JECE Journal of Electrical and Computer Engineering 2090-0155 2090-0147 Hindawi Publishing Corporation 10.1155/2016/4501046 4501046 Research Article A Model for Analyzing a Five-Phase Fractional-Slot Permanent Magnet Tubular Linear Motor with Modified Winding Function Approach http://orcid.org/0000-0003-1308-5050 Zhang Bo 1,2 2 Qi Rong 1 Lin Hui 1 http://orcid.org/0000-0001-7221-1219 Mwaniki Julius 1 Liu Bin-Da 1 Department of Electrical Engineering School of Automation Northwestern Polytechnical University Xi’an 710129 China nwpu.edu.cn 2 Department of Electrical Engineering Electronic Information College Xi’an Polytechnic University Xi’an 710048 China xpu.edu.cn 2016 27102016 2016 16 03 2016 13 06 2016 20 06 2016 2016 Copyright © 2016 Bo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a model for analyzing a five-phase fractional-slot permanent magnet tubular linear motor (FSPMTLM) with the modified winding function approach (MWFA). MWFA is a fast modeling method and it gives deep insight into the calculations of the following parameters: air-gap magnetic field, inductances, flux linkages, and detent force, which are essential in modeling the motor. First, using a magnetic circuit model, the air-gap magnetic density is computed from stator magnetomotive force (MMF), flux barrier, and mover geometry. Second, the inductances, flux linkages, and detent force are analytically calculated using modified winding function and the air-gap magnetic density. Finally, a model has been established with the five-phase Park transformation and simulated. The calculations of detent force reveal that the end-effect force is the main component of the detent force. This is also proven by finite element analysis on the motor. The accuracy of the model is validated by comparing with the results obtained using semianalytical method (SAM) and measurements to analyze the motor’s transient characteristics. In addition, the proposed method requires less computation time.

1. Introduction

Permanent magnet linear synchronous motors (PMLSM) have been developed for many years . Compared with traditional rotary-to-linear electric actuators, these direct linear electric-mechanical energy conversion devices have no mechanical gears and transmission systems; hence, they possess higher dynamic performance and reliability . Permanent magnet tubular linear motors (PMTLM) are a class of PMLSMs and are particularly attractive owing to zero attractive force between the stator and armature, high force density, excellent servo characteristics, and higher fault-tolerant performance . PMTLM has been widely used in many linear motion fields, for example, transportation, manufacturing, health care, electromagnetic launch in space applications , ECO-pedal system , and so forth.

An accurate model is important in analyzing and controlling PMTLMs. In this regard, the calculation of air-gap magnetic field is critical in modelling the motor. This is because the air gap flux density distribution has a deep influence on the thrust ripples . There are many methods to calculate the air-gap flux density distribution, for example, finite element method (FEM) , analytical method (AM) , or SAM . FEM is an accurate numerical prediction method but it is time-consuming; thereby, it is not suitable for the simulation of a controlled machine . AM can decrease the time consumed. It uses Laplace’s and Poisson’s equations to solve the scalar magnetic potential or the vector magnetic potential. However, the complicated boundary conditions increase the difficulty of the solving process . SAM can balance the accuracy of the model and the time consumed. In , a 5-phase PMTLM has been modelled with this method where the calculation of the magnetic field distribution has been done with FEM in advance. Consequently, modelling the motor is time-saving. However, once the motor power supply changes or faults arise, recalculation of the magnetic field distribution still needs using FEM. Hence, it is still time-consuming.

Modified winding function approach is another analytical method which gives insight into the calculations of parameters without considering the complicated boundary conditions. It is a simple, fast modelling method for motors. This paper shows the detailed calculations of the parameters for a five-phase FSPMTLM, such as the detailed calculation of the detent force, which are not expressed distinctly in [12, 15, 16]. The model of the studied motor is established with MWFA and Park transformation theory, and the simulations results obtained by the mathematical model are compared with the ones obtained by SAM and measurements.

2. Modified Winding Function Analysis 2.1. Description of the Machine

Figure 1 shows the physical structure of a five-phase PMTLM and the schematic outline of the halved motor cross section. The stator is separated into five sections. Each section forms one phase and a magnetic separation of a nonferromagnetic ring made of stainless steel, which is used as a flux barrier. The mover is assembled from permanent magnets (PMs) and ferromagnetic rings; the PMs are characterized by axial magnetization and are situated alternately with the ferromagnetic rings. Table 1 gives details of the machine specifications.

