Numerical Analysis and Parametric Study of a 7 Kw Tubular Permanent Magnet Linear Alternator

sustainability

Article Numerical Analysis and Parametric Study of a 7 kW Tubular Permanent Magnet Linear Alternator

Chin-Hsiang Cheng 1,* and Surender Dhanasekaran 2

1 Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan City 70101, Taiwan 2 International Doctoral Degree Program on Energy Engineering, National Cheng Kung University, Tainan City 70101, Taiwan; [email protected] * Correspondence: [email protected]; Tel.: +886-6-2757575 (ext. 63627)

Abstract: Free-Piston Stirling Engines (FPSEs) are known for their easy maintenance, longer lifetimes, high reliability, quiet operation due to no crankshafts, and having fewer seals compared to the traditional Stirling engine. Free-piston systems are popular in the conversion of thermal energy into electrical energy and are compatible with many types of heat sources. This research paper concen- trates on the development of a Permanent Magnet Linear Alternator (PMLA) and parametrically analyzing it to predict its limitations and performance over variable operable conditions and material choices. Operable conditions including stroke length and frequency of the translator, and material choice for the stator and magnets, are varied in this study to analyze the machine and put it to test for its extreme limitations. Spacing between slots is introduced to reduce the overall mass of the

stator and increase the power density. The load test is carried out with varied parameters. It induces a load EMF of 2.4 kV, yields a power of 7 kW, and has a power density of 314 W/kg by FEM analysis

Citation: Cheng, C.-H.; in peak variations. This study enumerates the performance variation of a PMLA over these varied Dhanasekaran, S. Numerical Analysis conditions and illustrates the limitations of such power-dense machines. and Parametric Study of a 7 kW Tubular Permanent Magnet Linear Keywords: permanent magnet linear alternator; free-piston Stirling engine; ﬁnite element analysis; Alternator. Sustainability 2021, 13, parametric study; quasi-Halbach array 7192. https://doi.org/10.3390/ su13137192

Academic Editors: Luca Cioccolanti, 1. Introduction Electo Eduardo Silva Lora and Global warming has been continuously rising in recent years and will continue to grow Mauro Villarini if no necessary actions are taken. Sustainable growth meeting the global demands for power production and reduction in greenhouse gas emissions should be brought under control Received: 14 May 2021 by 2030 or catastrophic disasters will happen with irreversible consequences. Limiting Accepted: 22 June 2021 ◦ Published: 26 June 2021 the global temperature rise within 1.5 C could slow down this process and make way for sustainable growth [1]. The demand for global energy supply has been consistently rising

Publisher’s Note: MDPI stays neutral in the recent past. Environmental pollution has been a major concern over the conventional with regard to jurisdictional claims in method of power production, and researchers across the globe are continuously working published maps and institutional afﬁl- on perfecting green energy technology to reduce pollution and ﬁnd alternative methods iations. of energy harvesting techniques in the power production sector. Conventional modes of power production have mostly used rotational motion. Oscillatory motion power sources have been converted to rotary motion ones for power production before the familiarity with linear generators, which resulted in acute mechanical power losses. The perk of a linear alternator is reversible; it can be used as an actuator for multiple applications which Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. requires two machines to start and continue to produce power [2]. In the recent past, linear This article is an open access article electric actuators and generators have become popular in domains such as FPSE, wave distributed under the terms and energy harvesting, hybrid electric vehicles, and sub-domaining applications such as waste conditions of the Creative Commons heat recovery, concentrated solar power, and space applications [3–6]. Attribution (CC BY) license (https:// Stirling engines have been advancing in the past 50 years and have been widely used ◦ creativecommons.org/licenses/by/ in solar energy harvesting. Having a temperature as low as 300 C, Stirling engines are 4.0/). even used in waste heat recovery systems in industries for power production with only

Sustainability 2021, 13, 7192. https://doi.org/10.3390/su13137192 https://www.mdpi.com/journal/sustainability Sustainability 2021, 13, 7192 2 of 15

