Research Article A Model for Analyzing a Five-Phase Fractional-Slot...

Research ArticleA Model for Analyzing a Five-Phase Fractional-SlotPermanent Magnet Tubular Linear Motor with ModifiedWinding Function Approach

Bo Zhang12 Rong Qi1 Hui Lin1 and Julius Mwaniki1

1Department of Electrical Engineering School of Automation Northwestern Polytechnical University Xirsquoan 710129 China2Department of Electrical Engineering Electronic Information College Xirsquoan Polytechnic University Xirsquoan 710048 China

Correspondence should be addressed to Bo Zhang paul8899126com

Received 16 March 2016 Revised 13 June 2016 Accepted 20 June 2016

Academic Editor Bin-Da Liu

Copyright copy 2016 Bo Zhang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits 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) withthemodified winding function approach (MWFA)MWFA is a fast modelingmethod and it gives deep insight into the calculationsof the following parameters air-gapmagnetic field inductances flux linkages and detent force which are essential in modeling themotor First using a magnetic circuit model the air-gap magnetic density is computed from stator magnetomotive force (MMF)flux barrier andmover geometry Second the inductances flux linkages and detent force are analytically calculated usingmodifiedwinding function and the air-gap magnetic density Finally a model has been established with the five-phase Park transformationand simulated The calculations of detent force reveal that the end-effect force is the main component of the detent force This isalso proven by finite element analysis on the motor The accuracy of the model is validated by comparing with the results obtainedusing semianalytical method (SAM) and measurements to analyze the motorrsquos transient characteristics In addition the proposedmethod requires less computation time

1 Introduction

Permanent magnet linear synchronous motors (PMLSM)have been developed for many years [1] Compared withtraditional rotary-to-linear electric actuators these directlinear electric-mechanical energy conversion devices haveno mechanical gears and transmission systems hence theypossess higher dynamic performance and reliability [2] Per-manentmagnet tubular linearmotors (PMTLM) are a class ofPMLSMs and are particularly attractive owing to zero attrac-tive force between the stator and armature high force densityexcellent servo characteristics and higher fault-tolerant per-formance [3] PMTLM has been widely used in many linearmotion fields for example transportation manufacturinghealth care electromagnetic launch in space applications [4]ECO-pedal system [5] and so forth

An accuratemodel is important in analyzing and control-ling PMTLMs In this regard the calculation of air-gap mag-netic field is critical in modelling the motor This is because

the air gap flux density distribution has a deep influence onthe thrust ripples [6]There aremanymethods to calculate theair-gap flux density distribution for example finite elementmethod (FEM) [7] analytical method (AM) [8ndash11] or SAM[12] FEM is an accurate numerical predictionmethod but it istime-consuming thereby it is not suitable for the simulationof a controlled machine [13] AM can decrease the time con-sumed It uses Laplacersquos and Poissonrsquos equations to solve thescalar magnetic potential or the vector magnetic potentialHowever the complicated boundary conditions increase thedifficulty of the solving process [14] SAM can balance theaccuracy of the model and the time consumed In [12] a 5-phase PMTLM has been modelled with this method wherethe calculation of the magnetic field distribution has beendone with FEM in advance Consequently modelling themotor is time-saving However once the motor power supplychanges or faults arise recalculation of the magnetic fielddistribution still needs using FEM Hence it is still time-consuming

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2016 Article ID 4501046 13 pageshttpdxdoiorg10115520164501046

2 Journal of Electrical and Computer Engineering

① Armaturersquos segment② Nonferromagnetic ring

with stainless steel③ Armature cover④ Linear sliding bearing⑤ Nonferromagnetic tube

① ② ③ ④ ⑤

(a)

⑥ Ferromagnetic ring⑦ Permanent magnet

x

L

A

g

B D EC

0

r

Stator

Mover

① ②

⑤

⑥ ⑦

120591

120591m120591p

Ro Ri

ds

ℎsℎc

wss

wc

bowtTt

① Armaturersquos segment② Nonferromagnetic ring with stainless steel⑤ Nonferromagnetic tube

(b)

Figure 1 Physical structure of PMTLM and the schematic outline of the halved motor cross section (a) motor physical (b) cross section ofthe motor

Table 1 Specifications of the PMTLM

Symbol Quantity Value119908ss Axial width of module 18mm119887119900 Axial width of slot open 12mm119908119888

Axial width of armature 10mm119908119905

Axial thickness of module core (leg) 3mm119879119905 Axial space of slot pitch 21mm120591 Pole pitch 15mmℎ119888 Height of armature coil 30mmℎ119904 Height of module 35mm119892 Air gap (mechanical clearance) 1mm119889119904 Axial magnetic separation distance 3mm120591119901 Axial ferromagnetic core width between PMs 7mm120591119898 Axial PM width 8mm119877119900 Outer radius of tubular reaction rail 15mm119877119894 Inner radius of tubular reaction rail 9mm119871 Axial width of stator 102mm119873 Number of turns 280slot

Modified winding function approach is another ana-lytical method which gives insight into the calculations ofparameters without considering the complicated boundaryconditions It is a simple fast modelling method for motorsThis paper shows the detailed calculations of the parametersfor a five-phase FSPMTLM such as the detailed calculationof the detent force which are not expressed distinctly in[12 15 16]Themodel of the studiedmotor is established withMWFA and Park transformation theory and the simulationsresults obtained by the mathematical model are comparedwith the ones obtained by SAM and measurements

2 Modified Winding Function Analysis

21 Description of the Machine Figure 1 shows the physicalstructure of a five-phase PMTLM and the schematic outlineof the halved motor cross sectionThe stator is separated intofive sections Each section forms one phase and a magneticseparation of a nonferromagnetic ringmade of stainless steelwhich is used as a flux barrier The mover is assembled frompermanent magnets (PMs) and ferromagnetic rings the PMsare characterized by axial magnetization and are situatedalternately with the ferromagnetic rings Table 1 gives detailsof the machine specifications

22 Modified Winding Function on the Motor MWFA wasfirst proposed for solving air-gap eccentricity in rotarymachines [17] The modified winding function (MWF) wasdeduced from the MMF drops in a magnetic circuit betweenthe stator and the rotor which was obtained under thefollowing assumptions the iron in the stator and mover hasinfinite permeability the magnetic saturation is neglectedthe mover length is infinite so that the structure of the motoris symmetrical there is no leakage flux in the shaft radial 119877119894and the magnetic permeability of the PMs is deemed as themagnetic permeability of air 1205830 It is defined as follows

119872(120593 120579) = 119899 (120593 120579) minus int21205870

119899 (120593 120579) 119892minus1 (120593 120579) 1198891205932120587 ⟨119892minus1 (120593 120579)⟩ (1)

where 120593 and 120579 represent the position of a stationary coil andangular position of the rotor with respect to stator respec-tively 119872(120593 120579) 119899(120593 120579) and 119892minus1(120593 120579) represent the modifiedwinding function turns function and air-gap functionrespectively The symbol ⟨sdot⟩ represents the average value ofldquosdotrdquo If the rotor is not eccentric (1) reduces to [17]

119872(120593 120579) = 119899 (120593 120579) minus ⟨119899⟩ (2)

where ⟨119899⟩ is the dc value of the turns function of the winding

Journal of Electrical and Computer Engineering 3

ds

0 x

N

minusN

wc2

wc2

120576Na(x)

5Tt

(a)

a

b

cd

e120572

(b)

Figure 2 Winding function of phase ldquo119886rdquo and winding space distribution (a) winding function (b) spatial structure of windings

