Multi-objective Optimal Design of a Five-Phase Fault-Tolerant Axial Flux PM Motor.
Transportation sectors are seeking for electrical actuator systems offering compactness, reduced weight, high reliability, fault-tolerance capability and simple maintenance requirements . PMSMs offer interesting advantages including compactness, high efficiency, high torque density, high power factor or rugged designs. Therefore, PMSMs are being applied in high performance applications where variable speed is required, including servo drives, robotics, or the transportation sector . In particular, AFPMs show smaller axial length than radial-flux permanent magnet machines  while offering enhanced torque and power densities for specific applications.
During the last decade an extensive research work about fault-tolerant operation of three-phase electrical motors and drives has been carried out [4-6]. Fault-tolerance can be ensured by using multiphase, and in special five-phase motor drives, with fractional-slot single-layer concentrated windings with physical, thermal, electrical, and magnetic isolation among phases [7,8]. It has been reported that five-phase machines are intrinsically more advantageous for fault-tolerant operation than three-phase machines, since in the event of a fault, by using the remaining healthy phases, these machines can offer continue operation [9,10]. Additional advantages of multiphase motor drives include reduced amplitude and increased frequency of the torque pulsations, reduction of the stator current per phase at the same phase voltage, reduction of the dc link current harmonics or higher torque per ampere for a given volume of the machine among others .
Electric motor manufacturers have always pursued the development of optimized motors, mainly aimed to minimize manufacturing costs and materials, while maintaining overall features and performance. The automotive sector is taking advantage of these improvements, thus enhancing fuel economy in electric and hybrid vehicles and reducing greenhouse gasses emissions. However, the optimal design of electrical machines is a complex issue, since it depends of several constrained design parameters which often are not independent and present nonlinear correlations due to the nonlinear behavior of the active materials . Motor design objectives are often difficult to meet, as for example low-cost and high power density motors with high efficiency are often pursued. Therefore, the optimal design is not unique and a compromised final solution is often selected .
Optimization algorithms seek to find the optimal or satisfactory design from a broad set of feasible designs by iteratively analyzing an objective function. Optimization algorithms often analyze the possible values of the design variables, which must be confined within the feasible design space. To this end, the design variables are usually constrained within reasonable values to ensure the achievement of an optimal solution . Optimization algorithms are classified into classical or deterministic and stochastic methods. Whereas the former ones use specific rules to move from one solution to the next one, stochastic algorithms apply probabilistic transition rules, thus usually being faster in locating the final solution . Among stochastic algorithms, genetic algorithms (GA) are applied in a wide spectrum of applications, since GAs are recognized as unbiased optimization methods to sample a large space of solutions .
This paper deals with the optimal design of a five-phase fault-tolerant axial-flux PMSM which has to provide the tractive force of an electric scooter. As initial design constraints, it has a NN configuration and non-overlapping fractional-slot concentrated windings. For optimization purpose a multi-objective design strategy is applied, in which the variables to be optimized are the motor efficiency and power density and ten input geometric and electric parameters are considered, with their respective bounds and constraints. As explained, this is a challenging task, due to the large sets of possible solutions and the non-linear behavior of the electric motor. The motor is sized by applying an iterative method based on a set of analytical equations, which allow computing the two output variables from the input parameters values. Next, by applying a multi-objective genetic algorithm (MoGA) approach, the input parameters are modified to find out the Pareto front of the broad set of solutions explored, and from it the best solution attained is identified based on the maximum Euclidean distance to the origin point. To increase the accuracy of the AFPM design, a further refinement loop is applied, in which the air gap flux density of the final solution attained by means of the optimization process based on the analytical equations is substituted with the air gap flux density obtained through 3D-FEM simulations of the same motor. At the last, the final solution obtained through the applied procedure is validated by means of a 3D-FEM analysis. For this purpose, the values of four relevant parameters which define the machine performance, obtained through the proposed optimization process and the 3D-FEM model are compared. It is worth noting that this paper combines the optimal design of a five-phase AFPM by means of a hybrid system combining analytical equations and 3D-FEM simulations, which has been scarcely analyzed in the technical literature, with special focus on the design of fault-tolerant motors, being these the main novelties.
