1. MAGNETISM AND ELECTRIC CURRENT 1. A magnet pole attracting (or repelling) another magnet pole of equal strength at unit distance with unit force1 is called a unit magnet pole. The space surrounding a magnet pole is called a magnetic field of force, or magnetic field. The magnetic field at unit distance from a unit magnet pole is called a unit magnetic field, and is represented by one line of magnetic force (or shortly "one line") per square centimeter, and from a unit magnet pole thus issue a total of 4 TT lines of magnetic force. The total number of lines of force issuing from a magnet pole is called its magnetic flux. The magnetic flux $ of a magnet pole of strength m is, <£ = 4 irm. At the distance I from a magnet pole of strength m, and therefore of flux $> = 4 xw, assuming a uniform distribution in all directions, the magnetic field has the intensity, n - since the 3> lines issuing from the pole distribute over the area of a sphere of radius I, that is, the area 4 irl2. A magnetic field of intensity H exerts upon a magnet pole of strength m the force, mH. Thus two magnet poles of strengths mi and mz, and distance I from each other, exert upon each other the force, 1 That is, at 1 cm. distance with such force as to give to the mass of 1 gram the acceleration of 1 cm. per second. 1 2 *:LE' VENTS OF ELECTRICAL ENGINEERING 2. Electric currents produce magnetic fields also; that is, the space surrounding the conductor carrying an electric current is a magnetic field, which appears and disappears and varies with the current producing it, and is indeed an essential part of the phenomenon called an electric current. Thus an electric current represents a' magnetomotive force (m.m.f.). The magnetic field of a straight conductor, whose return conductor is so far distant as not to affect the field, consists of lines of force surrounding the conductor in concentric circles. The intensity of this magnetic field is directly proportional to the current strength and inversely proportional to the dis- tance from the conductor. Since the lines of force of the magnetic field produced by an electric current return into themselves, the magnetic field is a magnetic circuit. Since an electric current, at least a steady current, can exist only in a closed circuit, electricity flows in an electric circuit. The magnetic circuit produced by an electric current surrounds the electric circuit through which the electricity flows, and inversely. That is, the electric circuit and the magnetic circuit are interlinked with each other. Unit current in an electric circuit is the current which produces in a magnetic circuit of unit length the field intensity 4?r, that is, produces as many lines of force per square centimeter as issue from a unit magnet pole. In unit distance from an electric conductor carrying unit current, that is, in a magnetic circuit of length 2?r, the field 4-7T intensity is ~ — — 2, and in the distance 2 the field intensity is unity; that is, unit current is the current which, in a straight conductor, whose return conductor is so far distant as not to affect its magnetic field, produces field intensity 2 in unit distance from the conductor. One-tenth of unit current is the practical unit, called one ampere. 3. One ampere in an electric circuit or turn, that is, one ampere-turn, thus produces in a magnetic circuit of unit length the field intensity 0.4 w, and in a magnetic circuit of length 0.4 TT I the field intensity — '-j — , and F ampere-turns produce in a magnetic circuit of length I the field intensity: „ 0.4 irFv tf H = — j — lines of force per sq. cm. MAGNETISM AND ELECTRIC CURRENT 3 regardless whether the F ampere-turns are due to F amperes F in a single turn, or 1 amp. in F turns, or — amperes in n turns. F, that is, the product of amperes and turns, is called magneto- motive force (m.m.f.). The m.m.f. per unit length of magnetic circuit, or ratio, _ _ m.m.f. _ ' " " length of magnetic circuit » is called the magnetizing force, or magnetic gradient. Hence, m.m.f. is expressed in ampere-turns; magnetizing force in ampere-turns per centimeter (or in practice frequently ampere-turns per inch), field intensity in lines of magnetic force per square centimeter. At the distance I from the conductor of a loop or circuit of F ampere-turns, whose return conductor is so far distant as not to affect the field, assuming the