Design refinement of synchronous reluctance motors through finite-element analysis PDF

Preview

Summary

1094 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 4, JULY/AUGUST 2000 Design Refinement of Synchronous Reluctance Motors Through Finite-Element Analysis Alfredo Vagati, Fellow, IEEE, Aldo Canova, Mario Chiampi, Michele Pastorelli, and Maurizio Repetto, Member, IEEE Abstract—In this paper...

Description

1094

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 4, JULY/AUGUST 2000

Design Refinement of Synchronous Reluctance Motors Through Finite-Element Analysis Alfredo Vagati, Fellow, IEEE, Aldo Canova, Mario Chiampi, Michele Pastorelli, and Maurizio Repetto, Member, IEEE

Abstract—In this paper, the key points in the design of synchronous reluctance motors are first evidenced and discussed, that is, the choice of rotor type, the stator–rotor joint design, and the optimization of the rotor structure. A purposely designed finite-element code is then introduced and validated, on the basis of properly obtained experimental data. Measured and computed torques are compared, with emphasis on the evaluation of the torque ripple. Last, the finite-element method code is used to illustrate some aspects of the stator–rotor design and to show the torque-ripple performance of different types of machine structure. Index Terms—Finite-element code, synchronous reluctance motors, torque-ripple performance.

I. INTRODUCTION

T

HE adoption of modern synchronous reluctance motors in controlled drives offers many advantages, with respect to other ac motor drives. Remarkable points are the low cost and high efficiency of this type of machine, together with its inherent suitability to position sensorless control. In spite of the quite recent industrial application (synchronous-reluctance-based servo drives just appeared last year on the market), the design of this type of motor can be considered quite mature, at least with reference to the transverse-laminated type of rotor construction. Nevertheless, some problems are still open, regarding optimization performance (e.g., torque ripple) or application-specific machine design. Although a reasonably good magnetic design can be obtained without using numerical techniques, e.g., the finite-element method (FEM), the design must be referred to these techniques when the nonlinear magnetic behavior plays a key role, e.g., to predict overload performance. Anyway, the availability of a specific FEM code can constitute an essential tool of design optimization, provided that some basic design choices are properly made. As a consequence, the aim of this paper is twofold. First, the basic choices of the synchronous reluctance motor design are summarized and discussed, also on the basis of the related literature. Second, a specific FEM code is introduced, to design refinement and optimization. This code is validated on the basis of purposely made experimental measurements. Paper IPCSD 00–007, presented at the 1999 Industry Applications Society Annual Meeting, Phoenix, AZ, Octboer 3–7, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review June 15, 1999 and released for publication February 28, 2000. The authors are with the Dipartimento di Ingegneria Elettrica Industriale, Politecnico di Torino, I-10129 Turin, Italy (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Publisher Item Identifier S 0093-9994(00)04772-1.

Fig. 1.

Schematic of (a) transverse and (b) axially laminated rotors.

II. TRANSVERSE VERSUS AXIALLY LAMINATED ROTORS Transverse and axially laminated types of rotor are available, both being adequately described in the literature [5], [9]–[12], [16]. They are schematically shown in Fig. 1. In spite of the much work which has been done on both types of structure, a ultimate comparison between two is not available, in the literature. However, it is the opinion of the authors that the transverse type of structure is largely preferable, for the following theoretical and practical reasons. In practice, the better suitability of the structure in Fig. 1(a) to industrial manufacturing is evident; in this case, the rotor lamination can be punched as a whole, like the other more traditional machines. The axially laminated structure [Fig. 1(b)], on the other hand, is theoretically appealing, because it looks the nearest to an ideal “distributed anisotropy structure.” However, , while, for this is only true for a two-pole structure , it has been shown in [1] that the ideal structure should have a variable ratio between the depths of magnetic and non magnetic laminations. Moreover, the above considerations only apply to an ideal slotless stator, while the usual toothed stator structure enhances in a practical axially laminated motor both torque ripple and additional iron losses.

