Research on the Non-Magnetic Conductor of a PMSM Based on the Principle of Variable Exciting Magnetic Reluctance

energies

Article Research on the Non-Magnetic Conductor of a PMSM Based on the Principle of Variable Exciting Magnetic Reluctance

Chunyan Li 1,* , Fei Guo 1, Baoquan Kou 2 and Tao Meng 1

1 Department of Mechanical and Electrical Engineering, Heilongjiang University, Harbin 150080, China; [email protected] (F.G.); [email protected] (T.M.) 2 Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150080, China; [email protected] * Correspondence: [email protected]

Abstract: A permanent magnet synchronous motor (PMSM) based on the principle of variable exciting magnetic reluctance (VMRPMSM) is presented. The motor is equipped with symmetrical non-magnetic conductors on both sides of the tangential magnetized permanent magnets (PMs). By placing the non-magnetic conductor (NMC), the magnetic reluctance in the exciting circuit is adjusted, and the ﬂux weakening (FW) of the motor is realized. Hence, the NMC is studied comprehensively. On the basis of introducing the motor structure, the FW principle of this PMSM is described. The shape of the NMC is determined by analyzing and calculating the electromagnetic force (EF) acting on the PMs. We calculate the magnetic reluctance of the NMC and research on the effects of the NMC on electromagnetic force, d-axis and q-axis inductance and FW performance. The critical speeds from the test of the no-load back electromotive force (EMF) verify the correctness of the NMC design. The analysis is corresponding to the test result which lays the foundation of design for this kind of new PMSM.

Keywords: permanent magnet synchronous motor; variable magnetic reluctance; non-magnetic conductor; flux weakening Citation: Li, C.; Guo, F.; Kou, B.; Meng, T. Research on the Non-Magnetic Conductor of a PMSM Based on the Principle of Variable 1. Introduction Exciting Magnetic Reluctance. Energies 2021, 14, 318. https://doi.org/ Permanent magnet (PM) materials, electronic power devices and converter techniques 10.3390/en14020318 have developed rapidly in recent years [1–3]. The permanent magnet synchronous motor (PMSM) has been widely studied and applied in many industries [4–7]. However, PMSM Received: 3 December 2020 is excited by PM and its magnetic field cannot be adjusted. This defect constrains the Accepted: 4 January 2021 applications of PMSM in wide speed range. Flux weakening (FW) control is a way to solve Published: 8 January 2021 the problem [8–10]. The traditional FW method of PMSM is accomplished by applying a large d-axis demagnetizing current. The FW range of PMSM is widened by this method, but Publisher’s Note: MDPI stays neu- the winding copper losses are increased. In addition, a risk of irreversible demagnetization tral with regard to jurisdictional clai- of PM is increased by a large negative d-axis demagnetizing current. Consequently, scholars ms in published maps and institutio- throughout the world have done a lot of work on FW. There are two ways to improve the nal affiliations. FW ability of PMSM, as shown below. One way is to design from the point of view of the motor structure. A lot of special PMSMs are proposed [11–18]. For example, in [13], a novel two-part rotor variable flux memory motor was proposed. The rotor comprises an AlNiCo part and a NdFeB part that Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. are adjacent to each other on the same shaft. The AlNiCo can be adjusted by a temporary This article is an open access article current pulse to control the magnetic flux. The flux adjustable capabilities are good and distributed under the terms and con- flexible. Relative independence of two rotor parts leads to more iron losses, especially in ditions of the Creative Commons At- the high-speed region under the negative magnetization state. In [14], a new segmented tribution (CC BY) license (https:// interior PM machine with a very wide constant power operation range was proposed. The creativecommons.org/licenses/by/ PM is divided into several pieces and the magnetic flux is adjusted by the magnetic bridge 4.0/). between two adjacent segmented PMs. A wide constant power speed ratio is achievable

Energies 2021, 14, 318. https://doi.org/10.3390/en14020318 https://www.mdpi.com/journal/energies Energies 2021, 14, 318 2 of 27

in such machines due to the magnetic bridge. Meanwhile, the magnetic bridge results in more leakage flux. The mechanical method is used to realize FW in [15]. The magnetic flux can be decreased by forming an axial difference gap between the stator and the rotor by a mechanical method. This novel field-weakening control can be used even with a surface-type PMSM. The axial movement of the rotor requires auxiliary mechanical support. In [16], there is an independent axial flux regulator for the PM, which provides a shunted magnetic circuit to the radial stator. Benefitting from the larger d-axis inductance, the presented motor is able to provide a large capability of magnetic flux regulation. The axial flux regulator makes the motor structure complex. In [17], a novel hybrid excitation PM motor with an independent ac excitation port to widen the speed range was proposed. An axial stator is placed on one side of the rotor to provide a shunted flux path of the radial stator. This independent axial flux regulator occupies some space, but it provides a shunted magnetic circuit to the primary radial stator. The ability of torque and flux regulation is superior to conventional PMSM especially in the deep flux-weakening region. Other methods include an axial magnetic-field-modulated brushless double-rotor motor, as in [18]. The other way is to design from the point of view of the FW control method [19–21]. The research on FW control method mainly focuses on vector control [22]. For example, in [23], a voltage feedback FW control scheme for vector-controlled interior PMSM driving systems was proposed. The control variable is the phase angle of reference current space vector. Based on a modification of the vector current reference angle, this scheme intrinsi- cally provides smooth transition to and from the maximum-torque-per-voltage operating region. The main advantage of this method provides maximization of the dynamical performances. In [24], an improved continuous-time model predictive control of PMSM for a wide speed range was proposed. This method includes the constant torque region and the FW region. The control method can realize good steady-state performance and switch between the constant torque and FW regions smoothly. In [25], the author took into account the influence of the resistive voltage drop in the stator windings for the FW control of a surface-mounted PMSM. The resistive voltage drop is usually neglected in similar studies. Direct torque control method of FW has also been greatly developed [26–32]. Direct torque control directly controls the motor torque to control the speed. This method does not require complex transformation of coordinate and saves a lot of computation time. For example, in [30], the author incorporated maximum torque per voltage trajectory into the conventional FW algorithm of direct torque and flux control. This method can extend the motor speed to the deep flux-weakening region without violating the maximum load angle condition. In [31], a direct torque and flux control method of a voltage-limited finite-settling step with a constant switching frequency for torque-controlled PMSM was proposed. Two independent voltage truncation rules are developed to facilitate the possible voltage vector choices. This method does not depend on the classical over-modulation methods at voltage limits. In [32], the direct torque and flux control of a dual-air gap axial-flux PM motor was studied. The proposed observer eliminates the rotary encoder and reduces the total weight and cost of the control system. Based on the above analysis, the FW problem of PMSM is still one of the important research issues in the variable frequency driving system today. In this paper, a new PMSM based on the principle of variable exciting magnetic reluctance (VMRPMSM) is proposed without additional control difficulty and auxiliary mechanical equipment. The FW method of this special VMRPMSM is different from the traditional PMSM. The idea of FW is to reduce the no-load back electromotive force (EMF) at high speed, instead of applying a demagnetizing current. The effective magnetic reluctance in the magnetic circuit is changed due to the non-magnetic conductor (NMC) by the movement of the permanent magnets, thus the no-load back EMF is changed. Compared with the traditional FW method, the q-axis current and the torque of the VMRPMSM are not reduced and the demagnetization risk of PM is improved. Solving the FW problem for this special motor is realized by the NMC in the rotor, so the NMC is studied comprehensively. First, the FW principle is described in Section2. By placing Energies 2021, 13, x FOR PEER REVIEW 3 of 29

Energies 2021, 14, 318 for this special motor is realized by the NMC in the rotor, so the NMC is studied compre-3 of 27 hensively. First, the FW principle is described in section II. By placing the NMC, the magnetic reluctance is adjusted and the air-gap magnetic flux is changed. Second, the approx- imatelythe NMC, triangular the magnetic NMC reluctanceshould be isdesigned adjusted theoretically and the air-gap by analyzing magnetic the flux force is changed. on the PMSecond, in section the approximately III. Third, the triangular relationship NMC between should the be designedmagnetic theoretically reluctance and by analyzing the geo- metricalthe force size on theof NMC PM in is Section analyzed3. Third, in section the relationship IV. Fourth, between the influences the magnetic of the reluctance NMC on electromagneticand the geometrical force size(EF), of inductance NMC is analyzed and FW in ability Section are4. studied Fourth, in the section influences V; the of FW the validityNMC on of electromagnetic the motor is verified force (EF), by inductanceload test. The and critical FW ability speeds are studiedfrom the in no-load Section5 back; the EMFFW validitytest verify of thethe motorcorrectness is verified of the by NMC load test.design The in critical section speeds VI. In fromthis paper, the no-load the NMC back isEMF studied test verifyin detail the so correctness we can better of the unde NMCrstand design and in design Section this6. Intype this of paper, motor. the NMC is studied in detail so we can better understand and design this type of motor. 2. Flux-Weakening Principle 2. Flux-Weakening Principle This FW method of the VMRPMSM is different from the traditional PMSM. The idea of FWThis is to FWreduce method the no-load of the VMRPMSM back EMF at is high different speed, from instead the traditionalof applying PMSM. demagnetizing The idea current.of FW is Based to reduce on the the introduction no-load back of EMF the atmoto highr structure, speed, instead the air-gap of applying magnetic demagnetizing flux equa- tioncurrent. is derived Based to on describe the introduction the FW principle. of the motorThe no-load structure, and therated air-gap load magnetic flux flux distributionsequation is derived are used to to describe show the the FW effect principle. directly The in no-load this section. and rated load magnetic flux distributions are used to show the FW effect directly in this section. 2.1. The Rotor Structure 2.1. The Rotor Structure The 8 poles-18 slots and fractional-slot stator for this PMSM is the same as in the The 8 poles-18 slots and fractional-slot stator for this PMSM is the same as in the common PMSM. The rotor is composed of the main and secondary PMs, the PM slots and common PMSM. The rotor is composed of the main and secondary PMs, the PM slots the non-magnetic conductors (NMCs), as shown in Figure 1. The material of the NMC is and the non-magnetic conductors (NMCs), as shown in Figure1. The material of the stainless steel. The main PMs can be fixed or move in the PM slots according to motor NMC is stainless steel. The main PMs can be fixed or move in the PM slots according to speed. motor speed.

(a)

(b)

FigureFigure 1. 1. TheThe new new permanent permanent magnet magnet synchronous synchronous motor motor (PMSM): (PMSM): (a ()a )the the prototype prototype of of the the motor; motor; ((bb)) the the cross cross section section of of the the rotor. rotor.

The main PMs start to move in the slot once the motor speed exceeds the rated speed. Due to the large magnetic reluctance of the NMC, the effective magnetic ﬂux area provided by the main PMs decreases with the increase in the moving distance of the main PMs.