II. THE ANALYZED MACHINE
In this paper an AFPM synchronous machine is selected to be optimally designed. This kind of actuator is known to be very convenient for some specific applications such as direct drive in-wheel applications  due to the high power and torque densities . It is worth noting that direct drive machines simplify the vehicle mechanical structure, thus the overall vehicle efficiency and weight are minimized. The studied machine has a torus geometry containing an internal stator and two outer rotor disks, which include the magnets. The rotor disks present a NN configuration, i.e. a north pole N facing another north pole N, which is placed at the other side of the stator . Fig. 1 shows the AFPM dealt with in this paper.
The stator core of the AFPM is made of strip wound steel and includes concentrated back-to-back connected windings, which are wounded inside the slots in the radial direction . Back-to-back windings allow enhancing the overall machine efficiency, since they offer reduced end windings length, thus presenting lower copper losses . In addition, both rotors contain permanent magnets with trapezoidal geometry to minimize the torque ripple .
Non-overlapping windings provide minimum mutual inductance between phases, thus minimizing interactions between the faulty phase and the others, so they are highly recommended in fault-tolerant machines. In addition, the design should ensure low mutual coupling among phases to limit the effect of the short circuit in one phase on other phases. The use of concentrated stator windings (i.e. windings encircling a single stator tooth, thus eliminating end-winding overlap among phase windings) offer several advantages compared to distributed windings, including a reduction of the copper volume of the end windings, which is especially significant when the axial length of motor is small, thus minimizing copper losses and improving the motor efficiency , compared to stator windings with integer number of slots per pole and per phase. They also allow reducing the total length of the machine and manufacturing costs since concentrated windings are easier to realize. Finally, an important advantage is that compared to distributed windings, concentrated windings tend to provide higher inductance when the magnetic flux linkage is the same, which allows extending the flux-weakening region . Since in this particular design the fault-tolerance is a must, a concentrated non-overlapping fractional-slot single-layer winding has been selected, as shown in Fig. 2, since it provides enough magnetic and electric insulation to avoid a major propagation of a short circuit in the event of a short circuit fault.
III. SIZING EQUATIONS OF THE FIVE PHASE MACHINE
This section develops the sizing equations to design the five-phase AFPM as described in ,
The mechanical and electrical speeds in rad/s are, respectively,
[mathematical expression not reproducible] (1)
[mathematical expression not reproducible] (2)
f and p being, respectively, the electrical frequency and the number of pole pairs.
The torque can be obtained from the output mechanical power as,
The number of slots per pole and per phase is,
Q being the number of slots and m the number of phases. The coil-pole fraction is defined as,
[a.sub.cp] =2p/Q (5)
The angular pole and slot pitches are as follows,
[[theta].sub.p] = [pi]/p (6)
The slot pitch in electrical radians is,
[[theta].sub.se] =2[pi]*p/Q (8)
The inside and outside pole pitches are,
Ri and Ro being, respectively, the inner and outer rotor radius. The inside and outside coil pitches are, respectively, [mathematical expression not reproducible] (11)
[mathematical expression not reproducible] (12)
And the inner slot pitch is calculated as,
[mathematical expression not reproducible] (13)
The distribution factor required to calculate the winding factor is as follows,
[mathematical expression not reproducible] (14)
When q is a fractional number less than 1, the pitch factor kp can be defined as the ratio between the vectorial and the arithmetic sum of the EMFs per coil side ,
k =sin([pi]y/Q) (15)
In addition the skew factor is,
k =1-[[theta].sub.m] /(2[pi]) = 1-p/Q (16)
And the magnet fraction is,
where [[tau].sub.m] and [[tau].sub.f] are, respectively, the permanent magnet and spacer widths shown in Fig. 3.
