Detent Force Reduction of a C-Core Linear Flux-Switching Permanent Magnet Machine with Multiple Additional Teeth

Abstract: C-core linear flux-switching permanent magnet (PM) machines (LFSPMs) are attracting more and more attention due to their advantages of simplicity and robustness of the secondary side, high power density and high torque density, in which both PMs and armature windings are housed in the primary side. The primary salient tooth wound with a concentrated winding consists of C-shaped iron core segments between which PMs are sandwiched and the magnetization directions of these PMs are adjacent and alternant in the horizontal direction. On the other hand, the secondary side is composed of a simple iron core with salient teeth so that it is very suitable for long stroke applications. However, the detent force of the C-core LFSPM machine is relatively high and the magnetic circuit is unbalanced due to the end effect. Thus, a new multiple additional tooth which consists of an active and a traditional passive additional tooth, is employed at each end side of the primary in this paper, so that the asymmetry due to end effect can be depressed and the detent force can be reduced by adjusting the passive additional tooth position. By using the finite element method, the characteristics and performances of the proposed machine are analyzed and verified.


Introduction
Cableless elevator systems based on linear machines are very attractive, since they can eliminate the drawbacks of conventional traction-geared cable elevators, such as the vertical oscillation and height limit caused by the strength and weight of cable [1], especially in case of the skyscrapers of 200-400 m height.Moreover, multicar elevators in one hoistway can be achieved more easily when the elevator vehicles are driven by independent linear machines [2], so a large proportion of the building space can be reclaimed.Hence, several kinds of linear machines have been investigated for cableless elevators.In [3], a linear permanent magnet (PM) synchronous machine (LPMSM) is adopted for cableless elevators because of its high efficiency and high power density, in which the armature windings are laid along the hoistway and can be controlled section by section according to the elevator vehicle position.PMs are installed on the elevator vehicle.However, the cost of LPMSM is very high because of the long laid armature windings and the expensive power electronic equipment used to divide the armature windings into sections.Due to the simplicity and robustness of the secondary side, the linear switched reluctance machine (LSRM) is regarded as the preferred machine for cableless elevators [4].However, the thrust force ripple of LSRMs is relatively higher due to the complex and nonlinear magnetic circuit and the control difficulty of LSRM is consequently increased [5].
In recent years, stator PM machines, namely the double salient PM (DSPM) machine, flux reverse PM (FRPM) machine and flux-switching PM (FSPM) machine have been widely investigated because of their advantages such as the simple and robust structure of the rotor [6,7].Both the PMs and armature windings of stator PM machines are housed in the stator, and the rotor is composed of a simple and cheap iron core with salient teeth, so the linear counterpart of the stator PM machine, namely the linear primary PM machine, is very suitable for long stroke applications, such as cableless elevators and urban rail transit systems, in which the primary and the secondary can be designed as the short side and long side, respectively, to integrate the high efficiency and power density performances of LPMSMs and the low cost merit of LSRMs.
Compared with the DSPM machine and the FRPM machine, the FSPM machine possesses the merits of higher power density and torque density [8], sinusoidal no-load EMF [9] and fault-tolerance capability [10][11][12][13][14], so FSPMs are attracting more and more attention and numerous novel rotary FSPM machines structures of have been proposed [15,16], such as the five-phase modular FSPM machine [17] and the hybrid excitation FS machine [18].Meanwhile lots of linear versions of FSPM (LFSPM) machines also have been presented for linear industry applications [19].In [20], E-core and C-core LFSPM machines were optimized and compared.Due to the improvement of the armature winding slot area compared with traditional U-core LFSPM machines, the C-core LFSPM machines can achieve a higher thrust force density and power density, in spite of the lesser usage of the PMs.
