AnAnalyticalSolutiontotheElectromagnetic PerformanceofanUltra …downloads.hindawi.com/journals/jece/2019/1469637.pdf · 2019-09-11 · inthecounterclockwisedirection.c isananglebetweenthe

Research ArticleAn Analytical Solution to the ElectromagneticPerformance of an Ultra-High-Speed PM Motor

Wenjie Cheng 1 Zhikai Deng1 Ling Xiao1 Bin Zhong2 and Yanhua Sun3

1College of Science Xirsquoan University of Science and Technology No 58 Yanta Road Xirsquoan China2School of Mechanical Engineering Xirsquoan University of Science and Technology No 58 Yanta Road Xirsquoan China3State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University No 28Xianning West Road Xirsquoan China

Correspondence should be addressed to Wenjie Cheng cwj20070807163com

Received 21 February 2019 Revised 23 July 2019 Accepted 18 August 2019 Published 12 September 2019

Academic Editor Gorazd Stumberger

Copyright copy 2019Wenjie Cheng et al-is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Bipolar rotors with cylindrical permanent magnets (PMs) which are magnetized in parallel are generally used in ultra-high-speedpermanent magnet synchronous motors (PMSMs) Aiming for this type of PMSM two methods namely the Poynting theoremand the Maxwell stress tensor were used to deduce the analytic expression of the electromagnetic driving torque A locked-rotorsimulation by the finite element (FE) method verifies the analytical solution In addition the cogging torque back-electromotiveforce (emf) armature reaction inductance etc were calculated compared and discussed-e no-load back-emf was checked by a60000 rpm free coast experiment -e proposed analytical solution is very convenient to guide the design of the ultra-high-speedPMSM and check its performance

1 Introduction

With advancements in new materials electrical and elec-tronics control technology etc the speed limits of rotatingmachines are being pushed higher and ultra-high-speedmotors are increasingly used in industry applications [1]Particularly in recent times the high-speed PMSM hasattracted more attention due to its high efficiency simplestructure high power density and small size [2 3] forexample it can be used as the driving motor for the fuel cellair blower [4 5] as an air centrifugal compressor [6] or asthe generator of a micro-gas turbine [7]

It is very significant to calculate the electromagnetictorque because it can be used to check the motor powerestimate noise and vibration and provide further guidancefor the motor control It is well known that the electro-magnetic torque is generated from the interaction of thePM field and the armature reaction field therefore how todetermine the above two magnetic fields becomes keyPMSMs often adopt a surface-mounted PM rotor and thiskind of rotor as seen in most relevant literatures [8 9] has

a common characteristic which is that the magnetic polearc is less than a pole pitch (the pole embrace is less than1) which means the air space between the magnets isoccupied by unmagnetized materials such as air or alu-minum alloy Rotors of the ultra-high-speed PMSM arealways subject to extreme conditions such as high speedand high temperature To improve the bending stiffnessand strength of the rotor ultra-high-speed PMSMsgenerally use a special rotor construction in which acylindrical PM magnetized in parallel is interference-fitted into a metallic sleeve therefore the rotor has singlepole pairs and each half of the magnet has a 180deg electricpole arc (the pole embrace is equal to 1) as shown inFigure 1 As mentioned in the literature [10] existinganalytical methods are successful only in certain cir-cumstances and are likely to fail in others for examplewhen the magnets have a 180deg electric pole arc In theliterature [11] study an open-circuit magnetostatic fieldsolution to a 2-pole rotor with a diametrically magnetizedsintered NdFeB magnet was given for a 20000 rpm 3-phase brushless PM dc motor Recently the literature [12]

HindawiJournal of Electrical and Computer EngineeringVolume 2019 Article ID 1469637 11 pageshttpsdoiorg10115520191469637

gave analytical solutions to the PM field of bipolar rotorswith ring or cylindrical PMs that are magnetized inparallel Nevertheless most relevant literature studies onanalytical models for predicting the armature reactionfield distribution are successful for the motor owing tothe high permeability shaft [13] however few studiesinvolve the hollow stator bore as shown in Figure 2(b)

For ultra-high-speed PMSMs including a stator withoverlapping windings and 2-pole parallel magnetizedcylindrical PMs analytical solutions to the magnetic fieldand corresponding electromagnetic torque are rarely re-ported at present which has resulted in a prolonged motordesign cycle to evaluating the motor performance Due tothe need of cooling and reduction of pulse torque theultra-high-speed PMSM generally adopts a large air gaphereupon the slot effect is very weak -erefore the as-sumption of ldquoslotless stator + equivalent current sheetrdquocan be introduced and some convenient formulas ofestimating motor performance are expected to beobtained

Hence this paper will first deduce the magnetic fieldthen obtain the electromagnetic driving torque by using thePoynting theorem andMaxwell stress tensor and then verifythe correctness by using the FE method In addition thecogging torque the back-electromotive force (emf) andarmature reaction inductance are calculated compared anddiscussed A 10 kW 120000 rpm prototype is designed andmanufactured and the no-load back-emf is checked by a60000 rpm free coast experiment

2 Analytical Model

-e cross section of the ultra-high-speed PMSMs is shown inFigure 1 in which slotted stator and distribution winding areused and the rotor is composed by installing a cylindricalparallel polarized PM into the metallic sleeve

For the sintered NdFeB magnet the demagnetizationcurve is approximately a straight line and the relativerecoil permeability μr is approximate to that of air μr waschosen from 1029ndash1100 in most literatures To simplifythe field calculation the following assumptions aremade

(1) -e PM has a linear second-quadrant de-magnetization curve and the permeability of thestator iron is infinite

(2) -e winding end effect and rotor eddy current areneglected

(3) -e PM field is calculated with a 2D solution byassuming a smooth stator surface that is the slottingis neglected as shown in Figure 2(a)

(4) Conductors in the stator slot can be equivalent to thecurrent sheet distributing evenly along the slotopening as shown in Figure 2(b)

In the calculation of the PM field the slot-opening effectis neglected therefore the calculation model of the PM fieldis shown in Figure 2(a) where R1 and R2 are the outer radius

and the inner radius of the PM and I and II represent thePM and the air region respectively

-e sleeve applies to a low permeability alloy which canbe physically equivalent to the air therefore in the calcu-lation model of the armature reaction field the stator boreseems to be full of air as shown in Figure 2(b) where b0 isthe width of the slot opening

-e loaded magnetic field is the superposition of theopen-circuit PM and the armature reaction field and theloaded air-gap flux density is defined as follows

