CHAPTER X. f EFFECnVH BSSISTANCi: Ain> BJEACTANOB. 72. The resistance of an electric circuit is determined : — 1.) By direct comparison with a known resistance (Wheat- stone bridge method, etc.). This method gives what may be called the true ohmic resistance of the circuit. 2.) By the ratio : Volts consumed in circu it Amperes in circuit In an alternating-current circuit, this method gives, not the resistance of the circuit, but the impedance, z = V/^ + x\ 3.) By the ratio : __ Power consumed __ (E.M.F.)' . (Current)* Power consumed ' where, however, the "power*' and the "E.M.F." do not include the work done by the circuit, and the counter E.M.Fs. representing it, as, for instance, in the case of the counter E.M.F. of a motor. In alternating-current circuits, this value of resistance is the energy coefficient of the E.M.F., — Energy compon ent of E.M.F. Total current It is called the effective resistance of the circuit, since it represents the effect, or power, expended by the circuit. The energy coefficient of current, _ Energy component of current ^ Total E.M.F. is called the effective conductance of the circuit. § 733 EFFECTIVE RESISTANCE AND REACTANCE. 105 In the same way, the value, _ Wattless component of E.M.F. Total current is the effective reactance, and , _ Wattless compo nent of current ■" Total E.M.F. is the effective susceptance of the circuit. While the true ohmic resistance represents the expendi- ture of energy as heat inside of the electric conductor by a current of uniform density, the effective resistance repre- sents the total expenditure of energy. Since, in an alternating-current circuit in general, energy is expended not only in the conductor, but also outside of it, through hysteresis, secondary currents, etc., the effective resistance frequently differs from the true ohmic resistance in such way as to represent a larger expenditure of energy. In dealing with alternating-current circuits, it is necessary, therefore, to substitute everywhere the values "effective re- sistance," "effective reactance," "effective conductance,'* and " effective susceptance," to make the calculation appli- cable to general alternating-current circuits, such as ferric inductances, etc. While the true ohmic resistance is a constant of the circuit, depending only upon the temperature, but not upon the E.M.F., etc., the effective resistance and effective re- actance are, in general, not constants, but depend upon the E.M.F., current, etc. This dependence is the cause of most of the difficulties met in dealing analytically with alternating-current circuits containing iron. 73. The foremost sources of energy loss in alternating- current circuits, outside of the true ohmic resistance loss, are as follows : 1.) Molecular friction, as, a.) Magnetic hysteresis; b) Dielectric hysteresis. 106 ALTERNATING-CURRENT PHENOMENA. [§ 74 2.) Primary electric currents, as, a.) Leakage or escape of current through the in- sulation, brush discharge ; b.) Eddy currents in the conductor or unequal current distribution. 3.) Secondary or induced currents, as, a,) Eddy or Foucault currents in surrounding mag- netic materials ; b.) Eddy or Foucault currents in surrounding con- ducting materials ; r.) Secondary currents of mutual inductance in neighboring circuits. 4.) Induced electric charges, electrostatic influence. While all these losses can be included in the terms effective resistance, etc., only the magnetic hysteresis and the eddy currents in the iron will form the subject of what follows. Magnetic Hysteresis, 74. In an alternating-current circuit surrounded by iron or other magnetic material, energy is expended outside of the conductor in the iron, by a kind of molecular friction, which, when the energy is supplied electrically, appears as magnetic hysteresis, and is caused by the cyclic reversals of magnetic flux in the iron in the alternating magnetic field of force. To examine this phenomenon, first a circuit may be con- sidered, of very high inductance, but negligible true ohmic resistance ; that is, a circuit entirely surrounded by iron, as, for instance, the primary circuit