Utafiti huu unatengeneza hybrid analytical model ili kutabiri electromagnetic performance ya high-torque-density three-segment Halbach-array permanent-magnet electric machine bila kulazimika kutumia detailed finite-element analysis yenye muda mrefu wa computation katika kila design step. Model inaunganisha Subdomain Method (SM) na Magnetic Equivalent Circuit (MEC), na pamoja na special magnetization direction ya Halbach magnets, inazingatia iron-core nonlinearity na axial end leakage flux. Analytical model ililinganishwa na 2B na 3B finite-element method, kisha physical prototype ya 15 kW yenye three-segment Halbach rotor ikatengenezwa na kutestiwa experimentally. Finding muhimu zaidi ya study ni kwamba torque capability katika high current inaweza kuoverestimated sana ikiwa end leakage flux itaachwa: source inatoa maximum torque ya 353 na 351 N·m mtawalia kwa 2B FEM na analytical models zisizojumuisha end leakage flux, huku 262 na 275 N·m zikipatikana katika 3B FEM na hybrid analytical model zinazojumuisha end leakage flux.

Hybrid model haijilengi kwenye torque estimation pekee. Radial na tangential magnetic-flux densities katika air gap, no-load back electromotive force (back-EMF), cogging torque na electromagnetic torque under load zinahesabiwa kutoka mathematical framework ileile. Katika comparison ya nominal operating condition, average torque ilikuwa karibu 101 N·m katika 3B FEM na karibu 100 N·m katika analytical model, na error iliripotiwa kuwa chini ya %2.

Experimental validation ilitumia 15 kW prototype yenye three-segment Halbach rotor. Katika no-load test ya 2400 r/min, measured line back-EMF RMS value iliripotiwa kuwa karibu 213 V na kuonekana kuwa consistent na analytical/FEM results. Torque-current experiments pia zilionyesha close agreement kati ya hybrid model na physical prototype. Hata hivyo, source inaeleza wazi kwamba experimental back-EMF amplitude ina deviations zinazotokana na differences kati ya ideal model assumptions na real machine.

Kwa mtazamo wa Uturuki, study haikufanywa directly katika electric-motor production line nchini Uturuki wala kwenye local vehicle-drive system. Kwa transfer kwenda electric-vehicle, aviation, defense au high-power-density motor-development work nchini Uturuki, local validation inahitajika kwa magnetic materials zilizotumiwa, lamination characteristics, magnet tolerances, cooling structure, manufacturing tolerances na motor geometry. Results za study haziwezi kuhamishwa nje ya specific prototype geometry kama direct performance guarantee.

Main problem ya research ni nini?

Halbach-array permanent-magnet machines ni special surface-magnet machine structures ambapo magnets kwenye rotor surface hupangwa kwa different magnetization directions. Katika study hii, purpose ya rotor structure hii ni kufanya magnetic-field distribution katika air gap iwe suitable kwa high torque density.

Hata hivyo, magnetization geometry ya Halbach array ni complex zaidi kuliko ordinary surface-magnet rotor structure. Kulingana na researchers, kujenga detailed finite-element model na recalculating kila design change kwa FEM husababisha long computational cycle, hasa katika optimization studies.

Ingawa analytical subdomain method inatoa faster alternative, traditional linear models zina important problem: iron core inapokaribia saturation katika high flux density na magnetic leakage flux kutoka axial rotor ends, effects hizi zinaweza zisiwe represented adequately katika simplified models. Hybrid SM–MEC approach ya study inalenga exactly problem hii.

Physical structure ya motor iliyochunguzwa ni ipi?

Kila Halbach pole kwenye rotor imeundwa na magnet segments tatu: main pole katikati na auxiliary poles mbili pembeni. Angle kati ya magnetization directions za main na auxiliary poles inaonyeshwa kwa \(\theta_m\). Ili kurahisisha prototype manufacturing, study ilichagua:

\[ \theta_m = 90^\circ \]

.