several watts [7]. A waste heat recovery system is used to pump the heat to be used as an additional source to the prime movers such as thermoacoustic engines and FPSEs [8,9]. NASA has been consistently developing free-piston energy systems continuously for space applications. A 12.5 kW electrical output with piston displacement of 14 mm at 70 Hz and a 6 kW electrical output with piston displacement of 16 mm at 60 Hz were developed in the years 1999 and 2010, respectively, by NASA’s Stirling space engine programs in multiple phases [10,11]. Linear alternators are categorized based on topology, phase, and type of mover. By topology, PMLAs are categorized into flat and tubular types; by phase, they are categorized into single and three-phase; by type of mover, they are categorized as moving magnet, moving iron, and moving coil [12]. Linear crankless internal combustion engines with only one moving part with a compact structure, higher efficiency, and reliability for remote power production applications integrate PMLAs in one linear system [13]. In a similar system, the PMLAs are used in the power generation to be used as an auxiliary power unit for a Hybrid Electric Vehicle (HEV) [14]. These integrated systems are used as range extenders in the HEVs apart from the main driving mechanism [15]. Wave energy is another key application of linear alternators. The fundamental difference observed in the applications of PMLAs in wave energy harvesting and combustion (internal and external) engines from a bird’s-eye view is Wave Energy Converters (WECs), which have a long stroke length and low frequency, and combustion inputs have a smaller stroke length and a higher frequency [4,13]. However, the construction and operation fea- tures of both of the alternator applications are different. WECs are power-dense machines ranging from a few to several hundred kilowatts of electrical power deliverability [16]. Their stroke lengths are about hundred times that of combustion applications. Continuous research is conducted on this front to reach sustainable growth and generalize this source of power. Recent developments in the research and manufacturing abilities of rare-earth magnets have provided the possibility to pack multiple combinations of NdFeB magnets for various utilizations [17], especially extending the demagnetization temperature to a maximum of 180 ◦C for special applications [18]. The Quasi-Halbach (QH) magnetization pattern of arranging the magnets creates a rotating pattern of magnetization. For the tubular PMLAs, the strong and weak sides of the arrangement are decided based on the application and type of the mover [19]. Axial and radial array magnets are placed depending on the number of strong and weak sections required by the specific application that the PMLA is designed for. In the constructional arrangement of the stator in a PMLA, the stator can be divided into multiple segments for structural independence. A tubular PMLA has a single coil accommodated in a 12-segment split stator or a flat-type PMLA with 18 coils accommodated in a four-segment squared-off stator to ensure structural independence and is applied in different applications [20–22]. Similarly, slotless generators are becoming popular for their high efficiency and negligible cogging force on the translator [23]. However, the flux linkage on the stator is minimal, and thus a lower EMF is induced in the coil. Several modifications are made to the permanent magnet to give satisfactory results. The utilization of the QH array over axial magnets proves to be more efficient in slotless generators [24]. In a free-piston linear system simulation, the integration of thermodynamic, dynamic, and electromagnetic models is necessary to predict the behavior of the system in the real- time working state. For operating the system at its full efficient state, a parametric study of the PMLA is necessary to understand the ability to work in a stable engine operation state [24]. For post-parametric analysis on FEA software, before the integration of the PMLA onto an engine, an experimental set-up is set to test the ability to match the numerical analysis results [25,26]. A rotary motor is set to operate at the operating frequency of the PMLA. Cogging force is a locking mechanism acting on the translator due to the magnetic field of the permanent magnet with the stator [27]. Cogging force is responsible for the noise and vibration created on the translator and causes excessive maintenance for the Sustainability 2021, 13, 7192 3 of 15

PMLA. Several techniques are used in the reduction in the cogging force, mostly by altering the shape of the magnet [28]. Using these techniques, the cogging force is reduced by 40%, also affecting the performance of the PMLA. These techniques are used on radial and axial magnets. Due to the utilization of a Soft Magnetic Composite (SMC) in most PMLAs, the eddy-current losses are negligible even by using a solid core [29]. Having considered all the constraints from the limitations of having a lighter machine to operate at a higher frequency, the overall mass of the machine should be reduced to have a higher specific power. Hence, the volume of the machine should be reduced to attain a higher efficient state of the PMLA [30]. The induced EMF is not readily usable and requires a control system to manage the starting and operation of the integrated system [31]. Linear alternators are designed and executed in different applications. However, their limitations are not tested or shown. Considering a proven alternator design [5], a few changes were made to the design to test the limitations of the PMLA numerically on a commercial software JMAG® Designer V18. To test these limitations, four parametric changes are made in this study. The specific power is one important parameter to be considered for remote applications such as space applications. By increasing the number of turns in a coil and reducing the overall mass of the stator, the specific power is boosted.

2. PMLA Design Due to the structural merits, the tubular type of linear alternator is more reliable than the flat-type [3]. The alternator design considered in this paper is an adaptation from a conventional model [5]. In convergence with the research aim, the stator of the PMLA was redesigned to accommodate a greater number of turns than the original model. For the design changes, firstly, the back iron thickness was reduced by 30% from the original dimensions. Secondly, slot spaces were included between the poles. The original design consisting of a five-pole arrangement was retained to not make any major change including the outer diameter and stack length of the stator and the air gap between the translator and the stator. By making these two major changes, the slot became wider and deeper. The introduction of the slot space reduced the mass of the stator from 15.055 kg to 7.6 kg—a 50% reduction in the overall mass of the stator. This resulted in accommodating more coil turns per slot, which raised from 2024 turns to 2990 turns per slot—a 30% raise per slot. According to Faraday’s law (1), the induced electromotive force (e) is directly dependent on the number of coil turns (N) and the magnetic flux (φ) linked to the coil.

dφ e = −N (1) dt The geometrical design data of the altered model presented in this research are mentioned in Table1. The complete PMLA structure considered for the simulation is shown in Figure1. The stacking factor of the stator is kept at 96% for the stator and the translator shaft is prescribed by the manufacturer of laminated steel. The mass of each part of PMLA is mentioned in Table2. The overall mass of the PMLA is 22.7 kg.

