{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T03:45:32Z","timestamp":1775619932046,"version":"3.50.1"},"reference-count":64,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T00:00:00Z","timestamp":1749600000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T00:00:00Z","timestamp":1749600000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Circuit Theory &amp; Apps"],"published-print":{"date-parts":[[2026,2]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>\n                    Incorporating the magnetic core material in a lumped\u2010element transformer model generally results in a nonlinear network with hysteresis. Such a network can be solved subsequently in the time domain using time\u2010stepping methods. The transient response reveals the inrush, whose decay time is generally enormous compared to the time constant of the excitation period (e.g., 20\u2009ms for a voltage source with a frequency of 50\u2009Hz). To directly obtain the steady\u2010state solution of the lumped\u2010element transformer model, the harmonic balance method is superior to such an analysis, because it is a frequency domain approach inherently yielding the steady\u2010state solution. The great advantage of harmonic balance is that it can handle nonlinearities by evaluating them in the time domain and using their spectral components in the frequency domain. This paper theoretically derives the harmonic balance algorithm and further solves a lumped\u2010element single\u2010phase transformer model based on a mutual and leakage flux approach utilizing\n                    <jats:italic>Hopkinson<\/jats:italic>\n                    's law. The nonlinear network element is represented by an energy\u2010based dry friction\u2010like hysteresis model depicting the core material of the transformer. The hysteresis parameters are fitted to no\u2010load current measurements, and the no\u2010load case, the short\u2010circuit case, and a chosen load case are simulated and compared to measurements. The algorithm's efficiency in terms of iterations and computational time is demonstrated by simulating with several excitation frequencies and for several load cases and comparing it to a transient analysis revealing the transformer's inrush.\n                  <\/jats:p>","DOI":"10.1002\/cta.70012","type":"journal-article","created":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T05:29:36Z","timestamp":1749619776000},"page":"625-637","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Method of Harmonic Balance for Lumped\u2010Element Transformer Models"],"prefix":"10.1002","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-4596-2168","authenticated-orcid":false,"given":"Alexander","family":"Sauseng","sequence":"first","affiliation":[{"name":"Institute of Fundamentals and Theory in Electrical Engineering Graz University of Technology  Graz Austria"}]},{"given":"Alice","family":"Reinbacher\u2010K\u00f6stinger","sequence":"additional","affiliation":[{"name":"Institute of Fundamentals and Theory in Electrical Engineering Graz University of Technology  Graz Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8916-851X","authenticated-orcid":false,"given":"Dennis","family":"Albert","sequence":"additional","affiliation":[{"name":"OMICRON electronics  Klaus Austria"}]},{"given":"Peter","family":"Hamberger","sequence":"additional","affiliation":[{"name":"SIEMENS Energy  Linz Austria"}]},{"given":"Manfred","family":"Kaltenbacher","sequence":"additional","affiliation":[{"name":"Institute of Fundamentals and Theory in Electrical Engineering Graz University of Technology  Graz Austria"}]},{"given":"Klaus","family":"Roppert","sequence":"additional","affiliation":[{"name":"Institute of Fundamentals and Theory in Electrical Engineering Graz University of Technology  Graz Austria"}]}],"member":"311","published-online":{"date-parts":[[2025,6,11]]},"reference":[{"key":"e_1_2_8_2_1","unstructured":"EEUG \u201cATPEMTP \u201d (2025) https:\/\/www.atp\u2010emtp.org(visited on 01\/16\/2025)."},{"key":"e_1_2_8_3_1","unstructured":"EMTP \u201cEMTP\u2014Electromagnetic Transients Program \u201d (2025) https:\/\/www.emtp.com\/products\/emtp(visited on 01\/16\/2025)."},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/tmag.2013.2285239"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/tia.2011.2126031"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/tmag.2019.2900690"},{"key":"e_1_2_8_7_1","volume-title":"Theory and Calculation of Alternating Current Phenomena","author":"Charles Proteus Steinmetz","year":"1900"},{"key":"e_1_2_8_8_1","doi-asserted-by":"publisher","DOI":"10.1108\/COMPEL\u201006\u20102021\u20100190"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/tmag.2006.879145"},{"key":"e_1_2_8_10_1","volume-title":"Transformer Engineering: Design, Technology, and Diagnostics","author":"Kulkarni S. 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