Optimal design of slotless tubular linear brushless PM machines using metaheuristic optimization techniques

Abstract

In this investigation, an optimal design procedure for tubular linear brushless permanent magnet (PM) machines has been proposed. A two-dimensional analytical magnetic field model in cylindrical coordinates has been employed to quickly and accurately calculate the magnetic flux density distribution and electromagnetic force. The objectives of the optimization problem are to simultaneously minimize the power losses and machine volume and the constraints are to ensure to develop the required level of electromagnetic force and to avoid saturation in the back-iron. The optimization problem has been solved using two types of metaheuristic optimization techniques, i.e., particle swarm optimization (PSO), and fuzzy adaptive PSO (FAPSO). The design procedure has been implemented on a case study to illustrate the capability of the proposed approach.

Keywords

Analytic solution electromagnetic force linear machines metaheuristic optimization two-dimensional model

1 Introduction

Although the rotational movement of rotary electric motors can be converted to the translational movement using transformational devices such as gear, ball screw, cram and crank shaft, linear electric motors, which directly provide the translational movement, are preferred due to their higher efficiency, higher reliability, more agility, less maintenance, and higher power density compared to the rotary electric motors combined with the transformational devices. Linear electric machines are categorized in terms of their geometry as flat (or planar) and tubular (or cylindrical). Tubular linear electric machines have some advantages over their flat counterparts such as the absence of end-windings which results in lower ohmic losses, higher force density, and almost no radial forces. Brushless PM tubular machines are classified in terms of the relative position of the armature and PM as: inner PM (Fig. 1a) and outer PM (Fig. 1b); in terms of the armature structure as: slotless and slotted; in terms of the magnetization direction of PM as: radial, axial, Halbach and quasi-Halbach magnetization patterns; in terms of relative axial lengths of the armature and PM as: equal length, shorter armature and longer armature; in terms of core materials as: iron-cored armature and PM, air-cored armature and PM, air-cored armature and iron-cored PM and iron-cored armature and air-cored PM. In this study, the aim is to optimally design an inner PM tubular machine with slotless stator structure, equipped with radial magnetization magnets, having the armature and slider with equal length, and iron-cored armature and PM.

Both open-circuit and armature reaction magnetic field distributions have been analytically calculated for tubular PM machines with three types of PM magnetizations: radial, axial and Halbach [1]. The axial lengths of the stationary and movable parts are assumed to be infinite and therefore the fringing effects have been neglected. The formulation is for slotless armature structures; however a Carter coefficient has been used to incorporate the slotting effect. Based on the analytical magnetic field calculations, the expressions for the thrust force, winding emf and self- and mutual-inductances have been presented.

The open-circuit magnetic field distribution of a tubular slotless linear motor with surface mounted PM under radial magnetization has been investigated [2]. Both open-circuit and armature reaction magnetic field distributions have been analytically calculated for tubular PM machines with three types of PM magnetizations: radial, axial and Halbach [1]. The axial lengths of the stationary and movable parts are assumed to be infinite and therefore the fringing effects have been neglected. The formulation is for slotless armature structures; however a Carter coefficient has been used to incorporate the slotting effect. Based on the analytical magnetic field calculations, the expressions for the thrust force, winding emf and self- and mutual-inductances have beenpresented. The open-circuit magnetic field distribution of a tubular slotless linear motor with surface mounted PM under radial magnetization has been investigated [2]. To analytically solve the problem, the magnetization has been slightly modified without having significant effect on the accuracy. Wang et al. [3] have used the analytical solutions proposed in [1] for optimal design of tubular PM machines. The thrust force and force ripple have been established as functions of motor geometric parameters to investigate their effects. The fringing effect associated with the finite armature length of slotless tubular PM machines having radial or Halbach magnetization pattern has analytically been investigated [4, 5]. Armature reaction magnetic field calculations have been reported in [6] for tubular machines equipped with axially magnetized magnet. The slotting effects have been accounted for using a Carter coefficient and the fringing effects have been neglected. Based on the analytical solution, the machine inductance and reluctance force have been predicted. Optimal design of radially magnetized, iron-cored, tubular PM machines and drive systems has been investigated in [7]. To this end, the analytical expressions for the calculations of the thrust force, emf, iron losses, winding inductances and resistance and the converter losses have been presented. Lu et al. [8] have presented the development of a slotless tubular PM micromotor with axially magnetized magnet. Both analytical and numerical methods have been used to establish important design criteria.

