Class Notes 7: Permanent Magnet “Brushless DC” Motors

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.685 Electric Machines Class Notes 7: Permanent Magnet “Brushless DC” Motors c 2005 James L. Kirtley Jr. 1 Introduction This document is a brief introduction to the design evaluation of permanent magnet motors, with an eye toward servo and drive applications. It is organized in the following manner: First, we describe three different geometrical arrangements for permanent magnet motors: 1. Surface Mounted Magnets, Conventional Stator, 2. Surface Mounted Magnets, Air-Gap Stator Winding, and 3. Internal Magnets (Flux Concentrating). After a qualitative discussion of these geometries, we will discuss the elementary rating parameters of the machine and show how to arrive at a rating and how to estimate the torque and power vs. speed capability of the motor. Then we will discuss how the machine geometry can be used to estimate both the elementary rating parameters and the parameters used to make more detailed estimates of the machine performance. Some of the more involved mathematical derivations are contained in appendices to this note. 2 Motor Morphologies There are, of course, many ways of building permanent magnet motors, but we will consider only a few in this note. Actually, once these are understood, rating evaluations of most other geometrical arrangements should be fairly straightforward. It should be understood that the “rotor inside” vs. “rotor outside” distinction is in fact trivial, with very few exceptions, which we will note. 2.1 Surface Magnet Machines Figure 1 shows the basic magnetic morphology of the motor with magnets mounted on the surface of the rotor and an otherwise conventional stator winding. This sketch does not show some of the important mechanical aspects of the machine, such as the means for fastening the permanent magnets to the rotor, so one should look at it with a bit of caution. In addition, this sketch and the other sketches to follow are not necessarily to a scale that would result in workable machines. This figure shows an axial section of a four-pole (p = 2) machine. The four magnets are mounted on a cylindrical rotor “core”, or shaft, made of ferromagnetic material. Typically this would simply be a steel shaft. In some applications the magnets may be simply bonded to the steel. For applications in which a glue joint is not satisfactory (e.g. for high speed machines) some sort of rotor banding or retaining ring structure is required. 1 Rotor Core (Shaft) Stator Winding in Slots Stator Core Air−Gap Rotor Magnets Figure 1: Axial View of a Surface Mount Motor The stator winding of this machine is “conventional”, very much like that of an induction motor, consisting of wires located in slots in the surface of the stator core. The stator core itself is made of laminated ferromagnetic material (probably silicon iron sheets), the character and thickness of the sheets determined by operating frequency and efficiency requirements. They are required to carry alternating magnetic fields, so must be laminated to reduce eddy current losses. This sort of machine is simple in construction. Note that the operating magnetic flux density in the air-gap is nearly the same as in the magnets, so that this sort of machine cannot have air-gap flux densities higher than that of the remanent flux density of the magnets. If low cost ferrite magnets are used, this means relatively low induction and consequently relatively low efficiency and power density. (Note the qualifier “relatively” here!). Note, however, that with modern, high performance permanent magnet materials in which remanent flux densities can be on the order of 1.2 T, air-gap working flux densities can be on the order of 1 T. With the requirement for slots to carry the armature current, this may be a practical limit for air-gap flux density anyway. It is also important to note that the magnets in this design are really in the “air gap” of the machine, and therefore are exposed to all of the time- and space- harmonics of the stator winding MMF. Because some permanent magnets have electrical conductivity (particularly the higher performance magnets), any asynchronous fields will tend to produce eddy currents and consequent losses in the magnets. 2.2 Interior Magnet or Flux Concentrating Machines Interior magnet designs have been developed to counter several apparent or real shortcomings of surface mount motors: • Flux concentrating designs allow the flux density in the air-gap to be higher than the flux density in the magnets themselves. 2 • In interior magnet designs there is some degree of shielding of the magnets from high order space harmonic fields by the pole pieces. • There are control advantages to some types of interior magnet motors, as we will show anon. Essentially, they have relatively large negative saliency which enhances “flux weakening” for high speed operation, in rather direct analogy to what is done in DC machines. • Some types of internal magnet designs have (or claim) structural advantages over surface mount magnet designs. Rotor Pole Pieces Armature in Slots Rotor Magnets Non−magnetic Rotor Core (shaft) Stator Core Figure 2: Axial View of a Flux Concentrating Motor The geometry of one type of internal magnet motor is shown (crudely) in Figure 2. The permanent magnets are oriented so that their magnetization is azimuthal. They are located between wedges of magnetic material (the pole pieces) in the rotor. Flux passes through these wedges, going radially at the air- gap, then azimuthally through the magnets. The central core of the rotor must be non-magnetic, to prevent “shorting out” the magnets. No structure is shown at all in this drawing, but quite obviously this sort of rotor is a structural challenge. Shown is a six-pole machine. Typically, one does not expect flux concentrating machines to have small pole numbers, because it is difficult to get more area inside the rotor than around the periphery. On the other hand, a machine built in this way but without substantial flux concentration will still have saliency and magnet shielding properties. A second morphology for an internal magnet motor is shown in Figure 3. This geometry has been proposed for highly salient synchronous machines without permanent magnets: such machines would run on the saliency torque and are called synchronous reluctance motors. however, the saliency slots may be filled with permanent magnet material, giving them some internally generated flux as well. The rotor iron tends to short out the magnets, so that the ’bridges’ around the ends of the permanent magnets must be relatively thin. They are normally saturated. 3 Stator Core Stator Slots Air Gap Rotor Saliency Slots Figure 3: Axial View of Internal Magnet Motor At first sight, these machines appear to be quite complicated to analyze, and that judgement seems to hold up. 2.3 Air Gap Armature Windings Shown in Figure 4 is a surface-mounted magnet machine with an air-gap, or surface armature winding. Such machines take advantage of the fact that modern permanent magnet materials have very low permeabilities and that, therefore, the magnetic field produced is relatively insensitive to the size of the air-gap of the machine. It is possible to eliminate the stator teeth and use all of the periphery of the air-gap for windings. Not shown in this figure is the structure of the armature winding. This is not an issue in “conventional” stators, since the armature is contained in slots in the iron stator core. The use of an air-gap winding gives opportunities for economy of construction, new armature winding forms such as helical windings, elimination of “cogging” torques, and (possibly) higher power densities. 