This paper presents a vector control of a 5-phase drive composed of a 5-leg Pulse Width Modulation (PWM) Voltage Source Inverter (VSI) supplying a permanent-magnet Brushless DC (BLDC) machine with trapezoidal waveform of the back-electromotive force (EMF). To achieve this control a Multi-machine Multi-converter model is used: the 5-phase machine is transformed into a set of two 2-phase fictitious machines which are each one controlled in a (d,q) frame as 3-phase machines with sine waveform back-EMF. In comparison with the 3-phase BLDC drives, the 5-phase ones present one particularity: a high sensitivity to the 7th harmonic of back-EMF. Experimental results show that the 7th harmonic of back-EMF, which represents only 5% of RMS back-EMF, induces high amplitude parasitic currents (29 % percent of RMS current). The model allows to explain the origin of this sensitivity and how to modify simply the control algorithm. Experimental improvements of the drive are presented.