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Investigation of Force Generation in a Permanent Magnet Synchronous Machine W. Zhu, Student Member, IEEE, S. Pekarek, Member, IEEE, B. Fahimi, Senior Member, IEEE and B. Deken, Student Member, IEEE Traditional analysis of permanent magnet synchronous machines has focused upon establishing a relationship between the quadrature (q) and direct (d) axis stator current (or voltage) and the electromagnetic force created to establish rotation (torque). In this paper, an alternative analysis of electromagnetic force production is considered. Specifically, the influences of q- and d-axis stator current on both the radial and tangential components of the airgap flux densities are first evaluated. Using a Maxwell Stress Tensor approach, the fields are then used to evaluate both the radial and tangential component of force density created in the airgap of the machine. From this perspective several interesting observations are made. First, it is shown that the d-axis current has zero influence on the average tangential force (torque), as predicted using traditional analysis, but it has a significant influence on the average radial component of force. Second, it is shown that the q-axis current contributes to both the average radial and average tangential components of force. Interestingly, it is also shown that under standard operating conditions, the average radial force far exceeds that of the average tangential component of force. Therefore, one can conclude that the magnetic fields established create a significant component of force in a direction that cannot produce torque. Abstract Index Terms Permanent Magnet Synchronous Motor, Maxwell Stress Tensor, Force Density, Torque Generation. I INTRODUCTION Over the past half-century, advances in the design of electric machinery have been relatively modest when compared to quantum leaps made in the design of controlled switches (semiconductors). Although advances in electric machinery have been relatively modest, the control of machines has changed dramatically due to increased computing power and the power handling capability of high power semiconductor devices. For each class of electric machinery, controls now exist to achieve a variety of desired responses (high-bandwidth current/torque transduction, minimal torque ripple for a desired average torque, maximum torque/amp, maximum efficiency, etc). New materials, manufacturing techniques, computer technologies, and semiconductor devices all promise to ensure further progress is made along existing drive system design guidelines. The motivation for this research has started with a question being asked by several researchers [1,2]–are there paths for new machine design/excitation strategies that could lead to significant changes in how electrical/mechanical energy conversion is achieved? To begin to address this question, an analysis of an existing surface-mount PM machine is performed. However, in contrast to a traditional ‘macroscopic’ approach in which the electromagnetic torque is the focus of the energy conversion process, in this research an alternative view of force generation in the PM machine is considered. Specifically, the radial and tangential components of magnetic flux density in the airgap are evaluated and used to calculate the electromagnetic forces (radial and tangential components). This so-called microscopic view of the force generation is used to view force production from a different perspective and helps to further understand how magnetic fields lead to force production. Based upon this analysis, several observations have been made. First, it is shown that the d-axis current has zero influence on the average tangential force (torque), as predicted using traditional analysis (in the absence of saturation). However, it has a significant influence on the average radial component of force. Specifically, it is shown that the relationship between daxis current and average radial force is a quadratic function. Second, it is shown that the relationship between q-axis current and average tangential force is linear (as predicted using traditional analysis), while the relationship between the q-axis current and average radial force is also a quadratic function – although a different function than for the d-axis. Interestingly, it is also shown that under standard operating conditions, the radial force far exceeds that of the tangential component of force. Therefore, one can conclude that the magnetic fields established create a significant component