None
Then from the preceding discussion, the e.m.f. consumed by re- sistance, r, inductive reactance, x} and condensive reactance, xc, is E = ie-"* | cos (0 - 0)1 r - ax - °. 2xc1 - sin (</> - 0) f x* 1 1 Lx - ITPJ I = izae-a* cos (0 — 0 -f- 5), where OSCILLATING CURRENTS Xc 349 x — tan 5 = 1 + a r — ax — a *• = substituting 0 -j- 5 for 0, and e = iza we have E = ee-a+ cos (</> - 0), . ( cos 5 , . sin 5 . , = ee-a* - cos (<f> — 0) + - - sin (<£ — I Za 2a hence in complex quantities, E = e(cos 6 — j sin 6) dec a, or, substituting, /-•* T „ f cos 5 . sin 5 , , I = e{- J — \ dec a; ( Za Za r — ax — 1 + a Xc dec a. 189. Thus in complex quantities, for oscillating currents, we have: conductance, r — ax — xc susceptance, x — Xc b = 1+a2 admittance, in absolute values. 350 ELECTRIC CIRCUITS in symbolic expression, Since the impedance is z - (r ~ ax ~ rrr.' we have V . . ^a z, Xa z' y = 2~; ° = T» b = r*; ^ &a "a &a that is, the same relations as in the complex quantities in alter- nating-current circuits, except that in the present case all the constants, ra, xa, za, g, z, y, 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, x, and condensive reactance, xet the impedance was represented in symbolic expression by Z = r.+jx. = (r - a* - j-J-j *. or numerically by Thus the inductive reactance, x, as well as the condensive reactance, xc, do not represent wattless e.m.fs. as in an alternating- current circuit, but introduce power components of negative sign, 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. 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-«*-ribx' = 0;x-rr^ = 0' substituting in this equation, *- 2 «L; xc = ; and expanding, we have a = 2aL That is, if in an oscillating-current circuit, the decrement, 1 and the frequency / = r — j , 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, / = j — F> an<^ the decrement, a = m-i 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 a = If r = 0, that is, in a circuit without resistance, we have a = 0, / = - -- ~7='> that is, the currents are alternating with no decre- 2 ment, and the frequency is that of resonance. .If 2r, J 1 < 0, that is, r > 2\/^, a and / become imaginary; T O ~~ JL \ O that is, the discharge ceases to be oscillatory. An electrical discharge assumes an oscillating nature only, if r < 2*f|v In the case r — 2 */~ we have a = oo , / = 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 dec a. Since / = (ii — jiz) dec a, $r = If dec a, Ex = - xf(a - j) dec a, flc = j-qf^i /(- « - i) dec «> we have r ~ ax ~ Xc = 0> hence, by substitution, $xe = xf (- a - j) dec a. OSCILLATING CURRENTS 353 The two constants, z'i and i2, of the discharge, are determined by the initial conditions — that is, the e.m.f. and the current at the time, t = 0. 194. Let a condenser of capacity, C, be discharged through a circuit of resistance, r, and inductance, L. Let e = e.m.f. at the condenser in the moment of closing the circuit — that is, at the time t = 0 or <f> = 0. At this moment the current is zero — that is, / = jit, ii = 0. Since $Xc = xf (— a — j) doc a = e at <£ = 0, y? dec a, we have xiz\/ L + az = e or iz = Substituting this, we have, / --jf dec a, EXc = - — —— - (1 -ja) dec a, , 2 the equations of the oscillating discharge of a condenser of initial voltage, e. Since 'r2C r - 1 we have JL r rL ~ 2a~ 2 Vr2C hence, by substitution, /S, = - je *- dec a, 4L the final equations of the oscillating discharge, in symbolic ex- pression. 23 INDEX Admittance, with oscillating cur- rents, 348 Air gap in magnetic circuit reducing wave distortion, 145 Alloys, resistance, 2 Alternating component of power of general system, 317 current electromagnet, 95 magnetic characteristic, 51 Alternations by capacity inductance shunt to arc, 187 Aluminum cell as condenser, 10 Amorphous carbon resistance, 23 Annealing, magnetic effect, 78 Anode, 6 Anthracite, resistance, 23 Apparatus economy of constant po- tential, constant current transformation, 281 of monocyclic -square, 276 of T connection, 265 Arc as alternating current power generator, 187 characteristics, 34 condition of stability on con- stant current, 173 on constant voltage, 169 conduction, 28, 31, 42 constants, 36 effective negative resistance, 191 equations, 35 as oscillator, 189 parallel operation on constant current, 175 shunted by capacity, 178, 184 and inductance, 184 by resistance on constant current, 172 singing