Stator ina fractional-slot concentrated-winding structure. Study inaeleza kwamba structure hii inaweza kupunguza winding-end length na kutoa advantage kwa efficiency.

Main geometric na electrical parameters za prototype machine ni:

Subdomain model inajengwaje?

Researchers waligawanya machine katika five separate analytical regions kulingana na magnetic medium na excitation source:

1. Halbach-array permanent-magnet region
2. Air gap
3. Stator tooth tip
4. Stator tooth
5. Armature winding

Cross-section katika Figure 2 ya study inaonyesha regions hizi tano kwenye motor geometry ileile. Hivyo vector magnetic potential ya kila region hutatuliwa kwa boundary conditions zake, kisha total solution hupatikana kwa kutumia magnetic-field continuity kati ya regions.

Main assumptions za model ni zipi?

Five main assumptions zilitumiwa kuanzisha initial subdomain solution:

• Electrical conductivities za machine parts zilipuuzwa.
• Magnetic permeability ya core material ilichukuliwa kuwa infinite katika initial subdomain model.
• Permanent-magnet material ilimodeliwa kwa linear demagnetization characteristic.
• Halbach magnets ziliwakilishwa kama equivalent arc-shaped magnets.
• Current density ilichukuliwa kuwa na component katika z-axis direction pekee.

Accuracy problem inayosababishwa na second assumption katika high-load conditions inapunguzwa baadaye kwa kutumia magnetic equivalent circuit kurudisha iron-core magnetic reluctance na saturation effect ndani ya model.

Angular positions za stator slots zinafafanuliwaje?

Position ya i-th stator-slot opening imefafanuliwa kwa:

\[ \alpha_i = \frac{2\pi i}{Q_s} - \frac{\pi}{Q_s}, \qquad 1\leq i\leq Q_s \]

. Hapa \(Q_s\) ni number ya stator slots na ni 18 katika motor iliyochunguzwa.

Position ya k-th main N-pole magnet kwenye rotor ni:

\[ \gamma_k = \frac{2\pi(k-1)}{p}+\delta, \qquad 1\leq k\leq p \]

. \(p\) ni number ya pole pairs, na \(\delta\) inawakilisha angle kati ya first N-pole magnet na first stator-slot opening.

Field inayotengenezwa na current imemodeliwaje?

Distribution ya vector magnetic potential katika stator slots imeelezwa kwa Poisson equation:

\[ \frac{\partial^2 A_{5j}}{\partial r^2} + \frac{1}{r}\frac{\partial A_{5j}}{\partial r} + \frac{1}{r^2}\frac{\partial^2 A_{5j}}{\partial\theta^2} = -\mu_{\mathrm{copper}}J_{jc} \]

. Hapa \(A_{5j}\) ni vector magnetic potential katika j-th stator slot, \(J_{jc}\) ni current density, \(r\) radial coordinate na \(\theta\) angular coordinate.

Periodic structure ya winding current density iligawanywa kwa Fourier series. Approach hii inaruhusu field distribution ndani ya slot kuwakilishwa kwa harmonic components. Study inaeleza kwamba higher number ya harmonics inaweza kuongeza accuracy lakini pia huongeza number ya unknown coefficients na computational burden; kwa hiyo maximum harmonic order ilichaguliwa ili kusawazisha computation na accuracy.

Three-segment Halbach magnetization inaingiajije kwenye model?

Magnetization vector katika Halbach array iligawanywa katika radial \(M_r\) na tangential \(M_\theta\) components katika two-dimensional polar coordinates. Different magnetization directions za main pole na auxiliary poles mbili zilifafanuliwa kwa piecewise functions zinazotegemea rotor angle.