However, the performances of LFSPM machines, including U-core, E-core and C-core LFSPM machines are usually severely deteriorated due to the asymmetrical characteristics of the magnetic field as a result of the end effect, although two additional teeth can be added on the each end of the primary side.In order to obtain the symmetrical 3-phase flux linkage and no-load EMF, a complementary and modular E-core LFSPM machine is proposed for urban rail transit and the corresponding mathematical model is established [21].
In this paper, a C-core LFSPM machine with two new multiple additional teeth is proposed for cableeless elevator direct drive systems.As shown in Figure 1, two LFSPM machines are located in the two gaps, respectively, between the hoistway and the elevator vehicle to achieve a balanced lifting force.The primary side is installed on the outside of the elevator car and the secondary side is laid along the hoistway.In Section 2, the machine configuration will be described and the operation principle will be introduced briefly.Then, the key parameters of the proposed machine will be optimized to achieve balanced electromagnetic performances and low detent force in Section 3. In Section 4, the characteristics of the C-core LFSPM machine will be analyzed by the 2-D finite element method (FEM).Finally, conclusions will be drawn in Section 5.
In recent years, stator PM machines, namely the double salient PM (DSPM) machine, flux reverse PM (FRPM) machine and flux-switching PM (FSPM) machine have been widely investigated because of their advantages such as the simple and robust structure of the rotor [6,7].Both the PMs and armature windings of stator PM machines are housed in the stator, and the rotor is composed of a simple and cheap iron core with salient teeth, so the linear counterpart of the stator PM machine, namely the linear primary PM machine, is very suitable for long stroke applications, such as cableless elevators and urban rail transit systems, in which the primary and the secondary can be designed as the short side and long side, respectively, to integrate the high efficiency and power density performances of LPMSMs and the low cost merit of LSRMs.
Compared with the DSPM machine and the FRPM machine, the FSPM machine possesses the merits of higher power density and torque density [8], sinusoidal no-load EMF [9] and fault-tolerance capability [10][11][12][13][14], so FSPMs are attracting more and more attention and numerous novel rotary FSPM machines structures of have been proposed [15,16], such as the five-phase modular FSPM machine [17] and the hybrid excitation FS machine [18].Meanwhile lots of linear versions of FSPM (LFSPM) machines also have been presented for linear industry applications [19].In [20], E-core and C-core LFSPM machines were optimized and compared.Due to the improvement of the armature winding slot area compared with traditional U-core LFSPM machines, the C-core LFSPM machines can achieve a higher thrust force density and power density, in spite of the lesser usage of the PMs.
However, the performances of LFSPM machines, including U-core, E-core and C-core LFSPM machines are usually severely deteriorated due to the asymmetrical characteristics of the magnetic field as a result of the end effect, although two additional teeth can be added on the each end of the primary side.In order to obtain the symmetrical 3-phase flux linkage and no-load EMF, a complementary and modular E-core LFSPM machine is proposed for urban rail transit and the corresponding mathematical model is established [21].
In this paper, a C-core LFSPM machine with two new multiple additional teeth is proposed for cableeless elevator direct drive systems.As shown in Figure 1, two LFSPM machines are located in the two gaps, respectively, between the hoistway and the elevator vehicle to achieve a balanced lifting force.The primary side is installed on the outside of the elevator car and the secondary side is laid along the hoistway.In Section 2, the machine configuration will be described and the operation principle will be introduced briefly.Then, the key parameters of the proposed machine will be optimized to achieve balanced electromagnetic performances and low detent force in Section 3. In Section 4, the characteristics of the C-core LFSPM machine will be analyzed by the 2-D finite element method (FEM).Finally, conclusions will be drawn in Section 5.  Energies 2017, 10, 318 3 of 14