3 Zeroth Order Rating In determining the rating of a machine, we may consider two separate sets of parameters. The first set, the elementary rating parameters, consist of the machine inductances, internal flux linkage and stator resistance. From these and a few assumptions about base and maximum speed it is possible to get a first estimate of the rating and performance of the motor. More detailed performance estimates, including efficiency in sustained operation, require estimation of other parameters. We will pay more attention to that first set of parameters, but will attempt to show how at least some of the more complete operating parameters can be estimated. 4 Rotor Core (Shaft) Stator Winding Stator Core 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 Rotor Magnets Figure 4: Axial View of a PM Motor With an Air-Gap Winding 3.1 Voltage and Current: Round Rotor To get started, consider the equivalent circuit shown in Figure 5. This is actually the equivalent circuit which describes all round rotor synchronous machines. It is directly equivalent only to some of the machines we are dealing with here, but it will serve to illustrate one or two important points. X ∩∩∩∩  + − Ea  Ia  + Vt - Figure 5: Synchronous Machine Equivalent Circuit What is shown here is the equivalent circuit of a single phase of the machine. Most motors are three-phase, but it is not difficult to carry out most of the analysis for an arbitrary number of phases. The circuit shows an internal voltage Ea and a reactance X which together with the terminal current I determine the terminal voltage V . In this picture armature resistance is ignored. If the machine is running in the sinusoidal steady state, the major quantities are of the form: Ea = ωλa cos (ωt + δ) Vt = V cos ωt 5 I 1    BM  ψ V   ? HH BM 6 HH B δ HH B HH B HH B jXI HH B HH B H jB H Ea Figure 6: Phasor Diagram For A Synchronous Machine Ia = I cos (ωt − ψ) The machine is in synchronous operation if the internal and external voltages are at the same frequency and have a constant (or slowly changing) phase relationship (δ). The relationship between the major variables may be visualized by the phasor diagram shown in Figure 3.1. The internal voltage is just the time derivative of the internal flux from the permanent magnets, and the voltage drop in the machine reactance is also the time derivative of flux produced by armature current in the air-gap and in the “leakage” inductances of the machine. By convention, the angle ψ is positive when current I lags voltage V and the angle δ is positive then internal voltage Ea leads terminal voltage V . So both of these angles have negative sign in the situation shown in Figure 3.1. If there are q phases, the time average power produced by this machine is simply: q P = V I cos ψ 2 For most polyphase machines operating in what is called “balanced” operation (all phases doing the same thing with uniform phase differences between phases), torque (and consequently power) are approximately constant. Since we have ignored power dissipated in the machine armature, it must be true that power absorbed by the internal voltage source is the same as terminal power, or: q P = Ea I cos (ψ − δ) 2 Since in the steady state: ω P = T p where T is torque and ω/p is mechanical rotational speed, torque can be derived from the terminal quantities by simply: q T = p λa I cos (ψ − δ) 2 In principal, then, to determine the torque and hence power rating of a machine it is only necessary to determine the internal flux, the terminal current capability, and the speed capability of the rotor. In fact it is almost that simple. Unfortunately, the model shown in Figure 5 is not quite complete for some of the motors we will be dealing with, and we must go one more level into machine theory. 6 3.2 A Little Two-Reaction Theory The material in this subsection is framed in terms of three-phase (q = 3) machine theory, but it is actually generalizable to an arbitrary number of phases. Suppose we have a machine whose three-phase armature can be characterized by internal fluxes and inductance which may, in general, not be constant but is a function of rotor position. Note that the simple model we presented in the previous subsection does not conform to this picture, because it assumes a constant terminal inductance. In that case, we have: λph = Lph I ph + λR (1) where λR is the set of internally produced fluxes (from the permanent magnets) and the stator winding may have both self- and mutual- inductances. Now, we find it useful to do a transformation on these stator fluxes in the following way: each armature quantity, including flux, current and voltage, is projected into a coordinate system that is fixed to the rotor. This is often called the Park’s Transformation. For a three phase machine it is:     ud ua      uq  = udq = T uph = T  ub  u0 uc (2) Where the transformation and its inverse are: cos θ cos(θ − 2π cos(θ + 2π 3 ) 3 ) 2  2π T =  − sin θ − sin(θ − 3 ) − sin(θ + 2π 3 )  3 1 1 1   2 T −1 2  (3) 2  cos θ − sin θ 1   2π =  cos(θ − 2π ) − sin(θ − ) 1  3 3 2π 2π cos(θ + 3 ) − sin(θ + 3 ) 1 (4) It is easy to show that balanced polyphase quantities in the stationary, or phase variable frame, translate into constant quantities in the so-called “d-q” frame. For example: Ia = I cos ωt 2π ) 3 2π Ic = I cos(ωt + ) 3 θ = ωt + θ0 Ib = I cos(ωt − maps to: Id = I cos θ0 Iq = −I sin θ0 Now, if θ = ωt + θ0 , the transformation coordinate system is chosen correctly and the “d-” axis will correspond with the axis on which the rotor magnets are making positive flux. That happens 7 if, when θ = 0, phase A is linking maximum positive flux from the permanent magnets. If this is the case, the internal fluxes are: λaa = λf cos θ 2π ) 3 2π = λf cos(θ + ) 3 λab = λf cos(θ − λac Now, if we compute the fluxes in the d-q frame, we have: λdq = Ldq I dq + λR = T Lph T −1 I dq + λR (5) Now: two things should be noted here. The first is that, if the coordinate system has been chosen as described above, the flux induced by the rotor is, in the d-q frame, simply:   λf   λR =  0  0 (6) That is, the magnets produce flux only on the d- axis. The second thing to note is that, under certain assumptions, the inductances in the d-q frame are independent of rotor position and have no mutual terms. That is: Ldq = T Lph T −1   Ld 0 0   Lq 0  = 0 0 0 L0 (7) The assertion that inductances in the d-q frame are constant is actually questionable, but it is close enough to being true and analyses that use it have proven to be close enough to being correct that it (the assertion) has held up to the test of time. In fact the deviations from independence on rotor position are small. Independence of axes (that is, absence of mutual inductances in the d-q frame) is correct because the two axes are physically orthogonal. We tend to ignore the third, or “zero” axis in this analysis. It doesn’t couple to anything else and has