Piecewise magnetization distributions hizi kisha ziligawanywa kwa Fourier transform kuwa:

\[ M_r(\theta)= \sum_{n=-N}^{N} \hat{M}_{r,n}e^{-jn\theta} \]

na

\[ M_\theta(\theta)= \sum_{n=-N}^{N} \hat{M}_{\theta,n}e^{-jn\theta} \]

. Hivyo field excitation ya magnet segments tatu tofauti inaingizwa mathematically kwenye subdomain solution.

Air-gap field inapatikanaje?

Katika air gap isiyo na external field source, homogeneous Laplace equation hutatuliwa. General form ya solution ni:

\[ A_{2j} = \sum_{n=1}^{\infty} \left[ a_{2n} \left(\frac{r}{R_s}\right)^n + b_{2n} \left(\frac{r}{R_r}\right)^{-n} \right]\cos(n\theta) \]

\[ + \sum_{n=1}^{\infty} \left[ c_{2n} \left(\frac{r}{R_s}\right)^n + d_{2n} \left(\frac{r}{R_r}\right)^{-n} \right]\sin(n\theta) \]

. Hapa \(R_s\) ni stator inner radius, \(R_r\) rotor outer radius; \(a_{2n}\), \(b_{2n}\), \(c_{2n}\) na \(d_{2n}\) ni coefficients zinazopatikana kutoka boundary conditions.

Kwa nini end leakage flux ni muhimu sana?

Main scientific point ya study inaonekana hapa. Two-dimensional magnetic model haiwezi naturally kuwakilisha kikamilifu magnetic-flux paths zinazofunga kupitia axial ends za rotor. Load inapoongezeka, magnetic reluctance ya main magnetic circuit huongezeka na influence ya leakage flux kwenye torque prediction huwa more important.

FEM field lines katika Figure 3 ya study zinaonyesha both main flux na flux inayovuja kupitia rotor-end region. Katika sehemu ya pili ya figure hiyo, physical paths hizi hubadilishwa kuwa equivalent circuit ya magnetic reluctances.

Kwa rotor axial leakage permeance, source inatumia:

\[ P_{sen}=0.264\mu_0H_{sn} \]

. Stator-tooth axial leakage permeance pia imefafanuliwa kwa separate relation inayojumuisha geometric dimensions.

Katika hybrid approach, stator-tooth magnetic reluctance inaunganishwa na air-gap magnetic reluctance na equivalent air-gap permeability kwa kila stator tooth huhesabiwa kama:

\[ \mu_{eqj} = \frac{\delta}{W_tL_{ef}} \frac{1}{R_{agj}+R_{tj}} \]

. Hapa \(\delta\) ni air-gap length, \(W_t\) stator-tooth width, \(L_{ef}\) effective axial length, \(R_{agj}\) air-gap magnetic reluctance na \(R_{tj}\) stator-tooth magnetic reluctance.

SM na MEC zinaunganishwaje?

Air-gap flux density inayohesabiwa kutoka subdomain model ina-integrated kando ya stator inner circumference ili kubaini magnetic flux inayoingia katika kila stator tooth. Fluxes hizi kisha hutumiwa kama excitation sources za magnetic equivalent circuit.

MEC solution imeandikwa katika nodal-matrix form inayotegemea Kirchhoff laws:

\[ G\,U=\Phi \]

. \(G\) ni magnetic conductance matrix, \(U\) node magnetic potentials, na \(\Phi\) magnetic-flux excitations. Mutual data transfer hii inaruhusu analytical air-gap solution kusahihishwa kulingana na iron-core saturation na end-leakage effect.

Electromagnetic outputs zinahesabiwaje?

Radial na tangential air-gap flux density hupatikana kutoka vector magnetic potential:

\[ B_r = \frac{1}{r} \frac{\partial A}{\partial\theta} \]

\[ B_\theta = - \frac{\partial A}{\partial r} \]

. Field components hizi mbili kisha hutumika katika back-EMF na torque calculations.

No-load cogging torque imehesabiwa kwa Maxwell stress tensor method:

\[ T_{ct} = \frac{L_{ef}r^2}{\mu_{air}} \int_0^{2\pi}B_rB_\theta\,d\theta \]

.

