Brushless Permanent Magnet Motor Design P2
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Brushless Permanent Magnet Motor Design P2
Brushless Permanent Magnet Motor Design P2: You've just picked up another book on motors. You've seen many others, but they all assume that you know more about motors than you do. Phrases such as armature reaction, slot leakage, fractional pitch, and skew factor are used with little or no introduction. You keep looking for a book that is written from a more basic, yet rigorous, perspective and you're hoping this is it.
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Nội dung Text: Brushless Permanent Magnet Motor Design P2
 Magnetic Modeling 21 X • < dx T~ Figure 2.9 Circulararc, straight line permeance model. In this equation, the extent that the fringing permeance extends up the sides of the blocks, is the only unknown. In those cases where X is not fixed by geometric constraints, it is commonly chosen to be some multiple of the air gap length. The exact value chosen is not that critical because the contribution of differential permeances decreases as one moves farther from the air gap. Thus as X increases beyond about 10g, there is little change in the total air gap permeance. Slot modeling " , 4 * Often electrical machines have slots facing an air gap which hold cur rentcarrying windings. Since the windings are nonmagnetic, flux crossing an air gap containing slots will try to avoid the low relative permeability of the slot area. This adds another factor that must be considered in determining the permeance of an air gap. To illustrate this point, consider Fig. 2.10a, where slots have been placed in the lower block of highly permeable material. Considering just one slot and the tooth between the slots, there are several ways to approximate the air gap permeance. The simplest and crudest method is to ignore the slot by assuming that it contains material of permeability equal to that of the rest of the block. In this case, the permeability is simply Pg = /¿oA/g, where A is the total crosssectional area facing the gap. Obviously, this is a poor approximation because the relative permeability of the slot is orders of magnitude lower than that of block material. Another crude approximation is to ignore the flux crossing the gap over the slot, giving a permeance of Pg = IAq(A  As)/g, where As is the crosssectional area of the slot facing the air gap. Neither of these methods is very accurate, but they do represent upper and lower bounds on the air gap permeance, respectively.
 22 Chapter T i g J (a) (b) Figure 2.10 A slotted structure. There are two more accurate ways of determining air gap permeance in the presence of slotting. The first is based on the observation that the flux crossing the gap over the slot travels a further distance before reaching the highly permeable material across the gap. As a result, the permeance can be written as Pg = /¿oA/ge, where ge = gkc is an effective air gap length. Here kc > 1 is a correction factor that increases the entire air gap length to account for the extra flux path distance over the slot. One approximation for kc is known as Carter's coefficient (Mukheiji and Neville, 1971; Qishan and Hongzhan, 1985). By apply ing conformal mapping techniques, Carter was able to determine an analytic magneticfield solution for the case where slots appear on both sides of the air gap. Through symmetry considerations it can be shown that the Carter coefficient for the aligned case, i.e., when the slots are directly opposite each other, is an acceptable approximation to the geometry shown in Fig. 2.10a. Two expressions for Carter's coefficient are 1  hi = II 5 — + 1 (2.12) ws ws given by Nasar (1987), and 2~ kco = ( 1  ^ ^ \ tan 1 g . /In 1 + (2.13) 777V given by Ward and Lawrenson (1977).
 Magnetic Modeling 23 The other more accurate method for determining the air gap perme ance utilizes the circulararc, straightline modeling discussed earlier. This method is demonstrated in Fig. 2.106. Following an approach similar to that described in (2.11), the permeance of the air gap can be written as rs  ws 4 7TWS\ Pg = Pa+Pb + Pc= MoL H— In 1 + g TT 4£/J where L is the depth of the block into the page. With some algebraic manipulation, this solution can also be written in the form of an air gap length correction factor, as described in the preceding paragraph. In this case, kc is given by ws ^ . 1 kC3 = 1 + — In 1 + TTW< (2.14) Te 7TT, *gJ J A comparison of (2.12), (2.13), and (2.14) shows that all produce similar air gap length correction factors. As illustrated in Fig. 2.11, kc2 gives a larger correction factor than &c3 and kcz gives a larger cor rection factor than k c i, with the deviation among the expressions in creasing as g/rs decreases and WS/TS increases. One important consequence of slotting shown in Fig. 2.12 is that the presence of slots squeezes the air gap flux into a crosssectional area (1  ws/rs) times smaller than the crosssectional area of the entire air gap. Thus the averageflux density at the base of the teeth is greater Figure 2.11 A comparison of various carter coefficients.
