{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,2]],"date-time":"2024-08-02T00:26:55Z","timestamp":1722558415442},"reference-count":27,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T00:00:00Z","timestamp":1656374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51875510"],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"The structural discontinuities in the form of air gaps in transformer cores cause the concentration of electromagnetic force, which is an important source of transformer vibration and noise. In this paper, an engineering model of magnetic flux density and electromagnetic force density on transformer core discontinuities is analytically developed. Based on a reasonable structural simplification and assumptions, magnetic flux density and electromagnetic force density are deduced as explicit functions of the geometric, material, and electrical excitation characteristics of the gap region and the transformer core. The accuracy of the established model is validated by the finite element method (FEM) combined with a magnetic measurement experiment. According to this engineering model, the electromagnetic force density can be reduced by decreasing the gap ratio and increasing the gap thickness to a reasonable level. The outcome of this paper can help to understand the physical mechanism of the electromagnetic force generated by core air gap discontinuities, which is meaningful for noise control and the condition monitoring of transformers.<\/jats:p>","DOI":"10.3390\/s22134869","type":"journal-article","created":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T05:48:38Z","timestamp":1656481718000},"page":"4869","source":"Crossref","is-referenced-by-count":1,"title":["An Engineering Model of Magnetic Flux Density and Electromagnetic Force Density at the Structural Discontinuity within Transformer Cores"],"prefix":"10.3390","volume":"22","author":[{"given":"Lingzhi","family":"Li","sequence":"first","affiliation":[{"name":"Key Laboratory of Intelligent Manufacturing Quality Big Data Tracing and Analysis of Zhejiang Province, China Jiliang University, Hangzhou 310018, China"}]},{"given":"Xuhao","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, The University of Western Australia, Perth, WA 6009, Australia"}]},{"given":"Jie","family":"Pan","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, The University of Western Australia, Perth, WA 6009, Australia"},{"name":"Department of Instrument Science and Engineering, Zhejiang University, Hangzhou 310027, China"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-3507-3360","authenticated-orcid":false,"given":"Xishan","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Instrument Science and Engineering, Zhejiang University, Hangzhou 310027, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shilyashki, G., Pf\u00fctzner, H., Palkovits, M., Windischhofer, A., and Giefing, M. (2017). 3D-Printed Detector Band for Magnetic Off-Plane Flux Measurements in Laminated Machine Cores. Sensors, 17.","DOI":"10.3390\/s17122953"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kim, M., and Lee, S. (2022). Power Transformer Voltages Classification with Acoustic Signal in Various Noisy Environments. Sensors, 22.","DOI":"10.3390\/s22031248"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7916","DOI":"10.1109\/TPEL.2016.2636572","article-title":"Influence of Material Properties and Geometric Shape of Magnetic Cores on Acoustic Noise Emission of Medium-Frequency Transformers","volume":"32","author":"Shuai","year":"2016","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1618","DOI":"10.1109\/TMAG.2003.810414","article-title":"Magnetic forces and magnetostriction in electrical machines and transformer cores","volume":"39","author":"Vandevelde","year":"2003","journal-title":"IEEE Trans. Magn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"8403406","DOI":"10.1109\/TMAG.2015.2469643","article-title":"Vibration Reduction of Inductors under Magnetostrictive and Maxwell Forces Excitation","volume":"51","author":"Rossi","year":"2015","journal-title":"IEEE Trans. Magn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1109\/TMAG.1972.1067540","article-title":"Power loss and flux density distributions in the T-joint of a three phase transformer core","volume":"8","author":"Moses","year":"1972","journal-title":"IEEE Trans. Magn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1109\/TMAG.1973.1067577","article-title":"Flux distribution and power loss in the mitered overlap joint in power transformer cores","volume":"9","author":"Jones","year":"2010","journal-title":"IEEE Trans. Magn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1109\/TMAG.1973.1067705","article-title":"Measurement of rotating flux in silicon iron laminations","volume":"9","author":"Moses","year":"1973","journal-title":"IEEE Trans. Magn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1186","DOI":"10.1109\/20.706473","article-title":"Comparison of transformer loss prediction from computed and measured flux density distribution","volume":"34","author":"Moses","year":"1998","journal-title":"IEEE Trans. Magn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"047607","DOI":"10.1063\/1.4994133","article-title":"Comparison between measured and computed magnetic flux density distribution of simulated transformer core joints assembled from grain-oriented and non-oriented electrical steel","volume":"8","author":"Shahrouzi","year":"2018","journal-title":"AIP Adv."