The permanent magnet leakage factor is,
[mathematical expression not reproducible] (18)
where lpm is the permanent magnet length, [mu]R is the relative permeability of the magnet and g the airgap length. The effective air gap to calculate the Carter coefficient is, [g.sub.c]=2g+[l.sub.pm] /[u.sub.R] (19)
And the Carter coefficient is calculated as,
[mathematical expression not reproducible] (20)
where ws is the slot opening, as show in Fig. 3. The air gap area is as follows,
[mathematical expression not reproducible] (21)
The average value of the air gap flux density is,
[mathematical expression not reproducible] (22)
Br being the remanence of the permanent magnets and Ce and Pc, respectively, the flux concentration factor and the permeance coefficient,
[mathematical expression not reproducible] (23)
The air gap flux is calculated as, The stator back iron width is, [k.sub.st] being the lamination stacking factor.
The stator tooth width calculated at the inner radius is,
[mathematical expression not reproducible] (26)
Bmax being the maximum allowable flux density in the stator yoke.
The slot bottom width is,
[w.sub.sb] =[[tau].sub.si]-[w.sub.tbi] (27)
The slot aspect ratio at the bottom width,
The shoe depth split between d1 and d2,
[d.sub.1] = [a.sub.sd] [w.sub.tbi] - [d.sub.2] (29)
asd being the shoe depth fraction, which is defined as,
[mathematical expression not reproducible] (30)
The number of turns per slot is,
[mathematical expression not reproducible] (31)
The peak value of the back-EMF,
[mathematical expression not reproducible] (32)
The peak value of the slot current,
[mathematical expression not reproducible] (33)
The peak value of the phase current is,
The conductor slot depth is obtained from the area required to fit the conductors according to the allowable current density,
[d.sub.1] = [I.sub.s]/([k.sub.cp][w.sub.sb][J.sub.max]) (35)
Assuring rectangular slots, the slot area is,
[A.sub.s] =[w.sub.ab][d.sub.3] (36)
The conductor current density is
[J.sub.c] = [I.sub.s]/([k.sub.cp] [A.sub.s]) (37)
where [k.sub.cp] is the slot fill factor, typically less than 60% (set to 50%) and [A.sub.load] is the stator electrical loading, which can be calculated as,
[mathematical expression not reproducible] (38)
According to Fig. 3, the stator tooth height is,
[d.sub.s] = [d.sub.1] + [d.sub.2] + [d.sub.3] (39)
AFPMs present protrusions in both the axial and radial directions, whose magnitude depends on the stator electrical loading [A.sub.load] current density [J.sub.c], and copper fill factor [k.sub.cp].
The inner protrusions along the axial length can be calculated as ,
[mathematical expression not reproducible] (40)
The effective stack or axial length of the AFPM depends on the rotor and stator axial lengths as,
[l.sub.t]=[l.sub.s] + 2[l.sub.s] + 2g (41)
From manufacturer's experience, the effective axial length of the stator is related to the stator core axial length and the amplitude of the inner protrusions Wcui as,
[l.sub.s] = [l.sub.sc] + [1.6W.sub.cu] (42)
The minimum value of the stator core axial length also may be calculated from the flux densities in the air gap [B.sub.g] and in the stator core Bsc and the geometric ratio [lambda] = R/[R.sub.o] as,
[l.sub.sc,min] = [pi][B.sub.g][R.sub.o](1 + [lambda])/([2B.sub.sc]p) (43)
This minimum value of [l.sub.sc ,min] is imposed to avoid a possible saturation and extra losses in the stator laminations. The new stator axial length is then computed from the total slot depth and the stator yoke width as,
[1.sub.sc] = 2[d.sub.s] = [W.sub.bi] [greater tham equal to] [l.sub.sc, min] (44)
Since the value of [l.sub.sc] calculated in (44) must be greater or equal than that in (43) to avoid stator laminations saturation, when [l.sub.sc] < [l.sub.sc, min] the value of wbi must be increased to satisfy (44), so it starts a new iteration from (25). The axial length of the rotor lr is calculated from the axial length of the rotor core lrc and the length of the permanent magnets lpm as,
[l.sub.r] = [l.sub.rc] + [1.sub.pm] (45)
The axial length of the rotor core lrc may be expressed as,
[mathematical expression not reproducible] (46)
Brc being the flux density in the rotor core. It is worth noting that Brc is almost stationary since it is mostly due to the permanent magnets placed in the rotor discs, whereas the ac magnetic flux mainly flows through the stator core.