 24 Chapter T Base of Figure 2.12 Flux squeezing at Tooth the base of a tooth. Flux by a factor of (1 — w s lr s )~ l . The importance of this phenomenon cannot be understated. For example, if the average flux density crossing the air gap is 1.0 T and slot fraction AS = WS/TS is 0.5, then the average flux density in the base of the teeth is (1.0)(1  0.5K 1 = 2.0 T. Since thisflux density level is sufficient to saturate (i.e., dramatically reduce the effective permeability of) most magnetic materials, there is an upper limit to the achievable air gap flux density in a motor. Later this will be shown to be a limiting factor in motor performance. Example The preceding discussion embodies the basic concepts of magnetic cir cuit analysis. Application of these concepts requires making assump tions about magnetic field direction, flux path lengths, and flux uni formity over crosssectional areas. To illustrate magnetic circuit analysis, consider the wound core shown in Fig. 2.13a and its corre sponding magnetic circuit diagram in Fig. 2.136. Assuming that the permeability of the core is much greater than that of the surrounding air, the magnetic field is essentially confined to the core, except at the air gap. Comparing Figs. 2.13a with 2.136, the coil is represented by the mmf source of value NI. The reluctance of the core material is modeled by the reluctance Rc = IJyA, where lc is the average length of the core from one side of the air gap around to the other, ¡x is the permeability of the core material, and A is the crosssectional area of the core. This modeling approximates the flux path length around bends as having median length. It also assumes that theflux density is uniform over the cross section. Rg, the reluctance of the air gap, is given by the inverse of the air gap permeance discussed earlier. Table 2.1 shows solutions of this magnetic circuit example for the three air gap models discussed earlier. The first row corresponds to the model shown in Fig. 2.8a, the second row to Fig. 2.86, and the third
 Magnetic Modeling 25 4/\IV " S (a) (b) Figure 2.13 A simple magnetic structure and its magnetic circuit model. row to Fig. 2.8c, with the fringe permeance having a width ten times larger than the air gap. The second column in the table is the air gap reluctance, the third column is the core reluctance, the fourth is the flux density in the core, B = and the fifth is the percentage of the applied mmf that appears across the air gap. Based on the information in the table, several statements can be made. First, the core reluctance is small with respect to the air gap reluctance. This follows because the permeability of the core material is several orders of magnitude greater than that of the air gap. As a result, the core reluctance has little effect on the solution, and more accurate modeling of the core is not necessary. Second, the reluctance of the air gap decreases as more fringing flux is accounted for. This increases the flux density in the core because the net circuit reluctance decreases with the decreasing air gap reluctance. Last, both methods which account for fringing flux lead to nearly identical solutions. The fact that the air gap dominates the magnetic circuit has profound implications in practice. It implies that the majority of the applied mmf appears across the air gap as shown in Table 2.1. For analytic work, it allows one to neglect the reluctance of the core in many cases, thereby TABLE 2.1 Magnetic Circuit Solutions Air gap Core flux Percentage air permeance model Rg( H' 1 ) Rc( H 1 ) density (T) gap mmf (%) Figure 2.8a 3.98e6 4.18e5 0.91 90.5 Figure 2.86 3.29e6 4.18e5 1.08 88.7 Figure 2.8c, X = 10£ 3.26e6 4.18e5 1.09 88.6
 26 Chapter T simplifying the analysis considerably. The dominance of the air gap also implies that the exact magnetic characteristics of the core do not have a great effect on the solution provided that the permeability of the core remains high. This is fortunate because the core is commonly made from materials having highly nonlinear magnetic properties. These properties are discussed next. Magnetic Materials Permeability As stated earlier in (2.1), in linear materials B and H are related by B = ¡xH, where ¡x is the permeability of the material. For convenience, it is common to express permeability with respect to the permeability of free space, fx — /x0 = Att • 10" 7 H/m. In doing so, a nondimension relative permeability is defined as M = — r (2.15) Mo and (2.1) is rewritten as B = fx^H. Using this relationship, materials having /xr = 1 are commonly called nonmagnetic materials, while those with greater permeability are called magnetic materials. Permeability as defined by (2.1) and (2.15) applies strictly to materials that are linear, homogeneous (have uniform properties), and isotropic (have the same properties in all directions). Despite this fact, however, (2.1) and (2.15) are used extensively because they approximate the actual prop erties of more complex magnetic materials