{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T17:04:08Z","timestamp":1792602248100,"version":"4.5.0"},"reference-count":40,"publisher":"The Electrochemical Society","issue":"5","license":[{"start":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T00:00:00Z","timestamp":1746748800000},"content-version":"vor","delay-in-days":8,"URL":"https:\/\/iopscience.iop.org\/page\/copyright"},{"start":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T00:00:00Z","timestamp":1746748800000},"content-version":"tdm","delay-in-days":8,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["J. Electrochem. Soc."],"published-print":{"date-parts":[[2025,5,1]]},"abstract":"\n In this work we derive and study the analytical solution of the voltage and current diffusion equation for the case of a finite-length resistor-constant phase element (CPE) transmission line (TL) network that can represent a model for porous electrodes in the absence of any Faradic processes. The energy storage component is considered to be an elemental CPE per unit length of impedance\n z<\/jats:italic>\n \n c<\/jats:italic>\n <\/jats:sub>\n (\n s<\/jats:italic>\n )\u00a0=\u00a01\/(\n c<\/jats:italic>\n \n \u03b1<\/jats:italic>\n <\/jats:sub>\n s<\/jats:italic>\n \n \u03b1<\/jats:italic>\n <\/jats:sup>\n ) with constant parameters (\n c<\/jats:italic>\n \n \u03b1<\/jats:italic>\n <\/jats:sub>\n ,\u00a0\n \u03b1<\/jats:italic>\n ) instead of the ideal capacitor of impedance\n z<\/jats:italic>\n (\n s<\/jats:italic>\n )\u00a0=\u00a01\/(\n c<\/jats:italic>\n s<\/jats:italic>\n ) usually assumed in TL modeling. The problem becomes a time-fractional diffusion equation for the voltage that we solve under galvanostatic charging, and derive from it a reduced impedance function of the form\n \n \n \n <\/jats:tex-math>\n \n \n \n z<\/mml:mi>\n <\/mml:mrow>\n \n \u03b1<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n \n \n s<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n s<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n \n \u2212<\/mml:mo>\n \u03b1<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n coth<\/mml:mi>\n (<\/mml:mo>\n \n \n s<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n \n \u03b1<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n , where\n s<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n \u00a0=\u00a0\n j\u03c9<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n is a normalized frequency. We also derive the system\u2019s step response, and the distribution function of relaxation times associated with it. The analysis can be viewed and used as a support for the fractal finite-length Warburg model.\n <\/jats:p>","DOI":"10.1149\/1945-7111\/add212","type":"journal-article","created":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T18:52:02Z","timestamp":1745952722000},"page":"056502","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":20,"title":["On the Distributed Resistor-Constant Phase Element Transmission Line in a Reflective Bounded Domain"],"prefix":"10.1149","volume":"172","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6044-9158","authenticated-orcid":true,"given":"Anis","family":"Allagui","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1400-5916","authenticated-orcid":false,"given":"Enrique H.","family":"Balaguera","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chunlei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"77","published-online":{"date-parts":[[2025,5,9]]},"reference":[{"key":"jesadd212bib1","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1016\/S0022-0728(00)00497-6","article-title":"Theory of the electrochemical impedance of anomalous diffusion","volume":"499","author":"Bisquert","year":"2001","journal-title":"Electroanal. 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Published on behalf of ECS by IOP Publishing Limited. All rights, including for text and data mining, AI training, and similar technologies, are reserved.","name":"copyright_information","label":"Copyright Information"},{"value":"2024-12-23","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2025-05-09","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}