘ x eee Se to

• & & ns | Ocoee o be . SS BS oo, ey aa ae —Seeen| oc —=— = Ire AEE te Ce! = wa + me ~ oy ery Tet ge are) = woe oS Q. ae - eet ne caees n a “a, wot at eau eer = a a c> Ea _ = BE: cre (jo) a naar aS ,. Sm pies ane o 3 = wn, = S op ae — BAB 8 on — ae ee Toa i ~~ me ro) “= soe RP o:s “ ~s = ee So . Gos. weet = 3 | O--, iin ; To S —_> > ~ . 4 et yes et - a an Meee ; ot ew ; | > — > ae >: ee Se | | SS haere) = = . . : =

84 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . . Substituting gp = 1 in equations (30) and (31) gives tg = Ao, 1 f= x do w= aan 22, Ao : and the natural impedance of the line then becomes, in velocity measure, Lo 1 1 €o ° Co ° Co Yo 7) ( ) where @é9 = maximum voltage, 79 = maximum current. That is, the natural impedance is the inductance, and the natural admittance is the capacity, per velocity unit of length, and is the main cliaracteristic constant of the line. The equations of the current and voltage of the line oscillation then consist, by (19), of trigonometric terms COS ¢ COS a, sin ¢ COS w, cos ¢ sin w, sin ¢ sin a, multiplied with the transient, e~“, and would thus, in the most general case, be given by an expression of the form , i= e—“Jfa,cos¢cosw+bhsin¢gcosw+ccos¢sinw +d.sin¢gsin w}, (42) e=e«—“{a,’cos¢ cos w +b,’ sing cos w + ¢’ cos ¢ sin w +d,'sin ¢sin w}, and its higher harmonics, that is, terms, with 2¢, 2u, 39, 3a, 44, 4a, etc. In these equations (42), the coefficients a, b, c, d, a’, b’, c’, d’ . are determined by the terminal conditions of the problem, that is, by the values of 7 and e at all points of the circuit w, at the |

TRAVELING WAVES. 105 Such simple traveling waves frequently are called “7zmpulses.” When such an impulse passes along the line, at any point of the line, the wave energy is zero up to the moment where the wave front of the impulse arrives. The energy then rises, more or less rapidly, depending on the steepness of the wave front, reaches a maximum, and then decreases again, about as shown in Fig. 50. The impulse thus is the combination of two waves, ; Vig. 50. — Traveling Wave. one, which decreases very rapidly, e~“+*)4, and thus preponder- ates in the beginning of the phenomenon, and one, which decreases slowly, e~“—5)4. Hence it may be expressed in the form:

• Do = ye Tet sll eh2em 4 (ge 24 - s)bg—2ar, (20) where the value of the power-transfer constant s determines the “ steepness of wave front.” Figs. 51 to 53 show oscillograms of the propagation of sueh an mS impulse over an (artificial) transmission line of 130 miles,* of the constants: r = 93.6 olims, L = 0.3944 henrys, . C = 1.135 microfarads, ; /L / thus of surge impedance zy = \ a= 590 ohms. The impulse is produced by a transformer charge. . Its duration, as measured from the oscillograms, is JT’) = 0.0036 second, . In Fig. 51, the end of the transmission line was connected to a noninduetive resistance equal to the surge impedance, so as to

• For description of the line see ‘‘ Design, Construction, and Test of an Arti- ficial Transmission Line,” by J. H. Cunningham, Proceedings A.LLE.E., January,

{ In the manner as described in “ Disruptive Strength of Air and Oil with Transient Voltages,” by J. L. R. Hayden and C. P. Steinmetz, Transactions A.I.E.E., 1910, page 1125. The magnetic energy of the transformer is, however, larger, about 4 joules, and the transformer contained an air gap, to give constant inductance. .

86 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ; That is, the voltage e can contain no even harmonics. If, however, the voltage contains no even harmonics, the current ; produced by this voltage also can contain no even harmonics. That is, it must be Co = 0, do = QO, aa’ => 0, be" = 0, y= 0, dy = 0, ag’ = 0, b,’ = 0, (46) cs = 0, ds=0, as = 0, &’ = 0, ete.

The complete expression of the stationary oscillation in a circuit open at the end w = 0 and short-circuited at the end w =5 thus would be 2 =e“! S(c, cos 6 + dsin ¢) sin w + (c3 cos3.¢ + d3sin 3 ¢)

sndw+... f, (47) e=e—"! S(a,’ cosd6+by’ sin d) cos w+ (a3’ cos 3 6 + b3’ sin 3 ¢) cos8a@+... f.