Machine Configuration and Operation Principle
The performance and characteristics of LFSPM machines have been widely investigated.Figure 2a shows an original C-core LFSPM machine, named "Machine I" in this paper, which can be obtained by splitting a rotary 6/13-pole C-core FSPM machine along the radial direction and unrolling it.It should be notice that the 6/13 mover and stator pole combination is selected in this paper because it can achieve the maximum no-load EMF and thrust force [20].

Machine Configuration and Operation Principle
The performance and characteristics of LFSPM machines have been widely investigated.Figure 2a shows an original C-core LFSPM machine, named "Machine I" in this paper, which can be obtained by splitting a rotary 6/13-pole C-core FSPM machine along the radial direction and unrolling it.It should be notice that the 6/13 mover and stator pole combination is selected in this paper because it can achieve the maximum no-load EMF and thrust force [20].Figure 3 shows the operation principle of the 6/13-pole C-core LFSPM machine.Assuming the PM magnetizes towards the left-hand side, at the relative position of the stator tooth and the mover tooth as shown in Figure 3a, the PM flux linkage of the coil nearly reaches the positive maximum.When the mover moves a quarter of the tooth pitch towards the left-hand side, as shown in Figure 3b, the mover module is symmetrical about the central axis of the stator slot.At this moment, the PM flux linkage of the coil equals zero.When the mover moves 1/4 tooth pitch again, as shown in Figure 3c, the PM flux linkage nearly reaches the negative maximum value.In Figure 3d, the mover module is symmetrical about the central axis of the stator tooth.Thus the PM flux linkage of the coil equals zero again.Assuming an ideally sinusoidal PM flux linkage, Figure 3e shows the PM flux linkage waveform when the mover travels one stator tooth pitch, in which positions A, B, C and D correspond to the positions in Figure 3a-d, respectively.
In order to reduce the detent force, two passive additional teeth are adopted [22] in a U-core LFSPM machine and the width of the side-slot has been optimized for the minimum detent force.Thus, the traditional passive additional teeth also can be employed in the C-core LFSPM machine, as shown in Figure 2b, which is named as "Machine II" in this paper, to reduce the detent force.However, the other electromagnetic performances, such as the flux linkage and the inductance, are unbalanced due to the end effect of the linear machine not only in the Machine I but also in the Machine II. Figure 3 shows the operation principle of the 6/13-pole C-core LFSPM machine.Assuming the PM magnetizes towards the left-hand side, at the relative position of the stator tooth and the mover tooth as shown in Figure 3a, the PM flux linkage of the coil nearly reaches the positive maximum.When the mover moves a quarter of the tooth pitch towards the left-hand side, as shown in Figure 3b, the mover module is symmetrical about the central axis of the stator slot.At this moment, the PM flux linkage of the coil equals zero.When the mover moves 1/4 tooth pitch again, as shown in Figure 3c, the PM flux linkage nearly reaches the negative maximum value.In Figure 3d, the mover module is symmetrical about the central axis of the stator tooth.Thus the PM flux linkage of the coil equals zero again.Assuming an ideally sinusoidal PM flux linkage, Figure 3e shows the PM flux linkage waveform when the mover travels one stator tooth pitch, in which positions A, B, C and D correspond to the positions in Figure 3a-d, respectively.
In order to reduce the detent force, two passive additional teeth are adopted [22] in a U-core LFSPM machine and the width of the side-slot has been optimized for the minimum detent force.Thus, the traditional passive additional teeth also can be employed in the C-core LFSPM machine, as shown in Figure 2b, which is named as "Machine II" in this paper, to reduce the detent force.However, the other electromagnetic performances, such as the flux linkage and the inductance, are unbalanced due to the end effect of the linear machine not only in the Machine I but also in the Machine II.In this paper, 2-D FEM models of the four machines shown in Figure 2 have been built.Thus the machine characteristics can be obtained.Figure 4   In order to balance the magnet circuit, two active additional teeth [23] are added at each end of the primary side of Machine I, as shown in Figure 2c, which is named as Machine III in this paper.It can be noticed that the active additional tooth is composed of two primary tooth iron cores and a PM block.Figure 5 shows the no-load magnetic field distribution of Machine III.It can be seen that the flux contributed by the active additional tooth runs through the mover teeth, thus balancing the magnetic field asymmetry caused by the end effect of the linear machine.The 3-phase flux linkage waveforms of Machine III are expected to be more balanced than the other two counterparts.As shown in Figure 6, the 3-phase flux linkage waveforms are nearly symmetrical in the positive and negative half-cycle.In this paper, 2-D FEM models of the four machines shown in Figure 2 have been built.Thus the machine characteristics can be obtained.Figure 4  In this paper, 2-D FEM models of the four machines shown in Figure 2 have been built.Thus the machine characteristics can be obtained.Figure 4  In order to balance the magnet circuit, two active additional teeth [23] are added at each end of the primary side of Machine I, as shown in Figure 2c, which is named as Machine III in this paper.It can be noticed that the active additional tooth is composed of two primary tooth iron cores and a PM block.Figure 5 shows the no-load magnetic field distribution of Machine III.It can be seen that the flux contributed by the active additional tooth runs through the mover teeth, thus balancing the magnetic field asymmetry caused by the end effect of the linear machine.The 3-phase flux linkage waveforms of Machine III are expected to be more balanced than the other two counterparts.As shown in Figure 6, the 3-phase flux linkage waveforms are nearly symmetrical in the positive and negative half-cycle.In order to balance the magnet circuit, two active additional teeth [23] are added at each end of the primary side of Machine I, as shown in Figure 2c, which is named as Machine III in this paper.It can be noticed that the active additional tooth is composed of two primary tooth iron cores and a PM block.Figure 5 shows the no-load magnetic field distribution of Machine III.It can be seen that the flux contributed by the active additional tooth runs through the mover teeth, thus balancing the magnetic field asymmetry caused by the end effect of the linear machine.The 3-phase flux linkage waveforms of Machine III are expected to be more balanced than the other two counterparts.As shown in Figure 6, the 3-phase flux linkage waveforms are nearly symmetrical in the positive and negative half-cycle.The merit of the traditional passive additional teeth in Machine II is that the detent force can be optimized through adjusting the width of the side-slot.However, the side slot width cannot be changed arbitrarily in Machine III, which will deteriorate the phase and the amplitude of the flux linkage and no-load EMF, especially of phase A and phase B which are arranged to be close to the additional teeth, so a new multiple additional tooth, which is composed of an active additional tooth and a traditional passive tooth, is employed in the C-core LFSPM machine.As shown in Figure 2d, in the proposed C-core LFSPM, which is named as Machine IV, both the PMs and armature windings are located in the primary side.The primary salient tooth wound with a concentrated winding consists of C-shaped laminated segments between which the PMs are sandwiched and the magnetization directions of these PMs are adjacent alternant in horizontal direction.And the secondary side is composed of a simple iron core with salient teeth so that it is very suitable to be used in long stroke applications, such as cableless elevators.Due to the use of two multiple additional teeth, the magnet circuit can be balanced and the detent force can be optimized through adjusting the width of the side slot, w ss .