The analytical open-circuit and armature reaction magnetic field calculations have been reported for 3-phase tubular modular PM machines having quasi-Halbach magnets [9, 10]. The analytical solution of the magnetic field problem has been used to derive the expressions for thrust force, emf and inductances. Wang et al. [11] presented a design procedure for a tubular linear PM generator equipped with a modular stator winding and a quasi-Halbach magnetization for application in a free-piston energy convertor. Ibrahim et al. [12] described the analysis of a short-stroke single-phase quasi Halbach tubular PM motor with nonmagnetic support tube. Gysen et al. [13, 14] presented the analytical and numerical methods to compute the magnetic field distribution of tubular PM actuators. Meessen et al. [15] investigated the effects of PM shapes in slotless tubular machines with Halbach array. Gysen et al. [16] proposed a semi-analytical magnetic field calculation due to armature reaction in tubular PM actuators with rectangular slots. Boroujeni et al. [17] presented a new tubular PM machine with square-shaped core section and its analytical magnetic model. Prudell et al. [18] presented a novel tubular PM generator buoy system to convert the linear motion of ocean waves into electrical energy. An exact 2-D analytical magnetic field calculation has been presented for tubular PM machines with slotted structures [19]. An optimal design of short-stroke single-phase tubular PM motor has been presented with the application to drive a reciprocating vapor compressor [20]. A 3-D analytical model based on Fourier analysis has been proposed to calculate the magnetic field distribution of tubular PM machines with skewed PMs [21]. Amara et al. [22] presented a 2-D analytical model to predict the armature reaction magnetic field of tubular PM machine with surface-inset magnet. Direct driven tubular PM generators with air-cored stator and slider have been proposed for wave energy conversion [23]. Huang et al. [24] proposed a tubular PM generator in which both the magnets and armature windings are located in the primary. The generator is employed for the sea-wave energy conversion. Yan et al. [25] presented an analytical model to compute the armature reaction field and inductances of a coreless tubular PM machine with dual Halbach array. Souissi et al. [26] presented the modeling and sizing approach for tubular PM machines based on magnetic equivalent circuit. The combinations of the particle swarm optimization technique with other artificial intelligence approaches are widely reported in the literature, e.g. [27 –30].

The remainder of the paper is organized as follows: in Section 2 the analytical model of tubular PM machine has been presented. Section 3 dedicated to the formulation of the optimization problem. Metaheuristic optimization techniques are briefly explained in Section 4. The results and conclusions are presented in Sections 5 and 6, respectively.

2 Tubular brushless PM machine model

Figure 2 shows the illustrative representation of a slotless tubular PM machine. Based on the magnetostatic assumption, the time variation of electric fields is negligible and therefore Maxwell’s equations can be expressed as follows if the eddy current reaction field is also neglected: $\nabla \times H = J$ (1) $\nabla \cdot B = 0$ (2) where H , B and J are, respectively, the magnetic field intensity, magnetic flux density and armature current density vectors. In linear media with magnetization B = μ₀μ_r H + μ₀ M where μ₀ is the free space permeability, μ_r is the relative permeability and M is the magnetization vector. Using the above mentioned relations and B = ∇ × A the following expression is derived: $\nabla^{2} A = - μ_{0} μ_{r} J - μ_{0} \nabla \times M$ (3) where A is the magnetic vector potential. Due to the infinite permeability assumption of the stator and slider yoke, the slotless tubular PM machine consists of three subdomains: magnets, air-gap, winding with the following Poisson’s and Laplace’s equations: $\nabla^{2} A^{w} = - μ_{0} J$ (4) $\nabla^{2} A^{m} = - μ_{0} \nabla \times M$ (5) $\nabla^{2} A^{a} = 0$ (6) where superscripts w, m and a indicate winding, magnet and air-gap regions.

