CHAPTER XII EFFECTIVE RESISTANCE AND REACTANCE 89. 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 resist- ance of the circuit. 2. By the ratio: Volts consumed in circuit Amperes in circuit In an alternating-current circuit, this method gives, not the resistance of the circuit, but the impedance, z = \/f^ + x^. 3. By the ratio: Power consumed, (Current) 2 where, however, the "power" does 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 power coefficient of the e.m.f.. Power component of e.m.f. Total current It is called the elective resistance of the circuit, since it represents the effect, or power, expended by the circuit. The power coeffi- cient of current, Power component of current ^ " Total e.m.f. ' is called the effective conductance of the circuit. Ill 112 ALTERNATING-CURRENT PHENOMENA In the same way, the value, Wattless component of e.m.f. X = Total current is the effective reactance, and Wattless component of current Total e.m.f. is the effective suscepta7ice of the circuit. While the true ohmic resistance represents the expenditure of power as heat inside of the electric conductor b}^ a current of uniform density, the effective resistance represents the total expenditure of power. Since in an alternating-current circuit, in general power is expended not only in the conductor, but also outside of it, through hysteresis, secondary currents, etc., the effective resist- ance frequently differs from the true ohmic resistance in such way as to represent a larger expenditure of power. In dealing with alternating-current circuits, it is necessarj-, therefore, to substitute everywhere the values "effective re- sistance," "effective reactance," "effective conductance," and "effective susceptance," to make the calculation applicable to general alternating-current circuits, such as inductive reactances containing iron, 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 reactance are, in gen- eral, 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. 90. 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; (6) Dielectric hysteresis. 2. Primary electric currents, as, (a) Leakage or escape of current through the insulation, brush discharge, corona. (6) Eddy currents in the conductor or unequal current distribution. EFFECTIVE RESISTANCE AND REACTANCE 113 -» 3. Secondary or induced currents, as, (a) Eddy or Foucault currents in surrounding magnetic materials; (b) Eddy or Foucault currents in surrounding conducting materials ; (c) Secondary currents of mutual inductance in neighboring circuits. 4. Induced electric charges, electrostatic induction or influence. While all these losses can be included in the terms effective resistance, etc., the magnetic hysteresis and the eddy currents are the most frequent and important sources of energy loss. Magnetic Hysteresis 91. In an alternating-current circuit surrounded by iron or other magnetic material, energy is expended outside of the con- ductor 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. To examine this phenomenon, first a circuit may be con- sidered, of very high inductive reactance, 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 mag- netic flux which generates in the electric circuit an e.ni.f. — the counter e.m.f. of self-induction. If the ohmic resistance is negligible, that is, practically no e.m.f. consuzned by the resist- ance, all the impressed e.m.f. must be consumed by the counter e.m.f. of self-induction, that is, the counter e.m.f. equals the impressed e.m.f.; hence, if the impressed e.m.f. is a sine wave, the counter e.m.f., and, therefore, the magnetic flux which generates the counter e.m.f., must follow a sine wave 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 cur- rent wave in this case from the hysteretic cycle of magnetic flux. From the number of turns, n, of the electric circuit, the effective counter e.m.f., E, and the frequency, /, of the current, the maxi- mum magnetic flux, ^, is found by the formula: E = \/2 7r7(/$ 10-^; 8 114 ALTERNATING-CURRENT PHENOMENA hence, ElO^ $ = —-= V 2 7rr?; A maximiiin flux, , and magnetic cross-section, A, give