Energies 20212021,, 1313,, xx FORFOR PEERPEER REVIEWREVIEW 44 ofof 2929

Energies 2021, 14, 318 The main PMs start to move in the slot once the motor speed exceeds the rated speed.4 of 27 Due to the large magnetic reluctance of the NMC, the effective magnetic flux area provided by the main PMs decreases with the increaseease inin thethe movingmoving distancedistance ofof thethe mainmain PMs. Hence, the air-gap magnetic field is weakened and the FW of the PMSM is realized. TheHence, applications the air-gap of the magnetic VMRPMSM field ispossibly weakened include and two the FWcases. of theOne PMSM is to replace is realized. the low- The powerapplications induction of the motor VMRPMSM which is possibly required include to work two in cases.a wide One speed is to range replace and the has low-power the ad- vantageinduction of energy motor whichconservation is required of PMSM. to work The in other a wide one speed is used range in andindustrial has the machinery advantage withof energy the requirements conservation of oflow-power PMSM. The and other wide one speed is used range. in industrial machinery with the requirements of low-power and wide speed range. 2.2. The FW Principle 2.2. The FW Principle ss isis defineddefined asas thethe distancedistance betweenbetween thethe mamainin PMPM andand thethe secondarysecondary PM.PM. ss isis relatedrelated s is defined as the distance between the main PM and the secondary PM. s is related toto thethe motormotor speed.speed. TheThe leakageleakage fluxflux isis usuallyusually veryvery smallsmall andand itit isis ignored.ignored. TheThe no-loadno-load to the motor speed. The leakage flux is usually very small and it is ignored. The no-load magnetic flux distribution is shown in Figure 2. magneticmagnetic flux flux distribution distribution is is shown shown in in Figure Figure 2.2.

((a)) ((b))

FigureFigure 2. 2. TheThe magnetic magnetic flux ﬂux path: path: (a (aa)) ) operatingoperating operating atat at oror or belowbelow below thethe rated rated speedspeed (((sss === 1111 mm);mm); ( ((bb))) operating operat-operat- ingingabove aboveabove the thethe rated ratedrated speed speedspeed (s =((ss 0 == mm). 00 mm).mm).

TheThe leakage leakage flux ﬂux and and the the magnetic magnetic reluctance reluctance of of the the stator stator tooth, tooth, the the stator stator yoke yoke and and thethethe rotorrotor rotor yokeyoke yoke areare are ignoredignored ignored duedue due toto to thethe the highhigh high magneticmagnetic magnetic permeabilitypermeability permeability ofof of thethe the rotorrotor rotor core.core. core. TheThe The no-loadno-load equivalent equivalent magnetic magnetic circuit circuit diagram diagram is is shown shown in in Figure Figure 3.3.

((a)) ((b)) Figure 3. The equivalent no-load magnetic circuit: (a) s = 11 mm; (b) 0 mm ≤ s < 11 mm (0 < m ≤ 1). FigureFigure 3. 3. TheThe equivalent equivalent no-load no-load magnetic magnetic circuit: circuit: (a (a) )ss == 11 11 mm; mm; ( (bb)) 0 0 mm mm ≤≤ s s<< 11 11 mm mm (0 (0 < < mm ≤≤ 1).1 ). Fcc isis thethe magneticmagnetic potentialpotential ofof thethe PM.PM. Φmm isis thethe magneticmagnetic fluxflux providedprovided byby thethe PMsPMs toto thethe externalexternal Fc is the magnetic potential of the PM. Φm is the magnetic flux provided by the PMs to the external magnetic circuit. Φδδ0101 andand Φδδ0202 areare thethe air-gapair-gap magneticmagnetic fluxflux whenwhen ss == 1111 mmmm andand 00 mmmm ≤≤ ss << 1111 mm,mm, magnetic circuit. Φδ01 and Φδ02 are the air-gap magnetic flux when s = 11 mm and 0 mm ≤ s < 11 mm, respectively.respectively. R00,, Rδδ andand Rnmnm areare thethe magneticmagnetic reluctancesreluctances ofof ththee PM,PM, thethe airair gapgap andand thethe wholewhole NMC,NMC, respectively. R , R and R are the magnetic reluctances of the PM, the air gap and the whole NMC, respectively.respectively. m is0is aa δcoefficientcoefficientnm fromfrom 00 toto 1.1. respectively. m is a coefficient from 0 to 1. Operating at or below the rated speed, the main PMs are fixed in the PM slot and ss isis Operating at or below the rated speed, the main PMs are fixed in the PM slot and s is 11 mm. The air-gap magnetic flux Φδδ0101 inin FigureFigure 3a3a isis shownshown asas EquationEquation (1):(1): 11 mm. The air-gap magnetic flux Φδ01 in Figure3a is shown as Equation (1): F Φ = cc δδ 01 F (1)(1) 01 ()RR+c Φδ01 = ()RRδδ 00 (1) (Rδ + R0)

Energies 2021, 13, x FOR PEER REVIEW 5 of 29

Energies 2021, 14, 318 5 of 27 The main PMs start to move once the speed exceeds the rated speed and operate in a new position steadily. s varies from 11 to 0 mm. Φδ02 in Figure 3b is shown as Equation (2): The main PMs start to move once the speed exceeds the rated speed and operate in a Fc new position steadily. s varies from 11 toΦ 0 mm.= Φδ02 in Figure3 b is shown as Equation (2): δ 02 ++ (2) RRmRδ 0 nm Fc Φδ02 = (2) The higher motor speed correspondsRδ + Rto0 +themR biggernm m, thus the mRnm is bigger, so Φδ02 is smaller. Φδ02 is obviously less than Φδ01 due to the large mRnm. Hence, the FW is realized. The higher motor speed corresponds to the bigger m, thus the mR is bigger, so The no-load magnetic flux distribution of the motor is shown innm Figure 4a. The mag- Φδ02 is smaller. Φδ02 is obviously less than Φδ01 due to the large mRnm. Hence, the FW neticis realized. flux lines on the right of Figure 4a are obviously sparse compared with that on the left ofThe Figure no-load 4a. The magnetic effectiveness flux distribution of the FW of for the the motor motor is shown can be in verified. Figure4 a.ω is The defined as themagnetic angle between flux lines the on themagnetomotive right of Figure force4a are and obviously the direct sparse axis. compared ω is set as with 0°, that45° and 90°, respectively.on the left of The Figure rated4a. Theload effectiveness magnetic flux of thedistribution FW for the of motor the motor can be is verified. shown inω Figureis 4b. ◦ Thedefined air-gap as the magnetic angle between field of the the magnetomotive motor is distor forceted anddue theto the direct effect axis. ofω theis setarmature as 0 , field. ◦ ◦ The45 andload 90 magnetic, respectively. field Thelines rated are distorted load magnetic in Figure flux distribution 4b compared of the with motor the is no-load shown distri- in Figure4b . The air-gap magnetic field of the motor is distorted due to the effect of the bution. The bigger angle results in more distortion when ω varies from 0° to 90°. The mag- armature field. The load magnetic field lines are distorted in Figure4b compared with neticthe no-load field lines distribution. of small Thes (s bigger= 0 mm) angle are resultsobviously in more sparse distortion compared when withω varies that from of large s (s =0 ◦11to mm). 90◦. TheThe magnetic load magnetic field lines field of smalldistributions (s = 0 mm)can also are obviously indicate sparsethe effectiveness compared of the FWwith for that this of largemotor.s ( s = 11 mm). The load magnetic field distribution can also indicate the effectiveness of the FW for this motor.

(a)

Figure 4. Cont.

Energies 2021, 13, x FOR PEER REVIEW 6 of 29

Energies 2021, 14, 318 6 of 27

Figure 4. Cont.

Energies 2021, 13, x FOR PEER REVIEW 7 of 29 Energies 2021, 14, 318 7 of 27

(b)

Figure 4.4.The The magnetic magnetic ﬂux flux distributions distributions (n = ( 1500n = 1500 r/min): r/min): (a) the (a no-load) the no-load distribution; distribution; (b) the load (b) the load distribution with with different different angle angleω. ω.

3. The ElectromagneticElectromagnetic Force Force Analysis Analysis The ﬂux-weakening ability of this VMRPMSM is related to the design of the non- The flux-weakening ability of this VMRPMSM is related to the design of the non- magnetic conductor. However, the design principle of the NMC shape depends on the magnetic conductor. However, the design principle of the NMC shape depends on the required electromagnetic force (EF) acting on the main PMs. The relationship between the EF and the motor speed should agree with the variation law of EF required by this flux- weakening principle. Hence, it is necessary to analyze the electromagnetic force.

Energies 2021, 14, 318 8 of 27

Energies 2021, 13, x FOR PEER REVIEW 8 of 29 required electromagnetic force (EF) acting on the main PMs. The relationship between the EF and the motor speed should agree with the variation law of EF required by this ﬂux-weakening principle. Hence, it is necessary to analyze the electromagnetic force. 3.1. The Electromagnetic Force Trend that Meets the Requirement of Flux Weakening 3.1. TheThe Electromagnetic premise of realizing Force Trend FW thatis that Meets the theelectromagnetic Requirement of force Flux Weakening(EF) must be the in- creasingThe function premise of realizingspeed, thus FW the is thatEF has the three electromagnetic cases, as shown force (EF)in Figure must 5. be the increasing function of speed, thus the EF has three cases, as shown in Figure5.

(a)

(b)

FigureFigure 5. TheThe force force versus motor speed: ( a)) the the centrifugal centrifugal force Fce;;( (bb)) the the electromagnetic electromagnetic force force Fe (the(the curves curves from 1 to 3).

TheThe centrifugal force force on the main PMs increases with the motor speed slowly in the beginningbeginning and and then rapidly. The EF should balance with the centrifugalcentrifugal force.force. The EF shownshown in in curve curve 1 1 of of Figure Figure 55 isis closestclosest toto thethe centrifugalcentrifugal force,force, andand curvecurve 11 isis ideal.ideal. Hence, we should design thethe shapeshape ofof thethe NMCNMC soso that that the the EF EF on on the the main main PMs PMs looks looks like like curve curve 1. 1.

Energies 2021, 13, x FOR PEER REVIEW 9 of 29

Energies 2021, 14, 318 3.2. The Electromagnetic Force Calculation 9 of 27 The forces acting on the main PMs mainly include the electromagnetic force (EF), the centrifugal force and the friction force. The first main PM in Figure 1 is an example; the forces3.2. The are Electromagnetic shown in Figure Force 6. Calculation TheThe movement forces acting of on the the main main PM PMs is mainly determined include by the force. electromagnetic The force balance force (EF), equation the on thecentrifugal main PM force in Figure and the 6 frictionis: force. The ﬁrst main PM in Figure1 is an example; the forces are shown in Figure6. =+α μ α The movement of the main PMFF isce determined ecos by force. F e sin The force balance equation on (3) the main PM in Figure6 is: where Fce and Fe are the centrifugal force and the EF, respectively. µ is the friction coeffi- Fce = Fe cos α + µFe sin α (3) cient. α is the angle between the EF and the radial center line of the slot. The tangential where F and F are the centrifugal force and the EF, respectively. µ is the friction coefﬁcient. componentce of thee EF is the positive pressure of the friction force. The radial component of α is the angle between the EF and the radial center line of the slot. The tangential component theof theEF is EF opposite is the positive to the pressure centrifugal of the force. friction force. The radial component of the EF is opposite to the centrifugal force.

(a) (b)

(c)

FigureFigure 6. 6. TheThe forces on on the the main main permanent permanent magnets magnets (PMs): (PMs): (a) in the(a) centerlinein the centerline of the PM of slot; the (PMb) in slot; (b) inthe the non-centerline non-centerline of the of the PM PM slot; slot; (c) the (c force) the diagram.force diagram.

