Design and Optimization of a Surface-Mounted Permanent-Magnet Linear Synchronous Motor with Maxwell ANSYS

Linear motor offers several features in many applications that require linear motion. Nevertheless, the presence of cogging force can deteriorate the thrust of a permanent magnet linear motor. Using several methodologies, a design of synchronous single sided linear iron-core motor was proposed. According to exact formulas with surface-mounted magnets and concentrated winding specification, which are relying on geometrical parameters. Two-dimensional performance analysis of the designed model and its multi-objective optimization were accomplished as a method to reduce the motor cogging force using MAXWELL ANSYS. The optimum model design results showed that the maximum force ripple was approximatrly reduced by 81.24%compared to the original model with a smaller ripple coefficient of 0.22. Likewise, the model was redesigned taking into consideration two cases; laminated core and solid core. It was found that the error between the analytical and numerical results of the output force did not exceed 0.0967%.


Introduction
Permanent magnet linear motors (PMLMs) are used in wide industrial applications; particularly for high positioning accuracy of concerned implementations. The linear motors (LMs) are mainly based on the electromagnetic concept where they delivere direct linear movement without using gears, cams, belts or other mechanical accessories. This is when the mechanical motion synchronizes the magnetic field. The PMLSMs can be classified into categories: flat or tubular, single sided or double sided, slotted or slot-less, air cored or iron-cored, and longitudinal flux or transverse flux. The LM consists of stator and the translator. Since, it can present direct linear motion with no need for any rotation to translation conversion of equipment; this leads to simplifying system structure and improving system efficiency [1]. On the other hand, the more predominant nonlinear effect of linear motors is force ripples (detent and reluctance forces) [2]. The detent (or cogging) force is generated by the attraction of the magnets to the LM's core. This force is produced even the winding current is absent. The force demonstrates a periodic relationship with regard to the position of the translator relative to the permanent magnets (PMs) mounted on the stator [3]. Moreover, the reluctance force is generated by the mutation in the winding self-inductance with respect to the position between the translator and stator [4, ٥]. Hence, the cogging forces generally cause an undesired vibration and noise; depending on the thrust magnitude [6].
In order to minimize the force ripple, different methods in the literature were developed. One of the most effective methods is the arrangements of the PMs. Thus, it must be optimized to reduce the cogging force, which is more effective than reluctance force as represented by Lim et al. [7]. Also, the method could be conducted by adding some parts as presented by Baker et al. [8]. It's found that adding a Flux Gathering Ring (FGR) to the outer stator teeth of a linear tubular generator could be an effective method to reducing the end effect. Hence, the cogging reduction happens as resultant. Chung et al. [9] proposed designing and analyzing a flux reversal linear synchronous motor (FRLSM). Three various design parameters were investigated and compared in motor characteristics e.g, back-EMF, detent force, normal force, thrust, and phase inductance by using Maxwell ANSYS. The investigation greatly showed low force ripple and good thrust force linearity via PMs' skewing and three-phase sinusoidal control. Cao et al. [10] investigated the effect of some leading design parameters (mover tooth width, the slot open width the width of the slot under PM, mover height and the motor stack length) on the force performance of a complementary and modular linear flux-switching permanent-magnet (LFSPM) motor. That is using finite element method (FEM). Jalal et al. [11] presented the FEA optimization techniques followed by development of a Matlab/Simulink model to investigate the effects of electrical machine inductance and the combined electromagnetic loading of the machine on the resulting force and cogging force. Palomino et al. [12] proposed a technique to combine the FEM and the statistical regression, which permit to implement the optimal skew angle of the PM. That causes a greater reduction of force ripple with the minimum thrust diminution. Ghasime [13] adopted a magnet segmentation method to reduce the cogging torque. Wheras, each surface PM was split into eight parts with symmetrical structure and equal angular widths. The angular gaps between them were utilized for reducing a number of optimization parameters.
Lazov and Uzunov [14] used two methods to reduce the detent force in one innovative PMLSM for 2D laser marking system. Two additional end teeth with chamfers in the magnetic circuit of the movable part were used while the other method modified the teeth of the ferromagnetic core with different lengths.
This paper suggests an optimization method to estimate the optimum dimensions of the PMLSM based on regular steps with the aid of the Maxwell ANSYS simulation. The optimum dimensions required are the magnet length, magnet width, magnet height, air gap length, motor length at zdirection and the motor ends' width.

