Theoty, Operation, Applications, Agjustinent, aud Conivel Magnetics, Electromagnetic Forces, Generated Voltage, and Energy Conversion 1.1 INTRODUCTION ‘This chapter stants with a brief review of electromagnetism and magnetic circuits, which are normally included in 2 basic circuits or physics course, This review is followed by a discussion ofthe development of the mechanical forces that are caused by the interaction of magnetic fields and that form the basis for all motor ection. Faraday’s law provides the basis from which all magnetically induced voltages are <etived. The relationship between applied torave and countertorque is developed and visualized through the application of Ler2’s law and the “Aux bunching” rule 1.2 MAGNETIC FIELD A magnetic field is a condition resulting from electric charges in modon. The mag- relic Held of a permanent magret is atinbuiec to the uncernpensatec spinning of elec- trons about their owa axis within the atomic structure of the material and to the par- allel alignment of these electrons with similar uncompensated electron spirs in the adjaccat stoms. Groups of adjacent atoms with parallel magnetic spins arc called domains, The magnetic fleld surrounding a current-carrying conductor is caused hy the movement of elecisie charges inthe form of an electric cutrent. For convenience in visualization wel analysis, magnetic fields are represented an diagrams by closed loops. These loops, called magnet tux lines, have been assigned a specific direction thats related to the polarity of a magnet, or the direction of current in acoil ora conducter. - ‘The direction of the magnetic field around a current can be determined by the right-hand rule: Grasp the conductor with the right hand, with te thumb pointing in the direction of conventional current, and the fingers will curl in the direction of the magnetic field. Tais can be visualized in Figure 1.1(a). 21 Chapter 1 FIGURE 1.1 Diroction of magnetic flux: fa) around a current-carrying conductor; in a col! bout a magnet. » In asimilar manner, to determine the direction ofthe magnetic field generated by current through a coil of wire, grasp the coil with the right hard, with the fingers curled in the éirection of the current, and the thumb will point in the dircetion of the ‘magnetic field. This can be visualized in Figure 1.1(b). ‘The direction of the magnetic field supplied by a magnet is out from the north pole ancl into the south pole, but is south-io-narh within the magnet, as shown in Figure 1.1(c). 1.3 MAGNETIC CIRCUIT DEFINED ‘Each magretic circuit shown in Figure 1.2 isan arrangement of feromagaetc maesi- als elled a core that forms a path to conain and guide the magnetic fuk in a specific ‘direction. The core shape shown in Figure 1.2{a) is used in transformers. Figure 1.206) ‘Magnetics, Electromagnetic Forces, Generated Voltage, and Eneray Conversion 3 Rowe iva o) FIGURE 1.2 Magnetic crit) fora tensforrar (al fore simple tworpole motor. shows the magnetic circuit of a simple two-pole motor; it includes a stator core, a ‘olor core, and two zir gaps. Note tha the flux always takes the shortest path across an sir gop. Magnetomotive Force ‘The ampere-tums (A-1) of the respective coils in Figure 1.2 represent the driving force, ealled magnetomoive force ot mf, that causes a magnetic field to appear inthe ‘corresponding magnetic circuits, Expressed in equation form, FaNT an where: F = mognetomotive force (mm!) in ampere-turns (A-t) N= umber of urns in coil 1 current in coil (A) Magnetic Field Intensity Magnetic field intensity, also called mmf gradient, is defined as the magnetomotive foree per unit lengyh of magneitecirent, and it may vary from point to pola ikrmughout the magnetic eireit. The average magnitude of the field intensity in. a homogeneous section of a magnetic cicuit is numerically equal tothe munf across the section divided by the effective length of the magnetic section. Thats, a 41 Chapter 1 where: _H = magnetic fickd intensity (A-t/m) ‘mean length of the magnesi eircul or section (m) F = mma ‘Note that in a homogeneous magnetic circuit o uniform cross section, the fed inten sity is the same at all points in the magnctc circuit. [a composite magnetic circuits, consisting of sections of different materials and/or diferent cross-sectional areas, however, the mageetic field intensity differs from section to seston. ‘Magnetic field intensity has many useful applications in magnetic circuit clea lations. One spexific application ie caleulaing the muagnetc-potentialdijerence, also called magreric drop or mmf drop, aeros a section of a magnetic eitout, The magnetic drop in ampere-tams per meter of magnetic core length in: magnede c:euit is analo- ‘us to the voltage drop in volts per meter of conductor teagth in an electric circuit Flux Density ‘The flux density is a measure of the concentration of lines of flux in a particular sec- tion of a magnetic circuit. Expressed mathematically, and referring to the hamoge- ‘neous core in Figure 1.2(0), = a» where: flux, webers (Wb) A = cross-sectional area (m1?) 