Different from the rotary machines the analysis of themodified winding function requires creating an appropriatecoordinate system on the stator As shown in Figure 1(b) theorigin of coordinates is on the left schematic outline of thehalved motor cross section and the distance between originand the center of the phase ldquoCrdquo is 25 times axial space of slotpitch119879119905Thepole pairs in the longitudinal section of the statorare 35 no matter whether the mover is run or not Hencethe motor has a fractional-slot structure and the slot-per-phase-per-pole (SPP) is equal to 27 Seen from Figure 1(b)and Table 1 along the shaft of the motor the relation of theslot number 119876119904 slot pitch 119879119905 pole pitch 120591 and pole pairs 119901 iswritten by

119876119904119879119905 = 2119901120591 (3)

Due to the influence of the fractional slot on the windingfunction distribution the modified winding function of thefive-phase FSPMTLM is defined as the product of windingfunction and the winding factor 119910119908 [18] Similar to the rotary5-phase PMSM [19] the longitudinal section of the five-phaseFSPMTLM windings is assumed to be symmetrical Figure 2shows the winding function of the phase ldquo119886rdquo and the spatialstructure of windings in five-phase FSPMTLMThe Fourierrsquosexpansion of the winding function of phase ldquo119886rdquo is calculatedfrom

119873119886 (119909)= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (4)

In Figure 2(a) 119873119886(119909) is the winding function of phaseldquo119886rdquo the letter ldquo120576rdquo represents the distance between the originand the radial center line of phase ldquo119886rdquo and 119873 representsthe stator turns number In Figure 2(b) ldquo120572rdquo is the windingspace geometric angle of the adjacent phases Because thestator structure is symmetrical the plane is evenly dividedinto five blocks and each angle is equal to 21205875 Combiningthe all-order winding factor 119910119908119896 (119896 is odd number and

x

C

g

B D EAD

r

0120591 120591m 120591p

ds

wss

wtTt

bo

Di

120579

Figure 3 Flux paths due to the stator slot and mover saliency

119896 = 1 3 5 ) and (3) yields the all-phase modified windingfunction119872119894(119909)

119872119894 (119909) = 4119873119901120587 sum119896=135

119910119908119896119896 sin(1199081198881198961205872120591 )sdot cos(119896120587120591 (119909 minus 120576 minus 119894120572))

(5)

where the subscript 119894 (119894 = 0 1 2 3 4) represents the ordinalnumber of phase 119886 119887 119888 119889 and 119890 respectively For example1198721(119909) shows themodifiedwinding function of phase ldquo119887rdquoThedetailed calculation of (5) is as shown in Appendix A

23 Air Gap Flux Density The generation of air gap fluxdensity is due to the current flowing in each phase Hence theflux density in phase ldquo119886rdquo and1198610(119909 120579) is defined as the productof the modified winding function 1198720(119909) and the inverse airgap function 119892minus1(119909 120579) [20]

1198610 (119909 120579) = 1205830119892minus1 (119909 120579)1198720 (119909) 119894119886 (6)

where 120579 is the distance which themover has covered from theorigin of the stator coordinates and 119894119886 is the current of phaseldquo119886rdquo

Calculation of the inverse air gap function 119892minus1(119909 120579)requires modeling the flux paths through the air gap regionswith straight lines and circular arc segments [20] The fluxpaths due to the mover saliency are shown in Figure 3 Theinverse air gap function 119892minus1(119909 120579) is obtained by the unslotted


2p120591

Air gap

Ro

R

Rse

Figure 4 The schematic diagram of computational volume in theair gap

air-gap flux density [21] and the relative air-gap permeance[22] The calculation of 119892minus1(119909 120579) in the Fourier series form is

119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(119899120587120591 (119909 minus 120579 minus 120576)) (7)

where the expression of the coefficient 119866119899 is119866119899 = 425119879119905 int

119879119905+120579+120576

120579+120576119892minus1 (119909 120579) cos(119899120587120591 (119909 minus 120579 minus 120576)) 119889119909

+ 225119879119905 int120579+120576

120579119892minus1 (119909 120579) cos(119899120587120591 (119909 minus 120579 minus 120576)) 119889119909

(8)

The detailed computations of the inverse air-gap function119892minus1(119909 120579) are shown in Appendix B

24 Calculations of Inductances and Permanent Magnet FluxLinkages In terms of the modified winding function theorythe inductances and the permanent magnet flux linkages arederived from the air-gap function and the modified windingfunctions Both are computed for the volume of the air-gapsection of the motor with the effective air-gap radius 119877se [23]as is depicted in Figure 4

In Figure 4 119877119900 and 119877 are the outer radius of tubularreaction rail and the radius of the stator tooth respectively(see Section 21) 119877se is the effective air-gap computationalradius and the length of the computational volume is 2p120591 (seeSection 22)

The self and mutual inductances are computed in thecomputational volume as shown by [17 23]

119871 119894119894 (120579) = 2120587119877se1205830 int21199011205910

1198722119894 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 0 1 2 3 4) 119871 119894119895 (120579) = 2120587119877se1205830 int2119901120591

0119872119894 (119909)119872119895 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 119895 119894 119895 = 0 1 2 3 4)

(9)

where 119871 119894119894(120579) and 119871 119894119895(120579) are the self-inductances and mutualinductances which are the function of mover position 120579119872119894(120579) and 119872119895(120579) are the modified winding functions of the119894th phase and the 119895th phase respectively

The permanent magnet flux linkages are derived from theair-gap flux density produced by permanent magnets and themodified winding functions as computed by

120595119898119894 (120579) = 21205871205830119877se int21199011205910

119872119894 (119909) 119861119892 (119909 120579) 119889119909 (10)

where 119861119892(119909 120579) represents the air-gap flux density which isthe product of PM flux density 119861pm(119909 120579) and relative air gappermeance 120582ag(119909 120579) [22]

The detailed calculations of the effective air-gap compu-tational radius 119877se and the air-gap flux density 119861119892(119909 120579) areshown in Appendix C

3 Mathematical Model

31 Basic Model of the PMLMs The stator voltage equationsand mechanical thrust equations compose the mathematicalmodel of PMLMs [24]

119881119904 = 119877119904119868119904 + 119889120595119904119889119905 (11)

where 119881119904 119877119904 and 119868119904 represent the stator voltage the statorresistance and the stator current respectively 120595119904 representsthe air-gap flux linkages produced by the permanent magnetand the stator currents It can be calculated from the followingformulae

120595119904 = 120595119898 + 119871 119904119868119904 (12)

where120595119898 is the flux linkages produced by PMs 119871 119904 representsthe inductances matrix including self- and mutual induc-tances matrix as computed by (9)

119865119890 = 119898119889]119889119905 + 119863] + 119865119897 + 119865detent (13)

where 119865119890 is the electromagnetic thrust force 119898 is the totalmass on the mover 119863 is the dynamical friction coefficient] is the mover velocity 119865119897 is the load force and 119865detent is thedetent force including the end-effect force 119865end and slot-effectforce 119865slot32 Calculations of the Detent Force The detent force is theinteraction force between the mover magnets and stator slotswithout the stator currents flowing [25] Due to the openingstator slots and the bilateral ends of the stator the detent forceis the sum of the end-effect force 119865end and slot-effect force119865slot

119865detent = 119865slot + 119865end (14)

Using the Virtual Work Method (VWM) 119865end and slot-effect force 119865slot are obtained by

119865 = minus120597119882120597120579 (15)

where119882 is the air gap magnetic field energy produced by thecomputational volume 119881 of each section as is given by [26]

119882 = 121205830 int119881 1198611198922 (119909 120579) 119889119881 (16)


Slot

-effe

ct fo

rce (

N)

Slot force using MWFA

minus3

minus2

minus1

0

1

2

3

0020 004 006minus004 minus002minus006Axial displacement (mm)

times10minus3

(a)