In the case of 2D problems in cylindrical coordinates, it is assumed that A = [0, A_θ (r, z) , 0], i.e. the vector magnetic potential has only tangential component; for radial magnetization magnets, the magnetization vector has only radial component, i.e. M = [M_r (z) , 0, 0], also the current density has only tangential component J = [0, J_θ (z, t) , 0]. Besides, from B = ∇ × A , the following relations are obtained:

$\begin{matrix} B_{r} (r, z) & = & - \frac{\partial A_{θ}}{\partial z} \\ B_{z} (r, z) & = & \frac{1}{r} \frac{\partial {rA}_{θ}}{\partial r} \end{matrix}$ (7)

Hence, in the case of 2D field analysis, (4–6) are expressed as $\frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial {rA}_{θ}^{w}}{\partial r}) + \frac{\partial^{2} A_{θ}^{w}}{\partial z^{2}} = - μ_{0} J_{θ}$ (8) $\frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial {rA}_{θ}^{m}}{\partial r}) + \frac{\partial^{2} A_{θ}^{m}}{\partial z^{2}} = - μ_{0} \frac{\partial M_{r}}{\partial z}$ (9) $\frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial {rA}_{θ}^{a}}{\partial r}) + \frac{\partial^{2} A_{θ}^{a}}{\partial z^{2}} = 0$ (10)

For the radial magnetization pattern, the Fourier series of the radial component of the magnetization vector is expressed as follows $M_{r} (z) = \sum_{n = 1, 3, 5}^{\infty} 4 \frac{B_{rem}}{μ_{0}} \frac{sin n \frac{π}{2} α_{p}}{n π} cos (n π z / τ_{p})$ (11) where α_p = τ_m/τ_p is the magnet length (τ_m) to pole pitch (τ_p) ratio. Also the current density distribution can be represented by the following Fourier series expansion:

$\begin{matrix} J_{θ} (t, z) & = & \sum_{n = 1, 3, 5}^{\infty} \frac{4 J_{0} (t)}{n π} sin (n \frac{π}{2} α_{wp}) sin (n \frac{π}{2} α_{w}) \\ \times sin (n π z / τ_{p}) \end{matrix}$ (12) where α_wp = τ_wp/τ_p is the coil pitch (τ_wp) to pole pitch (τ_p) ratio and α_w = τ_w/τ_p is the coil side span (τ_w) to pole pitch (τ_p) ratio as shown in Fig. 3.

The perpendicular component of the magnetic flux density vector (B_⊥) is continuous at the interface between two adjacent media. If the interface is source-free, then the parallel component of the magnetic field intensity vector (H_||) on one side of the boundary is equal to that of the other side. Therefore the boundary conditions are as follows: ${H_{z}^{w} |}_{r = R_{s}} = 0$ (13) ${H_{z}^{m} |}_{r = R_{r}} = 0$ (14) ${B_{r}^{w} |}_{r = R_{w}} = {B_{r}^{a} |}_{r = R_{w}}$ (15) ${H_{z}^{w} |}_{r = R_{w}} = {H_{z}^{a} |}_{r = R_{w}}$ (16) ${B_{r}^{a} |}_{r = R_{m}} = {B_{r}^{m} |}_{r = R_{m}}$ (17) ${H_{z}^{a} |}_{r = R_{m}} = {H_{z}^{m} |}_{r = R_{m}}$ (18)

