| : CIRCUITS WITH DISTRIBUTED LEAKAGE — 335 176. Consider an instance: it has been proposed, for the pur- pose of effectively grounding the overhead ground wire used for protection of transmission lines, to run a bare underground con- ductor, a few feet below the ground surface, and to connect the overhead ground wire to the underground wire at every pole. Assuming the underground conductor to be a bare copper wire having 0.41 cm. diameter, the overhead ground wire a steel cable equivalent in conductivity to a copper wire of 0.52 cm. diameter. What is the effective ground resistance of the underground wire alone, what that of the underground and overhead wire together? Assuming the leakage resistance of the underground wire to be 3 X 10-* mhos per meter? The resistance of the underground wire is 1.3 X 10-* ohms, that of the overhead ground wire is 0.82 X 10-3 ohms per meter. The effective resistance of one underground wire then is ; _ J "Ng r=1.3 X 10-3; g = 3 X 107, hence, ro = 0.66 ohm thus, two underground wires in multiple, in the two different di- rections, give an effective ground resistance of 0.33 ohm, including the overhead ground wire, the resistance is r= 4+, = 0.5 X 10-859 = 3 X 10°, 13x 107 + O82 x 10% hence, , ro = 0.41 ohm, thus, the two underground and two overhead wires together give an effective resistance of 0.205 ohm. This is a very much lower ground resistance than most local grounds possess. Assuming that 7 is the current which enters this ground wire at one point, J = 0, then the equation of current distribution, by (7), is r = 0.25 X 10-3; g = 6 X 10? (two in multiple, in the two opposite directions)

336 ELECTRIC CIRCUITS hence _ r= 2 = 0.205 g . a= Wrg = 1.225 X 107 thus, G = Spe ! 228 € = 0.205 toe—}-228 where I is given in kilometers.

At various distances from the starting point, the current in the conductors thus is: distance: 0. .| 0.6 | 1.0 | 1.5 | 2.0 2.5 3.0 4.0 5 km. t X to X 1. .| 0.54 | 0.204) 0.155) 0.086 | 0.046 | 0.025 | 0.0074} 0.0021

As seen, beyond 2 km. distance, the current in the conductor is practically nothing. :

177. If the current, i, is an alternating current, and the con- dition such that inductance and capacity are negligible, the equations (7), (9), (10), (11) and (13) remain the same, except that t,e and A are vector quantities, or general numbers: J, £, A.

Considering thus the more general case, where a voltage is in- duced in the leaky conductor. Such for instance is the case in the lead armor of a single-conductor alternating-current cable.

Let, then, r = resistance per unit length, g = shunted conductance per unit length, Eo = voltage induced in the conductor, per unit length. It is, then, in a line element, dl, d aE =r[f — Eo (14) alt . a 7 9" (15) Differentiating (15) and substituting into (14) gives d? E.

o m( 2) as

This is integrated by é [— = ew a7)

CIRCUITS WITH DISTRIBUTED LEAKAGE — 337 and by substituting (17) into (16), we get a=rg (18) hence, the current, z T= Ase! + Aget! + > ~ (19) where , a= +779 and A, and Az are complex imaginary integration constants. Substituting (18) into (15) gives the voltage, E= rol Aye“! — A gto} (20) where n= it (21) g 178. Suppose now no voltage is impressed upon the conductor, but the only existing voltage is that induced in the conductor, as for instance the cable armor. (a) Suppose the conductor is open at both ends: 1 = +l) and 1 = — lo, having the length 2 lo. It then is {=0 forl = +l Substituting this in (19) gives . Are + Agetee + 7? = 0 Ey Anetele + Aseaie +7 =0 hence, .

• __— Bo __ Ai= A: r(etate + e-ele) and é — Ho ett + et P= 3|1- Sapo U —al (22) =" (S45 E a r Eo etate + Pm in the center of the conductor, for / = 0, it is _ Bo 2 I= - sper] £=0 | 22 .

338 ELECTRIC CIRCUITS at the ends of the conductor, for 1 = +lo, it is T=0 _ To etele _. ¢-alo B= r Bol Sa + eh! hence, if the conductor is long, so that e~* is negligible compared with e+, it is To Eo = +-— EF, = +—= E + r 0 + 79 For an infinitely long conductor, lb) = © , equations (22) become _ Be t=" | (23) E=0 as was to be expected.

(b) Suppose the conductor is grounded at one end, | = 0, and

open at the other end, 2 = Jo. It is, then,

E =0 forl=0

I = 0 forl = h, hence, the equations are the same as (22). That is, a conductor grounded at one end and open at the other is the same as a con- ductor of twice the length, open at both ends.

A conductor grounded at both ends gives the same equation as an infinitely long conductor (23).

Suppose aly is large, so that «~ “ is negligible compared with e+ in equation (22). Then for all values of J, except those very close to Io, £ and the exponential term of J are negligible.