neither flux nor current anyway. Note that the direct- and quadrature- axis inductances are in principle straightforward to compute. They are direct axis the inductance of one of the armature phases (corrected for the fact of multiple phases) with the rotor aligned with the axis of the phase, and quadrature axis the inductance of one of the phases with the rotor aligned 90 electrical degrees away from the axis of that phase. Next, armature voltage is, ignoring resistance, given by: d d λph = T −1 λdq dt dt and that the transformed armature voltage must be: V ph = 8 (8) V dq = T V ph d = T (T −1 λdq ) dt d d (9) = λdq + (T T −1 )λdq dt dt The second term in this expresses “speed voltage”. A good deal of straightforward but tedious manipulation yields: T  d −1  T = dt 0 dθ dt 0 − dθ dt 0 0  0  0  0 The direct- and quadrature- axis voltage expressions are then: dλd Vd = − ωλq dt dλq Vq = + ωλd dt where dθ ω= dt Instantaneous power is given by: P = V a Ia + V b Ib + V c Ic (10) (11) (12) (13) Using the transformations given above, this can be shown to be: 3 3 P = Vd Id + Vq Iq + 3V0 I0 (14) 2 2 which, in turn, is: 3 3 dλd dλq dλ0 P = ω (λd Iq − λq Id ) + ( Id + Iq ) + 3 I0 (15) 2 2 dt dt dt Then, noting that ω = pΩ and that (15) describes electrical terminal power as the sum of shaft power and rate of change of stored energy, we may deduce that torque is given by: q T = p(λd Iq − λq Id ) (16) 2 Note that we have stated a generalization to a q- phase machine even though the derivation given here was carried out for the q = 3 case. Of course three phase machines are by far the most common case. Machines with higher numbers of phases behave in the same way (and this generalization is valid for all purposes to which we put it), but there are more rotor variables analogous to “zero axis”. Now, noting that, in general, Ld and Lq are not necessarily equal, then torque is given by: λ d = L d Id + λ f (17) λ q = L q Iq (18) q T = p (λf + (Ld − Lq ) Id ) Iq 2 9 (19) 3.3 Finding Torque Capability For high performance drives, we will generally assume that the power supply, generally an inverter, can supply currents in the correct spatial relationship to the rotor to produce torque in some reasonably effective fashion. We will show in this section how to determine, given a required torque (or if the torque is limited by either voltage or current which we will discuss anon), what the values of Id and Iq must be. Then the power supply, given some means of determining where the rotor is (the instantaneous value of θ), will use the inverse Park’s transformation to determine the instantaneous valued required for phase currents. This is the essence of what is known as “field oriented control”, or putting stator currents in the correct location in space to produce the required torque. Our objective in this section is, given the elementary parameters of the motor, find the capability of the motor to produce torque. There are three things to consider here: • Armature current is limited, generally by heating, • A second limit is the voltage capability of the supply, particularly at high speed, and • If the machine is operating within these two limits, we should consider the optimal placement of currents (that is, how to get the most torque per unit of current to minimize losses). Often the discussion of current placement is carried out using, as a tool to visualize what is going on, the Id , Iq plane. Operation in the steady state implies a single point on this plane. A simple illustration is shown in Figure 7. The thermally limited armature current capability is represented as a circle around the origin, since the magnitude of armature current is just the length of a vector from the origin in this space. In general, for permanent magnet machines with buried magnets, Ld < Lq , so the optimal operation of the machine will be with negative Id . We will show how to determine this optimum operation anon, but it will in general follow a curve in the Id , Iq plane as shown. Finally, an ellipse describes the voltage limit. To start, consider what would happen if the terminals of the machine were to be short-circuited so that V = 0. If the machine is operating at sufficiently high speed so that armature resistance is negligible, armature current would be simply: Id = − λf Ld Iq = 0 Now, loci of constant flux turn out to be ellipses around this point on the plane. Since terminal flux is proportional to voltage and inversely proportional to frequency, if the machine is operating with a given terminal voltage, the ability of that voltage to command current in the Id , Iq plane is an ellipse whose size “shrinks” as speed increases. To simplify the mathematics involved in this estimation, we normalize reactances, fluxes, currents and torques. First, let us define the base flux to be simply λb = λf and the base current Ib to be the armature capability. Then we define two per-unit reactances: xd = xq = 10 L d Ib λb L q Ib λb (20) (21) Voltage Limit Loci Optimal Torque Locus Speed = Base Speed > Base i q Armature Current Limit id Short Circuit Point Figure 7: Limits to Operation Next, define the base torque to be: q Tb = p λb Ib 2 and then, given per-unit currents id and iq , the per-unit torque is simply: te = (1 − (xq − xd ) id ) iq (22) It is fairly straightforward (but a bit tedious) to show that the locus of current-optimal operation (that is, the largest torque for a given current magnitude or the smallest current magnitude for a given torque) is along the curve: id iq v u u 2 ui t = − a +2 2 v u u u ia2 = −t − 2 2 1 4 (xq − xd ) !2 1 4 (xq − xd ) !2 1 − 2 (xq − xd ) 1 + 2 (xq − xd ) v u u t v u u t 1 4 (xq − xd ) !2 + ia2 2 (23) 1 4 (xq − xd ) !2 + ia2 2 (24) The “rating point” will be the point along this curve when ia = 1, or where this curve crosses the armature capability circle in the id , iq plane. It should be noted that this set of expressions only works for salient machines. For non-salient machines, of course, torque-optimal current is on the q-axis. In general, for machines with saliency, the “per-unit” torque will not be unity at the rating, so that the rated, or “Base Speed” torque is not the “Base” torque, but: Tr = Tb × te where te is calculated at the rating point (that is, ia = 1 and id and iq as per (23) and (24)). 