Table 1. Geometrical parameters of PMLA.

Parameters Dimensions (mm)

Stator Outer Diameter, So 136 Slot Width, Sw 21 Slot Height, Sh 30 Back Iron Thickness, Bt 5 Tooth Slot Thickness, Ss 5 Tooth Slot Gap, Ts 10.5 Stator Stack Length, Sl 230.5 Magnet Outer Diameter, Md 63 QH (Axial) Magnet Thickness, Ma 16.5 QH (Radial) Magnet Thickness, Mr 25 Sustainability 2021, 13, 7192 4 of 16

Table 1. Geometrical parameters of PMLA.

Parameters Dimensions (mm) Stator Outer Diameter, So 136 Slot Width, Sw 21 Slot Height, Sh 30 Back Iron Thickness, Bt 5 Tooth Slot Thickness, Ss 5 Tooth Slot Gap, Ts 10.5 Sustainability 2021, 13, 7192 4 of 15 Stator Stack Length, Sl 230.5 Magnet Outer Diameter, Md 63 Table 1. Cont.QH (Axial) Magnet Thickness, Ma 16.5 QH (Radial) Magnet Thickness, Mr 25 Parameters Dimensions (mm) Coil Inner Diameter, Ci 66 CoilCoil Inner Outer Diameter, Diameter, Ci Co 66 125 Coil Outer Diameter, C 125 Shaft Stack Length,o Hl 163.5 Shaft Stack Length, Hl 163.5 Shaft Inner Diameter, Hi 16 Shaft Inner Diameter, Hi 16 Shaft Outer Diameter, Hd 26 Shaft Outer Diameter, Hd 26 Air GapAir Gap 1.5 1.5

Table 2. Mass of parts in PMLA. Table 2. Mass of parts in PMLA. Part Mass (kg) Part Mass (kg) Stator 7.6 Stator 7.6 Shaft 4.1 Shaft 4.1 MagnetsMagnets 2.7 2.7 CoilsCoils 8.3 8.3

Figure 1. Slot spaced PMLA.

Figure 1. Slot spaced PMLA. 3. Baseline Case The PMLA is assigned with ﬁxed conditions for a baseline case. The baseline cases

for the variable parameters are mentioned in Table3. To ensure the continuity of the translator movement, a ten-cycle simulation has been carried out and found to have a stable movement over the period. Due to the QH array magnet arrangement, the strong and weak sections of the array form a uniform distribution of magnetic flux density (Figure2a) in the stator. The flux line (Figure2b) is uniformly distributed around the stator; the mark is kept at 10 mm from the start of the stator stack to understand the flow in the first slot.

Table 3. Baseline case parameters.

Baseline Parameters Parameter Stator material 35CS300 Magnet material N48H Stroke length (mm) 10 Frequency (Hz) 60 Sustainability 2021, 13, 7192 5 of 16

3. Baseline Case The PMLA is assigned with fixed conditions for a baseline case. The baseline cases for the variable parameters are mentioned in Table 3. To ensure the continuity of the translator movement, a ten-cycle simulation has been carried out and found to have a stable movement over the period. Due to the QH array magnet arrangement, the strong and weak sections of the array form a uniform distribution of magnetic flux density (Figure 2a) in the stator. The flux line (Figure 2b) is uniformly distributed around the stator; the mark is kept at 10 mm from the start of the stator stack to understand the flow in the first slot.

Table 3. Baseline case parameters.

Baseline Parameters Parameter Stator material 35CS300 Magnet material N48H Stroke length (mm) 10 Frequency (Hz) 60 Sustainability 2021, 13, 7192 5 of 15

(a)

(b)

Figure 2.FigureMagnetic 2. Magnetic flux density flux for density a no-load for a baseline no-load case: baseline (a) Vector case: ( plot;a) Vector (b) flux plot; line (b at) flux 10 mm line from at 10 the mm stack. from the stack. The baseline case is run in an open-circuit condition. Figure3 shows the transient The baselinevariations case is of run no-load in an voltage open-circuit and displacement condition. Figure (D) for 3 shows two cycles. the transient A peak open-circuit variations ofvoltage no-load Voc voltage of 3052 and V anddisplacement RMS voltage (D) offor 2052 two V cycles. was obtained A peak atopen-circuit the displacement dead voltage Voc ofpoint. 3052 AV sinusoidaland RMS voltage representation of 2052 V of was the obtained induced EMFat the such displacement as the displacement dead profile is observed. This slotted PMLA has a QH array; therefore, there is no compromise on performance and no-load cogging force (Cn), with a similar profile to the displacement exerted on the translator, which further reduces the vibration on the translator, bringing stability to the machine. This profile is observed throughout the parametric variations made. For the baseline case, a peak cogging force of 1568 N is induced on the translator which can be observed in Figure4. Sustainability 2021, 13, 7192 6 of 16 Sustainability 2021, 13, 7192 6 of 16

