Optimal Design of Large Permanent Magnet ... - IEEE Xplore

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 1, JANUARY 2013

Optimal Design of Large Permanent Magnet Synchronous Generators Juan A. Tapia

, Juha Pyrhönen , Jussi Puranen , Pia Lindh , and Sören Nyman

Electrical Engineering Department, University of Concepcion, Concepción, Chile Department of Electrical Engineering, Lappeenranta University of Technology, 53850 Lappeenranta, Finland The Switch Drive Systems, 53850 Lappeenranta, Finland Wärtsilä Finland Oy, 65101 Vaasa, Finland High power machine has become a large market for wind power and ship propulsion electric, among other applications. Since the size of these machines is much larger than conventional industrial ones, optimum design must be considered in order to reduce the material cost and increase profitability. In this paper, a simple analytic optimization algorithm is used to maximize the apparent airgap power transferred under tangential stress constraint. In this approach, close related expressions between the main design variables, operational restrictions, and external dimensions are derived to build the mathematical structure of the optimization process. To improve the torque capacity estimation of the designed machine, a correction procedure, based on the previous result, is used to remove the idealizations considered for the initial design. Close agreement with the finite element analysis results are found with this approach, which is based on analytical method. Index Terms—Optimization methods, permanent magnet (PM) generator design, tangential stress, wind power applications.

NOMENCLATURE

Number of turns per phase.

Peak linear current density value.

Number of pole pairs.

Peak airgap flux density.

Total number of the stator slots. Terminal voltage.

Conductor cross section.

Displacement angle.

Stator slot area.

Airgap length.

Stator inner diameter.

Torque angle.

Stator outer diameter. Stator non load induced voltage (max and rms).

I. INTRODUCTION

L

Stator induced voltage. Frequency. rms stator current. rms current density. Copper fill factor. Carter coefficient. Lamination stack factor. Electrical power waveform factor. Winding factor. Iron losses factor correction. Active stack stator length. Leakage inductance. Magnetizing inductance. Stator number of phases.

Manuscript received September 19, 2011; revised November 10, 2011, December 14, 2011, February 29, 2012, and May 14, 2012; accepted June 25, 2012. Date of publication July 10, 2012; date of current version December 19, 2012. Corresponding author: J. A. Tapia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2012.2207907

OW maintenance, high reliability and efficiency requirements for wind power generator tend to use gearless direct drive or semi geared machines. These types of machine are characterized by the large dimensions and weight but also with requirement on high efficiency and compactness. Similar situation is found in modern propulsion drive systems for large navy vessels, where efficiency and compactness is required in large synchronous machines [1], [2]. At this point, it is clear that permanent magnet (PM) used on electric machinery offers these characteristics for a wide range of applications. Optimal machine design must be oriented to maximize power production with adequate operational condition for the PMs excitation systems based. In high power applications, such as wind power and ship propulsion, the torque capacity is limited by the ability to remove the heat from the machine. In order to keep the interior temperature in a safe range, air or other fluid is forced to flow through the active material, to remove the heat created by the losses during the machine operation. Main sources of heat are copper and iron losses which are closely related with the linear current density and the airgap magnetic load. The relation between these two variables per unit of airgap perimeter defines the tangential stress and this eventually with the torque production capability of the machine [3], [4]. There are a number of cooling methods used on high power density machines to improve performance and security [5]. Therefore, ultimately, cooling issues have direct relation with this design variable.