Specifications of the PMTLM.

Symbol Quantity Value
w s s Axial width of module 18 mm
b o Axial width of slot open 12 mm
w c Axial width of armature 10 mm
w t Axial thickness of module core (leg) 3 mm
T t Axial space of slot pitch 21 mm
τ Pole pitch 15 mm
h c Height of armature coil 30 mm
h s Height of module 35 mm
g Air gap (mechanical clearance) 1 mm
d s Axial magnetic separation distance 3 mm
τ p Axial ferromagnetic core width between PMs 7 mm
τ m Axial PM width 8 mm
R o Outer radius of tubular reaction rail 15 mm
R i Inner radius of tubular reaction rail 9 mm
L Axial width of stator 102 mm
N Number of turns 280/slot

Physical structure of PMTLM and the schematic outline of the halved motor cross section: (a) motor physical; (b) cross section of the motor.

2.2. Modified Winding Function on the Motor

MWFA was first proposed for solving air-gap eccentricity in rotary machines . The modified winding function (MWF) was deduced from the MMF drops in a magnetic circuit between the stator and the rotor, which was obtained under the following assumptions: the iron in the stator and mover has infinite permeability; the magnetic saturation is neglected; the mover length is infinite so that the structure of the motor is symmetrical; there is no leakage flux in the shaft radial, Ri; and the magnetic permeability of the PMs is deemed as the magnetic permeability of air μ0. It is defined as follows(1)Mφ,θ=nφ,θ-02πnφ,θg-1φ,θdφ2πg-1φ,θ,where φ and θ represent the position of a stationary coil and angular position of the rotor with respect to stator, respectively. M(φ,θ), n(φ,θ) and g-1φ,θ represent the modified winding function, turns function, and air-gap function, respectively. The symbol · represents the average value of “·”. If the rotor is not eccentric, (1) reduces to (2)Mφ,θ=nφ,θ-n,where n is the dc value of the turns function of the winding.

Different from the rotary machines, the analysis of the modified winding function requires creating an appropriate coordinate system on the stator. As shown in Figure 1(b), the origin of coordinates is on the left schematic outline of the halved motor cross section, and the distance between origin and the center of the phase “C” is 2.5 times axial space of slot pitch Tt. The pole pairs in the longitudinal section of the stator are 3.5, no matter whether the mover is run or not. Hence, the motor has a fractional-slot structure and the slot-per-phase-per-pole (SPP) is equal to 2/7. Seen from Figure 1(b) and Table 1, along the shaft of the motor, the relation of the slot number Qs, slot pitch Tt, pole pitch τ, and pole pairs p is written by (3)QsTt=2pτ.

Due to the influence of the fractional slot on the winding function distribution, the modified winding function of the five-phase FSPMTLM is defined as the product of winding function and the winding factor yw . Similar to the rotary 5-phase PMSM , the longitudinal section of the five-phase FSPMTLM windings is assumed to be symmetrical. Figure 2 shows the winding function of the phase “a” and the spatial structure of windings in five-phase FSPMTLM. The Fourier’s expansion of the winding function of phase “a” is calculated from(4)Nax=4Npπk=1,3,5,1ksinwckπ2τcoskπτx-ε.

Winding function of phase “a” and winding space distribution: (a) winding function; (b) spatial structure of windings.

In Figure 2(a), Nax is the winding function of phase “a,” the letter “ε” represents the distance between the origin, and the radial center line of phase “a” and N represents the stator turns number. In Figure 2(b), “α” is the winding space geometric angle of the adjacent phases. Because the stator structure is symmetrical, the plane is evenly divided into five blocks and each angle is equal to 2π/5. Combining the all-order winding factor ywk (k is odd number and k=1,3,5,) and (3) yields the all-phase modified winding function, Mi(x): (5)Mix=4Npπk=1,3,5,ywkksinwckπ2τcoskπτx-ε-iα,where the subscript i(i=0,1,2,3,4) represents the ordinal number of phase a, b, c, d, and e, respectively. For example, M1(x) shows the modified winding function of phase “b.” The detailed calculation of (5) is as shown in Appendix A.