of an alternating-current transformer with open secondary circuit. The wave of current produces in the iron an alternating magnetic flux which induces in the electric circuit an E.M.F., — the counter E.M.F. of self-induction. If the ohmic resistance is negligible, the counter E.M.F. equals the impressed E.M.F. ; hence, if the impressed E.M.F. is §75] EFFECTIVE RESISTANCE AND REACTANCE, 107 a sine wave, the counter E.M.F., and, therefore, the mag- netic flux which induces the counter E.M.F. must follow sine waves also. The alternating wave of current is not a sine wave in this case, but is distorted by hysteresis. It is possible, however, to plot the current wave in this case from the hysteretic cycle of magnetic flux. From the number of turns, «, of the electric circuit, the effective counter E.M.F., E, and the frequency, iV, of the current, the maximum magnetic flux, *, is found by the formula: ^= V2 7r//iV*10-8; hence, . ^}^^ -s/2irnN A maximum flux, *, and magnetic cross-section, 5, give the maximum magnetic induction, (SS = ^ / S. If the magnetic induction varies periodically between + (B and — (B, the M.M.F. varies between the correspond- ing values + IF and — 5^, and describes a looped curve, the cycle of hysteresis. If the ordinates are given in lines of magnetic force, the abscissae in tens of ampere-turns, then the area of the loop equals the energy consumed by hysteresis in ergs per cycle. From the hysteretic loop the instantaneous value of M.M.F. is found, corresponding to an instantaneous value of magnetic flux, that is, of induced E.M.F. ; and from the M.M.F., IF, in ampere-turns per unit length of magnetic cir- cuit, the length, /, of the magnetic circuit, and the number of turns, «, of the electric circuit, are found the instantaneous values of current, /, corresponding to a M.M.F., SF, that is, as magnetic induction (B, and thus induced E.M.F. e, as : n 75. In Fig. 65, four magnetic cycles are plotted, with maximum values of magnetic inductions, (B = 2,000, 6,000, 10,000, and 16,000, and corresponding maximum M.M.Fs., 108 AL7ERNATING-CURRENT PHENOMENA. [S7& JF = 1.8, 2.8, 4.3, 20.0. They show the well-known hys- teretic loop, which becomes pointed when magnetic satu- ration is approach^. These magnetic cycles correspond to average good sheet iron or sheet steel, having a hysteretic coefficient, ij = .0033, and are given with ampere-turns per cm as abscissae, and kilo-lines of magnetic force as ordinates. 1 •:::: - " /, ,^' 1 .» 1 rJ 1^ h 1 „ 1 1 „ 1 - ^ ^ ^ y' ^ - =: ' Fig. eS. Hiitttrtfh Cytit of Bt In Figs. 66, 67, 68, and 69, the .sine curve of magnetic induction as derived from the induced E.M.F. is plotted in dotted lines. For the different values of magnetic induction of this sine curve, the corresponding values of M.M.F., hence of current, are taken from Fig. 66, and plotted, giving thus the exciting current required to produce the sine wave of magnetism ; that is, the wave of current which a sine wave of impressed E.M.F. will send through the circuit. ! 76] EFFECTIVE RESISTANCE ASD REACTANCE. 109 As shown in Figs. 66, 67, 68, and 69. these waves of alternating current are not sine waves, but are distorted by the superposition of higher harmonics, and are complex harmonic waves. They reach their maximum value at the same time with the maximum of magnetism, that is, 90° Fijt. Mai ' «7. Olllorllon of CUf ahead of the maximum induced E.M,F., and hence about 90° behind the maximum impressed E.M.F., but pass the zero line considerably ahead of the zero value of magnet- ism, or 42°, 52°, 50°, and 41°, respectively. The general character of these current waves is, that the maximum point of the wave coincides in time with the max- 110 AL TERNA TING-CURRENT PHEXOMENA (8 75 imum point of the sine wave of magnetism ; but the current wave is bulged out greatly at the rising, and hollowed in at the decreasing, side. With increasing magnetization, the maximum of the current wave becomes more pointed, as shown by the curve of Fig, 68, for = 10,000 ; and at still c -? / \ ce 10 Loo -V \ fF 4, \ / \1 3 4. \ ^ \ -^ / ^ N \ y ~ •N K / \ ^ ~ \ ^ •^ / '\ it \ \ X M L \ /• 1 "V (8 16( oo r\ / y * J 20 \ / \ / \ c 13 / \ / \ / ~^ VJ / w \ ^ N, ^ ' "\ \l flit. ttaaMM. Olitoriiiui of Cym I Wart ty Hyitemli. higher saturation a peak is formed at the maximum point, as in the curve of Fig. 69, for (B = 16,000. This is the case when the curve of magnetization remains within the range of magnetic saturation, since in the proximity of satu- ration the current near the maximum point of magnetization has to rise abnormally to cause even a small increase of magnetization. §§76,77] EFFECTIVE RESISTANCE AND REACTANCE, 111 The four curves, Figs. 66, 67, 68, and CO, are not drawn to the same scale. The maximum values of M.M.F., cor- responding to the maximum values of magnetic induction, > / / \ \ \ / / M ^ ^ A / -■ ■"-. A\ V / / •■ -. ^ -^ L' -\ '. ^ ^ — ' J / -^ \ ~1 — — ^ f \ \ -\ // \ / \ ^ \ ^ rift. 70 aia 71. Dlttortlan vf Cum lent sine waves and the wattless complex higher harmonics, which together form the distorted current wave. The equivalent sine wave of M,M.F. or of current, in Figs. 66 to 69, leads the magnetism by 34°, 44°, S8°, and 15°.5, S 78] EFFECTIVE RESISTANCE AND REACTANCE. 113 respectively. In Fig. 71 the equivalent sine wave almost coincides with the distorted curve, and leads the magnetism by only 9°. It is interesting to note, that even in the greatly dis- torted curves of Figs. 66 to 68, the maximum value of the equivalent sine wave is nearly the same as the maximum value of the original distorted wave of M.M.F., so long as magnetic saturation is not approached, being 1.8, 2.9, and 4.2, respectively, against 1.8, 2.8, and 4.3, the maximum values of the distorted curve. Since, by the definition, the effective value of the equivalent sine wave is the same as that of the distorted wave, it follows, that the distorted wave of exciting current shares with the sine wave the feature, that the maximum value and the effective value have the ratio of V2 -=- 1. Hence, below saturation, the maximum value of the distorted curve can be calculated from the effective value — which is given by the reading of an electro-dynamometer — by using the same ratio that applies to a true sine wave, and the magnetic characteris- tic can thus be determined by means of alternating cur- rents, with sufficient exactness, by the electro-dynamometer method. 78. In Fig. 72 is shown the true magnetic character- istic of a sample of good average sheet iron, as found by the method of slow reversals with the magnetometer ; for comparison there is shown in dotted lines the same char- acteristic, as determined with alternating currents by the electro-dynamometer, with ampere-turns per cm as ordi- nates, and magnetic inductions as abscissae. As repre- sented, the two curves practically coincide up to (B = 10,000 to 14,000 ; that is, up to the highest inductions practicable in alternating-current apparatus. For higher saturations, the curves rapidly diverge, and the electro-dynamometer curve shows comparatively small M.M.Fs. producing appar- ently very high magnetizations. AL TERNA Tr.\'G-CURRF..VT PllEXQMF.NA. The same Fig. 72 gives the curve of hysteretic loss, in ergs per cm^ and cycle, as ordinates, and magnetic induc- tions as abscissx. / ' / ; / / J /I / 1 / / 1 / / / 1 / / / ^ ,' / / Ul / // ^ / ' / / ' / / / / / ^' ' / ^ '-' " ^ •r- ^ ^ y i^ ^n; rr ST »M ri3 .- • M OS ns inr sns i»iS 5o« ma. SeiT wTF W " F!g. 12. HajBttliatlBH ant Hytttmli Can*. The elect r<>-dynamometcr methoii of determining the magnetic characteristic is preferable for use with alter- nating-current apparatus, since it is not affected by the phenomenon of magnetic "creeping," which, especially at § 79] EFFECTIVE RESISTANCE AND REACTANCE, 115 low densities, brings in the magnetometer tests the magnet- ism usually very much higher, or the M.M.F. lower, than found in practice in alternating-current apparatus. So far as current strength and energy consumption are concerned, the distorted wave can be replaced by the equi- valent sine wave, and the higher harmonic neglected. All the measurements of alternating currents, with the single exception of instantaneous readings, yield