End force using MWFA

End-

effec

t for

ce (N

)

minus004 minus002 0 002 004 006minus006Axial displacement (mm)

minus50minus40minus30minus20minus10

01020304050

(b)

Det

ent f

orce

(N)

Detent force using MWFADetent force using FEM


minus60

minus40

minus20

0

20

40

60

(c)

Figure 5 Calculations of slot force end force and detent force distribution versus mover position

The detailed calculations of the detent force obtained byusing the MWFA are shown in Appendix D

Figure 5 shows the calculation results of the slot-effectforce the end-effect force and the detent force In Figures5(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 usingMWFAandFEM Figure 6establishes the finite element analysis model of the studiedmotor in a cylindrical coordinate system The fractional-slot structure is adopted in the model and also the flux linedistribution of the five-phase PMTLM is shown in Figure 6

Seen from Figures 5(a)ndash5(c) the following conclusionsare obtained (1) since the amplitude of the end-effect forceis 463911 N yet the slot-effect force is 25mN the end-effectforce is the largest component of detent force That is to saythe slot-effect force has been weakened in such a motor (2)the results using MWFA to compute the detent force are inaccordance with results using FEM Hence the accuracy of

the detent force using the proposed method is validated byusing FEM

33 Mathematical Model of the Five-Phase FSPMTLM Asseen from Section 31 the basic mathematical model ofPMLMs is a multivariable system with strong coupling Itis hard to analyze the motor characteristics and control themotorThemotor system is decoupled by applying Park trans-formation theory (PTT) which has been widely applied inmotor vector control (VC) [27]Using PTT tomodel the stud-iedmotor can ease the analysis of the transient characteristicsIn [28] using a Park transformation matrix 119879(120579) to model afive-phase permanent magnet synchronous Motor (PMSM)was reported The studied motor has the same five-phasepower supply however it has the structure of the linearmotorwhich is different from the five-phase PMSMWhen the otherharmonic components are ignored the Park transformationmatrix becomes


23694e minus 00515797e minus 00579006e minus 00640309e minus 009minus78926e minus 006minus15789e minus 005minus23686e minus 005minus31582e minus 005minus39479e minus 005minus47376e minus 005minus55272e minus 005

31590e minus 005

47384e minus 00539487e minus 005

55280e minus 005

A (Wbm)

Figure 6 Five-phase FSPMTLMmodel using finite element method

119879 (120579) = 25

[[[[[[[[[[[[[[[[

cos(120587120591 120579) cos(120587120591 120579 minus 120572) cos(120587120591 120579 minus 2120572) cos(120587120591 120579 + 2120572) cos(120587120591 120579 + 120572)minus sin(120587120591 120579) minus sin(120587120591 120579 minus 120572) minus sin(120587120591 120579 minus 2120572) minus sin(120587120591 120579 + 2120572) minus sin(120587120591 120579 + 120572)cos(120587120591 120579) cos(120587120591 120579 + 2120572) cos(120587120591 120579 minus 120572) cos(120587120591 120579 + 120572) cos(120587120591 120579 minus 2120572)minus sin(120587120591 120579) minus sin(120587120591 120579 + 2120572) minus sin(120587120591 120579 minus 120572) minus sin(120587120591 120579 + 120572) minus sin(120587120591 120579 minus 2120572)

1radic21radic2

1radic21radic2

1radic2

]]]]]]]]]]]]]]]]

(17)

where 120579 is the position of the mover 120572 is the winding spacegeometric angles of adjacent phases and 120572 = 04120587 (seeSection 22)

Using (17) to transform (11)ndash(13) themathematicalmodelwith the Park transformation form is written as

119881119889 = 119877119904119894119889 minus 120587120591 ]119871119902119894119902 + 119871119889 119889119894119889119889119905 119881119902 = 119877119904119894119902 + 120587120591 ] (119871119889119894119889 + 1205951015840119898) + 119871119902 119889119894119902119889119905 119865119890 = 52 120587120591 119901 [1205951015840119898 119894119902 + (119871119889 minus 119871119902) 119894119889119894119902]

119898119889]119889119905 = 119863] + 119865119897 + 119865detent minus 119865119890

(18)

where 119881119889 and 119881119902 are the direct-axis and quadrature-axisvoltage119877119904 is the stator resistance 119894119889 and 119894119902 are direct-axis andquadrature-axis currents ] is the mover velocity 119871119889 and 119871119902are the direct-axis and quadrature-axis inductances1205951015840119898 is theamplitude of the PM flux linkage the remaining parametersare the same as (13) and the detailed calculations of 119881119889 119881119902119894119889 119894119902 and 1205951015840119898 are shown in Appendix E

4 The Simulation Analysis ofthe Motor Transients

In papers [12 15] a model using semianalytical methodwas validated by the measurements results of the transient

characteristics Likewise in order to verify the accuracy of theproposedmodel model (18) has been established on theMat-labSimulink platformwhere it has been simulated to analyzethe transient characteristics of the studied motor in thissection In addition to the basic motor parameters the statorresistance 119877119904 (119877119904 = 5Ω) the friction coefficient 119863 (119863 =3000) the nominal value of the winding current 119868 (119868 = 8A)and themass of themover119898 (119898 = 92 kg) which were shownin [12 15] and the following parameters of the motor model119871119889 119871119902 1205951015840119898 and 119865detent need to be calculated by using MWFAThe computed results are as follows 119871119889 = 36mH 119871119902 =68mH1205951015840119898 = 02261Wb and the detent force119865detent has beenshown in Appendix D Figure 7 shows the proposed mathe-matical model established on the MatlabSimulink platform

In Figure 7 the model established on the MatlabSim-ulink platform is composed by the current excitation thecurrent balance subsystem and mechanical balance subsys-tem The current excitation implemented using the PWMconverter is the same as the ones in [12 15] which are thefive-phase symmetric cosine current sources 119894119886 119894119887 119894119888 119894119889 and119894119890 while the mover velocity ] is derived from the frequenciesof the current sources as shown by

119891 = ]2120591 (19)

119865load represents the load force on the mover Thetarepresents position along the mover movements directionand 119896 is the amplification factorwhich canmagnify themoverposition 1000 timesThe corresponding unit of the position is


Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000

Current balance Mechanical balance

Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-

Figure 7 Establishment of the studied motor model on MatlabSimulink platform

in millimeters The current balance subsystem and mechani-cal balance subsystem are shown in Figures 8(a) and 8(b)

In Figure 8(a) the current subsystem is derived fromthe 3rd equation of (18) In the DQ transformation moduleshown in this figure which is established from the Park trans-formation matrix 119879(120579) is given in (17) 119865119898 is the amplitudeof the PM flux linkages which represents 1205951015840119898 in (18) InFigure 8(b) themechanical subsystem is derived from the 4thequation of (18)

The simulations using the proposed model to analyze themotor transients are carried out This is done for the threecases given in [12]The results are then compared with resultsfrom SAM and measurementsThe three cases are as follows

Case 1 Thevelocity of themover is set to 18mms and there isno load The simulation results of the mover position (versustime) and the velocity of the mover (versus time) are shownin Figures 9 and 10

Figure 9 shows the simulation results using the proposedmodel have a close agreement with the results from SAMand measurements in [12] Figure 10 shows slight oscillationsof the mover velocity around the set 18mms However theposition increases linearly with time without any oscillations

Case 2 The velocity of the mover is set to 1ms and noload is installed on the mover The simulation results of themover position (versus time) and the velocity (versus time)are shown in Figures 11 and 12

Figure 11 shows the simulation results using the proposedmodel are almost identical to the results from measurementsand SAM The results show some variances at the beginningof the mover running however these die out after 004 sFigure 12 shows the mover velocity around the setting of

1ms The slight oscillations of the mover velocity have smallvariations after 004 s Hence these results are consistent

