We present a comparison between a simple but general model of solar wind-magnetosphere-ionosphere coupling (the Hill model) and the output of a global magnetospheric MHD code, the Integrated Space Weather Prediction Model (ISM). The Hill model predicts transpolar potential and region 1 currents from environmental conditions specified at both boundaries of the magnetosphere: at the solar wind boundary, electric field strength, ram pressure, and interplanetary magnetic field direction; at the ionospheric boundary, conductance and dipole strength. As its defining feature, the Hill model predicts saturation of the transpolar potential for high electric field intensities in the solar wind, which accords with observations. The model predicts how saturation depends on boundary conditions. We compare the output from ISM runs against these predictions. The agreement is quite good for non-storm conditions (differences less than 10%) and still good for storm conditions (differences up to 20%). The comparison demonstrates that global MHD codes (like ISM) can also exhibit saturation of transpolar potential for high electric field intensities in the solar wind. We use both models to explore how the strength of solar wind-magnetosphere-ionosphere coupling depends on the strength of Earth's magnetic dipole, which varies on short geological timescales. As measured by power into the ionosphere, these models suggest that magnetic storms might be considerably more active for high dipole strengths.