11 (25) For sufficiently low speeds, the power electronic drive can command the optimal current to produce torque up to rated. However, for speeds higher than the “Base Speed”, this is no longer true. Define a per-unit terminal flux: V ψ= ωλb Operation at a given flux magnitude implies: ψ 2 = (1 + xd id )2 + (xq iq )2 which is an ellipse in the id , iq plane. The Base Speed is that speed at which this ellipse crosses the point where the optimal current curve crosses the armature capability. Operation at the highest attainable torque (for a given speed) generally implies d-axis currents that are higher than those on the optimal current locus. What is happening here is the (negative) d-axis current serves to reduce effective machine flux and hence voltage which is limiting q-axis current. Thus operation above the base speed is often referred to as “flux weakening”. The strategy for picking the correct trajectory for current in the id , iq plane depends on the value of the per-unit reactance xd . For values of xd > 1, it is possible to produce some torque at any speed. For values of xd < 1, there is a speed for which no point in the armature current capability is within the voltage limiting ellipse, so that useful torque has gone to zero. Generally, the maximum torque operating point is the intersection of the armature current limit and the voltage limiting ellipse: id = iq = v u u xd xd −t 2 2 2 xq − x2d xq − x d !2 + x2q − ψ 2 + 1 x2q − x2d q 1 − i2d (26) (27) It may be that there is no intersection between the armature capability and the voltage limiting ellipse. If this is the case and if xd < 1, torque capability at the given speed is zero. If, on the other hand, xd > 1, it may be that the intersection between the voltage limiting ellipse and the armature current limit is not the maximum torque point. To find out, we calculate the maximum torque point on the voltage limiting ellipse. This is done in the usual way by differentiating torque with respect to id while holding the relationship between id and iq to be on the ellipse. The algebra is a bit messy, and results in: id = − iq = 3xd (xq − xd ) − 4x2d (xq − xd ) 1 xq q x2d v ! u u 3x (xq − x ) − x2 2 (xq − xd ) (ψ 2 − 1) + xd d d t d − + 4x2d (xq − xd ) ψ 2 − (1 + xd id )2 2 (xq − xd ) x2d (28) (29) Ordinarily, it is probably easiest to compute (28) and (29) first, then test to see if the currents are outside the armature capability, and if they are, use (26) and (27). These expressions give us the capability to estimate the torque-speed curve for a machine. As an example, the machine described by the parameters cited in Table 1 is a (nominal) 3 HP, 4-pole, 3000 RPM machine. The rated operating point turns out to have the following attributes: 12 Table 1: Example Machine D- Axis Inductance Q- Axis Inductance Internal Flux Armature Current 2.53 mHy 6.38 mHy 58.1 mWb 30 A Table 2: Operating Characteristics of Example Machine Per-Unit D-Axis Current At Rating Point Per-Unit Q-Axis Current At Rating Point Per-Unit D-Axis Reactance Per-Unit Q-Axis Reactance Rated Torque (Nm) Terminal Voltage at Base Point (V) id iq xd xq Tr -.5924 .8056 1.306 3.294 9.17 97 The loci of operation in the Id , Iq plane is shown in Figure 8. The armature current limit is shown only in the second and third quadrants, so shows up as a semicircle. The two ellipses correspond with the rated point (the larger ellipse) and with a speed that is three times rated (9000 RPM). The torque-optimal current locus can be seen running from the origin to the rating point, and the higher speed operating locus follows the armature current limit. Figure 9 shows the torque/speed and power/speed curves. Note that this sort of machine only approximates “constant power” operation at speeds above the “base” or rating point speed. 4 Parameter Estimation We are now at the point of estimating the major parameters of the motors. Because we have a number of different motor geometries to consider, and because they share parameters in not too orderly a fashion, this section will have a number of sub-parts. First, we calculate flux linkage, then reactance. 4.1 Flux Linkage Given a machine which may be considered to be uniform in the axial direction, flux linked by a single, full-pitched coil which spans an angle from zero to π/p, is: φ= Z 0 π p Br Rldφ where Br is the radial flux through the coil. And, if Br is sinusoidally distributed this will have a peak value of 2RlBr φp = p 13 PM Brushless Machine Current Loci 60 40 Q-Axis Current (A) 20 0 -20 -40 -60 -80 -60 -40 -20 D-Axis Current (A) 0 20 Figure 8: Operating Current Loci of Example Machine PM Brushless Machine 10 Torque, N-m 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 Speed, RPM 6000 7000 8000 9000 Power, Watts 4000 3000 2000 1000 0 0 Figure 9: Torque- and Power-Speed Capability 14 Now, if the actual winding has Na turns, and using the pitch and breadth factors derived in Appendix 1, the total flux linked is simply: λf = 2RlB1 Na kw p (30) where kw = kp kb α kp = sin 2 sin m γ2 kb = m sin γ2 The angle α is the pitch angle, α = 2πp Np Ns where Np is the coil span (in slots) and Ns is the total number of slots in the stator. The angle γ is the slot electrical angle: 2πp γ= Ns Now, what remains to be found is the space fundamental magnetic flux density B1 . In Appendix 2 it is shown that, for magnets in a surface-mount geometry, the magnetic field at the surface of the magnetic gap is: B1 = µ0 M1 kg (31) where the space-fundamental magnetization is: M1 = Br 4 pθm sin µ0 π 2 where Br is remanent flux density of the permanent magnets and θm is the magnet angle. and where the factor that describes the geometry of the magnetic gap depends on the case. For magnets inside and p 6= 1, kg = Rsp−1 Rs2p − Ri2p     p  p+1 p R2 − R1p+1 + Ri2p R11−p − R21−p p+1 p−1  For magnets inside and p = 1, kg = 1 Rs2 − Ri2   1 2  R22 − R12 + Ri2 log R2 R1  For the case of magnets outside and p 6= 1: kg = Rip−1 Rs2p − Ri2p     p  p+1 p R2 − R1p+1 + Rs2p R11−p − R21−p p+1 p−1 and for magnets outside and p = 1, 15  1 kg = 2 Rs − Ri2   1 2 R22 − R12  + Rs2 log R2 R1  Where Rs and Ri are the outer and inner magnetic boundaries, respectively, and R2 and R1 are the outer and inner boundaries of the magnets. Note that for the case of a small gap, in which both the physical gap g and the magnet thickness hm are both much less than rotor radius, it is straightforward to show that all of the above expressions approach what one would calculate using a simple, one-dimensional model for the permanent magnet: kg → hm g + hm This is the whole story for the winding-in-slot, narrow air-gap, surface magnet machine. For airgap armature windings, it is necessary to take into account the radial dependence of the magnetic field. 4.2 Air-Gap Armature Windings With no windings in slots, the conventional definition of winding factor becomes difficult to apply. If, however, each of the phase belts of the winding occupies an angular extent θw , then the equivalent to (31) is: sin p θ2w kw = p θ2w Next, assume that the “density” of conductors within each of the phase belts of the armature winding is uniform, so that the density of turns as a function of radius is: N (r) = 2Na r 2 − Rwi 2 Rwo This just expresses the fact that there is more azimuthal room at larger radii, so with uniform density the number of turns as a function of radius is linearly dependent on radius. Here, Rwo and Rwi are the outer and inner radii, respectively, of the winding. Now it is possible to compute the flux linked due to a magnetic field distribution: λf = Z Rwo Rwi 2lNa kw r 2r 2 µ0 Hr (r)dr 2 p Rwo − Rwi (32) Note the form of the magnetic field as a function of radius expressed in 80 and 81 of the second appendix. For the “winding outside” case it is:  Hr = A r p−1 + Rs2p r −p−1  Then a winding with all its turns concentrated at the outer radius r = Rwo would link flux: λc =   2lRwo kw 2lRwo kw p−1 −p−1 µ0 Hr (Rwo ) = µ0 A Rwo + Rs2p Rwo p p 16 Carrying out (32), it is possible, then, to express the flux linked by a thick winding to the flux that would have been linked by a radially concentrated winding at its outer surface by: kt = λf λc where, for the winding outside, p 6= 2 case: 1 − x2+p ξ 2p 1 − x2−p + 2+p 2−p
2 kt = 2 (1 − x ) (1 + ξ 2p ) ! (33) where we have used the definitions ξ = Rwo /Rs and x = Rwi /Rwo . In the case of winding outside, p = 2, !