Design Procedure
The PMLSM is a linear motor where the mechanical motion is synchronizing the magnetic field. The PMLSMs can be classified into flat or tubular, single sided or double sided, slotted or slot-less, air cored or iron-cored, and longitudinal flux or transverse flux. In this paper, the design of a single-sided PMLSM with slotted armature core and surface PMs is proposed. A structure model of a three-phase flat type single sided PMLSM according to the MAXWELL ANSYS is depicted as in Fig. (1).
The core is composed of 13 slots and 13 teeth, 12 slots are filled with single layer concentrated three phase coil windings in a laminated 1010 steel core [15] as a solid core. Each phase in the motor has four slots. These magnetic cores may be made of thin laminations in order to minimize the eddy currents flowing through them. As for all electrical machines, the core losses are composed of hysteresis losses and eddy current losses. The specifications of PMs (NdFeB, Class N42SH) [16] used in the motor design are listed in Table 1. Generally, thrust production in the PMLSM is restricted by electrical and magnetic sources. Electrical source depends on the motor coils current. According to Lorentz law [17]: ‫ܨ‬ ⃗ ൌ ‫ܫܮ‬ ⃗ ൈ ‫ܤ‬ ሬ⃗ (vector version) and F= IBL sinθ (strength only) ...(1) at the same time, for magnetic components, the magnetic flux density and inductance effect on the thrust. While the flux of the PMs and coils effect on the magnetic flux density and inductance. This leads to generate a sequence of attracting and repelling forces between the salient poles and the PMs, expressed by a thrust force in the translator [18]. Fig. 2 depicts the working principle of the designed linear motor. As the armature moves, an EMF is induced in the stator windings. By differentiating the PM flux of the phase winding with time, a no-load rms voltage Ef is excited. Ef is excited by the PMs' flux and induced in each phase of the armature winding, it can be given as [17]: The magnetic circuit of an LSM with surface PMs as illustrated in the equivalent magnetic circuit is shown in Fig. 3.

Fig. 3. Equivalent magnetic circuit of a LM with surface PMs [17].
While, the magnetic flux linked with the armature winding may be given as: The relation between the velocity and frequency represented by [17] υs= 2τƒ …(9) The frequency could be estimated using Eq. (9), is found to be equal to 17.36Hz. That is according to the parameters listed in Table 2 with the following design parameters: Fmax = 350N mtranslator ≈ 6.6kg mload = 17kg υmax = 1m/s amax = 1.5m/s 2 Table 2, LM base line design parameters [20].
Based on these dimensions and assuming initial teeth lengths and PM volume followed by applying a special algorithm with multiple iterations to obtain the final main design dimensions of the PMs and the geometric dimensions of a single stator slot illustrated in Fig.  4. In the case of using semi-closed slot in the stator core, the detent force depends on the slot aperture width because of the minimizing effect on the slot harmonic components [21]. The armature winding resistance as a function of the winding parameters, dimensions, and electric conductivity can be represented by [17]: In the case of salient pole synchronous motor, the d-axis and q-axis synchronous reactances, respectively can be expressed as: ܺ ௦ௗ ൌ ܺ ଵ ܺ ௗ and ܺ ௦ ൌ ܺ ଵ ܺ …(11) Here, X1 = 2ߨfL1 is the armature leakage reactance, and Xad and Xaq are the d-axis and qaxis armature reaction reactance's, respectively [21] which are calculated as follows: is the d-axis equivalent air gap, Ksat>1 I and Ksatq≈ 1.
Demagnetization curve of the PM is sensitive to temperature changes. Both Br and Hc reduce as the magnet temperature elevates [17]: …(14) Where Bc20 and Hc20 are the flux density and coercive force of the PM at 20 o C, and ߙ < 0 and ߙ ு < 0 are the temperature coefficients in %/ o C for Br and Hc, respectively. The relative recoil magnetic permeability at PM temperature ߠ ெ of 50 o C can be calculated as: Where µrrec = 1.161. The following equations are the rms d-axis and q-axis armature currents as a function of V1, Xsd, Xsq, R1, Ef, and δ as shown in Fig. 5. The peak line current density of the armature or the specific electric loading is determined as: The input power Pi delivered to the armature winding is given by: The copper losses in the core windings is: ‫ݑܿ∆ܲ‬ ൌ ݉ ଵ ‫ܽܫ‬ ଶ ܴ ଵ . The electromagnetic power (P elm ) and thrust force (Fdx), respectively can be given as [17,22]: So, theforce ripple coefficient kr could be calculated by [23]:

Finite Element Simulation A. Solving and Post-Processing
Finite element method is one of the most accepted and widely used tools for the solution and optimization of linear and nonlinear partial differential equations [24]. In the current work, the analysis was performed using MAXWELL ANSYS that ignored end windings due to using 2D model. The structure parameters of the motor were presented. The solution was divided into two stages, load and no-load transient magnetic solution. Time-stepping FEA method depending on Maxwell equations was applied for analyzing and optimizing the magnetic and force characteristics of the LM. In the process of solving nonlinear transient magnetic problem, the calculated values of voltage V1 is 45V (at load) and frequency f= 17.36Hz, whereas the solution would be divided into 36 steps with 0.0016sec time period at 0.0001 nonlinear residual using backward Euler time integration method.
Mover mesh of the original motor is illustrated in Fig. 6 where the mesh should be dense enough to minimize the error calculation, whereas the model was designed with three regions of meshing. Each region contains 1200 elements.