the ... $ maximum magnetic induction, B = ^^ If the magnetic induction varies periodically between + B and — B, the magnetizing force varies between the corresponding values + / and — /, and describes a looped curve, the cycle of hysteresis. If the ordinates are given in lines of magnetic force, the abscissas 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 magnetiz- ing force is found, corresponding to an instantaneous value of magnetic flux, that is, of generated e.m.f. ; and from the mag- netizing force, /, in ampere-turns per units length of magnetic circuit, the length, I, of the magnetic circuit, and the number of turns, n, of the electric circuit, are found the instantaneous values of current, i, corresponding to a magnetizing force, /, that is, magnetic induction, B, and thus generated e.m.f., e, as: n 92. In Fig. 79, four magnetic cycles are plotted, with maximum values of magnetic induction, B = 2,000, 6,000, 10,000, and 16,- 000, and corresponding maximum magnetizing forces, / = 1.8, 2.8, 4.3, 20.0. They show the well-known hysteretic loop, which becomes pointed when magnetic saturation is approached. These magnetic cycles correspond to sheet iron or sheet steel, of a hysteretic coefficient, -q — 0.0033, and are given with ampere-turns per centimeter as abscissas, and kilolines of mag- netic force as ordinates. In Figs. 80 and 81, the curve of magnetic induction as derived from the generated e.m.f. is a sine wave. For the different values of magnetic induction of this sine curve, the corresponding values of magnetizing force /, hence of current, are taken from Fig. 79, 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 establish in the circuit. EFFECTIVE RESISTANCE AND REACTANCE 115 + 16,000 4 14,0(X) ;^ === " 12,009^ ■^ y ^ ^ (0,000 , / / / + p/xJo y 7 / A 6,000 / / / 4,00( // / // /-t- 2,000/ / , 0 -1 S _] B -1 4 _: 3 -1 3 - _i i /. :i + + + 0 +1 ! +1 1 1 1 4 +16 +l!Si +20 I /. '/I i/m '/ ^ 6,qCfl / / /J /, X^ imw / / ,^ 12,0( 0 / / 14,0( 0 — ___. ^=^ ^ 16 0( 0 Fig. 79.- — Hysteretic cycle of sheet iron > ^ =»^ S^ / /^ S \ B = 2000 /=1.8 1= 1.8 / \ ^. N /A / i . — — , ^ j^ 3^ k ^ / ^ /, ■^ "■ -a^ ^ y \^ ^ / i '/ ^ \ ^ :> I , k \ >< ^ ^ r^ N V s. \ B = 6000 /=2.8 1=2.9 /I S \ ^"^ / y r' \ / [/ _^ 2 -^ s 1 / ^ "^-- \ / \ ^ ^ y -N / \ \ '^ \, j '^ \ k / 1 \ // ^ ^/ / X- Fit!. 80. 116 ALTERNATING-CURRENT PHENOMENA As shown in Figs. 80 and 81, these waves of alternating current are not sine waves, but are distorted by the super-position 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° ahead of the maximum generated 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 magne- tism of 42°, 52°, 50° and 41°, respectively. I -^^^ f / ^ ><:^ \^ \ B = 10000 / = 4.3 1=4.2 // ^^ \ ^ / / — i - \ \ \ / / ^ \ / \ y "^ / \ \ 1 -^ y N /^ \: \ / ^ ^^ ^^ / / /\ f \ \ ^^ -^^ •^ J- \ ^^ \y / \ 8 = 16000 / = 20 1=13 A / ^ y \ ^ "^ V ^ y K. / \ \ N / \ / /. y \ %* 1 s ^ / / > /^ \J \ ^ N / / X \ \ "^ / -\ vy ^ Fig. 81. The general character of these current waves is, that the maxi- mum point of the wave coincides in time with the maximum 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 cur- rent wave becomes more pointed, as shown by the curves of Fig. 81, for B = 10,000; and at still higher saturation a peak is EFFECTIVE RESISTANCE AND REACTANCE 117 formed at the maximum point, as in the curve for B = 16,000. This is the case when the curve of magnetization reaches within the range of magnetic saturation, since in the proximity of saturation the current near the maximum point of magnetization has to rise abnormally to cause even a small increase of magneti- zation. The four curves, Figs. 80 and 81 are not drawn to the same scale. The maximum values of magnetizing force, corre- sponding to the maximum values of magnetic induction, B = 2,000, 6,000, 10,000, and 16,000 lines of force per square centi- meter, are/ = 1.8, 2.8, 4.3, and 20.0 ampere-turns per centimeter. In the different diagrams these are represented in the ratio of 8:6:4:1, in order to bring the current curves to approximately the same height. The magnetizing force, in c.g.s. units, is H = 47r//10 = 1.257/. 