0018-9464/$31.00 © 2012 IEEE

TAPIA et al.: OPTIMAL DESIGN OF LARGE PERMANENT MAGNET SYNCHRONOUS GENERATORS

Limiting on its value assures adequate operation of the generator and secure temperature of the materials, particularly PMs. Optimal machine design should maximize the energy delivered by the generator which provides revenue for the investor. It also minimizes material use to reduce cost and to guarantee safe operation for the machine. Optimization algorithms are available to improve the electromagnetic (EM) design based on algebraic relationship, physical quantities, and energy balance; some of them are strictly mathematical approximation and some are other procedures with more physical meaning. Different techniques with extended and documented results in the literature give us a chance to be applied on electrical machines design [6]–[12]. However, proper selection of the procedure has to combine simplicity precision and resources consumption to achieve valid and useful results. In [13] and [14], authors address the split ratio or inner to outer ratio optimization for surface and interior PM motors, using analytical approach. The torque to rotor volume ratio is maximized under fixed copper losses constrain. Slot area expressions for different design parameters influence are obtained considering equal iron magnetic loading on the stator yoke and tooth. Results presented are valid for low power motors where such an assumption is applicable; however in large PM machines where stator teeth are more magnetically stressed than the stator yoke, therefore a different value for each section of the magnetic circuit must be considered. This effect will help to increase heating problems, which is relevant, particularly when permanent magnets are used for excitation, since their properties are very sensitive to the temperature. Selection of the proper iron magnetic loading has to be part of the design process considering operational conditions and material properties. In this paper, an application of a widely know mathematical method is used to optimize the stator machines size based on maximizing the airgap Apparent Power value under a maximal tangential stress constrain. This solution provides the algebraic relationship between physical unknowns (stator dimensions), operational electric and magnetic values and transcendental function to optimize (inner to outer diameter ratio). The proposed method was developed for the particularities present on high power PM machines where the mechanical and thermal issues are more demanding, compared to traditional low power machines. The algorithm provides the optimal outer to inner diameter ratio for a given value of the operational constrain. Using these results, a finite element analysis (FEA) validation is carried out and a correction procedure is developed to improve the torque estimation. Hypotheses established for the ideal conditions to calculate the optimal diameter ratio are removed to include more realistic situations such as flux leakage and magnetomotive force drop on the magnetic circuit. Finally a flow chart summarizes the design procedure to achieve optimal design and electromagnetic torque requirement. II. ANALYTICAL RELATIONSHIPS In order to establish the analytical relationship between the electromagnetic variables, stator geometrical dimensions, and constrain functions, the airgap apparent power expression is developed in terms of the induced voltage and current on the stator

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winding as follows [15], [16]. The expression for the tangential stress is derived using the same quantities. A. Induced Voltage The induced voltage over one phase of a distributed stator AC winding can be expressed as (1) where voltage peak value

is (2)

For the present case, a sinusoidal airgap flux density is assumed as peak value. with B. Current Expression Since the typical load of a generator is a power electronic converter type (nonlinear load), stator current waveform can be far from sinusoidal. Therefore, considering any current waveform, it is possible to define the current waveform, , factor as (3) Introducing the linear stator current density expression, the maximum value of the current waveform is

, (4)

where (5) and finally, the current expression becomes (6) C. Airgap Power Expression The average apparent power transferred through the airgap over one period of the power source can be written in terms of the instantaneous induced voltage and the current equations as (7) replacing (1) and (6) in (7) results (8) The maximum apparent power available to transfer electromechanically from the stator to the rotor (motor operation) or vice versa (generator operation) is expressed in (8). D. Tangential Stress Tangential stress establishes the relation between the total electromagnetic torque developed by the machine per unit of rotor volume. This merit index has intimate relation with the machine construction, type of excitation source and cooling method [3]. The linear current density and the magnetic flux density (and their angle) define the maximum value achievable torque per rotor volume. For regular commercial silicon iron,

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maximum flux density is limited to values no larger than 2 teslas; therefore to increase the tangential stress, stator current must be increased with the corresponding copper losses and heating problems associated. Different types of electrical machines have distinct stress range for proper operation, but it can be increased as the cooling is improved. The relation between the peaks linear current density and airgap flux density magnetic load values define the tangential stress as (9) The preceding equation results from the application of a simple and basic electromagnetic principle , for the particular case where current is expressed in terms of a sinusoidal laminar current distribution over the stator surface interacting with a equally sinusoidal airgap flux distribution. Under these conditions, several idealizations are considered that, at the end, will affect the torque prediction quality. These issues are discussed in detail in later sections. Considering the total amount of copper on the stator slots, is equal to the total number of conductor sections on the machine; this means

Fig. 1. Section of an stator lamination with rectangular slots.

(14) where

and

are defined as (15) (16)

(10) therefore peak linear current density can be written using (5) and (10) becomes (11) Consequently, (9) can be rewritten as (12) This last expression relates the linear current density with the total slot area available. Optimization procedure uses this relafor a given outer machine tionship to achieve maximum diameter. At the same time, this index is used to establish a operational constraint to the optimization process.