Case 3 An additional mass 119898Δ (119898Δ = 192 kg) was linkedto the mover The average velocity of the mover has beenassumed 50mms The results of the position (versus time)and velocity (versus time) are shown in Figures 13 and 14respectively

The additional mass brings out more ripples in the moverposition than in Cases 1 and 2 The two causes of the ripplesare the resistance offered by the higher inertia to the motorand that by the detent force In Figure 13 the variation trendsbetween the simulation results using the proposedmodel andthe ones obtained by using SAM are similar although thereare some slight differences Also it is noted that the highervelocity changes seen in Figure 14 indicate the presence ofthrust ripples

5 Conclusions

In this paper a method based on modified winding functiontheory was applied to model a five-phase fractional-slot per-manent magnet tubular linear motor The proposed methodhas provided the detailed computational expressions of theair-gap flux density the inductance flux linkages and thedetent force Due to the fact that it has a property of fast mod-eling no matter whether the motor is healthy or not it canmake up for the time-consuming remodeling of the motorusing finite element analysis and semianalytical methodwhen the faults arise or supply power changes The analyticalresults showed that the slot-effect force had been greatlyreduced by this structural design and the main component


Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract

Five-phase DQ transformation

LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)

Figure 8 Subsystem of the proposed model (a) current balance subsystem (b) mechanical balance subsystem

of the detent force is the end-effect force This is as had beendemonstrated by using the finite element model In this pro-posed approach amodelwith five-phase Park transformationhad been established and simulated The simulation resultsfor transients on the motor were analyzed and were close tothose of semianalytical method and measurements In thisway the accuracy of the model using MWFA was validated

Appendix

A Calculations of the WindingFunction Winding Factor andModified Winding Function

A1 Winding Function Computation As shown in Fig-ure 2(a) on moving the coordinate system by a distance ldquo120576rdquoalong the 119909-axis the winding function of phase ldquo119886rdquo becomesan even function Using Fourierrsquos expansion method thewinding function of phase ldquo119886rdquo in form of Fourier expansionis written by

119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)

where 119873119886(119909) represents the winding function of phase ldquo119886rdquo119896 is the order of the coefficients of the Fourierrsquos expansion

119879119905 and 119908119888 represent the axial space of slot pitch and the axialwidth of armature (see Figure 1(b)) respectively 119901 representsthe pole pairs ldquo120576rdquo is the distance between the former originand the new origin (see Figure 2(a)) ldquo119886119896rdquo is the 119896th Fourierrsquosexpansion coefficient as calculated by

119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882

minus119873 cos(21199011198961205875119879119905 119909)119889119909) = 4119873119901119896120587sdot sin(1199081198881199011198961205875119879119905 )

(A2)

Plugging (A2) into (A1) and combining with (3) thewinding function of phase ldquo119886rdquo in form of Fourier expansionis given by

119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)

Then the expression (4) is obtained


Simulation results using the proposed modelSimulation results of the model using SAM [12]Measurement results [12]

1 2 3 4 50Time (s)

0

10

20

30

40

50

60

70

80

90

Posit

ion

(mm

)

Figure 9 Transient characteristic of the mover position under no-load for ] = 18mms

00175

0018

00185

Mov

er sp

eed

(ms

)

1 2 3 4 50Time (s)

Figure 10 Simulation results of the mover velocity under no-loadfor ] = 18mms

Simulation results using the proposed modelSimulation results using SAM [12]Measurement results [12]

0

10

20

30

40

50

60

70

80

90

100

Posit

ion

(mm

)

002 004 006 008 010 012000Time (s)

Figure 11 Transient characteristics of the mover position under no-load for ] = 1ms

002 004 006 008 01 0120Time (s)

0

02

04

06

08

1

12

Mov

er sp

eed

(ms

)

Figure 12 Simulation results of the mover velocity under no-loadfor ] = 1ms

05 10 15 2000Time (s)

0

10

20

30

40

50

60

70

80

90

100

Posit

ion

(mm

)


Figure 13 Transient characteristic of the mover position under anadditional inertial loadwith themass119898Δ = 192 kg for ] = 50mms

0

002

004

006

008

01

012

014

Mov

er sp

eed

(ms

)

05 1 15 20Time (s)

Figure 14 Simulation results of the mover velocity under anadditional inertial loadwith themass119898Δ = 192 kg for ] = 50mms


A2 Winding Factor Computations

Definition of 119889119904 119910119908119896 119910119910119896 and 119910119889119896 119889119904 shown in Figure 2(a) isthe axial magnetic separation distance between two adjacentphase 119910119908119896 119910119910119896 and 119910119889119896 represent the 119896th winding factor 119896thpitch factor and 119896th winding distribution factor respectivelyIn the light of the papers [29 30] 119910119908119896 is the product of 119910119910119896and 119910119889119896

119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where

119910119910119896 = sin(119896 (119879119905 minus 119889119904)120591 sdot 1205872 ) 119910119889119896 = sin (1198961205791198892)(119873 sin (1198961205791198892119873))

(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)

A3 All-Phase Modified Winding Function ComputationsAll-phase modified winding function119872119894(119909) is the product ofwinding function and winding factor Therefore combining

the spatial structure of windings in Figure 2(b) with all-orderwinding factor 119910119908119896 yields (3)

119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135

119910119908119896119896 sin(119908c1198961205872120591 ) cos(119896120587120591 (119909 minus 120576 minus 119894120572)) (A7)


B Analysis of Inverse Air Gap Function

The air gap magnetic field is the product of the PM field withunslotted stator and the relative air gap permeance [22] Paper[21] has shown the expression of unslotted PM field 119861119892

119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)

where 119867119888 = 950 kAm 120583119903 = 1048 119863 is the diameter of theinner stator core and119863 = 2(119877119900 + 119892)119863119894 is the diameter of theinner tubular reaction rail and 119863119894 = 2119877119894 120591119898 is the axial PMwidth and 120591119901 is the axial ferromagnetic core width betweenPMsWe can suppose 120591119901 is equal to 120591119898 (in Table 1 120591119898 = 8mmand 120591119901 = 7mm) thereafter along the 119909-axis the air gapfunction 119892(119909 120579) in one pitch 119879119905 is

119892 (119909 120579) asymp

1198921015840 + 1205872 (1198871199002 minus |119909 minus 120579 minus 120576|) |119909 minus 120579 minus 120576| le 12059111990121198921015840 1205911199012 lt |119909 minus 120579 minus 120576| le 119908ss2infin 119908ss2 lt |119909 minus 120579 minus 120576| le 119879119905 + 11988911990421198921015840 119879119905 + 1198891199042 lt |119909 minus 120579 minus 120576| le 119879119905 + 1198891199042 + 1199081199051198921015840 + 1205872 (|119909 minus 120579 minus 120576| minus (119879119905 + 1198891199042 + 119908119905)) 119879119905 + 1198891199042 + 119908119905 lt |119909 minus 120579 minus 120576| le 1205911199012 + 120591119892119888 1205911199012 + 120591 lt |119909 minus 120579 minus 120576| le 119879119905

(B2)

By using Fourier series expansion theory the inverse airgap function 119892minus1(119909 120579) is shown by

119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)

where 119866119899 is computed by

119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)

Combining (3) the above two formulas (B3) and (B4)are written by (5) and (6) respectively

C Definition of the Relative Air-GapPermeance 120582ag(119909 120579) the Air-Gap FluxDensity 119861119892(119909 120579) and Effective Air-GapComputational Radius 119877se

In light of the paper [22] the relative air-gap permeance120582ag(119909 120579) is computed by

120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)

Because air gap 119892 is small (see Table 1 119892 = 1mm) theair-gap flux density 119861119892(119909 120579) is deduced by the product of