93. The distortion of the current waves, /, in Figs. 80 and 81, is almost entirely due to the magnetizing current, and is caused by the disproportionality between magnetic induction, B, and magnetizing force, /, as exhibited by the magnetic characteristic or saturation curve, and is very little due to hysteresis. Resolving these curves, /, of Figs. 80 and 81 into two com- ponents, one in phase with the magnetic induction, B, or sym- metrical thereto, hence in quadrature with the induced e.m.f., and therefore wattless: the magnetizing current, ?'„; and the other, in time quadrature with the magnetic induction, B, hence in phase, or symmetrical, with the generated e.m.f., that is, representing power: the hysteresis power-current, ih. Then we see that the hysteresis power-current, ih, is practically a sine wave, while the magnetizing current, im, differs considerably from a sine wave, and tends toward peakedness — the more the higher the magnetic induction, B, that is, the more magnetic saturation is approached, so that for B = 16,000 a very high peak is shown, and the wave of magnetizing current, im, does not resemble a sine wave at all, but at the maximum value is nearly four times higher than a sine wave of the same instan- taneous values near zero induction would have. These curves of hysteresis power-current, ih, and magnetiz- ing current, i„„ derived by resolving the distorted current curves, /, of Figs. 80 and 81, are plotted in Fig. 82, the last one, corresponding to B = 16,000, with one-quarter the ordinates of the first three. 118 ALTERNATING-CURRENT PHENOMENA As curves, symmetrical with regard to the maximum value of B — im — , and to the zero value of 5 — 4 — , these curves are constructed thus: Let h = B sin (f) = sine wave of magnetic induction, 2.0 ~ " >^ ^ 1.0 i-h ^ ^ ps X. m -- > < V ■^ 0 ^ ■^ ^ *~~, -^ ' ^ ^ ^ ■^^ -1.0 ^ ^ > < , ^ ■^ -" Jh^ 2.0 J" / \ ^ — ~ / / S 1.0 /" \ S s /' ^ \ k, 0 ^ ^ -^ ^ ^ ^ /^ '' ^ N ^ -1.0 / ^ 1-m.^ \ \ V. 'h / N 2.0 M^ ■^ \ / s / > K V 1.0 / y \ \ / X \ \ / ^ \ S s / - y \ N s -1.0 / y \ \ / / 'in/ \ k \ -2.0 / / ^ \ \ \ / , ^ ^ 12.0 10.0 8.0 6.0 4.0 2.0 0 -2.0 -4.0 -6.0 -8.0 -10 0 \ / 1 ^ V / > s / \, V / / s if, / / \ —- y «i-. .^ \ ^ '' ^ ■^ — •^ ^ ■^ ~^ ■^ V. / — — Li. C \ / \ t / HYSTERESIS POWER CURRENT AND MAGNETIZING CURRENT B = 2000 6000. 10000, 16000 /•=1.8 2.8 4.3 20.0 / ^ -14.0 -16.0 -18.0 -20.0 \ J V \l 1 _ then Fig. 82. — X^ ih — X4. That is, im is the average value of / for an angle 0, and its supplementary angle 180 — <\>, ih the average value of / for an angle ^ and its negative angle — ^. EFFECTIVE RESISTANCE AND REACTANCE 119 94. The distortion of the wave of magnetizing current is as large as shown here only in an iron-closed magnetic circuit expending power by hysteresis only, as in an ironclad trans- former on open secondary circuit. As soon as the circuit ex- pends power in any other way, as in resistance or by mutual // V^ s V v \ / \ ^ f \ "'"'-. 1 • V \^ I /^ ^^ ^ ^\^ / —] "^ — \ \^ / \ s / ^ > \ <■-/ f __^- '' / ^ V ' \ V / \ \ / / Y ^ \ K \ ^ / / ^. --"' ~--J N ^ \ Vj / / ^x \ — ^ ) / ^ \- [ — — ■ ^"^^o — // / ^-^ — ' Vi !> V ' / \ \ s ^x '/ \ \ ^^^ / N / \ y/ / V y ^c:^ Fig. 83. — Distortion of current wave by hysteresis. inductance, or if an air-gap is introduced in the magnetic circuit, the distortion of the current wave rapidly decreases and practi- cally disappears, and the current becomes more sinusoidal. That is, while the distorting component remains the same, the sinusoidal component of the current greatly increases, and ob- 120 ALTERNATING-CURRENT PHENOMENA sciires the distortion. For example, in Fig. 83, two waves are shown corresponding in magnetization to the last curve of Fig. 80, as the one most distorted. The first curve in Fig. 83 is the current wave of a transformer at 0.1 load. At higher loads the distortion is correspondingly still less, except where the magnetic flux of self-induction, that is, flux passing between primary and secondary and increasing in proportion to the load, is so large as to reach saturation, in which case a distortion appears again and increases with increasing load. The second curve of Fig. 83 is the exciting current of a magnetic circuit containing an air-gap whose length equals 3^:400 the length of the magnetic circuit. These two curves are drawn to one-third the size of the curve in Fig. 80. As shown, both curves are practi- cally sine waves. The sine curves of magnetic flux are shown dotted as <^. 