(c) Assuming now as instance that, in such a stationary oscilla- tion as given by equation (47), the current in the circuit is zero at the starting moment of the transient for @ = 0. Then the equation of the current can contain no terms with cos ¢, as these would not be zero for @ = 0.

That is, it must be

qQ = 0,

cs = 0, (48)

c, = 0, ete. : At the moment, however, when the current is zero, the voltage

of the stationary oscillation must be a maximum. Asi = 0 for @ = 0, at this moment the voltage e must be a maximum, that is, the voltage wave can contain no terms with sin ¢, sin 34, etc. This means

by’ = 0, )

b;’ = 0, (49)

bs’ => 0, etc. j

Substituting (48) and (49) into equation (47) gives . t=e“ fd, sin¢dsnw+d;sin3¢sin30+d;,sn5¢sin5 w e = «"! fa)’ cos ¢ cos w+a;’ cos3 ¢cos3w+a; cos5¢ cos hw

+... 4.

TRAVELING WAVES. 107 give no reflection. The upper curve shows the voltage of the impulse at the beginning, the middle curve in the middle, and the lower curve at the end of the line. Fig. 52 gives the same three voltages, with the line open at the

end. This oscillogram shows the repeated reflections of the vol- tage impulse from the ends of the line,—the open end and the transformer inductance at the beginning. It also shows the in- crease of voltage by reflection.

| ,__$§_

| i, |

Vig. 538. —cp11153. — Reproduction of Oscillogram of Propagation of Im-

pulse Over Transmission Line; Reflection from Open End of Line.

Current.

Fig. 53 gives the current impulses at the beginning and the mid-

dle of the line, corresponding to the voltage impulses in Fig. 52, together with the primary current of the transformer, 7. This oscillogram shows the reversals of current by reflection, and the formation of a stationary oscillation by the successive reflections . of the traveling wave from the ends of the line.

LECTURE VIII. TRAVELING WAVES. | 33- In a stationary oscillation of a circuit having uniformly distributed capacity and inductance, that is, the transient of a circuit storing energy in the dielectric and magnetic field, current and voltage are given by the expression i = ine“ cos (6 F w — ), (1) e = ee“ sin (¢ F w — ¥); where ¢ is the time angle, w the distance angle, u the exponential - decrement, or the “power-dissipation constant,” and % and é) the maximum current and voltage respectively. . The power flow at any point of the circuit, that is, at any dis- tance angle w, and at any time ¢, that is, time angle ¢, then is p = él, = €pioe “ cos (6 F w — ¥) sin (¢ Fw— ¥), . = Ferm sin 2 (6 w — 7), | (2) and the average power flow is Po = avg p, (3) = 0.

Hence, in a stationary oscillation, or standing wave of a uni- form circuit, the average flow of power, po, is zero, and no power flows along the circuit, but there is a surge of power, of double frequency. That is, power flows first one way, during one-quarter cycle, and then in the opposite direction, during the next quarter- cycle, ete.

Such a transient wave thus is analogous to the permanent wave- of reactive power.

As in a stationary wave, current and voltage are in quadrature with each other, the question then arises, whether, and what

88

OSCILLATIONS OF THE COMPOUND CIRCUIT. 109 power-dissipation constant of the circuit, and u that of any section, this section must have a second exponential time decrement, . § = Up — U, (2)

which represents power transfer from the section to other sections, | or, if s is negative, power received from other sections. The oscil- lation of every individual section thus is a traveling wave, with a power-transfer constant s. ;

As wy is the average <issipation constant, that is, an average of the power-dissipation constants u of all the sections, and s = wy — u the power-transfer constant, some of the s must be positive, soine negative.

In any section in which the power-dissipation constant w is less than the average uo of the entire circuit, the power-transfer con- stant s is positive; that is, the wave, passing over this section, in- creases in intensity, builds up, or in other words, gathers energy, which it carries away from this section into other sections. In any seetion in whieh the power-dissipation constant u is greater than the average wy of the entire circuit, the power-transfer con- stant s is negative; that is, the wave, passing over this section, decreases in intensity and thus in energy, or in other words, leaves some of its energy in this section, that is, supplies energy to the section, which energy it brought from the other sections.