m.m.f. = F, since the length of the magnetic circuit = 2 irl, we obtain as the magnetizing force, : '-« and as the field intensity, 0 2 F H = 0.4 TT = 2^- 4. The magnetic field of an electric circuit consisting of two parallel conductors (or any number of conductors, in a poly- phase system), as the two wires of a transmission line, can be considered as the superposition of the separate fields of the conductors (consisting of concentric circles). Thus, if there are I amperes in a circuit consisting of two parallel conductors (conductor and return conductor), at the distance li from the first and h from the second conductor, the respective field intensities are, - l ~ ~TT and "-T and the resultant field intensity, if r = angle between the direc- tions of the two fields, H = \/# i2 + #22 + 2 #!#2 COST, COST. 4 ELEMENTS OF ELECTRICAL ENGINEERING In the plane of the conductors, where the two fields are in the same or opposite direction, the resultant field intensity is, 7 7 M2 where the plus sign applies to the space between, the minus sign the space outside of the conductors. The resultant field of a circuit of parallel conductors con- sists of excentric circles, interlinked with the conductors, and crowded together in the space between the conductors as shown in Fig. 1 by drawn lines. FIG. 1. — Magnetic field of parallel conductors. The magnetic field in the interior of a spiral (solenoid, helix, coil) carrying an electric current consists of straight lines. 5. If a conductor is coiled in a spiral of I centimeter axial N length of spiral, and N turns, thus n = j- turns per centimeter length of spiral, and / = current, in amperes, in the conductor, the m.m.f. of the spiral is F = IN, and the magnetizing force in the middle of the spiral, assuming, the latter of very great length, T N T f = nI = TI, thus the field intensity in the middle of the spiral or solenoid, H = 0.4 TT/ = 0.4 MAGNETISM AND ELECTRIC CURRENT 5 Strictly this is true only in the middle part of a spiral of such length that the m.m.f. consumed by the external or mag- netic return circuit of the spiral is negligible compared with the m.m.f. consumed by the magnetic circuit in the interior of the spiral, or in an endless spiral, that is, a spiral whose axis curves back into itself, as a spiral whose axis is curved in a circle. Magnetomotive force F applies to the total magnetic circuit, or part of the magnetic circuit. It is measured in ampere- turns. Magnetizing force / is the m.m.f. per unit length of mag- netic circuit. It is measured in ampere-turns per centimeter. Field intensity H is the number of lines of force per square centimeter. If I = length of the magnetic circuit or a part of the magnetic circuit, F = V, f = j, H = 0.4 »/ / = H 0.47T' = 1.257/ /= 0.796 #. 6. The preceding applies only to magnetic fields in air or other unmagnetic materials. If the medium in which the magnetic field is established is a "magnetic material," the number of lines of force per square centimeter is different and usually many times greater. (Slightly less in diamagnetic materials.) The ratio of the number of lines of force in a medium, to the number of lines of force which the same magnetizing force would produce in air (or rather in a vacuum), is called the permeability or magnetic conductivity /* of the medium. The number of lines of force per square centimeter in a mag- netic medium is called the magnetic induction B. The number of lines of force produced by the same magnetizing force in air, or rather, in the vacuum, is called the field intensity H. In air, magnetic induction B and field intensity H are equal. As a r-ule, the magnetizing force in a magnetic circuit is changed by the introduction of the magnetic material, due to the change of distribution of the magnetic flux. The permeability of air = 1 and is constant. 6 ELEMENTS OF ELECTRICAL ENGINEERING . The permeability of iron and other magnetic materials varies with the magnetizing force between a little above 1 and values beyond 10,000 in soft iron. The magnetizing force / in a medium of permeability /* pro- duces the field intensity H = 0.4 irf and the magnetic induction B = 0.4 TTflf. EXAMPLES 7. (1) A pull of 2 grams at 4 cm. radius is required to hold a horizontal bar magnet 12 cm. in length, pivoted at its center, in a position at right angles to the magnetic meridian. What is the intensity of the poles of the magnet, and the number of lines of magnetic force issuing from each pole, if the horizontal intensity of the terrestrial magnetic field H = 0.2, and the acceleration of gravity = 980? The distance between the poles of the bar magnet may be assumed as five-sixths of its length. Let m = intensity of magnet poles. I = 5 is the radius on which the terrestrial magnetism acts. Thus 2mHl = 2'm = torque exerted by the terrestrial magnetism. 