2.3. Air Gap Flux Density

The generation of air gap flux density is due to the current flowing in each phase. Hence, the flux density in phase “a” and B0x,θ is defined as the product of the modified winding function M0x and the inverse air gap function g-1x,θ :(6)B0x,θ=μ0g-1x,θM0xia,where θ is the distance which the mover has covered from the origin of the stator coordinates and ia is the current of phase “a.”

Calculation of the inverse air gap function g-1x,θ requires modeling the flux paths through the air gap regions with straight lines and circular arc segments . The flux paths due to the mover saliency are shown in Figure 3. The inverse air gap function g-1x,θ is obtained by the unslotted air-gap flux density  and the relative air-gap permeance . The calculation of g-1x,θ in the Fourier series form is (7)g-1x,θ=n=0Gncosnπτx-θ-ε,where the expression of the coefficient Gn is(8)Gn=42.5Ttθ+εTt+θ+εg-1x,θcosnπτx-θ-εdx+22.5Ttθθ+εg-1x,θcosnπτx-θ-εdx.

Flux paths due to the stator slot and mover saliency.

The detailed computations of the inverse air-gap function g-1x,θ are shown in Appendix B.

2.4. Calculations of Inductances and Permanent Magnet Flux Linkages

In terms of the modified winding function theory, the inductances and the permanent magnet flux linkages are derived from the air-gap function and the modified winding functions. Both are computed for the volume of the air-gap section of the motor with the effective air-gap radius Rse , as is depicted in Figure 4.

The schematic diagram of computational volume in the air gap.

In Figure 4, Ro and R are the outer radius of tubular reaction rail and the radius of the stator tooth, respectively (see Section 2.1). Rse is the effective air-gap computational radius and the length of the computational volume is 2pτ (see Section 2.2).

The self and mutual inductances are computed in the computational volume, as shown by [17, 23]:(9)Liiθ=2πRseμ002pτMi2xg-1x,θdxi=0,1,2,3,4,Lijθ=2πRseμ002pτMixMjxg-1x,θdxij,i,j=0,1,2,3,4,where Lii(θ) and Lij(θ) are the self-inductances and mutual inductances, which are the function of mover position θ; Mi(θ) and Mj(θ) are the modified winding functions of the ith phase and the jth phase, respectively.

The permanent magnet flux linkages are derived from the air-gap flux density produced by permanent magnets and the modified winding functions, as computed by(10)ψmiθ=2πμ0Rse02pτMixBgx,θdx,where Bgx,θ represents the air-gap flux density, which is the product of PM flux density Bpmx,θ and relative air gap permeance λagx,θ .

The detailed calculations of the effective air-gap computational radius Rse and the air-gap flux density Bgx,θ are shown in Appendix C.

3. Mathematical Model 3.1. Basic Model of the PMLMs

The stator voltage equations and mechanical thrust equations compose the mathematical model of PMLMs : (11)Vs=RsIs+dψsdt,where Vs, Rs, and Is represent the stator voltage, the stator resistance, and the stator current, respectively. ψs represents the air-gap flux linkages produced by the permanent magnet and the stator currents. It can be calculated from the following formulae:(12)ψs=ψm+LsIs,where ψm is the flux linkages produced by PMs; Ls represents the inductances matrix including self- and mutual inductances matrix, as computed by (9):(13)Fe=mdνdt+Dν+Fl+Fdetent,where Fe is the electromagnetic thrust force, m is the total mass on the mover, D is the dynamical friction coefficient, ν is the mover velocity, Fl is the load force, and Fdetent is the detent force including the end-effect force Fend and slot-effect force Fslot.

3.2. Calculations of the Detent Force

The detent force is the interaction force between the mover magnets and stator slots without the stator currents flowing . Due to the opening stator slots and the bilateral ends of the stator, the detent force is the sum of the end-effect force Fend and slot-effect force Fslot:(14)Fdetent=Fslot+Fend.

Using the Virtual Work Method (VWM), Fend and slot-effect force Fslot are obtained by (15)F=-Wθ,where W is the air gap magnetic field energy produced by the computational volume V of each section, as is given by (16)W=12μ0VBg2x,θdV.

The detailed calculations of the detent force obtained by using the MWFA are shown in Appendix D.

