[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84912-en":3,"doc-seo-84912-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84912,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Understanding Small-Signal Impedance Matrices in Different Reference Frames","Systematic analysis explains relationships among dq-domain, αβ-domain, and sequence-domain representations used for small-signal impedance modeling of voltage-source converters. AC impedance matrices in dq-complex and αβ-complex variables yield distinct sequence-domain formulations. For asymmetric systems, rotating and stationary reference frames produce different physical phenomena, making frame transformations physically inconsistent. The modified sequence-domain impedance matches the universal impedance model in the frequency domain, resolves notation issues in literature, and provides physical guidance for stability analysis using stationary vs rotating frames.","Understanding Small-Signal Impedance Matrices in Different Reference Frames  \nJ. Pedra  \nAbstract—This paper systematically analyzes the relationships among the dq-domain, αβ-domain, and sequence-domain representations used in small‑signal impedance modeling of voltage-source converters (VSCs). It is shown that the AC impedance matrix expressed with dq‑complex and αβ‑complex variables leads to different formulations in the sequence domain. The study demonstrates that asymmetric systems exhibit different physical phenomena in the rotating and stationary reference frames; therefore, the transformations between these frames are not physically consistent in such cases. It is also demonstrated that the so-called modified sequence-domain impedance is equivalent to the universal impedance model in the frequency domain. The analysis clarifies several notational inconsistencies found in the literature. Finally, a physical interpretation is presented highlighting the implications of using stationary and rotating reference frames for stability analysis of power converters.  \nIndex Terms—Complex and real transformations, symmetrical components, small-signal impedance, voltage-source converters.  \nI. INTRODUCTION  \nTas  \nhe application of time-dependent transformations, such the Park transformation, has been largely restricted to  \nelectrical machine theory and control during the past century [1]–[3] . In recent decades, however, stability studies of VSCHVDC systems in the dq and sequence domains have become a widely used analysis method. Both state-space and frequency-domain approaches are commonly applied to assess instability phenomena in these systems [4] . Frequency-domain methods based on system impedance characterization are particularly attractive because they can be applied using both analytical models and experimental measurements. These approaches originate from techniques originally developed for DC systems [5] . Impedance modeling of three-phase VSCHVDC systems is normally performed in the dq-real domain [6] or in the sequence domain [7]–[9], resulting in a 2 × 2 impedance matrix. The off-diagonal terms of the impedance matrix describe the positive-and negative-sequence coupling. This effect is not considered in [7] . A comprehensive description of the dq-complex formulation can be found in [10], [11] . The relationship between the impedance matrix Zdqin the dq-real frame and the sequence-domain matrix Zpn is presented in [9], giving rise to the so-called modified sequence-domain impedance (MSDI), which incorporates the  \nJ. Pedra is with the Dep. of Elect. Eng., UPC, Av. Diagonal 647, 08028 Barcelona, Spain (mail: [joaquin.pedra@upc.edu](joaquin.pedra@upc.edu)).  \nFig. 1. Overview of transformations.  \nmirror frequency effect (MFE) . A rigorous derivation based on the dq-complex frame is provided in [12] . Reference [8] shows that the sequence-domain impedance Zpn has the same marginal stability conditions as the dq-domain impedance matrix Zdq. The transformation from the αβ–real frame to the dq–real frame is presented in [13] . This transformation is nontrivial because it involves transitioning from a stationary reference frame to a rotating reference frame. This results from the application of the Park transformation to the voltage and current variables. Examples of such frame transformations can be found in [14], [15] . Transformations from the dq-complex domain to the αβ-complex domain are described in [16], [17] . These voltage–current relationships are referred to as the unified impedance model (UIM) and are claimed to be formulated in a stationary reference frame. However, as demostrated in Section V-C, the UIM is equivalent to the MSDI, consequently, both are formulated in a rotating reference frame. An additional case of interest, described in [18], [19], includes the DC side in the analysis, resulting in a 3 × 3 impedance/admittance matrix. A global view of the relationships among different frames has been presented","cbCaiaqT9t6xHK9G","https://ap.wps.com/l/cbCaiaqT9t6xHK9G","pdf",475078,1,10,"English","en",105,"# Introduction\n## Motivation and Background\n## Frequency-Domain Impedance Modeling\n## dq, αβ, and Sequence-Domain Formulations\n# Reference Frame Relationships\n## Stationary vs Rotating Frames\n## Symmetry Requirement for Physical Consistency\n# Key Contributions\n## Transformation Diagram and pn Frame Distinction\n## Physical Justification for Asymmetry Effects","[{\"question\":\"Which reference-frame domains are connected in this work for small-signal impedance modeling?\",\"answer\":\"The paper relates dq-domain, αβ-domain, and sequence-domain representations used to model small-signal impedance of voltage-source converters.\"},{\"question\":\"Why can frame transformations become physically inconsistent in asymmetric systems?\",\"answer\":\"Because rotating and stationary reference frames lead to different physical phenomena when the impedance (or admittance) matrix is asymmetric, so transforming between these frames lacks physical consistency.\"},{\"question\":\"How does the modified sequence-domain impedance relate to the universal impedance model?\",\"answer\":\"The modified sequence-domain impedance is shown to be equivalent to the universal impedance model in the frequency 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reference-frame domains are connected in this work for small-signal impedance modeling?","Question",{"text":75,"@type":76},"The paper relates dq-domain, αβ-domain, and sequence-domain representations used to model small-signal impedance of voltage-source converters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why can frame transformations become physically inconsistent in asymmetric systems?",{"text":80,"@type":76},"Because rotating and stationary reference frames lead to different physical phenomena when the impedance (or admittance) matrix is asymmetric, so transforming between these frames lacks physical consistency.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the modified sequence-domain impedance relate to the universal impedance model?",{"text":84,"@type":76},"The modified sequence-domain impedance is shown to be equivalent to the universal impedance model in the frequency 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