2 grams weight = 2 X 980 = 1960 units of force. These at 4 cm. radius give the torque 4 X I960 = 7840 g cm. Hence 2m = 7840. m = 3920 is the strength of each magnet pole and 3> = 4 Trm = 49,000, the number of lines of force issuing from each pole. 8. (2) A conductor carrying 100 amp. runs in the direc- tion of the magnetic meridian. What position will a compass needle assume, when held below the conductor at a distance of 50 cm., if the intensity of the terrestrial magnetic field is 0.2? The intensity of the magnetic field of 100 amp., 50 cm. 027 100 from the conductor, is H = — ^ — = 0.2 X -^r = 0.4, the direc- tion is at right angles to the conductor, that is, at right angles to the terrestrial magnetic field. If T = angle between compass needle and the north pole of the magnetic meridian, 10 = length of needle, m = intensity of its magnet pole, the torque of the terrestrial magnetism is Hmlo sin T = 0.2 mlo sin r, the torque of the current is 0.2/raZocosr cos r = - — — - = 0.4 MAGNETISM AND ELECTRIC CURRENT 7 In equilibrium, 0.2 mlQ sin r = 0.4 mlQ cos r, or tan r = 2, r = 63.4°. 9. (3) What is the- total magnetic flux per I = 1000 m. length, passing between the conductors of a long distance transmission line carrying 7 amperes of current, if Id = 0.82 cm. is the diam- eter of the conductors (No. 0 B. & S.), 18 = 45 cm. the spacing or distance between them? FIG. 2. — Diagram of transmission line for inductance calculation. At distance lr from the center of one of the conductors (Fig. 2), the length of the magnetic circuit surrounding this conductor is 2irlr) the m.m.f., 7 ampere-turns; thus the magnetizing force / = s— r> and the field intensity H = 0.4 irf = -^ — > and the A TTlr Lr flux in the zone dlr is d$ = - — j -> and the total flux from the surface of the conductor to the next conductor is, ).2 Ildlr 0.27/Floge Zrj£ = 0.2/nog.^- The same flux is produced by the return conductor in the same direction, thus the total flux passing between the trans- mission wires is, 2 $ = 0.4 II log, p I'd or per 1000 m. = 105 cm. length, QO 2 $ = 0.4 X 105 / log, TT^ = 0.4 X 105 X 4.70 I = 0.188 X 106 7, 8 ELEMENTS OF ELECTRICAL ENGINEERING or 0.188 / megalines or millions of lines per line of 1000 m. of which 0.094 / megalines surround each of the two conductors. 10. (4) In an alternator each pole has to carry 6.4 millions of lines, or 6.4 megalines magnetic flux. How many ampere- turns per pole are required to produce this flux, if the magnetic 2 8 10 2 14 6 FIG. 3. — Magnetization curves of various irons. circuit in the armature of laminated iron has the cross section of 930 sq. cm. and the length of 15 cm., the air-gap between stationary field poles and revolving armature is 0.95 cm. in length and 1200 sq. cm. in section, the field pole is 26.3 cm. in length and 1075 sq. cm. in section, and is of laminated iron, MAGNETISM AND E.M.F. 9 and the outside return circuit or yoke has a length per pole of 20 cm. and 2250 sq. cm. section, and is of cast iron? The magnetic densities are: BI = 6880 in the armature, B2 = 5340 in the air-gap, B3 = 5950 in the field pole, and B4 = 2850 in the yoke. The permeability of sheet iron is m = 2550 at 5i = 6880, MS = 2380 at B3 = 5950. The permeability of cast iron is /z4 = 280 at B4 = 2850. Thus the field intensity/ H = - j is: Hi = 2.7, #2 = 5340, H3 = 2.5, H, = 10.2. The magnetizing force (/ = {TT— ) is> fl = 2'15> ^2 = 4250' /3 = 1.99, /4 = 8.13 ampere-turns per centimeter. Thus the m.m.f. (F = fl) is: Fi = 32, F2 = 4040, F3 = 52, F4 = 163, or the total m.m.f. per pole is F = Fi + F2 + F3 + ^4 = 4290 ampere-turns. The permeability p of magnetic materials varies with the density B, thus tables or curves have to be used for these quan- tities. Such curves are usually made out for density B and magnetizing force /, so that the magnetizing force / correspond- ing to the density B can be derived directly from the curve. Such a set of curves is given in Fig. 3.