and rasping, 188, 189 tending to unstability, 164 transient characteristic, 192 as unstable conductor, 167 Arcing ground on transmission lines, 199 Area of BH relation, 53 Armature flux of alternator, 233 reactance flux of alternator, 232 reaction of alternator, 236 Attenuation constant, leaky con- ductor, 334 of synchronous machine oscil- lation, 213 B Balance of quarterphase system on singlephase load, 322 of singlephase load, 319 of threephase system on single- phase load, 325 of unbalanced power of system, 319 Bends in magnetic reluctivity curve, 49 Bismuth, diamagnetism, 77 Bridged gap in magnetic circuit, wave distortion, 148 Cable armor as circuit, 330 equation of induced current, 336 Capacity, 1 and inductance shunting circuit, 181 inductance shunt to arc pro- ducing alternations, 187 with oscillating current, 347 and reactance as wave screen, 154 in series regulating for constant current, 247 shunt to arc, 178, 184 to circuit, 178 Carbon, resistance, 21 Cathode, 6 Cell, 7 355 356 INDEX Characteristic, magnetic, 50 Chemical action in electrolytic con- duction, 6 Chromium, magnetic properties, 83 Circuit with distributed leakage, 330 magnetic, 43 Closed magnetic circuit, wave dis- tortion, 139 Cobalt iron alloy, magnetic, 78 magnetic properties, 80 Coefficient of hysteresis, 61 Coherer action of pyroelectric con- ductor, 19 Compensating voltage balancing un- balanced power, 320 Condenser, electrostatic, 9 power equation, 319 tending to instability, 164. See Capacity. Conductance with oscillating cur- rents, 349 Conduction, electric, 1 Conductors, mechanical magnetic forces, 106 Constant component of power in general system, 317 current arc, stability condition, 172 constant potential transfor- mation, 243, 286 reactance, 134 transformer and regulator, 250 magnetic, 77, 87, 88 potential constant current transformation, 243, 286 reactance, 133 term and even harmonics, 158 voltage arc, stability condition, 168 series operation, 297 Continuous conduction, 32 Corona conduction, 29, 42 Creepage, magnetic, 57 Critical points of reluctivity curve, 46 Cumulative oscillation, cause, 166 produced by arc, 188 in transformer, 199 surge, 166 Current wave distorted by mag- netism, 126 Damping power in synchronous motor oscillation, 210 winding in synchronous ma- chines, 211 Danger of higher harmonics, 121 Decrement of oscillating wave, 343 Demagnetization by alternating cur- rent, 54 temperature, 78 Diffusion current of polarization, 8 Direct current producing even har- monics, 159 Discharges, oscillating, 352 Discontinuous conduction, 29 Displacement of field poles eliminat- ing harmonics, 120 of position in synchronous ma- chine, 210 Disruptive conduction, 29, 42 Distortion of wave improving regu- lation in series circuits, 311 of voltage by bridged magnetic gap, 148 in constant potential con- stant current transforma- tion, 290 Distributed leakage of circuit, 330 winding, eliminating harmonics, 116 Double frequency armature reac- tion, 240 peaked wave, 113 Economy, apparatus, 281 Efficiency of electromagnet, 99 of monocyclic square, 277 of T-connection, 268 Electrodes, 6 Electrolytic cell, 8 condenser, 9 conductor, 442 INDEX 357 Electromagnet, 91 constant current, 93 potential, 98 efficiency, 99 Electronic conduction, 28, 40 Elimination of harmonics by alter- nator design, 116 Energy of hysteresis, 57 storage in constant potential constant current transfor- mation, 280 Even harmonics, 114, 153, 157 Excessive very high harmonics in distortion by magnetic sat- uration, 140 Exciting current of transformer de- pending on wave shape, 137 Exponent of hysteresis, 66 Face conductor in alternator, 114 Faraday's law of electrolytic con- duction, 6 Ferrites, magnetic, 80 Ferromagnetic density, 45 Field flux of alternator, 232 Film cutout in series circuits, 298 Flat top wave, 111 Flicker of lamps and wave shape, 124 Flux distribution of alternator field, 114 Fluxes, magnetic of alternator, 232 Forces, mechanical magnetic, 91, 107 Form factor of magnetic wave dis- tortion, 127 Fractional pitch armature winding eliminating harmonics, 119 Frequency conversion in cumulative surge, 166 of synchronous machine oscil- lation, 213 Friction molecular magnetic, 56 Frohlich's law, 43 Gap in magnetic circuit reducing wave distortion, 145 Gas pipes as circuits, 330 vapor and vacuum conduction, 28, 41 Geissler tube conduction, 29, 42 Gem filament