These last two parameters represent the saturation ratio of the stator teeth and stator yoke iron respect of the airgap flux density. Adequate selection of the iron magnetic load should be based on operational frequency value and iron magnetic properties in order to tolerate a reasonable amount of losses. At 50 and 60 Hz machine frequency operation, the author in [16] suggests values for and of 0.55 and 0.62, respectively, for regular silicon iron. For higher frequencies, there are some suggestions for the magnitude of the flux density in different part of the magnetic circuit [18] that can be considered. B. Total Stator Slot Area From the stator geometry depicted in Fig. 1, the following two expressions can be written: (17) (18)

III. GEOMETRICAL RELATIONS In general, for simplicity, the stator windings of low and medium power electrical machinery are unsorted and allocated by hand or other type of mechanism on the slots. They can easily be fixed using tooth tip performed during the punching process. In this manner, stator slots are trapezoidal and teeth are rectangular. On the other hand, in large machines, windings are prefabricated with solid or holed conductors with transposition and proper insulation. As a result, manufacturing and repair process is simple and less time consuming. A schematic representation of a larger synchronous machine stator lamination is depicted in Fig. 1, for the present analysis only the active magnetic portion of the iron is considered.

therefore the total stator slot area becomes as (19) where the function

is defined by (20)

with the

and

constant are (21) (22)

A. Magnetic Loading Based on the magnetic flux continuity law and the geometry described in Fig. 1, it is possible to express the inner stator tooth width and yoke depth as (13)

It is noticed that the total slot area, (19), is related with the linear current density, and ultimately, with the tangential stress, (12). Expression indicated on (20) is a function which defines the total slot area available for a given outer stator diameter. This function is determined by the inner to outer diameter ratio, the


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magnetic machine loading, and the number of poles. Equation (20) has difference from that encountered for trapezoidal slots [16]; therefore, optimization based on this estimation is affected by the dissimilar expressions. Later, differences from these two approaches are pointed out. C.

Function:

With (5), (10), and (19), the expression for the airgap apparent power, (8), can be expressed as (23) and function where related with the previous defined (20) as

is

(24) Equation (23) states that for given values of current density and airgap flux density, current and voltage waveforms, number of pole and operational frequency, the is maximal when , defined by (24), is maximal. This equafunction tion is only dependant on the diameter ratio and the iron magnetic loading across the magnetic circuit. At this point, maximum power is transferred through the airgap for a specified outer stator diameter and axial active stack length. In order to evaluate this function, a relation of 0.8125 is used to ratio. This value indicates that stator teeth are apas a proximately 23% more saturated than the stator yoke, which is a reasonable approximation for synchronous machines. In Fig. 2, functions defined by (24) are plotted in terms of the several inner to outer diameter ratio (solid lines). Additionally, function [16], [17] for trapezoidal slot geometry is depicted in the same figure (Fig. 2, dotted line). It is clear that the differences on the optimum values found using each approach will reach to dissimilar stator geometry. Optimum values for the function for the present case are located between 0.6 to parameter varies. These re0.67 of the diameter ratio as the sults indicate that the optimum inner diameter is approximately one half of the outer diameter with deep slots and thin tooth. However, this diameter ratio leads in a high value of stator slot area, resulting in an unacceptable high tangential stress compared with those recommended values [3] for the optimal design area, as is shown in Fig. 3. Consequently severe heating problems removal arises, a limit on the tangential stress value must be imposed to reduce the problem.

Fig. 2. Optimal diameter ratio for airgap VA for teeth magnetic loading . Solid line: rectangular slot optimization approach. Dotted line: trapezoidal slot optimization approach [16], [17].

Fig. 3. Tangential stress as a function of the diameter ratio with teeth magnetic as a parameter. loading

stress is limited to a maximum value specified by constraint function can be written as

. Then,

D. Constrain Function Introducing (19) and (20) in (12), the tangential stress equation, in terms of the diameter ratio expression, becomes

(25) On Lagrange multiplier formulation, a constrain function, , must be defined in terms of the maximum value allowed for the restricted variable. In this case, tangential

(26) The extended function to be optimized is formulated as (27), and are the function and where the constraint function, respectively, expressed by (24) and (26) and is the Lagrange multiplier: (27)

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TABLE I OPTIMAL DESIGN STATOR LAMINATION

Fig. 4. New optimal diameter ratio for airgap VA under tangential stress conas a parameter. strain with teeth magnetic loading

Fig. 5. Resulting stator lamination from the optimization process. a. Without constrain; b. with constrain.