Key Parameter Optimization
The proposed LFSPM machine with multiple additional teeth is operated based on the flux switching principle, which is same as the conventional FSPM machine.In this section, several key parameters will be optimized using the FEM in order to obtain the expected performances, such as the symmetry and sinusoidal 3-phase flux linkage and no-load EMF waveforms, the minimum detent force and the maximum average thrust force.It should be noted that the mover tooth tip width b mtt and the slot under the PM b sPM are designed to be equal to the one-eighth of the mover pitch τ m in the optimization process, as shown in Figure 7, which is similar with the traditional FSPM machine [9], thus the mover slot width b ms equals 5/8 times τ m , namely b mtt = b sPM = 1/8τ m = 1/5b ms .The merit of the traditional passive additional teeth in Machine II is that the detent force can be optimized through adjusting the width of the side-slot.However, the side slot width cannot be changed arbitrarily in Machine III, which will deteriorate the phase and the amplitude of the flux linkage and no-load EMF, especially of phase A and phase B which are arranged to be close to the additional teeth, so a new multiple additional tooth, which is composed of an active additional tooth and a traditional passive tooth, is employed in the C-core LFSPM machine.As shown in Figure 2d, in the proposed C-core LFSPM, which is named as Machine IV, both the PMs and armature windings are located in the primary side.The primary salient tooth wound with a concentrated winding consists of C-shaped laminated segments between which the PMs are sandwiched and the magnetization directions of these PMs are adjacent alternant in horizontal direction.And the secondary side is composed of a simple iron core with salient teeth so that it is very suitable to be used in long stroke applications, such as cableless elevators.Due to the use of two multiple additional teeth, the magnet circuit can be balanced and the detent force can be optimized through adjusting the width of the side slot, w ss .

Key Parameter Optimization
The proposed LFSPM machine with multiple additional teeth is operated based on the flux switching principle, which is same as the conventional FSPM machine.In this section, several key parameters will be optimized using the FEM in order to obtain the expected performances, such as the symmetry and sinusoidal 3-phase flux linkage and no-load EMF waveforms, the minimum detent force and the maximum average thrust force.It should be noted that the mover tooth tip width b mtt and the slot under the PM b sPM are designed to be equal to the one-eighth of the mover pitch τ m in the optimization process, as shown in Figure 7, which is similar with the traditional FSPM machine [9], thus the mover slot width b ms equals 5/8 times τ m , namely b mtt = b sPM = 1/8τ m = 1/5b ms .The merit of the traditional passive additional teeth in Machine II is that the detent force can be optimized through adjusting the width of the side-slot.However, the side slot width cannot be changed arbitrarily in Machine III, which will deteriorate the phase and the amplitude of the flux linkage and no-load EMF, especially of phase A and phase B which are arranged to be close to the additional teeth, so a new multiple additional tooth, which is composed of an active additional tooth and a traditional passive tooth, is employed in the C-core LFSPM machine.As shown in Figure 2d, in the proposed C-core LFSPM, which is named as Machine IV, both the PMs and armature windings are located in the primary side.The primary salient tooth wound with a concentrated winding consists of C-shaped laminated segments between which the PMs are sandwiched and the magnetization directions of these PMs are adjacent alternant in horizontal direction.And the secondary side is composed of a simple iron core with salient teeth so that it is very suitable to be used in long stroke applications, such as cableless elevators.Due to the use of two multiple additional teeth, the magnet circuit can be balanced and the detent force can be optimized through adjusting the width of the side slot, w ss .

Key Parameter Optimization
The proposed LFSPM machine with multiple additional teeth is operated based on the flux switching principle, which is same as the conventional FSPM machine.In this section, several key parameters will be optimized using the FEM in order to obtain the expected performances, such as the symmetry and sinusoidal 3-phase flux linkage and no-load EMF waveforms, the minimum detent force and the maximum average thrust force.It should be noted that the mover tooth tip width b mtt and the slot under the PM b sPM are designed to be equal to the one-eighth of the mover pitch τ m in the optimization process, as shown in Figure 7, which is similar with the traditional FSPM machine [9], thus the mover slot width b ms equals 5/8 times τ m , namely b mtt = b sPM = 1/8τ m = 1/5b ms .
Moreover, the individual parameter optimization method [20] is adopted in the key parameter optimization.On the other hand, the magnetic flux density is considered to ensure the iron core to be unsaturation.Moreover, the individual parameter optimization method [20] is adopted in the key parameter optimization.On the other hand, the magnetic flux density is considered to ensure the iron core to be unsaturation.
As shown in Figure 8, due to the passive additional teeth, the flux density values in the secondary side yoke nearby the two end teeth of primary side in Machine II and Machine IV are obviously higher than those in Machine I and Machine III  As shown in Figure 8, due to the passive additional teeth, the flux density values in the secondary side yoke nearby the two end teeth of primary side in Machine II and Machine IV are obviously higher than those in Machine I and Machine III.
Energies 2017, 10, 318 6 of 14 Moreover, the individual parameter optimization method [20] is adopted in the key parameter optimization.On the other hand, the magnetic flux density is considered to ensure the iron core to be unsaturation.
As shown in Figure 8

Width of Stator Teeth
First of all, the width of the stator (the secondary side) tooth tip and tooth root, which are shown in Figure 7, are optimized.Figure 9 shows the detent force peak-to-peak value of the proposed machine with different c stt and c stb , where c stt is defined as the ratio of the width of stator tooth tip b stt to the stator tooth pitch τ s , and c stb is defined as the ratio of the width of stator tooth root b stb to the stator tooth pitch τ s .It can be found that the detent force achieves the minimum value when c stt and c stb equal 0.39 and 0.55, respectively.Actually, c stb does not have a significant effect on the detent force.Figure 10 shows the change of the no-load EMF THD with respect to cstt and cstb.It indicates that the no-load EMF THD increases with the rise of cstt.And, the no-load EMF THD is not sensitive to cstb either.Consequently, the no-load EMF peak-to-peak values with the different cstt and cstb are calculated.As shown in Figure 11, the no-load EMF peak-to-peak value reduces with respect to the increase of cstt and cstb, due to the flux leakage around the stator teeth.

