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Variable speed multi-phase (more than three-phase) drive systems are seen as serious contender to the existing three-phase drives due to their distinct advantages. Supply to the multi-phase drives are invariably given from a voltage source inverter. However, this paper proposes an alternative solution for supplying multi-phase drive system using direct ac-ac converter called a matrix converter. The paper proposes the matrix converter topology with three-phase grid input and five-phase variable voltage and variable frequency output. The control technique developed and presented in the paper is based on space vector approach. The paper presents the complete space vector model of the proposed matrix converter topology. The model suggests 215 total switching combinations which reduce to 243 states considering the imposed constraints, out of which 240 are active and 3 are zero vectors. However, for space vector PWM (SVPWM) implementation only 90 active and 3 zero vectors can be used. The SVPWM algorithm is presented in the paper. The viability of the proposed solution is proved using analytical, simulation and experimental approach.