8 = fx density (Whim), or tesias (1) 1.4 RELUCTANCE AND THE MAGNETIC CIRCUIT EQUATION A very useful equation that expresses the relationship between magnetic fx, mmf, and the reluctance of the magnetic cireuit is a) F = magnctomotive force (A-1) & = reluctance of magnetic circuit (A-t/Wb) Reluctance Bis a measure of the opposition the magnetic circuit offers to the flax and 4s analogous to resistance in an elect circuit. The reluctance of a magnetic iret, ee section of a magnetic circuit is related to its length, cross-sectional area. and perme ability, Solving Eq. (1-4) for i, dividing numerator and denominator by ¢, and re- aranging ters, MN € Roe Tae WB ae” GA Magnetics, Eleewomagnetlc Forces, Generated Voit ye, and Energy Conversion | 5 Defining a8) a0 where: B= flux density (Wbin"), or teslas CD) H = magnatic field intensity (A-Um) = mesa length of magnetic circuit (on) ‘A = cross-sectional area (m*) = permeability of material (Wh/a.t » m) Equation (1-6) applies to a homogeneous section of & magnetic circuit oF uniform ens section Magnetic Permaability ‘The ratio = BIM is called magnetic permeability and has different values for differ- cent degrees of magnetization ofa specific magnetic core material. - 1.5 RELATIVE PERMEABILITY AND MAGNETIZATION CURVES Relative permeability is the retlo of the permeability of a material to the permeability Of free space; it is, in effect, a figure of merit that is very useful for comparing the magaetizabilily cf differert magnetic matenals whose relative permeabilities :1e ‘known, Expressed in equation form, a7 where: ty = permeability of itee space = 410°? (WE/A-t m) 2, = felative permeability, 2 dimenstontess constant 1 permeability of material (Wb/A+ rn) Representative graphs of Eq. (1-5) for some commonly used ferromagnetic materials, ‘are shown in Figure 1.3, The graphs, eailed B-H curves, magnetization carves, oF ssuarition curves, ave vety useful in Cesign, avd inthe analysis of machine and tins former behavior. “The four principal sections of a typical magretization curve are lustraed in Figure 1.4. The curve is concave up for “low” values of magnetic field intensity, ex- horizontal line for “very high” in‘ensitics. ‘The part of the curve that is concave down is knawn a8 the knee of the curve, anc the “Patened” section is the saruration region. 6 | Chapter ‘sorry SA + cueugrenceus_anais peeane? 2éenzes! Et : t up At t = Ht sana [fT THe s t + t rit i t 7 : : i tH es tH Peery a ith a Son Mago Ftd ney (Ac) FIGURE 1.3 Fopresentatve 8-H curves ‘or some commonly used ferromagnetic materials “Magnetic scturaion is complcte when all of the magnetic domains of the material are Coriented in the cirection of the applied magnetomotive force. Saturation begins at the ‘4271 of the Ence region and is essentially compleie when the curve starts to later Depending on the specific application, the magnetic core of an apparatus may be ‘operated in the linear region, and/or the saturation region, For example, transformers ‘and AC machines ace operated in the linear region and lower end of the kree: self excited DC generators and DC motors are operated in the upper end of the knee rogion, extending into the saturation region; separately excited DC generators are ‘operated in the incar and lower end of the kace region. Magnetization curves supplied oy manutacturers ter specific electrical steel sheets or cesting are usually plotted on semilog paper, and often include & curve of relative permeability ¥s. field intersicy, as shown in Figure 1.5." "Flgwe 13.2 ished by he manufac. ashe aged Bell iesay expresed nosed aps a. “ecemsert t9 Am milipy ty 19 S77 See Ape Kfrer carver etre AT ot am, Me ini vale of = (Dan Racca wen rain i complet, cing Magnetice, Electromagnetic Forces, Ganerstad Voltage, and Enargy Conversion | 7 Taw Medi Fan Ven High Magretic Fie Inert (7, Avid FIGURE 1.4 Exaggeraced ragnetzaton curvo ilustrating the four erincioal sections. The rlationship between the relaive permeability andthe reluctanes of a mag. atic core fs obtain by sovieg Eq (1-1) frp ard hen substaing ino Eq, (=) The results é t a=t 1-8) BAT ed oa nical thatthe reluctance of a maguetie cine ative permeability of the material, which, as shown in Figure 15 is dependert on the magnetization, and hence ig not constant, EXAMPLE (a) Letervine the voltage that mus! be applieé to the magnoiizing coil in Figure 14 6c) in onder w produce a us density of 0.200 T in the ait gap. Flur fringig, which always occurs along the sides of an airgap, 2s shown in Figure 1.6(b) will be essumed nogligible, Assume the magnesizaton curve for the core material (which is hemoge- nous) is that given in Figure 1.5. The eoil has 80 turns and a resistance of 0.08 £2. The cross sestion of the core materia] is 0.0400 m?, 1000 000 000 000 cco wo 20 260 1000 6 ‘Magee Fle itenshy (H,cerstods) (Qerstods X79.577 = At) Fs = f FEE HH sersecenaet 1 eres ee (LW fase ee FIGURE 15 ‘Magnetization end permeability curves or elctica sheet steel used in magrete ‘aplcatons. (Courtesy USX Com.| Magnetic, Electromagnetic Forest, Generated Voltage, and Energy Conversion | 9 » [psa © FIGURE 1.6 Megnete circuit for Example 1,': (a physica layout and dimensions; i) fx fringing: 10 flux distnbution. ues obtained from the permeability curve in Figure 15. Solution (2) The physical layout and dimensions ofthe magretic circuit shown in Figure 1.6¢a) are used in conyunetion withthe B-H curve 1o determine the magnetic field intensity in