Figure 5 shows the calculation results of the slot-effect force, the end-effect force, and the detent force. In Figures 5(a) and 5(b), the results of the slot-effect force and the end-effect force are obtained using MWFA, while Figure 5(c) shows the calculation results using MWFA and FEM. Figure 6 establishes the finite element analysis model of the studied motor in a cylindrical coordinate system. The fractional-slot structure is adopted in the model and also the flux line distribution of the five-phase PMTLM is shown in Figure 6.

Calculations of slot force, end force and detent force distribution versus mover position.

Five-phase FSPMTLM model using finite element method.

Seen from Figures 5(a)5(c), the following conclusions are obtained: (1) since the amplitude of the end-effect force is 46.3911 N yet the slot-effect force is 2.5 mN, the end-effect force is the largest component of detent force. That is to say, the slot-effect force has been weakened in such a motor; (2) the results using MWFA to compute the detent force are in accordance with results using FEM. Hence, the accuracy of the detent force using the proposed method is validated by using FEM.

3.3. Mathematical Model of the Five-Phase FSPMTLM

As seen from Section 3.1, the basic mathematical model of PMLMs is a multivariable system with strong coupling. It is hard to analyze the motor characteristics and control the motor. The motor system is decoupled by applying Park transformation theory (PTT) which has been widely applied in motor vector control (VC) . Using PTT to model the studied motor can ease the analysis of the transient characteristics. In , using a Park transformation matrix T(θ) to model a five-phase permanent magnet synchronous Motor (PMSM) was reported. The studied motor has the same five-phase power supply; however, it has the structure of the linear motor which is different from the five-phase PMSM. When the other harmonic components are ignored, the Park transformation matrix becomes (17)Tθ=25cosπτθcosπτθ-αcosπτθ-2αcosπτθ+2αcosπτθ+α-sinπτθ-sinπτθ-α-sinπτθ-2α-sinπτθ+2α-sinπτθ+αcosπτθcosπτθ+2αcosπτθ-αcosπτθ+αcosπτθ-2α-sinπτθ-sinπτθ+2α-sinπτθ-α-sinπτθ+α-sinπτθ-2α1212121212,where θ is the position of the mover, α is the winding space geometric angles of adjacent phases, and α=0.4π (see Section 2.2).

Using (17) to transform (11)–(13), the mathematical model with the Park transformation form is written as(18)Vd=Rsid-πτνLqiq+Lddiddt,Vq=Rsiq+πτνLdid+ψm+Lqdiqdt,Fe=52πτpψmiq+Ld-Lqidiq,mdνdt=Dν+Fl+Fdetent-Fe,where Vd and Vq are the direct-axis and quadrature-axis voltage; Rs is the stator resistance; id and iq are direct-axis and quadrature-axis currents, ν is the mover velocity; Ld and Lq are the direct-axis and quadrature-axis inductances. ψm is the amplitude of the PM flux linkage; the remaining parameters are the same as (13); and the detailed calculations of Vd, Vq, id, iq, and ψm are shown in Appendix E.

4. The Simulation Analysis of the Motor Transients

In papers [12, 15], a model using semianalytical method was validated by the measurements results of the transient characteristics. Likewise, in order to verify the accuracy of the proposed model, model (18) has been established on the Matlab/Simulink platform where it has been simulated to analyze the transient characteristics of the studied motor in this section. In addition to the basic motor parameters: the stator resistance Rs(Rs=5Ω); the friction coefficient D(D=3000); the nominal value of the winding current I(I=8A); and the mass of the mover m (m=9.2 kg), which were shown in [12, 15], and the following parameters of the motor model Ld, Lq, ψm and Fdetent need to be calculated by using MWFA. The computed results are as follows: Ld=3.6 mH; Lq=6.8 mH; ψm=0.2261 Wb, and the detent force Fdetent has been shown in Appendix D. Figure 7 shows the proposed mathematical model established on the Matlab/Simulink platform.

Establishment of the studied motor model on Matlab/Simulink platform.

In Figure 7, the model established on the Matlab/Simulink platform is composed by the current excitation, the current balance subsystem, and mechanical balance subsystem. The current excitation implemented using the PWM converter is the same as the ones in [12, 15], which are the five-phase symmetric cosine current sources, ia, ib, ic, id, and ie, while the mover velocity ν is derived from the frequencies of the current sources, as shown by(19)f=ν2τ.