Setting the main PMs in the center line of the slot is ideal. The total gap distance L Setting the main PMs in the center line of the slot is ideal. The total gap distance L between the main PM and the slot is set as 0.01, 0.02, 0.1 and 0.2 mm, respectively. The EF betweenon the ﬁrst the main main PM PM is shownand the in slot Figure is set7. as 0.01, 0.02, 0.1 and 0.2 mm, respectively. The EF on the first main PM is shown in Figure 7. The EF gradually increases with the decrease in s. The angle of the EF is around 270°. This means the direction of the EF points to the center of the rotor is exactly opposite to the centrifugal force. Meanwhile, the angle of the centerline of the slot is also 270°, thus the angle between the EF and the center line is very small. It means that the positive pressure of friction force is very small due to the small tangential component of the EF. Hence,

Energies 2021, 13, x FOR PEER REVIEW 10 of 29

the friction force can be ignored approximately if the main PMs are located in the centerline of the slot. In fact, the main PMs will rotate to one side of the PM slot, but the surface of the PM slot cannot be completely smooth. This means there is an air gap between two contact surfaces, so the main PMs are not closely attached to the surface of the PM slot. The devi- Energies 2021, 14, 318 10 of 27 ation ratio is defined as the ratio of the big air-gap distance and the small air-gap distance. The EF on the main PM with different deviation ratios is calculated as shown in Figure 8.

68 L = 0.01 L = 0.02 51 L = 0.1 Energies 2021, 13, x FOR PEER REVIEW 10 of 29 L = 0.2 (N) 34 F

17 the friction force can be ignored approximately if the main PMs are located in the centerline of the slot. 0 In fact, 1197531the main PMs will rotate to one side of the PM slot, but the surface of the PM slot cannot be completelys (mm) smooth. This means there is an air gap between two contact (a) surfaces,285 so the main PMs are not closely attached to the surface of the PM slot. The deviation ratio is defined as the ratio ofL =the 0.01 big air-gap distance and the small air-gap distance. The EF280 on the main PM with differentL = 0.02 deviation ratios is calculated as shown in Figure 8. L = 0.1 )

˚ 275 68 L = 0.2 L = 0.01 270 L = 0.02 Angle ( Angle 51 265 L = 0.1 L = 0.2 (N) 34

F 260 1197531 17 s (mm)

0 1197531 (b) s (mm) FigureFigure 7. The electromagneticelectromagnetic force force (EF) (EF) versus versus(sa(in) s the (in centerline the centerline of the PMof the slot): PM (a )slot): the amplitude; (a) the ampli- 285 tude;(b) the (b phase.) the phase L = 0.01 280 L = 0.02 ◦ The EF gradually increasesL = 0.1 with the decrease in s. The angle of the EF is around 270 . )

˚ 275 This means68 the direction of theL = EF 0.2 points to the center280 of the rotor is exactly opposite to the centrifugal270 force. Meanwhile, the angle of the centerline of the slot isratio also = 2702 ◦, thus the

Angle ( Angle ratio = 2 ratio = 4 angle265 between51 the EF and the center line is very small.270 It means that the positive pressure of friction force is veryratio small = 4 due to the small tangential component of the EF. Hence, the ) 260 ˚ friction(N) 34 force can be ignored approximately if the main260 PMs are located in the centerline of

F 1197531 the slot. s (mm)

In17 fact, the main PMs will rotate to one side ( Angle of250 the PM slot, but the surface of the PM slot cannot be completely smooth. This means(b) there is an air gap between two contact 240 Figuresurfaces, 7. 0The so electromagnetic the main PMs force are (EF) not versus closely s (in attached the centerline to the ofsurface the PM slot): of the (a) PMthe ampli- slot. The 1197531 1197531 tude;deviation (b) the ratio phase is defineds (mm) as the ratio of the big air-gap distance and thes small(mm) air-gap distance. The EF on the main PM with different deviation ratios is calculated as shown in Figure8. (a) (b) 68 280 ratio = 2 Figure 8. The EF versusratio = 2s (in the non-centerline of the PM slot, L = 0.1 mm): (a) the amplitude; (b) 51 270 ratio = 4 the phase. ratio = 4 ) ˚

(N) 34 260

F The deviation angle in Figure 8b between the EF and the center line is larger than that

in Figure17 7b. The deviation angle reduces ( Angle 250 as s decreases. The larger deviation ratio increases the deviation angle, and so increases the positive pressure of the friction force. 0 240 Notably, the1197531 friction force is relatively small corresponding1197531 to the radial component of the s (mm) s (mm) EF. Theoretically, the smaller friction force is good for reducing the calculating speed error according to the force balance. (a) (b)

FigureFigure 8. 8. TheThe EF EF versus versus s (ins (inthe thenon-centerline non-centerline of the of PM the slot, PM slot,L = 0.1L =mm): 0.1 mm):(a) the ( aamplitude;) the amplitude; (b) the(b )phase. the phase.

The deviation angle in Figure 8b between the EF and the center line is larger than that in Figure 7b. The deviation angle reduces as s decreases. The larger deviation ratio increases the deviation angle, and so increases the positive pressure of the friction force. Notably, the friction force is relatively small corresponding to the radial component of the EF. Theoretically, the smaller friction force is good for reducing the calculating speed error according to the force balance.

Energies 2021, 14, 318 11 of 27

The deviation angle in Figure8b between the EF and the center line is larger than that in Figure7b. The deviation angle reduces as s decreases. The larger deviation ratio Energies 2021, 13, x FOR PEER REVIEWincreases the deviation angle, and so increases the positive pressure of the friction11 force.of 29 Energies 2021, 13, x FOR PEER REVIEW 11 of 29 Notably, the friction force is relatively small corresponding to the radial component of the EF. Theoretically, the smaller friction force is good for reducing the calculating speed error according to the force balance. The influence of the air gap on the EF is shown in Figure 9. The big air gap increases TheThe influence inﬂuence of of the air gap onon thethe EFEF is is shown shown in in Figure Figure9. The9. The big big air air gap gap increases increases the magnetic reluctance in the magnetic circuit and reduces the air-gap magnetic flux den- thethe magnetic magnetic reluctance reluctance in in the the magnetic magnetic circ circuituit and and reduces reduces the the air-gap air-gap magnetic magnetic flux ﬂux density, thus the EF is reduced. sity,density, thus thusthe EF the is EF reduced. is reduced.

68 280 68 280 airgap = 0.4 mm airgap = 0.4 mm 51 airgap = 1 mm 270 51 airgap = 1 mm 270 ) ˚ ) (N) 260 34 ˚ F (N) 34 260 airgap = 0.4 mm F 17 ( Angle 250 airgapairgap = 1 mm = 0.4 mm 17 ( Angle 250 airgap = 1 mm 0 240 1197531 2401197531 0 s (mm) s (mm) 1197531 1197531 s (mm) s (mm) (a) (b) Figure 9. The effect of the ( aair) gap on EF (in the non-centerline, ratio = 2, (Lb =) 0.03 mm): (a) the amplitude curve; (b) the phase curve. FigureFigure 9. 9. TheThe effect effect of of the the air air gap gap on on EF EF (in (in the the non-centerline, non-centerline, ratio ratio = 2, = L 2, =L 0.03= 0.03 mm): mm): (a) the (a) theam- plitudeamplitude curve; curve; (b) the (b) thephase phase curve. curve. The winding current is set as 0.5, 1, 3 and 6 times the rated current; the influence of the windingTheThe winding winding current current current on EF is setshown asas 0.5,0.5, in 1,Figure1, 33 and and 10. 6 6 times Thetimes EF the the is rated significantlyrated current; current; thedecreased the influence influence and of of thethe deviationwinding winding current angle increases onon EF EF is is showndue shown to insaturati in Figure Figureon 10 .once 10. The Thethe EFis currentEF significantly is significantly exceeds decreased the decreasedvalue and of the6 and thetimesdeviation deviation the rated angle angle current. increases increases The due amplitude todue saturation to ofsaturati the onceEFon does theonce not current the change current exceeds clearly, exceeds the while value the the of value 6motor times of 6 timesusuallythe rated the runs rated current. at lesscurrent. The than amplitude oneThe timesamplitude of the the value EF of doesthe of EFthe not doesrated change not curre clearly, changent, so while clearly,it is thenot motorwhilenecessary usuallythe motorto consider the influence of the current on EF too much. usuallyruns at runs less thanat less one than times one the times value the of thevalue rated of current,the rated so curre it is notnt, necessaryso it is not to necessary consider to considerthe influence the influence of the current of the on current EF too on much. EF too much.

68 280 0.5*In 0.5*In 1*In 1*In 5168 275280 3*In0.5*In 3*In0.5*In 6*In 1*In ) 6*In1*In ˚ (N) 3451 270275 F 3*In 3*In 6*In ) ˚

6*In ( Angle (N) 1734 265270 F

170 ( Angle 260265 1197531 1197531 s (mm) s (mm) 0 260 1197531 1197531 (as) (mm) (b) s (mm)

FigureFigure 10. 10. TheThe effect effect of of the the current current on on EF EF (in (in the the non-centerline non-centerline of of the the PM PM slot, slot, LL == 0.05 0.05 mm): mm): (a ()a )the the amplitude curve; (b) the(a phase) curve. (b) amplitude curve; (b) the phase curve. Figure 10. The effect of the current on EF (in the non-centerline of the PM slot, L = 0.05 mm): (a) the amplitudeTheThe influence curve; inﬂuence (b )of the of the thephase PM PM curve.on on EF EF is is shown shown in in Figure Figure 11. 11 .The The thicker thicker PM PM and and the the better better PMPM material material can can provide provide more more magnetic magnetic flux ﬂux de density,nsity, so so the the EF EF on on the the main main PM PM is is bigger. bigger. The influence of the PM on EF is shown in Figure 11. The thicker PM and the better PM material can provide more magnetic flux density, so the EF on the main PM is bigger.

EnergiesEnergies 20212021, 13, 14, x, 318FOR PEER REVIEW 1212 of of 29 27

(N) 40 F

20 PM thickness = 4 mm PM thickness = 7 mm 0 1197531 s (mm)

280

270 ) ˚ 260 Angle ( Angle 250 PM thickness = 4 mm PM thickness = 7 mm 240 1197531 s (mm)

(a) 60

NdFeB30 45 NdFeB35

(N) 30 F

0 1197531 s (mm)

280 NdFeB30

275 NdFeB35 ) ˚

270 Angle ( Angle 265

260 1197531 s (mm)

(b)

FigureFigure 11. 11. TheThe effect effect of the of thePM PMthickness thickness and andmaterial material on EF: on (a EF:) the (a PM) the thickness PM thickness (in the (in non- the non- centerline,centerline, ratio ratio = =2, 2,L L= 0.03= 0.03 mm); mm); (b ()b the) the PM PM material material (in (in the the non-centerline, non-centerline, L L= 0.05= 0.05 mm). mm).