120582ag(119909 120579) and flux density produced by PMs with unslottedstator 119861pm-slotless(119909 120579)

119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)

where 119861pm-slotless(119909 120579) can be expanded by Fourier seriesalong the shaft of the motor

119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135

1119899 sin(1198991205872 )sdot cos(1198991205871205911198982120591 )sdot cos(119899120587120591 (119909 minus 120579 minus 120576))

(C3)

As shown in Figure 3 for an axially magnetized internalmagnet machine topology the effect of the slot openings maybe accounted for by introducing a Carter factor 119870119862 given by[10]

119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)

where 119879119905 is the armature slot pitch (see Table 1) 1198921015840 has beenderived from (B1) and the slotting coefficient 120574 is computedby

120574 = 4120587

11988711990021198921015840 tanminus1 ( 11988711990021198921015840) minus lnradic1 + ( 11988711990021198921015840)2

(C5)

where 119887119900 is the width of the armature slot openings There-after the effective airgap 119892119890 and effective air-gap computa-tional radius 119877se are given respectively by

119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)

D Definition of the Slot-EffectElectromagnetic Energy 119882slot and the End-Effect Electromagnetic Energy 119882end

119882slot is derived from the sum of the electromagnetic energyon the each tooth as computed by

119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905

+int119879119905minus1198891199042+119894119879119905119879119905minus1198891199042minus119882119905+119894119879119905

)sdot (119861119892 (119909 120579))2 119889119909

(D1)

where 119894 is defined in (5) and119861119892(119909 120579) is the air-gap flux densitywhich is calculated by (C1)ndash(C3)

119882end is derived from the sum of the electromagneticenergy on the two ends of the stator as calculated by

119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871

)sdot (120582end (119909) 119861pm-slotless (119909 120579))2 119889119909

(D2)

where 119871 (see Table 1) is the length of the stator sectionas is shown in Figure 1(b) 120582end(119909) is the air-gap relativepermeance on the two axially ends of the stator as is shownby

120582end (119909) =

119890(119909minus1198891199042)2119892119890 119909 le 11988911990421 1198891199042 lt 119909 lt 1198891199042 + 119871119890minus((119909+1198891199042+119871)2119892119890) 119909 ge 1198891199042 + 119871

(D3)

Then combining (15) with (D1)ndash(D3) the analytical formu-lae of detent force is computed by

119865detent (120579) = 119865slot (120579) + 119865end (120579)= minus019478 times 10minus3

times cos(12566 times (120579 + 1198791199052 )) minus 09739times 10minus3 times sin(12566 times (120579 + 1198791199052 ))minus 69245 times 10minus9times sin(10472 times (120579 + 1198791199052 )) minus 16441times 10minus8 times sin(12566 times (120579 + 1198791199052 ))+ 99405 times 10minus9times sin(94248 times 10minus8 times (120579 + 1198791199052 ))minus 5296 times 10minus10times sin(31416 times 10minus8 times (120579 + 1198791199052 ))minus 5177 times 10minus10times sin(20944 times (120579 + 1198791199052 )) minus 2698times 10minus9times sin(2199 times 10minus7 times (120579 + 1198791199052 ))minus 0292 times 10minus5


times cos(1885 times (120579 + 1198791199052 )) minus 0292times 10minus5 times cos(1885 times (120579 + 1198791199052 ))minus 039 times 10minus3times sin(41888 times (120579 + 1198791199052 )) minus 09739times 10minus3 times sin(8378 times (120579 + 1198791199052 ))minus 09739 times 10minus3times sin(8378 times (120579 + 1198791199052 )) minus 039times 10minus3 times cos(20944 times (120579 + 1198791199052 ))+ 314 times 10minus9times sin(20944 times (120579 + 1198791199052 )) minus 466466times sin(2120587 (120579 + 1198791199052)120591 )

(D4)

E Detailed Calculations for the MathematicalModel Parameters

Seen from the five-phase Park transformation matrix 119879(120579)the following relation can be obtained

119879minus1 (120579) = 52119879119879 (120579) (E1)

where 119879minus1(120579) and 119879119879(120579) represent the inverse matrix andmatrix transposition of 119879(120579) respectively

Applying 119879(120579) to (11) the direct-axis and quadrature-axisvoltage 119881119889 and 119881119902 and the direct-axis and quadrature-axiscurrents 119894119889 and 119894119902 are derived from

119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)

When the harmonics of the inductances are ignored thedirect-axis and quadrature-axis inductances are as follows

119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)

The permanent magnet flux linkages using the five-phasePark transformation matrix 119879(120579) are obtained by

119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)

The amplitude of the permanent magnet flux linkages isgiven by

1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)

where119872 represents the fundamental amplitude of the mod-ified winding function and it is given by

119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] I Boldea and S A Nasar ldquoLinear electric actuators andgeneratorsrdquo IEEETransactions on Energy Conversion vol 14 no3 pp 712ndash717 1999

[2] K-H Shin M-G Park H-W Cho and J-Y Choi ldquoCompara-tive study of armature reaction field analysis for tubular linearmachine with axially magnetized single-sided and double-sided permanent magnet based on analytical field calculationsrdquoJournal of Magnetics vol 20 no 1 pp 79ndash85 2015

[3] J Wang and D Howe ldquoTubular modular permanent-magnetmachines equipped with quasi-Halbachmagnetizedmagnetsmdashpart I magnetic field distribution EMF and thrust forcerdquo IEEETransactions on Magnetics vol 41 no 9 pp 2470ndash2478 2005

[4] F A B Kou S B L Li and T C C Zhang ldquoAnalysisand optimization of thrust characteristics of tubular linearelectromagnetic launcher for space-userdquo IEEE Transactions onMagnetics vol 45 no 1 pp 1ndash6 2008

[5] Y-K Kim B-G Gu I-S Jung S-HWon and J Hur ldquoAnalysisand design of slotted tubular linear actuator for the eco-pedalsystem of a vehiclerdquo IEEE Transactions on Magnetics vol 48no 2 pp 939ndash942 2012


[6] N R Tavana and A Shoulaie ldquoPole-shape optimization ofpermanent-magnet linear synchronous motor for reduction ofthrust ripplerdquo Energy Conversion and Management vol 52 no1 pp 349ndash354 2011

[7] D Deas P Kuo-Peng N Sadowski A M Oliveira J L Roeland J P A Bastos ldquo2-D FEM modeling of the tubular linearinduction motor taking into account the movementrdquo IEEETransactions on Magnetics vol 38 no 2 pp 1165ndash1168 2002

[8] N Bianchi ldquoAnalytical field computation of a tubular perma-nent-magnet linear motorrdquo IEEE Transactions on Magneticsvol 36 no 5 pp 3798ndash3801 2000

[9] J Wang D Howe and G W Jewell ldquoAnalysis and design opti-mization of an improved axially magnetized tubular perma-nent-magnet machinerdquo IEEE Transactions on Energy Conver-sion vol 19 no 2 pp 289ndash295 2004

[10] J Wang G W Jewell and D Howe ldquoA general framework forthe analysis and design of tubular linear permanent magnetmachinesrdquo IEEE Transactions on Magnetics vol 35 no 3 pp1986ndash2000 1999

[11] JWang Z Lin andDHowe ldquoAnalysis of a short-stroke single-phase quasi-Halbach magnetised tubular permanent magnetmotor for linear compressor applicationsrdquo IET Electric PowerApplications vol 2 no 3 pp 193ndash200 2008

[12] B Tomczuk and A J Waindok ldquoA coupled field-circuit modelof a 5-phase permanent magnet tubular linear motorrdquo Archivesof Electrical Engineering vol 60 no 1 pp 5ndash14 2011

