Linear Hybrid Reluctance Motor with High Density Force

: Linear switched reluctance motors are a focus of study for many applications because of their simple and sturdy electromagnetic structure, despite their lower thrust force density when compared with linear permanent magnet synchronous motors. This study presents a novel linear switched reluctance structure enhanced by the use of permanent magnets. The proposed structure preserves the main advantages of the reluctance machines, that is, mechanical and thermal robustness, fault tolerant, and easy assembly in spite of the permanent magnets. The linear hybrid reluctance motor is analyzed by ﬁnite element analysis and the results are validated by experimental results. The main ﬁndings show a signiﬁcant increase in the thrust force when compared with the former reluctance structure, with a low detent force.


Introduction
Currently, there are many applications that use hydraulic or pneumatic drives that are being substituted by linear electric actuators in manufacturing industries and robotic systems because of their more stable, precise, and controllable force, along with a higher energy efficiency [1]. In this context, reluctance machines present good environmental behavior due to their high efficiency and inherent ease of assembly and dismantling [2]. For these reasons, among others, there are several studies that have focused on new magnetic structures [3][4][5], in order to enhance their force performance [6] and increase force density by adding permanent magnets [7][8][9][10][11][12][13][14]. As an example, linear switched reluctance motors (LSRM) and linear permanent magnet synchronous motors (LPMSMs) have been proposed for propelling a ropeless elevator [15,16], for an automotive suspension system [17], and for a linear generator in direct drive wave-power converter [18]. Up to now, LPMSMs have a higher power density and efficiency when compared with LSRMs, despite their inherent detent or cogging force, which can be reduced or eliminated by applying several techniques [19,20]. Many current research papers deal with developing and optimizing linear motors in order to increase power density and efficiency [21][22][23].
With this aim, this work presents and analyzes a four-phase LSRM, whose novelty is in the mover. A series of vertically magnetized permanent magnets (PMs) are inserted into the slots of the mover of the LSRM. The resulting actuator is analyzed by finite element analysis (FEA) and a prototype was built and tested. The FEA results and the experimental results are in good agreement and reveal an approximately 100% increase in the peak thrust force; a moderate ripple factor operating in single pulse; and a relative low detent force, about 10% of peak thrust force. The results obtained anticipate an interesting actuator for high force density applications, such as those cited above.