By the power-transfer constant s, sections of low energy dissi- pation supply power to sections of high energy dissipation.

39. Let for instance in Fig. 43 be represented a circuit consist- ing of step-up transformer, transmission line, and load. (The load, consisting of step-down transformer and its secondary cir- cuit, may for convenience be considered as one circuit section.) Assuie now that the circuit is disconnected from the power sup- ply by low-tension switches, at A. This leaves transformer, line, and load as a compound oscillating circuit, consisting of four sections: the high-tension coil of the step-up transformer, the two lines, and the load.

Let then A, = length of line, A: = length of transformer circuit, and 3 = length of load circuit, in velocity measure.* If then

• If 1, = length of circuit section in any measure, and LZ, = inductance, Cy = capacity per unit of length 1, then the length of the circuit in velocity measure is Ay = gol1, Where oo = V TnCu. ;

Thus, if LD = induetanee, C = capacity~per transformer coil, n = number of transformer coils, for the transformer the unit of length is the coil; hence the

90 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ; Such a flow of power must occur in a circuit containing sec- tions of different dissipation constants u. For instance, if ina ~ circuit consisting of an unloaded transformer and a transmission line, as indicated in Fig. 40, at no load on the step-down trans- _ Line 2 Transformer : : a 2 _ : Fig. 40. ° former, the high-tension switches are opened at the generator end of the transinission line. The energy stored magnetically and © dielectrically in line and transformer then dissipates by a transient, as shown in the oscillogram Fig. 41. This gives the oscillation : . of a circuit consisting of 28 miles of line and 2500-kw. 100-kv. step-up and step-down transformers, and is produced by discon- necting this circuit by low-tension switches. In the transformer, the duration of the transient would be very great, possibly several . . seconds, as the stored magnetic energy (L) is very large, the dis- sipation of power (r and g) relatively small; in the line, the tran- - sient is of fairly short duration, as r (and. g) are considerable. . Left to themselves, the line oscillations thus would die out much . inore rapidly, by the dissipation of their stored energy, than the transformer oscillations. Since line and transformer are connected together, both must die down simultaneously by the same tran- ‘sient. It then follows that power must flow during the transient from the transformer into the line, so as to have both die down together, in spite of the more rapid energy dissipation in the line. Thus a transient in a compound circuit, that is, a circuit comprising sections of different constants, must be a traveling wave, that is, inust be accompanied by power transfer between the sections of the circuit.* A traveling wave, equation (4), would correspond to the case of effective power in a permanent alternating-current circuit, while the stationary wave of the uniform circuit corresponds to the case of reactive power. Since one of the most important applications of the traveling wave is the investigation of the compound circuit, it is desirable —

• In oscillogram Fig. 41, the current wave is shown reversed with regard to the voltage wave for greater clearness.

OSCILLATIONS OF THE COMPOUND CIRCUIT. 111 in the load: p = pre, the energy of the wave decreases rapidly. . Here the cuefficients of 7, pe, ps must be such that the wave at the beginning of one section has the same value as at the end of the preceding section.

In general, two traveling waves run around the circuit in opposite direction.

Each of the two waves reaches its maximum intensity in this circuit at the point where it leaves the transformer and enters the line, since in the transformer it increases, while in the line it again decreases, in intensity.

, en ererernet 0 1 = 900 4, W=100 Ag LL = 90 As AY OO = 1600 i ) . Step/up Transmission Lme Transformer Transmission Line 1.5 X10-3 1.0/x10-3 1,5 x 10-3 5x IQ We / = +700 S =-10 §=-800
U,= 800 Fig. 54. — Isnergy Distribution in Compound Oscillation of Closed Circuit; High Line Loss.

Assuming that the maximum value of the one wave is 6, that of the opposite wave 4 inegawatts, the power values of the two waves then are plotted in the upper part of Fig. 54, and their difference, that is, the resultant flow of power, in the lower part of Fig. 54. As seen from the latter, there are two power nodes, or points over which no power flows, one in the transformer and‘one in the load, and the power flows from the transformer over the line into the

~ load; the transformer acts as generator of the power, and of this

92 ELECTRIC DISCHARGES, WAVES AND IMPULSES. to introduce, when dealing with traveling waves, the velocity unit as unit of length, that is, measure the length with the distance of propagation during unit time (8 X 10!° cm. with a straight con- ductor in air) as unit of length. This allows the use of the same distance unit through all sections of the circuit, and expresses the wave length A» and the period 7’) by the same numerical values, ho = To= and makes the time angle @ and the distance angle w directly comparable: ¢$=2nft=2r LY do x (8) w= 2r— = 2d. Xo

34. If power flows along the circuit, three cases may occur:

(a) The flow of power is uniform, that is, the power remains constant in the direction of propagation, as indicated by A in. Fig. 42.