95. The distorted wave of current can be resolved into two components: A true sine ivave of equal effective intensity and equal 'power to the distorted wave, called the equivalent sine wave, and a wattless higher harmonic, consisting chiefly of a term of triple frequency. In Figs. 80, 81 and 83 are shown, as /, the equivalent sine waves, and as i, the difference between the equivalent sine wave and the real distorted wave, which consists of wattless complex higher harmonics. The equivalent sine wave of m.m.f. or of current, in Figs. 80 and 81, leads the magnetism in time phase by 34°, 44°, 38°, and 15.5°, respectively. In Fig. 83 the equivalent sine wave almost coincides with the distorted curve, and leads the magnetism by only 9 degrees. It is interesting to note that even in the greatly distorted curves of Figs. 80 and 81 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 this distorted wave of exciting current shares with the sine wave the feature, that the maximum value and the effective value have the ratio of ■\/2 -^ 1. Hence, below saturation, the maxi- mum value of the distorted curve can be calculated from the effective value — which is given by the reading of an electro- EFFECTIVE RESISTANCE AND REACTANCE 121 dynamometer — by using the same ratio that appUes to a true sine wave, and the magnetic characteristic can thus be deter- mined by means of alternating currents, with sufficient exact- ness, by the electrodynamometer method, in the range below saturation, that is, by alternating-current voltmeter and ammeter. i=l,(X)0 8,000 '3,000 l,m 6,000 '6,000 '7,000 8,000 S.OOO 10,000 U.CWO 12,000 13,000 14,000 16.000 16,000 17,000. Fig. 84. — Magnetization and hysteresis curve. 96. In Fig. 84 is shown the true magnetic characteristic of a sample of average sheet iron, as found by the method of slow reversals with the magnetometer; for comparison there is shown in dotted lines the same characteristic, as determined with alternating currents by the electrodynamometer, with ampere- 122 ALTERNATING-CURRENT PHENOMENA turns per centimeter as ordinates and magnetic inductions as abscissas. As represented, the two curves practically coincide up to a value oi B = 13,000; that is, up to fairl}'- high inductions. For higher saturations, the curves rapidly diverge, and the elec- trodynamometer curve shows comparatively small magnetizing forces producing apparently very high magnetizations. The same Fig. 84 gives the curve of hysteretic loss, in ergs per cubic centimeter and cycle, as ordinates, and magnetic inductions as abscissas. The electrodynamometer method of determining the magnetic characteristic is preferable for use with alternating-current apparatus, since it is not affected by the phenomenon of mag- netic "creeping," which, especially at low densities, may in the magnetometer tests bring the magnetism very much higher, or the magnetizing force lower, than found in practice in alter- nating-current apparatus. So far as current strength and power consumption are con- cerned, the distorted wave can be replaced by the equivalent sine wave and the higher harmonics neglected. All the measurements of alternating currents, with the single exception of instantaneous readings, yield the equivalent sine wave only, 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 most practical applications it is permissible to neglect the higher harmonics 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 dis- turbing factor which may become noticeable in those cases where the frequency of the higher harmonic is near the frequency of resonance of the circuit, that is, in circuits containing conden- sive as well as inductive reactance, or in those circuits in which the higher harmonic of currrent is suppressed, and thereby the voltage is distorted, as discussed in Chapter XXV. 97. 