with sufficient accuracy over a sufficiently wide operating range. Ferromagnetic materials, especially electrical steels, are the most common magnetic materials used in motor construction. The perme ability of these materials is described by (2.1) and (2.15) in an ap proximate sense only. The permeability of these materials is nonlinear and multivalued, making exact analysis extremely difficult. In addition to the permeability being a nonlinear, saturating function of the field intensity, the multivalued nature of the permeability means that the flux density through the material is not unique for a given field in tensity but rather is a function of the past history of the field intensity. Because of this behavior, the magnetic properties of ferromagnetic materials are often described graphically in terms of their BH curve, hysteresis loop, and core losses. Ferromagnetic materials Figure 2.14 shows the BH curve and several hysteresis loops for a typical ferromagnetic material. Each hysteresis loop is formed by ap
 Magnetic Modeling 27 plying ac excitation of fixed amplitude to the material and plotting B vs. H. The BH curve is formed by connecting the tips of the hysteresis loops together to form a smooth curve. The BH curve, or dc magnet ization curve, represents an average material characteristic that re flects the nonlinear property of the permeability but ignores its mul tivalued property. Two relative permeabilities are associated with the BH curve. The normalized slope of the BH curve at any point is called the relative differential permeability and is given by 1_ dB In addition, the relative amplitude permeability is simply the ratio of B to H at a point on the curve, }_B M ~ a TJ M H o Both of these permeability measures are useful for describing the rel ative permeability of the material. Over a significant range of oper ating conditions, they are both much greater than 1. As is apparent from Fig. 2.14, the relative differential permeability is small for low excitations, increases and peaks at medium excitations, and finally decreases for high excitations. At very high excitations, ¡xd approaches 1, and the material is said to be in hard saturation. For common elec
 28 Chapter T trical steels, hard saturation is reached at a flux density between 1.7 and 2.3 T, and the onset of saturation occurs in the neighborhood of 1.0 to 1.5 T. Core loss When ferromagnetic materials are excited with any timevarying ex citation, energy is dissipated due to hysteresis and eddy current losses. These losses are difficult to isolate experimentally; therefore, their combined losses are usually measured and called core losses. Figure 2.15 shows core loss density data of a typical magnetic material. These curves represent the loss per unit mass when the material is exposed uniformly to a sinusoidal magnetic field of various amplitudes. Total core loss in a block of material is therefore found by multiplying the mass of the material by the appropriate data value read from the graph. In brushless PM motors, different parts of the motor ferromagnetic material are exposed to different flux density amplitudes, different waveshapes, and different frequencies of excitation. Therefore, core loss data such as those shown in Fig. 2.15 are difficult to apply accu rately to brushless PM motors. However, because more accurate com putation of actual core losses is much more difficult to compute (Slemon and Liu, 1990), traditional core loss data are considered an adequate approximation. Hysteresis loss results because energy is lost every time a hysteresis loop is traversed. This loss is directly proportional to the size of the hysteresis loop of a given material, and therefore by inspection of Fig. Figure 2.15 Typical core loss characteristics of ferromagnetic material.
 Magnetic Modeling 29 2.14, it is proportional to the magnitude of the excitation. In general, hysteresis power loss is described by the equation Ph = hfB n m where kh is a constant that depends on the material type and dimen sions, f is the frequency of applied excitation, Bm is the maximum flux density within the material, and n is a materialdependent exponent between 1.5 and 2.5. Eddy current loss is caused by induced electric currents within the ferromagnetic material under timevarying excitation. These induced eddy currents circulate within the material, dissipating power due to the resistivity of the material. Eddy current power loss is approxi mately described by the relationship Pe = KfBl where ke is a constant. In this case, the power lost is proportional to the square of both frequency and maximum flux density. Therefore, one would expect hysteresis loss to dominate at low frequencies and eddy current loss to dominate at higher frequencies. The most straightforward way to reduce eddy current loss is to in crease the resistivity of the material. This is commonly done in a number of ways. First, electrical steels contain a small percentage of silicon, which is a semiconductor. The presence of silicon increases the resistivity of the steel substantially, thereby reducing eddy current losses. In addition, it is common to build an apparatus using lamina tions of material as shown in Fig. 2.16. These thin sheets of material are coated with a thin layer of nonconductive material. By stacking these laminations together, the resistivity of the material is dramat Ferromagrietic Laminations Figure 2.16 Laminated ferro magnetic material. Insulation