The general solutions of the partial differential equations in Equations (8–10) due to only the magnets can be written as follows: $\begin{matrix} A_{θ}^{w} (r, z) & = & \sum_{n = 1, 3, 5}^{\infty} [a_{n}^{w} {BI}_{1} (m_{n} r) + b_{n}^{w} {BK}_{1} (m_{n} r)] \\ \times sin (m_{n} z) \end{matrix}$ (19) $\begin{matrix} A_{θ}^{a} (r, z) & = & \sum_{n = 1, 3, 5}^{\infty} [a_{n}^{a} {BI}_{1} (m_{n} r) + b_{n}^{a} {BK}_{1} (m_{n} r)] \\ \times sin (m_{n} z) \end{matrix}$ (20) $\begin{matrix} A_{θ}^{m} (r, z) & = & \sum_{n = 1, 3, 5}^{\infty} {a_{n}^{m} {BI}_{1} (m_{n} r) + b_{n}^{m} {BK}_{1} (m_{n} r) \\ + \frac{1}{2} \frac{π {StruveL}_{1} (m_{n} r)}{m_{n}^{2}} P_{n}} sin (m_{n} z) \end{matrix}$ (21) where m_n = nπ/ - τ_p and P_n = (4B_rem/ - τ_p) sin(nπα_p/2). Based on the general solution of magnetic vector potential in Equations (19–21) and using Equation (7) the radial and axial components of the magnetic flux density vector in the three regions are obtained: $\begin{matrix} B_{r, PM}^{w} (r, z) & = & - \sum_{n = 1, 3, 5}^{\infty} m_{n} [a_{n}^{w} {BI}_{1} (m_{n} r) \\ + b_{n}^{w} {BK}_{1} (m_{n} r)] \times cos (m_{n} z) \end{matrix}$ (22) $\begin{matrix} B_{z, PM}^{w} (r, z) & = & \sum_{n = 1, 3, 5}^{\infty} m_{n} [a_{n}^{w} {BI}_{0} (m_{n} r) \\ - b_{n}^{w} {BK}_{0} (m_{n} r)] \times sin (m_{n} z) \end{matrix}$ (23) $\begin{matrix} B_{r, PM}^{a} (r, z) & = & - \sum_{n = 1, 3, 5}^{\infty} m_{n} [a_{n}^{a} {BI}_{1} (m_{n} r) \\ + b_{n}^{a} {BK}_{1} (m_{n} r)] \times cos (m_{n} z) \end{matrix}$ (24) $\begin{matrix} B_{z, PM}^{a} (r, z) & = & \sum_{n = 1, 3, 5}^{\infty} m_{n} [a_{n}^{a} {BI}_{0} (m_{n} r) \\ - b_{n}^{a} {BK}_{0} (m_{n} r)] \times sin (m_{n} z) \end{matrix}$ (25) $\begin{matrix} B_{r, PM}^{m} (r, z) & = & - \sum_{n = 1, 3, 5}^{\infty} \\ [- 2.5 pt] & m_{n} {a_{n}^{m} {BI}_{1} (m_{n} r) + b_{n}^{m} {BK}_{1} (m_{n} r) \\ + \frac{1}{2} \frac{π {StruveL}_{1} (m_{n} r)}{m_{n}^{2}} P_{n}} cos (m_{n} z) \end{matrix}$ (26) $\begin{matrix} B_{z, PM}^{m} (r, z) & = & \sum_{n = 1, 3, 5}^{\infty} \\ [- 2.7 pt] & m_{n} {a_{n}^{m} {BI}_{0} (m_{n} r) - b_{n}^{m} {BK}_{0} (m_{n} r) \\ + \frac{1}{2} \frac{π {StruveL}_{0} (m_{n} r)}{m_{n}^{2}} P_{n}} sin (m_{n} z) \end{matrix}$ (27)

Finally by imposing the boundary conditions mentioned in Equations (13–18), the integral coefficients are obtained as expressed in the Appendix. Similarly the magnetic flux density components due to only the armature current in the three regions can be written as follows: $\begin{matrix} B_{r, AR}^{w} (r, z) \\ [- 2.5 pt] & = - \sum_{n = 1, 3, 5}^{\infty} {[{\hat{a}}_{n}^{w} + F_{n}^{w} (m_{n} r)] {BI}_{1} (m_{n} r) \\ [{\hat{b}}_{n}^{w} + G_{n}^{w} (m_{n} r)] {BK}_{1} (m_{n} r)} cos (m_{n} z) \end{matrix}$ (28) $\begin{matrix} B_{z, AR}^{w} (r, z) \\ [- 2.5 pt] & = \sum_{n = 1, 3, 5}^{\infty} {[{\hat{a}}_{n}^{w} + F_{n}^{w} (m_{n} r)] {BI}_{0} (m_{n} r) \\ [- {\hat{b}}_{n}^{w} + G_{n}^{w} (m_{n} r)] {BK}_{0} (m_{n} r)} sin (m_{n} z) \end{matrix}$ (29) $\begin{matrix} B_{r, AR}^{a} (r, z) \\ [- 2 pt] & = - \sum_{n = 1, 3, 5}^{\infty} [{\hat{a}}_{n}^{a} {BI}_{1} (m_{n} r) + {\hat{b}}_{n}^{a} {BK}_{1} (m_{n} r)] \\ \times cos (m_{n} z) \end{matrix}$ (30) $\begin{matrix} B_{z, AR}^{a} (r, z) \\ = \sum_{n = 1, 3, 5}^{\infty} [{\hat{a}}_{n}^{a} {BI}_{0} (m_{n} r) - {\hat{b}}_{n}^{a} {BK}_{0} (m_{n} r)] \\ \times sin (m_{n} z) \end{matrix}$ (31) $\begin{matrix} B_{r, AR}^{m} (r, z) \\ = - \sum_{n = 1, 3, 5}^{\infty} [{\hat{a}}_{n}^{m} {BI}_{1} (m_{n} r) + {\hat{b}}_{n}^{m} {BK}_{1} (m_{n} r)] \\ \times cos (m_{n} z) \end{matrix}$ (32) $\begin{matrix} B_{z, AR}^{m} (r, z) \\ = \sum_{n = 1, 3, 5}^{\infty} [{\hat{a}}_{n}^{m} {BI}_{0} (m_{n} r) - {\hat{b}}_{n}^{m} {BK}_{0} (m_{n} r)] \\ \times sin (m_{n} z) \end{matrix}$ (33)