BloadIIr BW

IIr + BPMIIr

BloadIIθ BW

IIθ + BPMIIθ

⎧⎨

⎩ (1)

In equation (1) load PM and W represent the loadedPM and winding magnetic fields respectively r and θrepresent the radial and circumferential directionsrespectively

21 Open-Circuit Magnetic Field -e literature [12] hasdeduced the solution to the PM field shown in Figure 2(a)which is given as follows

BPMIIr

minus μ0MrR21 1 + R2

2rminus 2( 1113857

R21 μr minus 1( 1113857 minus R2

2 μr + 1( 1113857cos θ

BPMIIθ

μ0MrR21 minus 1 + R2

2rminus 2( 1113857


2 μr + 1( 1113857sin θ

(2)

where Mr is the amplitude of the residual magnetization ofthe PM μr is the relative recoil permeability of the PM μ0 isthe vacuum permeability and r and θ represent the radialand circumferential positions respectively

22 Armature Reaction Field -e governing equation of theair-gap region is as follows

zϕ2

z2r+

zϕrzr

+zϕ2

r2zθ2 0 (3)

where ϕ is a scalar magnetic potential and the relation of ϕand H (magnetic field intensity) is expressed as follows

Hθ minuszϕ

r zθ

Hr minuszϕzr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(4)

In the inner radius of the stator

Hθ1113868111386811138681113868 rR2

minus J (5)

where J is the Fourier series expansion of the current sheetAt the center of the stator the H should meet the following

Hθ1113868111386811138681113868 r⟶0neinfin

Hr

1113868111386811138681113868 r⟶0neinfin(6)

Taking into account the symmetry of the magneticfield the single conductor field can be solved according toequations (3)ndash(6) -e method to solve the armature

2 Journal of Electrical and Computer Engineering

reaction field is as follows the single conductor field issolved first and then the single-turn coil field can beobtained by adding two single conductor fields at a polardistance apart from each other Whereupon the single-phase winding field is superpositioned onto fields that areexcited by the multiple single-turn coils and finally thearmature reaction field can be calculated by super-imposing the fields of three-phase winding

For distributed winding multiple single-turn coil fieldsthat are a pitch apart from each other can be equivalent to aconcentrated winding by taking into account the distribu-tion coefficient and the pitch-shortening coefficient -eabove-mentioned treatment will lightly change higherharmonic components of the armature reaction field butvastly simplify the solution of the single-phase winding fieldwith good accuracy In addition it is assumed that in-stantaneous three-phase currents (ia ib ic) only involvefundamental components and they can be written asfollows

ia Im cos(Pωt + β)

ib Im cos Pωt + β minus2π3

1113874 1113875

ic Im cos Pωt + β minus4π3

1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where Im is the current amplitude of a single conductor P isthe number of pole pairs ω is the synchronous angularvelocity and β is the current phase

-e armature reaction field in Figure 2(b) is expressed asfollows

BWIIr

minus 2qNsμ0π

1113944

infin

v11113896Kdpv(2v minus 1)Ksov(2v minus 1)

middot Fo(2v minus 1 r) middot (minus 1)v

middot 1113890ia cos[(2v minus 1)(α minus c)]

+ ib cos (2v minus 1) α minus c minus2π3

1113874 11138751113876 1113877

+ ic cos (2v minus 1) α minus c minus4π3

1113874 11138751113876 111387711138911113897

BWIIα

minus 2qNsμ0π

1113944

infin


middot Fo(2v minus 1 r) middot (minus 1)vminus 1

middot 1113890ia sin[(2v minus 1)(α minus c)]

+ ib sin (2v minus 1) α minus c minus2π3

1113874 11138751113876 1113877

+ ic sin (2v minus 1) α minus c minus4π3

1113874 11138751113876 111387711138911113897

(8)

where Ksov(v) sin(vb0(2R2))(vb0(2R2)) is the slot-opening coefficient which is a function of the spatial har-monic order vFo(r v) rvminus 1Rv

2 reflects the influence of theair-gap thickness on the radialcircumferential flux densityof the v order harmonic wave α is the angle relative to thehorizontal direction as shown in Figure 2(b) and is positive

Mrθ

Mr

Mrr

R1

R2

θ

I (PM)

II (Air gap)

(a)

Stator bore

II

Air b0Slot

α = 0

Current sheetR2

(b)

Figure 2 Models of the electromagnetic field of the PMSMs (a) Open-circuit PM field (b) Armature reaction field

Stator iron

Winding

Sleeve

Magnet

Magnetizationdirection

Figure 1 Cross section of the ultra-high-speed PMSM

Journal of Electrical and Computer Engineering 3

in the counter clockwise direction c is an angle between theaxis of winding A and the d-axis q is the number of slots perphase per pole and Ns is the number of conductors per slot

23 Electromagnetic Torque andCoggingTorque In this parttwo methods namely the Poynting theorem and theMaxwell stress tensor were used to deduce the analyticexpression of the electromagnetic torque For the 2D fieldthe electric field strength vector E in the air-gap region onlyincludes the axial component Ez therefore the Poyntingvector is given by the following

1113937r

minus EzHθ

1113937θ

EzHr

1113937z

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)

where Ez μ0 1113938(zHθzt)dr -e electromagnetic powertransmitting from the stator to the rotor is mainly dependenton the radial component of the Poynting vector Πr whichcan be calculated from equation (10) as follows

PM Temω lef 11139462π

01113945

r

r dθ (10)

where lef is the length of the stator iron Taking the hori-zontal axis in Figure 2(b) as the reference position the PMfield is changed as follows

BPMIIr minus

μ0MrR21 1 + R2

2rminus 2( 1113857


2 μr + 1( 1113857middot cos(φ)

BPMIIθ


2rminus 2( 1113857


2 μr + 1( 1113857middot sin(φ)

(11)

where φ θ minus π2 α It is assumed that the electromag-netic power is mainly transmitted by the fundamentalmagnetic field of armature reaction field When the numberof pole pairs equals 1 by substituting equation (7) intoequation (8) the armature reaction field will become asfollows using the properties of trigonometric functions

BWIIr μ0G1 cos(ωt minus φ + δ)

BWIIθ μ0G1 sin(ωt minus φ + δ)

(12)

where δ β + c represents the angle between the axis of ar-mature reaction field and the axis of PM field andG1 (3qNsKdp1Kso1Im)(πR2) Equation (12) shows that thearmature reaction field is a travelling wave but the PM field is astanding wave shown by equation (11) To make the PM fieldrotate counterclockwise synchronously relative to the armaturereaction field equation (11) can written as the following

