Energy-Based Interpretation of the Dispersion Coefficient of the Constant Phase Element
Bibliographic record
Abstract
The dispersion coefficient of the constant phase element (CPE) is typically treated as an empirical fitting parameter in the analysis of impedance spectroscopy data, with no clear physical meaning. Here, we present an energy-based interpretation for this coefficient by linking it to the ratio of the dissipated or stored energy in the CPE relative to that supplied by the input source. Using the RC network equivalency of a CPE, we decompose the total input energy into a contribution stored in the capacitive modes and another dissipated in the resistive modes. Analytical expressions of these energy quantities are derived for three test examples: (i) a constant voltage, (ii) a voltage ramp, and (iii) a quadratic input of the form v ( t ) = λ t 2 . In all cases, we found that the ratios of any two of these energy quantities reduce to pure functions of the dispersion coefficient of the CPE, independent of excitation amplitude or material parameters. This result provides a new perspective on the CPE’s dispersion coefficient from a thermodynamic/energetic basis, with direct implications for supercapacitor characterization, battery modeling, as well as for the analysis of other electrochemical devices and systems exhibiting the CPE behavior.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".