Solving this problem according to the Lagrange formulation, the solution for optimal diameter ratio under tangential stress constrain is reached as

(28) In this case, optimum is displaced to higher values compared with the solution without restriction, shown in Fig. 2. Feasible optimal diameter ratios now are located between 0.8 and 0.9, where the maximum tangential stress meets the restriction imposed. A graphical representation of the new displacement is depicted in Fig. 4, both and functions are shown and the optimal are highlighted. Slot area has direct relation with the maximum tangential stress allowed on the design; therefore, significant reduction is found when a constraint condition is applied. Stator geometries obtained using the unrestricted and restricted approach, on scaled dimensions, is depicted on Fig. 5. The algorithm intends to reach a thinner structure when a restriction is imposed. In Table I, a set of numerical values for the stator lamination is listed for both conditions. Results on the machine dimensions from the application of the previous algorithm are presented in Fig. 6. These results are in terms of the stator tooth magnetic load (values are highlighted for ). As expected, with the tangential stress constrains imposed to the airgap apparent power function , optimal diameter ratio becomes closer to the unity [solid line in

Fig. 6. Optimal diameter ratio to maximize airgap VA and stator main magnitudes comparison for optimal diameter ratio.

Fig. 6(a)] with respect to the calculated with no constrains situation (dotted line in the same figure); as a result, stator becomes thin with large stator yoke depth. Some of the consequences of this fact are larger inner diameter and shorter stator tooth when the tangential stress is limited. As a result, since inner diameter is larger on the constrained case and airgap flux density is constant for both cases, flux per pole increases proportional to the ; therefore stator yoke becomes larger on the constraint situation [Fig. 6(b)]. At the same time, stator tooth is significantly short [Fig. 6(c)] which leads to an increment on the slot width [Fig. 6(d)]. IV. FINITE ELEMENT ANALYSIS VALIDATION To validate the performance of the torque prediction using the preceding design procedure, an FEA is carried out. Since the tangential stress approach, used as a constraint, is derived from an ideal current and airgap flux distribution, some corrections must be done based on the optimized structure obtained. Leakage and magnetizing inductance parameters are calculated for both ideal and non-ideal case to estimate the actual torque value. In this section, detailed numerical analyses and analytical results comparison on the ideal and non-ideal design case are presented.


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TABLE II FEA INPUT DESIGN DATA

Fig. 8. Results comparison between ideal and FEA solutions.

Fig. 7. Optimal stator design lamination for constant airgap diameter. Airgap diameter constant at 1562 mm.

where and are the flux intensity on the teeth and stator yoke, respectively, calculated from the BH iron characteristic (Appendix A). For this particular evaluation, stator and rotor yoke depth are considered to be equal. Electromagnetic torque obtained from the optimized design (considering the restricted tangential stress value imposed to the design procedure) is calculated using FEA. This result is depicted in Fig. 8. At the same time, torque calculated by (29) is also presented in this figure exhibiting a proportional dependence with the tangential stress. As is indicated, these two torque values came from the same tangential stress. Significant differences between the ideal case (29) and FEA are found as the tangential stress increases, which leads to a poor torque estimation based on (29). Several factors not considered on the ideal case torque calculation represented in (29) are the main source of this difference. Detailed discussion about these non-ideal conditions are presented in the next section.