3.3.3.3. Design Design Idea Idea of ofthe the Non-Magnetic Non-Magnetic Conductor Conductor (NMC) (NMC) According According to to EF EF WhenWhen s sisis 11 11 mm, mm, the the motor motor speed speed calculated calculated by by the the balance balance of of the the centrifugal centrifugal force force andand the the sum sum of of the the EF EF and and the the friction friction force force is is approximately approximately equal equal to to 1500 1500 r/min r/min (rated (rated speed).speed). The The motor motor runs runs more more than than 1500 1500 r/mi r/min,n, and and the the relatively relatively low low speed speed corresponds corresponds toto the the small small centrifugal centrifugal force, force, meaning meaning that that the the thin thin NMCs NMCs are are required. required. The The EF EF on on the the mainmain PM PM corresponding corresponding to to the the increasing increasing centrifugal centrifugal force force should should increase increase synchronously. synchronously. TheThe change change rate rate of of the the effective effective magnetic magnetic reluctance reluctance provided provided by by the the NMC NMC must must be be larger larger asas speed speed increases, increases, and and thus thus the the EF EF can can be be increased. increased. Therefore, Therefore, the the width width of of the the NMC NMC shieldingshielding the the main main PMs PMs should should gradually gradually increase. increase. To sum To sum up, the up, design the design principle principle of the of NMCthe NMC should should conform conform to the to following the following two two conditions: conditions: one one is, is,at atlow low speed, speed, the the NMC NMC

EnergiesEnergies2021 2021, 14, 13, 318, x FOR PEER REVIEW 1313 of of 27 29

shouldshould be be narrow narrow in in width. width. The The other other one one is, is, as theas the motor motor speed speed increases, increases, the widththe width of the of NMCthe NMC should should gradually gradually increase. increase.

4.4. TheThe CalculationCalculation ofof thethe MagneticMagnetic ReluctanceReluctance ofof thethe Non-MagneticNon-Magnetic Conductor Conductor TheThe adjustableadjustable excitingexciting reluctance of of the the motor motor is is mainly mainly provided provided by by the the NMCs, NMCs, so sothe the magnetic magnetic resistance resistance value value of the ofthe NMC NMC is ve isry very important. important. The Theeffect effect of the of dimensional the dimen- sionalparameters parameters on magnetic on magnetic reluctance reluctance of the ofNMC the NMCis studied, is studied, which which is beneficial is beneﬁcial for motor for motordesign. design. InIn orderorder to to calculate calculate thethe magnetic magnetic reluctancereluctance ofof thethe NMC NMC conveniently, conveniently, the the NMC NMC is is equivalentequivalent to to an an approximate approximate shape, shape, as as shown shown in in Figure Figure 12 12..

b O x dx C e h c x y B

f g θ

FigureFigure 12. 12.The The shapeshape ofof thethe non-magneticnon-magnetic conductor. conductor.

θθis is deﬁneddefined asas thethe angleangle between OAOA andand AB.AB. θθ cancan change change from from 0˚ 0 ◦toto 90 90˚. ◦The. The hy- hypotenusepotenuse of ofthe the actual actual NMC NMC shape shape is replaced is replaced by bythe the line line ABAB. a,. ba ,andb and c arec are defined deﬁned as the as thelengths lengths of OA of OA, OC, OC andand CB,CB respectively., respectively. Take Take a small a small section section of dx of dxon onthe the x-axis,x-axis, and and the theheight height is y is. Accordingy. According to the to the geometric geometric relation, relation, y cany can be beobtained obtained by byEquation Equation (4). (4). a a y =y =−(a tan(tanaxθ −θ x) ) (4)(4) b b

TheThe magneticmagnetic reluctancereluctance of of the the cross-section cross-sectionefgh efghin in unit unit axial axial length length is isdx dx/(/(µµ0µ0µnmnmyy).). Thus,Thus, the the magnetic magnetic reluctance reluctance of of the the cross-section cross-sectionOABC OABCin axialin axial length length of l efofis lef obtained is obtained by Equationby Equation (5). (5).

Z b dx b b Rnm = b dx= − bln 1 − b ctanθ (5) R 0 ==−−µ0µnmyle f aµ0µnmle f ln 1a ctanθ nm 0 μμ μμ (5) 00nmyl ef a nm l ef a where µ0 and µnm are the magnetic permeability of air and the relative magnetic perme- abilitywhere ofµ0 the and NMC, µnm are respectively. the magnetic permeability of air and the relative magnetic permea- bilityThe of the magnetic NMC, reluctancerespectively. of the NMC is greater than zero, so we can get Equation (6) as follows.The magnetic reluctance of the NMC is greater than zero, so we can get Equation (6) b as follows. 0 < 1 − ctanθ < 1 (6) a b S is deﬁned as the cross-section area01<− of the ctan1 NMC,θ < assuming that the S remains un-(6) changed and it is a constant. Since the length ofa a is several times greater than the length of c, thus c is designed as 2 mm in this paper. According to the geometry relationship, the S S is defined as the cross-section area of the NMC, assuming that the S remains un- and the θ are derived by Equation (7). changed and it is a constant. Since the length of a is several times greater than the length of c, thus c is designed as 2 mm in this( paper.a According+c to the geometry relationship, the S = ( 2 )b S and the θ are derived by Equation (7). b (7) tan θ = a−c +  = ac Sb() The lengths of a and b are derived from Equation2 (8) as follows.  (7) b tanθ =  ac− The lengths of a and b are derived from Equation (8) as follows.

Energies 2021, 13, x FOR PEER REVIEW 14 of 29

Energies 2021 14 , , 318  2S 14 of 27 ac=+2  tanθ  (8)  2S bcc=+−()tan2 θ  q θ  2S tan 2  a = tan θ + c (8) k is defined as the length of b dividedq by the length of a, which is shown in Equation  b = ( 2S + c2 − c) tan θ (9). tan θ k is deﬁned as the length of b divided by the length of a, which is shown in Equation (9). b k = (9) b a k = (9) a The lengths of a and b are given according to Equation (8). The parameters of a, b and θ restrictThe lengths each other of a and underb are the given condition according of tosame Equation S. For (8). example, The parameters the increase of a, b inand θ will leadθ restrict to the each increase other under in b, and thecondition the decrease of same in aS at. For the example, same time. the increaseThe magnetic in θ will reluctance lead derivedto the increase from Equation in b, and the (5) decrease versus the in a parametersat the same time.of a, b The, θ and magnetic k are reluctanceshown in derivedFigure 13. from Equation (5) versus the parameters of a, b, θ and k are shown in Figure 13.

(a) (b)

7 9x10

7 6x10 S =50 S =60 7 3x10 S =70 S =80 S =90 0 02468 k (c) (d)

Figure 13.13. TheThe magneticmagnetic reluctance reluctance versus versusθ, aθ, ,b aand, b andk:(a k): the(a) angle;the angle; (b) the (b) length the lengtha;(c) thea; (c length) the lengthb; (bd; )(d the) the ratio ratiok. k.

The largerlargerS Sof of the the NMC NMC provides provides larger larger magnetic magnetic reluctance. reluctance. Given Given the samethe sameS, the S, the larger θθ,, the the wider widerb band and the the shorter shortera are a are good good for for providing providing larger larger magnetic magnetic reluctance. reluctance. R is inversely proportional to the length of a. R increases as k increases, but the Rnmnm is inversely proportional to the length of a.nm Rnm increases as k increases, but the increment of the magnetic reluctance slows down. Rnm is approximately proportional to increment of the magnetic reluctance slows down. Rnm is approximately proportional to the length of b. The length of b cannot be chosen as too long for rotor space constraints. In the length of b. The length of b cannot be chosen as too long for rotor space constraints. In addition, the shape design of the NMC is also related to the EF. Hence, the design of the addition, the shape design of the NMC is also related to the EF. Hence, the design of the NMC should be considered comprehensively, and reasonable values of a, b and θ should NMCbe selected should as tradeoffs.be considered comprehensively, and reasonable values of a, b and θ should be selected as tradeoffs. 5. The Finite Element Analysis of the NMC 5. TheThe Finite above Element theoretical Analysis analysis of of the the EFNMC draws a conclusion that the width of the NMC shouldThe be above designed theoretical to be narrow analysis at low of speed the EF and draws then widen a conclusion gradually that with the the width increase of the NMCin speed. should The effectbe designed of the NMC to be shape narrow on at EF low is analyzed speed and by thethen ﬁnite widen element gradually method with to the determine whether it is consistent with the theory. In addition, the effect of the NMC on inductance and ﬂux-weakening ability is also studied in this section.

Energies 2021, 13, x FOR PEER REVIEW 15 of 29

increase in speed. The effect of the NMC shape on EF is analyzed by the finite element methodincrease to determinein speed. Thewhether effect it ofis consistentthe NMC shapwithe the on theory. EF is analyzed In addition, by the effectfinite ofelement the Energies 2021, 14, 318 NMCmethod on inductance to determine and whether flux-weakening it is consistent ability with is also the studied theory. inIn this addition, section. 15the of effect 27 of the NMC on inductance and flux-weakening ability is also studied in this section. 5.1. The Influence of the NMC on EF 5.1.5.1. The The Inﬂuence width Influence of of the the of NMCthe NMC on should EF on EF be designed to be narrow at low speed and then widen graduallyTheThe width with width ofthe theof theoretical the NMC NMC should should increase be designed be in designed speed. to be narrowIn to orderbe narrow at lowto verify speed at low this, and speed then Figure and widen 14 then shows widen thegraduallygradually different with withshape the theoreticalthe models theoretical of increase the increaseNMC in speed. with in In speed.the order same toIn verifySorder. this, to verify Figure this,14 shows Figure the 14 shows differentthe different shape modelsshape models of the NMC of the with NMC the samewithS the. same S.

Figure 14. The shape of the non-magnetic conductor. FigureFigure 14. 14.The The shape shape of the of non-magneticthe non-magnetic conductor. conductor. The influence of the NMC shape on EF is shown in Figure 15. TheThe inﬂuence influence of the of NMCthe NMC shape shape on EF on is shownEF is shown in Figure in 15Figure. 15.

68 280 trapezoid 68 280 trapezoid subtriangulartrapezoid 51 275 subtriangulartrapezoid subtriangular

51 ) 275 subtriangular ˚ (N)

34 ) 270 ˚ F

(N) 34 270 F 17 ( Angle 265 17 ( Angle 265 0 260 1197531 2601197531 0 s (mm) s (mm) 1197531 1197531 s (mm) s (mm)

(a) (b) (a) (b) FigureFigure 15. 15. The influenceinfluence of of the the NMC NMC shape shape on EFon (in EF the (in centerline): the centerline): (a) the ( amplitudea) the amplitude curve; (b curve;) the (b) thephaseFigure phase curve. 15.curve. The influence of the NMC shape on EF (in the centerline): (a) the amplitude curve; (b) the phase curve. TheThe EF ofof thethe model model for for the the trapezoidal trapezoidal NMC NM matchesC matches curve curve 3 in Figure 3 in Figure5, which 5, is which is notnot adapted adaptedThe EF toto theofthe the design design model idea idea for of of thethe the NMC. trapezoidal NMC. As aAs comparison, a NM comparison,C matches the EF the curve increases EF increases 3 in first Figure slowly first 5, slowly which is and then rapidly for the sub-triangular NMC, which matches curve 1 in Figure5 and meets andnot then adapted rapidly to thefor designthe sub-triangular idea of the NMC. NMC, As which a comparison, matches thecurve EF increases1 in Figure first 5 slowlyand theand requirements. then rapidly Hence, for thethe above sub-triangular analysis from NMC, Figure which 15 verifies matches that the curve sub-triangular 1 in Figure 5 and meetsshape the should requirements. be the ideal shape.Hence, the above analysis from Figure 15 verifies that the sub- meets the requirements. Hence, the above analysis from Figure 15 verifies that the sub- triangularThe top shape horizontal should width be the of ideal the sub-triangular shape. NMC in Figure 14 varies from 4 to 8triangular mm,The and top the shapehorizontal EF on should the mainwidth be PM the of is theideal shown sub-triangul shape. in Figure ar16 .NMC in Figure 14 varies from 4 to 8 mm, andTheThe topthe top widthEF horizontal on ofthe the main NMC width PM affects isof shown the EF sub-triangul little in Figure under thear16. NMC condition in Figure of the large14 variess. As froms 4 to 8 decreases,mm, and the the EF EF difference on the betweenmain PM the is width shown of 4in and Figure 8 mm 16. of the NMC begins to increase. The wider NMC provides the larger magnetic reluctance and the EF becomes larger. The ◦ ◦ maximum deviation angles are 6 and 3 for 4 and 8 mm of the NMC, respectively.