of force in a direction that does not produce motion. In our opinion, the main contributions of this research are two-fold. First, the analysis provides the community with a more complete understanding of how typical excitation adjusts the fields in the airgap and influences the overall force profile. Second, the results point to a general direction for further research in the area. Specifically, the fact that the majority of force produced does not lead to motion raises a question as to whether alternative geometries/excitation schemes can be developed to yield a more productive force profile for electric machines. Prior to proceeding with the analysis, it is important to highlight the works of researchers whose contributions are closely related. Specifically, several researchers have used analytical techniques to arrive at the magnetic field distribution in the airgap of electric machines [3-12]. Among the early efforts was Hague [3], who provided analytical solutions for the magnetic fields in the airgap of a machine due to current imbedded in iron or located inside the airgap of a machine. 1 φr m φs m More recently, analytical solutions for the fields in the airgap of PM machines have been provided in [4-11]. Although these works are related, differences exist in the methods used to account for stator slots, the co-ordinate system applied (i.e. polar/rectangular), and the solution method. Although very useful for understanding field behavior and in design, none have explored the link between q- and d-axis current excitation and the tangential component of the field created by the stator windings. Most recently, [12] has used analytical techniques to explore the force profile under q- and d- excitation. The primary goal of their effort was to consider the effect of using d-axis field weakening on force ripple. Two key assumptions in their analysis was 1) that the stator windings produce a pure sinusoidally distributed field in the airgrap, and 2) that this field contains a negligible tangential component - i.e. only a radial flux density exists from stator to rotor. Herein, to be more general, the fields created by the stator windings are not assumed to be a pure sinusoid – and in fact through FEA are confirmed to contain harmonics. Moreover, it is shown that in general one cannot assume that the stator windings produce a unidirectional field. In fact the vector nature of the fields created by the stator windings is critical to the production of both the tangential and radial components of force. C1 A2 A1' θrm S B1 C2' N as-axis N B2 C1' S A2' A1 C2 between the q- and as-axis ( rm ). Herein, so-called electrical angles s , r , and r are defined by multiplying the mechanical angles by the number of pole pairs. The relationship (1) s = r + r exists between electrical angles: The following assumptions have been made for the analysis provided: • The stator teeth and permanent magnets are rigid; no deformation due to radial and tangential force is experienced by these components. • The stator windings are concentrated and are wound at a full-pitch. • The permanent magnets are parallel-magnetized. • The permanent magnets are not demagnetized by the flux introduced by the phase currents. • The flux density in the z-axis is assumed zero (no end effects). • Hysteresis and Eddy currents are neglected. III MACROSCOPIC VIEW OF FORCE PRODUCTION q-axis B1' purpose of analysis that is used in Section V, the angle sm is defined as the position on the stator relative to the midpoint of the phase-a stator slot shown and the angle rm is defined as the position on the rotor relative to the midpoint of the permanent magnet shown. Although not shown on the diagram, the separation between angles rm and sm is the same as the angle B2' Prior to investigating force production under a so-called ‘microscopic’ view, it is useful to consider the analysis of the machine from a more traditional perspective. In traditional lumped-parameter-based machine analysis, a closed-form expression for the electromagnetic torque is established using an energy balance approach. This yields the expression [13] P Wc Te = (2) 2 r Where P is the number of magnetic poles and Wc is the coenergy of the coupling field that can be represented in a form Wc = ias aspm + ibs bspm + ics cspm + Wcpm ( r ) (3) In (3), Wcpm ( r ) represents energy that is due to the permanent magnets being attracted to stator iron and the terms Figure 1. A cross-sectional view of the PM machine and coordinate system. II MACHINE MODEL The PM machine studied in this