A. Optimal Design and Finite Element Evaluation

B. Ideal and Non-Idealities Issues on Torque Production

Using the optimization algorithm, a set of machine stators is designed for different tangential stress values considering stator inner diameter as a design input. In general, in a situation when there is no restriction on space, airgap diameter is considered as a design data which leads, using this procedure, to determine the outer diameter with the optimal diameter ratio defined by (28). Under this circumstance, output torque is proportional to the tangential stress as expressed in (29) for a given stack length [3]:

For the ideal case, Fig. 9(a), torque is the resultant effect of a continuous superficial sinusoidal current distribution, interacting with an equally sinusoidal airgap flux distribution and the shift angle between them . This entire phenomenon takes place in a smooth, continuous, and homogeneous airgap surface and infinity permeable stator and rotor iron [19]. The optimal torque production occurs when both distributions are in phase . The corresponding phasor diagrams that represent these ideal conditions are presented in Fig. 10(a) and (b), respectively. With this, there are not leakage flux and mmf drops along the magnetic circuit and all the current components contribute to torque production, and terminal voltage corresponds to the airgap voltage, induced from the resultant flux. From the same equation and (2) and (4), it can be easily demonstrated that electromagnetic torque for the ideal case is given by (31) [see (30) at the bottom of the next page]:

(29) In Table II, design variables are listed for the optimal design to calculate the machine dimensions for the FEA solution. The resultant stator lamination geometries achieved from this approach are depicted in Fig. 7. It is clear that, as the tangential stress increases, more slot area is required to allocate higher stator current; therefore slots become deeper and deeper (rising in Fig. 1), and since the iron magnetic loading is the same in each evaluation, teeth have the same airgap width ( in Fig. 1). Magnet geometry is determined in order to establish an airgap taking into account the BH curve and flux density value magnetic loading of the iron. PM height is calculated by (31),

(31) and the torque angle from Fig. 10(b) (32)

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phasor diagram shown in Fig. 10(c), non-ideal torque is calculated by (36) where the new torque angle is (37) Fig. 9. 80 kPa optimal design. a) Fields distributions for torque production on the airgap: ideal case. b) Non-ideal magnet and stator current allocation.

, the slotted stator surface is inFor the non-ideal case cluded by the Carter coefficient; at the same time, the increment of the magnetic circuit reluctance due to the mmf drop effect along the magnetic circuit and the nonlinear BH characteristic of the magnetic steel can be reflected as an additional airgap length on top of the physical one: (38)

Fig. 10. Equivalent phasor diagram. a) Arbitrary linear current density and airgap flux distribution. b) Optimal torque production conditions. c) Non-ideal condition considering effective airgap and leakage flux.

In this manner the equivalent non-ideal magnetizing inductance, , becomes (39)

Under these circumstances, the ideal magnetizing inductance, , corresponds to the relation between armature reaction flux linkages and the required current to impose this linkages in an ideal magnetic circuit with no saturation, infinite iron permeability, and no leakage. In this case, an analytical expression for this inductance can be written as (33) The equivalent airgap length, without slots are present on the stator and the magnetic material is perfect, can be written as (34) where is the magnet height to provide the excitation required on the airgap for ideal conditions, which means neglecting mmf drops on (30): (35) The non-idealities found on the real machines define elements that are not considered on the torque expression given by (29) or (31). Slotted stator surface, nonlinear magnetizing characteristic of the iron, and coils allocated in slots modify the predicted torque value calculated in ideal conditions. From

On the other hand, magnet height calculated by (30) maintains the airgap flux density value invariable for all tangential stress value; therefore the no load induced voltage, , varies with respect to the ideal case. This phasor diagram of condition is depicted in Fig. 10(c). Based on the previous work, it is possible to estimate as a proportion of the PM height values as (40) Conversely, winding coils allocated on slots, as is illustrated on Fig. 9(b), will introduce a leakage flux, which will increase as the tangential stress increases (deep slots), affecting torque production. In fact, for the same amount of current presented on the ideal case, portion of it creates flux that does not cross the airgap, which results in a lower resultant flux, and large angle between the induced voltage and current distribution. From the phasor diagram point of view, torque production is reduced since the torque angle results are larger compared to the ideal case. Similar effect introduces iron saturation since the total reluctance of the magnetic circuit varies. Indeed, as the tangential stress rises, stator tooth become longer and the mmf drop along them also increases; therefore, total excitation flux is reduced in each case. FEA solution takes into account these two effects on torque calculation.

(30)


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TABLE III BH CURVE

where the corresponding torque angles are obtained from (31) and (36), respectively. Using (39), torque estimation is calculated and compared with FEA and ideal case results, depicted in Fig. 8. It can be noticed that there is a better agreement with the numerical value when the non-idealities are included on the analytical torque estimation. Differences between the non-ideal solution and FEA results are explained due to the other leakage component not included (zigzag, belt, etc.). D. Improved Design Flow Chart

Fig. 11. Flow chart for optimum machine design and torque requirement calculation.