0093–9994/00$10.00 © 2000 IEEE

VAGATI et al.: DESIGN REFINEMENT OF SYNCHRONOUS RELUCTANCE MOTORS

Regarding torque ripple, this is due to the rotor magnetic reaction to stator slot harmonics, as explained in [2]. Of course, a torque ripple is present also in the transverse-type of motor. However, in this case, the rotor can be easily skewed, while this is clearly unpractical for an axially laminated rotor. On the other hand, stator skewing is normally avoided, because it is against automatic winding facility. Regarding the additional iron losses in the rotor of the axially laminated type, they have been found in [3] and confirmed in [5]. They can be explained in different ways. In [4], a simpli, which justifies fied analytical model is given, valid for these losses by flux oscillations in the deep rotor iron due to the effect of stator teeth. On the other hand, a different explanation is given in [5], where these losses are referred to eddy current induced in the rotor laminations by harmonic fields. Apart from the explanation, the amount of additional loss is considerable and represents a further drawback of the axially laminated type of rotor construction. The above-cited reasons are largely sufficient to prefer the transverse-laminated type of rotor. However, the persistent interest in the other type is probably due to the believe that the axially laminated rotor gives a better saliency. This is not correct; comparable anisotropy values are obtained from both rotors, of course, when the pole number is the same. Ten is a typical, nonsaturated value for four-pole machines [6], [7], while it can grow up to 20, for a two-pole rotor [11]. However, the unsaturated saliency ratio clearly gives insufficient information about motor performance. This is due to both the highly nonlinear magnetic behavior [6], [7] and the existence of a tradeoff between rotor magnetic insulation and allowed stator MMF, at fixed power dissipation [8], [12]. This point will be discussed later. In conclusion, the transverse-laminated type of rotor appears to be definitely preferable to the other one, from both theoretical and practical points of view. We will refer to this structure, only, in the following.

1095

Fig. 2.

Tradeoff between stator MMF and rotor anisotropy.

the basic rotor structure. These choices can heavily affect the performance, for example, the torque-ripple behavior. The other approach, followed by Vagati et al.[8], [12], is based on a generalized lumped-parameter modeling of the rotor magnetic circuit. The adopted model is, of course, approximate, since it refers to an applied MMF of purely sinusoidal shape. Moreover, the iron saturation is taken into account only for -axis excitation, thus disregarding the cross-saturation effect. In this way, however, the “best” stator can be defined without a preliminary definition of the rotor structure. It is supposed of the various flux barriers in the [12] that the permeances rotor are related to each other, in order to minimize the -axis flowthrough flux, according to the other constraints. It has also been shown that, for four-pole machines, this minimum-inducpermeances practically coincides with tance distribution of a constant-permeance distribution, at least if the rotor is complete [2]. As a consequence, the anisotropy ratio can be written as in (1), disregarding zig-zag inductance, rib leakage, and accounting for iron MMF drop by a Carter coefficient

III. STATOR AND ROTOR JOINT DESIGN Once the transverse-laminated type of machine is chosen, the designer’s job is paradoxically worsened, because of the too many degrees of freedom involved. On the other hand, since the torque is due to the anisotropic behavior, optimization of the lamination shape is clearly mandatory. The above problem can be faced by two different approaches. The first approach, followed by Kamper et al.[9], [10], is based first on definition of a basic rotor structure (i.e., number of flux barriers, eventual cutoff, etc.). Then, this structure is referred to a fixed number of parameters (barrier widths, rib widths, etc.), in order to be numerically optimized. The optimization procedure makes a direct use of a specific FEM program, to calculate at each step the relevant performance indexes (torque, torque/kVA, etc.). The advantage of this approach is that complex phenomena like the magnetic cross saturation are inherently taken into account in the optimization process. A disadvantage is that the result may heavily depend on the preliminary choices regarding

(1) coefficient can be conIt is shown in [8] and [12] that the sidered constant, for a fixed number. Thus, the ratio (1) is nearly inversely proportional to the total length , measured along the axis, which is shown in Fig. 2. If the outer diameter and the relative power dissipation are fixed, it follows from Fig. 2 and (1) that a tradeoff does exist between the allowed stator MMF and the rotor anisotropy. The related main design variables (per unit) are the inner to outer diameter ratio and the air gap to yoke flux-density ratio . If a large anisotropy is wanted, we need a large insulation. It follows that the inner diameter must be increased and, consequently, the air gap to yoke flux density has to decrease. At last, the slot area is reduced and the torque, too, as a consequence. In other words, a maximum-torque design implies a quite low anisotropy ratio, , and a large air-gap flux-density a low inner diameter