Linear Hybrid Reluctance Motor Structure and Operation Principle
LSRMs are classified according to the relation between the planes that contain the flux lines and the axis of movement. Longitudinal flux LSRM is when these planes are parallel, and transverse flux LSRM is when they are perpendicular. Longitudinal flux LSRM may be tubular or flat, with single or double side for the flat case. The LSRM has two main parts: the active part, which contains the concentrated windings, also called primary part; and the passive or secondary part, made of an empty slotted iron structure. This work focuses on the double-sided longitudinal flux LSRM, referred to from now on as LSRM.
The proposed structure is shown in Figure 1. The iron lamination structure is the same as that of a conventional four-phase double-sided longitudinal flux LSRM (see Figure 1 in grey), the modelling, simulation, and test of which are reported in the work of [24]. From this reluctance structure, a series of permanent magnets type NdFeB N32 (in black/white, see Figure 1) have been placed in the secondary slotted lamination. The PMs are vertically magnetized following the arrangement shown in Figure 1. Each phase has four coils per pole connected in series, in such a way that only one flux path is produced. The main dimensions of the primary (active part) are the pole length , pole width , and slot width , the sum of which defines the primary pole pitch = + . For the secondary (passive) part, the pole dimensions are the length , the width , the slot width , and the mover pole pitch, defined as = + . The magnet dimensions are the length and width . The stack lamination width is denoted as .  Figure 2 shows the LHRM operating principle in which the four coils of phase A are colored in red and blue, the blue side represents the current entering the paper and the red side is the current flowing out of the paper. The red line of Figure 2a represents the main flux line (Φ + ) of phase A at pole-alignment position, when it is fed with a positive current (I + ). When phase A current is inverted (I − ), (see Figure 2b) the blue line represents the main flux line at the same aligned position. In both cases, a significant force appears at pole alignment positions. The force is to the right , + when the current is I + , and to left , − when the current is I − . This phenomenon does not appear in an LSRM, where the force is null at aligned and unaligned pole positions.  Figure 2 shows the LHRM operating principle in which the four coils of phase A are colored in red and blue, the blue side represents the current entering the paper and the red side is the current flowing out of the paper. The red line of Figure 2a represents the main flux line (Φ A+ ) of phase A at pole-alignment position, when it is fed with a positive current (I A+ ). When phase A current is inverted (I A− ), (see Figure 2b) the blue line represents the main flux line at the same aligned position. In both cases, a significant force appears at pole alignment positions. The force is to the right F X,A+ when the current is I A+ , and to left F X,A− when the current is I A− . This phenomenon does not appear in an LSRM, where the force is null at aligned and unaligned pole positions.   The LSRM propulsion force versus position x and current i, ( ( , )) has the same direction (right/left) in the interval ∈ [0, /2], and for whatever current sign. In the LHRM, the propulsion force is null at the unaligned pole positions, or when the stator poles are aligned with any of the PM poles (N/S) for any current, see Figure 3a and 3c. The LHRM propulsion force has a positive direction (left to right) in the interval ∈ [0, ], (see Figure 3b) and a negative direction (right to left) in the interval ∈ [ , 2 · ] (see Figure 3d). This implies a current sign dependency of the LHRM over the propulsion force. The LSRM propulsion force versus position x and current i, (F(x, i)) has the same direction (right/left) in the interval x ∈ [0, τ S /2], and for whatever current sign. In the LHRM, the propulsion force is null at the unaligned pole positions, or when the stator poles are aligned with any of the PM poles (N/S) for any current, see Figure 3a,c. The LHRM propulsion force has a positive direction (left to right) in the interval x ∈ [0, τ S ], (see Figure 3b) and a negative direction (right to left) in the interval x ∈ [τ S , 2·τ S ] (see Figure 3d). This implies a current sign dependency of the LHRM over the propulsion force.   In summary, the LHRM is, similar to the LSRM, a position dependent machine, so an encoder is necessary to operate this actuator, along with an H-bridge inverter, which has to be used instead of the asymmetric bridge converter because of its current sign dependence.

Mathematical Model Equations
The presence of the PM allows one to split the total phase flux linkage (ψ) as the sum of the flux due to the phase-current (ψ i ) and the PMs flux-linkage (ψ PM ), as follows: The electrical phase circuit equation is as follows: where I, V, and R are the phase current, voltage, and resistance, respectively. Expressing the phase-current flux linkage as the product of the phase inductance and the current, that is: ψ i (i, x) = L(i, x)·I, and combining Equations (1) and (2), it yields the electrical circuit equation of the LHRM given by the following: In which the mover velocity is denoted as v. The motion equation neglecting friction is given as follows: where F x is the electromagnetic force, F d is the detent force, F l is the load, and M is the mover mass. The instantaneous electromagnetic force (F x ) is computed from the differential change of co-energy respect to position for a given phase current, as follows: where m is the number of phases. This set of Equations (1)-(5) performs the dynamic and static behavior of the LHRM, which can be implemented in MATLAB-Simulink software for their solution as in LSRM. The apparent power per phase, S VA or volt-ampere requirement per phase, is defined as the product of peak voltage (V p ) and peak current (I p ) multiplied by the number of switching devices per phase (n). From the area of the energy conversion loop (W), the average output power is given as follows: where k d is the magnetic duty cycle factor defined as k d = 2·x o f f /τ s . The ratio between the power output and the apparent power or W/VA ratio is given by the following:

Finite Element Analysis Results
The electromagnetic propulsion force is obtained from a 2D-FEM solver [25]. It uses the weighted Maxwell stress tensor technique [26], which is one of the most reliable approaches for force and torque computations [27]. It consists of computing the Maxwell stress tensor over a set of concentric surfaces of thickness δ and contour Γ(δ) wrapping the mover (see Equation (8)), with n y being a normal unit vector in the y-direction; and B x and B y are the magnetic field density in the x and y directions, respectively.
The main dimensions of the FEM model of LHRM shown in Figure 4 are reported in Table A1, see Appendix A. Figure 4 shows the flux lines obtained from the FEA for the zero current excitation phase. This position (see Figure 4) is set as the initial position x = 0 mm, with this position being equivalent to the position shown in Figure 3c, that is, the upper left phase A primary pole is aligned with an S-N magnet.
Energies 2018, 11, x FOR PEER REVIEW 5 of 14 phase. This position (see Figure 4) is set as the initial position x = 0 mm, with this position being equivalent to the position shown in Figure 3c, that is, the upper left phase A primary pole is aligned with an S-N magnet. The electromagnetic force is evaluated for a sequence of even positions in the interval ∈ [0, 2 · ] and for a set of constant current density values J ∈ [± 5, ± 10, ± 15] A/mm 2 . The LSRM has the same iron laminations and coils, but without magnets.
As can be observed, there are two force profiles for the LHRM (see Figure 5); one for positive current feeding (continuous line, see Figure 5) and one for the negative current (discontinuous line). The LHRM force profiles are substantially higher than in the LSRM. The main FEM comparison results obtained from phase A are summarized in Table 1  The electromagnetic force is evaluated for a sequence of even positions in the interval x ∈ [0, 2·τ s ] and for a set of constant current density values J ∈ [±5, ±10, ±15] A/mm 2 . The LSRM has the same iron laminations and coils, but without magnets.
As can be observed, there are two force profiles for the LHRM (see Figure 5); one for positive current feeding (continuous line, see Figure 5) and one for the negative current (discontinuous line). The LHRM force profiles are substantially higher than in the LSRM. The main FEM comparison results obtained from phase A are summarized in Table 1. [0, 2 · ] and for a set of constant current density values J ∈ [± 5, ± 10, ± 15] A/mm . The LSRM has the same iron laminations and coils, but without magnets.
As can be observed, there are two force profiles for the LHRM (see Figure 5); one for positive current feeding (continuous line, see Figure 5) and one for the negative current (discontinuous line). The LHRM force profiles are substantially higher than in the LSRM. The main FEM comparison results obtained from phase A are summarized in Table 1.  The phase-force profiles are shifted τ s mm when feeding with positive or negative phase current, that is, F X (x + τ S , −I) = F X (x, I), as can be seen in Figure 5. Table 2 shows the comparison results between phases, when feeding with a positive current density of J = +15 A/mm 2 (I = +2.95 A), along with the cogging force values whose force profile is depicted in Figure 6. As is usual in the linear machines, the border phases (i.e., phases A and D) are expected to have slightly different values because of the loss of magnetic symmetry in the borders.   Figure 7a, red thick line) is the total propulsion force when the motor phases are excited following the above-mentioned positive sequence for a flat current waveform and without overlapping phase currents. The optimal conduction interval for phase A is also depicted in Figure 7a, in which for the LHRM dimensions  Figure 7a, red thick line) is the total propulsion force when the motor phases are excited following the above-mentioned positive sequence for a flat current waveform and without overlapping phase currents. The optimal conduction interval for phase A is also depicted in Figure 7a, in which for the LHRM dimensions given in Table A1   In order to assess the demagnetization of the permanent magnets, the airgap flux density has been drawn in the y-direction, which is the PM-magnetization direction, and then comparing these field distributions with zero current and when exciting phase A.
