{"id":"https://openalex.org/W1908606840","doi":"https://doi.org/10.1002/cta.1864","title":"Extension of the variational equation to analog/digital circuits: numerical and experimental validation","display_name":"Extension of the variational equation to analog/digital circuits: numerical and experimental validation","publication_year":2012,"publication_date":"2012-10-11","ids":{"openalex":"https://openalex.org/W1908606840","doi":"https://doi.org/10.1002/cta.1864","mag":"1908606840"},"language":"en","primary_location":{"id":"doi:10.1002/cta.1864","doi":"https://doi.org/10.1002/cta.1864","is_oa":false,"landing_page_url":"https://doi.org/10.1002/cta.1864","pdf_url":null,"source":{"id":"https://openalex.org/S92132303","display_name":"International Journal of Circuit Theory and Applications","issn_l":"0098-9886","issn":["0098-9886","1097-007X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","poland-70","russia-white-list-2"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Circuit Theory and Applications","raw_type":"journal-article"},"volume":null,"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/A5086093589","display_name":"Federico Bizzarri","orcid":"https://orcid.org/0000-0002-6568-6491","observed_orcids":["https://orcid.org/0000-0002-6568-6491"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Federico Bizzarri","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080022239","display_name":"Angelo Brambilla","orcid":"https://orcid.org/0000-0001-9422-6446","observed_orcids":["https://orcid.org/0000-0001-9422-6446"]},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Angelo Brambilla","raw_affiliation_strings":["Politecnico di Milano Dipartimento di Elettronica e Informazione (DEI)  P.za. Leonardo da Vinci, n. 32 I\u201020133 Milano Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Politecnico di Milano Dipartimento di Elettronica e Informazione (DEI)  P.za. Leonardo da Vinci, n. 32 I\u201020133 Milano Italy","institution_ids":["https://openalex.org/I93860229"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064664899","display_name":"Giancarlo Storti Gajani","orcid":"https://orcid.org/0000-0001-8182-7891","observed_orcids":["https://orcid.org/0000-0001-8182-7891"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Giancarlo Storti Gajani","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5080022239"],"corresponding_institution_ids":["https://openalex.org/I93860229"],"apc_list":{"value":2780,"currency":"USD","value_usd":2780},"apc_paid":null,"fwci":2.6276,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.88424125,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":98},"biblio":{"volume":"41","issue":"7","first_page":"743","last_page":"752"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.2921999990940094,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.2921999990940094,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10187","display_name":"Radio Frequency Integrated Circuit Design","score":0.23180000483989716,"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/T11417","display_name":"Advancements in PLL and VCO Technologies","score":0.20419999957084656,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/variational-equations","display_name":"variational equations","score":0.9340000152587891},{"id":"https://openalex.org/keywords/analog-digital-circuits","display_name":"analog-digital circuits","score":0.8180000185966492},{"id":"https://openalex.org/keywords/shooting-method","display_name":"shooting method","score":0.8119999766349792},{"id":"https://openalex.org/keywords/floquet-theory","display_name":"Floquet theory","score":0.7929999828338623},{"id":"https://openalex.org/keywords/fundamental-matrix","display_name":"fundamental matrix","score":0.8840000033378601},{"id":"https://openalex.org/keywords/phase-noise","display_name":"phase noise","score":0.8679999709129333},{"id":"https://openalex.org/keywords/oscillators","display_name":"oscillators","score":0.6570000052452087},{"id":"https://openalex.org/keywords/relaxation-oscillator","display_name":"relaxation oscillator","score":0.5820000171661377},{"id":"https://openalex.org/keywords/steady-state-solutions","display_name":"stationary solutions","score":0.5839999914169312},{"id":"https://openalex.org/keywords/limit-cycles","display_name":"limit cycles","score":0.5830000042915344}],"concepts":[{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.6438256502151489},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5961000323295593},{"id":"https://openalex.org/C15627037","wikidata":"https://www.wikidata.org/wiki/Q541961","display_name":"Classification of discontinuities","level":2,"score":0.5751024484634399},{"id":"https://openalex.org/C29074008","wikidata":"https://www.wikidata.org/wiki/Q174925","display_name":"Analogue