[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82509-en":3,"doc-seo-82509-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82509,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Sinusoidality Index","Sinusoidality Index introduces a new metric for assessing how accurately AC voltage vectors follow an ideal circular trajectory under any periodic operating conditions. Instead of relying on Fourier spectral decomposition, the metric quantifies instantaneous trajectory deviation, revealing distortion characteristics that conventional Fourier-based estimations may miss. The approach uses geometric frequency concepts and estimates the quasi steady-state (QSS) frequency as the fundamental component. Examples demonstrate the metric’s ability to capture harmonic, interharmonic distortion, and unbalance effects.","Sinusoidality Index  \nJoan Gutiérrez-Florensa, Student, IEEE, Álvaro Ortega, Member, IEEE, Lukas Sigrist, Member, IEEE, and  \nFederico Milano, Fellow, IEEE  \narXiv :2607 .00419v1 [ ee ss . SY] 1 Jul 2026  \nAbstract—Maintaining sinusoidal or near-sinusoidal operating conditions in electrical systems is essential, as is their accurate assessment. This letter proposes a novel metric, namely the sinusoidality index, which quantifies the instantaneous deviation of the trajectory of an ac voltage vector with respect to a circle under any periodic operating conditions. This metric differs from conventional Fourier-based estimations by accounting for the trajectory of the waveform rather than its spectral decomposition. A variety of examples illustrates the properties of the proposed metric and highlights insights that may not be captured by conventional approaches.  \nIndex Terms—Harmonic distortion, power quality, differential geometry, Fourier transform.  \nI. INTRODUCTION  \nOperating at sinusoidal conditions is crucial in electric systems to ensure equipment protection, power quality, and overall system reliability, among other operational considerations. For this reason, international standards define methods to evaluate waveform distortion and impose limits within which the system can operate safely, e.g., [1] .  \nHowever, existing standards estimate the waveform distortion based on its decomposition from the Fourier series and define metrics, such as the Total Harmonic Distortion (THD), that rely on the summation of the contribution of each harmonic. This is the common approach utilized for most power quality indexes proposed in the literature [2], which are widely used in different applications to evaluate and improve power quality [3], [4] .  \nWhile conventional Fourier-based metrics are useful in many applications, they do not account for other conditions that might distort the fundamental signal, e.g., unbalanced conditions, and therefore cannot serve as a generalized metric to evaluate the deviation of a signal from an ideal sinusoid. Moreover, by the use of Fourier series, these metrics: (i) assume that the signal is periodic at the nominal frequency – 50 or 60 Hz, depending on the region–; and (ii) cannot handle well interharmonics and, more in general, transient conditions. This letter uses the geometric and fluid dynamics framework introduced in [5] to propose a novel metric, called sinusoidality index, to evaluate the deviation of the trajectory of voltage vectors from an ideal circle. The metric relies on the estimation of the Quasi Steady-State (QSS) frequency, a concept introduced in [6] and that represents the fundamental frequency of the measured voltage. The proposed metric is able to account for  \nJ. Gutiérrez-Florensa and F. Milano are with the School of Elec. & Electron. Eng., University College Dublin, Dublin, D04V1W8, Ireland. emails: joan.gutierrezflorensa1@ucdconnect.ie, federico.milano@ucd.ie,  \nÁ . Ortega and L. Sigrist are with School of Engineering, Comillas Pontifical University, 28015, Madrid, Spain. e-mails: {aortega, [lsigrist}@comillas.edu](lsigrist}@comillas.edu)  \nThis work is supported by Science Foundation Ireland (SFI) by funding J. Gutiérrez-Florensa and F. Milano under NexSys project, Grant No. 21/SPP/3756 .  \nany type of distortion, harmonic and interharmonic, as well as of unbalances. The proposed index can be also applied to analytic signals if they are properly transformed into planar space vectors.  \nII. BACKGROUND AND PERIODICAL CONDITIONS  \nThis letter builds on the geometric interpretation of the frequency introduced in [7] . Leveraging on this interpretation, the work in [5], uses the Lagrange derivative to decompose the geometric frequency, ωυ , of a vector representing the voltage at a node of three-phase circuit, υ, into different terms with a clear physical meaning borrowed from fluid mechanics:  \nυ × υ′ωυ =  \n|υ| 2  \nυ × ∂t υ   υ × (Rυ)  1 (w · υ)υ   1  \n= + − + (∇ ×υ)|υ| 2 |υ|","cbCaidI1yMKjaCZI","https://ap.wps.com/l/cbCaidI1yMKjaCZI","pdf",1058594,3,1,4,"English","en",105,"# Introduction\n# Background and Periodical Conditions","[{\"question\":\"What does the sinusoidality index measure in AC electrical systems?\",\"answer\":\"It quantifies the instantaneous deviation of the trajectory of an AC voltage vector from an ideal circle under periodic operating conditions.\"},{\"question\":\"How does the proposed metric differ from conventional Fourier-based distortion metrics?\",\"answer\":\"Conventional metrics estimate distortion from Fourier spectral decomposition by summing harmonic contributions, while sinusoidality index evaluates the waveform trajectory itself.\"},{\"question\":\"What frequency component does the method rely on, and why?\",\"answer\":\"It relies on the quasi steady-state (QSS) frequency, estimated from geometric concepts, to represent the fundamental frequency of the measured voltage and serve as a reference for trajectory 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does the sinusoidality index measure in AC electrical systems?","Question",{"text":74,"@type":75},"It quantifies the instantaneous deviation of the trajectory of an AC voltage vector from an ideal circle under periodic operating conditions.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed metric differ from conventional Fourier-based distortion metrics?",{"text":79,"@type":75},"Conventional metrics estimate distortion from Fourier spectral decomposition by summing harmonic contributions, while sinusoidality index evaluates the waveform trajectory itself.",{"name":81,"@type":72,"acceptedAnswer":82},"What frequency component does the method rely on, and why?",{"text":83,"@type":75},"It relies on the quasi steady-state (QSS) frequency, estimated from geometric concepts, to represent the fundamental frequency of the measured voltage and serve as a reference for trajectory 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