BPMIIr μ0 minus G2r

minus 2minus G31113872 1113873 middot cos(φ minus ωt)

BPMIIθ μ0 G2r

minus 2minus G31113872 1113873 middot sin(φ minus ωt)

(13)

where G2 MrR21R

22[R2

1(μr minus 1) minus R22(μr + 1)] and G3

MrR21[R2

1(μr minus 1) minus R22(μr + 1)] -erefore the loaded gap

magnetic intensity can be obtained as follows

HIIr G1 cos(ωt minus φ + δ) + minus G2rminus 2

minus G31113872 1113873 middot cos(φ minus ωt)1113960 1113961

HIIθ G1 sin(ωt minus φ + δ) + G2rminus 2

minus G31113872 1113873 middot sin(φ minus ωt)1113960 1113961

(14)

Furthermore Ez can be obtained byHIIθ and is expressedas the following

Ez μ0ωr G1 cos(ωt minus φ + δ) + G2rminus 2

+ G31113872 1113873cos(φ minus ωt)1113960 1113961

(15)

By substituting equation (15) into equation (9)Πrwill becalculated and according to equation (10) the electro-magnetic torque can be obtained as follows

Tem minus 2πμ0lefG1G2 sin δ δ isin [0 π] (16)

In the 2D electromagnetic field fθ BrBθμ0 and fθ isthe circumferential electromagnetic force density acting onthe stator or rotor -erefore the electromagnetic torque ofthe rotor which is shown in Figure 1 can be written as thefollowing

Tem 2μ0lef 1113946π

0HIIrHIIθr

2dθ (17)

By substituting equation (14) into equation (17) theexpression of the electromagnetic torque Tem is as follows

Tem minus 2πμ0lefG1G3r2sin δ δ isin [0 π]

asymp 6πμ0qKdp1Kso1NsIm

2πR2

R1

R21113888 1113889

2

Mrlefr2sin δ

(18)

In equation (18) when rR2 equation (18) is equal toequation (16) which indicates that the torque acting on therotor surface generated by the Maxwell stress equals thetorque acting on the current sheet generated by the ampereforce -e results show that the electromagnetic torque isproportional to the length of stator iron lef the ampere turnsper unit length of the stator inner surface NsIm2πR2 andthe magnetization Mr while it is proportional to the squareof the ratio (R1R2)

2 (PM radius to stator inner radius)-e cogging torque arises from the interaction of the

magnetomotive force (mmf) harmonics and the airgappermeance harmonics hence it exists in almost all types ofmotors in which the airgap permeance is not constant -ecogging torque will cause undesirable speed pulsationsvibrations possible resonance and acoustic noise [10] Forthe motor shown in Figure (1) because the radial andtangential components of the magnetic flux density vectoras presented in equation (11) are pure sinusoidal the mmfharmonics only contain the fundamental component andthe cogging torque is caused by stator slots In fact thecogging torque can also be calculated by equation (17)however HIIr and HIIθ should be replaced by HPM

IIr and HPMIIθ

respectively In the paper HPMIIr and HPM

IIθ are obtained by thehypothesis of a smooth stator surface hereupon they cannotreveal the slot effects However slot effects can be consideredanalytically by the subdomain technique Works in theliterature [10] have proposed a simple method to predict the


cogging torque based on the analytical calculation of the air-gap field distribution and the net lateral force acting on theteeth

24 Back-Electromotive Force -e flux through a phasewinding excited by PM and by winding respectively can bewritten as follows

Φpm 2 BPM

IIr( 1113857maxπ

τplef

ΦW 2 BW

IIr( 1113857maxπ

τplef

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

where τp is the pole pitch and lef is the length of stator iron-e back-emf amplitude in each phase winding assuming a120-degree coil span is then obtained as follows

E 444fkdpvNΦload (20)

where f is the angular frequency of the rotor kdpv is thewinding coefficient N is the number of turns per phase andΦload Φpm +ΦW When ΦW 0 it corresponds to anopen-circuit situation and the back-emf becomes the no-load back-emf E0

25 Armature Reaction Inductance -e constant-powertransformation from the three-phase coordinates to the d-qcoordinate is given by the following

id

iq

i0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

23

1113970

cos θ cos θ minus2π3

1113874 1113875 cos θ +2π3

1113874 1113875

minus sin θ minus sin θ minus2π3

1113874 1113875 minus sin θ +2π3

1113874 1113875

12

1113970 12

1113970 12

1113970

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where θ 1113938ω dt represents the rotor position which is anangle between the d-axis and winding axis of phase A Whenthe above two axes coincide θ 0Whereupon according toequation (21) there are

id

iq

⎡⎣ ⎤⎦ [C]

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22a)

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ [C]

Tid

iq1113890 1113891 (22b)

where [C] 23

radic 1 minus 12 120

3

radic2 minus

3

radic21113890 1113891 and [C]T and [C]minus 1

are transposed matrix and inverse matrix of [C] re-spectively Notably there is [C]T [C]minus 1 According toequation (22a) we have

id

23

1113970

iA minus12

iB minus12

iC1113874 1113875

iq

23

1113970 3

radic

2iB minus

3

radic

2iC1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

θ 0 which means Pωt 0 in equation (7)Substituting (7) into equation (23) yields

id 3

radicIRmscos(β)

iq 3

radicIRmssin(β)

⎧⎨

⎩ (24)

where IRms is the RMS of phase current IRms Im2

radic and β

denotes the electrical angle of the phase current vectorbeyond the d-axis Let β 0 by substituting equation (24)into equation (22b) to get the three-phase AC [iA iB iC]when the three-phase AC flows into the stator windings itwill generate the d-axis armature reaction field Obviouslythe stator iron saturation is most likely to occur when thedirection of the armature reaction field is in line with that ofthe PM field accordingly we only consider this situation

Denoting the inductance matrix in the abc coordinatesystem by

LABC1113858 1113859

LAA LAB LAC

LBA LBB LBC

LCA LCB LCC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)

-e flux linkage in the abc coordinate system can berepresented as

λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ LABC

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)

-e flux linkage in the dq coordinate system is alsoapplied to equation (22a) that is

λd

λq

⎡⎣ ⎤⎦ [C]

λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)

Substituting equation (26) and equation (22b) intoequation (27) yields the following

