#switching state vector

A power inverter is fed by dc-link voltage Vd; it generates a balanced, three-phase PWM voltage of fundamental period T1. At time instant t in [0, T1], the instantaneous discrete values of the potentials of phases a, b, and c are ua(t), ub(t), and uc(t), respectively. The switching state vector that the power inverter produces at this time instant is

uk(t) = 1.ua(t) + a.ub(t) + a2.uc(t), where a = exp{j.2.pi/3}.

An example follows: a three-phase PWM voltage of fundamental period T1 = 20 ms is produced by a three-level power inverter that is fed by a constant dc-link voltage Vd. Assume that, at t = 2.5 ms, phase a is connected to the positive dc rail, phase b is connected to the negative dc rail, and phase c is connected to the neutral point potential. It follows that the potentials of phase a, b, and c are ua(t) = +Vd/2, ub(t) = –Vd/2, and uc(t) = 0, respectively. With this configuration, the switching state vector that is selected at t = 2.5 ms is

uk(t) = 1.(+Vd/2) + a.(–Vd/2) + a2.0,

or 

uk(t) = (+Vd/2).sqrt(3).exp{ –j.pi/6}.

A common way to represent the switching state vector of this example is