research is shown in Fig. 1. It is a 1 Hp, 2000 rpm, 3-phase, 4-pole, 12-slot, surface-mounted permanent magnet machine that is used in home appliance applications. In Fig. 1, the phase-a winding magnetic axis and the q-axis of the rotor are shown, respectively. The mechanical rotor position ( rm ) is defined as the angle between the two axes. For the asm , bsm , represent the influence of the flux of the permanent magnets on the respective stator windings. For initial analysis, the flux linkages are assumed of the form: (4) asm = mag sin ( r ) csm bsm = mag sin ( r 120 ) (5) csm = mag sin ( r + 120 ) (6) Substituting (3) into (2) yields 2 Te = P ias 2 aspm bspm + ibs r + ics r cspm + Wcpm ( r ) r r (7) Wcpm ( r ) where represents the torque due to the r permanent magnets being attracted to the stator (cogging torque). When the equations of the PM machine are expressed in terms of physical (abc) variables, the stator flux linkages are functions of rotor position (due to (4)-(6)) and thus the lumped-parameter model is time-varying. To eliminate time-varying components a transformation of variables is applied in which the dynamics of the machine are represented in the rotor frame of reference [3]. The resulting dynamic model of the machine expressed in the rotor reference frame is of the form vqsr = rs iqsr + r dsr + p qsr (8) vdsr = rs idsr r qs r r qs =L i r ds = Lss idsr + +p r ds (9) r ss qs (10) (11) m 3P r m iqs + Tcog 2 2 where rs represents the stator resistance, Te = r qs angular velocity, v iqsr voltages, currents, r qs and and r ds r ds and v idsr (12) r the rotor the transformed stator the transformed stator the transformed stator flux linkages, and Tcog the cogging torque. Equations (8)-(12) represent the standard dynamic equations that are used to analyze machine performance. Their derivation is described in texts on PM machines and the response predicted using the model is known to be accurate, provided that the parameters are accurate. One item of interest is that from (12) it can be observed that the electromechanical torque generated by the machine is only associated with q-axis current, the d-axis current has no contribution to average torque. Based upon this observation, the d-axis current is typically set to zero or a minimum value to achieve a maximum torque per ampere ratio and minimum copper loss. There are instances where the d-axis current is introduced to have the effect of weakening the field created by the magnet to facilitate operating at higher machine speeds. Although the standard equations offer the basic principle to design the control of PMSMs, it is observed that the information provided is limited from the sense that there is little information on the overall force production. Specifically, it is known that within the airgap there are both radial and tangential components of force. To date, there has been very little focus on how the q- and d-axis components of current contribute to radial components of force, with the exception of [12], wherein the focus was on force ripple and assumed that the fields created by the stator winding are unidirectional (i.e. radial from stator to rotor). Thus an alternative microscopic investigation on torque and force characteristics is described in the following section to address both of these issues. IV MICROSCOPIC VIEW OF FORCE PRODUCTION An alternative to deriving the electromagnetic torque using an energy balance approach is to derive the components of force from the magnetic field using a Maxwell Stress Tensor method [14]. Specifically, within the airgap of the machine the local radial and tangential components of force density can be expressed [14] ft = fr = 1 µ0 (13) Br Bt 1 2µ0 (B 2 r Bt2 ) (14) where ft is the tangential component of force density (N/m^2), f r is the radial component of force density (N/m^2), Br is the radial component of the magnetic flux density and Bt is the tangential component of magnetic flux density. µ0 is the permeability of free space. Equations (13) and (14) provide the basis for a microscopic investigation of the force production within the machine. As a first step in viewing the force under the light of (13)(14) it is useful to consider the magnetic flux densities in the airgap of the machine subject to alternative stator excitation. To facilitate this study, the machine was modeled using a finite element approach wherein the radial and tangential components of flux density were evaluated using a FEA model created using the commercial package Maxwell [15]. A mesh with 21934 triangles was used in