F load represents the load force on the mover, Theta represents position along the mover movements direction, and k is the amplification factor which can magnify the mover position 1000 times. The corresponding unit of the position is in millimeters. The current balance subsystem and mechanical balance subsystem are shown in Figures 8(a) and 8(b).

Subsystem of the proposed model: (a) current balance subsystem, (b) mechanical balance subsystem.

In Figure 8(a), the current subsystem is derived from the 3rd equation of (18). In the DQ transformation module shown in this figure, which is established from the Park transformation matrix, T(θ) is given in (17). Fm is the amplitude of the PM flux linkages, which represents ψm in (18). In Figure 8(b), the mechanical subsystem is derived from the 4th equation of (18).

The simulations using the proposed model to analyze the motor transients are carried out. This is done for the three cases given in . The results are then compared with results from SAM and measurements. The three cases are as follows.

Case 1.

The velocity of the mover is set to 18 mm/s and there is no load. The simulation results of the mover position (versus time) and the velocity of the mover (versus time) are shown in Figures 9 and 10.

Figure 9 shows the simulation results using the proposed model have a close agreement with the results from SAM and measurements in . Figure 10 shows slight oscillations of the mover velocity around the set 18 mm/s. However, the position increases linearly with time without any oscillations.

Transient characteristic of the mover position under no-load for ν=18 mm/s.

Simulation results of the mover velocity under no-load for ν=18 mm/s.

Case 2.

The velocity of the mover is set to 1 m/s and no load is installed on the mover. The simulation results of the mover position (versus time) and the velocity (versus time) are shown in Figures 11 and 12.

Figure 11 shows the simulation results using the proposed model are almost identical to the results from measurements and SAM. The results show some variances at the beginning of the mover running; however, these die out after 0.04 s. Figure 12 shows the mover velocity around the setting of 1 m/s. The slight oscillations of the mover velocity have small variations after 0.04 s. Hence, these results are consistent.

Transient characteristics of the mover position under no-load for ν=1 m/s.

Simulation results of the mover velocity under no-load for ν=1 m/s.

Case 3.

An additional mass mΔ (mΔ=19.2 kg) was linked to the mover. The average velocity of the mover has been assumed 50 mm/s. The results of the position (versus time) and velocity (versus time) are shown in Figures 13 and 14, respectively.

Transient characteristic of the mover position under an additional inertial load with the mass mΔ=19.2 kg for ν=50 mm/s.

Simulation results of the mover velocity under an additional inertial load with the mass mΔ=19.2 kg for ν=50 mm/s.

The additional mass brings out more ripples in the mover position than in Cases 1 and 2. The two causes of the ripples are the resistance offered by the higher inertia to the motor and that by the detent force. In Figure 13, the variation trends between the simulation results using the proposed model and the ones obtained by using SAM are similar, although there are some slight differences. Also, it is noted that the higher velocity changes seen in Figure 14 indicate the presence of thrust ripples.

5. Conclusions

In this paper, a method based on modified winding function theory was applied to model a five-phase fractional-slot permanent magnet tubular linear motor. The proposed method has provided the detailed computational expressions of the air-gap flux density, the inductance, flux linkages, and the detent force. Due to the fact that it has a property of fast modeling no matter whether the motor is healthy or not, it can make up for the time-consuming remodeling of the motor using finite element analysis and semianalytical method when the faults arise or supply power changes. The analytical results showed that the slot-effect force had been greatly reduced by this structural design, and the main component of the detent force is the end-effect force. This is as had been demonstrated by using the finite element model. In this proposed approach, a model with five-phase Park transformation had been established and simulated. The simulation results for transients on the motor were analyzed and were close to those of semianalytical method and measurements. In this way, the accuracy of the model using MWFA was validated.

Appendix A. Calculations of the Winding Function, Winding Factor, and Modified Winding Function A.1. Winding Function Computation

As shown in Figure 2(a), on moving the coordinate system by a distance “ε” along the x-axis, the winding function of phase “a” becomes an even function. Using Fourier’s expansion method, the winding function of phase “a” in form of Fourier expansion is written by (A.1)Nax=k=1,3,5,akcos2pkπ5Ttx-ε,where Na(x) represents the winding function of phase “a”; k is the order of the coefficients of the Fourier’s expansion; Tt and wc represent the axial space of slot pitch and the axial width of armature (see Figure 1(b)), respectively; p represents the pole pairs; “ε” is the distance between the former origin and the new origin (see Figure 2(a)); “ak” is the kth Fourier’s expansion coefficient, as calculated by(A.2)ak=25Tt0wc/2Ncos2pkπ5Ttxdx+5Tt-wc/25Tt-Ncos2pkπ5Ttxdx=4Npkπsinwcpkπ5Tt.