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 current lags behind the e.m.f. by the time angle (90° — a), and the power is, therefore, P = IE cos (90° - a) ^- IE sin a. EFFECTIVE RESISTANCE AND REACTANCE 123 Thus the exciting current, /, consists of a power component, I sin a, called the hysteretic or magnetic power current, and a wattless component, I cos a, which is called the magnetizing current. Or, conversely, the e.m.f. consists of a power compo- nent, E sin a, the hysteretic power component, and a wattless component, E cos a, the e.m.f. consumed by self-induction. Denoting the absolute value of the impedance of the circuit, E J, by z — where z is determined by the magnetic characteristic of the iron and the shape of the magnetic and electric circuits — the impedance is represented, in phase and intensity, by the symbolic expression, Z — r -{- jx =^ z '&\n a -\- jz cos a; and the admittance by, 1 ^ g — JO = - Bin a — J- cos a = y sm a — jy cos a. The quantities z, r, x, and y, g, h 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 considerable, 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. 98. The loss of energy by hysteresis due to molecular magnetic friction is, with sufficient exactness, proportional to the 1.6th power of magnetic induction, B. Hence it can be expressed by the formula: Wh = vB'-' where — TF//= loss of energy per cycle, in ergs or (c.g.s.) units (= 10~^ joules) per cubic centimeter, B = maximum magnetic induction, in lines of force per sq. cm., and 77 = the coefficient of hysteresis. This I found to vary in iron from 0.001 to 0.0055. As a safe mean, 0.0033 can be accepted for common annealed sheet iron or sheet steel, 0.002 for silicon steel and 0.0010 to 0.0015 for specially selected low hysteresis steel. In gray cast iron, rj averages 124 ALTERNATING-CURRENT PHENOMENA 0.013; it varies from 0.0032 to 0.028 in cast steel, according to the chemical or physical constitution; and reaches values as high as 0.08 in hardened steel (tungsten and manganese steel). Soft nickel and cobalt have about the same coefficient of hysteresis as gray cast iron; in magnetite I found -q = 0.023. In the curves of Figs. 79 to 84, y] = 0.0033. At the frequency, /, the loss of power in the volume, V, is, by this formula, p = ^/FB'« 10-7 watts = vfV [jj 10-7 ^^atts, where A is the cross-section of the total magnetic flux, . The maximum magnetic flux, $, depends upon the counter e.m.f. of self-induction, E = V27r/?i$ 10-8, or ^ ElO^ 2 7r/n' where n = number of turns of the electric circuit, / = frequency. Substituting this in the value of the power, P, and canceling, we get, p _ E^ VAO^ .o^ ^10' . ^ - V J0.6 20.H^1.6^1.6,,j^l.6 - ^^'n j-0.6 ^1.6,^1.6'' or KE^ ^ V 10^-« _ 710=^ , ^ ~ fo.6 > wUere A — ■'?20.8 ^^.1.6^^1.6^^1.6 — '^"'?^ 1.6.^1. 6' V or, substituting 77 = 0.0033, we have A' = 191.4-j-|-^j-g; or, substituting V = Al, where / = length of magnetic circuit. 20.8^1.6^0.6^^1.6 ^0.6,,^1.6 -----^0.6^1.6' and 58r,E'-H10' _ 191AE''-H ^ ~~ /"O-e^o.e^i.e — yo. 6^0,6^1. 6 In Figs. 85, 86, and 87 is shown a curve of hysteretic loss, with the loss of power as ordinates, and in curve 85, with the e.m.f., E, as abscissas, for ^ = 6, A = 20, / = 100, and n = 100; in curve 86, with the number of turns as abscissas, for I = 6, A - 20, / = 100, and E = 100; EFFECTIVE RESISTANCE AND REACTANCE 125 Rl LATION BETWt EN Eand P F OR 1 = 5,A =20, / = = 10 D.n = 1 00 / A / a. m / o n / / i / / v / 4 y ^ /- ^ ^ ^ [T E.^ I.F. 20 40 GO 60 100 130 HO 100 180 200 220 210 2G0 230 300 320 310 300 380 400 120 410 Fig. 85. — Hysteresis loss as function of E.M.F. RELATION BETWEEN n AND P FOR i = 6.A=20, /=100.E = 100. 150 200 250 300 n = NUMBER OF TURNS Fig. 86. 350 400 126 AL TERN A TING-C URREN T PHENOMENA in curve 87, with the frequency, /, or the cross-section, A, as abscissas, for I = Q, n ^ 100, and E = 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 0.6* power of the frequency and of the cross-section. 