Energies 2021, 13, x FOR PEER REVIEW 16 of 29

68 4 mm 51 Energies 2021, 13, x FOR PEER REVIEW 8 mm 16 of 29 Energies 2021, 14, 318 16 of 27

(N) 34 F

17 68 0 4 mm 51 1197531 8s mm(mm) (a) (N) 34280 F

17270 ) ˚ 0260 1197531 s (mm)

Angle ( Angle 4 mm 250 (a) 280 8 mm 240 270 1197531

) s (mm) ˚ 260 (b)

Angle ( Angle 4 mm 250 8 mm Figure 16. The effect of the top horizontal width of the sub-triangular NMC on EF (in the non- center line240 of the slot, ratio = 2, L = 0.03 mm): (a) the amplitude curve; (b) the phase curve. 1197531 s (mm)

The top width of the NMC affects EF little under the condition of the large s. As s decreases, the EF difference between the (widthb) of 4 and 8 mm of the NMC begins to increase. The wider NMC provides the larger magnetic reluctance and the EF becomes FigureFigure 16. 16. TheThe effect effect of of the the top top horizontal horizontal width width of of the the sub-triangular sub-triangular NMC NMC on EF (in the non- non-center larger. The maximum deviation angles are 6° and 3° for 4 and 8 mm of the NMC, respec- centerline of line the of slot, the ratioslot, =ratio 2, L == 2, 0.03 L = mm):0.03 mm): (a) the (a) amplitude the amplitude curve; curve; (b) the (b phase) the phase curve. curve. tively. TheThe topbottom bottom width horizontal of the NMC widthwidth affects of of the the EF sub-triangular sub-triangular little under NMCthe NMC condition in Figure in Figure of14 thevaries 14 large varies from s. fromAs 0.5 tos 0.5 todecreases,2 2 mm, mm, and and the the theEF EF differenceEF is shownis shown between in Figurein Figure the17. width17. of 4 and 8 mm of the NMC begins to increase. The wider NMC provides the larger magnetic reluctance and the EF becomes larger. 68The maximum deviation angles are 6° and300 3° for 4 and 8 mm of the NMC, respec- 0.5 mm tively. 1 mm The51 bottom horizontal2 mm width of the sub-triangular280 NMC in Figure 14 varies from 0.5 )

to 2 mm, and the EF is shown in Figure 17. ˚

(N) 34 260 F 0.5 mm 68 300 17 0.5 mm ( Angle 240 1 mm 1 mm 2 mm 51 2 mm 280 0 220 ) 1197531 ˚ 1197531 (N) 34 s (mm) 260 s (mm) F 0.5 mm 17 ( Angle 240 1 mm (a) 2 mm (b) 220 FigureFigure 17.0 17. TheThe effect effect of of the the bottom bottom size of the sub-triangularsub-triangular NMC NMC on on EF: EF: (a ()a the) the amplitude amplitude curve; curve; 1197531 1197531 (b) the phase curve. s (mm) s (mm) (b) the phase curve. TheThe wider wider bottom bottom width ofof thethe NMCNMC(a) makes makes the the EF EF larger. larger. Once Once the the bottom bottom(b width) width exceedsFigureexceeds 17. 1 The 1 mm, mm, effect the the of EF the doesbottom notnot size increaseincrease of the sub-triangular slowlyslowly in in the theNMC beginning beginning on EF: (a and) theandthen amplitude then rapidly rapidly curve; with with increasing(bincreasing) the phase speed,curve. whichwhich does does not not meet meet the requirementthe requirement of the of FW. the Hence, FW. theHence, bottom the width bottom widthis designed is designed to be noto morebe no than more 1 mm.than 1 mm. TheThe widerbottom bottom bottom width width widthaffects of thethe the EFEF NMC moremore thanmakes than the the the top top EF width widthlarger. under under Once the the condition conditionbottom ofwidth largeof large sexceeds.s Hence,. Hence, 1 the mm, the bottom bottom the EF width width does is not more increase importantimportant slowly and and in should theshould beginning be be designed designed and specially.then specially. rapidly The The with rated rated speed is a critical speed at which the main PM starts to move. Hence, the bottom width speedincreasing is a criticalspeed, whichspeed doesat which not meetthe main the requirement PM starts to of move. the FW. Hence, Hence, the the bottom bottom width widthshould is bedesigned designed to tobe satisfyno more the than balance 1 mm. of its EF and the centrifugal force corresponding to theThe rated bottom speed width under affects this width.the EF more The bottom than the width top width is chosen under as 1the mm condition when s isof 11 large mm s.by Hence, calculation, the bottom meanwhile, width is the more EF important exactly matches and should the centrifugal be designed force. specially. The rated speed is a critical speed at which the main PM starts to move. Hence, the bottom width

Energies 2021, 14, 318 14 of 23

68 300 0.5 mm 1 mm 51 2 mm 280

(N) 34 260 F 0.5 mm

17 (˚) Angle 240 1 mm 2 mm 0 220 11 9 7 5 3 1 11 9 7 5 3 1 s (mm) s (mm)

(a) (b) Figure 17. The effect of the bottom size of the sub-triangular NMC on EF: (a) the amplitude curve; (b) the phase curve.

The wider bottom width of the NMC makes the EF larger. Once the bottom width exceeds 1 mm, the EF does not increase slowly in the beginning and then rapidly with increasing speed, which does not meet the requirement of the FW. Hence, the bottom width is designed to be no more than 1 mm. The bottom width affects the EF more than the top width under the condition of large s. Hence, the bottom width is more important and should be designed specially. The rated speed is a critical speed at which the main PM starts to move. Hence, the bottom width should be designed to satisfy the balance of its EF and the centrifugal force corresponding to the rated speed under this width. The bottom width is chosen as 1 mm when s is 11 mm by calculation, meanwhile, the EF exactly matches the centrifugal force.

5.2. The Influence of the NMC on Inductance The effect of the NMC on inductance is shown in Figure 18. The q-axis inductance (Lq) is larger than that of the d-axis inductance (Ld) whether we put the NMCs in the rotor or not. Meanwhile, the large current results in reducing both the d-axis and the q-axis inductance. The reason is that the large current increases the saturation of the rotor core. Energies 2021, 14, 318 Figure 18b shows that the NMC has little effect on the d-axis inductance and q17-axis of 27 inductance. The area of the NMC is not large compared to the rotor and it does not affect saturation too much. Meanwhile, the FW of this motor is less dependent on inductance than the common PMSM is. Therefore, in the design of the NMC, it is not necessary to 5.2. The Inﬂuence of the NMC on Inductance emphasize a consideration of the d-axis inductance and q-axis inductance. The effect of the NMC on inductance is shown in Figure 18.

0.05 0.05 with the NMC 0.04

without the NMC 0.04 (H)

0.03 (H) 0.03

Ld Lq 0.02 0.02 with the NMC without the NMC Energies 2021, 14, 318 0.01 0.01 15 of 23 0.6 1 1.4 1.8 0.6 1 1.4 1.8 Rated current times Rated current times

(a)

40 40 35 35

30 30

(mH) (mH)

Ld L

L Ld 25 25 Lq Lq 20 20 4 6 8 0.5 1 2 Bottom horizontal width (mm) Top horizontal width (mm) (b)

FigureFigure 18. 18. ThThee effect effect of ofthe the NMC NMC on on inductance: inductance: (a) (thea) the existence existence of the of theNMC NMC in the in therotor rotor;; (b) (tbhe) the differentdifferent NMC NMC size sizess (rated (rated current). current).

5.3. TheThe Influence q-axis of inductance the NMC on (L FWq) is A largerbility than that of the d-axis inductance (Ld) whether weThe put no the-load NMCs back in EMF the rotor is derived or not. from Meanwhile, Equation the(10) largeas follow currents. results in reducing both the d-axis and the q-axis inductance. The reason is that the large current increases the saturation of the rotor core. np ENkBL0 = 4.44() dpef  (10) Figure 18b shows that the NMC has little60 effect on the d-axis inductance and q-axis inductance. The area of the NMC is not large compared to the rotor and it does not affect where n, p, N, Kdp, B, τ and Lef are the motor speed, the numbers of pole pairs, the winding saturation too much. Meanwhile, the FW of this motor is less dependent on inductance turns, the winding factor of the fundamental wave, the magnetic flux density, the pole than the common PMSM is. Therefore, in the design of the NMC, it is not necessary to pitch and the axial length of the motor, respectively. emphasize a consideration of the d-axis inductance and q-axis inductance. For common PMSMs, p, N, Kdp, B,  and Lef are close to constants. However, for the new5.3. PMSM, The Inﬂuence the above of the parameters NMC on FW are Ability constants except for B. The main PMs are located at differentThe no-load positions back in EMFthe slot, is derived and the from magnetic Equation reluctance (10) as follows.is different, so the air-gap magnetic flux density is different and it is related to s. The no-load magnetic flux density is shown in Figure 19. np E = 4.44( )Nk BτL (10) 0 60 dp e f

where n, p, N, Kdp, B, τ and Lef are the motor speed, the numbers of pole pairs, the winding 0.8 0.8 turns, the winding factor of the fundamental wave, the magnetic ﬂux density, the pole pitch0.7 and the axial length of the motor, respectively.0.7

For common PMSMs, p, N, Kdp, B, τ and Lef are close to constants. However, for the (T) (T) 0.6

common PMSM 0.6 B

newB PMSM, the above parameters are constants except for B. The main PMs are located at different0.5 positions in the slot, and the magnetic0.5 reluctancenew is different, PMSM so the air-gap magnetic ﬂux density is different and it is related to s. The no-load magnetic ﬂux density is shown0.4 in Figure 19. 0.4 0 1 2 3 4 5 6 7 8 9 10 11 10 9 8 7 6 5 4 3 2 1 0 Time (s) s (mm)

(a) (b) Figure 19. The no-load air-gap magnetic flux density (n = 1500 r/min): (a) the magnetic flux density versus time for the common PMSM; (b) the magnetic flux density versus s for the new PMSM.