The peak flux density in the slots is,
The slot resistance is,
[mathematical expression not reproducible] (48)
And the end-turn resistance is calculates as,
[mathematical expression not reproducible] (49)
The phase resistance is obtained from the slot and end-turn resistances as,
The air gap inductance,
[mathematical expression not reproducible] (51)
The slot leakage inductance is,
[mathematical expression not reproducible] (52)
and the end-turn inductance,
[mathematical expression not reproducible] (53)
Finally, the total phase inductance is,
[L.sub.ph] = [2N.sub.sp]([L.sub.s] + [L.sub.g] + [L.sub.e]) (54)
Both the slot opening ws and the slot depth d3 have a great impact on the leakage inductance, since when the first one lowers and/or the last one rises, the slot leakage inductance Ls in (52) increases. Under fault conditions, a high inductance limits the rate of change of the current while increasing the amount of time available to detect such a fault. However, as a high inductance increases the electric time constant, it makes the motor more difficult to drive. As a result, a tradeoff exists.
The Joule losses in the stator conductors are,
[mathematical expression not reproducible] (55)
The total core losses are,
[P.sub.cl] = [[rho].sub.bi][V.sub.st][p.sub.spec] (56)
[[rho].sub.bi]being the lamination mass density, pspec the steel laminations specific loss (see Fig. 4), and [V.sub.st] the volume of the stator steel laminations, which is calculated as,
[mathematical expression not reproducible] (57)
[k.sub.st] is the stacking factor of the magnetic core.
The overall efficiency is,
[mathematical expression not reproducible] (58)
where [P.sub.sl] are the stray losses, which are mainly composed of friction, windage, and other minor components. This paper assumes [P.sub.s] as 1% of the output power .
Finally, the power density is calculated from the mechanical output power Pout and the AFPM volume as,
[mathematical expression not reproducible] (59)
IV. SPECIFICATIONS FOR MAXIMIZING FAULT TOLERANCE
As explained, multiphase machines are well suited for fault-tolerant applications. However, to further improve fault-tolerant capability, other requirements must be fulfilled, including magnetic, electrical, physical and thermal isolation among phases  to minimize the possibility of phase-to-phase faults occurrence. This condition may be achieved by using non-overlapping single-layer fractional-slot concentrated windings around each tooth , because in this configuration the phase windings are arranged in independent modules. Whereas magnetic isolation minimizes the voltages induced in adjacent phases due to a fault current in the faulty phase, physical and thermal isolation allows reducing the risk of occurrence of faults between nearby phases. Finally, to ensure electrical isolation among phases it is also highly desirable to connect each phase to a distinct single-phase full-bridge PWM converter since each power switch has to withstand the phase voltage instead of the line voltage in star-connected systems .
To ensure a high level of fault-tolerance capability in the analyzed five-phase AFPM some restrictions are imposed as follows ,
1. It is suggested to use non-overlapping fractional-slot single -layer concentrated windings. These windings tend to present a low winding factor. In addition, both the MMF and EMF harmonic content in single-layer windings is higher than in double-layer windings, so torque pulsations are increased as well acoustic noise emissions. To minimize the abovementioned problems the number of pole pairs p and the number of stator slots Q must be similar but not equal. This condition is attained when Q [+ or -] 2 = 2p.