the equiv- alent sine wave only, and suppress the higher harmonic ; since all measuring instruments give either the mean square of the current wave, or the mean product of instantaneous values of current and E.M.F., which, by definition, are the same in the equivalent sine wave as in the distorted wave. Hence, in all practical applications, it is permissible to neglect the higher harmonic altogether, and replace the dis- torted wave by its equivalent sine wave, keeping in mind, however, the existence of a higher harmonic as a possible disturbing factor which may become noticeable in those very rare cases where the frequency of the higher harmonic is near the frequency of resonance of the circuit. 79. The equivalent sine wave of exciting current leads the sine wave of magnetism by an angle a, which is called the angle of hysteretic advance of phase. Hence the cur- rent lags behind the E.M.F by ^^ 90° — a, and the power is therefore, P= IE cos (90° ^a) = IE sin a. Thus the exciting current, /, consists of an energy com- ponent, / sin a, which is called the hysteretic energy current y and a wattless component, /cos a, which is called the mag- netizing current. Or, conversely, the E.M.F. consists of an energy component, E sin a, the hysteretic energy E.M.F., and a wattless component, E cos a, the E.M.F. of self induction. Denoting the absolute value of the impedance of the IIG AL TERNA TING-CURRENT PHENOMENA, [§ 80 circuit, -£"//, by ir, — where z is determined by the mag- netic characteristic of the iron, and the shape of the magnetic and electric circuits, — the impedance is repre- sented, in phase and intensity, by the symbolic expression, Z =^ r ^ jx = ;? sin a — jz cos a ; and the admittance by, K = ^ + y ^ = - sin a + y - cos a = >» sin a + jy cos a. z z The quantities, xr, r, ;r, and y^ gy 6, are, however, not constants as in the case of the circuit without iron, but depend upon the intensity of magnetization, (B, — that is, upon the E.M.F. This dependence complicates the investigation of circuits containing iron. In a circuit entirely inclosed by iron, a is quite consider- able, ranging from 30° to 50° for values below saturation. Hence, even with negligible true ohmic resistance, no great lag can be produced in ironclad alternating-current circuits. 80. The loss of energy by hysteresis due to molecular friction is, with sufficient exactness, proportional to the 1.6*** power of magnetic induction (B. Hence it can be ex- pressed by the formula : where — JVfg = loss of energy per cycle, in ergs or (C.G.S.) units (= 10"" "^ Joules) per cm', (B = maximum magnetic induction, in lines of force per cm^, and i; = the coefficient of hysteresis. This I found to vary in iron from .00124 to .0055. As a fair mean, .0033 can be accepted for good average annealed sheet iron or sheet steel. In gray cast iron, y\ averages .013 ; it varies from .0032 to .028 in cast steel, according to the chemical or physical constitution ; and reaches values as high as .08 in hardened steel (tungsten and manganese i 80] EFFECTIVE RESISTANCE AND REACTANCE. 117 Steel). Soft nickel and cobalt have about the same co- efficient of hysteresis as gray cast iron ; in magnetite I found i; = .023. In the curves of Fig. 62 to 69, i; = .0083. At the frequency, N, the loss of power in the volume, V^ is, by this formula, — P^nN r(B'« 10 - ^ watts = nNv(^'\o-'' watts, where 5 is the cross-section of the total magnetic flux, *. The maximum magnetic flux, *, depends upon the counter E.M.F. of self-induction, E= V2^iV/i*10-«, EW or * = V2 ir Nn where n = number of turns of the electric circuit. Substituting this in the value of the power, Py and canceling, we get, — p = i^ ^^io<^« ^ gg e;^ rio » . or ^= -^vM-' Where ^=, ^,,^,,^,^^^^ ==58,^^,^^^; or, substituting r) = .0033, we have A = 191.4 ^^ ^ ^ ; or, substituting y= SZ, where Z = length of magnetic circuit, _ r^Z 10 '-' _ 58,yZ10» _ . . , Z . and /> = 58 77 jg '-^ Z 10* ^ 191.4 ^>'^Z ^.6^.6,^1.6 iV-* 5 ••«*•• * In Figs. 73, 74, and 75, is shown a curve of hysteretic loss, with the loss of power as abscissae, and in curve 73, with the E.M.F., E, as ordinates, for Z = 1, 5 == 1, N= 100, and;/ = 100; 118 ALTERXAThXG-CURRENT PHENOMENA. [§80 J /' IIIUI / ,..,T P" rr NEj UN P f O. 1. 