C RK. a Sof Chao OY, a ee B ON SS Srtte se 1 Pe ae betes Cc . eee wee A! ~~ s ——>] Fig. 42. — Energy Transfer by Traveling Wave.

(6) The flow of power is decreasing in the direction of propaga- tion, as illustrated by B in Fig. 42.

(c) The flow of power is increasing in the direction of propaga- tion, as illustrated by C in Fig. 42.

Obviously, in all three cases the flow of power decreases, due to the energy dissipation by r and g, that is, by the decrement «+, Thus, in case (a) the flow of power along the circuit decreases at

OSCILLATIONS OF THE COMPOUND CIRCUIT. W3 That is, the power-transfer constant of the line has become posi- tive, s, = 33, and the line now assists the transformer in supplying power to the load. Assuming again the values of the two travel- - , ing waves, where they leave the transformer (which now are not the maximum values, since the waves still further increase in intensity in passing over the lines), as 6 and 4 megawatts respec- tively, the power diagram of the two waves, and the power dlia- gram of their resultant, are given in Fig. 55. Step up , Ul = 500 1 = 100 UL = 500 t1=1600 ——>- Trans sions Line Trai Wests Transinissioly ae 5 x3 S=-1067 U,= 533 , Vig. 55. — Energy Distribution in Compound Oscillation of Closed Circuit; Low Line Loss. , In a closed circuit, as here discussed, the relative intensity of the two component waves of opposite direction is not definite, but depends on the circuit condition at the starting moment of the transient. In an oscillation of an open compound circuit, the relative intensities of the two component waves are fixed by the condition that at the open ends of the circuit the power transfer must be . zero. As illustration inay be considered a circuit comprising the high- potential coil of the step-up transformer, and the two lines, which are assumed as open at the step-down end, as illustrated diagram- ; matically in Fig. 56.

94 ELECTRIC DISCHARGES, WAVES AND IMPULSES,

(b) If the flow of power decreases along the line, every line clement receives more power at one end than it gives off at the other ‘end. That is, energy is supplied to the line elements by the flow of power, and the stored energy of the line element thus decreases at a slower rate than corresponds to its power dissipation by rand g. Or, in other words, a part of the power dissipated in the line element is supplied by the flow of power along the line, and only a part supplied by the stored energy.

Since the current and voltage would decrease by the term e~“¢, if the line clement had only its own stored energy available, when receiving energy from the power flow the decrease of current and voltage would be slower, that is, by a term

ete sit; (10) hence the exponential decrement wu is decreased to (u — s), and s then is the exponential coefficient corresponding to the energy supply by the flow of power. . ; Thus, while wu is called the disszpation constant of the circuit, s may be called the power-transfer constant of the circuit.

Inversely, however, in its propagation along the circuit, 4, such a traveling wave must decrease in intensity more rapidly than corresponds to its power dissipation, by the same factor by which it increased the energy supply of the line elements over which it passed. That is, as function of the distance, the factor e~ * must enter.* In other words, such a traveling wave, in passing along the line, leaves energy behind in the line elements, at the rate

. e+st and therefore decreases faster in the direction of progress by e~®. That is, it scatters a part of its energy along its path of travel, and thus dies down more rapidly with the distance of travel. ‘

Thus, in a traveling wave of decreasing power flow, the time decrement is changed. to e~“~ 4, and the distance decrement e+ added, and the equation of a traveling wave of decreasing power flow thus is 1 =e (Ut E—™ COS (H— w— Y) = toe “ETS &—® Cos (6—w—Y), (11) e = ee He“ CoS (P6— wW— ¥) = ee“ ETSE- CoS (6 —w—Y);

• Due to the use of the velocity unit of length d, distance and time are given the same units, & = Ao; and the time decrement, e+ st, and the distance decrement, e-*, give the same coefficient s in the exponent. Otherwise, the velocity of propagation would enter as factor in the exponent.