point. A sinusoidal representation of the induced EMF such as the displacement profile is point. A sinusoidal representation of the induced EMF such as the displacement profile is observed. observed. This slotted PMLA has a QH array; therefore, there is no compromise on performance This slotted PMLA has a QH array; therefore, there is no compromise on performance and no-load cogging force (Cn), with a similar profile to the displacement exerted on the and no-load cogging force (Cn), with a similar profile to the displacement exerted on the translator, which further reduces the vibration on the translator, bringing stability to the translator, which further reduces the vibration on the translator, bringing stability to the machine. This profile is observed throughout the parametric variations made. For the machine. This profile is observed throughout the parametric variations made. For the baseline case, a peak cogging force of 1568 N is induced on the translator which can be baseline case, a peak cogging force of 1568 N is induced on the translator which can be observed in Figure 4. observed in Figure 4. Sustainability 2021, 13, 7192 6 of 15

4000 10 4000 10 Voc VocD D 2000 5 2000 5

0 0 0 0 Voltage (V) Voltage (V) Displacement (mm)

-2000 -5 Displacement (mm) -2000 -5

-4000 -10 -40000.000 0.006 0.011 0.017 0.022 0.028 0.033-10 0.000 0.006 0.011 Time0.017 (s) 0.022 0.028 0.033 Time (s)

Figure 3. Transient variation of no-load voltage and displacement. FigureFigure 3. 3. TransientTransient variation of no-load voltage and displacement.

2000 10 2000 10 Cn Cn D D 1000 5 1000 5

0 0 0 0 Cogging force (N) Displacement (mm) Cogging force (N) -1000 -5 Displacement (mm) -1000 -5

-2000 -10 -20000.000 0.006 0.011 0.017 0.022 0.028 0.033-10 0.000 0.006 0.011 Time0.017 (s) 0.022 0.028 0.033 Time (s)

Figure 4. Transient variation of cogging force andand displacement.displacement. Figure 4. Transient variation of cogging force and displacement. 4. Parametric Study 4. Parametric Study 4. ParametricThe PMLA Study consists of several physical parameters and conditions to be assigned for The PMLA consists of several physical parameters and conditions to be assigned for the numericalThe PMLA analysis consists to of be several conducted, physical out parameters of which, few and parameters conditions cannot to be assigned be changed for the numerical analysis to be conducted, out of which, few parameters cannot be changed thedue numerical to the direct analysis inﬂuence to be theyconducted, hold with out of the which, geometrical few parameters dimensions, cannot thus be changing changed due to the direct influence they hold with the geometrical dimensions, thus changing the duethe originalto the direct design. influence However, they the hold operating with the conditions geometrical assigned dimensions, to the thus translator changing are the original design. However, the operating conditions assigned to the translator are the fre- originalfrequency design. and strokeHowever, length the in operating the electromagnetic conditions assigned simulation. to the The translator choice of are materials the frequency and stroke length in the electromagnetic simulation. The choice of materials as- quencyassigned and for stroke the PMLA length can in be the varied electromagne to study thetic simulation. behavior change The choi andce to of choose materials the best assigned for the PMLA can be varied to study the behavior change and to choose the best signedcase for for the the PMLA. PMLA Predominantly, can be varied theto study system the consists behavior ofthree change major and components: to choose the stator, best coil, and the translator (shaft and magnets). The material choice for the coil is insulated copper and cannot be varied. The choice of magnet and stator material was chosen based on the previous research carried out by our research group [32]. The numerical simulations were carried out subject to the understanding of epistemic uncertainties and were assumed to be in near-perfect condition [33,34].

5. Results and Discussion All simulations were carried out with a purely resistive load of 1000 Ω. Due to the utilization of the SMC core, there is a negligible value of eddy-current losses. Hence, eddy- current losses were neglected from the study for all the variations made. All representations Sustainability 2021, 13, 7192 7 of 16

case for the PMLA. Predominantly, the system consists of three major components: stator, coil, and the translator (shaft and magnets). The material choice for the coil is insulated copper and cannot be varied. The choice of magnet and stator material was chosen based on the previous research carried out by our research group [32]. The numerical simulations were carried out subject to the understanding of epistemic uncertainties and were assumed to be in near-perfect condition [33,34].