2. To increase the phase inductances and reduce the short circuit currents, the slots in the stator should be deep enough . The phase impedance calculated as Zph = [Emax]/[I.sub.s] must be close to 1 p.u., i.e.
fb being the base value of the electrical frequency and Lb the base value of the phase inductance, which is calculated as,
[L.sub.b]=[w.sub.b] /[I.sub.N] (61)
The peak base value of the flux linkage [W.sub.b] is,
[W.sub.b]=[E.sub.max] /[[omega].sub.b] (62)
[E.sub.max] being the peak value of the back-EMF, and [[omega].sub.b] the base electrical angular frequency, defined as [[omega].sub.b] =2nfb. Both the air gap and slot impedances Lg and Ls in (51) and (52), respectively, are adjusted during the optimization process, in order to force the value of Lb to converge to Lph. Therefore if Lb differs from Lph more than a specified tolerance, the value of d3 is changed accordingly to fulfill Lb [approximately equal to] Lph, thus starting a new iteration from (35).
V. THE MULTI-OBJECTIVE OPTIMIZATION PROCESS
Multi-objective optimization problems involve optimizing simultaneously more than one objective function, where often trade-offs between conflicting objectives must be taken. Electric motors are known to be complex and non-linear systems but in most optimal motor designs, a single-objective function is selected . Typical objectives are minimum cost, highest efficiency or minimum weight. An electric machine is usually defined by a rather large number of parameters, including geometric dimensions, material characteristics, winding configurations  and some constrains. Therefore, to achieve an optimal design of the AFPM it seems reasonable to apply a multi-objective approach based on a genetic algorithm (GA) since GAs are mathematical methods well-suited for solving constrained and unconstrained single- or multi-objective optimization problems. These algorithms have been successfully applied in problems dealing with highly nonlinear, non-differentiable or stochastic objective functions  and specifically in the optimal design process of electric machines [29-31].
GAs are based on the concept of natural selection through the survival of the most suitable individuals, so that the individuals producing a least fitted solution have a small probability of reproduction whereas those producing the most fitted solutions have a greater chance . The iterative heuristic search produced by the GA randomly selects a set of individuals from the current population at each step, which represents the chromosomes and after some generations the solution tends to an optimum. The specimens which produce a local optimal are the solution selected at each step and the parents of the offspring individuals for the next generation. By repeating this process, a population of individuals which evolves toward an optimal solution is produced.
Genetic algorithms perform mathematical operations to imitate genetic reproduction mechanisms including individual's selection (parents for the next generation), crossover operation (combines two parents' solutions to produce children solutions for the subsequent generation), and mutation (applies random changes to parents and children solutions) with the aim of obtaining an optimal solution.
Due to the quite large number of parameters involved in the design process of an electric machine, to limit the complexity and computational burden of the optimization process it is required to identify a reduced set of parameters which most significantly affects the performance of the analyzed machine . This paper deals with ten parameters whose values are changed during the optimization process, which are shown in Table II. These parameters are coded into binary strings forming the chromosomes. The elitist scattered crossover operation is applied, which at each generation produces a random binary vector from which the genes of both parents are combined to form the offspring, as indicated in Fig. 5. Next, the mutation process is applied, by which the GA creates small random variations in each bit of the chromosomes, providing genetic diversity and enabling GA to search for wider spaces. The whole process is repeated until an established tolerance criterion is achieved; this could be either the ratio of change between consecutive solutions or the maximum number of generations which is previously settled.
VI. MODEL REFINEMENT THROUGH 3D FEM ASSISTANCE
Therefore, Once the an optimal solution has been obtained by means of the optimization method based on the analytical equations explained in sections 3-5 (step 1 in Fig. 6), and in order to improve the accuracy of the final design attained, a further refinement is carried out with the help of 3D-FEM simulations of the optimized motor (step 2 in Fig. 6). However, it is well known that 3D-FEM simulations have a high computational burden, so the number of simulations must be minimized.