The no-load back EMF of the common PMSM and the new PMSM is shown in Figure 20. The no-load back EMF of the common PMSM in Figure 20a is proportional to motor speed due to constant magnetic flux density. However, as the speed increases, the no-load air-gap magnetic density of the new PMSM is reduced with the increase in motor speed. As the motor speed increases, the distance s becomes smaller and the no-load back EMF is reduced, which is shown in Figure 20b. Thus, it can be seen that reducing the no-load back EMF is an alternative new way to realize FW except for applying a large negative d- axis current.

Energies 2021, 13, x FOR PEER REVIEW 19 of 29

Energies 2021, 13, x FOR PEER REVIEW 19 of 29 Energies 2021, 14, 318 18 of 27

0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7

(T) 0.6 (T) 0.6

B common PMSM B

(T) 0.6 (T) 0.6

B common PMSM B 0.5 0.5 new PMSM 0.5 0.5 new PMSM 0.4 0.4 012345678910 0.4 0.4 11109876543210 012345678910Time (s) s (mm) 11109876543210 Time (s) s (mm) (a) (b) (a) (b) Figure 19. The no-load air-gap magnetic flux density (n = 1500 r/min): (a) the magnetic flux density versusFigureFigure time19. 19. The Thefor no-loadthe no-load common air-gap air-gap PMSM; magnetic magnetic (b) the flux flux magnetic density density (flux (nn == 1500density 1500 r/min): r/min): versus (a () a sthe) for the magnetic the magnetic new PMSM.flux flux density density versusversus time time for for the the common common PMSM; PMSM; (b (b) )the the magnetic magnetic flux flux density density versus versus ss forfor the the new new PMSM. PMSM. The no-load back EMF of the common PMSM and the new PMSM is shown in Figure 20. TheTheThe no-load no-load no-load back back back EMF EMF EMF of of the the of common thecommon common PMSMPMSM PMSM inand Figure the and new 20a the PMSM is new proportional PMSM is shown is into shown Figuremotor in speed20.Figure The due no-load20 .to The constant back no-load EMFmagnetic back of the EMF flux common density. of the commonPMSM However, in PMSMFigure as the in20aspeed Figure is proportional increases, 20a is proportionalthe to no-load motor air-gapspeedto motor due magnetic speedto constant duedensity tomagnetic constant of the fluxnew magnetic density.PMSM flux isHowever, reduced density. aswith However, the the speed increase asincreases, the in speed motor the increases, no-load speed. Asair-gapthe the no-load motor magnetic air-gapspeed density increases, magnetic of the the densitynew distance PMSM of the s isbecomes newreduced PMSM smaller with is the reduced and increase the withno-load in themotor back increase speed. EMF in isAsmotor reduced, the motor speed. which speed As theis increases,shown motor speedin Figurethe increases,distance 20b. Thus, s the becomes distance it can smaller bes becomesseen and that smallerthe reducing no-load and the theback no-load no-load EMF backisback reduced, EMF EMF is which isan reduced, alternative is shown which new in isFigure way shown to 20b. realiz in Thus, Figuree FW it except20 canb. be Thus, for seen applying it that can reducing be a seen large that thenegative reducingno-load d- axisbackthe current. no-loadEMF is an back alternative EMF is an new alternative way to realiz new waye FW to except realize for FW applying except fora large applying negative a large d- axisnegative current. d-axis current.

(a) (a)

120 120 1 mm 90 31 mmmm 5 mm 90 3 mm 60 75 mmmm 97 mmmm

Back EMF (V) 60 (b) 119 mm mm

Back EMF (V) 30 (b) 11 mm 30 0 0 0 1.3 2.6 3.9 5.2 Time (ms) 0 1.3 2.6 3.9 5.2 Time (ms) Figure 20. The no-load back electromotive force (EMF): (a) the no-load back EMF versus speed; (b) Figurethe no-load 20. The back no-load EMF back of the electromotive new PMSM versusforce (EMF):s (n = 1500(a) the r/min). no-load back EMF versus speed; (b) theFigure no-load 20. The back no-load EMF of back the electromotivenew PMSM versus force s(EMF): (n = 1500 (a) r/min).the no-load back EMF versus speed; (b) the no-loadThe differenceback EMF of of the the new no-load PMSMback versus EMF s (n = between 1500 r/min). the two extreme positions of the mainThe PMs difference at the same of the speed no-load is used back to measureEMF between the FW the effect two brought extreme by positions the special of rotorthe mainstructureThe PMs difference at rather the same than of speed thethe negativeno-load is used d-axisback to measur EMF demagnetization ebetween the FW effectthe current.two brought extreme The by extreme thepositions special positions of rotor the structuremainof the PMs main rather at PMsthe than same corresponding the speed negative is used to d-axiss areto measur 11 demagnetization ande 0 the mm, FW respectively. effect current. brought GoodThe by extreme FW the abilityspecial positions usually rotor structuremeans the rather larger than no-load the negative back EMF d-axis difference demagnetization between the current. two extreme The extreme positions positions of the

Energies 2021, 13, x FOR PEER REVIEW 20 of 29

Energies 2021, 14, 318 of the main PMs corresponding to s are 11 and 0 mm, respectively. Good FW19 ability of 27 usually means the larger no-load back EMF difference between the two extreme positions of the main PMs at the same speed. Figure 21 shows the no-load back EMF waveforms with differentmain PMs sizes at theof the same NMC speed. at the Figure speed 21 showsof 1500 the r/min. no-load back EMF waveforms with different sizes of the NMC at the speed of 1500 r/min.

180 top = 4 mm 120 top = 6 mm 60 top = 8 mm 0

Back EMF (V) 0246810 -60 -120 -180 Time (ms)

180 bottom = 0.5 mm 120 bottom = 1 mm 60 bottom = 2 mm 0 0246810 -60 Back EMF (V) -120 -180 Time (ms)

(a)

90 top = 4 mm 60 top = 6 mm 30 top = 8 mm 0

Back EMF (V) -30 0246810 -60 -90 Time (ms)

90 bottom = 0.5 mm 60 bottom = 1 mm 30 bottom = 2 mm 0

Back EMF (V) -30 0246810 -60 -90 Time (ms)

(b)

FigureFigure 21. 21. TheThe no-load backback EMF EMF with with different different top andtop bottomand bottom widths widths of the NMC of the (n NMC= 1500 ( r/min):n = 1500 r/min):(a) s = ( 11a) mm;s = 11 (b mm;) s = 0 ( mm.b) s = 0 mm.

EnergiesEnergies2021 2021, 14, 13, 318, x FOR PEER REVIEW 2021 of of 27 29

6.6. ExperimentalExperimental VerificationVerification TheThe designdesign principleprinciple ofof thethe NMCNMC shapeshape dependsdepends onon thethe requiredrequired electromagneticelectromagnetic forceforce (EF)(EF) actingacting onon the the main main PMs. PMs. UnderUnder thethe force force condition condition that that determines determines this this NMC NMC shape,shape, thethe criticalcritical speedsspeeds areare calculatedcalculated byby thethecorresponding corresponding centrifugal centrifugal force force according according toto forceforce balancebalance onon thethe main main PMs PMs when when the the PMs PMs are are in in extreme extreme positions positions ( s(s= = 1111 mmmm and and ss= = 00 mmmm).). TheyThey areare thethe calculatedcalculated criticalcritical speedsspeeds relatedrelated toto thethe NMC.NMC. TheThe measuredmeasured two two criticalcritical speeds speeds are are obtained obtained by by the the test test of of no-load no-load back back EMF. EMF. Whether Whether the the calculated calculated critical criti- speedscal speeds according according to the to forcethe force balance balance are are consistent consistent with with the the measured measured critical critical speeds speeds is indirectlyis indirectly reflected reflected in in the the correctness correctness of of NMC NMC design. design. In In this this section, section, the the manufacturing manufacturing processprocess of of the the NMC NMC is is described. described. The The FW FW validity validity of of the the motor motor is is verified verified by by load load test. test. The The criticalcritical speedsspeeds fromfrom thethe no-loadno-load back back EMF EMF test test verify verify the the correctness correctness of of the the NMC NMC design. design.

6.1.6.1. TheThe ManufacturingManufacturing ProcessProcess ofof thethe Non-MagneticNon-Magnetic ConductorConductor andand thethe Rotor Rotor TheThe manufacturingmanufacturing processprocess ofof thethe non-magneticnon-magnetic conductorconductor (NMC)(NMC) andand thethe rotorrotor isis shownshown inin FigureFigure 22 22..

FigureFigure 22.22.The The prototypeprototype ofof thethe non-magneticnon-magnetic conductorconductor and and the the rotor. rotor.

Firstly,Firstly, the the two two NMCs NMCs on on both both sides sides of of the the PM PM slot slot are are cut cut out out as as a whole.a whole. Secondly, Secondly, a frocka frock clamp clamp is is made made to to install install and and fix fix on on both both sides sides of of the the rotor rotor core. core. When When cutting cutting some some slotsslots inin thethe frockfrock clampclamp andand thethe rotorrotor core,core, thethe shapeshape ofof thethe slotsslots isis justjust likelike the the shape shape of of thethe NMCs.NMCs. Thirdly,Thirdly, the the NMCs NMCs are are inserted inserted into into the the rotor rotor core core and and the the frock frock clamp clamp to to make make themthem aa subassembly.subassembly. We We cut cut the the PM PM slot slot in in the the subassemblysubassembly and and guarantee guarantee the the parallelism parallelism ofof thethe PM PM slot. slot. The The frock frock clamp clamp is thenis then removed removed and and the the surface surface of the of rotorthe rotor core iscore polished. is pol- Theished. PM The slots PM with slots good with parallelism good parallelism are processed. are processed. Fourthly, Fourthly, the main the and main the and secondary the sec- PMs are embedded into the PM slots. Lastly, the two baffles are installed on both sides of ondary PMs are embedded into the PM slots. Lastly, the two baffles are installed on both the rotor core, and the rotor components are finished. sides of the rotor core, and the rotor components are finished. 6.2. The Work Characteristics and the FW Test 6.2. The Work Characteristics and the FW Test The rated parameters and test results of the motor are shown in Table1. The rated parameters and test results of the motor are shown in Table 1.

Energies 2021, 14, 318 21 of 27

Energies 2021, 14, 318 Table 1. The rated parameters and test results. 18 of 23

Rated Parameters Test Results Power 600 W Out power 601 W TorqueSpeed 3.8 1500N.m r/min Output Rated torque speed 15003.82 r/minN.m EfficiencyTorque 89% 3.8 N·mEfficiency Output torque 3.8290.1% N·m MaxEfficiency speed 3000 r/min 89% Max Efficiency speed 3800 90.1% r/min Max speed 3000 r/min Max speed 3800 r/min The distance s is changed to adjust the magnetic field for the VMRPMSM. For comparison,The the distance torques–isspeed changed curves to adjustof the thetraditional magnetic PMSM field forand the the VMRPMSM. VMRPMSM For are compar- calcu- latedison, by the the torque–speed finite element curves method. of the The traditional traditional PMSM PMSM and with the VMRPMSMthe same size are means calculated that sby is set the as finite 11 m elementm and method.s remains The a constant. traditional The PMSM load test with system the same, its mechanical size means thatcharacter-s is set isticsas 11 and mm the and efficiencys remains tests a constant. are shown The in load Figure test system,23. its mechanical characteristics and the efficiency tests are shown in Figure 23.