That is, for the entire length of the leaky conductor, except very close to the ends, it is, approximately,

_ Be a, | (24) E=0

Near the ends of the conductor, where J is near to lo, e~® is negligible compared with e+”, and equations (22) thus assume the form, ;

l= Foty - ena} ro (25) = 1° Fe —ale—t) = Eve

CIRCUITS WITH DISTRIBUTED LEAKAGE 3389

179. Asan instance, consider thelead armor of asingle-conductor cable, 10 km. long, carrying an alternating current such that it induces 60 volts per kilometer. The armor is open at either end, and of internal diameter of 4.2 cm., external diameter of 4.6 cm. The leakage conductance from the cable armor to ground is 1 mho per kilometer. What is the voltage and current distribution in the cable? What is it with 10 mhos. what with 0.1 mho per kilometer leakage conductance?

A lead section of the armor of (2.3? — 2.17) + = 2.7 cm.?, at the specific resistance of lead p = 19 X 10-‘, gives: r = 0.7 ohm per kilometer.

It is, then, ; r=0.7 g= 1 b= 5 E.= 60 thus, a = 0.84 T = 0.84 hence, . I = 86{1 — 0.015 (et 0%! 4 <— 9-842) } 28 E= 1.08 {e+ 0.841 7 0.841) ( )

Thus the maximum current in the cable armor is, 1 = 0: J = 83.4 amp., and this current decreases very slowly, and is still, for l= 2:7 = 79 amp.

The maximum voltage between cable armor and ground is, for 1 = +5: FE = 72 volts, and decreases fairly rapidly, being, for l= +4: E = 81.1 volts.

If the cable is laid in very well-conducting soil,

, g = 10 it is a = 2.65 To = 0.265. T = 86{1 — 1.75 x 10° (e728 4 POD} | E =40X 1078 {_ +265! _ 7-65} ( ) in this case, the current is practically constant, J = 86, and the voltage zero over the entire cable armor except very near the ends, where it rises to E = 22.7 volts for! = 5. Within 1 km. from the ends, or for = 4, it is still: J = 80;E = 1.6. That is, over

340 ELECTRIC CIRCUITS most of the length, the cable armor already acts as an infinitely long conductor. Hence, for values of J near the end of the conductor, J] and E are more conveniently expressed by the equation (25), = _ .-2.65(5— D aa (2085-0 | (28) det TT TT TT TE TT fest | he Tote Tol | leo) | CE EEN | Iie I Tel T/A ee tN XL tel I | fol TTA Tt EL EAT Tal | tol TEV EE ET EE VNAL fool | tol AT EET AAA Tol Stool YT A ete KAD ET ool Phe 7 NA tal | tol VAT TT TAT LA | ol Chol YET TT YE rN tol | p | pee A tt a Pi tT yet yt tT TT TT bel | PTT ZL I Veal TT tt bel | PT Tye TAT TT TT TT bel | PT TTT VYE TELE TT UT tool | PTT TIAL TT TT TT bal | PT TT VET TTT TTT Tt bel I PTT TAT EEE TTT TT beet | PTT TART TT ET ET Tt bel PET YET TT TT TT Ty booed | PL Poet ete tbe Td Fig. 128. Inversely, if the cable is laid in ducts, which are fairly dry, and the leakage conductance thus is only g =0.1 it is ; a = 0.265 To = 2.65

. CIRCUITS WITH DISTRIBUTED LEAKAGE 341 hence, I = 86 { 1 _ 0.25 (et. 265! + €— 0-268) } E = 57 { e0-2681 — ¢-0.268} (29) In this case, the maximum voltage between cable armor and ground is, atl = +5: EF = 200. As illustrations are shown, in Fig. 128, with | as abscisse, the curves of I and E, calculated from equations (26), (28) and (29). 180. Considering now the case of a conductor, which is not connected to a source of voltage, nor has any voltage induced in it, but is laid in a ground in which a potential difference exists, due to stray currents passing through the ground. Such, for instance, may be a water pipe laid in the ground parallel with a poorly bonded railway circuit. Assuming the potential difference, eo, exists in the ground, per . unit length of conductor. The conditions obviously are the same, as if the ground were at constant potential, and the potential difference, —€o, existed in the conductor per unit length. Thus we get the same equations as (22) and (23). If the potential difference is continuous, as when due to a direct-current railway circuit, obviously the quantities J, Z, A, and A: are not alternat- ing vector quantities, but scalar numbers: i, e, etc. That is,

• €& etal 4 ¢-al i=9(1- gayest | (30) e =e, r etale + e~ ae Assuming thus as an instance a water pipe of 5 km. length: lo = 5, extending through a territory having 50 volts potential difference, or : €9 = 10. Assuming that it is connected with the return circuit so that there is no potential difference at one end: e=0 forl = 0. Let the resistance of the water pipe be r = 0.01 ohm per kilo- meter, and the leakage conductance be g = 10 mhos per kilometer. It is, then, ' . r = 0.01 . g = 10 ly = 5 eo = 10 thus, a = 0.316 To = 0.0316

. 342 ELECTRIC CIRCUITS hence, 4 = 1000 { 1 — 0.2 (ct0-210 + -0.218)} eo ey | ep hence, the maximum current, for 1 = 0:7 = 600 amp. the maximum voltage, for! = 5:e = 28.8 volts. Pit TTT TTT TT Ty yet ft PET tT TTT TT TT ET fel Ty PETE ETT TT TAL Tt desl Ted Cal Tol PNET TALL Tl fool PT TINE ET VE TT Toot [evel I ETT LIN | TALL Taal tocol] PET TT IN VET TT Tach Tecol | PT TT TT XE Ed Td Tol , PE ET ETA KELL Tal ovo] PT tT te | INT TA TL tol foo] _| PT TT VE LT bee LT I ol lool _| Pt TY lee iN tt del taal PT AY Lee VT dat tool PY ee tT NT tt Tl List sp ds das do dsb | [ol [ol | Fia. 129. As seen, a very considerable current may flow under these conditions. Fig. 129 shows, with I as abscisse, the current, i,and voltage, e, and the current which enters the conductor per unit length, S.