},{"key":"ref_11","first-page":"17","article-title":"Magnetic flux distribution in power transformer core with mitred joints","volume":"117","author":"Qi","year":"2015","journal-title":"J. Appl. Phys."},{"key":"ref_12","unstructured":"Moses, A.J., Anderson, P.I., Phophongviwat, T., and Tabrizi, S. (September, January 21). Contribution of magnetostriction to transformer noise. Proceedings of the Universities Power Engineering Conference, Cardiff, UK."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.ijmecsci.2016.09.022","article-title":"Spatial distributions of magnetostriction, displacements and noise generation of model transformer cores","volume":"118","author":"Shilyashki","year":"2016","journal-title":"Int. J. Mech. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3759","DOI":"10.1109\/20.908346","article-title":"Relevance of magnetostriction and forces for the generation of audible noise of transformer cores","volume":"36","author":"Weiser","year":"2000","journal-title":"IEEE Trans. Magn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1109\/20.312511","article-title":"Three dimensional flux distributions in transformer cores as a function of package design","volume":"30","author":"Pfutzner","year":"1994","journal-title":"IEEE Trans. Magn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1108\/03321649910296708","article-title":"Investigations of the audible noise of inductors with respect to different ferromagnetic materials","volume":"18","author":"Terorde","year":"1999","journal-title":"Compel"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2261","DOI":"10.1007\/s00202-018-0701-0","article-title":"Magnetic equivalent circuit of mf transformers: Modeling and parameter uncertainties","volume":"100","author":"Guillod","year":"2018","journal-title":"Electr. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Serigne, M., Nicolas, B., and Vincent, L. (2016). An Anisotropic Model for Magnetostriction and Magnetization Computing for Noise Generation in Electric Devices. Sensors, 16.","DOI":"10.3390\/s16040553"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kunicki, M., and Wotzka, D. (2019). A Classification Method for Select Defects in Power Transformers Based on the Acoustic Signals. Sensors, 19.","DOI":"10.3390\/s19235212"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1750","DOI":"10.1109\/TMAG.1984.1063495","article-title":"Numerical analysis of flux and loss distributions in electrical machinery","volume":"20","author":"Nakata","year":"1984","journal-title":"IEEE Trans. Magn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1109\/TMAG.1982.1061828","article-title":"Flux distribution in three phase transformer cores with various t-joint geometries","volume":"18","author":"Basak","year":"1982","journal-title":"IEEE Trans. Magn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1109\/63.388001","article-title":"Introduction to modeling of transformers and coupled inductors","volume":"10","author":"Witulski","year":"1995","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3701","DOI":"10.1109\/20.718532","article-title":"New transformer model including joint air gaps and lamination anisotropy","volume":"34","author":"Elleuch","year":"1998","journal-title":"IEEE Trans. Magn."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, L., Du, X., Pan, J., Adrian, K., David, M., Huang, H., and Zheng, J. (2019). Distributed magnetic flux density on the cross-section of a transformer core. Electronics, 8.","DOI":"10.3390\/electronics8030297"},{"key":"ref_25","unstructured":"Benedict, R.P. (1980). Fundamentals of Pipe Flow, Wiley."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4523","DOI":"10.1109\/TMAG.2011.2151201","article-title":"Rotational magnetization in transformer coresa review","volume":"47","author":"Pfutzner","year":"2011","journal-title":"IEEE Trans. Magn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1109\/20.312443","article-title":"Influence of air gaps in stacked transformer cores consisting of several packages","volume":"30","author":"Loffler","year":"1994","journal-title":"IEEE Trans. Magn."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4869\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,1]],"date-time":"2024-08-01T04:05:24Z","timestamp":1722485124000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4869"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,28]]},"references-count":27,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22134869"],"URL":"https:\/\/doi.org\/10.3390\/s22134869","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,6,28]]}}}