Plugging (A.2) into (A.1) and combining with (3), the winding function of phase “a” in form of Fourier expansion is given by(A.3)Nax=k=1,3,5,akcos2pkπ5Ttx-ε=4Npπk=1,3,5,1ksinwckπ2τcoskπτx-ε.

Then, the expression (4) is obtained.

A.2. Winding Factor Computations

Definition of d s , y w k , y y k , and ydk. ds shown in Figure 2(a) is the axial magnetic separation distance between two adjacent phase; ywk, yyk, and ydk represent the kth winding factor, kth pitch factor, and kth winding distribution factor, respectively. In the light of the papers [29, 30], ywk is the product of yyk and ydk:(A.4)ywk=yyk·ydk,where(A.5)yyk=sinkTt-dsτ·π2,ydk=sinkθd/2Nsinkθd/2N,where(A.6)θd=πτabsτ-Tt-ds.

A.3. All-Phase Modified Winding Function Computations

All-phase modified winding function Mi(x) is the product of winding function and winding factor. Therefore, combining the spatial structure of windings in Figure 2(b) with all-order winding factor ywk yields (3):(A.7)Mix=Nix·ywk=4Npπk=1,3,5,ywkksinwckπ2τcoskπτx-ε-iα.Then, the expression (5) is obtained.

B. Analysis of Inverse Air Gap Function

The air gap magnetic field is the product of the PM field with unslotted stator and the relative air gap permeance . Paper  has shown the expression of unslotted PM field Bg:(B.1)Bg=Hcτm2μ0g=Hcτm2μ0g+τmτpD/μrD2-Di2,where Hc=950 kA/m, μr=1.048, D is the diameter of the inner stator core and D=2(Ro+g), Di is the diameter of the inner tubular reaction rail and Di=2Ri, τm is the axial PM width, and τp is the axial ferromagnetic core width between PMs. We can suppose τp is equal to τm (in Table 1, τm=8 mm and τp=7 mm); thereafter, along the x-axis, the air gap function g(x,θ) in one pitch Tt is(B.2)gx,θg+π2bo2-x-θ-εx-θ-ετp2gτp2<x-θ-εwss2wss2<x-θ-εTt+ds2gTt+ds2<x-θ-εTt+ds2+wtg+π2x-θ-ε-Tt+ds2+wtTt+ds2+wt<x-θ-ετp2+τgcτp2+τ<x-θ-εTt.

By using Fourier series expansion theory, the inverse air gap function g-1x,θ is shown by (B.3)g-1x,θ=n=0Gncos2pnπ5Ttx-θ-ε,where Gn is computed by(B.4)Gn=22.5Ttθθ+εg-1x,θcos2pnπ5Ttx-θ-εdx+42.5Ttθ+εTt+θ+εg-1x,θcos2pnπ5Ttx-θ-εdx.

Combining (3), the above two formulas (B.3) and (B.4) are written by (5) and (6), respectively.

C. Definition of the Relative Air-Gap Permeance <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M235"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mtext>ag</mml:mtext></mml:mrow></mml:msub><mml:mfenced separators="|"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi /><mml:mi>θ</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>, the Air-Gap Flux Density <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M236"><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msub><mml:mfenced separators="|"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi /><mml:mi>θ</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>, and Effective Air-Gap Computational Radius <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M237"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mtext>se</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In light of the paper , the relative air-gap permeance λagx,θ is computed by(C.1)λagx,θ=g·g-1x,θ.

Because air gap g is small (see Table 1, g=1 mm), the air-gap flux density Bgx,θ is deduced by the product of λagx,θ and flux density produced by PMs with unslotted stator Bpm-slotlessx,θ, (C.2)Bgx,θ=λagx,θBpm-slotlessx,θ,where Bpm-slotlessx,θ can be expanded by Fourier series along the shaft of the motor:(C.3)Bpm-slotlessx,θ=4Bgπn=1,3,5,1nsinnπ2cosnπτm2τcosnπτx-θ-ε.