electronics","level":3,"score":0.5091106295585632},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42585402727127075},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.41456782817840576},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4134949743747711},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40842628479003906},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34875649213790894},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3283967077732086},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2926543354988098},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28088462352752686},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.1176544725894928},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.09908431768417358},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1002/cta.1864","doi":"https://doi.org/10.1002/cta.1864","is_oa":false,"landing_page_url":"https://doi.org/10.1002/cta.1864","pdf_url":null,"source":{"id":"https://openalex.org/S92132303","display_name":"International Journal of Circuit Theory and Applications","issn_l":"0098-9886","issn":["0098-9886","1097-007X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","poland-70","russia-white-list-2"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Circuit Theory and Applications","raw_type":"journal-article"},{"id":"pmh:oai:re.public.polimi.it:11311/684795","doi":null,"is_oa":false,"landing_page_url":"http://hdl.handle.net/11311/684795","pdf_url":null,"source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W1540356186","https://openalex.org/W1546490615","https://openalex.org/W1552277675","https://openalex.org/W1558079603","https://openalex.org/W1904904019","https://openalex.org/W1998369000","https://openalex.org/W2007702453","https://openalex.org/W2021673620","https://openalex.org/W2052773149","https://openalex.org/W2065367661","https://openalex.org/W2067694918","https://openalex.org/W2080673725","https://openalex.org/W2080856993","https://openalex.org/W2098315134","https://openalex.org/W2104741613","https://openalex.org/W2111079473","https://openalex.org/W2115477597","https://openalex.org/W2141960683","https://openalex.org/W2147673130","https://openalex.org/W2162767308","https://openalex.org/W2166299135","https://openalex.org/W2478030348","https://openalex.org/W3140474906","https://openalex.org/W4232610995","https://openalex.org/W4233863580","https://openalex.org/W4237636837","https://openalex.org/W4239509541","https://openalex.org/W4244001917","https://openalex.org/W4255914035"],"related_works":["https://openalex.org/W2141960683","https://openalex.org/W2097007944","https://openalex.org/W2104741613","https://openalex.org/W1996731978","https://openalex.org/W1479773997","https://openalex.org/W2189131994","https://openalex.org/W2147673130","https://openalex.org/W2101436064","https://openalex.org/W2166299135","https://openalex.org/W2123014999"],"abstract_inverted_index":{"ABSTRACT":[0],"Conventional":[1],"shooting":[2,75],"methods":[3,43],"cannot":[4],"be":[5],"used":[6],"to":[7,48,83],"determine":[8],"the":[9,27,40,50,55,58,93,97,105],"steady\u2010state":[10],"solution":[11],"of":[12,29,42,57,96,104],"circuits":[13,30],"whose":[14],"model":[15,52],"is":[16],"characterised":[17],"by":[18,81],"a":[19,33,110,128,133],"vector":[20],"field":[21],"exhibiting":[22],"zero":[23],"order":[24],"discontinuities.":[25],"In":[26,68],"analysis":[28],"working":[31],"on":[32,127],"stable":[34],"limit":[35],"cycle,":[36],"this":[37,69,79,89],"limitation":[38],"prevents":[39],"use":[41,72],"that":[44,77],"exploit":[45],"Floquet":[46],"theory":[47],"compute":[49],"variational":[51],"and":[53,86],"thus":[54],"properties":[56],"fundamental":[59,106],"matrix":[60],"and,":[61],"for":[62],"example,":[63],"phase":[64,115],"noise":[65],"in":[66,113],"oscillators.":[67],"paper,":[70],"we":[71],"an":[73],"improved":[74],"method":[76,90],"solves":[78],"drawback":[80],"resorting":[82],"saltation":[84],"matrices":[85],"show":[87],"how":[88],"makes":[91],"possible":[92],"correct":[94],"computation":[95],"first":[98],"left":[99],"eigenfunction":[100],"(known":[101],"as":[102,132],"ppv)":[103],"matrix.":[107],"ppvs":[108],"are":[109,123],"key":[111],"aspect":[112],"determining":[114],"noise.":[116],"The":[117],"results":[118],"obtained":[119],"through":[120],"numerical":[121],"simulations":[122],"compared":[124],"with":[125],"measurements":[126],"relaxation":[129],"oscillator":[130],"serving":[131],"simple":[134],"but":[135],"significant":[136],"comparison":[137],"vehicle.":[138],"Copyright":[139],"\u00a9":[140],"2012":[141],"John":[142],"Wiley":[143],"&":[144],"Sons,":[145],"Ltd.":[146]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}
