HTS Dynamo Modeling Comparison Using The <b>J</b>-<b>A</b>-<i>φ</i> and <b>H</b>-<b>A</b> Formulations Considering Two Distinct Operating Scenarios

High-temperature superconducting (HTS) dynamos enable contact-free injection of large direct currents into superconducting circuits, and their multiphysics electromagnetic-thermal coupling and practical circuit interactions requires<br/>models that are both accurate and computationally efficient. This work implements a hybrid <b>J</b>–<b>A</b>–<i>φ</i> formulation coupled to thermal model. It solves the scalar magnetic potential <i>φ</i>, the magnetic vector potential <b>A</b>, the current density <b>J</b>, and the temperature <i>T</i>. To consider the temperature dependence, the critical current density is a function of temperature that is computed by a thermal model. Two operating scenarios are examined: (i) an open-circuit case with zero-net transport current and (ii) a charging case in which the HTS tape is coupled to a series <i>R</i>–<i>L</i> circuit. Simulations at 10 Hz and 500 Hz evaluate global variables: instantaneous voltage between the terminal of tape <i>V</i>_sc, cumulative time-averaged voltage <i>V</i>_DC, average temperature <i>T</i>_av, and charging current <i>I</i>_sc using a segregated <i>H</i>-formulation as the reference and benchmarking against <b>J</b>-<b>A</b>, <b>H</b>-<b>A</b> and <b>J</b>-<b>A</b>-<i>φ</i> formulations. The <b>J</b>–<b>A</b>–<i>φ</i> formulation delivers substantial reductions in computational cost marked decreases the time footprint relative to <b>H</b>-<b>A</b> while preserving the accuracy of the reference solution. These results indicate that <b>J</b>–<b>A</b>–<i>φ</i> is a reliable and efficient tool for design-oriented studies of HTS dynamos.