Width of Stator Teeth
First of all, the width of the stator (the secondary side) tooth tip and tooth root, which are shown in Figure 7, are optimized.Figure 9 shows the detent force peak-to-peak value of the proposed machine with different c stt and c stb , where c stt is defined as the ratio of the width of stator tooth tip b stt to the stator tooth pitch τ s , and c stb is defined as the ratio of the width of stator tooth root b stb to the stator tooth pitch τ s .It can be found that the detent force achieves the minimum value when c stt and c stb equal 0.39 and 0.55, respectively.Actually, c stb does not have a significant effect on the detent force.

Width of Stator Teeth
First of all, the width of the stator (the secondary side) tooth tip and tooth root, which are shown in Figure 7, are optimized.Figure 9 shows the detent force peak-to-peak value of the proposed machine with different c stt and c stb , where c stt is defined as the ratio of the width of stator tooth tip b stt to the stator tooth pitch τ s , and c stb is defined as the ratio of the width of stator tooth root b stb to the stator tooth pitch τ s .It can be found that the detent force achieves the minimum value when c stt and c stb equal 0.39 and 0.55, respectively.Actually, c stb does not have a significant effect on the detent force.Figure 10 shows the change of the no-load EMF THD with respect to cstt and cstb.It indicates that the no-load EMF THD increases with the rise of cstt.And, the no-load EMF THD is not sensitive to cstb either.Consequently, the no-load EMF peak-to-peak values with the different cstt and cstb are calculated.As shown in Figure 11, the no-load EMF peak-to-peak value reduces with respect to the increase of cstt and cstb, due to the flux leakage around the stator teeth.

Width of Stator Teeth
First of all, the width of the stator (the secondary side) tooth tip and tooth root, which are shown in Figure 7, are optimized.Figure 9 shows the detent force peak-to-peak value of the proposed machine with different c stt and c stb , where c stt is defined as the ratio of the width of stator tooth tip b stt to the stator tooth pitch τ s , and c stb is defined as the ratio of the width of stator tooth root b stb to the stator tooth pitch τ s .It can be found that the detent force achieves the minimum value when c stt and c stb equal 0.39 and 0.55, respectively.Actually, c stb does not have a significant effect on the detent force.Figure 10 shows the change of the no-load EMF THD with respect to cstt and cstb.It indicates that the no-load EMF THD increases with the rise of cstt.And, the no-load EMF THD is not sensitive to cstb either.Consequently, the no-load EMF peak-to-peak values with the different cstt and cstb are calculated.As shown in Figure 11, the no-load EMF peak-to-peak value reduces with respect to the increase of cstt and cstb, due to the flux leakage around the stator teeth.Consequently, the no-load EMF peak-to-peak values with the different c stt and c stb are calculated.As shown in Figure 11, the no-load EMF peak-to-peak value reduces with respect to the increase of c stt and c stb , due to the flux leakage around the stator teeth.Based on Figures 10 and 11, a conclusion can be drawn that the better performances of no-load EMF can be achieved when the relatively smaller cstt and cstb are adopted.However, the excessive narrow stator teeth will result in serious magnetic saturation when the currents are fed into the armature windings.Thus the thrust force will be restrained and the efficiency of the machine will be reduced.So, the thrust force has to be analyzed to obtain the optimal cstt and cstb values taking into account the magnetic field in iron core caused by both of the armature current and the PMs.
Figure 12 shows the average thrust force values with the different cstt and cstb.It can be found that the thrust force improves firstly and then reduces with the increase of cstt, and reduces with the increase of cstb.When cstt = 0.37 and cstb = 0.37, the thrust force achieves the maximum value which is equal to be about 225 N.