 30 Chapter T ically increased in the direction of the stack. Since the nonconductive material is also nonmagnetic, it is necessary to orient the lamination edges parallel to the desired flow of flux. It can be shown that eddy current loss in laminated material is proportional to the square of the lamination thickness. Thus thin laminations are required for lower loss operation. Laminations decrease the amount of magnetic material available to carry flux within a given crosssectional area. To compensate for this in analysis, a stacking factor is defined as ^ _ cross section occupied by magnetic material (2 16) total cross section This factor is important for the accurate calculation of flux densities in laminated magnetic materials. Typical stacking factors range from 0.5 to 0.95. Though not extensively used in motor construction, it is possible to use powdered magnetic materials to reduce eddy current loss to a min imum. These materials are composed of powdered magnetic material suspended in a nonconductive resin. The small size of the particles used, and their electrical isolation from one another, dramatically in creases the effective resistivity of the material. However, in this case the effective permeability of the material is decreased because the nonmagnetic resin appears in all flux paths through the material. Permanent magnets Many different types of PM materials are available today. The types available include alnico, ferrite (ceramic), rareearth samariumcobalt, and neodymiumironboron (NdFeB). Of these, ferrite types are the most popular because they are cheap. NdFeB magnets are more popular in higherperformance applications because they are much cheaper than samarium cobalt. Most magnet types are available in both bonded and sintered forms. Bonded magnets are formed by suspending pow dered magnet material in a nonconductive, nonmagnetic resin. Mag nets formed in this way are not capable of high performance, since a substantial fraction of their volume is made up of nonmagnetic ma terial. The magnetic material used to hold trinkets to your refrigerator door is bonded, as is the magnetic material in the refrigerator door seal. Sintered magnets, on the other hand, are capable of high per formance because the sintering process allows magnets to be formed without a bonding agent. Overall, each magnet type has different prop erties leading to different constraints and different levels of perform ance in brushless PM motors. Rather than exhaustively discuss each of these magnet types, only generic properties of PMs will be discussed.
 Magnetic Modeling 31 Those wishing more indepth information should see references such as McCaig and Clegg (1987). Stated in the simplest possible terms, PMs are magnetic materials with large hysteresis loops. Thus the starting point for understanding PMs is their hysteresis loop, the first and second quadrant of which are shown in Fig. 2.17. For convenience, the field intensity axis is scaled by ¡jlq, giving both axes dimensions in tesla. (Note: This also visually compresses the field intensity axis. The uncompressed slope of the line in the second quadrant is approximately /x0, which is very small.) The hysteresis loop shown in the figure is formed by applying the largest possible field to an unmagnetized sample of material, then shutting it off. This allows the material to relax, or recoil, along the upper curve shown in the figure. The final position attained is a func tion of the magnetic circuit external to the magnet. If the two ends of the magnet are shorted together by a piece of infinitely permeable material (an infinite permeance) as shown in Fig. 2.18a, the magnet is said to be keepered, and the final point attained is H = 0. The flux density leaving the magnet at this point is equal to the remanence, denoted B r . The remanence is the maximum flux density that the mag net can produce by itself. On the other hand, if the permeability sur rounding the magnet is zero (a zero permeance) as shown in Fig. 2.18b, no flux flows out of the magnet and the final point attained is B = 0. At this point, the magnitude of the field intensity across the magnet is equal to the coercivity, denoted Hc. For permeance values between zero and infinity, the operating point lies somewhere in the second quadrant, i.e., between the remanence and coercivity. The absolute value of the slope of the load line formed from the operating point to B (T) Second First Quadrant Quadrant VH (T) Figure 2.17 The BH loop of a permanent magnet.
