Analytical Modeling of Cogging Torque in Closed-Slot PMSMs Incorporating Slot-Bridge Saturation Effects

Analytical Modeling of Cogging Torque in Closed-Slot PMSMs Incorporating Slot-Bridge Saturation Effects | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Analytical Modeling of Cogging Torque in Closed-Slot PMSMs Incorporating Slot-Bridge Saturation Effects Gaoyuan Fan, Luyao Wang, Tianxiang Zhu, Xiaohua Bao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9404011/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract The closed-slot stator structure is an effective way of suppressing cogging torque in permanent magnet synchronous machines (PMSMs). However, a closed-form analytical relationship between the slot bridge thickness and the resulting cogging torque has not yet been established. To address this issue, this paper derives, for the first time, a closed-slot structure analytical expression that relates cogging torque to the closed-slot bridge thickness while accounting for localized magnetic saturation. The cogging torque under linear conditions is first obtained using the energy method in conjunction with the closed-slot geometry. To improve computational accuracy, an iterative nonlinear magnetic circuit model(NMCM) is then introduced. This method iteratively updates the bridge permeance based on the material's actual B – H characteristics, thereby accurately incorporating the saturation-dependent permeability variation into the analytical model. The analysis reveals distinct physical boundaries for the suppression efficacy of the slot bridge. When the bridge thickness is below the critical saturation threshold, the permeability collapses to near unity. The bridge then behaves magnetically as an extended air gap, and the suppression effect is lost. When the thickness exceeds the optimal suppression range, excessive flux shunting reduces the main magnetic flux. Based on this analytical relationship, a predictive criterion for optimal bridge thickness selection is proposed, significantly reducing the need for extensive parametric finite element analysis (FEA). A 30-pole, 90-slot submersible PMSM is taken as a case study. The analytically predicted optimal thickness range (0.3–0.6 mm) agrees well with FEA. Additionally, the no-load test conducted on the closed-slot prototype also verified the accuracy of the NMCM. The analytical framework established in this paper provides a generalized theoretical foundation for the direct design of low-torque-ripple PMSMs with closed slots. closed-slot structure permanent magnet synchronous motors cogging torque suppression nonlinear magnetic circuit method (NMCM) Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 17 Apr, 2026 Editor assigned by journal 16 Apr, 2026 Submission checks completed at journal 14 Apr, 2026 First submitted to journal 13 Apr, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9404011","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":625215350,"identity":"2110e9a0-a9fe-4160-916e-5fbf0462e309","order_by":0,"name":"Gaoyuan Fan","email":"","orcid":"","institution":"Hefei University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Gaoyuan","middleName":"","lastName":"Fan","suffix":""},{"id":625215351,"identity":"2cacf237-3788-490c-b568-6243bb90ff91","order_by":1,"name":"Luyao Wang","email":"","orcid":"","institution":"Hefei University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Luyao","middleName":"","lastName":"Wang","suffix":""},{"id":625215352,"identity":"3229a363-90e4-49c2-a518-fabfe2285c16","order_by":2,"name":"Tianxiang Zhu","email":"","orcid":"","institution":"Hefei University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Tianxiang","middleName":"","lastName":"Zhu","suffix":""},{"id":625215354,"identity":"b82d8968-3a84-40bd-b182-6b0a4fadbec6","order_by":3,"name":"Xiaohua Bao","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwUlEQVRIiWNgGAWjYBACAyBmZqg4wA/m8RCv5cwByQY2krQwtpGixZz98DHpwnl3JAzuNzA+eNvGIG9OSItlT1qy8cxtzyQMjjEwG85tYzDc2UDIYQdyDB/zbjtcB9TCJs3bxpBgcICQlvNvDA7zzjkMsoX9N3FaboBsaQBrYWMmSovljGfJxjzHDktIHktslpxzTsJwAyEt5vzJx6R5ag5L8B0+fPDDmzIbeYK2IAHGBiAhQbz6UTAKRsEoGAW4AQCH/D7nU9TARQAAAABJRU5ErkJggg==","orcid":"","institution":"Hefei University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Xiaohua","middleName":"","lastName":"Bao","suffix":""}],"badges":[],"createdAt":"2026-04-13 12:23:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9404011/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9404011/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109068279,"identity":"240ed281-ae37-412e-9705-0e4d12d5c250","added_by":"auto","created_at":"2026-05-12 10:05:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1471521,"visible":true,"origin":"","legend":"","description":"","filename":"AnalyticalModelingofCoggingTorqueinClosedSlotPMSMsIncorporatingSlotBridgeSaturationEffects.