the calculations. To minimize error, the contour of integration was established in the middle of the airgap [16]. Hence the distribution of the flux density (and force density) components was calculated in the middle of the airgap. A convergence test was conducted to ensure accuracy. Specifically, the number of triangles was increased to roughly 130000 and the difference between the forces calculated was within 0.5%. The tangential and radial flux density distribution along the airgap for a machine in which the stator is de-energized and the rotor is located at r = 0 is shown in Fig. 2. In Fig. 2, the horizontal axis represents the position of an observer inside the airgap of the machine. From the waveforms it can be seen that the permanent magnet creates a significant radial component of flux density (Brpm). There are changes in the radial component around the regions directly below stator slots. In contrast, the tangential distribution of the flux density (Btpm) has relatively 3 small amplitude and only occurs at the location immediately around the stator slots. The tangential flux density is attributed to flux from the magnet traveling to the iron wall of the slots. Through visual inspection, it can be seen that multiplication of Br and Bt will yield a tangential component of force density that has equal positive and negative components. Therefore, integration of the tangential component of force density will yield a value of 0 (confirmed numerically), meaning zero average torque (as expected). In contrast, at zero stator current, the radial component of force density will be nonzero and is dominated by the radial component of the flux density. 0.5 Br Bt 0.4 0.3 Flux density [T] 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 50 100 150 200 250 angular position in airgap (φsm) 300 350 Figure 2. Tangential and radial flux density in airgap generated by PMs. As a second study, the overall tangential and radial flux density distribution for a case in which current was r r = 0 and iqs = 4.6 A (rated), and the applied in a manner ids permanent magnets are maintained in the rotor, is shown in Fig. 3. It is noted that the figure contains the distribution at a single rotor position ( r = 0 ). Similar curves are obtained at all positions of the rotor. Br Bt 0.5 Comparing the results shown in Fig. 3 with the flux densities of the de-energized machine (Fig. 2), it can be seen that the q-axis stator current influences both the radial and tangential components of flux density. As one might expect, this influence is most clear in the vicinity of stator slots. In contrast to the de-energized machine, in each pole-pitch the tangential component of flux density is positive over the region that the radial component is positive. It is negative over the region that the radial component of flux density is negative. Therefore, the result is that the tangential component of force has a positive average value (i.e. average torque is produced). It is also interesting to consider that between stator slots, the radial components of flux density are larger than the values observed in the de-energized machine. This leads to an increase in the average radial component of force. The force densities computed at a single rotor position for r r ids = 0 and iqs = 4.6 A are shown in Fig. 4. For clarity, the negative value of the tangential component of force is plotted. From these plots it can be observed that the force density distributions along the airgap are far from uniform. This is due to the existence of slots and a spatial current distribution that is discontinuous. Considering the two waveforms of force density, it is interesting to note that the peak values of both the radial and tangential force densities are roughly the same. However, the tangential component of force is created in the regions near the stator slots. In contrast, the radial components are distributed over an entire pole-pitch. Therefore, upon integration it is clear that the radial component of force will be greater than the tangential component. It is also interesting to note that a relatively small percentage of the airgap space actively participates in the torque generation process at a given rotor position. It would appear that there is an opportunity to improve torque density of a PMSM through design modifications (geometries, winding configurations, excitation) that yield a nonzero tangential force density over a wider region – similar to the radial force density). Methods to search for such alternatives using numerical techniques are being pursued in ongoing research. 