1096

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 4, JULY/AUGUST 2000

value. On the contrary, when a low -axis reactance is wanted, to reduce the kilovoltampere needs, a larger diameter must be , together with lower air-gap flux-density designed values. Of course, the torque is lower, in this case. The above-cited results, obtained by the simplified analysis previously described, have been confirmed and validated by Kamper et al.[9], [10], through their optimal FEM-based approach, which also takes into account the effect of cross saturation. In conclusion, the ultimate choice of the main machine pais always a compromise between opposite exirameters gencies, thus needing to be solved by the designer’s skill and, also, in consideration of the requirements of the specific application. The low sensitivity of machine performance to small variations allows adoption of simplified lumped-parameters models, in this case. On the other hand, the situation can be different when the effect of cross saturation becomes important, as discussed in [7]. It is shown there that this kind of machine is particularly suited to large overload. As a consequence, the magnetic design must be accurately tuned to optimize the motor performance over a quite wide current range. In this case, the use of an FEM code is strongly suggested. A main degree of freedom is represented by the relative saturation level of the rotor iron, with reference to the stator one. This problem will be faced in the following, for a defined rotor structure. IV. DEFINITION OF THE ROTOR STRUCTURE This argument has been treated in [2] through a theoretical approach verified by experiment. There, the attention was focused on the rotor optimization with respect to torque-ripple content. The main results are only summarized here. is • The number of rotor “equivalent slots” per pole pair preferably chosen accordingly with [13] (2) of the various flux barriers are de• The permeances signed in order to produce a -axis flux-density distribution at the air gap which has the same harmonic content as the rotor magnetic potential (rotor reaction to a sinusoidal -axis MMF). • The rib and web widths are designed in order to produce an air-gap flux-density distribution which is the nearest to a sinusoid, under sinusoidal -axis excitation. The experimental results confirmed the above guidelines, showing a very low ripple content, for skewed-rotor machines. Of course, adoption of an FEM approach allows us to confirm and extend the validity of the above-described design approach. However, an FEM-based calculation of low torque ripples is not a straightforward task and needs experimental validation. This is given by comparing the calculated ripple with that measured from a purposely built machine, without rotor skewing. In this way, a better comparison is possible, as shown in the following. After this validation, the FEM code can be an important tool for complete prediction of machine performance.

Fig. 3. Plot of flux lines, i

= 34 A, = 70

.

V. ADOPTED FEM CODE The analysis of the magnetic field distribution within synchronous reluctance motors reduces to the solution of two–dimensional (2-D) magnetostatic problems where the imposed current values are linked to the rotor position, according to the considered operating condition. Skewed rotors are treated by means of the usual slicing techniques. Each problem, expressed in terms of magnetic vector potential , is governed by (3) where is the known current density and is the nonlinear . relation Equation (3) is linearized following the iterative fixed-point (FP) technique, leading to (4) is the iteration index, is the FP coefficient, and is the residual to be iteratively evaluated starting from any trial value. Homogeneous Dirichlet boundary conditions are imposed on inner and outer boundaries; periodicity conditions are invoked on lateral boundaries in order to limit the domain under study to a two-poles fraction of the machine. The solution of the problem is performed by the FEM, using triangular elements and firstorder shape functions. The 2-D approach neglects three-dimensional (3-D) effects, such as end-leakage effect and rotor skewing. The first one can be included by adding to the computed fluxes the end-leakage fluxes estimated by experiment or by empirical relationships. The skewing effect is taken into account approximately, by averaging the results obtained with three or five rotor angular positions included within the skewing angle. The difficulties of the FEM approach previously described are due to the intricate geometry of the motor. This is evident from Fig. 3, where a flux-lines plot is shown, referred to a loaded condition, near to the rated value. The large amount of geometrical data and the need of rotor angular displacements with respect to the fixed stator make complicated and heavy both description and meshing of the domain. Rotor and stator are separately handled by a high-level preprocessor; in particular, the rotor structure, including iron ribs and flux barriers, is described by means of a specific code able