90 85 80 75 RELATION BETWEEN N AND P FOR A = 20, ; = 6,m = iOO.E=100. 65 60 55 |50 0 45 II 40 "35 30 25 \ \ 1 V \ 20 \ 15 10 "V, --. — . b 100 200 300 / = FREQUENCY Fig. 87. 400 99. If <7 = effective conductance, the power component of a current is / = Eg, and the power consumed in a conductance, g, is P = IE = E^g. Since, however, ^1.6 ^1.6 P = ^'^"luT' we have K -^^^ ^ ^'^J it is: , = „^ = 58 ""0' = 191 4 J y fO.ejjJOA ^0.4A).6^0.6^I.6 "^^^'^ ^0.4/0. 6^0.6,^^1.6 From this we have the following deduction : The effective conductance due to magnetic hysteresis is propor- tional to the coefficient of hysteresis, rj, and to the length of the mag- netic circuit, I, and inversely proportional to the 0.4 power of the e.m.f., to the O.e''' power of the frequency, f, and of the cross-section EFFECTIVE RESISTANCE AND REACTANCE 127 of the magnetic circuit, A, and to the 1.6 power of the number of turns, n. Hence, the effective hysteretic conductance increases with decreasing e.m.f., and decreases with increasing 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 power consump- tion, it can, within a limited range of variation — as, for instance, in constant-potential transformers — be assumed as constant without serious error. 36 34 32 30 28 26 24 22 , 20 218 " 16 14 12 10 8 6 4 2 0 &5 11 1 1 1 1 1 1 1 1 RELATION BETWEEN 9 AND E FOR; = 6,/=100 A=20,»=100 y \ ^ N. \ s s. \ ^ ^~ — — — 50 100 150 200 250 Fig. 88. 300 350 400 In Figs. 88, 89, and 90, the hysteretic conductance, g, is plotted, for i = 6, £; = 100, / = 100, A = 20 and n = 100, respectively, with the conductance, g, as ordinates, and with E as abscissas in Curve 88. / as abscissas in Curve 89. n as abscissas in Curve 90. 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 variation in / or A by 50 per cent, causes a variation in g of 21 per cent. If (R = magnetic reluctance of a circuit, F^^ = maximum 128 ALTERNATING-CURRENT PHENOMENA 90 85 80 75 70 65 60 55 50 o45 r 40 35 30 25 20 15 10 5 0 t!5 1 1 1 1 1 1 1 1 1 1 RELATION BETWEEN 9 AND N FOR J = 6,E=100. A = 20, n = 100 1 1 \ \ ' I \ — \ \ *Vh ^ .^ — — . — , _ 50 100 150 200 250 /=CYCLES Fig. 89. 300 350 400 90 85 80 75 70 65 60 55 50 o45 ^40 35 30 25 20 15 10 5 0 r RELATION BETWEEN n AND g FOR i=6.E-100,/=100,A=20 1 1 1 1 \ \ \ ) ^ \ s s. \ s "^ ^ - — 50 100 150 200 250 300 n=NUMBER OF TURNS Fig. 90. 350 400 EFFECTIVE RESISTANCE AND REACTANCE 129 m.m.f., / — effective current, since I\/2 = maximum current, the magnetic flux, (R (R Substituting this in the equation of the counter e.m.f. of self- induction, E = V2 irfn^ 10"', we have „ 2 wnJI 10-« ^= ^ 5 hence, the absolute admittance of the circuit is y = ^^ -^^^E = 2^f^T where 10« , , a = ^ — 5, a constant. 2 Trrr Therefore, the absolute admittance, y, of a circuit of negligible resistance is proportional to the magnetic reluctance, (R, and in- versely proportional to the frequency, f, and to the square of the number of turns, n. 100. 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 mag- netic field exists entirely within iron, such as the magnetic cir- cuit of a well-designed alternating-current transformer — (R is the reluctance of the iron circuit. Hence, if /* = permeability since Fk and and ^= $ F IF ^ ^ IH ^ m.m.f., $ = A(B = fiAH =^ magnetic flux, 10 Z . (R = 4:Tr/JiA substituting this value in the equation of the admittance, (R 10^ 130 ALTERNATING-CURRENT PHENOMENA we have . y'SwVfjiAf-fu.' where c = 110' 127 ZIO^ 8 x^n^A w^A Therefore, in an ironclad circuit, the absolute admittance, y, is inversely proportional to the frequency, f, to the permeability, n, to the cross-section. A, and to the square of the number of turns, n; and directly proportional to the length of the magnetic circuit, I. The conductance is _ K . 