00 K: IOC / / / vm / / UMI DL -f / / / am / m y ,M f. n » f\9- 73. Hgtttmlt Leu (u Fmethn vf C. » u _X Li 1 H BETW M- P 11 'Ul '■ ,, f 0- •• \ s •v, „ MB in ,r UN — ■ T r" 1 H * M m 1 DO mi Flf. T4. HiftUruli lau — fmttlon of Humbtr of Taritt. £81] EFFECTIVE RESISTANCE AND REACTANCE. - 1 M 1 1 1 1 1 1 1 fe«S-I.L=l...-IOO.E-100 ' " " \ T \ ■'■ I \ ' ^ - — , " N- F« »" HC Fie. 7B. Hfitttntli lm> as fwicdon of Cgelit. in curve 74, with the number of turns as ordinates, for Z = i,S= l,Ar= 100, and-f^ 100; in curve 75, wiih the frequency, jV, or the cross-section, S, as ordinates, for /,=.!,« = 100, and £ — 100. As shown, the hysteretic loss is proportional to the 1.6''' power of the E.M.F,, inversely proportional to the 1.6"' power of the number of turns, and inversely proportional to the .6'*' power of frequency, and of cross-section. 81, Ji £■ = effective conductance, the energy compo- nent of a current is /= E^g; and the energy consumed in a conductance, ^, is P = IE = E'^g- Since, however : we have A - N-* = ^S\ From this we have the following deduction ; 120 ALTEENATJNG-CVMRENT PHENOMENA. [181 The effective conductance due to magnetic hysteresis is proportional to the coefficient of hysteresis, ij, and to the length of the magnetic circuit, L, and inversely proportional to the 4'^pO'wer of the E.M.F., to the .6'* power of tlie frequency, N, and of tlie .cross-section of the magnetic circuit, S, and to the 1.6'* power of the number oj turns, n. Hence, the effective hysteretic conductance increases with decreasing E.M.F,, and decreases with increasing M 1 1 1 1 1 1 1 1 ^ tVi^-T^^^Q^%-= if "1 00 '^ \ \ ^ ' ~ •~~ 2 ~- — . a - — _ E f\f. 19. Hfwitrula OontfifCtanM ni FUBtUon ef f.M.f. E.M.F. ; it varies, however, much slower than the E.M.F, so that, if the hysteretic conductance represents only a part of the total energy consumption, it can, within a limited range of variation — as, for instance, in constant potential transformers — be assumed as constant without serious error. In Figs. 76, 77, and 78, the hysteretic conductance.^, is plotted, for A = 1, .^ = 100, .V= 100, 5 = 1, and n = 100, respectively, with the conductance, g, as ordinates, and with S81] EFFECTIVE RESISTANCE AND REACTANCE. 121 M 1 1 1 1 1 1 1 ! 1 1 1 1 foV L-1°e"oo"''-I! n"-10O ' " ' 1 ' \ ■" \ ■-- ^ — 1 _ _ i -_ _ L ^ _ _ - Fig. TT. Hgttemlt Cmtfuclanca ■■ FaiKtlon of Ci IIU tU ION Br 1* DO FOI ^ H. 1. u .IL o a " \, IL , 's RO ■ — FIj. 78. Hiattmla CoBAwtomw oa ArnethM (>f Hambir of Twia. 122 AL TERN A TING-CURRENT PHENOMENA. [ § 82 E as abscissae in Curve 76. iVas abscissas in Curve 77. n as abscissae in Curve 78. As shown, a variation in the E.M.F. of 50 per cent causes a variation in g of only 14 per cent, while a varia- tion in iV or 5 by 50 per cent causes a variation in g of 21 per cent. If (R = magnetic reluctance of a circuit, IF^ = maximum M.M.F., / = effective current, since / V2 = maximum cur- rent, the magnetic flux, ^ IF^ «/V2 (R CR • Substituting this in the equation of the counter E.M.F. of self-induction, ^=V2 7r^«*10-», , J, 27r«^^Z10-» we have E = ; (R hence, the absolute admittance of the circuit is ^= VJM^ = -^ = (R10« a^ E 2irn^N N' , 10» where a = - — - , a constant. Therefore^ the absolute admittance^ y$ of a circuit of neg- ligible resistance is proportional to the magnetic reluctance^ (R, and inversely proportional to the frequency f N, and to the square of the number of tums^ n. 82. In a circuit containing iron, the reluctance, (R, varies with the magnetization; that is, with the E.M.F. Hence the admittance of such a circuit is not a constant, but is also variable. In an ironclad electric circuit, — that is, a circuit whose magnetic field exists entirely within iron, such as the mag- netic circuit of a well-designed alternating-current trans- J 82] EFFECTIVE RESISTANCE AND REACTANCE. 123 former, — (R is the reluctance of the iron circuit. Hence, if ;x = permeability, since — and $F^ = Z-F= :?