5 x 10 fr -ft 0.4 1 0.3 Force density [N/(m*m)] Flux density [T] 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0.5 0 -0.5 -0.5 -1 0 50 100 150 200 250 angular position in airgap (φsm) 300 350 Figure 3. Radial and tangential flux densities along the airgap when r r ids = 0 and iqs = 4.6 A. 0 50 100 150 200 250 angular position in airgap (φsm) 300 350 Figure 4. Tangential and radial force density distribution along the airgap at a single rotor position ( r = 0 ). 4 Br Bt 0.6 0.4 Flux density [T] A plot of the average component of radial and tangential components of force/length of the machine (N/m) as a function of q-axis current is shown in Fig. 5. Comparing values of radial and tangential components of force, it is clear that that radial component is much greater than the tangential component. It is also observed that the tangential component of force is linearly related to q-axis current (as predicted from the macro-scopic model). However, it can be seen that an increase in q-axis current also leads to an increase in the radial component of force. As will be shown in Section V, the relationship between radial force and q-axis current is a quadratic function. 0.2 0 -0.2 -0.4 0 50 6000 100 150 200 250 angular position in airgap (φsm) 300 350 Figure 6. Radial and tangential flux densities along the airgap when r r ids = 4.0 A and ( iqs = 4.6 A). 5000 4000 tangential force radial force 5 x 10 3000 fr -ft 1.5 2000 1 1000 0 0 1 2 3 4 5 6 q-axis current [A] 7 8 9 Figure 5. Average tangential and radial force with increasing q-axis r current ( ids = 0 A). The flux density distributions along the airgap for a r = 4.0 A case in which the d-axis phase current is set to ids r = 4.6 A) are shown in Fig. 6. Comparing Fig. 6 to Fig. ( iqs 3, it can be observed that the introduction of the d-axis current leads to a change in both the radial and tangential components of flux density. If one compares values of the radial component of flux density over a pole-pitch, there is a general increase in value. In contrast, the tangential component does not see a general increase in magnitude over a pole pitch. Rather, as shown by the tangential components that are encircled in the figure, there are places where the tangential component of flux changes r = 0 A. Therefore, in sign relative to their value when ids r contrast to the case in which ids = 0 A, the multiplication of Br and Bt leads to regions in which the tangential force density distribution is negative. The radial and tangential components of force density are shown in Fig. 7. Force density [N/(m*m)] Average value of radial and tangential force [N/m] -0.6 7000 0.5 0 -0.5 -1 -1.5 0 50 100 150 200 250 angular position in airgap (φsm) 300 350 Figure 7. Tangential and radial force density distribution along the airgap at a r single rotor position ids r = 4.0 A ( iqs = 4.6 A). Shown in Fig. 8 are the flux density distributions along the airgap for a case in which the d-axis phase current is set to r r ids = 4.0 A ( iqs = 4.6 A). The radial and tangential components of force density are shown in Fig. 9. Comparing the radial component of flux density over a pole-pitch to those shown in Fig. 3 and Fig. 6, there is a general decrease in value. In contrast, the magnitude of the tangential component increases slightly over a pole pitch. Similar to the case in which d-axis current is positive, there are locations where the tangential component of flux density changes sign relative to its value r = 0 A. Multiplication of Br and Bt leads to regions in when ids which the tangential force density distribution is negative. Comparing radial force density, it is seen that the decrease in radial flux density leads to a general decrease compared to the r r = 0 A and ids = 4.6 A. cases in which ids 5 15000 B Average value of radial and tangential force [N/m] r 0.5 Bt 0.4 0.3 Flux Density [T)] 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 tangential force radial force 10000 0 −0.5 -8 0 50 100 150 200 250 300 Angular Position in the Air Gap, fsm [degrees] 350 Figure 8. Radial and tangential flux densities along the airgap when r r ids = 4.0 A and ( iqs = 4.6 A). 