{"id":"https://openalex.org/W2977895693","doi":"https://doi.org/10.1109/tie.2019.2944055","title":"Torque Calculation of Permanent Magnet Spherical Motor Based on Virtual Work Method","display_name":"Torque Calculation of Permanent Magnet Spherical Motor Based on Virtual Work Method","publication_year":2019,"publication_date":"2019-10-02","ids":{"openalex":"https://openalex.org/W2977895693","doi":"https://doi.org/10.1109/tie.2019.2944055","mag":"2977895693"},"language":"en","primary_location":{"id":"doi:10.1109/tie.2019.2944055","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tie.2019.2944055","pdf_url":null,"source":{"id":"https://openalex.org/S58031724","display_name":"IEEE Transactions on Industrial Electronics","issn_l":"0278-0046","issn":["0278-0046","1557-9948"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Industrial Electronics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100615427","display_name":"Hongfeng Li","orcid":"https://orcid.org/0000-0002-7917-5182"},"institutions":[{"id":"https://openalex.org/I162868743","display_name":"Tianjin University","ror":"https://ror.org/012tb2g32","country_code":"CN","type":"education","lineage":["https://openalex.org/I162868743"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Hongfeng Li","raw_affiliation_strings":["School of Electrical and Information Engineering, Tianjin University, Tianjin, China"],"raw_orcid":"https://orcid.org/0000-0002-7917-5182","affiliations":[{"raw_affiliation_string":"School of Electrical and Information Engineering, Tianjin University, Tianjin, China","institution_ids":["https://openalex.org/I162868743"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100627032","display_name":"Zhao Yan-fen","orcid":"https://orcid.org/0000-0002-2193-1854"},"institutions":[{"id":"https://openalex.org/I162868743","display_name":"Tianjin University","ror":"https://ror.org/012tb2g32","country_code":"CN","type":"education","lineage":["https://openalex.org/I162868743"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yanfen Zhao","raw_affiliation_strings":["School of Electrical and Information Engineering, Tianjin University, Tianjin, China"],"raw_orcid":"https://orcid.org/0000-0002-2193-1854","affiliations":[{"raw_affiliation_string":"School of Electrical and Information Engineering, Tianjin University, Tianjin, China","institution_ids":["https://openalex.org/I162868743"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100365189","display_name":"Bin Li","orcid":"https://orcid.org/0000-0002-5430-8532"},"institutions":[{"id":"https://openalex.org/I162868743","display_name":"Tianjin University","ror":"https://ror.org/012tb2g32","country_code":"CN","type":"education","lineage":["https://openalex.org/I162868743"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Bin Li","raw_affiliation_strings":["School of Electrical and Information Engineering, Tianjin University, Tianjin, China"],"raw_orcid":"https://orcid.org/0000-0002-5430-8532","affiliations":[{"raw_affiliation_string":"School of Electrical and Information Engineering, Tianjin University, Tianjin, China","institution_ids":["https://openalex.org/I162868743"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038903047","display_name":"Guidan Li","orcid":"https://orcid.org/0000-0002-8979-4198"},"institutions":[{"id":"https://openalex.org/I162868743","display_name":"Tianjin University","ror":"https://ror.org/012tb2g32","country_code":"CN","type":"education","lineage":["https://openalex.org/I162868743"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Guidan Li","raw_affiliation_strings":["School of Electrical and Information Engineering, Tianjin University, Tianjin, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical and Information Engineering, Tianjin University, Tianjin, China","institution_ids":["https://openalex.org/I162868743"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102805478","display_name":"Lifeng Cui","orcid":null},"institutions":[{"id":"https://openalex.org/I162868743","display_name":"Tianjin University","ror":"https://ror.org/012tb2g32","country_code":"CN","type":"education","lineage":["https://openalex.org/I162868743"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lifeng Cui","raw_affiliation_strings":["School of Electrical and Information Engineering, Tianjin University, Tianjin, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical and Information Engineering, Tianjin