λd

λq

⎡⎣ ⎤⎦ [C] LABC1113858 1113859[C]T

id

iq

⎡⎣ ⎤⎦ Ldq1113960 1113961id

iq

⎡⎣ ⎤⎦ (28)

where [Ldq] Ld Lmd

Lmq Lq1113890 1113891 [C][LABC][C]T it represents

the inductance matrix in the dq coordinate system If the endleakage is not considered and other leakage factors such ashigher harmonics and slot are also ignored then Ld and Lq

will represent the d-axis and q-axis armature reaction in-ductances namely Lad and Laq respectively From the FEM[LABC] will be obtained and then it will turn into [Ldq] byusing the above transformation

Lad and Laq can also be calculated by analytical methodand the process is as follows


-e load flux linkage of phase A is calculated as

λA NR2lef 1113946αy2

minus αy2BloadIIr R2 θ( 1113857dθ (29)

where N is the number of turns per coil of one phasewinding and αy is the coil pitch angle -e load flux linkageof phase B and C is expressed as

λB λAcos2π3

λC λAcos4π3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)

According to the above-mentioned [λA λB λC] fromequation (27) we will get [λd λq] And then there is

Ld λd

id (31)

Similarly neglecting some leakage factors Ld calculatedby equation (31) corresponds to Lad

3 Example Validation

To verify the analytic solution to the electromagnetic torquea locked-rotor simulation is carried out by the FEMmethod-e motor parameters are listed in Table 1

In the simulation the windings are fed by a symmetricalthree-phase alternating current to produce a rotating ar-mature reaction field but the rotor speed is set to zero thatis the rotor is locked When the armature reaction fieldrotates relative to the rotor with the angular velocity ω thelocked-rotor torque will generate and vary periodically withδ (an angle between two axes of the armature reaction fieldand PM field and it changes periodically from 0 to 360degrees) -e locked-rotor phenomenon can be applied toverify the electromagnetic torque of equation (16) andequation (18)

When the motor is fed with currents as shown inequation (7) the locked-rotor torques are as shown inFigure 3 -ree kinds of input currents of 2000Hz withdifferent amplitudes such as 10A 30A and 50A areconsidered Figure 3(a) shows the three-phase currents witha 10A peak value in FEM -e locked-rotor torque curvesobtained by FEM are shown in Figure 3(b) which shows thatthe locked-rotor torque is also sinusoidal with time and itspeak value is increased with the increasing amplitude of thephase current A comparison of the results of FEM and theanalytical model is illustrated in Figure 3(c) in which it canbe found that the analytical solution is in good agreementwith the FEM solution

Peak torques obtained by the FE model are comparedwith those from the derived analytical equations-e resultswhich are listed in Table 2 show that the errors are less than5 which proves the correctness of equation (16) andequation (18)

-emagnetic field results that were calculated by FEM areshown in Figure 4 where red and blue conductors indicate the

current flows out of and into the cross section respectively Inaddition the PMhas a parallel magnetization (the direction ofthe magnetization is vertically upward)

Because the PM field is much larger than the armaturereaction field the magnetic fields of the motor in Figure 4 atdifferent times are approximately the same At 02ms thelocked-rotor torque in Figure 3(b) is negative and thecorresponding motor magnetic field is shown in Figure 4(a)in which the armature reaction field lags behind the PM fieldby approximately 135 degrees which just causes a negativetorque Moreover the armature reaction field weakens thePM field the maximum flux density occurs in the statortooth and its value is 138 T At 04ms the locked-rotortorque equals zero as shown in Figure 3(b) and the cor-responding motor magnetic field is shown in Figure 4(b)where the armature reaction field coincides with the PMfield which verifies the null torque At 04ms the maximumflux density of the stator tooth is 170 T At 05ms thelocked-rotor torque approaches the maximum as shown inFigure 3(b) and the corresponding motor magnetic field isshown in Figure 4(c) It can be seen that the armature re-action field direction is left (90 degrees ahead of the PMfield) which verifies the maximum torque At 05ms themaximum flux density of the stator tooth is 160 T

-ree methods namely the FM method the air-gappermeance method [10] and the subdomain method wereused to estimate the cogging torque and the results areshown in Figure 5 -e air-gap permeance method is basedon the assumption that the flux crosses the magnet and airgap in a straight line wherever a magnet faces a toothHowever the above assumption will no longer be suitablefor the motor mentioned in the paper because the flux willbend in the air gap as shown in Figure 5(a) the coggingtorque calculated by the air-gap permeance method is smallrelative to that from FEM and the subdomain method asshown in Figure 5(b) -e cogging torque peak is approx-imately 370 μNm and is 005 of the rated torque (08Nm)which is too small to be ignored In fact some degrees ofharmonic components will exist in the polarized PM thusthe cogging torque peak will increase

-e stator tooth tips in relation to the width of the slotopenings also influence the back-emf waveform When theheight of the tooth tips is small magnetic saturation existsand the emf waveform tends to be a trapezoid Howeverwhen the tooth tip height is too large the emf waveform willdeteriorate [11] -e motor consists of a stator having 12

Table 1 Parameters of the high-speed PMSM

Symbol Quantity ValuesR1 PM radius 12times10minus 3mR2 Inner radius of stator 174times10minus 3mQ Slot numbers 12lef Core length 55times10minus 3mM Magnetization 820000Amω Angular velocity 12560 radsNs Conductors per slot 20μr Relative recoil permeability 1038P Number of pole pairs 1


teeth which carry overlapping winding -e no-load back-emf waveform obtained from the FEM is displayed inFigure 6(a) and the emf waveform is almost purely sinu-soidal which is ideal for ultra-high-speed PMSMs As shownin Figure 5(a) a stator tooth has the maximum flux density1542 T which does not reach the saturation region of thematerial-e comparison of the no-load back-emf calculatedby equation (20) and that of FEM is shown in Figure 6(b)which manifests a good agreement of both methodsFigure 6(c) shows the prototype of the PMSM in which therotor is supported by gas foil bearings and the experimentalresults are shown in Figure 6(d) A coast-down test under60000 rmin is carried out that is the frequency converterwill shut down when the rotor is stable at 60000 rmin and

then the rotor speed will decrease to zero gradually byovercoming air friction In this process the stator has a back-emf but no current thereupon the rotor is not subjected tothe electromagnetic force Hereupon the curve of a no-loadback-emf vs speed can be achieved -e experimental anddesign values of the no-load back-emf are 570V and 596 Vat 746Hz respectively and the difference between them iswithin 46

-e results of Lad and Laq are shown in Figure 7 It canbe found that Lad and Laq are same when current is smalland both of them decrease with the increase of the currentBecause the d-axial magnetic circuit is more saturated Laddescends faster -e magnetic saturation is neglected inthe analytical method whereas Lad not varies with thecurrent