5 x 10 fr 1 Force Density [Nm−2] 5000 ft 0.5 0 −0.5 −1 0 50 100 150 200 250 300 Angular Position in the Air Gap, fsm [degrees] 350 Figure 9. Tangential and radial force density distribution along the airgap r at a single rotor position ids = r 4.0 A ( iqs = 4.6 A). Under both positive and negative d-axis current, changes in radial and tangential flux densities are such that the average value of the tangential component of force is not related to d-axis current (as predicted by the macroscopic model). The average value of the radial and tangential components of the force/length are plotted as a r = 0 A) in Fig. 10. It can be function of d-axis current ( iqs seen in Fig. 10 is that the radial component of force changes significantly with change in d-axis current. -6 -4 -2 0 2 d-axis current [A] 4 6 8 Figure 10. Average tangential and radial force under different d-axis current excitation (q-axis set to 0). V ANALYSIS OF OBSERVATIONS In order to provide some insight into the results observed in the previous section, the contributions of the airgap flux densities is explored analytically. For a PM machine, both Br and Bt are created by two sources: the stator windings and the permanent magnets. Since the PM machine usually has a relatively large effective airgap, for the purposes of analysis, saturation is neglected. Therefore, using superposition, Br = Brpm + Brs (15) Bt = Btpm + Bts (16) where Brpm, Btpm, Brs and Bts denote the radial and tangential flux density created by the PM and stator windings, respectively. Substituting the field flux densities (15)-(16) into (13)-(14) the tangential and radial force density can be expressed as: 1 ft = Bts + Btpm Brpm + Brs (17) µ0 fr = 1 2 µ0 [( Brpm + Brs )2 ( Bts + Btpm )2 ] (18) Using (17) and (18), the tangential and radial force can be derived from the design and excitation. Specifically, the overall tangential and radial force/length at a single rotor position can be expressed as: P 2 Ft = ft ( r ) Rd r (19) 2 0 P 2 Fr = f r ( r ) Rd r (20) 2 0 and the electromagnetic torque expressed as Te = Ft RLef (21) where R is the radius of the contour upon which the Maxwell Stress Tensor is calculated and Lef is the effective stack length (meaning the physical stack length multiplied by a stacking factor). From (17), ft can be divided into four parts: 6 (i) Brpm Btpm L Brs = (ii) B rs Bts (iii) Brpm Bts P 2 µ0 2 [Brpm Bts + Brs Btpm ]Rd s (22) 0 To consider the terms in (22) it is useful to expand each of the flux densities in terms of a series. Specifically, considering Fig. 2, one can see that Brpm can be expressed as a Fourier series of the form Brpm = Brpmk cos(k r ) (23) k =1 Similarly, it can be seen that Btpm can be expressed as a Fourier series of the form Btpm = + Br cs (25) Btsk = Bt + Bt as bs + Bt cs (26) Btpmk sin(k r ) where L is the number of stator slots, Br as , Br bs , Br cs , are the radial components of flux density created by the phase windings. Bt as , Bt bs , and Bt cs are the tangential components of flux density created by the phase windings. The tangential and radial flux density distribution along the airgap created by the current in a single slot (obtained using the FE model) is shown in Fig. 11. If both the stator and rotor iron are unsaturated, the local flux densities created by current in each slot are proportional to the current. Thus we can define: Btsk ( s ) = islot b1 ( s ) (27) Brsk ( s ) = islot b2 ( s ) (28) where b1 and b2 are associated with the geometry of the machine. From the numerical results it can be seen that for the machine studied, b1 is an even function and b2 is an odd function (with respect to sm ), respectively. This is consistent with the analytical derivations made on simplified machines. Specifically in [3] solutions of the Poisson equation in cylindrical coordinates are obtained for 1) a machine assuming stator currents embedded in stator iron and 2) a machine assuming stator currents in the airgap with a smooth stator inner surface. In both cases, the results indicate a solution with even and odd symmetry, respectively. Brsk [T] contribution of (ii) on average tangential force is due to the fact that Brs is an odd function and Bts is an even function, which will become clear later in the analytical development. Therefore, a conclusion is that only terms (iii) and (iv) can generate an average torque. For the analysis presented herein, the focus is on average values of force and therefore using (iii) and (iv) one can express the force/length due to these two terms as = bs k =1 Terms (i) and (ii) produce cogging and reluctance torque due to the existence of slots. However, both will yield zero average force. A zero average value of term (i) can be seen by considering Fig. 2. Specifically, integration of the Brpm Btpm shown in Fig. 2 over a pole pitch is zero. The iv ) + Br as L Bts = (iv) Brs Btpm Ft (iii Brsk = Br k =1 (24) k =1 and therefore one might expect the series to only include