Leakage inductance is now taken into account considering the leakage flux along stator slot. The analytical expression for the leakage inductance, , parameter is (41) where is the specific permeance derived for a rectangular slot, according to Fig. 1 as [8] (42) In this manner, non-idealities present on the actual stator lamination calculated are considered on the inductances expression. By using these expressions, more accurate torque estimation can be performed based on the result of the optimal design algorithm.

C. Torque Correction Factor In order to improve the torque estimation based on the result of the optimal diameter ratio procedure and the specified tangential stress value, an approximation of the non-ideal torque can be calculated using (31) and (35). In fact, the non-ideal to ideal torque ratio defines a correction factor based on these expressions

The design method presented suggests an iterative algorithm to meet the torque requirement based on optimal diameter ratio calculation with tangential stress constrain. To summarize the procedure, a flow chart is developed to highlight each step, as depicted in Fig. 11. With initial data input, optimal diameter ratio is calculated and stator diameter and slot dimensions are computed. Initial ideal torque is obtained from the rotor volume and maximum tangential stress specified. Inductance parameters for ideal and non-ideal case are calculated; based on that, torque correction factor and non-ideal torque is determined. If this result does not meet the torque requirement, airgap torque is increased and the procedure is repeated.

V. CONCLUSION In this paper, an optimization procedure based on a wellknown mathematical approach—Lagrange multiplier—is used to maximize the airgap apparent power under tangential stress constrain for large PM generator. This type of machine offers design particularities that are addressed on the analytical construction of the optimization functions which have a simple physical meaning for a proper understanding. Rectangular slots on the stator laminations, required to allocate preformed winding, describe a distinctive function expression. Cooling problem is considered by the introduction of limited value of the tangential stress; this limit can be modified by using proper cooling system. To improve the torque capacity of the designed lamination, a correction procedure is developed to take into account the non-ideal condition establishes by the slots-teeth group and iron characteristics. Results show close agreement of the presented analytical algorithm and FEA solution.

APPENDIX A (43)

Table III lists data about the BH curve.

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ACKNOWLEDGMENT The authors would like to thank TEKES, the Finnish funding agency for Technology and Innovation, and industrial partners for their support. REFERENCES [1] H. Li and Z. Chen, “Overview of different wind generator systems and their comparisons,” IET Renew. Power Gen., vol. 2, no. 2, pp. 123–138, 2008. [2] V. Moreno and A. Pizago, “Future trends in electric propulsion systems for commercial vessels,” J. Maritime Res., vol. IV, no. 2, pp. 81–100, 2007. [3] J. Pyrhönen, T. Jokinen, and V. Hrabovcova, Design of Rotating Electrical Machines. New York: Wiley, 2008. [4] J. Turunen, “Modeling of heat transfer of permanent magnet generator in wind power application,” Master’s thesis (in Finnish), Lappeenranta Univ. Technol., Lappeenranta, Finland, 2010. [5] W. Tong, S. Wu, Z. An, H. Zhang, and R. Tang, “Cooling system design and thermal analysis of multibrid permanent magnet wind generator,” presented at the Int. Conf. Electrical and Control Engineering (ICECE), Wuhan, China, Jun. 2010, pp. 3499–3502. [6] G. Liuzzi, S. Ludici, F. Parasiliti, and M. Villani, “Multiobjective optimization techniques for the design of induction motors,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1261–1264, May 2003. [7] J. Andersson, A Survey of Multiobjective Optimization in Engineering Design Linköping University, Linköping, Sweden, Dept. Mech. Eng., Tech. Rep.: LiTH-IKP-R-1097, 2001. [8] S. L. Ho, S. Yang, G. Ni, and H. C. Wong, “A tabu method to find the pareto solution of multiobjective optimal design problems in electromagnetics,” IEEE Trans. Magn., vol. 38, no. 2, pp. 1013–1016, Mar. 2002. [9] S. D. Sudhoff, J. Cale, B. Cassimere, and M. Swinney, “Genetic algorithm based design of a permanent magnet synchronous machine,” presented at the IEEE Int. Conf. Electric Machines and Drives (IEMDC), San Antonio, TX, May 2005, pp. 1011–1019. [10] H. Hong and J. Yoo, “Shape design of the surface mounted permanent magnet in a synchronous machine,” IEEE Trans. Magn., vol. 47, no. 8, pp. 2109–2117, Aug. 2011. [11] S. Sadeghi and L. Parsa, “Multiobjective design optimization of fivephase Halbach array permanent-magnet machine,” IEEE Trans. Magn., vol. 46, no. 8, pp. 3289–3292, Aug. 2011. [12] L. dos Santos Coelho, H. Ayala, and P. Alotto, “A multiobjective Gaussian particle swarm approach applied to electromagnetic optimization,” IEEE Trans. Magn., vol. 46, no. 8, pp. 3289–3292, Aug. 2010. [13] L. J. Wu, Z. Q. Zhu, J. T. Chen, Z. P. Xia, and G. W. Jewell, “Optimal split ratio in fractional-slot interior permanent-magnet machines with non-overlapping windings,” IEEE Trans. Magn., vol. 46, no. 5, pp. 1235–1242, May 2010. [14] Y. Pang, Z. Q. Zhu, and D. Howe, “Analytical determination of optimal split ratio for permanent magnet brushless motors,” IEE Proc.-Electr. Power Appl., vol. 153, no. 1, pp. 7–13, Jan. 2006. [15] J. A. Tapia, “Development of the consequent pole permanent magnet machines,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Wisconsin, Madison, WI, 2002. [16] T. A. Lipo, Introduction to AC Machine Design, Wisconsin Power Electronics Research Center, Univ. Wisconsin, 2007.