[13] T Lubin T Hamiti H Razik and A Rezzoug ldquoComparisonbetween finite-element analysis and winding function theoryfor inductances and torque calculation of a synchronous reluc-tance machinerdquo IEEE Transactions on Magnetics vol 43 no 8pp 3406ndash3410 2007

[14] S Saied K Abbaszadeh and A Tenconi ldquoImprovement towinding function theory for PM machine analysisrdquo in Pro-ceedings of the 3rd IEEE International Conference on PowerEngineering Energy and Electrical Drives (PowerEng rsquo11) pp 1ndash6Malaga Spain May 2011

[15] A Waindok and M Piegza ldquoTests of the controllers settingsfor a 5-phase permanent magnet tubular linear motor usingMatlabSimulink softwarerdquo in Proceedings of the InternationalSymposium on Electrodynamic and Mechatronic Systems (SELMrsquo13) pp 53ndash54 Zawiercie Poland May 2013

[16] A Waindok ldquoModeling and measurement of transients for a5-phase permanent magnet tubular linear actuator includingcontrol and supply systemrdquo Solid State Phenomena vol 214 pp121ndash129 2014

[17] N A Al-Nuaim and H A Toliyat ldquoA novel method for mod-eling dynamic air-gap eccentricity in synchronous machinesbased onmodified winding function theoryrdquo IEEE Transactionson Energy Conversion vol 13 no 2 pp 156ndash162 1998

[18] L Serrano-Iribarnegaray P Cruz-Romero and A Gomez-Exposito ldquoCritical review of the modified winding functiontheoryrdquo Progress in Electromagnetics Research vol 133 pp 515ndash534 2013

[19] S Jordan C DManolopoulos and J M Apsley ldquoWinding con-figurations for five-phase synchronous generators with dioderectifiersrdquo IEEE Transactions on Industrial Electronics vol 63no 1 pp 517ndash525 2016

[20] P C Krause O Wasynczuk S D Sudhoff and S PekarekAnalysis of Electric Machinery and Drive Systems vol 75Wiley-IEEE Press Piscataway NJ USA 2013

[21] N Bianchi S Bolognani D D A Corte and F Tonel ldquoTubularlinear permanent magnet motors an overall comparisonrdquo IEEE

Transactions on Industry Applications vol 39 no 2 pp 466ndash475 2003

[22] Z Q Zhu and D Howe ldquoInstantaneous magnetic field distri-bution in brushless permanent magnet DC motors III Effectof stator slottingrdquo IEEE Transactions on Magnetics vol 29 no1 pp 143ndash151 1993

[23] S Nandi H A Toliyat and A G Parlos ldquoPerformance analysisof a single phase induction motor under eccentric conditionsrdquoinProceedings of the IEEE-IASAnnualMeeting pp 174ndash181 1997

[24] W Xin-Huan Z Hong-Wei and K Run-Sheng ldquoResearchon modeling and simulation of permanent magnet linearsynchronous motor for vertical transportation systemrdquo in Pro-ceedings of the Chinese Control and Decision Conference (CCDCrsquo09) pp 3332ndash3336 July 2009

[25] G Remy G Krebs A Tounzi and P-J Barre ldquoFinite elementanalysis of a PMLSM (part 2)-cogging force and end-effect forcecalculationsrdquo in Proceedings of the 6th International Symposiumon Linear Drives for Industry Applications (LDIA rsquo07) LilleFrance September 2007

[26] D Zarko D Ban and T A Lipo ldquoAnalytical calculation ofmagnetic field distribution in the slotted air gap of a surfacepermanent-magnet motor using complex relative air-gap per-meancerdquo IEEE Transactions on Magnetics vol 42 no 7 pp1828ndash1837 2006

[27] P Pillay and R Krishnan ldquoModeling simulation and analysisof permanent-magnet motor drives I The permanent-magnetsynchronousmotor driverdquo IEEETransactions on Industry Appli-cations vol 25 no 2 pp 265ndash273 1989

[28] L Parsa and H A Toliyat ldquoFive-phase permanent-magnetmotor drivesrdquo IEEE Transactions on Industry Applications vol41 no 1 pp 30ndash37 2005

[29] G J Li Z Q Zhu W Q Chu M P Foster and D A StoneldquoInfluence of flux gaps on electromagnetic performance ofnovel modular PM machinesrdquo IEEE Transactions on EnergyConversion vol 29 no 3 pp 716ndash726 2014

[30] J Wang and D Howe ldquoTubular modular permanent-magnetmachines equipped with quasi-Halbach magnetized magnets-part I Magnetic field distribution EMF and thrust forcerdquo IEEETransactions on Magnetics vol 41 no 9 pp 2470ndash2478 2005

International Journal of

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



① Armaturersquos segment② Nonferromagnetic ring

with stainless steel③ Armature cover④ Linear sliding bearing⑤ Nonferromagnetic tube

① ② ③ ④ ⑤

(a)

⑥ Ferromagnetic ring⑦ Permanent magnet

x

L

A

g

B D EC

0

r

Stator

Mover

① ②

⑤

⑥ ⑦

120591

120591m120591p

Ro Ri

ds

ℎsℎc

wss

wc

bowtTt

① Armaturersquos segment② Nonferromagnetic ring with stainless steel⑤ Nonferromagnetic tube

(b)

Figure 1 Physical structure of PMTLM and the schematic outline of the halved motor cross section (a) motor physical (b) cross section ofthe motor

Table 1 Specifications of the PMTLM

Symbol Quantity Value119908ss Axial width of module 18mm119887119900 Axial width of slot open 12mm119908119888

Axial width of armature 10mm119908119905

Axial thickness of module core (leg) 3mm119879119905 Axial space of slot pitch 21mm120591 Pole pitch 15mmℎ119888 Height of armature coil 30mmℎ119904 Height of module 35mm119892 Air gap (mechanical clearance) 1mm119889119904 Axial magnetic separation distance 3mm120591119901 Axial ferromagnetic core width between PMs 7mm120591119898 Axial PM width 8mm119877119900 Outer radius of tubular reaction rail 15mm119877119894 Inner radius of tubular reaction rail 9mm119871 Axial width of stator 102mm119873 Number of turns 280slot

Modified winding function approach is another ana-lytical method which gives insight into the calculations ofparameters without considering the complicated boundaryconditions It is a simple fast modelling method for motorsThis paper shows the detailed calculations of the parametersfor a five-phase FSPMTLM such as the detailed calculationof the detent force which are not expressed distinctly in[12 15 16]Themodel of the studiedmotor is established withMWFA and Park transformation theory and the simulationsresults obtained by the mathematical model are comparedwith the ones obtained by SAM and measurements

2 Modified Winding Function Analysis

21 Description of the Machine Figure 1 shows the physicalstructure of a five-phase PMTLM and the schematic outlineof the halved motor cross sectionThe stator is separated intofive sections Each section forms one phase and a magneticseparation of a nonferromagnetic ringmade of stainless steelwhich is used as a flux barrier The mover is assembled frompermanent magnets (PMs) and ferromagnetic rings the PMsare characterized by axial magnetization and are situatedalternately with the ferromagnetic rings Table 1 gives detailsof the machine specifications

22 Modified Winding Function on the Motor MWFA wasfirst proposed for solving air-gap eccentricity in rotarymachines [17] The modified winding function (MWF) wasdeduced from the MMF drops in a magnetic circuit betweenthe stator and the rotor which was obtained under thefollowing assumptions the iron in the stator and mover hasinfinite permeability the magnetic saturation is neglectedthe mover length is infinite so that the structure of the motoris symmetrical there is no leakage flux in the shaft radial 119877119894and the magnetic permeability of the PMs is deemed as themagnetic permeability of air 1205830 It is defined as follows