When IRms 10A and 100A the errors of the twomethods are 9 and 2 respectively It shows that theresults of the analytical method and FEM are very close Inthe analytical method only the fundamental harmonic isused to calculate the flux linkage However in the FEMexcept for the fundamental harmonic high-order harmonics

009 020 040 060 080 100 120ndash1000

1000

XY Plot 3torque_Cal_10kw120krpm

ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)

Input current (phase A)

Input current (phase C)Input current (phase B)

Time (ms)

(a)


Torque Im = lsquo10ArsquoTorque_1 Im = lsquo30ArsquoTorque_2 Im = lsquo50Arsquo

Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A

Analytic 10AAnalytic 30AAnalytic 50A

(c)

Figure 3-e results of the locked-rotor (a)-e three-phase alternating currents (b)-e torque vs time (c) Result comparison of FEM andthe analytical model

Table 2 Peak torque comparison between FE and the theory

Im (A) FEM (Nmiddotm) Analytical (Nmiddotm) Error ()10 0266 0271 18730 0802 0815 16250 1333 1357 176


B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001

69212e ndash 00160561e ndash 00151910e ndash 00143259e ndash 00134608e ndash 00125957e ndash 001

86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000

Time = 00002sSpeed = 0000000rpmPosition = 0000000deg

0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000

12540e ndash 00410637e ndash 00121262e ndash 00131887e ndash 00142512e ndash 00153136e ndash 00163761e ndash 00174386e ndash 00185011e ndash 00195936e ndash 001

11689e + 00012751e + 000

14876e + 00015938e + 000

Speed = 0000000rpmPosition = 0000000deg

Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001

14749e ndash 00410049e ndash 00120083e ndash 00130117e ndash 00140152e ndash 00150186e ndash 00160220e ndash 00170254e ndash 00180288e ndash 001

10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000

Position = 0000000deg

Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)

Figure 4 Flux density vector field and current vector field of the FE model (a) Fields at 00002 s (b) Fields at 00004 s (c) Fields at 00005 s

B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000

96871e ndash 00219326e ndash 00128966e ndash 00138605e ndash 00148245e ndash 00157884e ndash 00167523e ndash 001

47663e ndash 004

77163e ndash 00186802e ndash 00196442e ndash 001

12536e + 000

(a)

10 20155 25 300Angle (degmech)

ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)

FEPermeanceSubdomain

(b)

Figure 5 PM field and cogging torque (a) PM field (b) Comparison of the cogging torque of FEM and the analytical model


are also taken into account in the flux linkage -erefore theresult of the analytical method is smaller than that of FEM

It is worth noting that the stator tooth has the mostserious saturation as shown in Figure 8(a) and the fluxdensities reach 2086 T and 1880 T when IRms 100A(28IN) and 50A (14IN) respectively IN denotes the ratedcurrent of the motor -e working points of stator toothcorresponding to the both working condition are illustratedin Figure 8(b) It shows that the working point is located atthe end of the linear segment of the B-H curve whenIRms 14IN It indicates that the magnetic saturation of thestator iron is not prominent when the prototype worksunder rated working conditions and it applies to the pro-posed analytical method in this paper

In addition in order to enhance the cooling of the rotorthe air gap should be chosen as large as possible so thatmagnetic saturation in the tooth tips rarely occurs Fur-thermore the teeth in a 12-slot stator are diametricallysymmetrical the air-gap field distribution is symmetrical

000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1

Curve info rms_1 maxInduced voltage (phase A) 1560441 2188502Induced voltage (phase B) 1563378 2186613Induced voltage (phase C) 1573086 2184715

(a)

0 01 02 03 04 05Time (ms)

FE (phase A)Analytic

Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV

FrequencyVoltage RMS

(d)

Figure 6 Results of the no-load back-emf (a) No-load back-emf of FEM at 120000 rpm (b) Comparison of no-load back-emf at120000 rpm (c) Prototype of the ultra-high-speed PMSM (d) Experimental curves of the phase voltage and frequency

0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA

Lad FEMLaq FEMLad Analytical

d- an

d q-

axis

indu

ctan

ces (

mH

)

Figure 7 Curves of Lad and Laq vs the current


and no unbalanced magnetic pull will exist if the rotor andstator axes are coincident Although the rotor rotates ec-centrically with respect to the stator in fact the eccentricityis too tiny due to the constraint of bearings to cause a notableunbalanced magnetic pull

4 Conclusion

For the ultra-high-speed PMSM including a stator withoverlapping windings and the 2-pole parallel magnetizedcylindrical PM electromagnetic torques are rarely reportedat present which leads to a prolonged motor design cycle toevaluating motor performance -erefore the work of thispaper is as follows

(1) -is paper deduces the analytical solution to theelectromagnetic driving torque of ultra-high-speedPMSMs which is verified by the FE methodaccording to locked-rotor simulation -e resultsshow that the electromagnetic torque is pro-portional to the length of stator iron the ampereturns per unit length of the stator inner surface andthe magnetization while it is proportional to thesquare of the ratio ie PM radius to stator innerradius

(2) -e cogging torque back-emf armature reactioninductance and so on are calculated compared anddiscussed For the PMSM studied in the paper theair-gap permeance method was shown not to beapplicable to the estimation of the cogging torquedue to the fact that the flux bends in the air gap andthus a high-accuracy method (eg the subdomainmethod) should be adopted

(3) A prototype is designed and manufactured and theno-load back-emf is checked by a 60000 rpm freecoast experiment -e experimental and designvalues of the no-load back-emf are 570 V and 596V

at 746Hz respectively and the difference betweenthem is within 46

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work is supported by National Natural ScienceFoundation of China (Grant nos 51705413 51705416 and11502196) and China Postdoctoral Science Foundation(Grant no 2017M613291XB)

References

[1] A Tenconi S Vaschetto and A Vigliani ldquoElectrical machinesfor high-speed applications design considerations andtradeoffsrdquo IEEE Transactions on Industrial Electronics vol 61no 6 pp 3022ndash3029 2014

[2] J H Ahn J Y Choi C H Park C Han C W Kim andT G Yoon ldquoCorrelation between rotor vibration and me-chanical stress in ultra-high-speed permanent magnet syn-chronous motorsrdquo IEEE Transactions on Magnetics vol 53no 11 Article ID 8209906 2017