The PM operating point is defined as the point where the permeance of the magnetic circuit meets the magnet BH characteristic. The irreversible demagnetization occurs when the PM's flux density decreases under magnet BH-characteristic knee value. For N32 grade, the residual induction is 1.2 T at 20 ℃ and the knee values are 0.3 T at 60 ℃, 0.4 T at 80 ℃, 0.5 T at 100 ℃, and 0.6 T at 120 ℃.
The critical points for phase A are found when the propulsion force reach its peak values, which occurs at mover positions of 6 mm and 28 mm for JA = −15A/mm 2 , and at mover positions of 10 mm and 22 mm for JA = + 15A/mm 2 (see Figure 7a). The points analyzed are at mover positions of 22 mm and JA = 15A/mm 2 , and mover at 28 mm and JA = −15A/mm 2 (see Figure 8). The PM critical areas are In order to assess the demagnetization of the permanent magnets, the airgap flux density has been drawn in the y-direction, which is the PM-magnetization direction, and then comparing these field distributions with zero current and when exciting phase A.
The PM operating point is defined as the point where the permeance of the magnetic circuit meets the magnet BH characteristic. The irreversible demagnetization occurs when the PM's flux density decreases under magnet BH-characteristic knee value. For N32 grade, the residual induction is 1.2 T at 20 • C and the knee values are 0.3 T at 60 • C, 0.4 T at 80 • C, 0.5 T at 100 • C, and 0.6 T at 120 • C. The critical points for phase A are found when the propulsion force reach its peak values, which occurs at mover positions of 6 mm and 28 mm for J A = −15A/mm 2 , and at mover positions of 10 mm and 22 mm for J A = +15A/mm 2 (see Figure 7a). The points analyzed are at mover positions of 22 mm and J A = 15A/mm 2 , and mover at 28 mm and J A = −15A/mm 2 (see Figure 8). The PM critical areas are highlighted (see Figure 8a,b). The B y (x,J) distribution field is computed along the airgap red line shown in Figure 8a,b, with the variable x being the position over this line, and J being the current density, whose values are zero in all phases J = 0, and J = J A = ±15 A/mm 2 (see Figure 8c,d in black/red lines, respectively). The results reveal that PM would be demagnetized when operating at J A = 15 A/mm 2 at mover position of 22 mm (see Figure 8c), and for magnet temperature exceeding 80 • C.
field distributions with zero current and when exciting phase A.
The PM operating point is defined as the point where the permeance of the magnetic circuit meets the magnet BH characteristic. The irreversible demagnetization occurs when the PM's flux density decreases under magnet BH-characteristic knee value. For N32 grade, the residual induction is 1.2 T at 20 ℃ and the knee values are 0.3 T at 60 ℃, 0.4 T at 80 ℃, 0.5 T at 100 ℃, and 0.6 T at 120 ℃.
The critical points for phase A are found when the propulsion force reach its peak values, which occurs at mover positions of 6 mm and 28 mm for JA = −15A/mm 2 , and at mover positions of 10 mm and 22 mm for JA = + 15A/mm 2 (see Figure 7a). The points analyzed are at mover positions of 22 mm and JA = 15A/mm 2 , and mover at 28 mm and JA = −15A/mm 2 (see Figure 8). The PM critical areas are highlighted (see Figure 8a,b). The By(x,J) distribution field is computed along the airgap red line shown in Figures 8a,b, with the variable x being the position over this line, and J being the current density, whose values are zero in all phases J = 0, and = = ± 15 A/mm 2 (see Figures 8c and 8d in black/red lines, respectively). The results reveal that PM would be demagnetized when operating at JA = 15 A/mm 2 at mover position of 22 mm (see Figure 8c), and for magnet temperature exceeding 80 °C.

