In the present work, intrinsic voltage rectification in graphene three-terminal junctions (GTTJs) is investigated, with the goal to advance understanding of physical mechanisms behind this effect. In the first part, finite element simulations based on a field-effect transistor model are made to calculate output-voltage characteristics for realistic diffusive GTTJs. Within this model, rectification is described as an electrostatic effect. The simulations fit well to a substantial number of reported experimental results and provide engineering guidelines for rectification efficiency enhancement, such as good graphene material quality and high capacitive gate coupling. According to the simulations, efficiency in difusive GTTJs is intrinsically limited to around 60%, as a consequence of the inability to pinch off conductivity in graphene. In the second part, etched GTTJs of different sub-micron constriction widths (down to 100 nm) are fabricated on Si/SiO2 substrates and characterized electrically at 296 K and 87 K. Reproducibility of the rectification effect is demonstrated. Typical room-temperature rectification efficiencies are 10 to 20% at 100 mV input voltage, whereas efficiencies at 87 K are below 10%. The highest room-temperature rectification efficiency measured in our devices is ~ 40% (at 400 mV input voltage), which is higher than most efficiencies reported in the literature. Experiments show higher efficiencies at room temperature than at 87 K, in contrast to the predictions of the field-effect simulations. This indicates that mechanisms other than the field effect contribute to the rectification effect. We propose an explanation based on Joule-heating and thermal voltages, as the devices are operated in regimes of high power dissipation. This thermal model predicts thermal voltages which show bias-voltage, gate-voltage, and temperature dependences in conformance with measured output voltages. We conclude that Joule-heating effects need to be considered for GTTJ devices. At the same time, possibilities for alternative GTTJ functionality open up.