 32 Chapter T [/ I I H = 0, B = Br B = 0, H = Hc (a) (b) Figure 2.18 Operation of a magnet at its (a) remanence and (6) coer civity. the origin, normalized by (Xq, is known as the permeance coefficient (PC) (Miller, 1989). Therefore, operating at the remanence is a PC of infinity, operating at the coercivity is a PC of zero, and operating halfway between these points is a PC of 1. Hard PM materials such as samariumcobolt and NdFeB materials have straight demagnetization curves throughout the second quadrant at room temperature, as shown in Fig. 2.19. The slope of this straight line is equal to /xr/xq, where ¡xR is the recoil permeability of the material. The value of fxR is typically between 1.0 and 1.1. At higher tempera tures, the demagnetization curve tends to shrink toward the origin, as shown in Fig. 2.19, with these changes often approximated as tem perature coefficients on Br and Hc. As this shrinking occurs, the flux available from the magnet drops, reducing the performance of the mag net. This performance degradation is reversible, however, as the de magnetization curve returns to its former shape as temperature drops. In addition to shrinking toward the origin as temperature increases, the knee of the demagnetization characteristic may move into the sec ond quadrant as shown in Fig. 2.19. This deviation from a straight line causes the flux density to drop off more quickly as  H c is ap proached. Operation in the area of the knee can cause the magnet to lose some magnetization irreversibly because the magnet will recoil along a line of lower magnetization, as shown by the dotted line in Fig. 2.19. If this happens, the effective Br and Hc drop, lowering the performance of the magnet. Since this is clearly undesirable, it is nec essary to assure that magnets operate away from the coercivity at a sufficiently large PC (denoted Pc in Fig. 2.19).
 Magnetic Modeling 33 magnets. In addition to the fundamental hysteresis characteristic of PM mag net material, PM material also exhibits a pronounced anisotropic be havior. That is, the material has a preferred direction of magnetization that gives it a permeability that is dramatically smaller in other di rections. This fact implies that care must be used when orienting and magnetizing magnets to be sure they follow the desired direction of magnetization with respect to the desired geometrical shape. Moreover, it implies that little flux leaks from the side of a magnet if the magnet is not terribly long. Before moving on, it is beneficial to define the maximum energy product, as this specification is usually the first specification used to compare magnets. The maximum energy product (BH) m a x of a magnet is the maximum product of the flux density and field intensity along the magnet demagnetization curve. This product is not the actual stored magnet energy (even though it has units of energy), but rather it is a qualitative measure of a magnet's performance capability in a magnetic circuit. By convention, {BH)m&x is usually specified in the English units of millions of gaussoersteds (MGOe). However, some magnet manufacturers do conform to SI units of joules per cubic meter (1 MGOe = 7.958 kJ/m 3 ). For magnets with ¡xR « 1, (BH) m a x occurs near the unity PC operating point. It can be shown that operation at
 34 Chapter T (.BH)max is the most efficient in terms of magnet volumetric energy density. Despite this fact, PMs in motors are almost never operated at (.BH)max because of possible irreversible demagnetization with increas ing temperature, as discussed in the previous paragraph (Miller, 1989). PM magnetic circuit model To move the magnet operating point from its static operating point determined by the external permeance, an external magnetic field must be applied. In a motor, the static operating point lies somewhere in the second quadrant, usually at a PC of 4 or more. When motor wind ings are energized, the operating point dynamically varies following minor hysteresis loops about the static operating point, as shown in Fig. 2.20. These loops are thin and have a slope essentially equal to that of the demagnetization characteristic. As a result, the trajectory closely follows the straightline demagnetization characteristic de scribed by Bm= Br + (XRtMfim (2.17) This equation assumes that the magnet remains in a linear operating region under all operating conditions. Driving the magnet past the remanence into the first quadrant normally causes no harm, as this is point.
 Magnetic Modeling 35 in the direction of magnetization. However, if the external magnetic field opposes that developed by the magnet and drives the operating point into the third quadrant past the coercivity, it is possible to ir reversibly demagnetize the magnet if a knee in the characteristic is encountered. Using (2.17), it is possible to develop a magnetic circuit model for a PM. Let the rectangular magnet shown in Fig. 2.21a be described by (2.17). Then the flux leaving the magnet is (f>m = BmAm = BrAm + AmHm where Am is the crosssectional area of the magnet face in the direction of magnetization. Using (2.4), (2.5), and (2.6), this equation can be rewritten as m = r + PmF„ (2.18) where 4>r = BrA, (2.19) is a fixed flux source, and where V
 36 Chapter T sumes that the physical magnet is uniformly magnetized over its cross section and is magnetized in its preferred direction of magnetization. When the magnet shape differs from the rectangular shape shown in Fig. 2.21a, it is necessary to reevaluate its magnetic circuit model. In brushless PM motors having a radial air gap, the magnet shape may appear as an arc, as shown in Fig. 2.22. The magnetic circuit model of this shape can be found by considering it to be a radial stack of differential length magnets, each having a model as given in Fig. 2.216. During magnetization the same amount of flux magnetizes each differential length. As a result, the achieved remanence decreases lin early with increasing radius because the same flux over an increasing area gives a smaller flux density (Hendershot, 1991). Therefore, in tegration of these differential elements gives a magnet magnetic circuit model of the same form as Fig. 2.216 with / W A ( 2 2 1 ) m ln(l + IJri) and r = BrLdpn (2.22) where Br is the remanence achieved at and L is the axial length of the magnet into the page. In the common case where lm « rL (2.21) can be simplified by approximating the permeance shape as rectangular with an average cross section. This approximation gives Pm = fxRtM)Ldp + £) (2.23) Example To illustrate the concepts presented in this chapter, consider the mag netic apparatus and circuit shown in Fig. 2.23. The apparatus consists Figure 2.22 An arcshaped mag net.