[3] M-S Lim S-H Chai J-S Yang and J-P Hong ldquoDesign andverification of 150 krpm PMSM based on experiment resultsof prototyperdquo IEEE Transactions on Industrial Electronicsvol 62 no 12 pp 7827ndash7836 2015

[4] D-K Hong B-C Woo J-Y Lee and D-H Koo ldquoUltra highspeed motor supported by air foil bearings for air blowercooling fuel cellsrdquo IEEE Transactions on Magnetics vol 48no 2 pp 871ndash874 2012

20864e + 00019560e + 00018256e + 00016953e + 00015649e + 00014345e + 00013041e + 00011737e + 00010433e + 00091289e ndash 00178250e ndash 00165210e ndash 00152171e ndash 00139132e ndash 00126092e ndash 00113053e ndash 00113431e ndash 004

B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)

Figure 8 Magnetic saturation condition of the stator iron (a) Distribution of flux density for IRms 100A (b) Working points of the statortooth


[5] D-K Hong B-C Woo Y-H Jeong D-H Koo andC-W Ahn ldquoDevelopment of an ultra high speed permanentmagnet synchronous motorrdquo International Journal of Pre-cision Engineering and Manufacturing vol 14 no 3pp 493ndash499 2013

[6] S-M Jang H-W Cho and S-K Choi ldquoDesign and analysisof a high-speed brushless DC motor for centrifugal com-pressorrdquo IEEE Transactions on Magnetics vol 43 no 6pp 2573ndash2575 2007

[7] D-K Hong D-S Joo B-CWoo D-H Koo and C-W AhnldquoUnbalance response analysis and experimental validation ofan ultra high speed motor-generator for microturbine gen-erators considering balancingrdquo Sensors vol 14 no 9pp 16117ndash16127 2014

[8] A B Proca A Keyhani A EL-Antably W Lu and M DaildquoAnalytical model for permanent magnet motors with surfacemounted magnetsrdquo IEEE Transactions on Energy Conversionvol 18 no 3 pp 386ndash391 2003

[9] F Dubas C Espanet and A Miraoui ldquoAn original analyticalexpression of the maximum magnet thickness in surfacemounted permanent magnet motorsrdquo De European PhysicalJournal Applied Physics vol 38 no 2 pp 169ndash176 2007

[10] Z Q Zhu and D Howe ldquoAnalytical prediction of the coggingtorque in radial-field permanent magnet brushless motorsrdquoIEEE Transactions on Magnetics vol 28 no 2 pp 1371ndash13741992

[11] Z Q Zhu K Ng and D Howe ldquoDesign and analysis of high-speed brushless permanent magnet motorsrdquo in Proceedings ofthe International Conference on Electrical Machines amp Drivespp 381ndash385 IET Cambridge UK September 1997

[12] W Cheng L Yu L Huang Y Lv L Li and Y Sun ldquoAn-alytical solution to magnetic field distribution of a parallelmagnetised rotor with cylindrical or ring-type permanentmagnetrdquo IET Electric Power Applications vol 9 no 6pp 429ndash437 2015

[13] Z Q Zhu and D Howe ldquoInstantaneous magnetic field dis-tribution in brushless permanent magnet DC motors IIArmature-reaction fieldrdquo IEEE Transactions on Magneticsvol 29 no 1 pp 136ndash142 1993


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gave analytical solutions to the PM field of bipolar rotorswith ring or cylindrical PMs that are magnetized inparallel Nevertheless most relevant literature studies onanalytical models for predicting the armature reactionfield distribution are successful for the motor owing tothe high permeability shaft [13] however few studiesinvolve the hollow stator bore as shown in Figure 2(b)

For ultra-high-speed PMSMs including a stator withoverlapping windings and 2-pole parallel magnetizedcylindrical PMs analytical solutions to the magnetic fieldand corresponding electromagnetic torque are rarely re-ported at present which has resulted in a prolonged motordesign cycle to evaluating the motor performance Due tothe need of cooling and reduction of pulse torque theultra-high-speed PMSM generally adopts a large air gaphereupon the slot effect is very weak -erefore the as-sumption of ldquoslotless stator + equivalent current sheetrdquocan be introduced and some convenient formulas ofestimating motor performance are expected to beobtained

Hence this paper will first deduce the magnetic fieldthen obtain the electromagnetic driving torque by using thePoynting theorem andMaxwell stress tensor and then verifythe correctness by using the FE method In addition thecogging torque the back-electromotive force (emf) andarmature reaction inductance are calculated compared anddiscussed A 10 kW 120000 rpm prototype is designed andmanufactured and the no-load back-emf is checked by a60000 rpm free coast experiment

2 Analytical Model

-e cross section of the ultra-high-speed PMSMs is shown inFigure 1 in which slotted stator and distribution winding areused and the rotor is composed by installing a cylindricalparallel polarized PM into the metallic sleeve

For the sintered NdFeB magnet the demagnetizationcurve is approximately a straight line and the relativerecoil permeability μr is approximate to that of air μr waschosen from 1029ndash1100 in most literatures To simplifythe field calculation the following assumptions aremade

(1) -e PM has a linear second-quadrant de-magnetization curve and the permeability of thestator iron is infinite

(2) -e winding end effect and rotor eddy current areneglected

(3) -e PM field is calculated with a 2D solution byassuming a smooth stator surface that is the slottingis neglected as shown in Figure 2(a)

(4) Conductors in the stator slot can be equivalent to thecurrent sheet distributing evenly along the slotopening as shown in Figure 2(b)

In the calculation of the PM field the slot-opening effectis neglected therefore the calculation model of the PM fieldis shown in Figure 2(a) where R1 and R2 are the outer radius

and the inner radius of the PM and I and II represent thePM and the air region respectively

-e sleeve applies to a low permeability alloy which canbe physically equivalent to the air therefore in the calcu-lation model of the armature reaction field the stator boreseems to be full of air as shown in Figure 2(b) where b0 isthe width of the slot opening

-e loaded magnetic field is the superposition of theopen-circuit PM and the armature reaction field and theloaded air-gap flux density is defined as follows

BloadIIr BW

IIr + BPMIIr

BloadIIθ BW

IIθ + BPMIIθ

⎧⎨

⎩ (1)

In equation (1) load PM and W represent the loadedPM and winding magnetic fields respectively r and θrepresent the radial and circumferential directionsrespectively

21 Open-Circuit Magnetic Field -e literature [12] hasdeduced the solution to the PM field shown in Figure 2(a)which is given as follows

