10. HYSTERESIS AND EFFECTIVE RESISTANCE 46. If an alternating current 01 = I, in Fig. 21, exists in a circuit of reactance x = 2 irfL and of negligible resistance, the HYSTERESIS AND EFFECTIVE RESISTANCE 49 magnetic flux produced by the current, 0$ = $, is in phase with the current, and the e.m.f. generated by this flux, or counter e.m.f. of self-inductance, OE'" = E'" = xl, lags 90 degrees be- hind the current. The e.m.f. consumed by self-inductance or impressed e.m.f. OE" = E" = xl is thus 90 degrees ahead of the current. Inversely, if the e.m.f. OE" = E" is impressed upon a circuit of reactance x = 2 irfL and of negligible resistance, the current E" 01 = I = - - lags 90 degrees behind the impressed e.m.f. x This current' is called the exciting or magnetizing current of the magnetic circuit, and is wattless. ' If the magnetic circuit contains iron or other magnetic mate- rial, energy is consumed in the magnetic circuit by a frictional resistance of the material against a change of magnetism, which is called molecular magnetic friction. If the alternating current is the only avail- able source of energy in the magnetic cir- cuit, the expenditure of energy by molec- ular magnetic friction appears as a lag of the magnetism behind the m.m.f. of the Q| r >i current, that is, as magnetic hysteresis, and can be measured thereby. Magnetic hysteresis is, however, a dis- tinctly different phenomenon from molec- ular magnetic friction, and can be more or less eliminated, as for instance by me- chanical vibration, or can be increased, without changing the molecular magnetic friction. 47. In consequence of magnetic hysteresis, if an alternating e.m.f. OE" = E" is im- pressed upon a circuit of negligible resistance, the exciting current, or current producing the magnetism, in this circuit is not a wattless current, or current of 90 degrees lag, as in Fig. 21, but lags less than 90 degrees, by an angle 90 — a, as shown by OI = I in Fig. 22. Since the magnetism 0$ = $ is in quadrature with the e.m.f. E" due to it, angle a is the phase difference between the magnet- ism and the m.m.f., or the lead of the m.m.f., that is, the exciting 4 FIG. 21.— Phase re- lations of magnetizing current, flux and self- inductive e.m.f. 50 ELEMENTS OF ELECTRICAL ENGINEERING current, before the magnetism. It is called the angle of hysteretic lead. In this case the exciting current 01 = I can be resolved in two components: the magnetizing current 01 2 — 1 2, in phase with the magnetism 0$ = $, that is, in quadrature with the e.m.f. OE" = E"j and thus wattless, and the magnetic power component of the current or the hysteresis current OIi = Ii, in phase jvvith the e.m.f. OE" = E", or in quadrature with the magnetism 0$ = $. Magnetizing current and hysteresis current are the two com- ponents of the exciting current. FIG. 22. — Angle of hysteretic lead. FIG. 23. — Effect of resistance on phase relation of impressed e.m.f. in a hysteresisless circuit. If the circuit contains besides the reactance x = 2 wfL, a re- sistance r, the e.m.f. OE" = E" in the preceding Figs. 21 and 22 is not the impressed e.m.f., but the e.m.f. consumed by self- inductance or reactance, and has to be combined, Figs. 23 and 24, with the e.m.f. consumed by the resistance, OE' = E' = Ir, to get the impressed e.m.f. OE = E. Due to the hysteretic lead a, the lag of the current is less in Figs. 22 and 24, a circuit expending energy in molecular mag- netic friction, than in Figs. 21 and 23, a hysteresisless circuit. As seen in Fig. 24, in a circuit whose ohmic resistance is not negligible, the hysteresis current and the magnetizing current are not in phase and in quadrature respectively with the im- pressed e.m.f., but with the counter e.m.f. of inductance or e.m.f. consumed by inductance. Obviously the magnetizing current is not quite wattless, since HYSTERESIS AND EFFECTIVE RESISTANCE 51 energy is consumed by this current in the ohmic resistance of the circuit. Resolving, in Fig. 25, the impressed e.m.f. OE = E into two components, OEi = EI in phase, and OE2 = E2 in quadrature with the current 01 = I, the power component of the e.m.f., EI, is greater than Er = Ir, and the reactive component E2 is less than E" OE, Ix. FIG. 24. — Effect of resistance on phase relation of impressed e.m.f. in a circuit having hys- teresis. FIG. 25. — Impressed e.m.f. resolved into components in phase and in quadrature with the exciting current. The value r' ance, and the value x' — -r EI power e.m.f. . ... . . . . -=- = *- - is called the effective resist- I total current E2 wattless e.m.f. I is called the ap- total current parent or effective reactance of the circuit. 