OSCILLATIONS OF THE COMPOUND CIRCUIT. 115 as tu give the same rate of energy dissipation in all circuit sections. As the result of this power transfer, the stored energy of the system inust be uniformly distributed throughout the entire circuit, and if it is not so in the beginning of the transient, local traveling waves redistribute the energy throughout the oscillat- ’ ing circuit, as stated before. Such local oscillations are usually of very high frequency, but sometimes come within the range of the oscillograph, as in Fig. 47. During the oscillation of the complex circuit, every circuit clement dd (in velocity measure), or every wave length or equal part of the wave length, therefore contains the same amount of stored energy. That is, if eg = niaxinium voltage, tp = maximum ; , EotoAg current, and XA» = wave length, the average energy > must be constant throughout the entire circuit. Since, however, in velocity measure, dp is constant and equal to the period 7p through- out all the seetions of the cireuit, the product of maximum voltage and of maximum current, eo’y, thus must be constant throughout the entire cireuit. The same applies to an ordinary traveling wave or impulse. Since it is the same energy which moves along the circuit at a constant rate, the energy contents for equal sections of the circuit must be the same except for the factor e~?"4, by which the energy decreases with the time, and thus with the distance traversed during this time. Maximum voltage ey and maximum current %, however, are related to each other by the condition. €o /Lo a = lig = wz) 3 | ip VG @) : and as the relation of Lo and C> is different in the different sections, and that very much so, 2, and with it the ratio of maximum voltage to maximum current, differ for the different sections of the circuit. If then e; and % are maximum voltage and maximum current : . . I, . . , respectively of one section, and 2 = vz is the “natural imped- 1 . ance”’ of this section, and e, 72, and z. = yz are the correspond- 2 ing values for another section, it is ; cle = €il1, (4)

96 ELECTRIC DISCHARGES, WAVES AND IMPULSES. that is, the power decreases with the time at a greater, but with the distance at a slower, rate than corresponds to the power dissipation. . For a wave moving in opposite direction, again the sign of X ' and thus of w would be reversed.

35. In the equations (10) to (15), the power-transfer constant s is assumed as positive. In general, it is more convenient to assume that s nay be positive or negative; positive for an increas- ing, negative for a decreasing, flow of power. The equations (13) to (15) then apply also to the case (}) of decreasing power flow, but in the latter case s is negative. They also apply to the case (a) for s = Q.

The equation of current, voltage, and power of a traveling wave

then can be combined in one expression: = Lge UTS € = Cos (P-Fw—y) = tye “em §F MOS (GF w—Y), ( e= ee *TNE=% COS (PF w— 7) =eoe™ “9 F™ Cos (PF w—7), | (16) yo = ap ent tal ered = Se ule ~28(t FA) , (17) where the upper sign applies to a wave traveling in the direction 7 toward rising values of A, the lower sign to a wave traveling in opposite direction, toward decreasing 4. Usually, waves of both directions of travel exist simultaneously (and in proportions de- pending on the terminal conditions of the oscillating system, as the values of i and e at its ends, etc.).

s = 0 corresponds to a traveling wave of constant power flow (ease (a)).

s > 0 corresponds to a traveling wave of increasing power flow, that is, a wave which drains the circuit over which it travels of some of its stored energy, and thereby increases the time rate of dying out (case (c)).

s < 0 corresponds to a traveling wave of decreasing power flow, ; that is, a wave whieh supplies energy to the circuit over which it travels, and thereby decreases the time rate of dying out of the transient.

If s is negative, for a transient wave, it always must be

. —s =u, since, if — s > u, u+s would be negative, and «~ (+4! would increase with the time; that is, the intensity of the transient would

OSCILLATIONS OF THE COMPOUND CIRCUIT. 117 42. At the transition point between two successive sections, | the current and voltage respectively must be the same in the oe two sections. Since the maximum values of current and voltage respectively are different in the two sections, the phase angles of the waves of the two sections must be different at the transition point; that is, a change of phase angle occurs at the transition point. This is illustrated in Fig. 58. Let zo = 200 in the first section ' (transmission line), 29 = 800 in the second section (transformer). The transformation ratio between the sections then is 300 =2; , that is, the maximum voltage of the second section is twice, and the maximum current half, that of the first section, and the waves of current and of voltage in the two sections thus may be as illustrated for the voltage in Fig. 58, by e1ee. | Fig. 58. — Effect of Transition Point on Traveling Wave. If then e’ and 7’ are the values of voltage and current respec- . tively at the transition point between two sections 1 and 2, and - ° e: and % the maximum voltage and maximum current respec- ° tively of the first, e, and 7 of the second, section, the voltage phase : , and current phase at the transition point are, respectively: : For the wave of the first section: , “F : © = cos 1 and = =cos 5}. a (9) For the wave of the second section: , t © = cos Y2 and ” = cos 6e. 2 ce)

104 ELECTRIC DISCHARGES, WAVES AND IMPULSES. circuit and measuring the voltage across the inductance by spark , gap. These traveling waves of very high frequency are extremely | local, often extending over a few hundred feet only.