^Z3C = M.M.F., * = 5CB = /A 5 JC = magnetic flux, and (R = ; ; 4 TT/XO substituting this value in the equation of the admittance, (R10« , Z10» z y = ;^ TTr» ^^ have where z = 2 7r«W Stt^z/V-STV^ ^ft' Z10» 127Z10« 8ir'^«^5 //^^ Therefore^ in an ironclad circuity the absolute admittance^ y^ is inversely proportional to the freqtiettcy, N^ to the pemie- ability, ^, to the cross-section, S, and to the square of the number of turns, n ; and directly proportional to the length of the magnetic circuit, Z. The conductance is ^ = ; and the admittance, y = Nil' hence, the angle of hysteretic advance is Sm a = -^ = ^^ ; y ZE'' ' or, substituting for A and z (p. 117), N'^ TiZ10»» %T»n^S sm a = u — '- , or, substituting E = 2^nJVnS(&10'^, we have sin a = —t-^ 124 AL TERNA TING-CURRENT PHENOMENA. [ § 83- which is independent of frequency, number of turns, and shape and size of the magnetic and electric circuit. Therefore^ in an ironclad inductance^ the angle of hysteretic advance^ a, dcpetids upon the magnetic constants^ permeability and coefficient of hysteresis^ atid upon the maximmn magnetic induction^ but is entirely independent of t/te frequency y of the sliape and other conditiotis of the magnetic and electric circuit ; andy therefore y all ironclad magnetic circuits constructed of t/ie same quality of iron and using the same magnetic density^ give the same angle of hysteretic advaftce. The angle of hysteretic advance^ o, in a closed circuit transformer, depends upon the quality of the iron, and upon the magftctic density only. The sine of the angle of hysteretic advatue equals Jf times the product of the permeability and coefficient of hysteresis, divided by the .^'* power of the magnetic density. 83. If the magnetic circuit is not entirely ironclad, and the magnetic structure contains air-gaps, the total re- luctance is the sum of the iron reluctance and of the air reluctance, or cR = (R^ + (R„ ; hence the admittance is Therefore, in a circuit containing iron, the admittattce ts the sum of the admittance due to the iron part of the circuit, y^ = a I N(^^, and of the admit tame due to the air part of the circuit, y^= a I iV(R„ , if the iron and the air are in series in the magfietic circuit. The conductance, g, represents the loss of energy in the iron, and, since air has no magnetic hysteresis, is not changed by the introduction of an air-gap. Hence the angle of hysteretic advance of phase is sm a = -^ = — ^ — = -^ s — , y yi-\-ya yi cR< + (Ha § 84] EFFECTIVE RESISTANCE AND REACTANCE. 125 and a maximum, ^/^,, for the ironclad circuit, but decreases with increasing width of the air-gap. The introduction of the air-gap of reluctance, (R^, decreases sin a in the ratio, ^- (R, + (R/ In the range of practical application, from (B = 2,000 to (B'= 12,000, the permeability of iron varies between 900 and 2,000 approximately, while sin a in an ironclad circuit varies in this range from .51 to .69. In air, fi = 1. If, consequently, one per cent of the length of the iron is replaced by an air-gap, the total reluctance only varies in the proportion of 1 J to Ig^^j or about 6 per cent, that is, remains practically constant ; while the angle of hysteretic advance varies from sin a = .035 to sin a = .064. Thus g is negligible compared with ^, and b is practically equal to y. Therefore, in an electric circuit containing iron, but forming an open* magnetic circuit whose air-gap is not less than yij> the length of the iron, the susceptance is practi- cally constant and equal to the admittance, so long as saturation is not yet approached, or, . (Rrt5 N b = — , or : a: = — . N (Rtf The angle of hysteretic advance is small, below 4**, and the hysteretic conductance is. The current wave is practically a sine wave. As an instance, in Fig. 71, Curve II., the current curve of a circuit is shown, containing an air-gap of only ^J^ of the length of the iron, giving a current wave much resem- bling the sine shape, with an hysteretic advance of 9°. 