5. Results and Discussion All simulations were carried out with a purely resistive load of 1000 Ω. Due to the utilization of the SMC core, there is a negligible value of eddy-current losses. Hence, eddy- Sustainability 2021, 13, 7192 current losses were neglected from the study for all the variations made. All representa-7 of 15 tions of Power (Pl), voltage (Vl), and cogging force (Cl) are Root Mean Square (RMS) values in the parametric study results. of Power (Pl), voltage (Vl), and cogging force (Cl) are Root Mean Square (RMS) values in 5.1.the Effects parametric of Frequency study results. Frequency is one of the operating conditions assigned for the simulation. It was var- 5.1. Effects of Frequency ied in steps of 10 from 10 Hz to 100 Hz. All other parameters were set in the baseline case to maintainFrequency uniformity. is one of theFigure operating 5 represents conditions the plot assigned between for the the simulation. induced power It was versus varied thein steps varied of frequency. 10 from 10 A Hz linear to 100 rise Hz. was All seen other over parameters a course wereuntil set60 Hz. in the After baseline 60 Hz, case satu- to rationmaintain was uniformity. observed in Figure the curve.5 represents Since mo thest plotengines between operate the in induced this range power from versus 60 Hz the to 100varied Hz, frequency. it was deemed A linear reasonable rise was to seen stop over at this a course level and until this 60 PMLA Hz. After also 60 agrees Hz, saturation with this was observed in the curve. Since most engines operate in this range from 60 Hz to 100 Hz, range [3]. It is analogous that the induced EMF has a similar profile to the power; Figure it was deemed reasonable to stop at this level and this PMLA also agrees with this range [3]. 6 shows the induced EMF for the frequency variation. It is analogous that the induced EMF has a similar proﬁle to the power; Figure6 shows the The cogging force on the translator is constant after 10 Hz. There is a steep rise in the induced EMF for the frequency variation. cogging force from 10 Hz to 20 Hz. Figure 7 represents the influence of frequency change The cogging force on the translator is constant after 10 Hz. There is a steep rise in the in cogging force on the translator. From 20 Hz to 100 Hz, there is a near-constant force cogging force from 10 Hz to 20 Hz. Figure7 represents the inﬂuence of frequency change with minute variations from 674 N to 681 N. in cogging force on the translator. From 20 Hz to 100 Hz, there is a near-constant force with minute variations from 674 N to 681 N.

1000

750

500 Power (W)

250

0 Sustainability 2021, 13, 7192 0 10 20 30 40 50 60 70 80 90 100 110 8 of 16 Frequency (Hz)

FigureFigure 5. 5. InfluenceInﬂuence of of frequency frequency on on power. power.

1000

750

500 Voltage (V)

250

0 0 10 20 30 40 50 60 70 80 90 100 110 Frequency (Hz)

FigureFigure 6. 6. InfluenceInﬂuence of of frequency frequency on on load load voltage. voltage.

800

600

400 Cogging Force (N) 200

0 0 10 20 30 40 50 60 70 80 90 100 110 Frequency (Hz)

Figure 7. Influence of frequency on the cogging force.

5.2. Effects of Stroke Length Stroke length is another operating condition assigned for the simulation. The movement of the translator from the top dead point to the bottom dead point is considered here as one complete stroke. This stroke length was varied from 5 mm to 40 mm in 5 mm increments. The influence of stroke length is strong on induced power. Figure 8 represents the influence of stroke length on power induced on the load. There is a linear growth in the power with the rise in stroke length. Due to the limitation of the design, it was limited to a 40 mm stroke for this analysis. The power induced on the resistive load varied from 210 W for a 5 mm stroke to 7116 W for a 40 mm one. A wide band of power deliverability is available for the stroke variation. Similarly, the induced load voltage, from the influence of stroke length, has the same profile as power, shown in Figure 9. The induced load voltage varied from 408 V to 2395 V the stroke varied from 5 mm to 40 mm. Sustainability 2021, 13, 7192 8 of 16

1000

750

500 Voltage (V)

250

0 0 10 20 30 40 50 60 70 80 90 100 110 Frequency (Hz)

Sustainability 2021, 13, 7192 Figure 6. Influence of frequency on load voltage. 8 of 15

800

600

400 Cogging Force (N) 200

0 0 10 20 30 40 50 60 70 80 90 100 110 Frequency (Hz)

FigureFigure 7. 7. InfluenceInﬂuence of of frequency frequency on on the the cogging cogging force. force.

5.2.5.2. Effects Effects of of Stroke Stroke Length Length StrokeStroke length length is is another another operating operating condition condition assigned assigned for for the the simulation. simulation. The The move- move- mentment of of the the translator translator from from the the top top dead dead poin pointt to the to thebottom bottom dead dead point point is considered is considered here here as one complete stroke. This stroke length was varied from 5 mm to 40 mm in 5 mm as one complete stroke. This stroke length was varied from 5 mm to 40 mm in 5 mm in- increments. The influence of stroke length is strong on induced power. Figure8 represents crements. The influence of stroke length is strong on induced power. Figure 8 represents the influence of stroke length on power induced on the load. There is a linear growth in the influence of stroke length on power induced on the load. There is a linear growth in the power with the rise in stroke length. Due to the limitation of the design, it was limited the power with the rise in stroke length. Due to the limitation of the design, it was limited to a 40 mm stroke for this analysis. The power induced on the resistive load varied from to a 40 mm stroke for this analysis. The power induced on the resistive load varied from 210 W for a 5 mm stroke to 7116 W for a 40 mm one. A wide band of power deliverability is Sustainability 2021, 13, 7192 210 W for a 5 mm stroke to 7116 W for a 40 mm one. A wide band of power deliverability9 of 16 available for the stroke variation. Similarly, the induced load voltage, from the influence of is available for the stroke variation. Similarly, the induced load voltage, from the influence stroke length, has the same profile as power, shown in Figure9. The induced load voltage of stroke length, has the same profile as power, shown in Figure 9. The induced load volt- varied from 408 V to 2395 V the stroke varied from 5 mm to 40 mm. age varied from 408 V to 2395 V the stroke varied from 5 mm to 40 mm. 8

4 Power (KW)

0 0 5 10 15 20 25 30 35 40 45 Stroke length (mm)

FigureFigure 8. 8.EffectEffect of of stroke stroke length length on on power. power.