119872(120593 120579) = 119899 (120593 120579) minus int21205870

119899 (120593 120579) 119892minus1 (120593 120579) 1198891205932120587 ⟨119892minus1 (120593 120579)⟩ (1)

where 120593 and 120579 represent the position of a stationary coil andangular position of the rotor with respect to stator respec-tively 119872(120593 120579) 119899(120593 120579) and 119892minus1(120593 120579) represent the modifiedwinding function turns function and air-gap functionrespectively The symbol ⟨sdot⟩ represents the average value ofldquosdotrdquo If the rotor is not eccentric (1) reduces to [17]

119872(120593 120579) = 119899 (120593 120579) minus ⟨119899⟩ (2)

where ⟨119899⟩ is the dc value of the turns function of the winding


ds

0 x

N

minusN

wc2

wc2

120576Na(x)

5Tt

(a)

a

b

cd

e120572

(b)



119876119904119879119905 = 2119901120591 (3)


119873119886 (119909)= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (4)


x

C

g

B D EAD

r

0120591 120591m 120591p

ds

wss

wtTt

bo

Di

120579



119872119894 (119909) = 4119873119901120587 sum119896=135


(5)



1198610 (119909 120579) = 1205830119892minus1 (119909 120579)1198720 (119909) 119894119886 (6)




2p120591

Air gap

Ro

R

Rse



119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(119899120587120591 (119909 minus 120579 minus 120576)) (7)


119879119905+120579+120576


+ 225119879119905 int120579+120576


(8)





119871 119894119894 (120579) = 2120587119877se1205830 int21199011205910

1198722119894 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 0 1 2 3 4) 119871 119894119895 (120579) = 2120587119877se1205830 int2119901120591

0119872119894 (119909)119872119895 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 119895 119894 119895 = 0 1 2 3 4)

(9)



120595119898119894 (120579) = 21205871205830119877se int21199011205910

119872119894 (119909) 119861119892 (119909 120579) 119889119909 (10)





119881119904 = 119877119904119868119904 + 119889120595119904119889119905 (11)


120595119904 = 120595119898 + 119871 119904119868119904 (12)


119865119890 = 119898119889]119889119905 + 119863] + 119865119897 + 119865detent (13)


119865detent = 119865slot + 119865end (14)


119865 = minus120597119882120597120579 (15)


119882 = 121205830 int119881 1198611198922 (119909 120579) 119889119881 (16)


Slot

-effe

ct fo

rce (

N)


minus3

minus2

minus1

0

1

2

3


times10minus3

(a)


End-

effec

t for

ce (N

)



01020304050

(b)

Det

ent f

orce

(N)



minus60

minus40

minus20

0

20

40

60

(c)









31590e minus 005

47384e minus 00539487e minus 005

55280e minus 005

A (Wbm)


119879 (120579) = 25

[[[[[[[[[[[[[[[[


1radic21radic2

1radic21radic2

1radic2

]]]]]]]]]]]]]]]]

(17)



119881119889 = 119877119904119894119889 minus 120587120591 ]119871119902119894119902 + 119871119889 119889119894119889119889119905 119881119902 = 119877119904119894119902 + 120587120591 ] (119871119889119894119889 + 1205951015840119898) + 119871119902 119889119894119902119889119905 119865119890 = 52 120587120591 119901 [1205951015840119898 119894119902 + (119871119889 minus 119871119902) 119894119889119894119902]

119898119889]119889119905 = 119863] + 119865119897 + 119865detent minus 119865119890

(18)






119891 = ]2120591 (19)



Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000


Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-












5 Conclusions



Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract


LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ds

0 x

N

minusN

wc2

wc2

120576Na(x)

5Tt

(a)

a

b

cd

e120572

(b)



119876119904119879119905 = 2119901120591 (3)


119873119886 (119909)= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (4)


x

C

g

B D EAD

r

0120591 120591m 120591p

ds

wss

wtTt

bo

Di

120579



119872119894 (119909) = 4119873119901120587 sum119896=135


(5)



1198610 (119909 120579) = 1205830119892minus1 (119909 120579)1198720 (119909) 119894119886 (6)




2p120591

Air gap

Ro

R

Rse



119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(119899120587120591 (119909 minus 120579 minus 120576)) (7)


119879119905+120579+120576


+ 225119879119905 int120579+120576


(8)





119871 119894119894 (120579) = 2120587119877se1205830 int21199011205910

1198722119894 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 0 1 2 3 4) 119871 119894119895 (120579) = 2120587119877se1205830 int2119901120591

0119872119894 (119909)119872119895 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 119895 119894 119895 = 0 1 2 3 4)

(9)



120595119898119894 (120579) = 21205871205830119877se int21199011205910

119872119894 (119909) 119861119892 (119909 120579) 119889119909 (10)





119881119904 = 119877119904119868119904 + 119889120595119904119889119905 (11)


120595119904 = 120595119898 + 119871 119904119868119904 (12)


119865119890 = 119898119889]119889119905 + 119863] + 119865119897 + 119865detent (13)


119865detent = 119865slot + 119865end (14)


119865 = minus120597119882120597120579 (15)


119882 = 121205830 int119881 1198611198922 (119909 120579) 119889119881 (16)


Slot

-effe

ct fo

rce (

N)


minus3

minus2

minus1

0

1

2

3


times10minus3

(a)


End-

effec

t for

ce (N

)



01020304050

(b)

Det

ent f

orce

(N)



minus60

minus40

minus20

0

20

40

60

(c)









31590e minus 005

47384e minus 00539487e minus 005

55280e minus 005

A (Wbm)


119879 (120579) = 25

[[[[[[[[[[[[[[[[


1radic21radic2

1radic21radic2

1radic2

]]]]]]]]]]]]]]]]

(17)



119881119889 = 119877119904119894119889 minus 120587120591 ]119871119902119894119902 + 119871119889 119889119894119889119889119905 119881119902 = 119877119904119894119902 + 120587120591 ] (119871119889119894119889 + 1205951015840119898) + 119871119902 119889119894119902119889119905 119865119890 = 52 120587120591 119901 [1205951015840119898 119894119902 + (119871119889 minus 119871119902) 119894119889119894119902]

119898119889]119889119905 = 119863] + 119865119897 + 119865detent minus 119865119890

(18)






119891 = ]2120591 (19)



Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000


Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-












5 Conclusions



Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract


LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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)

1 2 3 4 50Time (s)



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002 004 006 008 010 012000Time (s)


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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2p120591

Air gap

Ro

R

Rse



119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(119899120587120591 (119909 minus 120579 minus 120576)) (7)


119879119905+120579+120576


+ 225119879119905 int120579+120576


(8)





119871 119894119894 (120579) = 2120587119877se1205830 int21199011205910

1198722119894 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 0 1 2 3 4) 119871 119894119895 (120579) = 2120587119877se1205830 int2119901120591

0119872119894 (119909)119872119895 (119909) 119892minus1 (119909 120579) 119889119909

(119894 = 119895 119894 119895 = 0 1 2 3 4)

(9)



120595119898119894 (120579) = 21205871205830119877se int21199011205910

119872119894 (119909) 119861119892 (119909 120579) 119889119909 (10)





119881119904 = 119877119904119868119904 + 119889120595119904119889119905 (11)


120595119904 = 120595119898 + 119871 119904119868119904 (12)


119865119890 = 119898119889]119889119905 + 119863] + 119865119897 + 119865detent (13)


119865detent = 119865slot + 119865end (14)


119865 = minus120597119882120597120579 (15)


119882 = 121205830 int119881 1198611198922 (119909 120579) 119889119881 (16)


Slot

-effe

ct fo

rce (

N)


minus3

minus2

minus1

0

1

2

3


times10minus3

(a)


End-

effec

t for

ce (N

)



01020304050

(b)