Width of Side Slot
The multiple additional tooth is composed of an active additional tooth and a traditional passive additional tooth.And the width of the side-slot, which is between the active additional tooth and the passive additional tooth, can be adjusted to optimize the detent force of the C-core LFSPM machine.As shown in Figure 2d, the width of the side-slot is defined as: where wmss is the width of the side-slot, τs is the pitch of the stator tooth and css is the ratio between wmss and τs.
Figure 13 shows the detent force waveforms versus css when cstt = 0.37 and c stb = 0.5.And the peak-to-peak values with the different css are listed in Table 1.It can be seen that the detent force peak-to-peak value changes versus css.When css = 0.625, as listed using italic numbers, the detent force peak-to-peak value can achieve the minimum value which is 38.3% of that of Machine III.Based on Figures 10 and 11, a conclusion can be drawn that the better performances of no-load EMF can be achieved when the relatively smaller c stt and c stb are adopted.However, the excessive narrow stator teeth will result in serious magnetic saturation when the currents are fed into the armature windings.Thus the thrust force will be restrained and the efficiency of the machine will be reduced.So, the thrust force has to be analyzed to obtain the optimal c stt and c stb values taking into account the magnetic field in iron core caused by both of the armature current and the PMs.
Figure 12 shows the average thrust force values with the different c stt and c stb .It can be found that the thrust force improves firstly and then reduces with the increase of c stt , and reduces with the increase of c stb .When c stt = 0.37 and c stb = 0.37, the thrust force achieves the maximum value which is equal to be about 225 N.  Based on Figures 10 and 11, a conclusion can be drawn that the better performances of no-load EMF can be achieved when the relatively smaller cstt and cstb are adopted.However, the excessive narrow stator teeth will result in serious magnetic saturation when the currents are fed into the armature windings.Thus the thrust force will be restrained and the efficiency of the machine will be reduced.So, the thrust force has to be analyzed to obtain the optimal cstt and cstb values taking into account the magnetic field in iron core caused by both of the armature current and the PMs.
Figure 12 shows the average thrust force values with the different cstt and cstb.It can be found that the thrust force improves firstly and then reduces with the increase of cstt, and reduces with the increase of cstb.When cstt = 0.37 and cstb = 0.37, the thrust force achieves the maximum value which is equal to be about 225 N.

Width of Side Slot
The multiple additional tooth is composed of an active additional tooth and a traditional passive additional tooth.And the width of the side-slot, which is between the active additional tooth and the passive additional tooth, can be adjusted to optimize the detent force of the C-core LFSPM machine.As shown in Figure 2d, the width of the side-slot is defined as: where wmss is the width of the side-slot, τs is the pitch of the stator tooth and css is the ratio between wmss and τs.
Figure 13 shows the detent force waveforms versus css when cstt = 0.37 and cstb = 0.5.And the peak-to-peak values with the different css are listed in Table 1.It can be seen that the detent force peak-to-peak value changes versus css.When css = 0.625, as listed using italic numbers, the detent force peak-to-peak value can achieve the minimum value which is 38.3% of that of Machine III.

Width of Side Slot
The multiple additional tooth is composed of an active additional tooth and a traditional passive additional tooth.And the width of the side-slot, which is between the active additional tooth and the passive additional tooth, can be adjusted to optimize the detent force of the C-core LFSPM machine.As shown in Figure 2d, the width of the side-slot is defined as: where w mss is the width of the side-slot, τ s is the pitch of the stator tooth and c ss is the ratio between w mss and τ s .
Figure 13 shows the detent force waveforms versus c ss when c stt = 0.37 and c stb = 0.5.And the peak-to-peak values with the different c ss are listed in Table 1.It can be seen that the detent force peak-to-peak value changes versus c ss .When c ss = 0.625, as listed using italic numbers, the detent force peak-to-peak value can achieve the minimum value which is 38.3% of that of Machine III.Defining the active length of the proposed machine as the distance between the midlines of the two passive additional teeth, the active length of the proposed machine can be expressed as: where wmt, wPM and wmst are the width of the mover tooth, the width of the PM and the width of passive additional tooth, respectively, τm is the pitch of the mover tooth, and La is the active length of the proposed machine.Consequently, the width of the side-slot and the active length of the proposed machine can be obtained according to the dimension parameters.As listed in Table 1, when css = 0.625, the detent force peak-to-peak value equals 17.32 N, and the width of the side-slot and the active length of the proposed machine are equal to 11.538 mm and 323.077 mm, respectively.It should be noticed that the ratio between La and τs is 17.5.In fact, the detent force of the PM linear machine is composed of two components, namely, the slot effect and the end effect.And the adjustment of the width of side-slot is mainly to reduce the end effect detent force.When the active length of the short side of the LFSPM machine equals odd times of half of the stator tooth pitch, the phase difference of the two end effect detent force components caused by each end, respectively, equals 180°.Thus the resultant end effect detent force can be minimized, which is consistent with the FEM result.

Electromagnetic Performance Analysis
Based on the optimization design, a 3-phase 6/13-pole C-core LFSPM machine with the multiple teeth is designed.The corresponding key parameters are listed in Table 2. Then the static characteristics of the proposed machine are analyzed by using the FEM.Defining the active length of the proposed machine as the distance between the midlines of the two passive additional teeth, the active length of the proposed machine can be expressed as: where w mt , w PM and w mst are the width of the mover tooth, the width of the PM and the width of passive additional tooth, respectively, τ m is the pitch of the mover tooth, and L a is the active length of the proposed machine.Consequently, the width of the side-slot and the active length of the proposed machine can be obtained according to the dimension parameters.As listed in Table 1, when c ss = 0.625, the detent force peak-to-peak value equals 17.32 N, and the width of the side-slot and the active length of the proposed machine are equal to 11.538 mm and 323.077 mm, respectively.It should be noticed that the ratio between L a and τ s is 17.5.In fact, the detent force of the PM linear machine is composed of two components, namely, the slot effect and the end effect.And the adjustment of the width of side-slot is mainly to reduce the end effect detent force.When the active length of the short side of the LFSPM machine equals odd times of half of the stator tooth pitch, the phase difference of the two end effect detent force components caused by each end, respectively, equals 180 • .Thus the resultant end effect detent force can be minimized, which is consistent with the FEM result.