University, Tianjin, China","institution_ids":["https://openalex.org/I162868743"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5100615427"],"corresponding_institution_ids":["https://openalex.org/I162868743"],"apc_list":null,"apc_paid":null,"fwci":1.0896,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.78519967,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"67","issue":"9","first_page":"7736","last_page":"7745"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10278","display_name":"Electric Motor Design and Analysis","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10278","display_name":"Electric Motor Design and Analysis","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11525","display_name":"Piezoelectric Actuators and Control","score":0.9670000076293945,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11602","display_name":"Magnetic Bearings and Levitation Dynamics","score":0.9663000106811523,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.6862924098968506},{"id":"https://openalex.org/keywords/direct-torque-control","display_name":"Direct torque control","score":0.6746305823326111},{"id":"https://openalex.org/keywords/rotor","display_name":"Rotor (electric)","score":0.6491799354553223},{"id":"https://openalex.org/keywords/torque","display_name":"Torque","score":0.6028292775154114},{"id":"https://openalex.org/keywords/virtual-work","display_name":"Virtual work","score":0.5421470403671265},{"id":"https://openalex.org/keywords/torque-motor","display_name":"Torque motor","score":0.48812517523765564},{"id":"https://openalex.org/keywords/stator","display_name":"Stator","score":0.4818747937679291},{"id":"https://openalex.org/keywords/magnet","display_name":"Magnet","score":0.4379580020904541},{"id":"https://openalex.org/keywords/vector-control","display_name":"Vector control","score":0.43421903252601624},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.42584073543548584},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3991740345954895},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.3444693088531494},{"id":"https://openalex.org/keywords/induction-motor","display_name":"Induction motor","score":0.34154754877090454},{"id":"https://openalex.org/keywords/voltage","display_name":"Voltage","score":0.16415733098983765},{"id":"https://openalex.org/keywords/mechanical-engineering","display_name":"Mechanical engineering","score":0.156174898147583},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.12172132730484009},{"id":"https://openalex.org/keywords/electrical-engineering","display_name":"Electrical engineering","score":0.08928453922271729}],"concepts":[{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.6862924098968506},{"id":"https://openalex.org/C109871850","wikidata":"https://www.wikidata.org/wiki/Q1227571","display_name":"Direct torque control","level":4,"score":0.6746305823326111},{"id":"https://openalex.org/C17281054","wikidata":"https://www.wikidata.org/wiki/Q193466","display_name":"Rotor (electric)","level":2,"score":0.6491799354553223},{"id":"https://openalex.org/C144171764","wikidata":"https://www.wikidata.org/wiki/Q48103","display_name":"Torque","level":2,"score":0.6028292775154114},{"id":"https://openalex.org/C136560751","wikidata":"https://www.wikidata.org/wiki/Q1144923","display_name":"Virtual work","level":3,"score":0.5421470403671265},{"id":"https://openalex.org/C204054776","wikidata":"https://www.wikidata.org/wiki/Q2444432","display_name":"Torque motor","level":5,"score":0.48812517523765564},{"id":"https://openalex.org/C2776529397","wikidata":"https://www.wikidata.org/wiki/Q190312","display_name":"Stator","level":2,"score":0.4818747937679291},{"id":"https://openalex.org/C16389437","wikidata":"https://www.wikidata.org/wiki/Q11421","display_name":"Magnet","level":2,"score":0.4379580020904541},{"id":"https://openalex.org/C152060993","wikidata":"https://www.wikidata.org/wiki/Q1759392","display_name":"Vector control","level":4,"score":0.43421903252601624},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.42584073543548584},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3991740345954895},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.3444693088531494},{"id":"https://openalex.org/C80962145","wikidata":"https://www.wikidata.org/wiki/Q207450","display_name":"Induction