Analysis of a Discrete Stator Hybrid Excited Flux Switching Linear Machine

A discrete stator hybrid excited flux switching linear machine (DSHEFSLM) is proposed in this paper. The proposed DSHEFSLM uses a discrete stator to reduce the iron loss and overall cost of the machine. Assistant teeth are used at both ends of the mover to overcome the unbalance in the three phases, which is a global issue in linear machines. Field excitation (FE) is used, which adds field regulation capability to the proposed machine and makes it suitable for a wide speed operation range. A magnetic equivalent circuit model is used to find the suitable coil combination and no-load flux linkage. The multiobjective genetic global optimization is used to optimize the design parameters of the whole machine while keeping the slot area, electric and magnetic loadings constant. A correlation table is drawn to show the impact of different design parameters on average thrust force. The optimization has increased the peak-to-peak flux linkage by 11%, average thrust force by 34.60%, thrust force density by 34.60%, decreased thrust force ripples, and detent force by 21.05% and 8.58% respectively. The proposed machine has reduced the volume of the permanent magnet by 39.18% and offers 28.09% higher average thrust force and thrust force density compared to the flux switching permanent magnet machine proposed in the literature.

P m mover core permeance. P pm PM permeance. P s stator core permeance.

I. INTRODUCTION
Linear machines are in high demand in various applications, ranging from domestic applications to industrialization. Linear machines are preferred over rotary machines because they produce force directly along the x-axis with higher efficiency, reliability, and less energy loss. Machines with permanent magnet (PM) on a short mover are gaining popularity due to their advantages of high power density and efficiency [1]. PM flux switching linear (PMFSL), PM doubly salient linear (PMDSL), and PM flux reversal linear (PMFRL) machines are examples of mover-mounted PM machines. PMFSL machine combines the characteristics of PM synchronous linear machines (PMSLM) and switched reluctance linear machines (SRLM). PMFSL machine has high power density [2], bipolar flux linkage, and a rugged stator, making it suitable for high-speed long stroke applications [3]. Additionally, improved temperature control has reduced manufacturing costs [4] and is compatible with extreme environmental conditions [5].
However, the unregulated PM flux of the PMFSL machine limits its speed range. The alternate choice is the field excited flux switching linear (FEFSL) machine, which offers a wide speed range based on variation in field current. But the FEFSL machine suffers from low flux linkage and has lower thrust force density than PMFSL machines. To overcome the disadvantages of PMFSL and FEFSL machines, hybrid excited flux switching linear (HEFSL) machines are proposed, which possess better flux linkage, high thrust force density, and better flux controlling ability [4]. The authors in [1], [4], [6]- [22] proposed and investigated numerous hybrid excited rotary and linear topologies. Based on the rotating hybrid excited flux switching machine suggested in [4], [6] proposed and investigated a new type of HEFSL machine which replace the rare earth PMs with ferrite magnets and to improve flux modulation, additional field excitation (FE) windings are used. After that, the proposed HEFSL is transformed into a double-sided HEFSL machine [9]. Three possible types of hybrid excited flux switching (HEFS) rotary topologies with FE coils situated in stator slots are proposed and researched in [7]. Three FSHE machines based on the positioning of PMs at the bottom, middle and top are proposed in [11]. The investigation showed that the bottom PM has the best use of the PM and flux regulating ability.
A new partitioned primary, segmented stator HEFSL machine is proposed in [23], [24] having separate field and armature windings, which help in good temperature distribution. However, they have poor flux linkage due to large flux linking paths and have a flux circulation problem. In [25], a slot PM machine is proposed with concentrated field and armature winding hence low copper losses and good efficiency, but it has problems of magnetic saturation, flux leakage, and rare earth magnets are used in substantial volume. Another HEFSL machine is proposed in [17] in which the PM is sandwiched between two DC excitation sources and is converted into dual mover topology in [19].
This paper proposes a series hybridized discrete stator hybrid excited flux switching linear machine (DSHEFSLM), which is based on the PMFSL machine proposed in [26]. The proposed machine has reduced the volume of PM by 39.18% and offers a 28.09% higher average thrust force and thrust force density. Furthermore, the proposed design uses FE windings which adds flux regulation capability and make it suitable for wide-speed applications like electric power train. In the rest of the paper, section II introduces the machine topology and its working principle; the magnetic equivalent model is employed in section III to find suitable coil combinations and find the machine's initial performance. Genetic optimization is done in section IV, while simulation results obtained by finite element analysis (FEA) are discussed in section V. Finally, section VI concludes the paper.