Again by imposing the boundary conditions mentioned in Equations (13–18), the integral coefficients are obtained as expressed in the Appendix and $F_{n}^{w} (m_{n} r)$ and $G_{n}^{w} (m_{n} r)$ are also shown in the Appendix. The electromagnetic force exerted to the armature coil can be obtained by the following expression: $F_{em} (t) = \int_{V} [J (t, z) \times B_{PM} (r, z)] dV$ (34)

Since the current density vector has only the tangential component, the above expression for one coil can be rewritten as follows: $F_{em} (t) = - \int_{z - \frac{1}{2} τ_{w}}^{z + \frac{1}{2} τ_{w}} \int_{R_{w}}^{R_{s}} 2 π {rJ}_{θ} (t, z) B_{r, PM}^{w} (r, z) drdz$ (35)

3 Optimization problem

The optimization problem initiates with the determining the optimization variables which are the radial thickness of PMs, the radial thickness of the stator yoke, the radial thickness of the mover yoke, the radial thickness of the winding region, the pole pitch, number of pole-pairs and the magnet length per pole pitch ratio as listed in Table 1 along with their upper and lower bounds. The second step is to introduce the objective functions which, in this study, are the minimization of the power losses and volume. The third step is to consider the constraints (also known as primary objectives) which are to develop a required level of electromagnetic force, and to maintain the stator and slider yoke flux density below an appropriate level to avoid saturation. In the fourth step, the objectives and constraints need to be expressed as functions of optimization variables. To this end the 2-D analytical model presented in Section 2 has been used as the main core of the formulation.

The power losses consist of the copper losses, hysteresis losses and eddy current losses which are, respectively, calculated using the followingexpressions: $P_{cu} = 3 R_{ph} I_{rms}^{2}$ (36) $P_{h} = k_{h} ρ_{y} V_{s} {fB}_{sy}^{n}$ (37) $P_{h} = k_{e} ρ_{y} V_{s} f^{2} B_{sy}^{2}$ (38) where R_ph = ρl_ph/A_c is the per phase winding resistance, l_ph = πN_ph (R_w + R_s) is the length of each phase winding, N_ph = 2pfloor (k_fτ_pl_w/(3A_c)) is the number of winding turns per-phase. The motor total volume is also computed by: $V_{tot} = π R_{o}^{2} L_{st}$ (39) where R_o = R_s + l_sy. The stator and mover back-iron flux densities can, respectively, be obtained by: $B_{sy} = \frac{τ_{p}}{2 l_{sy}} max [B_{r, PM}^{w} (R_{s}, z)]$ (40) $B_{my} = \frac{τ_{p}}{2 l_{my}} max [B_{r, PM}^{m} (R_{r}, z)]$ (41)

Finally the optimization problem is mathematically expressed as follows: $x_{opt} = arg min_{x}$ (42)

Subject to $F_{em} (x) \geq 100$ (43) $B_{sy} < 1.5$ (44) $B_{my} < 1.5$ (45) where $x = [\begin{matrix} l_{pm} & l_{sy} & l_{my} & l_{w} & τ_{p} & α_{p} \end{matrix} p]$ , w_p and w_V are the weighting factors and P_losses = P_cu + P_h + P_e. Table 2 lists the constant parameters used for the optimization problem.