4 t » CHAPTER XVIII OSCILLATING CURRENTS ° Introduction

181. An electric current varying periodically between constant maximum and minimum values—that is, in equal time intervals repeating the same values—is called an alternating current if the arithmetic mean value equals zero; and is called a pulsating cur-

rent if the arithmetic mean value differs from zero.

Assuming the wave as a sine curve, or replacing it by the equivalent sine wave, the alternating current is characterized by the period or the time of one complete cyclic change, and the amplitude or the maximum value of the current. Period and amplitude are constant in the alternating current.

A very important class are the currents of constant period, but geometrically varying amplitude; that is, currents in which the amplitude of each following wave bears to that of the pre- ceding wave a constant ratio. Such currents consist of a series of waves of constant length, decreasing in amplitude, that is, in

‘ strength, in constant proportion. They are called oscillating currents in analogy with mechanical oscillations—for instance of the pendulum—in which the amplitude of the vibration de- creases in constant proportion.

Since the amplitude of the oscillating current varies, constantly decreasing, the oscillating current differs from the alternating current in so far that it starts at a definite time and gradually dies out, reaching zero value theoretically at infinite time, prac- tically in a very short time, short usually even in comparison with the time of one alternating half-wave. Characteristic con- stants of the oscillating current are the period, 7’, or frequency, f= r the first amplitude and the ratio of any two successive amplitudes, the latter being called the decrement of the wave. The oscillating current will thus be represented by the product of a periodic function, and a function decreasing in geometric proportion with the time. The latter is the exponential function, A/—*,

343

344 ELECTRIC CIRCUITS 182. Thus, the general expression of the oscillating current is I = A‘-* cos (2 xft — 6). Since Al-ot = ASA-0t = i, where « = basis of natural logarithms, the current may be expressed,

I = ie cos (2 aft — 0) = ie ** cos ( o— 8), Ne PRESET TT TT ET TE EE EE ET SNR See A ttt Lb MT tool T/L esol NY [oso] | A Tego SP foot = ones PT ATT TT | NE Pe ttt PT RET YT peepee tt TT SKU ae eee et ees Pr] ta TTT TT TI Face o2® | TTT TT PET TT PET PTT ET ETE ET Tt

Fia. 130. EISEN Me SSK Seceeas! PES SSN SY OOOO) Mafase LSE PROM BORE TRE RRN ONAN LAIR RSS ERRORS SON ROOT | HN LEPROSY tt fein Sa ea HN HHE SA ASR pees tH YIIIT | HAH Oy eet ALN coe eR TT ETT TI waue eenete! SOS TES Sa a, a4 VT HLL YT | 1 ASS Reet Nar e EMIT | ORR oe ALHT 7 TH | SRR SLIMY KES SH sesteeeee uly U/ SRO EE FEES AAPL PLY Fig. 131. where ¢ = 2-ft; that is, the period is represented by a complete revolution. In the same way an oscillating e.m.f. will be represented by E = ee cos (¢— 6).

OSCILLATING CURRENTS 345 Such an oscillating e.m.f. for the values, e = 5,a = 0.1485 ore ?™ = 0.4, 6 = 0,

\ is represented in rectangular codrdinates in Fig. 130, and in polar codrdinates in Fig. 131. As seen from Fig. 130 the oscillating wave in rectangular codrdinates is tangent to the two exponential curves,

. y= +ee~%

In polar codrdinates, the oscillating wave is represented in Fig. 131 by a spiral curve passing the zero point twice per period, and tangent to the exponential spiral,

y = tee”,

The latter are called the envelopes of a system of oscillating

‘ waves. One of them is shown separately, with the same con- stants as Figs. 130 and 131,in Fig. 132. Its characteristic feature is: The angle which any concentric circle makes with the curve, y = ee, is

tana = oe —a, Chine Fia. 132. Fia. 133.

which is, therefore, constant; or, in other words: ‘‘The envelope of the oscillating current is the exponential spiral, which is char- acterized by a constant angle of intersection with all concentric circles or all radii vectores.’’ The oscillating current wave is the product of the sine wave and the exponential or loxodromic spiral.

183. In Fig. 133 let y = ee~* represent the exponential spiral; let z =e cos (¢ — 6) represent the sine wave; and let E = ece~* cos (¢ — 6)

346 ELECTRIC CIRCUITS represent the oscillating wave.

We have then

dE tan B = Edd _ — sin (¢ — 6) — acos(¢ — 6) 7 cos (¢ — 0) = — {tan (¢ — 6) +a}; that is, while the slope of the sine wave, z = e cos (¢ — @), is represented by tan y = — tan (¢ — 9), the slope of the exponential spiral, y = ee~*, is tana = — a = constant, that of the oscillating wave, E = ee** cos (¢ — 8), is tan 8 = — {tan (¢ — 6) + a}.