Electromagnetic Performance Analysis
Based on the optimization design, a 3-phase 6/13-pole C-core LFSPM machine with the multiple teeth is designed.The corresponding key parameters are listed in Table 2. Then the static characteristics of the proposed machine are analyzed by using the FEM. Figure 14 shows the 3-phase no-load flux linkage waveforms of four machines.It can be seen that flux linkage peak-to-peak values of Machine III and Machine IV are obviously improved compared with those of the other machines due to the contribution of the active additional teeth PMs.Machine II exhibits the maximum DC components and the most serious asymmetry in its 3-phase flux linkage peak-to-peak values.Furthermore, the asymmetry of Machine IV is effectively restrained, although the DC components are a little bigger than those of Machine I and Machine III.In order to indicate the improvement clearly, the key parameters of the no-load flux linkage are listed in Table 3.  Figure 14 shows the 3-phase no-load flux linkage waveforms of four machines.It can be seen that flux linkage peak-to-peak values of Machine III and Machine IV are obviously improved compared with those of the other machines due to the contribution of the active additional teeth PMs.Machine II exhibits the maximum DC components and the most serious asymmetry in its 3-phase flux linkage peak-to-peak values.Furthermore, the asymmetry of Machine IV is effectively restrained, although the DC components are a little bigger than those of Machine I and Machine III.In order to indicate the improvement clearly, the key parameters of the no-load flux linkage are listed in Table 3.Then the no-load EMF can be obtained by differentiating the flux linkage with respect to time.As shown in Figure 15, the no-load EMF waveforms of machine IV are also very symmetrical, and the peak-to-peak values of the 3-phase are 43.3V, 43.1 V and 43 V, respectively.
When the id = 0 control strategy is adopted, namely, the armature currents are in phase with the 3-phase no-load EMF, the thrust force can be calculated.Figure 16 shows the thrust force waveforms of four machines.The detent force waveforms are plotted in Figure 17.Then the no-load EMF can be obtained by differentiating the flux linkage with respect to time.As shown in Figure 15, the no-load EMF waveforms of machine IV are also very symmetrical, and the peak-to-peak values of the 3-phase are 43.3V, 43.1 V and 43 V, respectively.
When the i d = 0 control strategy is adopted, namely, the armature currents are in phase with the 3-phase no-load EMF, the thrust force can be calculated.Figure 16 shows the thrust force waveforms of four machines.The detent force waveforms are plotted in Figure 17.Then the no-load EMF can be obtained by differentiating the flux linkage with respect to time.As shown in Figure 15, the no-load EMF waveforms of machine IV are also very symmetrical, and the peak-to-peak values of the 3-phase are 43.3V, 43.1 V and 43 V, respectively.
When the id = 0 control strategy is adopted, namely, the armature currents are in phase with the 3-phase no-load EMF, the thrust force can be calculated.Figure 16 shows the thrust force waveforms of four machines.The detent force waveforms are plotted in Figure 17.Then the no-load EMF can be obtained by differentiating the flux linkage with respect to time.As shown in Figure 15, the no-load EMF waveforms of machine IV are also very symmetrical, and the peak-to-peak values of the 3-phase are 43.3V, 43.1 V and 43 V, respectively.
When the id = 0 control strategy is adopted, namely, the armature currents are in phase with the 3-phase no-load EMF, the thrust force can be calculated.Figure 16 shows the thrust force waveforms of four machines.The detent force waveforms are plotted in Figure 17.The corresponding values are compared in Table 4.It can be seen that Machine IV can achieve the maximum average value and the minimum peak-to-peak value of thrust force.Although the detent force peak-to-peak value of Machine II is a little smaller than that of Machine IV, the thrust ripple is higher than that of Machine IV.The peak-to-peak value of the detent force of Machine IV is 17.32 N, which is less than 8% of the average thrust force.

Conclusions
In this paper, a C-core LFSPM machine is proposed for long stroke applications such as cableless elevators.The key of the proposed machine is that two multiple additional teeth are employed at each end of the mover side.The multiple additional teeth are composed of an active additional tooth and a traditional passive additional tooth, in which the function of the active additional tooth is to balance the asymmetric 3-phase magnetic circuit due to the end effect, and the passive additional tooth contributes to reduce the detent force through adjusting the width of side-slot.Based on FEM analysis, the width of stator tooth and the width of side-slot are optimized to achieve high thrust force, low detent force, high no-load EMF THD, and so on.Finally, the average thrust force of 224.99 N and the detent force peak-to-peak value of 17.32 N of proposed machine can be achieved.It should be mentioned that Machine IV possesses the minimum thrust force ripple of 14.88% and the most balanced 3-phase no-load flux linkage among the four machines.It confirms that the end effect of proposed machine is restricted greatly due to multiple additional teeth.However, the thrust force ripple is still high for the stable operation of elevator due to the double tooth structure of FSPM machines, so suitable control strategies, such as the harmonic current injection method can be adopted to provide a lower thrust force ripple.The prototype machine and its experiment results will be presented in a future paper.The corresponding values are compared in Table 4.It can be seen that Machine IV can achieve the maximum average value and the minimum peak-to-peak value of thrust force.Although the detent force peak-to-peak value of Machine II is a little smaller than that of Machine IV, the thrust ripple is higher than that of Machine IV.The peak-to-peak value of the detent force of Machine IV is 17.32 N, which is less than 8% of the average thrust force.

