Magnetic reconnection occurring between the interplanetary magnetic field and the dayside magnetopause causes a circulation of magnetic flux and plasma within the magnetosphere, known as the Dungey cycle. This circulation is transmitted to the ionosphere via field-aligned currents (FACs). The magnetic flux transport within the Dungey cycle is quantified by the cross-polar cap potential (CPCP or transpolar voltage). Previous studies have suggested that under strong driving conditions the CPCP can saturate near a value of 250 kV. In this study we investigate whether an analogous saturation occurs in the magnitudes of the FACs, using observations from the Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE). The solar wind speed, density and pressure, the Bz component of the interplanetary magnetic field, and combinations of these, were compared to the concurrent integrated current magnitude, across each hemisphere. We find that FAC magnitudes are controlled most strongly by solar wind speed and the orientation and strength of the interplanetary magnetic field. FAC magnitude increases monotonically with solar wind driving but there is a distinct knee in the variation around IMF Bz = -10 nT, above which the increase slows.