Hence, it is increased over that of the alternating sine wave by the constant, a.

The ratio of the amplitudes of two consequent periods is A= ts se 72 xa,

A is called the numerical decrement of the oscillating wave, a the exponential decrement of the oscillating wave, a the angu- lar decrement of the oscillating wave. The oscillating wave can be represented by the equation,

E = ee **" cos (¢ — 6). In the example represented by Figs. 130 and 131, we have A = 0.4, a = 0.1435, a = 8.2°. Impedance and Admittance 184. In complex imaginary quantities, the alternating wave, 8 = ecos (¢ — 4) is represented by the symbol, E = e(cos 6 — jsin 0) = e; — jer.

By an extension of the meaning of this symbolic expression, - the oscillating wave, E = ee~** cos (¢ — 6), can be expressed by the symbol,

E = e(cos 0 — j sin 6) deca = (€: — jes) deca, where a = tan a is the exponential decrement, a the angular decrement, e~?** the numerical decrement.

if 7 7 _ ‘ OSCILLATING CURRENTS 347 Inductance 185. Let r = resistance, L = inductance, and 2 = 22fL = reactance, in a circuit excited by the oscillating current, I = ie~** cos (@ — 0) = i(cos 6 +7 sin 6) deca = (t1 + jis) dec a, where 7, = 7008 6, i: = isin 6, a = tana. We have then, the e.m.f. consumed by the resistance, r, of the circuit, E, = rI dec a. The e.m.f. consumed due to the inductance, L, of the circuit, aI dI aI E,= La = 2 afl 75 = a6 Hence E, = — xie**{sin (6 — 6) + a cos (¢ — 6)} _ atest, 9 = — ose Bin (eo — + a). Thus, in symbolic expression, E, = — Sosa (7 Bin (8 — a) — jcos (8 — a)} deca = — xi(a — j) (cos @ — jain 6) dec a; that is, E, = — aI (a — j) deca. Hence the apparent reactance of the oscillating-current cir- cuit is, in symbolic expression, X = 2(a — j) deca. Hence it contains a power component, az, and the impedance is Z=(r—X) dec a= {r—2z(a—j)} deca = (r —ax + jz) dec a. Capacity 186. Let r = resistance, C = capacity, and x. = To = con- densive reactance. In a circuit excited by the oscillating current, I, the e.m.f. consumed due to the capacity, C, is 1 1 Bag = Gf lat = gapg [ tae = & [1a9;

348 ELECTRIC CIRCUITS or, by substitution, Ba = 2 {ie 008 (6 - 6) dd = Tp ate [sin (6 — w) — a 608 (6 — 6)| Lie . . ; = T+ 0%) cong 1M (O — 8 — @); . hence, in symbolic expression, xt : . E,, = (+ a4) cos a! (6+ a) —Jj cos (6 + a)} dec a = Tra @-A (cos 6 — j sin @) dec a; hence, x E., = Tsao a — j) [ deca; that is, the apparent capacity reactance of the oscillating circuit is, in symbolic expression, Le X.= ita (— a — j) dec a. 187. We have then: in an oscillating-current circuit of resistance, r, inductive re- actance, z, and condensive reactance, z., with an exponential decrement a, the apparent impedance, in symbolic expression, is, = _ — i) 4% (ga Z {+ x(a D+ pal a j) | dee = _ ze _ i(, — —te__ {r-a(2 + rita) + i(2 — pyr) |deo = 14 + jXa; and, absolute, __ 2a = V re? + ta Le 2 Ze 7? =ylr- (+ -Fa)] +[2#-rFal- Admittance 188. Let N I = ie~** cos (6— 6) = current.

Then from the preceding discussion, the e.m.f. consumed by re- sistance, r, inductive reactance, z, and condensive reactance, 2c, is = je-a6 - —arz —- —*—z,| — si _

E = te | cos (¢ 6)[r az—F i at| sin (¢ — 8)

Ze . [- “I+ al} = iz¢e~°* cos (@ — 8 + 8),

! OSCILLATING CURRENTS 349 where 2— —7 . 2 tan 6 = —+it_., ran Tp Git Ze ? a 2 t= (2-7 Ha) +(r- - pat) 5 substituting @ + 6 for @, and e = iz, we have E = ee~* cos (¢ — 8), I = £ e+ cos ($ — 6 — 8) = cca [285 05 (% — 6) + St? ain (g — 0) |; hence in complex quantities, E = e(cos 6 — jsin 6) deca, cosé =. sin 3 . [= Bi -i= | dee a; or, substituting, a r— az — ——,2 1+ a?°° T= | owe) Det (2 - ra) +(r — az — rpa*) ¢—- : 2 -j ite’ dec a. . 2-734) + (r - az — 52) | ( 1+ 2 1+a?°° | 189. Thus in complex quantities, for oscillating currents, we have: conductance, a ;