incandescent lamp, 22 Grounded leaky conductor, 333 H Half turn windings, 114 Hardness, magnetic, coefficient of, 44 Harmonics, effect of, 121 even, 153, 157 separation by wave screens, 157 Heusler alloys, magnetic properties, 81 High harmonics in alternator, 120 excessive in wave distortion by magnetic saturation, 140 by slot pitch, 120 temperature insulators, 26 Homogeneous magnetic materials, 55 Hunting of synchronous machines, 166, 208 Hysteresis, 56 loss and wave shape, 112 Impedance and admittance with oscillating currents, 346 of line in regulation of series circuits, 306 Induced current in leaky cable armor, 336 Inductance, 1 and capacity shunting circuit, 181 power equation, 316 as wave screen, 153 Induction motor magnetic circuits, 228 instability, 164, 201 Inefficiency of magnetic cycle, 60 Infinitely long leaky conductor, 332 Instability by capacity shunt, 180 of circuits, 165 358 INDEX Instability of induction motors, 201 of pyroelectric conductor, 16 of synchronous motor, 208 Instantaneous power of general sys- tem, 317 Insulators, 23, 42 as pyroelectric conductor, 25 Iron cobalt alloy, magnetic, 78 magnetic properties, 79 resistance, 4 Kennelly's law of reluctivity, 44 Lag of damping power in synchron- ous machine, 213 of synchronizing force, 212 Lamp circuits in series, 297 equivalent of line impedance in series circuits, 306 Law of hysteresis, 62 Leakage, distributed, of circuits, 330 flux of alternating current trans- formers, 217 reducing wave distortion, 145 Leaky conductor, 330, 332, 336 Load balance of polyphase system, 314 character determining stability in induction motor, 205 Loop of hysteresis, 56 Loss, percentage, in magnetic cycle, 60 Loxodromic spiral, 345 Luminescence in gas and vapor con- duction, 28 Luminous streak conduction in pyro- electric conductor, 18 M Magnetic circuits of induction motor, 228 elements, 77 friction, 56 mechanical forces, 107 Magnetism, 43 tables and data, 87, 88 wave distortion by saturation, 128 Magnetite arc, 36 hysteresis, 62 magnetic properties, 80 as pyroelectric conductor, 14 Magnetization curve, 48 Magnetkies, magnetic properties, 80 Manganese alloys, magnetic prop- erties, 81 steel, magnetic properties, 79 Mechanical magnetic forces, 91, 107 Mercury arc characteristic, 39 Metals, resistance, 2 Metallic carbon, resistance, 22 conductors, 142 induction, magnetic, 47 magnetic density, 45 Mixtures as pyroelectric conductors, 21 Molecular magnetic friction, 56 Monocyclic square, 261, 273, 283, 293 Mutual inductive flux of alternator armature reaction, 237 N Negative resistance of arc, effective, 191 Neodymium, magnetism, 77 Nernst lamp conductor, 13, 24 Nickel, magnetic properties, 81 steel, magnetic properties, 79 Nominal induced e.m.f. of alter- nator, 236 O Oils as insulators, 26 Open circuited leaky conductor, 332 magnetic circuit, wave shape distortion, 145 Organic insulators, 24 Oscillating approach to equilibrium condition, 210 currents, 343 discharges, 352 INDEX 359 Oscillations of arcing ground on transmission line, 197 in capacity inductance shunt to circuit, 181 cumulative, produced by arc, 188 which becomes permanent, 165 resistance of arc, 196 Outflowing current in leaky con- ductor, 334 Overshooting of alternator current at load change, 238 Oxygen, magnetism, 77 Pyroelectric conductor, 10, 42 classification, 20 resistance increase by high fre- quency, 19 tending to instability, 164 Pyroelectrolytes, 10, 18 Quarterphase system balanced on singlephase load, 322 Parallel operation of arc on constant current, 175 Peak of current wave by magnetic saturation, 126 reactance, 134 voltage used in arc starting, 152 by magnetic saturation, 128 Peaked wave, 111 Permanent instability, 165 magnetism, 43 Pitch deficiency of winding eliminat- ing harmonics, 120 Polarization cell, 8 voltage, 7 Polyphase constant current trans- formation, 284, 287 power equation, unbalanced, 317 systems, load balance, 314 Position change of synchronous motor with load, 209 Power component of reactance with oscillating currents, 347 equation of singlephase load, 315 of unbalanced polyphase load, 317 Primary cell, 7 Pulsating currents and wave screens, 156 magnetic flux and even har- monics, 159 Rail return circuit, 330 Railway return circuits, 330, 341 Rasping arc, 189 Reactance depending on wave shape, 132 on induction apparatus, 216 inductive, constant current regulation, 246, 281 of line