Device (DMT) arranged to provide an additional voltage, when the heat engine starts up, for the electrical power supply of the rotating electrical machine (AD). This additional voltage is added to a rated voltage of the on-board electrical system (RB) provided by a battery (BAT). The device comprises a second voltage source (Ucap) and electronic switches (K1, K2), each comprising at least one MOSFET transistor. At least one MOSFET transistor (K2) of the switches, mounted in series with the second voltage source, is controlled in linear mode at the end of the startup of the heat engine. Advantageously, the device comprises current control loops for controlling the transistors in linear mode.