Therefore, a 3D-FEM model of the AFPM is generated from the solution attained in step 1. Since the air gap flux density [B.sub.g] is one of the major design parameters in determining the AFPM performance, the value of [B.sub.g] obtained in step 1 is compared with the one obtained by means of the 3D-FEM model. In the case of discrepancy, a next iteration is started by adjusting the [B.sub.g] value in (22) to that obtained in the FEM simulation, therefore calculating a new solution of the AFPM design. This process is iterated until the error between the [B.sub.g] value obtained by means of the optimization process based on the analytical equations and the [B.sub.g] value obtained through the 3D-FEM simulations is less than a specified tolerance. However, even attaining the pre-established [B.sub.g] value, it does not ensure to fulfill the design specifications such as output power and torque, or power density among others, so a final validation of the obtained design is highly desirable (see Section VIII). It is worth noting that with only three iterations of this last loop, the convergence was attained.
This section shows the results attained by applying the MoGA with the fault-tolerance constraints.
In this paper there are some fixed parameters whose values don't change during the optimization process, which are detailed in Table I. They include the desired output power, base speed, number of phases, pole pairs, number of slots, and desired flux density in the stator and rotor cores, which greatly determine the AFPM size and performance, as well as the main properties of the materials dealt with.
The input or design variables, that is, the variables whose values are changed during the optimization phase to explore the whole set of possible solutions, are shown in Table II. The number of design variables is usually a tradeoff between model accuracy and complexity due to the large number of possible combinations to be analyzed during the optimization process. Therefore, this paper deals with the ten most influencing design variables.
The multi-objective optimization process considers two objective functions, i.e. the AFPM power density and efficiency. The main parameters of the MoGA include a population size of 1400 individuals, the maximum number of generations is 200, the probability of crossover is 0.8 and the probability of mutation is 0.01.
Fig. 7 shows all solutions explored by the MoGA algorithm and the Pareto front. From all this set of solutions, the one maximizing the Euclidean distance to the origin point is selected as the best solution.
Table III summarizes some of the main parameters of the best solution attained by means of the MoGA approach.
VIII. MODEL VALIDATION THROUGH FEM
As it is well known, although radial-flux machines are often simulated by means of two-dimensional finite elements analysis, an axial flux machine must be modeled by means of three-dimensional finite elements methods (3D-FEM) due to its inherent three-dimensional geometry. Therefore, an accurate three-dimensional FEM model of the analyzed AFPM with around 1.4*10 (6) tetrahedral volumetric elements was carefully prepared in the Flux-Cedrat[R] environment. This model includes all geometrical and physical characteristics of the machine components, including the electric circuit. Using this modeling system it is possible to obtain diverse type of electric and magnetic quantities of the AFPM obtained through the optimization process with very high accuracy, thus allowing to verify the excellent motor performance obtained by means of the method applied in this paper. The motor was supplied by means of five balanced voltage sources (72[degrees] phase shift between two consecutive phase voltages).
Fig. 8 plots the 3D-FEM model along with the permanent magnets and a partial view of the 3D-mesh applied in the FEM simulations.
Fig. 9 shows the magnetic flux distribution within the AFPM analyzed where it is proved that the flux density in the rotor laminations is almost stationary.
Fig. 10 shows the output torque, back-EMF and phase inductance obtained through 3D-FEM simulations
In this paper a five-phase fault-tolerant axial-flux permanent magnet synchronous motor has been designed by applying a multi-objective optimization approach based on a genetic algorithm. The design process has been assisted by means of a hybrid method combining a set of analytical equations and 3D-FEM simulations, taking care of reducing the demanding computational burden due to an intensive use of 3D-FEM simulations. Some fault-tolerance requisites have been considered during the optimization process to ensure further fault-tolerance capability. Two objective functions (the power density and efficiency of the machine) are considered in the multi-objective optimization process. The accuracy of the sizing method has been validated by means of three-dimensional finite element simulations, which show great accuracy when comparing four relevant parameters that define the machine performance. These results show the usefulness and feasibleness of the applied method to design electric motors with enhanced fault-tolerance capability.