II. MACHINE TOPOLOGY AND WORKING PRINCIPLE
The proposed DSHEFSLM is shown in Fig. 1. The proposed design has a discrete stator that reduces iron consumption and iron losses in the stator. Mover contains all the excitation sources. PMs are placed at the alternate pole tip of the mover. Three-phase concentrated armature windings are used and are wound on the iron teeth. Each phase is composed of two sets of armature coils. Field excitation windings are used and wound on the mover teeth containing PMs, which provides series hybridization. The leading design parameters are given in Table 1 and denoted in Fig. 2.   by FE and PM goes pass through the mover tooth containing PM crosses air gap enters stator body and leaves immediately, passes again across the air gap and enters the subsequent mover tooth i.e. the iron tooth, passes through mover's yoke and complete its circuit. This all can be evident from

III. MAGNETIC EQUIVALENT MODEL
To reduce the computational complexity and drive storage, MEM is used to find initial performance and suitable coil combinations for the proposed design. As MEM models all parts of the machine like stator, air gap, and mover, it gives  better results in terms of accuracy. In the case of the proposed machine, two magnetic equivalent models are developed based on the position of mover shown in Fig. 5. The models are developed considering one magnet magnetized and extended to the whole machine to reduce the computational time and drive storage. The model formulations are based on the following equations [21]: The permeance of iron parts can be calculated as: After calculating the permeance of all parts, and sources at nodes, nodal analysis is employed to calculate flux flowing out and flowing in to the corresponding nodes as: N ), the permeance matrix can be written as follows: Eq. (6) is solved in three steps [27], through iterative process to calculate the no-load flux linkage with considering the initial relative permeability of 4000. Firstly, H i in the mover and stator part is calculated using Eq. (8). Once H i is calculated, relative permeability is updated using Eq. (9). Finally, magnetic flux density (Eq. (10)) is calculated using the same iterative process. Fig. 6 shows the no-load flux linkage obtained by FEA analysis and calculated through MEM. There is a small error between the two but considering computational time, drive storage it is not that significant.
The MEM is validated by JMAG v20.1 and detailed comparison based on computation time and drive storage is given in Table 2. The MEM greatly reduces the drive storage and computational time as compared to FEA. It is to mention that both MEM and FEA are done using 64 bit operating Lenovo system with 8 GB RAM, Intel(R), Core(TM) i5-8500 CPU 3.00GHz, 3000Mhz.

IV. GENETIC OPTIMIZATION
Genetic optimization (GO) is a heuristic search method that mimics the natural evolutionary process. This method is often used to generate useful solutions for optimization and search problems. GO does not rely on the initial (starting) point of the search, does not require any derivative information of the objective function or the constraint function, and is least likely to fall into a local minimum. The flowchart of GO   used is shown in Fig. 7. Firstly, the initial design is made in geometry editor, and then the CAD parameters are imported to the designer. An objective function is defined in Eq. 11 comprises three sub objective functions, and the weighting factor is being specified based on the requirement. Ranges of varying CAD parameters and constraints are defined given in Table 3, to ensure constant electric and magnetic loadings. Number of generations and population size is initialized to achieve a global value of the objective function.    stack length, rated current, and armature turns are kept constant during optimization. The correlation of average thrust force with different design variables is shown in Table 4. Fig. 8 to Fig. 12 presents the TF avg versus different geometry variables defined. From the figures, it can be seen that the TF avg has achieved a global value of 313.5708N after the optimization, which is colored black and encircled red in all the figures. A detailed quantitative comparison Table 5 is drawn, which compares the initial and optimized design performances.

V. ELECTROMAGNETIC PERFORMANCE ANALYSIS A. NO-LOAD ANALYSIS
The no-load flux linkage of the proposed DSHEFSLM design is shown in Fig. 13a, whereas Fig. 13b shows the harmonics. The flux linkage of all three phases is bipolar and sinusoidal. The impact of FE on the no-load flux linkage is shown in Fig. 14. Fig. 15 depicts the proposed machine's flux weakening and boosting capabilities at various FE current densities J e = 0, ±4, ±8, ±12 (A/mm 2 ). The flux density in the air gap increases when the field current is positive, resulting in a stronger flux linkage; when the field current is negative, the flux density in the air gap decreases, resulting in a weaker flux linkage. The induced EMF can be calculated using Eq. 12 in which represents the no-load flux linkage. Fig. 16a illustrates the back-EMF of the proposed DSHEFSLM at the velocity of 2m/s. The harmonics of the back-EMF are given in Fig. 16b. The fundamental harmonic is important for  generating thrust force, while the other high-order harmonics produce only thrust ripple and gain losses.
The attraction between the stator's core and the mover's excitation causes detent force at no load (PM and FE). Detent force causes noise and vibration. If the attraction is stronger, noise and vibration will be increased, which in turn lower the TF avg . The detent force of the proposed DSHEFSLM initial and optimized designs is shown in Fig. 17.