An approximate estimate of the effective frequency of these very high frequency local traveling waves can often be made from their striking distance_across a simall inductance, by means of the relation = = ve = Zo, cliscussed in Lecture VI. _

0 0 For instance, in the 100,000-volt transmission line of Fig. 484, the closing of the high-tension oil switch produces a high-frequency . oscillation which at a point near its origin, that is, near the switch, jumps a spark gap of 3.3 cm. length, corresponding to e: = 35,000 volts, across the terminals of a small inductance consisting of 34 turns of 1.3 cm. copper rod, of 15 cm. mean diameter and 80 cm. length. The inductance of this coil is calculated as approxi- mately 13 mierohenrys. The line constants, from line to neutral,

are L = 0.3823 henry, C = 2.2 x 10-° farad; hence z = Vz = V0.1465 X 10? = 383 ohms.

The sudden change of voltage at the line terminals, produced by closing the switch, is ae = 57,700 volts effective, or a maximum of e) = 57,700 X V 2= 81,500 volts, and thus gives a maximum transient current in the impulse, of to =2= 212

0 amperes. 2% = 212 amperes maximum, traversing the inductance of 13 microhenrys, thus give the voltage, recorded by the spark gap, of e: = 35,000. If then f = frequency of impulse, it is Q = 2 rfLio. =, Or, t= 555i _ 35,000 ~ 27 X13 X 107-8 K 212 = 2,000,000 cycles.

37. A common form of traveling wave is the discharge of a | local accumulation of stored energy, as produced for instance by a direct or induced lightning stroke, or by the local disturbance eaused by a change of circuit conditions, as by switching, the blowing of fuses, etc.

100 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

that is, are not transient, but permanent or alternating currents

and voltages.

Writing the two waves in (18) separately gives

t= tet cos (6 — w — 1) — o'e" * (P+ w—72), 2 19) e = eet ® cos (6 — w — 71) + ey'e~™ (6 + w—), J (

and these are the equations of the alternating-current transmission

line, and reduce, by the substitution of the complex quantity for

the function of the time angle ¢, to the standard form given in

“Transient Phenomena,” Section IIT.

36. Obviously, traveling waves and standing waves may occur simultaneously in the same circuit, and usually do so, just as in alternating-current circuits effective and reactive waves occur simultaneously. In an alternating-current circuit, that is, in . permanent condition, the wave of effective power (current in phase with the voltage) and the wave of reactive power (current in quadrature with the voltage) are combined into a single wave, in which the current is displaced from the voltage by more than 0 hut less than 90 degrees. This cannot be done with transient waves. The transient wave of effective power, that is, the travel- ing wave,

1 = igem “eS cos (6 Fw — ¥), @ = epee“ eS EN Cos (PW Fw—y), eannot be combined with the transient wave of reactive power, that is, the stationary wave, t= tye“ cos (6 Fw —y’), e=ee “sin (@Fw— 7’), to form a transient wave, in which current and voltage differ in phase by more than 0 but less than 90 degrees, since the traveling wave contains the factor e«~¢%, resulting from its progression along the circuit, while the stationary wave does not contain this factor, as it does not progress.

This makes the theory of transient waves more’ complex than . that of alternating waves.

Thus traveling waves and standing waves can be combined only locally, that is, the resultant gives a wave in which the phase angle - between current and voltage changes with the distance \ and with the time ¢. .

LECTURE X. _ CONTINUAL AND CUMULATIVE OSCILLATIONS.

43. A transient is the phenomenon by which the stored energy readjusts itself to a change of circuit conditions. In an oscilla- tory transient, the difference of stored energy -of the previous and the after condition of the circuit, at a circuit change, oscillates between magnetic and diélectrie energy. As there always must be some cnergy dissipation in the circuit, the oscillating energy of the transient must steadily decline, that is, the transient must die out, at a rate depending on the energy dissipation in the cir- cult.