Electromagnetic torque under load hupatikana kutoka electromotive forces na currents za three phases:

\[ T_e = \frac{E_AI_A+E_BI_B+E_CI_C}{\Omega} \]

. \(E_A\), \(E_B\), \(E_C\) ni phase back-EMF values; \(I_A\), \(I_B\), \(I_C\) phase currents na \(\Omega\) mechanical angular speed.

No-load FEM comparison ilionyesha nini?

Katika two-dimensional FEM, maximum flux density katika stator core ilionekana kuwa karibu 2,1 T; source ilitathmini hii kuwa chini ya saturation flux density ya core iliyotumiwa.

Radial na tangential air-gap flux-density curves zilionyesha waveforms zilizo karibu sana kati ya analytical model na FEM. Amplitude ya tangential component ni distinctly smaller kuliko radial component na ina higher harmonic content.

Baadhi ya local dips katika radial flux density zilihusishwa na angular regions ambapo tangentially magnetized segments za Halbach array zinalingana na stator slots.

Source-internal inconsistency: Text inatoa maximum radial air-gap flux density ya FEM model kuwa karibu 1,15 T na analytical model karibu 1,13 T; lakini katika sentence ileile inasema FEM value ni lower kuliko analytical estimate. 1,15 T ni numerically greater kuliko 1,13 T. Statement hii imehifadhiwa bila silent correction.

Model ilikaribia kwa kiasi gani katika back-EMF na cogging torque?

Katika no-load back-EMF curve, FEM na analytical solution zinaenda karibu juu ya nyingine. Cogging-torque waveform pia ni similar, ingawa analytical model torque amplitude imebaki kidogo lower kuliko FEM result.

Researchers wanahusisha difference hii na accumulation ya small calculation differences katika air-gap flux density ndani ya Maxwell stress tensor integral, pamoja na calculation differences zinazotokana na FEM mesh structure.

Kuacha end leakage flux kunabadilisha vipi torque prediction?

Most prominent comparison ya study ni load torque-current analysis. Two-dimensional approaches ambazo hazizingatii end leakage flux zinapredict maximum torque karibu 350 N·m, huku 3B FEM na hybrid-model results zinazozingatia end leakage flux zikishuka hadi karibu 260–275 N·m.

Verianla Live: Scientific source-of-truth ni visible HTML table hapo juu. Interactive graph hutengenezwa kutoka table hii na site-level Verianla Live v0.2 engine.

Study inaeleza difference kati ya 2B FEM prediction ya 353 N·m na 3B FEM result ya 262 N·m inayojumuisha end leakage flux kuwa karibu %25,7. Comparison hii ni main evidence ya study kwamba end leakage flux haiwezi kupuuzwa katika torque prediction ya short-axial high-torque-density machines.

Torque agreement katika nominal operating point

Katika torque waveforms under load, 3B FEM ilitoa average torque ya karibu 101 N·m, huku hybrid analytical model inayojumuisha end leakage flux ikitoa karibu 100 N·m. Source inaeleza error katika comparison hii kuwa chini ya %2.

Result hii ni important kwa sababu inaonyesha kwamba hybrid model haibaki close na 3B FEM katika no-load field distribution pekee, bali pia katika load condition ambapo current inatumika na magnetic saturation/leakage-flux effects huwa more pronounced.

Physical prototype ilivalidated vipi?

Katika final stage ya study, 15 kW three-segment Halbach-rotor PM machine inayolingana na theoretical motor geometry ilitengenezwa. Carbon-fiber sleeve ilitumika kuhakikisha mechanical safety ya permanent magnets kwenye rotor.

Experimental platform ina prototype motor, drive motor, oscilloscope na torque sensor. Source inaonyesha photos za rotor, stator na completed test platform pamoja; hivyo validation haikutegemea FEM comparison pekee.