[17] V. B. Honsinger, “Sizing equation for electrical machinery,” IEEE Trans. Energy Convers., vol. EC-2, no. 1, pp. 116–121, Mar. 1987. [18] S. Huang, J. Luo, F. Leonardi, and T. A. Lipo, “A general approach to sizing and power density equations for comparison of electrical machines,” presented at the 31st IEEE Industry Applications Conf., San Diego, CA, Oct. 6–10, 1996, pp. 836–842, vol. 2. [19] A. Ivanov-Smolensky, Electrical Machines. Moscow, Russia: MIR Publishers, 1982, vol. 1.

Juan A. Tapia received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Concepcion, Concepcion, Chile, in 1991, 1997, respectively, and the Ph.D. degree from the University of Wisconsin, Madison, in 2002. Since 1992, he has worked with the Department of Electrical Engineering, University of Concepcion, where he is currently an Associate Professor. Since 2010, he has been a FiDiPro Fellow from the Academy of Finland at Lappeenranta University of Technology where he conducts research on PM machine on LUT-Energia Group. His primary research areas are electrical machine design, numerical method for electromagnetic field, DSP-based electric machine control, and renewable energy.

Juha Pyrhönen received the D.Sc. degree from Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 1991. He became an Associate Professor of Electrical Engineering at LUT in 1993 and a Professor of Electrical Machines and Drives in 1997. He is currently the Head of the Department of Electrical Engineering, LUT, where he is engaged in research and development of electric motors and electric drives. His current interests include different synchronous machines and drives, induction motors and drives, and solid-rotor high-speed induction machines and drives.

Jussi Puranen received the M.Sc. (technology) degree in electrical engineering and the D.Sc. (technology) degree from Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 2003 and 2006, respectively. He is currently working as an electric development engineer at The Switch Drive Systems, Lappeenranta, Finland, where his main responsibility is electromagnetic design of permanent magnet wind generators.

Pia Lindh received the M.Sc. degree in energy technology and the D.Sc. degree in electrical engineering (technology) from Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 1998 and 2004, respectively. She is currently a Researcher and Teacher with the Department of Electrical Engineering, LUT. Her research work focuses on drives and motors, especially permanent magnet motors.

Sören Nyman received the M.Sc. (technology) degree in electrical and energy engineering from the University of Vaasa, Vaasa, Finland, in 2008. He has experience from industrial applications within electrical machinery. Currently, he is working in engine automation development at Wärtsilä Finland Oy, Vaasa, Finland.