components that are multiples of the number of slots, it has a fundamental period of 360 (electrical) degrees, i.e. Btpm1 0 . This is due to the behavior of the flux around the transition region between magnets and can be seen in Fig. 2. To evaluate the effect of the stator windings, it is convenient to first consider that the net field created by current in the stator windings is the sum of the fields created by the current in each stator slot. Specifically, if Brsk and Btsk represent the radial and tangential flux density created by current in kth slot, then the net flux densities are expressed φsm Btsk [T] Prior to proceeding with the flux densities produced by the stator windings, it is interesting to note that although Btpm is due to interaction of the magnets with stator slots φsm Figure 11. Radial and tangential components of flux density introduced by current in a single stator slot. Based upon the result of current in a single stator slot, one can express the tangential components of flux density contributed by the individual phase windings in a form Bt _ as ( s ) = ias B1 ( s ) (29) Bt _ bs ( s ) = ibs B1 ( s 2 / 3) (30) Bt _ cs ( s ) = ics B1 ( s + 2 / 3) (31) 7 the focus is on average tangential force which is represented by the first term on the right hand side of the equal sign in (43). Considering the expression, it is clear that the average tangential force is only a function of the q-axis stator current. The derivation of radial force follows a similar set of steps, although the interaction of all components is a factor in the average radial force. Plugging (25) and (26) into (18) and the result into (20) yields where B1 ( s ) is an even (zero average) periodic function with respect to s . Specifically, B1 ( s ) = B1k cos(k s ) (32) k =1 Similarly, one can express the radial components of flux density due to individual phase windings in a form Br _ as ( s ) = ias B2 ( s ) (33) Br _ bs ( s ) = ibs B2 ( 2 / 3) s Br _ cs ( s ) = ics B2 ( s P Fr = 4µ0 (34) + 2 / 3) (35) B2 k sin(k s ) Btpm ( + i) ibs = is cos( r + Fr = (37) 120) i (38) + B2 ( + s 0 ) Btpm ( s 2 3 s 2 + 2 /3 2 3 r Btpm ( 2 /3 s is cos ( ) is cos 2 /3 P 2 µ0 + B2 2 2 / 3+ 2 P 2 µ0 + B2 + P 2 µ0 = iv ) i ) s + i 2 3 ) Rd s r r s + Rd r is cos Btpm ( r + r i ) + B1 ( ) Brpm ( B1 s B1 s s r + ) 2 3 Brpm ( s r ) 2 3 Brpm ( s r ) s r iqs = is cos( i ) (41) r ids = is sin( i ) (42) one can obtain a general form for the tangential force/length due to Brpm Bts and Btpm Brs as iv ) = 3P 8µ0 Br _ is i = a ,b,c (44) r)+ Bt _ is Rd s (B 2 2k k , k 3,9, etc B1k 2 ) ( iqsr 2 + idsr 2 ) + $ Brpm1 + B11 Btpm1 ) idsr % R + Fr ( r ) k & (45) where Fr ( r ) represents a component of radial force that is a function of rotor position (force ripple). Force ripple is evaluated in [12] for the special case in which the stator windings yield only a pure sinusoidal radial flux density. A value of average normal force is also provided in [12] based upon the sinusoidal-radial approximation. Equation (45) will simplify to the result provided in [12] if the same assumptions are applied – although based upon the findings herein, these cannot be made in general. Herein the focus is on the average force which is represented by the first three terms on the right hand side of the equal sign in (45). From (45) it is clear that the average radial force is a quadratic function of both q- and d-axis currents. Interestingly, the q- and d-axis expressions take slightly different forms. Specifically the third term on the right 2 Btpmk )+3 (B 21 r (40) Substituting (23), (24), (32), and (36) into (40), applying trigonometric identities, and using the relationship that Ft (iii s !9 " !# 4 2 rpmk s 2 3 Rd s P 4µ0 (B ics = is cos( r + i 120) (39) and substituting (37)-(39), into (29)-(31) and (33)-(35) and the net result into (22), one obtains the expression Ft (iii r)+ s Substituting the Fourier series representation of each of the components into (44) and using trigonomic identities to simplify, one obtains a result for the radial force: (36) k =1 r Brpm ( i = a ,b , c Assuming phase current excitation of the form: ias = is cos( 0 2 2 where B2 ( s ) is an odd (zero average) periodic function with respect to s , i.e. B2 ( s ) = 2 r Btpmk B2 k iqs R + Ft ( Brpmk B1k + k r ) k (43) where Ft ( r ) represents a component of tangential force that is a function of rotor position (i.e force ripple). Herein hand side of the equal sign only includes ids . This accounts for the more significant effect that the d-axis current has on the radial component of force (seen in