84. To determine the electric constants of a circuit containing iron, we shall proceed in the following way : Let — £ = counter E.M.F. of self-induction ; 126 AL TERN A TING-CURRENT PHENOMENA, [§ 84 then from the equation, ^ = V2ir«^*10-», where, N = frequency, n = number of turns, we get the magnetism, *, and by means of the magnetic cross section, 5, the maximum magnetic induction : (B = /^. From (B, we get, by means of the magnetic characteristic of the iron, the M.M.F., = $F ampere-turns per cm length, where if OC = M.M.F. in C.G.S. units. Hence, if Li = length of iron circuit, ^^z=z L^^ =. ampere-turns re- quired in the iron ; if La = length of air circuit, ^a = — -~ — -=* ampere-turns re- quired in the air ; hence, "5=^^^ -^^ "5^=^ total ampere-turns, maximum value, and $F / V2 = effective value. The exciting current is «V2 and the absolute admittance, E If $Fj is not negligible as compared with IF^, this admit- tance, j, is variable with the E.M.F., E, If — V = volume of iron, x) = coefficient of hysteresis, the loss of energy by hysteresis due to molecular magnetic friction is, hence the hysteretic conductance is ^ = IV / E^, and vari- able with the E.M.F., E. §85] EFFECTIVE RESISTANCE AND REACTANCE. 127 The angle of hysteretic advance is, — sin a = ^ ; the susceptance, b = -\/y^ — g^ \ the effective resistance, r = -^ ; and the reactance, ^ == — f 85. As conclusions, we derive from this chapter the following : — 1.) In an alternating-current circuit surrounded by iron, the current produced by a sine wave of E.M.F. is not a true sine wave, but is distorted by hysteresis. 2.) This distortion is excessive only with a closed mag- netic circuit transferring no energy into a secondary circuit by mutual inductance. 3.) The distorted wave of current can be replaced by the equivalent sine wave — that is a sine wave of equal effec- tive intensity and equal power — and the superposed higher harmonic, consisting mainly of a term of triple frequency, may be neglected except in resonating circuits. 4.) Below saturation, the distorted curve of current and its equivalent sine wave have approximately the same max- im lun value. 5.) The angle of hysteretic advance, — that is, the phase difference between the magnetic flux and equivalent sine wave of M.M.F., — is a maximum for the closed magnetic circuit, and depends there only upon the magnetic constants of the iron, upon the permeability, /x, the coefficient of hys- teresis, i;, and the maximum magnetic induction, as shown in the equation, » sin a = —i—l . (B* 6.) The effect of hysteresis can be represented by an admittance, Y = ^ +jb, or an impedance, Z = r —jx. 7.) The hysteretic admittance, or impedance, varies with the magnetic induction ; that is, with the E.M.F., etc. 1 28 AL TERN A TING-CURRENT PHENOMENA, [ § 85 8.) The hysteretic conductance,^, is proportional to the coefficient of hysteresis, % and to the length of the magnetic circuit, Z, inversely proportional to the .4*** power of the E.M.F., E, to the Jfi^^ power of frequency, N, and of the cross-section of the magnetic circuit, 5, and to the 1.6*** power of the number of turns of the electric circuit, ;/, as expressed in the equation, ^ 58 iy Z 10« 9.) The absolute value of hysteretic admittance, — is proportional to the magnetic reluctance : (R = (R, + (R„ , and inversely proportional to the frequency, JVy and to the square of the number of turns, ;/, as expressed in the 10.) In an ironclad circuit, the absolute value of admit- tance is proportional to the length of the magnetic circuit, and inversely proportional to cross-section, S, frequency, N, permeability, fi, and square of the number of turns, n, or . = l?iA10« 11.) In an open magnetic circuit, the conductance,^, is the same as in a closed magnetic circuit of the same iron part. 12.) In an open magnetic circuit, the admittance,/, is practically constant, if the length of the air-gap is at least yJu of the length of the magnetic circuit, and saturation be not approached. 13.) In a closed magnetic circuit, conductance, suscep- tance, and admittance can be assumed as constant through a limited range only. 14.) From the shape and the dimensions of the circuits, and the magnetic constants of the iron, all the electric con- stants, ^, 6,j^; r, X, s, can be calculated. §86] FOUCAULT OR EDDY CURRENTS. 129