5 95

4 76 the VMRPMSM

3 57 ） (N.m)

T1 calculation % T T2 test （

2 38 η T3 η 1 19 calculation for the traditional PMSM 0 0 300 1000 1700 2400 3100 3800 n (r/min)

(a) (b)

. 3

20 ms/div

v i d / A

5 7 . 3

10 ms/div

Figure 23. Cont. Figure 23. Cont. v i d / A

5 7 . 3

10 ms/div

Energies 2021, 14, 318 18 of 23

Torque 3.8 N.m Output torque 3.82 N.m Efficiency 89% Efficiency 90.1% Max speed 3000 r/min Max speed 3800 r/min

The distance s is changed to adjust the magnetic field for the VMRPMSM. For comparison, the torque–speed curves of the traditional PMSM and the VMRPMSM are calculated by the finite element method. The traditional PMSM with the same size means that s is set as 11 mm and s remains a constant. The load test system, its mechanical characteristics and the efficiency tests are shown in Figure 23.

5 95

4 76 the VMRPMSM

3 57 ） (N.m)

T1 calculation % T T2 test （

2 38 η T3 η 1 19 calculation for the traditional PMSM 0 0 300 1000 1700 2400 3100 3800 n (r/min)

(a) (b)

. 3

20 ms/div

v i d / A

5 7 . 3

10 ms/div Energies 2021 , 14 , 318 22 of 27

Figure 23. Cont. v i d / A

5 7 .

Energies 2021, 14, 318 3 19 of 23

10 ms/div

7 .

5 ms/div

(c)

Figure 23. The working characteristics test: (a) the load experiment test system; (b) the torque Figure 23. The working characteristics test: (a) the load experiment test system; (b) the torque and and the efficiency of the PMSM based on the principle of variable exciting magnetic reluctance the efficiency of the PMSM based on the principle of variable exciting magnetic reluctance (VMRPMSM) and the traditional PMSM; (c) the measured current and torque operated at 500, 1500, (VMRPMSM) and the traditional PMSM; (c) the measured current and torque operated at 500, 1500, 2500, 3500 r/min. 2500, 3500 r/min. The test of the rated efficiency for the VMRPMSM is 90%. The efficiency of the motor The test of the ratedin whole efficiency speed range for the is good.VMRPMSM The VMRPMSM is 90%. T canhe efficiency meet the demand of the motor of the rated output in whole speed rangepower. is good. The maximumThe VMRPMSM measured can speed meet of the the demand motor is 3800of th r/min.e rated The output FW ability of the power. The maximumVMRPMSM measured is verifiedspeed of by the the motor experimental is 3800 test. r/min The. The maximum FW ability motor of speed the is 2.5 times VMRPMSM is verifiedthat by of thethe ratedexperimental speed. The test. torque–speed The maximum curve motor calculated speed by is finite 2.5 elementtimes method is that of the rated speed.basically The consistent torque–speed with the curve measured calculated curve. by As finite shown element in Figure method 23b, the speedis range of the VMRPMSM is significantly wider than that of the traditional motor. basically consistent with the measured curve. As shown in Figure 23b, the speed range of the VMRPMSM is significantly6.3. The Validation wider Method than ofthat the of Correctness the traditional of NMC motor. Design The NMC of the PMSM is very important for realizing FW. The design of the NMC 6.3. The Validation Methodis closely of the related Correctness to the rated of NMC speed Design of the new PMSM. The EF on the main PM located The NMC of thein PMSM the initial is positionvery important as shown for in realizing Figure1 is FW calculated. The design by the of finite the element NMC method to is closely related to getthe therated corresponding speed of the centrifugal new PMSM force.. The This EF centrifugal on the main force PM on thelocated main in PM must keep the initial position asbalance shown with in Figure the sum 1 of is the calculated EF and the by friction the finite force. element The special method speed to corresponding get to this exactly calculated centrifugal force is selected as the value of the rated speed. the corresponding centrifugal force. This centrifugal force on the main PM must keep bal- The main PMs keep stationary at s = 11 mm operated below the rated speed, and the ance with the sum ofair-gap the EF magnetic and the fluxfriction density force is. closeThe special to a constant. speed corresponding The rated speed to of this the motor is the exactly calculated centrcriticalifugal speed. force The is mainselected PMs as begin the value to move of inthe the rated PM speed slot once. the speed of the motor The main PMs exceedskeep stationary the rated at speed. s = 11 The mm main operated PMs will below become the stationaryrated speed, again and once the the forces on air-gap magnetic fluxthem density reach ais new close balance to a andconstant. work steadily The rated at the speed new position of the motor in the PM is the slot. The air-gap critical speed. The mainmagnetic PMs flux begin density to move of the in motor the PM working slot once at this the new speed speed of is the reduced motor relative to the exceeds the rated speed.rated speed.The main This PMs movement will become process ofstationary the main PMsagain continues once the until forces the mainon PMs reach them reach a new balanceto the top and of thework PM steadily slot. The at main the PMsnew always position work in atthe the PM top slot. of the The PM air slot- and become stationary again once the speed of the motor exceeds a critical speed. This corresponding gap magnetic flux density of the motor working at this new speed is reduced relative to speed is another critical speed. The NMC shape is designed according to the balance of the the rated speed. Thiscentrifugal movement force process and the of sum the of main the EF PMs and continues the friction until force onthe the main main PMs PMs at the rated reach to the top of thespeed. PM Therefore,slot. The main the design PMs validity always of work the NMC at the can top be of indirectly the PM reflectedslot and by the test of become stationary againno-load once back the EMF speed and of the the critical motor speeds. exceeds a critical speed. This corresponding speed is another critical speed. The NMC shape is designed according to the balance of the centrifugal force and the sum of the EF and the friction force on the main PMs at the rated speed. Therefore, the design validity of the NMC can be indirectly reflected by the test of no-load back EMF and the critical speeds.

6.4. The Experimental Test of the NMC The no-load air-gap magnetic flux density is derived by the Equation (10) as follows.

60E0 B = (11) 4.44npNkdp L ef

The prototype is driven by the driving motor, which is shown in Figure 24a. Figure 24c shows the measured waveforms of the no-load back EMF at different speeds. The

Energies 2021, 14, 318 23 of 27 Energies 2021, 13, x FOR PEER REVIEW 25 of 29

6.4. The Experimental Test of the NMC The no-load air-gap magnetic ﬂux density= is derived60E0 by the Equation (10) as follows. B (11) 4.44npNkτ L 60E dp ef B = 0 (11) 4.44npNkdpτLe f

The prototypeprototype is drivenis driven by theby drivingthe driving motor, motor, which iswhich shown is in shown Figure 24ina. FigureFigure 24a.24c Figure 24cshows shows the measured the measured waveforms waveforms of the no-load of the backno-load EMF back at different EMF at speeds. different The speeds. average The av- erageeffective effective value of value the three of the phases three of phases the no-load of th backe no-load EMF at back 1500 EMF r/min at is 1500 70.4 V,r/min which is 70.4 V, whichis in agreement is in agreement with the simulationwith the simulation results in Figure results 20. in The Figure no-load 20. air-gap The no-load magnetic air-gap ﬂux mag- neticdensity flux test density curve in test Figure curve 24b isin derived Figure by24b testing is derived the no-load by testing back EMF the of no-load the prototype back EMF of theaccording prototype to Equation according (11). to Equation (11).

0.8 test 0.7 theory

0.6 (T)

B 0.5 critical speed 0.4

0.3 200 1400 2600 3800 n (r/min)

(a) (b)

Figure 24. Cont.

EnergiesEnergies 20212021, 13, 14, x, 318FOR PEER REVIEW 24 of 27 26 of 29

2500 r/min

2ms/div

(c)

Figure 24.24. The validity testtest ofof thethe NMCNMC according according to to the the no-load no-load back back EMF EMF test test (the (the speed-up speed-up process):process): (a) (thea) the no-load no-load back back EMF EMF test; test; (b (b) )the the no-load no-load air-gap magneticmagnetic ﬂux flux density density test; test; (c) ( thec) the no- loadno-load back back EMF EMF waveforms waveforms at at different different speeds.speeds.

Theoretically,Theoretically, the the no-load no-load air-gap air-gap magnetic magnetic density density varies varies with motor with speedmotor in speed three in three ways asas shownshown below: below: (1) InIn the the firstfirst case,case, the the speed speed is lessis less than than 1500 15 r/min.00 r/min. The centrifugalThe centrifugal force on force the mainon the main PMsPMs is is less thanthan the the sum sum of of the the EF EF and and the frictionthe friction force, force, and the and main the PMs main are PMs fixed are fixed in the position of s = 11 mm. The no-load air-gap magnetic flux density is close to a in the position of s = 11 mm. The no-load air-gap magnetic flux density is close to a constant. (2) constant.In the second case, the speed varies from 1500 to 3000 r/min. The main PMs can (2) Inmove the insecond the PM case, slot andthe speed keep in varies balance from dynamically, 1500 to 3000 so the r/min. no-load The air-gap main magnetic PMs can move indensity the PM is aslot variable and keep and it in is balance related to dynamically, the shape of the so NMC.the no-load The shape air-gap of the magnetic NMC den- sityis an is approximate a variable and triangle, it is andrelated the widthto the of shape the NMC of the narrows NMC. first The and shape then widensof the NMC is anlater approximate as the motor triangle, speed increases. and the Hence,width theof the no-load NMC air-gap narrows magnetic first and density then is widens laterdecreased as the first motor slowly speed and increases. then fast as Hence, the speed the increases no-load theoretically.air-gap magnetic density is de- (3) creasedIn the third first case,slowly the and speed then is morefast as than the 3000 speed r/min. increases The centrifugal theoretically. force on the (3) Inmain the PMsthird is case, larger the than speed the sumis more of the than EF and3000 the r/min. friction The force, centrifugal so the main force PM on is the main fixed in the position of s = 0 mm. Again, the no-load air-gap magnetic flux density is PMs is larger than the sum of the EF and the friction force, so the main PM is fixed in close to a constant. the position of s = 0 mm. Again, the no-load air-gap magnetic flux density is close to The test of the critical speeds for the movement of the main PMs in the speed-up processa constant. is shown in Table2. The test of the critical speeds for the movement of the main PMs in the speed-up processTable 2. The is shown critical speed in Table in the 2. speed-up process.

TableIn Theory, 2. The The critical Main speed PM (The in the Speed-Up speed-up Process) process. Calculation (r/min) Test (r/min) stationary 0–1500 0–1600 In Theory, The Mainmoveable PM (The Speed-Up Process) 1500–3000 Calculation (r/min) 1600–3100Test (r/min) stationarystationary >30000–1500 3100–3800 0–1600 moveable 1500–3000 1600–3100 The rated speed of thestationary motor is designed by the force balance. The>3000 shape of the NMC3100–3800 directly affects the EF on the main PMs, so the NMC shape is closely related to the rated speed. The rated speed is the critical speed at which the main PM starts to move. The The rated speed of the motor is designed by the force balance. The shape of the NMC critical speed is designed as 1500 r/min. The measured speed is 1600 r/min. The error directly affects the EF on the main PMs, so the NMC shape is closely related to the rated speed. The rated speed is the critical speed at which the main PM starts to move. The critical speed is designed as 1500 r/min. The measured speed is 1600 r/min. The error of the designed value and the measured value lies in that the friction force is difficult to

Energies 2021, 14, 318 25 of 27

of the designed value and the measured value lies in that the friction force is difficult to estimate exactly; however, the error is acceptable. This indicates that the shape design of the NMC of the prototype is consistent with the analysis. The experimental results in Figure 24b and Table2 show that the motor operates below 1600 r/min and the no-load air-gap magnetic flux density is a constant. In the speed range from 1600 to 3100 r/min, the no-load air-gap magnetic flux density declines slowly first and then rapidly with the increase in speed. The test curve of the no-load air-gap magnetic density is consistent with the shape analysis of the NMC. The speed is more than 3100 r/min and the no-load air-gap magnetic flux density remains a constant again. The measured no-load air-gap magnetic flux density is consistent with the theoretical analysis. The correctness and effectiveness of the NMC shape on EF and FW effect are verified.

