This thesis covers the investigation into the use of Space Vector Modulation for the control of a 4-leg matrix converter, which is capable of providing a 3-phase plus neutral supply from a standard balanced 3-phase source. Traditional 3x3 matrix converters have limited use in this application as they are only capable of supplying a balanced three-phase load. It would be desirable to be able to power unbalanced and non-linear loads, requiring that the converter provides a neutral connection. As with voltage source inverters, this goal can be achieved by extending the number of output legs in the matrix converter to four. In this thesis, a new Space Vector Modulation technique is proposed for this 4-leg, or 3x4, matrix converter. This technique is an extension of the method currently in use on 3x3 matrix converters, and so it allows the de-coupled control of both the input and output. The thesis then goes onto cover the build of a demonstration converter, looking at the different aspects which make up a converter, to finally go on to prove the theory, and a set of results are presented to validate this.