• rT — az - T+a?* 7 Ze ? a ” (- fa) +(r— a - rpa*) susceptance, Le pe Le 2 a 2) (@- ra) +(r- a2 — 7p ae) admittance, in absolute values, y= VETER = oo (@-7Fa) + (* - 2 — > pat)

350 ELECTRIC CIRCUITS in symbolic expression, (r - or - 7, 2.) - i(z - > ) ; ita) 3 1 +a? Y= g- jo = a oar a are | 3 a 7° z— ~~. +(r - az - 52) ( 1+ ra) 1+ a? Since the impedance is a . Ze _ ° Z= (r- ae - Pam) +5(2- rea) = Tet ja we have 1 1 To Za Y=giv=7i9 = 7a b = 75) that is, the same relations as in the complex quantities in alter- nating-current circuits, except that in the present case all the constants, Ta, a, Za, J, 2, ¥, depend upon the decrement, a. It is interesting to note that with oscillating currents, resist- ance as well as conductance have a negative term added, which depends on the decrement a. Such a negative resistance repre- sents energy production, and its meaning in the present case is, that with the decrease of the oscillating current and voltage, their stored magnetic and dielectric energy become available. Circuits of Zero Impedance 190. In an oscillating-current circuit of decrement, a, of resistance, r, inductive reactance, z, and condensive reactance, 2c, the impedance was represented in symbolic expression by . a . Le Bm rect ite = (r— or ~ Toe) +i(2 - Pa) or numerically by —————. a 2 x 2 = 2 2— — —_ —_ — —*}. = Vritad = (rac - rte) + (2-75) Thus the inductive reactance, z, as well as the condensive reactance, z., do not represent wattless e.m.fs. as in an alternating- current circuit, but introduce power components of negative sign, a ~ 8 — Tp ge i that means, in an oscillating-current circuit, the counter e.m.fs. of self-induction is not in quadrature behind the current, but lags less than 90°, or a quarter period, and the charging current of a condenser is less than 90°, or a quarter period, ahead of the im- pressed e.m.f.

if ee OO OSCILLATING CURRENTS 351

191. In consequence of the existence of negative power com- ponents of reactance in an oscillating-current circuit, a phe- nomenon can exist which has no analogy in an alternat- ing-current circuit; that is, under certain conditions the total impedance of the oscillating-current circuit can equal zero:

Z=0. In this case we have r— az —-—— 2, =0;2--. =0 oO Tae Tea substituting in this equation, ae t= 2afL; 2. = 2 afi and expanding, we have . a7 Vig 3 r [4b r afm 97 Vn ~~ ar

That is, if in an oscillating-current circuit, the decrement,

V6 -? and the frequency f = iar the total impedance of the circuit is zero; that is, the oscillating current, when started once, will continue without external energy being impressed upon the circuit.

192. The physical meaning of this is: If upon an electric circuit a certain amount of energy is impressed and then the circuit left to itself, the current in the circuit will become oscillat- ing, and the oscillations assume the frequency, f = tar and the decrement,

+a ng —}

That is, the oscillating currents are the phenomenon by which an electric circuit of disturbed equilibrium returns to equilibrium.

This feature shows the origin of the oscillating currents, and the means of producing such currents by disturbing the equi-

352 ELECTRIC CIRCUITS - librium of the electric circuit; for instance, by the discharge of a condenser, by make-and-break of the circuit, by sudden electro- static charge, as lightning, etc. Obviously, the most important oscillating currents are those in a circuit of zero impedance, representing oscillating discharges of the circuit. Lightning strokes frequently belong to this class. Oscillating Discharges 193. The condition of an oscillating discharge is Z = 0, that is, 1 r r [f4L 0= ep PS = ah = BEV PC ;

If r = 0, that is, in a circuit without resistance, we have a = 0, f= TEE ; that is, the currents are alternating with no decre- ment, and the frequency is that of resonance.

If aa < 0, that is, r > af, a and f become imaginary; that is, the discharge ceases to be oscillatory. An electrical discharge assumes an oscillating nature only, if r < aa/%, In the case r = NE we have a = o, f = 0; that is, the current dies out without oscillation.

From the foregoing we have seen that oscillating discharges —as for instance the phenomena taking place if a condenser charged to a given potential is discharged through a given circuit,

or if lightning strikes the line circuit—are defined by the equation, Z = 0 deca. Since T = (i: — juz) dec a, - E, = [r dec @, . Le . E.=—2[(a—j)deca, E., = ita I(— a — j) dec a, we have r—azr——"— 2, =0 | ; 1+a?"° , . Ze —tt+T pe = % . hence, by substitution, . E., = 2] (— a — j) deca.