Thus, the oscillation resulting from a change of circuit condi- tions ean become continual, that is, of constant amplitude, or cumulative, that is, of increasing amplitude, only if a steady supply of oscillating energy occurs.

Continual and cumulative oscillations thus involve a con- tinual energy supply to the oscillating system, therefore cannot be mere readjustments of circuit conditions by the dissipation of stored energy.

If the continual energy supply is less than the energy dlissipa- tion in the circuit, the oscillation dies out, that is, is transient, but with a lowered: attenuation constant. This for instance is the ease with the transient in those sections of a compound circuit, in which the energy transfer constant is negative. If the con- - tinual energy supply equals the energy dissipation, the oscillation is continual, and if the energy supply is greater than the energy dissipation, the oscillation becomes cumulative, that is, increas- ing in amplitude, until either the system breaks down or, by the increase of the energy dissipation, it becomes equal to the energy supply, and the oscillation becomes continual.

A continual or cumulative oscillation thus involves an energy . and frequency transformation, from the low-frequency or con- tinuous-current energy of the power supply of the system to the high-frequency energy of the oscillation.

119 :

106. BLECTRIC DISCHARGES, WAVES AND IMPULSES. I ——> Fig. 51. —cn11145.— Reproduction of Oscillogram of Propagation of Impulse Over Transmission Line; no Reflection, Voltage. — Fig. 52.—cp11152. — Reproduction of Oscillogram of Propagation of Im- pulse Over Transmission Line; Reflection from Open End of Line. Voltage.

hod | . LECTURE IX. OSCILLATIONS OF THE COMPOUND CIRCUIT. 38. The most interesting and most important application of the traveling wave is that of the stationary oscillation of a com- : pound circuit, as industrial circuits are never uniform, but consist of sections of different characteristics, as the generating system, transformer, line, load, ete. Oscillograms of such circuits have been shown in the previous leeture. If we have a circuit consisting of sections 1, 2,3... , of the , respective lengths (in velocity measure) Ay,» As... 4 this entire circuit, when left to itself, gradually dissipates its stored energy by a transient. As function of the time, this transient — must deerease at the same rate uv» throughout the entire circuit. Thus the time decreinent of all the sections must be : 7 em tt, Every section, however, has a power-dissipation constant, w, 22, uz3 ... , Which represents the rate at which the stored energy of the section would be dissipated by the losses of power in the section, eT Mil eT Mat eT ust |, But sinee as part of the whole cireuit each section must die down at the same rate «", in addition to its power-dissipation a , decrement e™, 74 . . . , each section must still have a second tine deerement, «~ %—")!, e~@o—m)t | + | This latter does not represent power dissipation, and thus represents power transfer. That is, §; = Ug — Uy, S. = Up — Ue, (1) It thus follows that in a compound circuit, if wo is the average exponential time decrement of the complete circuit, or the average 108

112. ELECTRIC DISCHARGES, WAVES AND IMPULSES. power a fraction is consumed in the line, the rest supplied to the load.

40. The diagram of this transient power transfer of the system ; thus is very similar to that of the permanent power transmis- sion by alternating currents: a source of power, a partial con- sumption in the line, and the rest of the power consumed in the load. However, this transient power-transfer diagram does not repre- sent the entire power which is being consumed in the circuit, as power is also supplied from the stored energy of the circuit; and the case may thus arise — which cannot exist in a permanent power transmission — that the power dissipation of the line is less than corresponds to its stored energy, and the line also supplies power to the load, that is, acts as generator, and in this case the power would not be a maximum at the transformer terminals, but would still further increase in the line, reaching its maxi- mum at the load terminals. This obviously is possible only

• with transient power, where the line has a store of energy from : which it can draw in supplying power. In permanent condition the line could not add to the power, but must consume, that is, cate the permanent power-transmission diagram must always be like Fig. 54. Not so, as seen, with the transient of the stationary oscillation. Assume, for instance, that we reduce the power dissipation in the line by doubling the conductor section, that is, reducing the resistance to one-half. As LZ thereby also slightly decreases, C increases, and g possibly changes, the change brought about in the constant u = 3(z + 5) is not necessarily a reduction to one- half, but depends upon the dimensions of the line. Assuming therefore, that the power-dissipation constant of the line is by the doubling of the line section reduced from wu = 900 to w = 500, this gives the constants: Line. Transformer. Line. Load. Sum. . A= 1.5X10-3 1X10-* = 1.5X10-") .5X10-* ~4.5x10-3 w= 500 100 500 1600 uA= 75 1 5 8 2.4 hence, uo = average u= a = 533, and: s= +33 +433 +33 — 1067