B. LOAD ANALYSIS
The thrust force of the proposed DSHEFSLM is maximized after the genetic optimization, as shown in Fig. 18. The TF avg of 232.965N in the initial design was increased to 313.571N after optimization under the same electric and magnetic loading. Compared to the initial design, the optimized design has an increase of 34.6% in the TF avg which is 31.2% more than the conventional design [26] average thrust force. The behavior of average thrust force versus armature and field current is observed in Fig. 19. The variation in TF avg is observed while varying the armature current (I AC ) from 0 to 15A and the field current (I FE ) from 0 to 6A. The flux density plot: Fig. 20 shows the distribution of flux along all parts of the machine, indicating no saturation region.  Fig. 18 shows the ripples are a little higher in the thrust force. Since skewing is widely used to reduce the ripples in the thrust force, hence 3D skewing is employed in this study to minimize the ripples. Different skewed stators shown in Fig. 21 are employed to the mover, and the performance is observed. Skewing reduce ripples at the cost of a slight decrement in the average thrust force. 3D finite element analysis is done, and the impact of skewing step number on the performance is enlisted in Table 6. A stator three-step skewing structure is utilized for further analysis since the step skewing number should be minimal to simplify the implementation. With 3-step skewing, the ripples are reduced by 26% at the cost of 11% decrement in the average thrust force shown in Fig. 22.

2) DYNAMIC AND EFFICIENCY ANALYSIS
The force and power versus velocity curves for the proposed machine are calculated using the method [28], [29]. The force-velocity curve and the power-velocity curve are shown VOLUME 10, 2022   in Fig. 23. From the analysis, it is clear that at the velocity of 3.59m/sec, the proposed DSHEFSLM achieved a maximum TF avg of 313.57N while the output power (P out ) reaches 1140W . As force is inversely proportional to speed, the thrust will decrease at higher speeds, thereby maintaining a constant output power. Fig. 23 shows that the proposed DSHEFSLM has an excellent constant power operation capability.
The P out of a linear machine is calculated by the product of the TF avg and the associated velocity. At the same time, the input power is the sum of resultant P out and the total losses, i.e., copper and iron losses. The losses are computed at various points on the force-velocity graph with different velocity and electric loading. The taken points are made up of current and starting angles of current, as shown in Fig. 24. The copper losses are calculated using Eq. 15, whereas iron losses are determined using 2D FEA at all points specified. At point 1, the thrust force is maximum, and speed is lower; hence copper losses are higher, and iron losses are lower, i.e.,   273W and 128W , respectively. At point 14, the iron losses are at a maximum (approximately 239W ) because iron losses directly relate to speed while the copper losses were 123W , and hence efficiency is lower. The efficiency at all points of  the proposed machine is shown in Fig. 25.
Also from [30], (13) becomes where I is current, ρ is resistivity ( −m), J is current density (A/mm 2 ), L is length of wire (mm), N is number of turns, and Q is number of slot pairs.

C. LOSSES ANALYSIS
Different losses like iron, copper, and eddy current losses in PM are considered, and their effect on efficiency is observed. For the detailed loss analysis, one point at the maximum force region (P1), one point in the high-speed region (P14), and   three points in the best operation region (P4, P5, P6) are considered. Since iron losses are related to speed, at point 1, the speed is lower, and hence iron losses are smaller, and copper losses are higher. While at high speed (point 14), iron losses are higher, and copper losses are lower. Hence efficiency is   lower at these points. The FE losses are calculated using Eq. (14) and are constant because a constant excitation source is provided. When the eddy current losses of all PMs are considered, the TF avg is decreased i.e. at point 1 the TF avg was 313.57N initially when PM losses are considered the TF avg dropped to 300.81N . The detailed losses analysis is given in Table 7. The power factor of the proposed machine is 0.76, lagging considering all the excitation sources.

VI. COMPARISON WITH REGULAR DESIGN
Finally, the proposed DSHEFSLM is compared with the conventional design, and the comparison results are shown in Table 8. Both the designs use discrete stators, the same rated current, and the number of turns. In the proposed design, the volume of PM is reduced by 39.18% while achieving better performance as compared to the conventional design [26]. Further, the proposed design uses field excitation winding which adds flux controlling capability.

VII. CONCLUSION
A three-phase discrete stator HEFLM is investigated, which combines the high thrust density feature of permanent magnet machines and flux controlling features of FE machines.
The proposed design has a single tooth armature and field excitation windings, thus minimizing copper losses. At both ends of the mover, auxiliary teeth are used to provide a path for the flux that is leaking from the ends of the mover.
The results obtained through MEM show great agreement with FEA having an accuracy of 94.7%, thereby reducing computation time and drive storage. Genetic optimization is used to globally optimize the design parameters of the machine. Thrust force ripples were minimized by using 3D skewing. Dynamic analysis of the machine is done to determine the machine's performance over a wide range of speeds. A performance comparison between the proposed and the conventional design is carried out to display the superiority of the proposed design over the design proposed in the literature.