7. Conclusions Qualitative analysis, quantitative calculation and testing of the non-magnetic conductor for the new PMSM indicate: (1) The idea of FW for the VMRPMSM is to reduce the no-load back EMF at high speed, instead of applying a demagnetizing current. The effective magnetic reluctance in the magnetic circuit is changed due to the NMC by the movement of the permanent magnets, thus the no-load back EMF is changed. The speed range is widened by 2.5 times that of the rated speed. (2) The shape of the NMC depends on the EF required to satisfy the FW. The centrifugal force increases with the speed slowly in the beginning and then rapidly. Theoretically, the width of the NMC should be designed to be narrow at low speed and then widen gradually with the increase in speed according to the EF. The sub-triangular shape is ideal to meet the requirement of the force. The effect of trapezoidal and sub- triangular NMC on EF by finite element method indicates that the EF increases with the speed slowly in the beginning and then rapidly for the motor of sub-triangular NMC. This verifies that the sub-triangular shape is ideal, which is consistent with the theoretical analysis. (3) The no-load air-gap magnetic density is a variable. It keeps constant, variable and constant when s = 11 mm, 0 mm < s < 11 mm and s = 0 mm, respectively. The critical speeds are calculated for s = 11 mm and s = 0 mm by the corresponding centrifugal force according to force balance on the main PMs. According to the force balance, the NMC shape should be designed to make sure that the critical movement speed of the main PMs is exactly equivalent to the rated speed. The measured two critical speeds are obtained by the test of no-load back EMF. The measured critical speeds agree with the calculated critical speeds and that verifies the correctness of the shape design of the NMC. (4) The NMC affects the EF on the main PMs greatly while the NMC has little effect on the inductance. The inductance design does not need to consider the NMC. The wider NMC is beneficial for the FW ability of the motor. The NMC is chosen to be as wide as possible to enlarge the FW under the condition of the requirements of the EF.

Author Contributions: Analysis, investigation and validation, C.L.; supervision, B.K.; writing— original draft preparation, F.G. and C.L.; writing—review and editing, T.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported in part by the National Natural Science Foundation of China under Grant 51307045, in part by the Heilongjiang Natural Science Foundation under Grant LH2019E075, in part by the Heilongjiang Provincial Education Department Foundation under Grant KJCX201915 and TSTAU-C2018014, in part by the Heilongjiang University Outstanding Youth Science Foundation under Grant JCL201705. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Energies 2021, 14, 318 26 of 27

Data Availability Statement: All data are given in the paper. Separately there is no other data. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Nishio, H.; Machida, K. Magnetic anisotropy effects on squareness ratio of Nd-Fe-B sintered magnets with different coercivity. IEEE Trans. Magn. 2020, 56, 2100604. [CrossRef] 2. Zhang, J.; Liu, J.; Yang, J.; Zhao, N.; Wang, Y.; Zheng, T. A modified DC power electronic transformer based on series connection of full-bridge converters. IEEE Trans. Power Electron. 2019, 34, 2119–2133. [CrossRef] 3. Ma, K.; Choi, U.M.; Blaabjerg, F. Prediction and validation of wear-out reliability metrics for power semiconductor devices with mission profiles in motor drive application. IEEE Trans. Power Electron. 2018, 33, 9843–9853. [CrossRef] 4. Chen, H.; Lee, C. Parametric sensitivity analysis and design optimization of an interior permanent magnet synchronous motor. IEEE Access 2019, 7, 159918–159929. [CrossRef] 5. Ekanayake, S.; Dutta, R.; Rahman, F.; Xuan, B.M. Performances of a fractional-slot concentrated-winding permanent magnet synchronous machine under position sensorless control in deep flux-weakening region. IEEE Trans. Ind. Appl. 2019, 55, 5938–5946. [CrossRef] 6. Wang, X.; Reitz, M.; Yaz, E.E. Field oriented sliding mode control of surface-mounted permanent magnet AC motors: Theory and applications to electrified vehicles. IEEE Trans. Veh. Technol. 2018, 67, 10343–10356. [CrossRef] 7. Zhang, Z.; Wang, C.; Zhou, M.; You, X. Parameters compensation of permanent magnet synchronous motor in flux-weakening region for rail transit. IEEE Trans. Power Electron. 2020, 35, 12509–12521. [CrossRef] 8. Zhang, Z.; Wang, C.; Zhou, M.; You, X. Flux-weakening in PMSM drives: Analysis of voltage angle control and the single current controller desgin. IEEE J. Emerg. Sel. Top. Power Electron. 2019, 7, 437–445. [CrossRef] 9. ELLoumi, N.; Bortolozzi, M.; Masmoudi, A.; Mezzarobba, M.; Olivo, M.; Tessarolo, A. Numerical and analytical approaches to the modeling of a spoke type IPM machine with enhanced flux weakening capability. IEEE Trans. Ind. Appl. 2019, 55, 4702–4714. [CrossRef] 10. Dang, L.; Bernard, N.; Bracikowski, N.; Berthiau, G. Design optimization with flux weakening of high-speed PMSM for electrical vehicle considering the driving cycle. IEEE Trans. Ind. Electron. 2017, 64, 9834–9843. [CrossRef] 11. Chai, F.; Zhao, K.; Li, Z.; Gan, L. Flux weakening performance of permanent magnet synchronous motor with conical rotor. IEEE Trans. Magn. 2017, 53, 8208506. [CrossRef] 12. Kong, Y.; Lin, M.; Jia, L. A novel high power density permanent-magnet synchronous machine with wide speed range. IEEE Trans. Magn. 2020, 56, 7505206. [CrossRef] 13. Zheng, Y.; Wu, L.; Fang, Y.; Huang, X.; Lu, Q. A hybrid interior permanent magnet variable flux memory machine using two-part rotor. IEEE Trans. Magn. 2019, 55, 8106008. [CrossRef] 14. Dutta, R.; Rahman, M.F. Design and analysis of an interior permanent magnet (IPM) machine with very wide constant power operation range. IEEE Trans. Energy Convers. 2008, 23, 25–33. [CrossRef] 15. Kim, K.C. A novel magnetic flux weakening method of permanent magnet synchronous motor for electric vehicles. IEEE Trans. Magn. 2012, 48, 4042–4045. [CrossRef] 16. Wang, D.; Peng, C.; Xue, D.; Zhang, D.; Wang, X. Performance assessment and comparative study of a permanent magnet machine with axial flux regulator. IEEE Trans. Energy Convers. 2019, 34, 1522–1531. [CrossRef] 17. Wang, D.; Zhang, D.; Xue, D.; Peng, C.; Wang, X. A new hybrid excitation permanent magnet machine with an independent AC excitation port. IEEE Trans. Ind. Electron. 2019, 66, 5872–5882. [CrossRef] 18. Wang, M.; Tong, C.; Song, Z. Performance analysis of an axial magnetic-field-modulated brushless double-rotor machine for hybrid electric vehicles. IEEE Trans. Ind. Electron. 2019, 66, 806–817. [CrossRef] 19. Wang, C.; Zhu, Z.Q. Fuzzy logic speed control of permanent magnet synchronous machine and feedback voltage ripple reduction in flux-weakening operation region. IEEE Trans. Ind. Appl. 2020, 56, 1505–1517. [CrossRef] 20. Bedetti, N.; Calligaro, S.; Petrella, R. Analytical design and autotuning of adaptive flux-weakening voltage regulation loop in IPMSM drives with accurate torque regulation. IEEE Trans. Ind. Appl. 2019, 56, 301–313. [CrossRef] 21. Xu, W.; Ismail, M.M.; Liu, Y.; Islam, M.R. Parameter optimization of adaptive flux-weakening strategy for permanent-magnet synchronous motor drives based on particle swarm algorithm. IEEE Trans. Power Electron. 2019, 34, 12128–12140. [CrossRef] 22. Chen, J.J.; Chin, K.P. Automatic flux-weakening control of permanent magnet synchronous motors using a reduced-order controller. IEEE Trans. Power Electron. 2000, 15, 881–890. [CrossRef] 23. Bolognani, S.; Calligaro, S.; Petrella, R. Adaptive flux-weakening controller for interior permanent magnet synchronous motor drives. IEEE J. Emerg. Sel. Top. Power Electron. 2014, 2, 236–248. [CrossRef] 24. Su, D.; Zhang, C.; Dong, Y. An improved continuous-time model predictive control of permanent magnetic synchrounous motors for a wide-speed range. Energies 2017, 10, 2051. [CrossRef] 25. Tursini, M.; Chiricozzi, E.; Petrella, R. Feedforward flux-weakening control of surface mounted permanent magnetic synchronous motors accounting for resistive voltage drop. IEEE Trans. Ind. Electron. 2010, 57, 440–448. [CrossRef] 26. Lin, X.; Huang, W.; Jiang, W. Position senseless direct torque control for six-phase permanent magnet synchronous motor under two-phase open circuit. IET Electr. Power Appl. 2019, 13, 1625–1637. [CrossRef] Energies 2021, 14, 318 27 of 27

27. Choi, Y.S.; Choi, H.H.; Jung, J.W. Feedback linearization direct torque control with reduced torque and flux ripples for IPMSM drives. IEEE Trans. Power Electron. 2016, 31, 3728–3737. [CrossRef] 28. Nasr, A.; Gu, C.; Bozhko, S.; Gerada, C. Performance enhancement of direct torque-controlled permanent magnet synchronous motor with a flexible switching table. Energies 2020, 13, 1907. [CrossRef] 29. Bao, G.; Qi, W.; He, T. Direct torque control of PMSM with modified finite set model predictive control. Energies 2020, 13, 234. [CrossRef] 30. Ekanayake, S.; Dukmi, R.; Rahman, F.; Xuan, B.M. Direct torque and flux control of interior permanent magnet synchronous machine in deep flux-weakening region. IET Electr. Power Appl. 2018, 12, 98–105. [CrossRef] 31. Kim, S.; Seok, J.K. Finite-settling-steps direct torque and flux control for torque-controlled interior PM motors at voltage limits. IEEE Trans. Ind. Appl. 2014, 50, 3374–3381. [CrossRef] 32. Nguyen, T.D.; Foo, G.; Tseng, K.J.; Vilathgamuwa, D.M. Modeling and sensorless direct torque and flux control of a dual air gap axial flux permanent magnet machine with field weakening operation. IEEE-ASME Trans. Mechatron. 2013, 19, 412–422. [CrossRef]