9 ~ fO.GJPO.i' and the admittance, fO.6^0 c hence, the angle of hysteretic advance is g Knf\ or, substituting for A and c (§119), J0.4 ^i 105.8 sin a - /i^o 4 20-»7ri-«A"-*'wi-« ao» ^^y^.4^0.4^0.4^0.4 22.2^ ^0.4 103-2 or, substituting E = 2»-^7r/nA(BlO~^ we have 4/X77 sin oc = ^> 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 ad- vance, a, depends upon the ynagnetic constants, permeability and coefficient of hysteresis, and upon the maximum magnetic induction, but is entirely independent of the frequency, of the shape and other conditions of the magnetic and electric circuit; and, therefore, all ironclad magnetic circuits constructed of the same quality of iron and using the same magnetic density, give the same angle of hys- teretic advance, and the same power factor of their electric energizing circuit. EFFECTIVE RESISTANCE AND REACTANCE 131 The angle of hysteretic advance, a, in a closed circuit trans- former and the no-load -power factor, depend upo7i the quality of the iron, and upon the magnetic density only. The sine of the angle of hysteretic advance equals 4 tiines the product of the permeability and coefficient of hysteresis, divided by the 0 . 4"* power of the magyietic density. 101. If the magnetic circuit is not entirely ironclad, and the magnetic structure contains air-gaps, the total reluctance is the sum of the iron reluctance and of the air reluctance, or (R = (Ri + (Raj hence the admittance is y = Vg' + h^ = ?((R. + (Ra). Therefore, in a circuit containing iron, the admittance is the sum of the admittance due to the iron part of the circuit, yi = — y-'' and of the admittance due to the air part of the circuit, ya = —^' if the iron and the air are in series in the magnetic circuit. The conductance, g, represents the loss of power 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 g G g (^i sm a = - = y Vi + Va yi (R,- + (Ra and a maximum, -, for the ironclad circuit, but decreases with increasing width of the air-gap. The introduction of the air- gap of reluctance, (Ra, decreases sin a in the ratio, (Ri (R, + (Ra In the range of practical application, from B = 2,000 to B = 14,000, the permeability of iron usually exceeds 1,000, while sin a in an ironclad circuit varies in this range from 0.51 to 0.69. In air, fj, = 1. If, consequently, 1 per cent, of the length of the iron consists of an air-gap, the total reluctance only varies by a few per cent., that is, remains practically constant; while the angle of hysteretic advance varies from sin a = 0.035 to sin a = 0.064. Thus g is negligible compared with b, and b is practically equal to y. 132 ALTERNATING-CURRENT PHENOMENA Therefore, in an electric circuit containing iron, but forming an open magnetic circuit whose air-gap is not less than Koo the length of the iron, the susceptance is practically constant and equal to the admittance, so long as saturation is not yet ap- proached, or, , (R. / b ^ J, or: .: = ^-- The angle of hysteretic advance is small, and the hysteretic con- ductance is JC 0 ~ ^0.4/0.6' The current wave is practically a sine wave. As an example, in Fig. 83, Curve II, the current curve of a circuit is shown, containing an air-gap of only 3-^00 of the length of the iron, giving a current wave much resembling the sine shape, with an hysteretic advance of 9°. 102. To determine the electric constants of a circuit con- taining iron, we shall proceed in the following way: Let E = counter e.m.f. of self-induction then from the equation, E = V2 7^/;/4>10-^ where / = frequency, n = number of turns, we get the magnetism, 4>, and by means of the magnetic cross- section. A, the maximum magnetic induction: B — -j- From B, we get, by means of the magnetic characteristic of the iron, the magnetizing force, = / ampere-turns per centimeter length where •' Air U H = magnetizing force in c.g.s. units. Hence, if h = length of iron circuit, Fi = /,/ = ampere-turns required in the iron; if la = length of air circuit, Fa = ■ . " - = ampere-turns required in the air; EFFECTIVE RESISTANCE AND REACTANCE 133 hence, F = Fi -\- Fa = total ampere-turns, maximum value, and F — y= = effective value. The exciting current is V2 7 = 4-. and the absolute admittance, If Fi is not negligible as compared with Fa, this admittance, y, is variable with the e.rn.f., E. U V = volume of iron, rj = coefficient of hysteresis, the loss of power by hysteresis due to molecular magnetic friction is P = vfVB'-'; P hence the hysteretic conductance is g = i^, and variable with the e.m.f., E. The angle of hysteretic advance is 9 . sm a = - , y the susceptance. b = Vif- - g^; the effective resistance, ^ _ fiL. and the reactance, h 103. 