Experimental Results
In order to validate the FEM results, an LHRM prototype (see Figures 9b and 9c) was built and tested, with the main dimensions shown in Table A1. The experimental setup (see Figure 9a) consists of a system for positioning the mover at a given position and a load cell for measuring the static thrust force. The force is measured in each millimeter, in the range ∈ [0, 2 · τ ], which means 32 measures for the 6 current densities J∈ [±5, ±10, ±15]. Considering the four phases, the total number of measures is 4 x 6 x 32 = 768. These results are compared with the FEM results in Figures 10-13. The experimental results are in agreement with the FEM results, which validates the FEM simulation results.
of a system for positioning the mover at a given position and a load cell for measuring the static thrust force. The force is measured in each millimeter, in the range ∈ [0, 2 · τ ], which means 32 measures for the 6 current densities J∈ [±5, ±10, ±15]. Considering the four phases, the total number of measures is 4 x 6 x 32 = 768. These results are compared with the FEM results in Figures 10-13. The experimental results are in agreement with the FEM results, which validates the FEM simulation results.

Conclusions and Discussion
From the experimental results, the operating principle and force profiles are validated and this enables the linear hybrid reluctance motor (LHRM) as a good candidate for applications with high force density. Unlike LSRM, this machine needs to operate through a four-phase full-bridge converter, in order to supply the phase current with the appropriate sign. As in LSRM, the mover position and currents must be acquired; therefore, a linear encoder and current sensors are required for implementing a force control. This force control can be designed for mitigating the force ripple

Conclusions and Discussion
From the experimental results, the operating principle and force profiles are validated and this enables the linear hybrid reluctance motor (LHRM) as a good candidate for applications with high force density. Unlike LSRM, this machine needs to operate through a four-phase full-bridge converter, in order to supply the phase current with the appropriate sign. As in LSRM, the mover position and currents must be acquired; therefore, a linear encoder and current sensors are required for implementing a force control. This force control can be designed for mitigating the force ripple (see Figure 7b) by adjusting the currents waveform [28], and thus the force can remain almost flat to its maximum value for each current density.
From the point of view of the cogging force (see Figure 13), this presents a series of peaks of about 10% of the peak force at J = 15 A/mm 2 , which is a reasonable value. Further research should be done to eliminate cogging force by phase-shifting, or by a sensitivity analysis in the pole's proportions and number of phases.
The LHRM actuator presented in this paper is conceived as a servomotor used in manufacturing lines or precision positioning applications whose velocity is relatively low (u b < 2 m/s) and where the force control is required. For this reason, a typical hysteresis current control is adopted, which is robust and fast [29], being the current controlled around a reference value (I ref ), and whose ripple depends on the hysteresis range. The shape of the controlled current and voltage are shown in Figure 14, and it is characterized by the turn on/off positions (see Table 3) and by the reference current, the value of which is directly linked to the desired propulsion force. (see Figure 7b) by adjusting the currents waveform [28], and thus the force can remain almost flat to its maximum value for each current density. From the point of view of the cogging force (see Figure 13), this presents a series of peaks of about 10% of the peak force at J = 15 A/mm 2 , which is a reasonable value. Further research should be done to eliminate cogging force by phase-shifting, or by a sensitivity analysis in the pole's proportions and number of phases.
The LHRM actuator presented in this paper is conceived as a servomotor used in manufacturing lines or precision positioning applications whose velocity is relatively low (ub < 2 m/s) and where the force control is required. For this reason, a typical hysteresis current control is adopted, which is robust and fast [29], being the current controlled around a reference value (Iref), and whose ripple depends on the hysteresis range. The shape of the controlled current and voltage are shown in Figure  14, and it is characterized by the turn on/off positions (see Table 3) and by the reference current, the value of which is directly linked to the desired propulsion force.    The inherent fault tolerance of the reluctance machines and thus of LSRMs is preserved in the LHRM, because once the damaged phase is removed, the LHRM keeps operating at a new diminished force, as can be inferred from Figure 7a.
The motor is easily assembled despite having to put permanent magnets in the mover slots (see Figure 9c), because flux lines attach the PM to the magnetic structure (see Figure 4) and there are no repulsion forces.
The thermal aspect is also good because PMs are in the secondary (passive) part, which allows high current densities in the primary (active) part for intermittent operation.
From all the findings reported above, this novel structure of linear hybrid reluctance motor can be an interesting option in applications, which requires a high thrust force for intermittent duty cycles, such as in manufacturing industries and robotic systems.

Conflicts of Interest:
The author declare no conflict of interest. Appendix A.