TheThe cogging cogging force force has has a alinear linear elevation elevation in in Figure Figure 10 10 due due to to a achange change in in stroke stroke length length andand attains attains saturation saturation after after 25 25 mm. mm. This This saturation saturation is issimilar similar to to the the frequency frequency variation variation fromfrom 20 20 Hz Hz to to100 100 Hz. Hz. This This level level of saturation of saturation leaves leaves a choice a choice to operate to operate the machine the machine with- outwithout a strong a strong influence inﬂuence of the of cogging the cogging force force after after a 25 a mm 25 mm stroke stroke for for an an application application that that requiresrequires more more power. power.

Voltage (KV) 1

0 0 5 10 15 20 25 30 35 40 45 Stroke length (mm)

Figure 9. Effect of stroke length on load voltage. Sustainability 2021, 13, 7192 9 of 16

4 Power (KW)

0 0 5 10 15 20 25 30 35 40 45 Stroke length (mm)

Figure 8. Effect of stroke length on power.

The cogging force has a linear elevation in Figure 10 due to a change in stroke length and attains saturation after 25 mm. This saturation is similar to the frequency variation from 20 Hz to 100 Hz. This level of saturation leaves a choice to operate the machine with- out a strong influence of the cogging force after a 25 mm stroke for an application that requires more power. Sustainability 2021, 13, 7192 9 of 15

Voltage (KV) 1

0 0 5 10 15 20 25 30 35 40 45 Sustainability 2021, 13, 7192 10 of 16 Stroke length (mm)

FigureFigure 9. 9.EffectEffect of ofstroke stroke length length on on load load voltage. voltage.

1500

1000

500 Cogging Force (N)

0 0 5 10 15 20 25 30 35 40 45 Stroke length (mm)

Figure 10.10. Effect of stroke lengthlength onon thethe coggingcogging force.force. 5.3. Effects of Magnet Material 5.3. Effects of Magnet Material Using NXXH magnets for their higher demagnetization temperature. For the case of Using NXXH magnets for their higher demagnetization temperature. For the case of magnet material variation, N30H, N40H, N48H, and N50H were considered. The influence magnet material variation, N30H, N40H, N48H, and N50H were considered. The influ- of magnet material change on power and cogging force is represented in Figure 11. There ence of magnet material change on power and cogging force is represented in Figure 11. was a change in induced power from 514 W to 826 W due to the magnetic material change. There was a change in induced power from 514 W to 826 W due to the magnetic material Similarly, the cogging force varied from 338 N to 542 N. However, the similarity of the change. Similarly, the cogging force varied from 338 N to 542 N. However, the similarity profile between the power and cogging force was uniform throughout the material change. of the profile between the power and cogging force was uniform throughout the material It is evident that the induced power comes with a similar cogging force, and they are change. It is evident that the induced power comes with a similar cogging force, and they proportional to each other. This similarity is found in the induced load voltage in Figure 12. Hence,are proportional choosing to the each right other. magnetic This similarity material is for found a specific in the induced application load and voltage maximum in Fig- coggingure 12. Hence, force handling choosing ability the right should magnetic be in equilibrium. material for a specific application and maximum cogging force handling ability should be in equilibrium.

1000 600

Pl Cl 500 800

400 600

300

Power (W) 400 200 Cogging force (N)

200 100

0 0 N30H N40H N48H N50H Magnet material

Figure 11. Effect of magnet material on power and cogging force. Sustainability 2021, 13, 7192 10 of 16

1500

1000

500 Cogging Force (N)

0 0 5 10 15 20 25 30 35 40 45 Stroke length (mm)

Figure 10. Effect of stroke length on the cogging force.

5.3. Effects of Magnet Material Using NXXH magnets for their higher demagnetization temperature. For the case of magnet material variation, N30H, N40H, N48H, and N50H were considered. The influence of magnet material change on power and cogging force is represented in Figure 11. There was a change in induced power from 514 W to 826 W due to the magnetic material change. Similarly, the cogging force varied from 338 N to 542 N. However, the similarity of the profile between the power and cogging force was uniform throughout the material change. It is evident that the induced power comes with a similar cogging force, and they are proportional to each other. This similarity is found in the induced load voltage in Fig- ure 12. Hence, choosing the right magnetic material for a specific application and maximum cogging force handling ability should be in equilibrium. Sustainability 2021, 13, 7192 10 of 15

1000 600

Pl Cl 500 800

400 600

300

Power (W) 400 200 Cogging force (N)

200 100

0 0 Sustainability 2021, 13, 7192 N30H N40H N48H N50H 11 of 16 Magnet material

FigureFigure 11. 11. EffectEffect of of magnet magnet material material on on power power and and cogging cogging force. force.