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Harold SAAVEDRA (1), Jordi-Roger RIBA (2), Luis ROMERAL (1)
(1) Universitat Politecnica de Catalunya, Electronic Engineering Department, Terrassa, 08222, Spain (2)Universitat Politecnica de Catalunya, Electric Engineering Department, Igualada, 08700, Spain email@example.com
This work was supported in part by the Spanish Ministry of Science and Technology under the TRA2013-46757-R research Project
Digital Object Identifier 10.4316/AECE.2015.01010
TABLE I. FIXED PARAMETERS AND REQUIREMENTS OF THE FIVE-PHASE AFPM Quantity Symbol Magnitude Output power [P.sub.out] 2000 W Base speed N 1100 rpm Output torque [T.sub.out] 17.36 Nm Number of phases m 5 Base electrical frequency f 165 Hz Pole pairs p 9 Number of slots Q 20 Laminations width [S.sub.lw] 0.35mm Saturation flux density [B.sub.sat] 1.5 T Slot fill factor [k.sub.cp] 0.5 Laminations type M330-35A - Lamination stacking factor [k.sub.st] 0.95 Steel laminations mass density [[rho].sub.steel] 7600 kg/m3 Wire insulation thickness [W.sub.i] 0.5-2 mm Conductors material Copper - Conductors conductivity [[rho].sub.cu] 16.78 n[OMEGA]*m at 25[degrees]C Permanent magnets (PM) [[rho].sub.m] 7400 kg/[m.sub.3] mass density PM relative permeability [M.sub.r] 1.05 PM remanent flux density [B.sub.r] 1.3 T Stator core flux density [B.sub.sc] 1.8 (peak value) Rotor core flux density [B.sub.rc] 1.7 (peak value) TABLE II. DESIGN VARIABLES OF THE FIVE-PHASE AFPM Quantity Symbol Magnitude Machine inner radius x(1)= Ri 0.03-0.1 m Machine outer radius x(2)= Ro 0.06-0.2 m Magnet spacer width x(3)= [tau]j 1-5 mm Magnet thickness x(4)= lpm 2-9 mm Air gap length x(5)= g 0.5-1.5 mm Slot tip length x(6)= d1 1-4 mm Slot tip shoe length x(7)= d2 1-2 mm Slot width/opening x(8)= ws 2-10 mm Peak value of the back-EMF x(8)= Emax 24-60 V Maximum conductor current density x(10)= Jmax 2-5 A/mm2 TABLE III. RESULTS ATTAINED BY APPLYING FAULT-TOLERANCE RESTRICTIONS Quantity Symbol Magnitude Power Density [P.sub.den] 72931 W/[m.sup.3] Efficiency [eta] 94.75% Rotor outer radius [R.sub.o] 0.10 m Rotor inner radius [R.sub.i] 50 mm Number of turns per phase [N.sub.ph] 52 Air gap length g 0.9 mm Axial length of machine [l.sub.t] 87 mm Magnet thickness [l.sub.pm] 4 mm Rotor thickness [l.sub.cr] 15.2 mm Stator thickness [l.sub.cs] 48.1 mm Phase inductance [L.sub.ph] 1.1 mH Phase resistance [R.sub.ph] 0.007 [OMEGA] Air gap flux density (peak value) [B.sub.g] 0.689 T Current per phase (peak value) [I.sub.ph] 18.5 A EMF per phase (peak value) [E.sub.max] 21.59 V TABLE IV. RESULTS ATTAINED Quantity Optimization 3D-FEM method [B.sub.g], air gap flux density (peak value) 0.689 T 0.711T [T.sub.out] , output torque 17.36 Nm 15.70 Nm [E.sub.max], back-EMF (peak value) 21.59 V 22.04 V [L.sub.ph] , phase inductance 1.10 mH 1.14 mH Quantity Relative error [B.sub.g], air gap flux density (peak value) 3.19% [T.sub.out] , output torque 9.56% [E.sub.max], back-EMF (peak value) 3.75% [L.sub.ph] , phase inductance 3.64%
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|Author:||Saavedra, Harold; Riba, Jordi-Roger; Romeral, Luis|
|Publication:||Advances in Electrical and Computer Engineering|
|Date:||Feb 1, 2015|
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