4 Metaheuristic optimization techniques

The applications of optimization techniques have recently been expanded in different realms and the optimal design of electromagnetic devices is not an exception. Nonetheless, some of these optimization problems can hardly be solved by conventional gradient based optimization techniques due to discontinuity of the objective functions and/or constraints. Heuristic and metaheuristic optimization techniques can bridge this gap. Among different metaheuristic optimization techniques, the particle swarm optimization (PSO), and fuzzy adaptive PSO have been used, in this paper, to solve the optimization problem defined in the previous section. To this end, each of these optimization techniques is briefly explained.

4.1 Particle swarm optimization

PSO is one of the population based metaheuristic optimization techniques inspired by the group movements of birds and fish to find food. In this process, the position of each particle is affected by the position of the best particle, known as leader, and the best position of the particle itself during the past movements. Therefore the velocity of each particle is updated by the following expression:

$\begin{matrix} v_{i}^{k + 1} & = & ω v_{i}^{k} + r_{1} C_{1} ({Gbest}^{k} - x_{i}^{k}) \\ + r_{2} C_{2} ({Pbest}_{i}^{k} - x_{i}^{k}) \end{matrix}$ (46) where $v_{i}^{k}$ and $x_{i}^{k}$ are the velocity and position of the ith particle at the kth iteration of the optimization process; Gbest^k is the position of the best particle till the kth iteration; ${Pbest}_{i}^{k}$ is the best personal position of the ith particle till the kth iteration; ω, C₁ and C₂ are the PSO tuning parameters; and r₁ and r₂ are two random values between 0 and 1. Based on the velocity of each particle, its position is calculated: $x_{i}^{k + 1} = x_{i}^{k} + v_{i}^{k + 1}$ (47)

It is noted that the position and velocity of each particle is a vector with N_v dimensions, where N_v is the number of optimization variables. At each iteration, the above two equations are used N_p times, where N_p is the number of particles.

4.2 Fuzzy adaptive PSO

The tuning parameters of the conventional PSO have a significant influence on the performance of the optimization process. Also the appropriate value of each of these tuning parameters is case-dependent and it is difficult to find a general value to work properly for all optimization problems. Therefore, it is wise to employ an approach to adaptively tune these parameters depending on the case study. In the conventional PSO technique, the tuning parameters are constant during the optimization procedure. The aim of the fuzzy adaptive PSO (FAPSO) algorithm is to improve the performance of the conventional PSO algorithm by adaptive tuning of the parameters ω, C₁ and C₂ using a fuzzy based technique [27]. FAPSO operates based on two concepts: best fitness (BF) and unchanged fitness (UF), which are normalized as follows: $NBF = \frac{BF - {BF}_{min}}{{BF}_{max} - {BF}_{min}}$ (48) $NUF = \frac{UF - {UF}_{min}}{{UF}_{max} - {UF}_{min}}$ (49)

The membership functions of NBF and NUF which are the inputs of the fuzzy system and those of ω, C₁ and C₂, which are the outputs of the fuzzy system are shown in Fig. 4. The fuzzy rule bases for the outputs are tabulated in Tables 3–5.

5 Results

The optimization problem has been solved with two metaheuristic optimization techniques which are particle swarm optimization (PSO), and adaptive fuzzy PSO (AFPSO). In both optimization problems, the number of particles is set to 500 and the maximum number of iterations is set to 50.

The geometry, characteristics and specifications of the designed machines using the two optimization techniques are listed in Table 6 in which the FAPSO outperforms the PSO. As evident from the results, the three constraints have been satisfied in both optimization techniques. Figure 5 shows the results during the optimization procedure when the PSO technique has been employed. Similar results for FAPSO have been presented in Fig. 6. The electromagnetic force of the machine using FAPSO is analytically calculated as shown in Fig. 7 and compared with those of FEM. Figures 8 and 9 show the phase back-emf and flux linkage of the machine using FAPSO.

6 Conclusion

The optimal design of tubular PM machine is a constrained multi-objective nonlinear optimization problem with a high-dimensional search space and therefore the conventional gradient-based optimization can hardly be employed to solve the problem. In this investigation, two metaheuristic optimization techniques, i.e. PSO, and FAPSO, have been used to optimally design a slotless brushless tubular PM machine and it is shown that the FAPSO technique outperforms the other rival due to fuzzy based tuning of the parameters. To express the objective functions and the constraints as functions of the optimization parameters, the 2-D analytical model has been used which has the accuracy of the 2-D numerical approaches and the speed of the analytical techniques.