OSCILLATING CURRENTS 353 The two constants, 7, and 13, of the discharge, are determined by the initial conditions—that is, the e.m.f. and the current at the » time, ¢ = 0. i 194. Let a condenser of capacity, C, be discharged through a 4 circuit of resistance, r, and inductance, L. Let e = e.m.f. at the i condenser in the moment of closing the circuit—that is, at the i timet = Oor¢? = 0. At this moment the current is zero—that is, . IT = fiz, i, = 0. . Since E., = 2[ (—a—j) dea=eat¢ =0, i . _ a ; we have Tix 1 + a? = 6 or ta = Fe \ Substituting this, we have, ; = —j—*“— d = —je ——— d Pa lyipa Oe , e é =. = === (1 + ja) dec a, F., = — —=== (1 — ja) dec a, . &. Via’ + ja) eca, £ Vida | ja) eC a ! the equations of the oscillating discharge of a condenser of initial h voltage, e. po Since ; 2=2nfL, i 1 k 0 i ———————— ty —=-1 rc r i 2 af = 2 aL’ ‘ . we have | *= 9479 Vno 7 hence, by substitution, : . |C . IC | T= — je a{S deo a, Be = — jerff deo, er IC fA L . . | B= SE ( aE 143) doo a er |C B L . E:, = ~ 58 ( ro 1 - 5) dec a, -! _ fA L a Jib-1 F . r re -1 rc 4nL the final equations of the oscillating discharge, in symbolic ex- pression. | 23 | \

Digitized by Goog le

INDEX A Arcing ground on transmission lines, 199 Admittance, with oscillating cur- Area of BH relation, 53 rents, 348 Armature flux of alternator, 233 Air gap in magnetic circuit reducing reactance flux of alternator, 232 wave distortion, 145 reaction of alternator, 236 Alloys, resistance, 2 Attenuation constant, leaky con- Alternating component of power of ductor, 334 general system, 317 of synchronous machine oscil- current electromagnet, 95 lation, 213 magnetic characteristic, 51 Alternations by capacity inductance B shunt to arc, 187 Aluminum cell as condenser, 10 Balance of quarterphase system on Amorphous carbon resistance, 23 singlephase load, 322 Annealing, magnetic effect, 78 of singlephase load, 319 Anode, 6 of threephase system on single- Anthracite, resistance, 23 phase load, 325 Apparatus economy of constant po- of unbalanced power of system, tential, constant current 819 transformation, 281 Bends in magnetic reluctivity curve, of monocyclic square, 276 49 ' of T connection, 265 Bismuth, diamagnetism, 77 Arc as alternating current power Bridged gap in magnetic circuit, generator, 187 wave distortion, 148 characteristics, 34 condition of stability on con- Cc stant current, 173 on constant voltage, 169 Cable armor as circuit, 330 conduction, 28, 31, 42 equation of induced current, 336 constants, 36 Capacity, 1 effective negative resistance, and inductance shunting circuit, 191 181 equations, 35 inductance shunt to arc pro- as oscillator, 189 ducing alternations, 187 parallel operation on constant with oscillating current, 347 current, 175 and reactance as wave screen, _ shunted by capacity, 178, 184 154 and inductance, 184 in series regulating for constant by resistance on constant current, 247 current, 172 shunt to arc, 178, 184 singing and rasping, 188, 189 to circuit, 178 tending to unstability, 164 Carbon, resistance, 21 transient characteristic, 192 Cathode, 6 as unstable conductor, 167 Cell, 7 355

356 INDEX Characteristic, magnetic, 50 Current wave distorted by mag- Chemical action in electrolytic con- netism, 126 . duction, 6 ‘ Chromium, magnetic properties, 83 D Circuit with distributed leakage, 330 . magnetic, 43 Damping power in synchronous Closed magnetic circuit, wave dis- motor oscillation, 210 ' tortion, 139 winding in synchronous ma- Cobalt iron alloy, magnetic, 78 chines, 211 magnetic properties, 80 Danger of higher harmonics, 121 Coefficient of hysteresis, 61 Decrement of oscillating wave, 343 Coherer action of pyroelectric con- Demagnetization by alternating cur- ductor, 19 rent, 54 Compensating voltage balancing un- temperature, 78 balanced power, 320 Diffusion current of polarization, 8 Condenser, electrostatic, 9 Direct current producing even har- power equation, 319 monics, 159 tending to instability, 164. See Discharges, oscillating, 352 Capacity. Discontinuous conduction, 29 Conductance with oscillating cur- Displacement of field poles eliminat- rents, 349 ing harmonica, 120 . Conduction, electric, 1 of position in synchronous ma- Conductors, mechanical magnetic chine, 210 forces, 106 Disruptive conduction, 29, 42 Constant component of power in Distortion of wave improving regu- general system, 317 lation in series circuits, 311 current arc, stability condition, of voltage by bridged magnetic 172 gap, 148 constant potential transfor- in constant potential con- mation, 243, 286 stant current transforma- reactance, 134 tion, 200 transformer and regulator,250 Distributed leakage of circuit, 330 magnetic, 77, 87, 88 winding, eliminating harmonics, potential constant current 116 transformation, 243, 286 Double frequency armature reac- reactance, 133 tion, 240 term and even harmonics, 158 peaked wave, 113 voltage arc, stability condition, 168 series operation, 297 E Continuous conduction, 32 ; Corona conduction, 29, 42 Economy, apparatus, 281 Creepage, magnetic, 57 Efficiency of electromagnet, 99 Critical points of reluctivity curve, of monocyclic square, 277 46 of T-connection, 268 Cumulative oscillation, cause, 166 Electrodes, 6 produced by arc, 188 Electrolytic cell, 8 in transformer, 199 condenser, 9 surge, 166 conductor, 442 |