As shown in Figure 3 for an axially magnetized, internal magnet machine topology, the effect of the slot openings may be accounted for by introducing a Carter factor KC given by (C.4)KC=TtTt-γg,where Tt is the armature slot pitch (see Table 1), g has been derived from (B.1), and the slotting coefficient γ is computed by (C.5)γ=4πbo2gtan-1bo2g-ln1+bo2g2,where bo is the width of the armature slot openings. Thereafter, the effective airgap ge and effective air-gap computational radius Rse are given, respectively, by(C.6)ge=g+KC-1g,Rse=Ro+ge.

D. Definition of the Slot-Effect Electromagnetic Energy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M258"><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mtext>slot</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and the End-Effect Electromagnetic Energy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M259"><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mtext>end</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

W slot is derived from the sum of the electromagnetic energy on the each tooth, as computed by(D.1)Wslot=12μ0RoRo+g2πrdri=0,1,2,3,4ds/2+iTtds/2+wt+iTt+Tt-ds/2-Wt+iTtTt-ds/2+iTtBgx,θ2dx,where i is defined in (5) and Bg(x,θ) is the air-gap flux density which is calculated by (C.1)–(C.3).

W end is derived from the sum of the electromagnetic energy on the two ends of the stator, as calculated by (D.2)Wend=12μ0RoRo+KOg2πrdr-ds/2+ds/2+LλendxBpm-slotlessx,θ2dx,where L (see Table 1) is the length of the stator section, as is shown in Figure 1(b). λendx is the air-gap relative permeance on the two axially ends of the stator, as is shown by(D.3)λendx=ex-ds/2/2ge;xds21;ds2<x<ds2+Le-x+ds/2+L/2ge;xds2+L.Then, combining (15), with (D.1)–(D.3), the analytical formulae of detent force is computed by(D.4)Fdetentθ=Fslotθ+Fendθ=-0.19478×10-3×cos1256.6×θ+Tt2-0.9739×10-3×sin1256.6×θ+Tt2-6.9245×10-9×sin1047.2×θ+Tt2-1.6441×10-8×sin1256.6×θ+Tt2+9.9405×10-9×sin9.4248×10-8×θ+Tt2-5.296×10-10×sin3.1416×10-8×θ+Tt2-5.177×10-10×sin2094.4×θ+Tt2-2.698×10-9×sin2.199×10-7×θ+Tt2-0.292×10-5×cos1885×θ+Tt2-0.292×10-5×cos1885×θ+Tt2-0.39×10-3×sin418.88×θ+Tt2-0.9739×10-3×sin837.8×θ+Tt2-0.9739×10-3×sin837.8×θ+Tt2-0.39×10-3×cos209.44×θ+Tt2+3.14×10-9×sin209.44×θ+Tt2-46.6466×sin2πθ+Tt/2τ.

E. Detailed Calculations for the Mathematical Model Parameters

Seen from the five-phase Park transformation matrix T(θ), the following relation can be obtained:(E.1)T-1θ=52TTθ,where T-1θ and TTθ represent the inverse matrix and matrix transposition of Tθ, respectively.

Applying Tθ to (11), the direct-axis and quadrature-axis voltage, Vd and Vq, and the direct-axis and quadrature-axis currents. id and iq, are derived from(E.2)TθVs=TθRsis+TθdΛsdt=TθRsT-1θTθis+ddtTθΛs-dTθdtT-1θTθΛs,where (E.3)TθΛs=TθLssT-1θTθis+Tθψm=Λdqs.

When the harmonics of the inductances are ignored, the direct-axis and quadrature-axis inductances are as follows:(E.4)TθLssT-1θ=LdLq000,where(E.5)Ld=52μ0RseM2pτ2G0+G2,Lq=52μ0RseM2pτ2G0-G2.

The permanent magnet flux linkages using the five-phase Park transformation matrix T(θ) are obtained by(E.6)Tθψm=ψm1,0,0,0,0T.

The amplitude of the permanent magnet flux linkages is given by(E.7)ψm=MB1pτπRseg2G0+G2,where M represents the fundamental amplitude of the modified winding function and it is given by(E.8)M=4Nyw1pπsinwcπ2τ.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this manuscript.

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