48. Due to the loss of energy by hysteresis (eddy currents, etc.), the effective resistance differs from, and is greater than, the ohmic resistance, and the apparent reactance is less than the true or inductive reactance. The loss of energy by molecular magnetic friction per cubic centimeter and cycle of magnetism is approximately W = r}B^, where B = the magnetic flux density, in lines per sq. cm. W = energy, in absolute units or ergs per cycle (= 10~7 watt-seconds or joules), and t\ is called the coef- ficient of hysteresis. 52 ELEMENTS OF ELECTRICAL ENGINEERING In soft annealed sheet iron or sheet steel and in silicon steel, rj varies from 0.60 X 10~3 to 2.5 X 10~3, and can in average, for good material, be assumed as 1.5 X 10~3. The loss of power in the volume, V, at flux density B and frequency /, is thus P = VfoB1'6 X 10"7, in watts, and, if / = the exciting current, the hysteretic effective resist- ance is P B1'6 r" =J-* = VfrW-^' If the flux density, B, is proportional to the current, /, sub- stituting for B, and introducing the constant k, we have rn V ~ 'PA' that is, the effective hysteretic resistance is inversely propor- tional to the 0.4 power of the current, and directly proportional to the frequency. 49. Besides hysteresis, eddy or Foucault currents contribute to the effective resistance. Since at constant frequency the Foucault currents are pro- portional to the magnetism producing them, and thus approxi- mately proportional to the current, the loss of power by Foucault currents is proportional to the square of the current, the same as the ohmic loss, that is, the effective resistance due to Foucault currents is approximately constant at constant frequency, while that of hysteresis decreases slowly with the current. Since the Foucault currents are proportional to the frequency, their effective resistance varies with the square of the frequency, while that of hysteresis varies only proportionally to the frequency. The total effective resistance of an alternating-current circuit increases with the frequency, but is approximately constant, within a limited range, at constant frequency, decreasing some- what with the increase of magnetism. EXAMPLES 50. A reactive coil shall give 100 volts e.m.f. of self-inductance at 10 amp. and 60 cycles. The electric circuit consists of 200 turns (No. 8 B. & S.) (= 0.013 sq. in.) of 16 in. mean length of turn. The magnetic circuit has a section of 6 sq. in. and a HYSTERESIS AND EFFECTIVE RESISTANCE 53 mean length of 18 in. of iron of hysteresis coefficient rj = 2.5 X 1CT3. An air gap is interposed in the magnetic circuit, of a section of 10 sq. in. (allowing for spread), to get the desired reactance. How long must the air gap be, and what is the resistance, the reactance, the effective resistance, the effective impedance, and the power-factor of the reactive coil? The coil contains 200 turns each 16 in. in length and 0.013 sq. in. in cross section. Taking the resistivity of copper as 1.8 X 10~6, the resistance is 1.8 X 10~6 X 200 X 16 0.013 X 2.54" im> where 2.54 is the factor for converting inches to centimeters. (1 inch = 2.54 cm.) Writing E = 100 volts generated, / = 60 cycles per second, and n = 200 turns, the maximum magnetic flux is given by E = 4.44 fn$; or, 100 = 4.44 X 0.6 X 200$, and 3> = 0.188 megaline. This gives in an air gap of 10 sq. in. a maximum density B = 18,800 lines per sq. in., or 2920 lines per sq. cm. Ten amperes in 200 turns give 2000 ampere-turns effective or F = 2830 ampere-turns maximum. Neglecting the ampere-turns required by the iron part of the magnetic circuit as relatively very small, 2830 ampere-turns have to be consumed by the air gap of density B = 2920. Since D 4?rF loT the length of the air gap has to be 47TX2830 To* =: ToxWo == L22 cm" or °'48 ln' With a cross section of 6 sq. in. and a mean length of 18 in., the volume of the iron is 108 cu. in., or 1770 cu. cm. OOO The density in the iron, BI = -- g -- = 31,330 lines per sq. in., or 4850 lines per sq. cm. With an hysteresis coefficient 77 = 2.5 X 10~3, and density BI = 4850, the loss of energy per cycle per cubic centimeter is W = i/fii1-6 = 2.5 X 10-3 X 48501-6 = 1980 ergs, 54 ELEMENTS OF ELECTRICAL ENGINEERING and the hysteresis loss at/ = 60 cycles and the volume V = 1770 is thus P = 60 X 1770 X 1980 ergs per sec. = 21.0 watts, which at 10 amp. represent an effective hysteretic resistance, 21.0 r2 = -y~j- — 0.21 ohm. Hence the total effective resistance of the reactive coil is r = n + r2 = 0.175 + 0.21 = 0.385 ohm the effective reactance is 777 x = ~j = 10 ohms; the impedance is z = 10.01 ohms; the power-factor is T cos 0 — - = 3.8 per cent.; z the total apparent power of the reactive coil is I2z = 1001 volt-amperes, and the loss of power, Pr = 38 watts.