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, and inversely, a sine wave of current requires waves of magnetism and e.m.f. differing from sine shape. 2. This distortion is excessive only with a closed magnetic circuit transferring no energy into a secondary circuit by mutual inductance. 3. The distorted wave of current can be replaced by the equiva- lent sine wave — that is, a sine wave of equal effective intensity and equal power — and the superposed higher harmonic, con- 134 ALTERNATING-CURRENT PHENOMENA sisting 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 maximum value. 5. The angle of hysteretic advance — that is, the phase dif- ference 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, ju, the coefficient of hysteresis, -q, and the maxi- mum magnetic induction, as shown in the equation, 4/ir7 6. The effect of hysteresis can be represented by an admittance Y — g — 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. 8. The hysteretic condtictance, g, is proportional to the coefficient of hysteresis, t), and to the length of the magnetic circuit, I, inversely proportional to the 0.4*^^ power of the e.m.f. E, to the 0.6*^ power of frequency, /, and of the cross-section of the magnetic circuit. A, and to the 1.6* power of the number of turns of the electric circuit, n, as expressed in the equation, _ 58 r?? 10^ • 9. The absolute value of hysteretic admittance, y = Vr/2 + h\ is proportional to the magnetic reluctance: (R, = (Jlj + (Ra, and inversely proportional to the frequency, /, and to the square of the number of turns, n, as expressed in the equation, ((Ri + (Ra) 10« y 2 Tjri' 10. In an ironclad circuit, the absolute value of admittance is proportional to the length of the magnetic circuit, and inversely proportional to cross-section, A, frequency, /, permeability, ^ and square of the number of turns, n, or _ 127 no^ ^'' " n'^Ajii ' 11. In an open magnetic circuit, the conductance, g, is the same as in a closed magnetic circuit of the same iron part. EFFECTIVE RESISTANCE AND REACTANCE 135 12. In an open magnetic circuit, the admittance, y, is prac- tically constant, if the length of the air-gap is at least 3^f oo o^ the length of the magnetic circuit, and saturation be not approached. 13. In a closed magnetic circuit, conductance, susceptance, 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 constants, g, h, y; r, x, z, can be calculated. 104. The preceding applies to the alternating magnetic circuit, that is, circuit in which the magnetic induction varies between equal but opposite limits: Bi = -\- Bq and B-i = — Bo- In a pulsating magnetic circuit, in which the induction B varies between two values Bi and Bo, which are not equal numerically, and which may be' of the same sign or of opposite sign, that is in which the hysteresis cycle is unsymmetrical, the law of the 1.6*^^ power still applies, and the loss of energy per cycle is pro- portional to the 1.6*^*^ power of the amplitude of the magnetic variation : but the hysteresis coefficient tj is not the same as for alternating magnetic circuits, but increases with increasing average value n of the magnetic induction. Such unsymmetrical magnetic cycles occur in some types of induction alternators, Mn which the magnetic induction does not reverse, but pulsates between a high and a low value in the same direction. Unsymmetrical magnetic cycles occasionally occur — and give trouble — in transformers by the entrance of a stray direct current (railway return) over the ground connection, or when an unsuit- able transformer connection is used on a synchronous converter feeding a three-wire system. Very unsymmetrical cycles may give very much higher losses than symmetrical cycles of the same amplitude. For more complete discussion of unsymmetrical cycles see "Theory and Calculation of Electric Circuits." * See "Theory and Calculation of Electric Apparatus."