1000

900

800

Voltage (V) 700

600

500 N30H N40H N48H N50H Magnet material

Figure 12. Effect of magnet material on load voltage. 5.4. Effects of Stator Material 5.4. Effects of Stator Material Electrical steel for the stator and translator shaft was chosen from the same material Electrical steel for the stator and translator shaft was chosen from the same material combination for the variation. Three different materials—35CS300, 35CS210 and 50CS290— werecombination selected forfor thisthe comparison. variation. TheThree 35 anddifferent 50 series materials material— varied35CS300, with 35CS210 the thickness and 50CS290of the sheet—were metal: selected 0.35 mm for andthis 0.5comparison. mm for 35 The and 35 50 and series 50 materials.series material Additionally, varied with the thedensity thickness of the of material the sheet varies metal: with 0.35 thickness. mm andSince 0.5 mm there for is 35 a and limitation 50 series of stackingmaterials. factor Ad- ditionally,for the sheet the metals density at 96%of the for material all the materialsvaries with from thickness. China Steel, Since consideration there is a limitation of denser of stackingmaterials factor was avoidedfor the sheet for this metals comparison. at 96% for The all influencethe materials of a from change China in stator Steel, material consid- erationon Pl and of Cdenserl is not materials significant was enough, avoided though, for this in comparison. comparison The to theinfluence profiles of of a Pchangel and Cinl, statorthe 35 material series steel on P hasl and a C betterl is not ratio significant than the enough, 50 series though steel, seenin comparison in Figure to13 the. Similarly, profiles ofthe P 35l and series Cl, the has 35 a better series V steell induction has a better capability ratio thanthan thethe 5050 seriesseries steel.steel seen This comparativein Figure 13. Similarly,analysis can the be 35 seen series in Figurehas a better14. Vl induction capability than the 50 series steel. This comparative analysis can be seen in Figure 14.

800 526

Cl 799 525

798 524 Power (W) Cogging force (N) 797 523

796 522 35CS300 35CS210 50CS290 Stator Material

Figure 13. Effect of stator material on power and cogging force. Sustainability 2021, 13, 7192 11 of 16

1000

900

800

Voltage (V) 700

600

500 N30H N40H N48H N50H Magnet material

Figure 12. Effect of magnet material on load voltage.

5.4. Effects of Stator Material Electrical steel for the stator and translator shaft was chosen from the same material combination for the variation. Three different materials—35CS300, 35CS210 and 50CS290—were selected for this comparison. The 35 and 50 series material varied with the thickness of the sheet metal: 0.35 mm and 0.5 mm for 35 and 50 series materials. Ad- ditionally, the density of the material varies with thickness. Since there is a limitation of stacking factor for the sheet metals at 96% for all the materials from China Steel, consideration of denser materials was avoided for this comparison. The influence of a change in stator material on Pl and Cl is not significant enough, though, in comparison to the profiles of Pl and Cl, the 35 series steel has a better ratio than the 50 series steel seen in Figure 13. Similarly, the 35 series has a better Vl induction capability than the 50 series steel. This Sustainability 2021, 13, 7192 comparative analysis can be seen in Figure 14. 11 of 15

800 526

Cl 799 525

798 524 Power (W) Cogging force (N) 797 523

Sustainability 2021, 13, 7192 796 522 12 of 16 35CS300 35CS210 50CS290 Stator Material

FigureFigure 13. 13. EffectEffect of of stator stator material material on on power power and and cogging cogging force. force.