Conclusions
In this paper, a C-core LFSPM machine is proposed for long stroke applications such as cableless elevators.The key of the proposed machine is that two multiple additional teeth are employed at each end of the mover side.The multiple additional teeth are composed of an active additional tooth and a traditional passive additional tooth, in which the function of the active additional tooth is to balance the asymmetric 3-phase magnetic circuit due to the end effect, and the passive additional tooth contributes to reduce the detent force through adjusting the width of side-slot.Based on FEM analysis, the width of stator tooth and the width of side-slot are optimized to achieve high thrust force, low detent force, high no-load EMF THD, and so on.Finally, the average thrust force of 224.99 N and the detent force peak-to-peak value of 17.32 N of proposed machine can be achieved.It should be mentioned that Machine IV possesses the minimum thrust force ripple of 14.88% and the most balanced 3-phase no-load flux linkage among the four machines.It confirms that the end effect of proposed machine is restricted greatly due to multiple additional teeth.However, the thrust force ripple is still high for the stable operation of elevator due to the double tooth structure of FSPM machines, so suitable control strategies, such as the harmonic current injection method can be adopted to provide a lower thrust force ripple.The prototype machine and its experiment results will be presented in a future paper.

Figure 2 .
Figure 2. C-core linear flux-switching permanent magnet machine configurations.(a) Without additional tooth, Machine I; (b) with traditional passive additional tooth, Machine II; (c) with active additional tooth, Machine III; (d) with multiple additional tooth, Machine IV.

Figure 2 .
Figure 2. C-core linear flux-switching permanent magnet machine configurations.(a) Without additional tooth, Machine I; (b) with traditional passive additional tooth, Machine II; (c) with active additional tooth, Machine III; (d) with multiple additional tooth, Machine IV.
shows the 3-phase PM flux linkage waveforms of Machine I and Machine II.It can be seen that the flux linkage waveforms of Machine I are asymmetrical in the positive and negative half-cycle.Furthermore, the flux linkage waveforms of Machine II are more deteriorated compared with those of Machine I due to the magnetic circuit of the traditional additional teeth.

Figure 4 .
Figure 4. 3-phase PM flux linkage waveforms of Machine I and Machine II.

Figure 4 .
Figure 4. 3-phase PM flux linkage waveforms of Machine I and Machine II.

Figure 4 .
Figure 4. 3-phase PM flux linkage waveforms of Machine I and Machine II.

Figure 6 .
Figure 6.3-phase flux linkage waveforms of Machine III.

Figure 9 .
Figure 9. Detent force peak-to-peak value with different cstt and c stb .

Figure 10 .
Figure 10.THD of no-load EMF with different cstt and c stb .

Figure 9 .
Figure 9. Detent force peak-to-peak value with different cstt and c stb .

Figure 10 .
Figure 10.THD of no-load EMF with different cstt and c stb .

Figure 9 .
Figure 9. Detent force peak-to-peak value with different c stt and c stb .

Figure 10 Figure 8 .
Figure10shows the change of the no-load EMF THD with respect to c stt and c stb .It indicates that the no-load EMF THD increases with the rise of c stt .And, the no-load EMF THD is not sensitive to c stb either.

Figure 9 .
Figure 9. Detent force peak-to-peak value with different cstt and c stb .

Figure 10 .
Figure 10.THD of no-load EMF with different cstt and c stb .

Figure 10 .
Figure 10.THD of no-load EMF with different c stt and c stb .

Figure 11 .
Figure 11.No-load EMF peak-to-peak value with different cstt and c stb .

Figure 12 .
Figure 12.Average thrust force value with different cstt and c stb .

Figure 11 .
Figure 11.No-load EMF peak-to-peak value with different c stt and c stb .

Figure 11 .
Figure 11.No-load EMF peak-to-peak value with different cstt and cstb.

Figure 12 .
Figure 12.Average thrust force value with different cstt and cstb.

Figure 12 .
Figure 12.Average thrust force value with different c stt and c stb .

Figure 13 .
Figure 13.Detent force waveforms with different c ss .

Table 1 .
Side-slot width and machine active length with different css.

Table 1 .
Side-slot width and machine active length with different c ss .

Table 2 .
Key parameters of the proposed machine.

Table 2 .
Key parameters of the proposed machine.

Table 4 .
Thrust force and detent force values (N).

Table 4 .
Thrust force and detent force values (N).