Dynamo-type flux pumps can be employed to energize high-temperature superconducting (HTS) magnets to maintain the steady current mode or in rotor windings of electrical HTS motors and generators to inject the current without the need for current leads. To understand the complicated operating working mechanism of the flux pump it is necessary to study both in open-circuit and connected to an HTS coil. In this work, we employ two models to analyze these configurations. The first one is the Minimum Electromagnetic Entropy Production (MEMEP). The second one is a segregated H-formulation finite-element model. For open-circuit mode, we investigate the effect of the airgap on DC voltage generation. Later, we also study the energizing process of an HTS coil modeled by a series resistance and an inductance. These models will also be compared to experiments.