B. Magnetic Field Distribution
On no-load case, considering the axial symmetry of the PMLSM, the designed motor was simplified to a 2D axial symmetry model in Cartesian coordinates ignoring core losses. An irregular distribution of the current density in the conductor was considered. The distribution of the magnetic lines is depicted as shown in Fig. 7. Various shades of rainbow color denoted magnetic flux density values at time equal to 0.0032 sec plus the leakage flux inside the slots was clearly noticed. Furthermore, the instantaneous PM flux linkages and induced back emf waveforms at noload magnetic field solutions were obtained as shown in Figs. 8 and 9, respectively. It was observed that the waveform of the flux linkage was approximate sinusoid. Since there was no harmonic magnetic flux, which might be induced by the current at the load case. The PMs flux waveforms were three phase symmetrical, i.e. they had the same magnitude with 120 electrical degrees shifting. This is a fundamental requirement of three-phase motor design. Unlike the flux linkage, the back emf has irregular sinusoid.     Figure 11, shows the PMLSM model's thrust force at load moving 2߬ distance to +x direction and the armature current waveforms for the original model with three-phase AC voltage source, neglecting the core losses of the laminated core. The thrust force with rms value was found to be 350.3387N (with 0.9176 ripple coefficient) at Ia≈4.32A. The thrust ripple depended on the field harmonic produced by the magnets (in the air gap) in addition to current harmonics. The optimization was accomplished based on the model in Sec. 2. Fig. 12, shows the results of two cases based on MAXWELL ANSYS analysis to predict best dimensions that lead to minimum cogging force at no load case. In Fig. 12-a, the analysis was accomplished based on Table 3-a (first design). the analysis has shown that the maximum cogging force was decreased by 73.9%. Fig. 12-b presents the second design results, which were done with another set of constant values. The cogging force results have decreased by 77.43%.The second design gave the best percentage value of the cogging force reduction. Fig. 12-c represents the ends widths effect on the cogging forces varation of second design at constant values of (em=5, g=1.5 and hM=3) with ends width of Table 3-b. The results showed that, using 2.5mm end length led to 81.24% reduction in cogging force.   Table 4 clarifies the orginal model dimensions and parameters plus the optimum design results from the steps above. That is to find the optimum output thrust as illustrated in Fig. 13 for the load case using MAXWELL ANSYS. The rms value of the output force was found to be equal to 320.4833N at the mean value for the rms motor currents of Ia =3.555A. The output force might be equal to 320N at mload = 17kg and optimum mtranslator ≈ 4.75kg. It could be seen that the thrust ripple reduced effectively for the optimal designed motor case. In fact, the optimized motor suffered from thrust ripple twice less than that for the original model. Therefore, the rms input voltage of three phase had to be increased by 11.56V to compensate the lost thrust caused by reducing the PM volume. (a) Thrust force as a function of time at rms current = 3.555A with optimum parameters.

C. Simulink Design Optimization of the PMLSM Model with and without Core Losses
(b) 3-phase of 2D simulated currents with optimum parameters. Fig.13. Motor performance with optimum parameters. The core loss could affect the machine flux linkages. As these measurements were done at low speed, consequently the core loss component was sort of small where it was equal to (25.568W) as shown in Fig. 14-a and the output force at Fig.  14-b was (253.4N) at mean value of the rms motor currents of Ia = 5.29 A. Therefore, errors in the flux linkage computation were small since magnetic losses were caused by harmonic main flux and harmonic leakage flux [26]. Thrust force constants, force ripple coefficient and the efficiency percentage [27] results are illustrated in Table. 5. The greater value of the force constant (kF = Fdx/ Ia) [28] led to least thrust ripple krand highest efficiency which represented the best model. Variant results were observed in the model's kF-RMS values. This disparity was located due to a few reasons whereas two of them were convergent unlike the rest. The convergence of the first two models' values was related to the error between the exact and the analytical equations. While the increment of the third model was due to applying the optimization method to enhance the performance, which is the goal of this paper. On the other hand, loss consideration in the fourth model caused the kF-RMS value to decrease. However, the ripple coefficient was equal to 0.319 that was considered an acceptable value to the practical designs in comparison with rest of the models that contain high efficiency or better ripple coefficient but remain a virtual design.

Conclusions
Towards enhancing the performance of the single sided PMLSM, a primary structure of 10/12 pole-slot three phase synchronous single-sided linear motor has been designed and modeled using 2D FEA simulation whereas a Maxwell ANSYS has been employed. The following conclusions can be presented: Greek letters δ the load angle between the terminal motor phase voltage V1 and Ef Φf the magnetic flux produced by the PM ∅ the motor power factor angle, the phase angle between V1 and Ia ØA, ØC, ØC the magnetic flux of three phases A, B and C, respectively ⍵p PM width θ the angle between the field direction and the conductor direction µo the free space magnetic permeability σ the electric conductivity of a conductor at an operating temperature τ pole pitch