1 − x4 ξ 4 2 − log x (34) kt = (1 − x2 ) (1 + ξ 2p ) 4 In a very similar way, we can define a winding factor for a thick winding in which the reference radius is at the inner surface. (Note: this is done because the inner surface of the inside winding is likely to be coincident with the inner ferromagnetic surface, as the outer surface of the outer winding ls likely to be coincident with the outer ferromagnetic surface). For p 6= 2: 2x−p kt = (1 − x2 ) (1 + η 2p ) 1 − x2+p 1 − x2−p + (ηx)2p 2+p 2−p ! (35) and for p = 2: 2x−2 kt = (1 − x2 ) (1 + η 2p ) 1 − x4 − (ηx)4 log x 4 ! (36) where η = Ri /Rwi So, in summary, the flux linked by an air-gap armature is given by: λf = 2RlB1 Na kw kt p (37) where B1 is the flux density at the outer radius of the physical winding (for outside winding machines) or at the inner radius of the physical winding (for inside winding machines). Note that the additional factor kt is a bit more than one (it approaches unity for thin windings), so that, for small pole numbers and windings that are not too thick, it is almost correct and in any case “conservative” to take it to be one. 4.3 Interior Magnet Motors: For the flux concentrating machine, it is possible to estimate air-gap flux density using a simple reluctance model. The air- gap permeance of one pole piece is: ℘ag = µ0 l where θp is the angular width of the pole piece. 17 Rθp g And the incremental permeance of a magnet is: ℘m = µ0 hm l wm The magnet sees a unit permeance consisting of its own permeance in series with one half of each of two pole pieces (in series) : ℘u = ℘ag Rθp wm = ℘m 4g hm Magnetic flux density in the magnet is: Bm = B0 ℘u 1 + ℘u And then flux density in the air gap is: Bg = 2hm 2hm wm Bm = B0 Rθp 4ghm + Rθp wm The space fundamental of that can be written as: B1 = 4 pθp wm sin B0 γm π 2 2g where we have introduced the shorthand: γm = 1 1+ w m θp R g 4 hm The flux linkage is then computed as before: λf = 4.4 2RlB1 Na kw p (38) Winding Inductances The next important set of parameters to compute are the d- and q- axis inductances of the machine. We will consider three separate cases, the winding-in-slot, surface magnet case, which is magnetically “round”, or non-salient, the air-gap winding case, and the flux concentrating case which is salient, or has different direct- and quadrature- axis inductances. 4.4.1 Surface Magnets, Windings in Slots In this configuration there is no saliency, so that Ld = Lq . There are two principal parts to inductance, the air-gap inductance and slot leakage inductance. Other components, including end turn leakage, may be important in some configurations, and they would be computed in the same way as for an induction machine. If magnet thickness is not too great, we may make the narrow air-gap assumption, in which case the fundamental part of air-gap inductance is: Ld1 = 2 lR q 4 µ0 Na2 kw s 2 π p2 (g + hm ) 18 (39) Here, g is the magnetic gap, including the physical rotational gap and any magnet retaining means that might be used. hm is the magnet thickness. Since the magnet thickness is included in the air-gap, the air-gap permeance may not be very large, so that slot leakage inductance may be important. To estimate this, assume that the slot shape is rectangular, characterized by the following dimensions: hs height of the main portion of the slot ws width of the top of the main portion of the slot hd height of the slot depression wd slot depression opening Of course not all slots are rectangular: in fact in most machines the slots are trapezoidal in shape to maintain teeth cross-sections that are radially uniform. However, only a very small error (a few percent) is incurred in calculating slot permeance if the slot is assumed to be rectangular and the top width is used (that is the width closest to the air-gap). Then the slot permeance is, per unit length:   1 hs hd + P = µ0 3 ws wd Assume for the rest of this discussion a standard winding, with m slots in each phase belt (this assumes, then, that the total number of slots is Ns = 2pqm), and each slot holds two halfcoils. (A half-coil is one side of a coil which, of course, is wound in two slots). If each coil has Nc turns (meaning Na = 2pmNc ) , then the contribution to phase self-inductance of one slot is, if both half-coils are from the same phase, 4lPNc2 . If the half-coils are from different phases, then the contribution to self inductance is lPNc2 and the magnitude of the contribution to mutual inductance is lPNc2 . (Some caution is required here. For three phase windings the mutual inductance is negative, so are the senses of the currents in the two other phases, so the impact of “mutual leakage” is to increase the reactance. This will be true for other numbers of phases as well, even if the algebraic sign of the mutual leakage inductance is positive, in which case so will be the sense of the other- phase current.) We will make two other assumptions here. The standard one is that the winding “coil throw”, s or span between sides of a coil, is N 2p − Nsp . Nsp is the coil “short pitch”. The other is that each phase belt will overlap with, at most two other phases: the ones on either side in sequence. This last assumption is immediately true for three- phase windings (because there are only two other phases. It is also likely to be true for any reasonable number of phases. Noting that each phase occupies 2p(m − Nsp ) slots with both coil halves in the same slot and 2pNsp slots in which one coil half shares a slot with each of two different phases, we can write down the two components of slot leakage inductance, self- and mutual: h Las = 2pl (m − Nsp ) (2Nc )2 + 2Nsp Nc2 Lam = −2plNsp Nc2 i For a three- phase machine, then, the total slot leakage inductance is: La = Las − Lam = 2plPNc2 (4m − Nsp ) For a uniform, symmetric winding with an odd number of phases, it is possible to show that the effective slot leakage inductance is: La = Las − 2Lam cos 19 2π q Total synchronous inductance is the sum of air-gap and leakage components: so far this is: Ld = Ld1 + La 4.4.2 Air-Gap Armature Windings It is shown in Appendix 1 that the inductance of a single-phase of an air-gap winding is: X La = Lnp n where the harmonic components are: Lk = 8 π + + 2 N2 µ0 lkwn a k(1 − x2 )2   1 − x2−k γ 2k  1 − x2+k (4 − k2 ) (1 − γ 2k ) ξ 2k 1 − xk+2 2 + (2 + k)2 (1 − γ 2k )   1 − γ −2k x2+k   ξ −2k 1 − x2−k  2 (2 − k)2 (γ −2k − 1) 1 − x2−k (4 − k2 ) (γ −2k − 1)  where we have used the following shorthand coefficients:  k 1 − x2  − 4 − k2 2 Rwi Rwo Ri Rs Rwo Rs x = γ = ξ = This fits into the conventional inductance framework: Ln = 2 4 µ0 Na2 Rs Lkwn ka π N 2 p2 g if we assign the “thick armature” coefficient to be: ka =   1 − x2−k γ 2k 1 − x2+k 2gk 1  Rwo (1 − x2 )2 (4 − k2 ) (1 − γ 2k ) + +  ξ 2k 1 − xk+2 2 + (2 + k)2 (1 − γ 2k )  1 − γ −2k x2+k  2 (2 − k)2 (γ −2k − 1) 1 − x2−k (4 − k2 ) (γ −2k − 1) 20  ξ −2k 1 − x2−k   − k 1− 4 − k2 2 x2   and k = np and g = Rs − Ri is the conventionally defined “air gap”. If the aspect ratio Ri /Rs is not too far from unity, neither is ka . In the case of p = 2, the fundamental component of ka is: "
2 # 2gk 1 1 − x4 2γ 4 + x4 1 − γ 4 γ4 ξ 4 1 − x4 2 ka = − log x + (log x) + 2 8 4 (1 − γ 4 ) ξ 4 (1 − γ 4 ) 16 (1 − γ 4 ) Rwo (1 − x2 )
For a q-phase winding, a good approximation to the inductance is given by just the first space harmonic term, or: 2 q 4 µ0 Na2 Rs Lkwn ka Ld = 2π n 2 p2 g 4.4.3 Internal Magnet Motor The permanent magnets will have an effect on reactance because the magnets are in the main flux path of the armature. Further, they affect direct and quadrature reactances differently, so that the machine will be salient. Actually, the effect on the direct axis will likely be greater, so that this type of machine will exhibit “negative” saliency: the quadrature axis reactance will be larger than the direct- axis reactance. A full- pitch coil aligned with the direct axis of the machine would produce flux density: Br = µ 0 Na I  2g 1 + Rθp wm 4g hm  Note that only the pole area is carrying useful flux, so that the space fundamental of radial flux density is: B1 = µ0 Na I 4 sin pθ2m 2g π 1 + wm Rθp hm 4g Then, since the flux linked by the winding is: λa = 2RlNa kw B1 p The d- axis inductance, including mutual phase coupling, is (for a q- phase machine): Ld = 2 q 4 µ0 Na2 Rlkw pθp γm sin 2 2π p g 2 The quadrature axis is quite different. On that axis, the armature does not tend to push flux through the magnets, so they have only a minor effect. What effect they do have is due to the fact that the magnets produce a space in the active air- gap. Thus, while a full- pitch coil aligned with the quadrature axis will produce an air- gap flux density: Br = µ0 N I g the space fundamental of that will be: 21 µ0 N I 4 pθt 1 − sin B1 = 2 g π   where θt is the angular width taken out of the pole by the magnets. So that the expression for quadrature axis inductance is: 2 q 4 µ0 Na2 Rlkw Lq = 2π p2 g 5 pθt 1 − sin 2   Current Rating and Resistance The last part of machine rating is its current capability. This is heavily influenced by cooling methods, for the principal limit on current is the heating produced by resistive dissipation. Generally, it is possible to do first-order design estimates by assuming a current density that can be handled by a particular cooling scheme. Then, in an air-gap winding:  2 2 − Rwi Na Ia = Rwo θ we Ja 2 and note that, usually, the armature fills the azimuthal space in the machine: 2qθwe = 2π For a winding in slots, nearly the same thing is true: if the rectangular slot model holds true: 2qNa Ia = Ns hs ws Js where we are using Js to note slot current density. Now, suppose we can characterize the total slot area by a “space factor” λs which is the ratio between total slot area and the annulus occupied by the slots: for the rectangular slot model: λs = Ns hs ws
2 − R2 π Rwo wi where Rwi = R+hd and Rwo = Rwi +hs in a normal, stator outside winding. In this case, Ja = Js λs and the two types of machines can be evaluated in the same way. It would seem apparent that one would want to make λs as large as possible, to permit high currents. The limit on this is that the magnetic teeth between the conductors must be able to carry the air-gap flux, and making them too narrow would cause them to saturate. The peak of the time fundamental magnetic field in the teeth is, for example, Bt = B1 2πR Ns wt where wt is the width of a stator tooth: wt = 2π(R + hd ) − ws Ns so that Bt ≈ B1 1 − λs 22 5.1 Resistance Winding resistance may be estimated as the length of the stator conductor divided by its area and its conductivity. The length of the stator conductor is: lc = 2lNa fe where the “end winding factor” fe is used to take into account the extra length of the end turns (which is usually not negligible). The area of each turn of wire is, for an air-gap winding : Aw = 2 − R2 θwe Rwo wi λw 2 Na where λw , the “packing factor” relates the area of conductor to the total area of the winding. The resistance is then just: 4lNa2
Ra = 2 − R2 λ σ θwe Rwo w wi and, of course, σ is the conductivity of the conductor. For windings in slots the expression is almost the same, simply substituting the total slot area: Ra = 2qlNa2 Ns hs ws λw σ The end turn allowance depends strongly on how the machine is made. One way of estimating what it might be is to assume that the end turns follow a roughly circular path from one side of the machine to the other. The radius of this circle would be, very roughly, Rw /p, where Rw is the average radius of the winding: Rw ≈ (Rwo + Rwi )/2 Then the end-turn allowance would be: fe = 1 + 6 πRw pl Appendix 1: Air-Gap Winding Inductance In this appendix we use a simple two-dimensional model to estimate the magnetic fields and then inductances of an air-gap winding. The principal limiting assumption here is that the winding is uniform in the ẑ direction, which means it is long in comparison with its radii. This is generally not true, nevertheless the answers we will get are not too far from being correct. The style of analysis used here can be carried into a three-dimensional, or quasi-three dimensional domain to get much more precise answers, at the expense of a very substantial increase in complexity. The coordinate system to be used is shown in Figure 10. To maintain generality we have four radii: Ri and Rs are ferromagnetic boundaries, and would of course correspond with the machine shaft and the stator core. The winding itself is carried between radii R1 and R2 , which correspond with radii Rwi and Rwo in the body of the text. It is assumed that the armature is carrying a current in the z- direction, and that this current is uniform in the radial dimension of the armature. If a single phase of the armature is carrying current, that current will be: Jz0 = θwe 2 N a Ia
