Hi, I'm actually trying to do some simulation on matrix converters for my university power electronics subject. I am reading through the different types of modulation techniques. One of them, the Venturini direct approach, with sinusoidal voltage output and input current. I'm having a little trouble figuring how did Venturini derived the two basic solution for the duty cycles. The paper I'm reading from is from here, http://eprints.nottingham.ac.uk/14034/1/430229.pdf, under segment 3.2 Mathematical model and modulation techniques. The particular duty cycle solution can be found on page 23 of the paper itself. Thanks in advance!