motor","level":3,"score":0.34154754877090454},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"score":0.16415733098983765},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.156174898147583},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.12172132730484009},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.08928453922271729},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tie.2019.2944055","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tie.2019.2944055","pdf_url":null,"source":{"id":"https://openalex.org/S58031724","display_name":"IEEE Transactions on Industrial Electronics","issn_l":"0278-0046","issn":["0278-0046","1557-9948"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Industrial Electronics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7","score":0.7300000190734863}],"awards":[{"id":"https://openalex.org/G359230534","display_name":null,"funder_award_id":"51877149","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6188998833","display_name":null,"funder_award_id":"51677130","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7206701508","display_name":null,"funder_award_id":"17JCTPJC46600","funder_id":"https://openalex.org/F4320326190","funder_display_name":"Tianjin Science and Technology Committee"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320326190","display_name":"Tianjin Science and Technology Committee","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W55921605","https://openalex.org/W1982213546","https://openalex.org/W1982579370","https://openalex.org/W2004565437","https://openalex.org/W2034086734","https://openalex.org/W2043276313","https://openalex.org/W2045512362","https://openalex.org/W2052733983","https://openalex.org/W2095278275","https://openalex.org/W2098667894","https://openalex.org/W2103514848","https://openalex.org/W2115188546","https://openalex.org/W2124519428","https://openalex.org/W2128431896","https://openalex.org/W2131615793","https://openalex.org/W2141779674","https://openalex.org/W2152603626","https://openalex.org/W2152796171","https://openalex.org/W2166785907","https://openalex.org/W2177290733","https://openalex.org/W2332659080","https://openalex.org/W2607314208","https://openalex.org/W2752165960","https://openalex.org/W2783766970","https://openalex.org/W4238540086","https://openalex.org/W6663736650","https://openalex.org/W6674858822","https://openalex.org/W6679120140","https://openalex.org/W6680735344","https://openalex.org/W6682054328"],"related_works":["https://openalex.org/W2808687568","https://openalex.org/W1970791636","https://openalex.org/W1546878080","https://openalex.org/W2381644008","https://openalex.org/W2098931593","https://openalex.org/W2336804855","https://openalex.org/W2810609448","https://openalex.org/W2356164680","https://openalex.org/W4247153582","https://openalex.org/W3043464003"],"abstract_inverted_index":{"A":[0],"double-layer":[1],"permanent":[2,33],"magnet":[3],"spherical":[4,21],"motor":[5],"(PMSM)":[6],"with":[7,132],"same":[8],"spatial":[9],"distribution":[10],"of":[11,54,57,64,69,77,86,117,134,149],"stator":[12,28,78],"poles":[13,16,58,79],"and":[14,39,80,101,107,115,146],"rotor":[15,81],"is":[17,48,59,95],"designed":[18],"based":[19,43],"on":[20,44,61],"harmonic":[22],"theory":[23,100],"in":[24,126],"this":[25,127],"article.":[26],"The":[27,122],"coils":[29],"are":[30,89,109],"equivalent":[31],"to":[32,35,111],"magnets":[34],"simplify":[36],"the":[37,40,51,62,66,70,74,83,92,105,113,118,143],"calculation,":[38],"torque":[41,52,87,93,120,124],"model":[42,53,94,125],"virtual":[45],"work":[46],"method":[47],"derived.":[49],"First,":[50],"single":[55],"pair":[56],"derived":[60],"premise":[63],"considering":[65],"fundamental":[67],"component":[68],"magnetic":[71],"field.":[72],"Then,":[73],"relative":[75],"positions":[76],"poles,":[82],"direction":[84],"angles":[85],"components":[88],"discussed.":[90],"Finally,":[91],"obtained":[96,123],"by":[97],"space":[98],"vector":[99],"superposition":[102],"theorem.":[103],"Moreover,":[104],"simulation":[106],"experiment":[108],"implemented":[110],"verify":[112],"feasibility":[114],"correctness":[116],"proposed":[119],"model.":[121],"article":[128],"has":[129],"similar":[130],"expression":[131],"that":[133],"traditional":[135],"motor,":[136],"which":[137],"lays":[138],"a":[139],"theoretical":[140],"foundation":[141],"for":[142],"structure":[144],"optimization":[145],"control":[147],"strategy":[148],"PMSM.":[150]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}