A Closed-Form Matrix Solution for High-Order Wave Reflection in an Open-Ended Coaxial Line for Rapid Dielectric Spectroscopy
Bibliographic record
Abstract
Permittivity spectroscopy using open-ended coaxial probes for material characterization has applications in various fields, including biomedical engineering. The frequency-dependent permittivity of a material is extracted from the measured reflection coefficient through a coaxial probe. Current models that relate the reflection coefficient to the dielectric properties of the material struggle to balance accuracy and computational efficiency, limiting their utility in near real-time applications. This article introduces a novel matrix-based closed-form solution of the reflection coefficient of an open-ended coaxial probe. The approach combines full-wave analysis with a Taylor series expansion, leading to a straightforward matrix calculation. By reformulating the forward problem to decouple the material properties from the geometric properties of the probe, the required numerical integral only needs to be calculated once for each probe geometry. This significantly reduces computational time while providing similar or greater accuracy than existing methods. The model has been validated experimentally using two coaxial probes and four reference liquids, achieving an average error of 3.15%. Further validation through 9600 simulations in Ansys HFSS demonstrated an average error of 2.9%. When applied to inverse problems for estimating material permittivity, the model exhibited an average error of 4.35% while being 376 times faster than existing state-of-the-art models, with similar or enhanced accuracy. These advancements facilitate real-time, full-wave permittivity spectroscopy, offering substantial benefits for medical diagnostics and monitoring.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.003 |
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".