797

796.5

796 Voltage (V)

795.5

795 35CS300 35CS210 50CS290 Stator Material

FigureFigure 14. 14. EffectEffect of of stator stator material material on on load load voltage. voltage. 6. Validation 6. Validation To validate the current results from the FEA simulation of the derived model, the To validate the current results from the FEA simulation of the derived model, the original model is designed and simulated assigning the original conditions from Ref. [5]. original model is designed and simulated assigning the original conditions from Ref. [5]. Figure 15 represents the comparison between the results from the paper (Figure 15a) and Figure 15 represents the comparison between the results from the paper (Figure 15a) and FEA simulation results (Figure 15b) carried out for validation purposes. The impacts of FEA simulation results (Figure 15b) carried out for validation purposes. The impacts of translator position on induced voltage for the Halbach array and radial array are compared translator position on induced voltage for the Halbach array and radial array are com- from the experimental results of Behrooz Rezaeealam [5]. In comparison to the original paredresults from from the Figure experimental 15a, the results simulated of Behrooz results Rezaeealam from Figure [5]. 15 bIn have comparison similar to profiles the orig- at a inalmaximum results from displacement Figure 15a, of the 42.5 simulated mm. Peak results voltages from of Figure 230 V 15b and have 205 Vsimilar were obtainedprofiles at in a Halbachmaximum and displacement radial array of configurations, 42.5 mm. Peak respectively, voltages of 230 are V seen and from 205 V Ref. were [5]. obtained These peak in Halbachvoltages and of 225radial V and array 200 configurations, V were also obtained respectively, for the are two seen configurations from Ref. [5]. by These the present peak voltagessimulation. of 225 Although V and 200 a smallV were variation also obtained of overlap for the is foundtwo configurations in the data from by the Ref. present [5], this simulation.is not seen Although in the present a small results variation due of to overlap ideal conditions. is found in the Hence, data therefrom Ref. is a reasonable[5], this is notagreement seen in the between present the results present due and to ideal existing conditions. results. Hence, there is a reasonable agree- ment betweenTo further the understand present and the existing advantage results. gained with the present model to achieve a higherTo further performance understand than thethe existingadvantage model gained considered with the in present Ref. [5 ],model an analysis to achieve was ex-a higherecuted performance at 21.27 Hz than and 42.5the existing mm stroke model for co thensidered Halbach in arrayRef. [5], with an 16.67 analysisΩ as was the exe- load cutedresistance. at 21.27 The Hz results and 42.5 obtained mm stroke from for this the analysis Halbach in comparison array with 16.67 with theΩ as existing the load model re- sistance.are represented The results inTable obtained4 to showfrom thethis relative analysis performance in comparison of the with present the existing PMLA. model In this are represented in Table 4 to show the relative performance of the present PMLA. In this table, data not provided are indicated with ‘-’. It is seen that by using the present model, one is able to achieve both higher power and higher voltage by properly modifying the design parameters. Sustainability 2021, 13, 7192 12 of 15

Sustainability 2021, 13, 7192 table, data not provided are indicated with ‘-’. It is seen that by using the present13 model, of 16 one is able to achieve both higher power and higher voltage by properly modifying the design parameters.

(a) Data from Ref. [5]

250

200

150

100

50 Halbach array Radial array 0

-50

Output Voltage (V) -100

-150

-200

-250 0 5 10 15 20 25 30 35 40 45 Translator position (mm)

(b) Present results

FigureFigure 15. 15. ComparisonComparison between between the the pres presentent and and existing existing results. results.

Table 4. Parameters and results comparison between reference [5] and present model.

Reference [5] Present Model Back Iron Thickness (mm) 14.7066 5 Tooth Slot Thickness (mm) - 5 Tooth Slot Gap (mm) - 10.5 Air Gap (mm) 1.651 1.5 Rms Force (N) 2465 1416 Voltage (V) 74 128 Power (W) 336 618

Sustainability 2021, 13, 7192 13 of 15

Table 4. Parameters and results comparison between reference [5] and present model.

7. Conclusions The developed model introduces a slot space between slots to reduce the use of excess material to achieve a higher specific power than the existing model. Adapted by proven models, the design modifications were found to be improved in the baseline case consideration. The major findings are summarized as follows: (1) A level of saturation was seen after 60 Hz in the frequency variation. However, the domestic supply of Taiwan is fixed at 60 Hz, and hence was found to be a suitable solution that avoids post-processing of the induced EMF; most FPSEs operate within this range. The cogging force on the translator had a minuscule change due to the frequency variation and 939 W of power on the load in frequency variation. (2) With stroke length variation, there is a linear rise in induced voltage, varying beyond 40 mm, which is a design limitation, and applications with such inappropriate lengths are hard to find. A peak power of 7.1 kW was obtained for a 40 mm stroke, with the highest cogging force of 1.36 kN acting on the translator. (3) There was a linear rise in the induced power in the variation of magnetic material, and thus the cogging force. Hence, a balance between the required power delivery and an application to withstand the maximum cogging force felt on the translator should be chosen for specific applications. (4) The 35 series material has proven to be a better choice for the stator and shaft material, of which 35CS300 was found to have better-induced EMF than 35CS210. The iron loss in 35CS210 is comparatively lower, but the cogging force does not make a significant difference; hence, 35CS300 has proven to be a better material choice. Though having a considerable number of material choices for comparison, it was limited to a small number due to similar previous research [32]. (5) Having values as low as 31 W and as high as 7116 W, the specific power varies from 1.4 W/kg to 313.5 W/kg. There is a wide band of power availability with this PMLA. Power-dense machines of such proportions have been possible in the past with suitable integrations. (6) Considering four parameters, a newer, lighter model from an existing conventional model has been developed which can provide more power under the same operating conditions. We came up with a PMLA that can produce a 7.1 kW peak power which is numerically analyzed and validated.

Author Contributions: Conceptualization, C.-H.C.; data curation, S.D.; formal analysis, C.-H.C.; funding acquisition, C.-H.C.; investigation, S.D.; resources, C.-H.C.; software, S.D.; supervision, C.-H.C.; writing—original draft, S.D.; writing—review and editing, C.-H.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. Sustainability 2021, 13, 7192 14 of 15

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