Fig. 8). Moreover, this same term also shows how applying a negative d-axis current reduces the radial component of force. In contrast, application of a qaxis current can only act to increase the radial component of force. From (45) one can also observe how the different components of the flux densities play a role in the production of average radial force. As noted in Section II, the effects of saturation, eddycurrents, and hysteresis have been neglected in the analysis of force production. In saturation, the analysis starting with (25) is invalid, since one cannot assume linearity. Initial research indicates that in saturation the odd/even symmetry nature of the radial and tangential components of flux density remain the same. However, since the flux density components are nonlinear 8 functions of stator current, the tangential force cannot be expressed as a linear function of q-axis current. The effects of hysteresis and eddy currents on the flux and force densities may also be of interest. However, since machine models that include hysteresis and eddy current effects are rarely necessary to predict the relationship between torque and stator current, these effects may not be significant (except possibly in high speed applications). VI CONCLUSION The magnetic flux density and force density distribution along the airgap of a PM machine have been investigated analytically by leveraging a field solution obtained from finite element analysis. The effect of both qand d-axis phase current on the radial and tangential field and force distribution has been analyzed. Based upon this analysis, several observations have been made. First, it is observed that under standard operation the majority of the force produced in the machine is in the radial direction, which cannot translate into rotor rotation. Second, it has been found that the q- and d-axis components of current both contribute to the average radial force – and that their relationship can be expressed as a quadratic function. It is also verified that the average component of tangential force (torque) is linearly related to q-axis stator current, while the d-axis current has no influence of average torque. Finally, through the analytical process an evaluation of how stator excitation influences flux densities to produce torque and radial force has been provided which helps to provide a so-called microscopic view of force production in the machine. [11] [12] [13] [14] [15] [16] Permanent-Magnet Machines,” IEEE Transactions on Magnetics, Vol. 38, No. 1, pp. 229-238, Jan. 2002. Wang X., Li Q., Wang S., Li Q., “Analytical Calculation of Air-Gap Magnetic Field Distribution and Instantaneous Characteristics of Brushless DC Motors,” IEEE Transactions on Energy Conversion, Vol. 18, No. 3, pp. 424-432, Sept. 2003. Jiao G. and Rahn C., “Field Weakening for Radial Force Reduction in Brushless Permanent Magnet DC Motors,” IEEE Transactions on Magnetics, Vol. 40, No. 5, Sept. 2004. P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery, IEEE Press, 1995. Belahcen, A., “Overview of the Calculation Methods for Forces in Magnetized Iron Cores of Electrical Machines,” Seminar on Modeling and Simulation of Multi-technological Machine Systems, 29 November 1999, Espoo, Finland, pp. 4.1-4.7. Maxewll 2D Field Simulator Manuals,” Ansoft Corporation, 2002. L. Chang, A.R. Eastham, G.E. Dawson, “Permanent Magnet Synchronous Motors: Finite Element Torque Calculations,” Industry Applications Society Annual Meeting 1989, Vol.1, October 1-5, 1989, pp.69–73. VII REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] Goals of the Grainger Center for Electric Machinery and Electromechanics, UIUC, http://machines.ece.uiuc.edu/. NASA Low Emissions Alternative Power Project CLEVELAND, OH 44135-3191 http://prod.nais.nasa.gov/.. B. Hague, The Principles of Electromagnetism Applied to Electrical Machines, New York : Dover Publications, 1962. N. Boules, “Two-Dimensional Field Analysis of Cylindrical Machines with Permanent Magnet Excitation,” IEEE Transactions on Industrial Applications, Vol. IA-20, pp. 1267-1277, 1984. Q. Gu an d H. Gao, “Airgap Field for PM Electrical Machines,” Electrical Machines and Power Systems, Vol. 10, pp. 459-470, 1985. Boules, N., “Prediction of No-Load Flux Density Distribution in PM Machines,” IEEE Transactions on Industrial Applications, Vol. IA-21, pp. 633-643, 1985. Q. Gu an d H. Gao, “Effect of Slotting in PM Electrical Machines,” Electrical Machines and Power Systems, Vol. 10, pp. 273-284, 1985. 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