INDEX 357 . Electromagnet, 91 Gas pipes as circuits, 330 constant current, 93 vapor and vacuum conduction, potential, 98 28, 41 ’ efficiency, 99 Geissler tube conduction, 29, 42 Electronic conduction, 28, 40 Gem filament incandescent lamp, 22 Elimination of harmonics by alter- Grounded leaky conductor, 333 nator design, 116 Energy of hysteresis, 57 H ‘ storage in constant potential constant current transfor- Half turn windings, 114 mation, 280 Hardness, magnetic, coefficient of, Even harmonics, 114, 153, 157 44 Excessive very high harmonics in Harmonics, effect of, 121 distortion by magnetic sat- even, 153, 157 uration, 140 separation by wave screens, 157 ' Exciting current of transformer de- Heusler alloys, magnetic properties, ‘ pending on wave shape, 137 81 Exponent of hysteresis, 66 High harmonics in alternator, 120 excessive in wave distortion by F magnetic saturation, 140 by slot pitch, 120 Face conductor in alternator, 114 temperature insulators, 26 Faraday’s law of electrolytic con- Homogeneous magnetic materials, duction, 6 55 Ferrites, magnetic, 80 Hunting of synchronous machines, Ferromagnetic density, 45 166, 208 Field flux of alternator, 232 Hysteresis, 56 ' Film cutout in series circuits, 208 loss and wave shape, 112 ' Flat top wave, 111 . Flicker of lamps and wave shape, 124 I Flux distribution of alternator field, 114 Impedance and admittance with Fluxes, magnetic of alternator, 232 oscillating currents, 346 Forces, mechanical magnetic, 91, 107 of line in regulation of series Form factor of magnetic wave dis- circuits, 306 _ tortion, 127 Induced current in leaky cable Fractional pitch armature winding armor, 336 eliminating harmonics, 119 Inductance, 1 Frequency conversion in cumulative and capacity shunting circuit, surge, 166 181 of synchronous machine oscil- power equation, 316 lation, 213 as wave screen, 153 Friction molecular magnetic, 56 Induction motor magnetic circuits, Frohlich’s law, 43 228 instability, 164, 201 G Inefficiency of magnetic cycle, 60 Infinitely long leaky conductor, 332 Gap in magnetic circuit reducing Instability by capacity shunt, 180 wave distortion, 145 of circuits, 165

358 INDEX Instability of induction motors, 201 Magnetism, 43 of pyroelectric conductor, 16 tables and data, 87, 88 of synchronous motor, 208 wave distortion by saturation, Instantaneous power of general sys- 128 tem, 317 Magnetite arc, 36 Insulators, 23, 42 hysteresis, 62 as pyroelectric conductor, 25 magnetic properties, 80 Iron cobalt alloy, magnetic, 78 as pyroelectric conductor, 14 magnetic properties, 79 Magnetization curve, 48 resistance, 4 Magnetkies, magnetic properties, 80 Manganese alloys, magnetic prop- K erties, 81 steel, magnetic properties, 79 Kennelly’s law of reluctivity, 44 Mechanical magnetic forces, 91, 107 Mercury arc characteristic, 39 L Metals, resistance, 2 Metallic carbon, resistance, 22 Lag of damping power in synchron- conductors, 142 ous machine, 213 induction, magnetic, 47 of synchronizing force, 212 magnetic density, 45 Lamp circuits in series, 207 Mixtures as pyroelectric conductors, equivalent of line impedance in 21 series circuits, 306 Molecular magnetic friction, 56 Law of hysteresis, 62 Monocyclic square, 261, 273, 283, Leakage, distributed, of circuits, 330 293 flux of alternating current trans- Mutual inductive flux of alternator formers, 217 armature reaction, 237 reducing wave distortion, 145 Leaky conductor, 330, 332, 336 N Load balance of polyphase system, Negative resistance of arc, effective, a sys 191 character determining stability Neodymium, magnetism, 17 in induction motor, 205 Nernst lamp conductor, 13, 24 yoo of hysteresis, 56 . al Nickel, magnetic properties, 81 88, pee in magnetic cycle, steel, magnetic properties, 79 Loxodromic spiral, 345 Nominal induced em, of alter- . Py ? Luminescence in gas and vapor con- duction, 28 re} Luminous streak conduction in pyro- electric conductor, 18 Oils as insulators, 26 Open circuited leaky conductor, 332 M magnetic circuit, wave shape distortion, 145 Magnetic circuits of induction Organic insulators, 24 motor, 228 Oscillating approach to equilibrium elements, 77 condition, 210 friction, 56 currents, 343 mechanical forces, 107 discharges, 352