Footnotes

Appendix

The integral coefficients of the magnetic flux density solution due to PM are obtained by solving the following algebraic simultaneous equations: $Λ x = b$ where $x = {[\begin{matrix} a_{n}^{w} & b_{n}^{w} & a_{n}^{a} & b_{n}^{a} & a_{n}^{m} & b_{n}^{m} \end{matrix}]}^{T}$ $b = \frac{π P_{n}}{2 m_{n}^{2}} {[\begin{matrix} 0 & - {StruveL}_{0} (m_{n} R_{r}) & 0 & 0 & {StruveL}_{1} (m_{n} R_{m}) & {StruveL}_{0} (m_{n} R_{m}) \end{matrix}]}^{T}$ $Λ = [\begin{matrix} {BI}_{0} (m_{n} R_{s}) & - {BK}_{0} (m_{n} R_{s}) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {BI}_{0} (m_{n} R_{r}) & - {BK}_{0} (m_{n} R_{r}) \\ {BI}_{1} (m_{n} R_{w}) & {BK}_{1} (m_{n} R_{w}) & - {BI}_{1} (m_{n} R_{w}) & - {BK}_{1} (m_{n} R_{w}) & 0 & 0 \\ {BI}_{0} (m_{n} R_{w}) & - {BK}_{0} (m_{n} R_{w}) & - {BI}_{0} (m_{n} R_{w}) & {BK}_{0} (m_{n} R_{w}) & 0 & 0 \\ 0 & 0 & {BI}_{1} (m_{n} R_{m}) & {BK}_{1} (m_{n} R_{m}) & - {BI}_{1} (m_{n} R_{m}) & - {BK}_{1} (m_{n} R_{m}) \\ 0 & 0 & μ_{rm} {BI}_{0} (m_{n} R_{m}) & - μ_{rm} {BK}_{0} (m_{n} R_{m}) & - {BI}_{0} (m_{n} R_{m}) & {BK}_{0} (m_{n} R_{m}) \end{matrix}]$

Similarly the integral coefficients of the magnetic flux density solution due to armature reaction are obtained by solving the following algebraic simultaneous equations: $Λ \hat{x} = \hat{b}$ where $\hat{x} = {[\begin{matrix} {\hat{a}}_{n}^{w} & {\hat{b}}_{n}^{w} & {\hat{a}}_{n}^{a} & {\hat{b}}_{n}^{a} & {\hat{a}}_{n}^{m} & {\hat{b}}_{n}^{m} \end{matrix}]}^{T}$

$\hat{b} = - [\begin{matrix} F_{n}^{w} (m_{n} R_{s}) {BI}_{0} (m_{n} R_{s}) + G_{n}^{w} (m_{n} R_{s}) {BK}_{0} (m_{n} R_{s}) \\ 0 \\ F_{n}^{w} (m_{n} R_{w}) {BI}_{1} (m_{n} R_{w}) - G_{n}^{w} (m_{n} R_{w}) {BK}_{1} (m_{n} R_{w}) \\ F_{n}^{w} (m_{n} R_{w}) {BI}_{0} (m_{n} R_{w}) + G_{n}^{w} (m_{n} R_{w}) {BK}_{0} (m_{n} R_{w}) \\ 0 \\ 0 \end{matrix}]$

and $\begin{matrix} F_{n}^{w} (m_{n} r) = - \frac{μ_{0} J_{n}}{m_{n}} \int_{m_{n} R_{w}}^{m_{n} r} \frac{{BK}_{1} (x) dx}{{BI}_{1} (x) {BK}_{0} (x) + {BK}_{1} (x) {BI}_{0} (x)} \\ G_{n}^{w} (m_{n} r) = - \frac{μ_{0} J_{n}}{m_{n}} \int_{m_{n} R_{w}}^{m_{n} r} \frac{{BI}_{1} (x) dx}{{BI}_{1} (x) {BK}_{0} (x) + {BK}_{1} (x) {BI}_{0} (x)} \\ J_{n} = \frac{4 J_{0}}{n π} sin (n \frac{π}{2} α_{wp}) sin (n \frac{π}{2} α_{w}) \end{matrix}$

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