BPMIIr

minus μ0MrR21 1 + R2

2rminus 2( 1113857


2 μr + 1( 1113857cos θ

BPMIIθ


2rminus 2( 1113857


2 μr + 1( 1113857sin θ

(2)

where Mr is the amplitude of the residual magnetization ofthe PM μr is the relative recoil permeability of the PM μ0 isthe vacuum permeability and r and θ represent the radialand circumferential positions respectively

22 Armature Reaction Field -e governing equation of theair-gap region is as follows

zϕ2

z2r+

zϕrzr

+zϕ2

r2zθ2 0 (3)

where ϕ is a scalar magnetic potential and the relation of ϕand H (magnetic field intensity) is expressed as follows

Hθ minuszϕ

r zθ

Hr minuszϕzr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(4)

In the inner radius of the stator

Hθ1113868111386811138681113868 rR2

minus J (5)

where J is the Fourier series expansion of the current sheetAt the center of the stator the H should meet the following

Hθ1113868111386811138681113868 r⟶0neinfin

Hr

1113868111386811138681113868 r⟶0neinfin(6)

Taking into account the symmetry of the magneticfield the single conductor field can be solved according toequations (3)ndash(6) -e method to solve the armature






1113874 1113875


1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)



BWIIr

minus 2qNsμ0π

1113944

infin





1113874 11138751113876 1113877


1113874 11138751113876 111387711138911113897

BWIIα

minus 2qNsμ0π

1113944

infin





1113874 11138751113876 1113877


1113874 11138751113876 111387711138911113897

(8)



Mrθ

Mr

Mrr

R1

R2

θ

I (PM)

II (Air gap)

(a)

Stator bore

II

Air b0Slot

α = 0

Current sheetR2

(b)


Stator iron

Winding

Sleeve

Magnet






1113937r

minus EzHθ

1113937θ

EzHr

1113937z

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)



01113945

r

r dθ (10)


BPMIIr minus

μ0MrR21 1 + R2

2rminus 2( 1113857



BPMIIθ


2rminus 2( 1113857



(11)




(12)




BPMIIθ μ0 G2r


(13)

where G2 MrR21R

22[R2


MrR21[R2







(14)



+ G31113872 1113873cos(φ minus ωt)1113960 1113961

(15)





0HIIrHIIθr

2dθ (17)




2πR2

R1

R21113888 1113889

2

Mrlefr2sin δ

(18)




IIr and HPMIIθ






Φpm 2 BPM

IIr( 1113857maxπ

τplef

ΦW 2 BW

IIr( 1113857maxπ

τplef

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)





id

iq

i0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

23

1113970


1113874 1113875 cos θ +2π3

1113874 1113875


1113874 1113875 minus sin θ +2π3

1113874 1113875

12

1113970 12

1113970 12

1113970

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


id

iq

⎡⎣ ⎤⎦ [C]

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22a)

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ [C]

Tid

iq1113890 1113891 (22b)

where [C] 23


3

radic2 minus

3



id

23

1113970

iA minus12

iB minus12

iC1113874 1113875

iq

23

1113970 3

radic

2iB minus

3

radic

2iC1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)


id 3

radicIRmscos(β)

iq 3

radicIRmssin(β)

⎧⎨

⎩ (24)


radic and β



LABC1113858 1113859

LAA LAB LAC

LBA LBB LBC

LCA LCB LCC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)


λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ LABC

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)


λd

λq

⎡⎣ ⎤⎦ [C]

λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)


λd

λq

⎡⎣ ⎤⎦ [C] LABC1113858 1113859[C]T

id

iq

⎡⎣ ⎤⎦ Ldq1113960 1113961id

iq

⎡⎣ ⎤⎦ (28)

where [Ldq] Ld Lmd










λB λAcos2π3

λC λAcos4π3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


Ld λd

id (31)



















009 020 040 060 080 100 120ndash1000

1000


ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)



Time (ms)

(a)



Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A


(c)





B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






1113874 1113875


1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)



BWIIr

minus 2qNsμ0π

1113944

infin





1113874 11138751113876 1113877


1113874 11138751113876 111387711138911113897

BWIIα

minus 2qNsμ0π

1113944

infin





1113874 11138751113876 1113877


1113874 11138751113876 111387711138911113897

(8)



Mrθ

Mr

Mrr

R1

R2

θ

I (PM)

II (Air gap)

(a)

Stator bore

II

Air b0Slot

α = 0

Current sheetR2

(b)


Stator iron

Winding

Sleeve

Magnet






1113937r

minus EzHθ

1113937θ

EzHr

1113937z

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)



01113945

r

r dθ (10)


BPMIIr minus

μ0MrR21 1 + R2

2rminus 2( 1113857



BPMIIθ


2rminus 2( 1113857



(11)




(12)




BPMIIθ μ0 G2r


(13)

where G2 MrR21R

22[R2


MrR21[R2







(14)



+ G31113872 1113873cos(φ minus ωt)1113960 1113961

(15)





0HIIrHIIθr

2dθ (17)




2πR2

R1

R21113888 1113889

2

Mrlefr2sin δ

(18)




IIr and HPMIIθ






Φpm 2 BPM

IIr( 1113857maxπ

τplef

ΦW 2 BW

IIr( 1113857maxπ

τplef

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)





id

iq

i0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

23

1113970


1113874 1113875 cos θ +2π3

1113874 1113875


1113874 1113875 minus sin θ +2π3

1113874 1113875

12

1113970 12

1113970 12

1113970

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


id

iq

⎡⎣ ⎤⎦ [C]

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22a)

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ [C]

Tid

iq1113890 1113891 (22b)

where [C] 23


3

radic2 minus

3



id

23

1113970

iA minus12

iB minus12

iC1113874 1113875

iq

23

1113970 3

radic

2iB minus

3

radic

2iC1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)


id 3

radicIRmscos(β)

iq 3

radicIRmssin(β)

⎧⎨

⎩ (24)


radic and β



LABC1113858 1113859

LAA LAB LAC

LBA LBB LBC

LCA LCB LCC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)


λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ LABC

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)


λd

λq

⎡⎣ ⎤⎦ [C]

λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)


λd

λq

⎡⎣ ⎤⎦ [C] LABC1113858 1113859[C]T

id

iq

⎡⎣ ⎤⎦ Ldq1113960 1113961id

iq

⎡⎣ ⎤⎦ (28)

where [Ldq] Ld Lmd










λB λAcos2π3

λC λAcos4π3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


Ld λd

id (31)



















009 020 040 060 080 100 120ndash1000

1000


ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)



Time (ms)

(a)



Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A


(c)





B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1113937r

minus EzHθ

1113937θ

EzHr

1113937z

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)



01113945

r

r dθ (10)


BPMIIr minus

μ0MrR21 1 + R2

2rminus 2( 1113857



BPMIIθ


2rminus 2( 1113857



(11)




(12)




BPMIIθ μ0 G2r


(13)

where G2 MrR21R

22[R2


MrR21[R2







(14)



+ G31113872 1113873cos(φ minus ωt)1113960 1113961

(15)





0HIIrHIIθr

2dθ (17)




2πR2

R1

R21113888 1113889

2

Mrlefr2sin δ

(18)




IIr and HPMIIθ






Φpm 2 BPM

IIr( 1113857maxπ

τplef

ΦW 2 BW

IIr( 1113857maxπ

τplef

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)





id

iq

i0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

23

1113970


1113874 1113875 cos θ +2π3

1113874 1113875


1113874 1113875 minus sin θ +2π3

1113874 1113875

12

1113970 12

1113970 12

1113970

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


id

iq

⎡⎣ ⎤⎦ [C]

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22a)

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ [C]

Tid

iq1113890 1113891 (22b)

where [C] 23


3

radic2 minus

3



id

23

1113970

iA minus12

iB minus12

iC1113874 1113875

iq

23

1113970 3

radic

2iB minus

3

radic

2iC1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)


id 3

radicIRmscos(β)

iq 3

radicIRmssin(β)

⎧⎨

⎩ (24)


radic and β



LABC1113858 1113859

LAA LAB LAC

LBA LBB LBC

LCA LCB LCC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)


λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ LABC

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)


λd

λq

⎡⎣ ⎤⎦ [C]

λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)


λd

λq

⎡⎣ ⎤⎦ [C] LABC1113858 1113859[C]T

id

iq

⎡⎣ ⎤⎦ Ldq1113960 1113961id

iq

⎡⎣ ⎤⎦ (28)

where [Ldq] Ld Lmd










λB λAcos2π3

λC λAcos4π3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


Ld λd

id (31)



















009 020 040 060 080 100 120ndash1000

1000


ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)



Time (ms)

(a)



Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A


(c)





B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Φpm 2 BPM

IIr( 1113857maxπ

τplef

ΦW 2 BW

IIr( 1113857maxπ

τplef

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)





id

iq

i0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

23

1113970


1113874 1113875 cos θ +2π3

1113874 1113875


1113874 1113875 minus sin θ +2π3

1113874 1113875

12

1113970 12

1113970 12

1113970

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


id

iq

⎡⎣ ⎤⎦ [C]

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22a)

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ [C]

Tid

iq1113890 1113891 (22b)

where [C] 23


3

radic2 minus

3



id

23

1113970

iA minus12

iB minus12

iC1113874 1113875

iq

23

1113970 3

radic

2iB minus

3

radic

2iC1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)


id 3

radicIRmscos(β)

iq 3

radicIRmssin(β)

⎧⎨

⎩ (24)


radic and β



LABC1113858 1113859

LAA LAB LAC

LBA LBB LBC

LCA LCB LCC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)


λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ LABC

iA

iB

iC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)


λd

λq

⎡⎣ ⎤⎦ [C]

λA

λB

λC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)


λd

λq

⎡⎣ ⎤⎦ [C] LABC1113858 1113859[C]T

id

iq

⎡⎣ ⎤⎦ Ldq1113960 1113961id

iq

⎡⎣ ⎤⎦ (28)

where [Ldq] Ld Lmd










λB λAcos2π3

λC λAcos4π3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


Ld λd

id (31)



















009 020 040 060 080 100 120ndash1000

1000


ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)



Time (ms)

(a)



Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A


(c)





B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






λB λAcos2π3

λC λAcos4π3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


Ld λd

id (31)



















009 020 040 060 080 100 120ndash1000

1000


ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)



Time (ms)

(a)



Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A


(c)





B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






009 020 040 060 080 100 120ndash1000

1000


ndash750

750500

ndash500ndash250

250000

Curr

ent (

A)



Time (ms)

(a)



Time (ms)

ndash150

150

ndash100

100

ndash050

050

000

020 040 060 080 120100 130000

Y1 (N

ewto

nMet

er)

(b)

0 02 04 06 08 1 12ndash15

ndash05

0

05

1

15

ndash1

2

t (ms)

Torq

ue T

emN

m

FE 10AFE 30AFE 50A


(c)





B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


B (tesla)13842e + 000

12112e + 00011247e + 000

95165e ndash 00186514e ndash 001


86550e ndash 00240351e ndash 005

17306e ndash 001

77863e ndash 001

10382e + 000

12977e + 000


0 40 80 (mm)

(a)

B (tesla)17001e + 000

13813e + 000

10626e + 000


11689e + 00012751e + 000

14876e + 00015938e + 000


Time = 00004s0 40 80 (mm)

(b)B (tesla)

16056e + 000

90322e ndash 001


10036e + 00011039e + 00012043e + 00013046e + 00014049e + 00015053e + 000


Time = 0000500000000000001sSpeed = 0000000rpm

40 80 (mm)0

(c)


B (tesla)15428e + 00014464e + 00013500e + 000

11572e + 00010608e + 000


47663e ndash 004


12536e + 000

(a)


ndash400

400

ndash200

200

ndash100

100

0

ndash300

300

Cog

ging

torq

ue (μ

Nm

)


(b)






000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





000 020 040 060 080 100Time (ms)

ndash25000

ndash12500

000

12500

25000

Y1 (V

)XY Plot 1


(a)

0 01 02 03 04 05Time (ms)


Vol

tage

(V)

ndash300

ndash200

ndash100

0

100

200

300

(b)

(c)

Time (s)0 50 100 150 200

0

500

1000

1500Fr

eque

ncy

(Hz)

0

50

100

150

Vol

tage

RM

SV


(d)


0 20 40 60 80 1000098

01

0102

0104

0106

0108

011

Phase current IRMSA


d- an

d q-

axis

indu

ctan

ces (

mH

)




4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



4 Conclusion






Data Availability




Acknowledgments


References






B (tesla)

(a)

375

250

125

000000E + 000 500E + 005

H (A_per_meter)

B (te

sla)

Working point 28IN

Working point 14IN

(b)















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

AnAnalyticalSolutiontotheElectromagnetic PerformanceofanUltra …downloads.hindawi.com/journals/jece/2019/1469637.pdf · 2019-09-11 · inthecounterclockwisedirection.c isananglebetweenthe