R22 − R12 23 Winding Outer Magnetic Boundary 11111111111111111111111111111 00000000000000000000000000000 00000000000000000000000000000 11111111111111111111111111111 R 00000000000000000000000000000 11111111111111111111111111111 s 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 R 00000000000000000000000000000 11111111111111111111111111111 i 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 R 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 1 00000000000000000000000000000 11111111111111111111111111111 R Inner Magnetic 00000000000000000000000000000 11111111111111111111111111111 2 Boundary 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 Figure 10: Coordinate System for Inductance Calculation over the annular wedge occupied by the phase. The resulting distribution can be fourier analyzed, and the n-th harmonic component of this will be (assuming the coordinate system has been chosen appropriately): 4 θwe 4 N a Ia Jzn = Jz0 sin n = kwn nπ 2 π R22 − R12 where the n-th harmonic winding factor is: kwn = sin n θwe 2 n θwe 2 and note that θwe is the electrical winding angle: θwe = pθw Now, it is easiest to approach this problem using a vector potential. Since the divergence of flux density is zero, it is possible to let the magnetic flux density be represented by the curl of a vector potential: B =∇×A Taking the curl of that:   ∇ × ∇ × A = µ0 J = ∇∇ · A − ∇2 A and using the Coulomb gage ∇·A =0 we have a reasonable tractable partial differential equation in the vector potential: ∇2 A = −µ0 J Now, since in our assumption there is only a z- directed component of J, we can use that one component, and in circular cylindrical coordinates that is: 1 ∂ ∂Az 1 ∂2 r + 2 2 Az = −µ0 Jz r ∂r ∂r r ∂θ 24 For this problem, all variables will be varying sinusoidally with angle, so we will assume that angular dependence ejkθ . Thus: 1 ∂ ∂Az k2 r − 2 Az = −µ0 Jz r ∂r ∂r r (40) This is a three-region problem. Note the regions as: i w o Ri < r < R 1 R1 < r < R 2 R2 < r < R s For i and o, the current density is zero and an appropriate solution to (40) is: Az = A+ r k + A− r −k In the region of the winding, w, a particular solution must be used in addition to the homogeneous solution, and Az = A+ r k + A− r −k + Ap where, for k 6= 2, Ap = − or, if k = 2, µ0 Jz r 2 4 − k2 µ0 Jz r 2 1 Ap = − log r − 4 4   And, of course, the two pertinent components of the magnetic flux density are: 1 ∂Az r ∂θ ∂Az = − ∂r Br = Bθ Next, it is necessary to match boundary conditions. There are six free variables and correspondingly there must be six of these boundary conditions. They are the following: • At the inner and outer magnetic boundaries, r = Ri and r = Rs , the azimuthal magnetic field must vanish. • At the inner and outer radii of the winding itself, r = R1 and r = R2 , both radial and azimuthal magnetic field must be continuous. These conditions may be summarized by: kAi+ Rik−1 − kAi− Ri−k−1 = 0 kAo+ Rsk−1 − kAo− Rs−k−1 = 0 µ0 Jz R2 k−1 −k−1 Aw + Aw − = Ao+ R2k−1 + Ao− R2−k−1 + R2 − R2 4 − k2 25 2µ0 Jz R2 4 − k2 µ0 Jz R1 k−1 −k−1 Aw + Aw − + R1 − R1 4 − k2 2µ0 Jz R1 k−1 −k−1 −kAw + kAw + + R1 − R1 4 − k2 k−1 −k−1 −kAw + kAw + + R2 − R2 = −kAo+ R2k−1 + kAo− R2−k−1 = Ai+ R1k−1 + Ai− R1−k−1 i = −kA+ R1k−1 + kAi− R1−k−1 Note that we are carrying this out here only for the case of k 6= 2. The k = 2 case may be obtained by substituting its particular solution in at the beginning or by using L’Hopital’s rule on the final solution. This set may be solved (it is a bit tedious but quite straightforward) to yield, for the winding region: Az = µ0 Jz 2k  + Rs2k R22−k − Ri2k R12−k R22+k − R12+k

+ (2 − k) Rs2k − Ri2k (2 + k) Rs2k − Ri2k " R22−k − R12−k  (2 − k) Ri−2k − Rs−2k + ! rk  Rs−2k R22+k − Ri−2k R12+k   (2 + k) Ri−2k − Rs−2k   2k 2  r −k − r 4 − k2 Now, the inductance linked by any single, full-pitched loop of wire located with one side at azimuthal position θ and radius r is: λi = 2lAz (r, θ) To extend this to the whole winding, we integrate over the area of the winding the incremental flux linked by each element times the turns density. This is, for the n-th harmonic of flux linked: 4lkwn Na λn = 2 R2 − R12 Z R2 R1 Az (r)rdr Making the appropriate substitutions for current into the expression for vector potential, this becomes: λn = 2 N 2I 8 µ0 lkwn a a
2 π k R2 − R12 2  " Rs2k R22−k − Ri2k R12−k R22+k − R12+k

+ (2 + k) R22k − Ri2k (2 − k) Rs2k − Ri2k ! R2k+2 − R1k+2 k+2   R−2k R22+k − Ri−2k R12+k  R22−k − R12−k 2k R24 − R14     + s − + 2−k 4 − k2 4 (2 − k) R−2k − Rs−2k (2 + k) R−2k − Rs−2k 7 R22−k − R12−k i i Appendix 2: Permanent Magnet Field Analysis This section is a field analysis of the kind of radially magnetized, permanent magnet structures commonly used in electric machinery. It is a fairly general analysis, which will be suitable for use with either surface or in-slot windings, and for the magnet inside or the magnet outside case. This is a two-dimensional layout suitable for situations in which field variation along the length of the structure is negligible. 