INDEX 359 Oscillations of arcing ground on Pyroelectric conductor, 10, 42 transmission line, 197 classification, 20 in capacity inductance shunt to resistance increase by high fre- ’ circuit, 181 quency, 19 cumulative, produced by arc, tending to instability, 164 188 Pyroelectrolytes, 10, 18 which becomes permanent, 165 resistance of arc, 196 Q : Outflowing current in leaky con- ductor, 334 Overshooting of alternator current Seer lerkae lood. 322" on at load change, 238 , Oxygen, magnetism, 77 R ; P Rail return circuit, 330 Railway return circuits, 330, 341 Parallel operation of arcon constant Rasping arc, 189 current, 175 Reactance depending on wave shape, Peak of current wave by magnetic 132 } saturation, 126 on induction apparatus, 216 reactance, 134 inductive, constant current voltage used in arc starting, 152 regulation, 246, 281 by magnetic saturation, 128 of line in regulation of series Peaked wave, 111 circuit, 306 Permanent instability, 165 with oscillating currents, 347 magnetism, 43 self inductive and mutual in- | Pitch deficiency of winding eliminat- ductive, of alternator arma- ing harmonics, 120 ture, 239 Polarization cell, 8 shunt in series circuit, 298 voltage, 7 regulating series circuit by Polyphase constant current trans- saturation, 302 } formation, 284, 287 of synchronous machines, 232 power equation, unbalanced, total, of transformer, 224 317 , of transformer, measurement, systems, load balance, 314 227 Position change of synchronous and short-circuit stress, 100 motor with load, 209 as wave screen, 153 Power component of reactance with Reactive power of system, total and oscillating currents, 347 resultant, 317 equation of singlephase load, Recovery of induction motor after 315 overload, 204 of unbalanced polyphase Rectification by arc, 32 load, 317 by electronic conduction, 40 Primary cell, 7 giving even harmonics, 159 Pulsating currents and wave screens, Rectifying voltage range of alter- 156 nating arc, 33 magnetic flux and even har- Reflected current in leaky conductor, monics, 159 334

360 INDEX Reflection at end of leaky conductor, Shunt protective device in series 334 circuits, 298 Regulating pole converter and wave Silicon as pyroelectric conductor, 13 shape, 123 steel, hysteresis, 62 Regulation of series circuits by react- magnetic properties, 79 ance shunt, 301 Sine wave as standard, 111 Regulator, constant current, 251 Singing arc, 188 Reluctivity, 43 Singlephase load, power equation, curve, 46 315 Remanent magnetism, 43 Spark conduction, 28 Resistance, 1 discharge producing oscillations, : effective, of leaky conductor, 197 333 Speed change of induction motor of line in series circuits, 306 with load, 209 negative effective, of arc, 191 instability of motor, 202 Resistivity, magnitude of different Stability characteristic’ of arc on . conductors, 42 constant current, 173 Resonance of transformer with har- on constant voltage, 169 monics of magnetic bridged condition of capacity shunting gap, 151 arc, 184 Resonant wave screens, 157 shunting circuit, 178 Resonating circuit, constant current of induction motor, 201 regulation, 256, 261, 282, of parallel operation of arc, 290 175 as wave screen, 154 of synchronous machine, 215 Resultant flux of alternator, 232 curves of arc, 36, 168 Rising magnetic characteristic, 51 of pyroelectric conductor, 20 _ Stable magnetic characteristic, 54 8 Storage battery, 8 Streak conduction of pyroelectric Saturation coefficient, magnetic, 44 conductor, 18, 42 magnetic, 77 Stream voltage of arc, 35

• equation of wave shape, 137 of Geissler tube, 29 shaping waves, 125 Susceptance with oscillating cur- of reactance shunting series rents, 350 circuit, 302 Symmetrical wave, 114 value, magnetic, 46 Synchronizing force and power, 210 Screen, wave-, 153 Synchronous reactance of alter- Secondary cell, 8 nator, 236 Self inductive armature flux of machines, hunting, 208 . alternator, 234 reactance, 232 Series operation, constant current, motor tending to instability, 297 164 constant voltage, 297 T Shape of hysteresis curve, 68 Short circuit stress in transformer, T-connection of constant current 99 transformation, 256, 261, third harmonic in alternator, 282, 290 244 as wave screen, 154

INDEX 361 Temperature coefficient of insula- Unipolar induction, 114 tors, 24 Unstable electrical equilibrium, 165 of electrolytes, 4 magnetic characteristic, 54 of pyroelectrics, 10 Unsymmetrical magnetic cycles, 73 of resistance, 2 Terminal drop of arc, 35 Vv of Geissler tube, 29 Third harmonic absent in balanced Vacuum arc characteristic, 39 three-phase alternator, 242 conduction, 28 present in unbalanced three- Vapor conduction, 28 phase alternator, 243 Voltage, wave distortion by bridged in three-phase winding, 118 magnetic gap, 148 Three-phase system balanced on by magnetic saturation, 128, three-phase load, 325 143 winding and third harmonic, 118 Ww ; Transformer, constant current, 250 Water pipes as circuits, 330 cumulative oscillation, 199 Wave screens, 153 short-circuit stress, 99 separating different harmon- Transient, 165 ics, 157 arc characteristic, 192 pulsating currents, 156 polarization current, 9 distortion in constant current reactance of alternator arma- transformation, 290 ture, 240 improving regulation in series Triple harmonic. See Third har- circuits, 311 monic. ; . shape distortion by magnetic frequency harmonic of single- saturation, 137 phase load, 241 shaping of, 111 True self-inductive flux of alternator transmission in leaky d-c. con- armature, 237 ductor, 334 Two frequency alternator, 116 Z U Zero impedance circuits with oscil- Unidirectional conduction of arc, 32 lating currents, 350

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