No-load prototype experiment ilionyesha nini?

Prototype iliendeshwa no-load kwa 2400 r/min na line back-EMF waveform ikapimwa. Measured signal ilionyesha approximately sinusoidal structure, na effective back-EMF iliripotiwa kuwa karibu 213 V.

Result hii ilikuwa in good agreement na FEM na analytical calculations. Hata hivyo, researchers wanaeleza kwamba remaining deviation kati ya experimental back-EMF amplitude na analytical model inatokana na unavoidable differences kati ya model assumptions na real physical machine.

Torque-current prototype experiment ilionyesha nini?

Torque sensor iliunganishwa kati ya prototype na drive motor, na electromagnetic torque dhidi ya phase current ikapimwa kwa 2400 r/min. Figure 10 ya study inalinganisha experimental points, 3B FEM result na hybrid analytical model kwenye graph ileile.

Trends za data sets zote tatu ziko close. Researchers wanahusisha agreement hii na model kuzingatia both core saturation na axial end leakage flux.

Phase-current waveform iliyopimwa katika 3000 r/min operating condition pia imewasilishwa separately kwa oscilloscope image.

Matokeo yanayoungwa mkono na utafiti

• Special magnetization ya three-segment Halbach rotor inaweza kumodeliwa analytically kwa subdomain method.
• Kuunganisha SM na magnetic equivalent circuit kunaruhusu iron-core nonlinearity na axial end leakage flux kuingizwa katika analytical prediction.
• Kuacha end leakage flux kulisababisha torque capability katika high current kuoverestimated kwa kiasi kikubwa kwa high-torque-density motor iliyochunguzwa.
• Average torque prediction under load ya hybrid analytical model inayojumuisha end leakage flux ilikaribia 3B FEM kwa error chini ya %2 katika condition iliyochunguzwa.
• Back-EMF na torque-current experiments kwenye 15 kW physical prototype ziliunga mkono general trend ya analytical model.

Matokeo ambayo utafiti hauungi mkono au haujatesti

• Haijaonyeshwa kwamba model itatoa error limit ileile katika all Halbach motor geometries.
• Halbach segmentations nyingine zaidi ya three segments hazijalinganishwa na physical prototype.
• Magnetization angles nyingine zaidi ya \(\theta_m=90^\circ\) hazijachunguzwa experimentally kwenye prototype hii.
• Study haitoi long-term durability, thermal-life au magnet-demagnetization test.
• Hakuna field validation under real duty cycle katika electric vehicle au aviation system.
• Ingawa analytical method inaelezwa kuwa faster computationally kuliko FEM, source haitoi direct computation-time table au numerical acceleration factor.

Mbinu na Matokeo ya Utafiti

Research design

Study inatumia three-stage validation structure:

1. Kujenga hybrid analytical model inayotegemea subdomain na magnetic equivalent circuit,
2. Electromagnetic validation kwa 2B na 3B FEM results,
3. Experimental validation kwa real 15 kW Halbach motor prototype.

Physical effects zinazofunikwa na analytical model

No-load comparisons

Radial na tangential air-gap flux-density waveforms zilionyesha close agreement kati ya analytical model na 2B FEM. No-load back-EMF curves pia zilioverlap kwa kiasi kikubwa. Katika cogging torque, analytical model ilifuata waveform lakini ikapredict amplitude kidogo lower kuliko FEM.

Main numerical results under load

Experimental validation conditions

Accuracy statements zinapaswa kutafsiriwaje?

Abstract ya study inaeleza kwamba analytical-model error ni chini ya %2 kwa FEM comparisons. Hasa katika average torque comparison under load, 3B FEM inatoa karibu 101 N·m na hybrid analytical model karibu 100 N·m.

Katika conclusion section, however, validation statement ya “within %5 error” inatumika kwa broader validation framework ya prototype, 3B FEM na analytical model. Kwa hiyo, si sahihi kusummarize all results za article kwa single universal “%2 error” claim.