26 8 Layout The assumed geometry is shown in Figure 11. Assumed iron (highly permeable) boundaries are at radii Ri and Rs . The permanent magnets, assumed to be polarized radially and alternately (i.e. North-South ...), are located between radii R1 and R2 . We assume there are p pole pairs (2p magnets) and that each magnet subsumes an electrical angle of θme . The electrical angle is just p times the physical angle, so that if the magnet angle were θme = π, the magnets would be touching. Magnets Outer Magnetic Boundary 11111111111111111111111111111 00000000000000000000000000000 00000000000000000000000000000 11111111111111111111111111111 R 00000000000000000000000000000 11111111111111111111111111111 s 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 R 00000000000000000000000000000 11111111111111111111111111111 i 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 R 00000000000000000000000000000 11111111111111111111111111111 000000000000 111111111111 1 00000000000000000000000000000 11111111111111111111111111111 R Inner Magnetic 00000000000000000000000000000 11111111111111111111111111111 2 Boundary 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 Figure 11: Axial View of Magnetic Field Problem If the magnets are arranged so that the radially polarized magnets are located around the azimuthal origin (θ = 0), the space fundamental of magnetization is: M = ir M0 cos pθ (41) where the fundamental magnitude is: M0 = 4 θme Brem sin π 2 µ0 (42) and Brem is the remanent magnetization of the permanent magnet. Since there is no current anywhere in this problem, it is convenient to treat magnetic field as the gradient of a scalar potential: H = −∇ψ (43) ∇2 ψ = −∇ · H (44) The divergence of this is: 27 Since magnetic flux density is divergence-free, ∇·B =0 (45) ∇ · H = −∇ · M (46) we have: or: 1 ∇2 ψ = ∇ · M = M0 cos pθ (47) r Now, if we let the magnetic scalar potential be the sum of particular and homogeneous parts: ψ = ψp + ψh (48) where ∇2 ψh = 0, then: 1 ∇2 ψp = M0 cos pθ (49) r We can find a suitable solution to the particular part of this in the region of magnetization by trying: ψp = Cr γ cos pθ (50) Carrying out the Laplacian on this:   1 ∇2 ψp = Cr γ−2 γ 2 − p2 cos pθ = M0 cos pθ r (51) which works if γ = 1, in which case: ψp = M0 r cos pθ 1 − p2 (52) Of course this solution holds only for the region of the magnets: R1 < r < R2 , and is zero for the regions outside of the magnets. A suitable homogeneous solution satisfies Laplace’s equation, ∇2 ψh = 0, and is in general of the form: ψh = Ar p cos pθ + Br −p cos pθ (53) Then we may write a trial total solution for the flux density as: A1 r p + B1 r −p cos pθ   M0 r ψ= A2 r p + B2 r −p + cos pθ 1 − p2
ψ = A3 r p + B3 r −p cos pθ
Ri < r < R 1 ψ = R1 < r < R 2 R2 < r < R s (54) (55) (56) The boundary conditions at the inner and outer (assumed infinitely permeable) boundaries at r = Ri and r = Rs require that the azimuthal field vanish, or ∂ψ ∂θ = 0, leading to: B1 = −Ri2p A1 (57) −Rs2p A3 (58) B3 = 28 At the magnet inner and outer radii, Hθ and Br must be continuous. These are: 1 ∂ψ r∂θ  ∂ψ = µ0 − + Mr ∂r Hθ = − (59) Br (60) These become, at r = R1 : M0 1 − p2     M0 + M0 −pA1 R1p−1 + Ri2p R1−p−1 = −p A2 R1p−1 − B2 R1−p−1 − 1 − p2  −pA1 R1p−1 − Ri2p R1−p−1    = −p A2 R1p−1 + B2 R1−p−1 − p (61) (62) and at r = R2 : M0 1 − p2     M0 −pA3 R2p−1 + Rs2p R2−p−1 = −p A2 R2p−1 − B2 R2−p−1 − + M0 1 − p2  −pA3 R2p−1 − Rs2p R2−p−1    = −p A2 R2p−1 + B2 R2−p−1 − p (63) (64) Some small-time manipulation of these yields: M0 1 − p2   M0 A1 R1p + Ri2p R1−p = A2 R1p − B2 R1−p + pR1 1 − p2   M0 A3 R2p − Rs2p R2−p = A2 R2p + B2 R2−p + R2 1 − p2   M0 A3 R2p + Rs2p R2−p = A2 R2p − B2 R2−p + pR2 1 − p2  A1 R1p − Ri2p R1−p  = A2 R1p + B2 R1−p + R1 (65) (66) (67) (68) Taking sums and differences of the first and second and then third and fourth of these we obtain: 1+p 1 − p2 p−1 = −2B2 R1−p + R1 M0 1 − p2 1+p = 2A2 R2p + R2 M0 1 − p2 p−1 = −2B2 R2−p + R2 M0 1 − p2 2A1 R1p = 2A2 R1p + R1 M0 2A1 Ri2p R1−p 2A3 R2p 2A3 Rs2p R2−p (69) (70) (71) (72) and then multiplying through by appropriate factors (R2p and R1p ) and then taking sums and differences of these, M0 p + 1 2 1 − p2  M p−1 0 = R1 R2−p − R2 R1−p 2 1 − p2 (A1 − A3 ) R1p R2p = (R1 R2p − R2 R1p )   A1 Ri2p − A3 Rs2p R1−p R2−p 29 (73) (74) Dividing through by the appropriate groups: A1 − A3 = A1 Ri2p − A3 Rs2p = R1 R2p − R2 R1p M0 1 + p R1p R2p 2 1 − p2 (75) R1 R2−p − R2 R1−p M0 p − 1 2 1 − p2 R1−p R2−p (76) and then, by multiplying the top equation by Rs2p and subtracting:  A1 Rs2p − Ri2p  = R1 R2p − R2 R1p M0 1 + p R1p R2p 2 1 − p2 ! Rs2p − R1 R2−p − R2 R1−p M0 p − 1 2 1 − p2 R1−p R2−p (77) This is readily solved for the field coefficients A1 and A3 : M0    p + 1  1−p p − 1  1+p 1−p 1+p 2p R − R R + R − R s 1 1 2 2 p2 − 1 p2 − 1 (78) M0    1  1−p 1  1+p R1 − R21−p Ri2p − R2 − R11+p 1−p 1+p (79)  A1 = −  2p 2 Rs − Ri2p  A3 = −  2p 2 Rs − Ri2p   Now, noting that the scalar potential is, in region 1 (radii less than the magnet), ψ = A1 (r p − Ri2p r −p ) cos pθ r < R1 ψ = A3 (r p − Rs2p r −p ) cos pθ r > R2 and noting that p(p + 1)/(p2 − 1) = p/(p − 1) and p(p − 1)/(p2 − 1) = p/(p + 1), magnetic field is: Hr = Hr = r < R1 M0 2  Rs2p − (80) Ri2p r > R2 M0 2  Rs2p − Ri2p     p p R11−p − R21−p Rs2p + R21+p − R11+p p−1 p+1    p p R11−p − R21−p Ri2p + R21+p − R11+p p−1 p+1       r p−1 + Ri2p r −p−1 cos pθ (81)    r p−1 + Rs2p r −p−1 cos pθ The case of p = 1 appears to be a bit troublesome here, but is easily handled by noting that:  p  1−p R2 R1 − R21−p = log p→1 p − 1 R1 lim Now: there are a number of special cases to consider. For the iron-free case, Ri → 0 and R2 → ∞, this becomes, simply, for r < R1 : Hr =  M0 p  1−p R1 − R21−p r p−1 cos pθ 2 p−1 30 (82) Note that for the case of p = 1, the limit of this is Hr = M0 R2 log cos θ 2 R1 and for r > R2 : Hr =  M0 p  p+1 R2 − R1p+1 r −(p+1) cos pθ 2 p+1 For the case of a machine with iron boundaries and windings in slots, we are interested in the fields at the boundaries. In such a case, usually, either Ri = R1 or Rs = R2 . The fields are: at the outer boundary: r = Rs : Hr = M0 Rsp−1 Rs2p − Ri2p     p  p+1 p R2 − R1p+1 + Ri2p R11−p − R21−p cos pθ p+1 p−1  or at the inner boundary: r = Ri : Hr = M0 Rip−1 Rs2p − Ri2p     p p  p+1 R2 − R1p+1 + Rs2p R11−p − R21−p cos pθ p+1 p−1  31