where

VAGATI et al.: DESIGN REFINEMENT OF SYNCHRONOUS RELUCTANCE MOTORS

to enforce, as constraints, fixed values of the bridge thickness and of barrier width and declination. The stator mesh is defined on a subdomain, constituted by one-half slot and tooth, which is then replicated on the whole machine; the program also automatically generates the winding distribution. The two submeshes are finally joined by a routine which recognizes the outer boundary of the rotor and the inner boundary of the stator and creates an even grid in the air gap in order to guarantee reliable torque estimation through the Maxwell tensor method. The chosen mesh of the whole motor includes about 20 000 elements with about 10 000 nodal unknowns. frame. Therefore, The FEM analysis is developed in the comthe supply currents are preliminary determined from ponents by means of an inverse Park transformation. Then, frame the computed phase flux linkages are reported in the through a Park transformation. VI. MEASURING SETUP AND AVAILABLE MOTORS Two types of measurement are provided for validation of the magnetic model of the motor is meaFEM code. First, the sured; second, the torque is measured, including torque ripple. The adopted magnetic model measuring setup is extensively described in [7]. The tested motor is current supplied by a current-regulated pulewidth-modulated (CRPWM) inverter and axes. The motor speed is imposed by a controlled on the point is obtained by coaxial speed-controlled motor. Each averaging two motoring and braking symmetrical situations ( and ), in order to compensate for resistive drop. A reasonably low speed is chosen (500–750 r/min), in order to neglect the residual influence of core loss on the measurement. and again are successively driven and first The currents and third results averaged, in order to overcome the temperature variation. The supply duration is quite short, in order to allow large overcurrents with respect to the rated value. A different setup is adopted for torque measurement, as already described in [2]. The motor under test is still supplied by current vector control, while the motor shaft is connected through a precision torque meter to the low-speed end of a one-way gear. The high-speed end of the gear is speed controlled by a small brushless drive. In this way, the speed is imposed by this drive through the gear and completely decoupled from the torque ripple of the motor under test. In rpm), addition, the motor can be tested at a very low speed ( thus allowing a good measurement of the harmonic content of torque ripple, without being constrained by the typically low torque meter bandwidth. Three motors were at disposal, making use of the same laminations. The stator lamination is four-pole, 36 slots. Its outer diameter is 148 mm, while the inner one is 93 mm. The rotor lamination has 44 equivalent slots, thus the three motors are 18/22 motors [2]. Two of the motors have the rotor skewed by one stator tooth pitch and differ only by the stack length. The short (skewed) motor has a stack length of 90 mm, while it is 240 mm for the long (skewed) motor. From the measured characteristics of these two motors, the effect of end leakage is indirectly measured. Thus, a 3-D correction factor can be eval-

1097

Fig. 4. Measured end-leakage inductance versus current, motor.

= 0; =2, long

uated, to be introduced into the (2-D) FEM computations of the magnetic model. The third motor is made by the same stator of the former long motor, together with a different rotor, without skewing. The torque ripple measured from this long (nonskewed) motor is used to validate the torque computations based on the FEM code. In fact, the torque ripple of the skewed motors is too low to constitute a good frame of comparison. Last, the stator windings are normal full-pitched windings, this to point out the worst conditions, in terms of torque ripple. The obtained results are shown in the following. VII. MAGNETIC MODEL VALIDATION From the measured flux linkage versus current characare derived, since the teristics, the flux versus MMF ones turn number of each motor is known. For each value of the arguare obtained, for ment of the vector , two relationships and short motors. If both these fluxes are both long split between main and end-leakage components, the latter can be imagined equal for the two motors, while the former can be supposed proportional to the stack lengths. As a consequence, (5) can be easily derived, giving the end leakage from the meafluxes and the known ratio of stack lengths sured (5) is obtained. However, this From (5), an inductance value value is practically a function of both current modulus and current argument , with respect to the axis. This is shown . As can be seen, in Fig. 4, for the long motor and inductance is more going from the axis to the axis the than doubled, at ...