Main methodological limits

• Subdomain solution inaanza kwa idealized material assumptions.
• Saturation na leakage-flux effects zinaongezwa tena approximately analytically kupitia MEC.
• Harmonic order imewekewa limit kwa kusawazisha computational burden na accuracy.
• Experimental validation ilifanywa kwenye single 15 kW three-segment Halbach prototype.
• Detailed processing time au numerical speedup ratio dhidi ya FEM haijaripotiwa kwa analytical method.

Maelezo ya Chanzo na Mbinu

Jina kamili asilia la utafiti: Analytical Modeling and Analysis of High-Torque-Density Three-Segment Halbach Array PM Machine by Considering Leakage Flux

Waandishi: Jinlin Huang; Qingfeng Sun; Chen Wang.

Author order: Original order katika source imehifadhiwa.

Equal contribution/co-first author: Hakuna equal-contribution au co-first-author statement katika source.

Corresponding author: Jinlin Huang.

Taasisi: Electrical Engineering College, Anhui Technical College of Mechanical and Electrical Engineering, Wuhu, China; Key Laboratory of Electric Drive and Control of Anhui Province, Wuhu, China; School of Electrical Engineering, Anhui Polytechnic University, Wuhu, China; Wuhu Overseas Students’ Entrepreneurship Park, Wuhu, China.

Jarida: Machines.

Mchapishaji: MDPI, Basel, Switzerland.

Volume / article number: 14, 683.

DOI: 10.3390/machines14060683.

Article received date: 29 April 2026.

Revision: 3 June 2026.

Acceptance: 5 June 2026.

Publication date: 12 June 2026.

Aina ya chanzo: Peer-reviewed research article; includes analytical electromagnetic modeling, 2B/3B FEM validation na physical-prototype experiment.

Official publication link:MDPI official article page.

Leseni: Creative Commons Attribution (CC BY).

Funding: Natural Science Research Project of Anhui Province of P. R. China, 2024AH050219 na 2023AH052697; pia Key Laboratory of Electric Drive and Control of Anhui Province, Anhui Polytechnic University, DQKJ202507.

Data availability: Imeelezwa kwamba original contributions za study ziko ndani ya article na additional questions zinaweza kuelekezwa kwa corresponding author.

Conflict of interest: Waandishi walitangaza kwamba hakuna conflict of interest.

Author contributions: Conceptualization na methodology Jinlin Huang; formal analysis Qingfeng Sun; investigation Jinlin Huang; resources Jinlin Huang na Chen Wang; first draft Jinlin Huang; review na editing Jinlin Huang na Chen Wang; supervision Qingfeng Sun zimeripotiwa.

Source-internal inconsistency note: Katika explanation ya radial air-gap flux density, ingawa approximately 1,15 T imetolewa kwa FEM na approximately 1,13 T kwa analytical model, text inasema FEM value ni lower kuliko analytical value. Numbers zilizotolewa haziendani na statement hii na source haijasahihishwa kimya kimya.

Accuracy-scope note: Abstract ya study inaeleza error kuwa chini ya %2 kwa specific FEM comparisons, wakati conclusion section inatumia broader validation statement ya within %5 kwa 3B FEM na experimental validation. Statements hizi mbili zimehifadhiwa katika contexts zake.

Scientific-content boundary: Electromagnetic mechanisms, equations, model assumptions, torque values na experimental results katika Verianla text hii zinategemea study iliyochunguzwa. Hakuna new scientific finding kutoka external sources iliyoongezwa; external check ilitumika only kuthibitisha bibliographic publication identity.

Main methodological boundary: Study imethibitisha specific three-segment 15 kW Halbach PM motor topology kwa 2B/3B FEM na physical prototype. Haijaonyeshwa experimentally katika study hii kwamba same error level itadumu kwa other segment counts, different magnetization angles, different axial aspect ratios na other high-power-density motor geometries.