Det

ent f

orce

(N)



minus60

minus40

minus20

0

20

40

60

(c)









31590e minus 005

47384e minus 00539487e minus 005

55280e minus 005

A (Wbm)


119879 (120579) = 25

[[[[[[[[[[[[[[[[


1radic21radic2

1radic21radic2

1radic2

]]]]]]]]]]]]]]]]

(17)



119881119889 = 119877119904119894119889 minus 120587120591 ]119871119902119894119902 + 119871119889 119889119894119889119889119905 119881119902 = 119877119904119894119902 + 120587120591 ] (119871119889119894119889 + 1205951015840119898) + 119871119902 119889119894119902119889119905 119865119890 = 52 120587120591 119901 [1205951015840119898 119894119902 + (119871119889 minus 119871119902) 119894119889119894119902]

119898119889]119889119905 = 119863] + 119865119897 + 119865detent minus 119865119890

(18)






119891 = ]2120591 (19)



Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000


Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-












5 Conclusions



Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract


LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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(mm

)



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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Slot

-effe

ct fo

rce (

N)


minus3

minus2

minus1

0

1

2

3


times10minus3

(a)


End-

effec

t for

ce (N

)



01020304050

(b)

Det

ent f

orce

(N)



minus60

minus40

minus20

0

20

40

60

(c)









31590e minus 005

47384e minus 00539487e minus 005

55280e minus 005

A (Wbm)


119879 (120579) = 25

[[[[[[[[[[[[[[[[


1radic21radic2

1radic21radic2

1radic2

]]]]]]]]]]]]]]]]

(17)



119881119889 = 119877119904119894119889 minus 120587120591 ]119871119902119894119902 + 119871119889 119889119894119889119889119905 119881119902 = 119877119904119894119902 + 120587120591 ] (119871119889119894119889 + 1205951015840119898) + 119871119902 119889119894119902119889119905 119865119890 = 52 120587120591 119901 [1205951015840119898 119894119902 + (119871119889 minus 119871119902) 119894119889119894119902]

119898119889]119889119905 = 119863] + 119865119897 + 119865detent minus 119865119890

(18)






119891 = ]2120591 (19)



Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000


Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-












5 Conclusions



Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract


LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











31590e minus 005

47384e minus 00539487e minus 005

55280e minus 005

A (Wbm)


119879 (120579) = 25

[[[[[[[[[[[[[[[[


1radic21radic2

1radic21radic2

1radic2

]]]]]]]]]]]]]]]]

(17)



119881119889 = 119877119904119894119889 minus 120587120591 ]119871119902119894119902 + 119871119889 119889119894119889119889119905 119881119902 = 119877119904119894119902 + 120587120591 ] (119871119889119894119889 + 1205951015840119898) + 119871119902 119889119894119902119889119905 119865119890 = 52 120587120591 119901 [1205951015840119898 119894119902 + (119871119889 minus 119871119902) 119894119889119894119902]

119898119889]119889119905 = 119863] + 119865119897 + 119865detent minus 119865119890

(18)






119891 = ]2120591 (19)



Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000


Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-












5 Conclusions



Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract


LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Current A

Current B

Current C

Current D

Current E

Theta

0 Velocity

k = 1000


Position

Mover velocity

Phase a

Phase b

Phase c

Phase d

Phase e

Theta

Fload

Fload

FeFe

ia

ib

ic

id

ie

k-












5 Conclusions



Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

Product

Add

Subtract


LdLd

Lq

Lq

Fe

Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Phase a

Phase b

Phase c

Phase d

Phase e

Theta

1

2

3

4

5

6

+

+

minus

+

k- 1

Scope

Product 1

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Add

Subtract


LdLd

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Fm

Fm

id

iq

times

times

(a)

1

2

f(u)

+

minus

minus

3e3

+

minus

1s 1s 1

2

Theta

Velocity

Subtract

ProductSubtract

92

DivideIntegrator

Integrator 1D

divideFload

Ftotal

Fdetent

Fe times

times

(b)



Appendix



119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576)) (A1)



119886119896 = 25119879119905 (int1199081198882

0119873 cos(21199011198961205875119879119905 119909)119889119909

+ int51198791199055119879119905minus1199081198882


(A2)


119873119886 (119909) = sum119896=135

119886119896 cos(21199011198961205875119879119905 (119909 minus 120576))= 4119873119901120587 sum

119896=135

1119896 sin(1199081198881198961205872120591 ) cos(119896120587120591 (119909 minus 120576)) (A3)




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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1 2 3 4 50Time (s)

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119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119910119908119896 = 119910119910119896 sdot 119910119889119896 (A4)

where


(A5)

where

120579119889 = 120587120591 abs (120591 minus (119879119905 minus 119889119904)) (A6)



119872119894 (119909) = 119873119894 (119909) sdot 119910wk = 4119873119901120587sdot sum119896=135





119861119892 = 1198671198881205911198982 12058301198921015840 = 1198671198881205911198982 1205830119892 + 120591119898120591119901119863120583119903 (1198632 minus 1198632119894 ) (B1)


119892 (119909 120579) asymp


(B2)


119892minus1 (119909 120579) = infinsum119899=0

119866119899 cos(21199011198991205875119879119905 (119909 minus 120579 minus 120576)) (B3)


119866119899 = 225119879119905sdot int120579+120576


+ 425119879119905sdot int119879119905+120579+120576


(B4)




120582ag (119909 120579) = 1198921015840 sdot 119892minus1 (119909 120579) (C1)




119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119861119892 (119909 120579) = 120582ag (119909 120579) 119861pm-slotless (119909 120579) (C2)


119861pm-slotless (119909 120579) = 4119861119892120587 sum119899=135


(C3)


119870119862 = 119879119905119879119905 minus 1205741198921015840 (C4)


120574 = 4120587


(C5)


119892119890 = 119892 + (119870119862 minus 1) 1198921015840119877se = 119877119900 + 119892119890 (C6)



119882slot = 121205830 int119877119900+119892

119877119900

2120587119903 119889119903sdot sum119894=01234

(int1198891199042+119908119905+1198941198791199051198891199042+119894119879119905


)sdot (119861119892 (119909 120579))2 119889119909

(D1)



119882end = 121205830 int119877119900+119870119874119892

119877119900

2120587119903 119889119903(int1198891199042minusinfin

+intinfin1198891199042+119871


(D2)


120582end (119909) =


(D3)






(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(D4)



119879minus1 (120579) = 52119879119879 (120579) (E1)



119879 (120579)119881119904 = 119879 (120579) 119877119904119894119904 + 119879 (120579) 119889Λ 119904119889119905= 119879 (120579) 119877119904119879minus1 (120579) 119879 (120579) 119894119904 + 119889119889119905 [119879 (120579) Λ 119904]

minus 119889119879 (120579)119889119905 119879minus1 (120579) [119879 (120579) Λ 119904] (E2)

where

119879 (120579) Λ 119904 = 119879 (120579) 119871 ss119879minus1 (120579) 119879 (120579) 119894119904 + 119879 (120579) 120595119898= Λ 119889119902119904 (E3)


119879 (120579) 119871 ss119879minus1 (120579) =[[[[[[[[[

119871119889 119871119902 00

0

]]]]]]]]] (E4)

where

119871119889 = 521205830119877se1198722119901120591 (21198660 + 1198662) 119871119902 = 521205830119877se1198722119901120591 (21198660 minus 1198662)

(E5)


119879 (120579) 120595119898 = 1205951015840119898 [1 0 0 0 0]119879 (E6)


1205951015840119898 = 1198721198611119901120591120587119877se1198921015840 (21198660 + 1198662) (E7)


119872 = 41198731199101199081119901120587 sin(1199081198881205872120591 ) (E8)

Competing Interests


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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