 Magnetic Modeling 37 of a PM, highly permeable ferromagnetic material, and an air gap. Given that the ferromagnetic material has very high permeability, its reluctance can be ignored, resulting in a magnetic circuit consisting of the magnet equivalent circuit and the air gap permeance as shown in Fig. 2.236. Since the flux leaving the magnet is equal to that crossing the air gap, the magnet and air gap flux densities are related by Bg  Bm Ag  BmC$ (2.24) where Am and Ag are the crosssectional areas of the magnet and air gap, respectively, and C$ = Am/Ag is the flux concentration factor. When C^ is greater than 1, the flux density in the air gap is greater than that at the magnet surface. The magnet flux is easily found by flux division as m = BmAtn = n ~4>r Brlm Fm = PM + PO «,ii V g ® (a) (b) Figure 2.23 A simple magnetic structure and its magnetic circuit model.
 38 Chapter T and the field intensity operating point of the magnet Hm = FJlm normalized by the magnet coercivity Hc = Br/{/xR¿¿o) is Hm _ 1 _ 1 _ Oil (O OC) Hc 1 + lJ(nRg)C^ Br ' Comparing (2.25) with (2.26), it is clear that there is an inverse relationship between the magnet flux density and its field intensity. As one increases the other decreases. Furthermore, from (2.25), the magnetflux density increases as theflux concentration factor decreases or as the ratio of the magnet length to air gap increases. Therefore, a longer relative magnet length increases the available air gap flux den sity. The exact operating point of the magnet is found by computing the permeance coefficient, PC = — ^ = ^ r * =  1, pushes the PC lower. The fundamental importance of (2.27) can be seen by considering what is required to maintain a constant PC as the concentration factor increases. Multiplying the numerator and denominator of (2.27) by AmAg and simplifying gives P V (2'28)   t k where Vm and Vg are the magnet and air gap volumes, respectively. Now if Crf, is doubled to 2C 6 and the air gap volume remains constant, the magnet volume must increase by a factor of 2 2 = 4 to maintain a constant PC. If the magnet crosssectional area remains constant, this implies that the magnet length must increase by a factor of 4. The implication of this analysis is that concentrating the flux of a PM does not come without the penalty of geometrically increasing magnet volume. Conclusion In this chapter, the basics of magnetic circuit analysis were presented. Starting with fundamental magnetic field concepts, the concepts of
 Magnetic Modeling 39 permeance, reluctance, flux, and mmf were developed. Permeance models for blocks of magnetic material, air gaps, and slotted magnetic structures were developed. The properties of ferromagnetic and per manentmagnet materials were discussed. A magnetic circuit model of a permanent magnet was introduced and the concept of flux con centration was illustrated. With this background it is now possible to discuss how magnetic fields interact with the electrical and mechanical parts of a motor. These concepts are discussed in the next chapter.
 Chapter Electrical and 3 Mechanical Relationships As stated in the first chapter, the operation of a brushless PM motor relies on the conversion of electrical energy to magnetic energy and from magnetic energy to mechanical energy. In this chapter, the con nections between magnetic field concepts, electric circuits, and me chanical motion will be explored to illustrate this energy conversion process. Flux Linkage and Inductance Self inductance To define flux linkage and selfinductance, consider the magnetic cir cuits shown in Fig. 3.1. This circuit is said to be singly excited since it has only one coil to produce a magnetic field. Theflux flowing around the core is due to the current I, and the direction of flux flow is clockwise because of the righthand rule. Using the magnetic circuit equivalent of Ohm's law, the flux produced is given by *  N I where R is the reluctance seen by the mmf source. Since thisflux passes through, or links, all N turns of the winding, the total flux linked by the winding is called the flux linkage, which is defined as A = N(f) (3.1) 41
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