in regulation of series circuit, 306 with oscillating currents, 347 self inductive and mutual in- ductive, of alternator arma- ture, 239 shunt in series circuit, 298 regulating series circuit by saturation, 302 of synchronous machines, 232 total, of transformer, 224 of transformer, measurement, 227 and short-circuit stress, 100 as wave screen, 153 Reactive power of system, total and resultant, 317 Recovery of induction motor after overload, 204 Rectification by arc, 32 by electronic conduction, 40 giving even harmonics, 159 Rectifying voltage range of alter- nating arc, 33 Reflected current in leaky conductor, 334 360 INDEX Reflection at end of leaky conductor, 334 Regulating pole converter and wave shape, 123 Regulation of series circuits by react- ance shunt, 301 Regulator, constant current, 251 Reluctivity, 43 curve, 46 Remanent magnetism, 43 Resistance, 1 effective, of leaky conductor, 333 of line in series circuits, 306 negative effective, of arc, 191 Resistivity, magnitude of different conductors, 42 Resonance of transformer with har- monics of magnetic bridged gap, 151 Resonant wave screens, 157 Resonating circuit, constant current regulation, 256, 261, 282, 290 as wave screen, 154 Resultant flux of alternator, 232 Rising magnetic characteristic, 51 S Saturation coefficient, magnetic, 44 magnetic, 77 equation of wave shape, 137 shaping waves, 125 of reactance shunting series circuit, 302 value, magnetic, 46 Screen, wave-, 153 Secondary cell, 8 Self inductive armature flux of alternator, 234 Series operation, constant current, 297 constant voltage, 297 Shape of hysteresis curve, 68 Short circuit stress in transformer, 99 third harmonic in alternator, 244 Shunt protective device in series circuits, 298 Silicon as pyroelectric conductor, 13 steel, hysteresis, 62 magnetic properties, 79 Sine wave as standard, 111 Singing arc, 188 Singlephase load, power equation, 315 Spark conduction, 28 discharge producing oscillations, 197 Speed change of induction motor with load, 209 instability of motor, 202 Stability characteristic of arc on constant current, 173 on constant voltage, 169 condition of capacity shunting arc, 184 shunting circuit, 178 of induction motor, 201 of parallel operation of arc, 175 of synchronous machine, 215 curves of arc, 36, 168 of pyroelectric conductor, 20 Stable magnetic characteristic, 54 Storage battery, 8 Streak conduction of pyroelectric conductor, 18, 42 Stream voltage of arc, 35 of Geissler tube, 29 Susceptance with oscillating cur- rents, 350 Symmetrical wave, 114 Synchronizing force and power, 210 Synchronous reactance of alter- nator, 236 machines, hunting, 208 reactance, 232 motor tending to instability, 164 T-connection of constant current transformation, 256, 261, 282, 290 as wave screen, 154 INDEX 361 Temperature coefficient of insula- tors, 24 of electrolytes, 4 of pyroelectrics, 10 of resistance, 2 Terminal drop of arc, 35 of Geissler tube, 29 Third harmonic absent in balanced three-phase alternator, 242 present in unbalanced three- phase alternator, 243 in three-phase winding, 118 Three-phase system balanced on three-phase load, 325 winding and third harmonic, 118 Transformer, constant current, 250 cumulative oscillation, 199 short-circuit stress, 99 Transient, 165 arc characteristic, 192 polarization current, 9 reactance of alternator arma- ture, 240 Triple harmonic. frequency harmonic of single- phase load, 241 True self-inductive flux of alternator armature, 237 Two frequency alternator, 116 IT Unidirectional conduction of arc, 32 Unipolar induction, 114 Unstable electrical equilibrium, 165 magnetic characteristic, 54 Unsymmetrical magnetic cycles, 73 Vacuum arc characteristic, 39 conduction, 28 Vapor conduction, 28 Voltage, wave distortion by bridged magnetic gap, 148 by magnetic saturation, 128, 143 W Water pipes as circuits, 330 Wave screens, 153 separating different harmon- ics, 157 pulsating currents, 156 distortion in constant current transformation, 290 improving regulation in series circuits, 311 shape distortion by magnetic saturation, 137 shaping of, 111 transmission in leaky d-c. con- ductor, 334 Zero impedance circuits with oscil- lating currents, 350 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $I.OO ON THE SEVENTH DAY OVERDUE. ENGINEERING LIBRARY KOV 1 4 1950 ore 4 i38o 10m-7, '44 (1064s) YC 33427 ^504. ' Engineering Library UNIVERSITY OF CALIFORNIA LIBRARY. 