114. ELECTRIC DISCUARGES, WAVES AND IMPULSES. Choosing the samme lengths and the same power-dissipation constants as in the previous illustrations, this gives: ; Line. Transformer, Line. Sum. A= 1.5xX 10-4 1X10-3 1.5x10-* 4x10-3 “= 900 100 900 ur= 1.35 1 1.35 2.8 Zur hence, ut) = average x= sy = 700, and: s= ~on +600 —200 Line . Transformer Line Fig. 56. The diagram of the power of the two waves of opposite direc- tions, and of the resultant power, is shown in Fig. 57, assuming 6 megawatts as the maximum power of each wave, whicli is reached at the point where it leaves the transformer. i ‘Lransmission Line Transformer Transmission Line ———- $$$ 8800 + u = 900 u —100 u = 900 , ——oO . Transmission Line Transformer Transmnissioh\Ling 5x 10-3 1.9°x 10-3 6x 10% fon+-600 S==- 200 TL, = 700 Fig. 57. — Energy Distribution in Compound Oscillation of Open Circuit. In this case the two waves must be of the same intensity, so as to give 0 as resultant at the open ends of the line. A power node then appears in the center of the transformer.
115. Astationary oscillation of a compound circuit consists of two traveling waves, traversing the circuit in opposite direction, and transferring power between the circuit sections in such a manner

CONTINUAL AND CUMULATIVE OSCILLATIONS 125 TTT ETT EET |, || Transient ae i Volt-Ampere Characteristic a . an High Temperature Metal Arc 160 | oe eT N \ ror PASSE . | — Sr Tig. 66.

116 ELECTRIC DISCHARGES, WAVES AND IMPULSES. , and since . ; = 22; ; = &, (5) | substituting €2 = 1222, ml . . €) = 1121, ~ into (4) gives 12°22 = 2721, or _ _ pa h@ , a” Va" Vad’ 7) and €? ei? 2 a or _ _ . @ oy. [BG (8) C1 41 C, Ly That is, in the same oscillating circuit, the maximum voltages . ey in the different sections are proportional to, and the maximum currents 7 inversely proportional to, the square root of the natural impedances 29 of the sections, that is, to the fourth root of the ae ratios of inductance to capacity a. : At every transition point between successive sections traversed by a traveling wave, as those of an oscillating system, a trans- formation of voltage and of current oceurs, by a transformation : ratio which is the square root of the ratio of the natural imped- ances, 29 = VE, of the two respective sections. When passing from a section of high capacity and low induc- tance, that is, low impedance 29, to a section of low capacity and high inductance, that is, high impedance zo, as when passing from a transmission line inte a transformer, or from a cable into a trans- mission line, the voltage thus is transformed up, and the current transformed down, and inversely, with a wave passing in opposite direction. A low-voltage high-current wave in a transmission line thus becomes a high-voltage low-current wave in a.transformer, and inversely, and thus, while it may be harmless in the line, may become destructive in the transformer, ete.

118 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Dividing the two pairs of equations of (9) gives

f cone 2 et y/2, COS Y1 Cy *2 (10) cos be hk _ y/2 cos 6; ln Z hence, multiplied, COSY y COS Bp | COS ¥1 COS 6; or (11) COSY. COSO, COSY) COS 6a” | or COS 71 COS 64 = COS Ye COS 693 that is, the ratio of the cosines of the current phases at the tran- sition point is the reciprocal of the ratio of the cosines of the voltage phases at this point. ;

Sinee at the transition point between two sections the voltage and current change, from ¢1, 4 to @, %, by the transforination ratio / at this change can also be represented as a partial reflection. That is, the current 7, can be considered as consisting of a compo- nent %, which passes over the transition point, is “ transmitted ”’ current, and a component 7)’ = 7) — %, which is “ reflected ” eurrent, ete.’ The greater then the change of circuit constants at the transition point, the greater is the difference between the currents and voltages of the two sections; that is, the more of current and voltage are reflected, the less transmitted, and if the change of constants is very great, as when entering from a trans- mission line a reactance of very low capacity, almost all the current is reflected, and very little passes into and through the | reactance, but a high voltage is produced in the reactance.