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9404011/v1_covered_e74ba2c9-c4d7-4aed-bea5-1bf028dbc652.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Analytical Modeling of Cogging Torque in Closed-Slot PMSMs Incorporating Slot-Bridge Saturation Effects","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"electrical-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"elen","sideBox":"Learn more about [Electrical Engineering](http://link.springer.com/journal/202)","snPcode":"202","submissionUrl":"https://submission.nature.com/new-submission/202/3","title":"Electrical Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"closed-slot structure, permanent magnet synchronous motors, cogging torque suppression, nonlinear magnetic circuit method (NMCM)","lastPublishedDoi":"10.21203/rs.3.rs-9404011/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9404011/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe closed-slot stator structure is an effective way of suppressing cogging torque in permanent magnet synchronous machines (PMSMs). However, a closed-form analytical relationship between the slot bridge thickness and the resulting cogging torque has not yet been established. To address this issue, this paper derives, for the first time, a closed-slot structure analytical expression that relates cogging torque to the closed-slot bridge thickness while accounting for localized magnetic saturation. The cogging torque under linear conditions is first obtained using the energy method in conjunction with the closed-slot geometry. To improve computational accuracy, an iterative nonlinear magnetic circuit model(NMCM) is then introduced. This method iteratively updates the bridge permeance based on the material's actual \u003cem\u003eB\u003c/em\u003e\u0026ndash;\u003cem\u003eH\u003c/em\u003e characteristics, thereby accurately incorporating the saturation-dependent permeability variation into the analytical model. The analysis reveals distinct physical boundaries for the suppression efficacy of the slot bridge. When the bridge thickness is below the critical saturation threshold, the permeability collapses to near unity. The bridge then behaves magnetically as an extended air gap, and the suppression effect is lost. When the thickness exceeds the optimal suppression range, excessive flux shunting reduces the main magnetic flux. Based on this analytical relationship, a predictive criterion for optimal bridge thickness selection is proposed, significantly reducing the need for extensive parametric finite element analysis (FEA). A 30-pole, 90-slot submersible PMSM is taken as a case study. The analytically predicted optimal thickness range (0.3\u0026ndash;0.6 mm) agrees well with FEA. Additionally, the no-load test conducted on the closed-slot prototype also verified the accuracy of the NMCM. The analytical framework established in this paper provides a generalized theoretical foundation for the direct design of low-torque-ripple PMSMs with closed slots.\u003c/p\u003e","manuscriptTitle":"Analytical Modeling of Cogging Torque in Closed-Slot PMSMs Incorporating Slot-Bridge Saturation Effects","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-12 06:01:51","doi":"10.21203/rs.3.rs-9404011/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-17T22:33:51+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-16T05:52:29+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-14T10:33:59+00:00","index":"","fulltext":""},{"type":"submitted","content":"Electrical Engineering","date":"2026-04-13T12:10:26+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"electrical-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"elen","sideBox":"Learn more about [Electrical Engineering](http://link.springer.com/journal/202)","snPcode":"202","submissionUrl":"https://submission.nature.com/new-submission/202/3","title":"Electrical Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"52473b3a-c468-4727-8f2b-9036bc9d47f0","owner":[],"postedDate":"May 12th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-12T06:01:52+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-12 06:01:51","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9404011","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9404011","identity":"rs-9404011","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}