  The high-$T_c$ superconducting (HTS) dynamo exploits the nonlinear
resistivity of an HTS tape to generate a DC voltage when subjected to a varying
magnetic field. This leads to the so-called flux pumping phenomenon and enables
the injection of DC current into a superconducting coil connected to the dynamo
without current leads. In this work, the process of charging a coil by an HTS
dynamo is examined in detail using two numerical models: the Minimum
Electromagnetic Entropy Production and the segregated $\textbf{H}$-formulation
finite element model. The numerical results are compared with an analytical
method for various airgaps and frequencies. Firstly, the I-V curves of the
modeled HTS dynamo are calculated to obtain the open-circuit voltage,
short-circuit current and internal resistance. Afterward, the process of
charging a coil by the dynamo including the charging current curve and its
dynamic behavior are investigated. The results obtained by the two models show
excellent quantitative and qualitative agreement with each other and with the
analytical method. Although the general charging process of the coil can be
obtained from the I-V curve of the flux pump, the current ripples within a
cycle of dynamo rotation, which can cause ripple AC loss in the HTS dynamo, can
only be captured via the presented models.
