This paper proposes two complex vector models to characterize unbalanced th
ree-phase four-wire systems that contain pulsewidth-modulated voltage-sourc
e converters (PWM-VSCs). In the first case, the three-phase system is decom
posed in orthogonal odq components. A complex vector model is then built fr
om the dq and o components. In the second case, a complex vector model is i
ntroduced by decomposing the system in three single-phase systems. In both
cases, an orthogonal, fictitious circuit is introduced to handle a single-p
hase system, such as the homopolar system or each one of the three single-p
hase systems that compose a three-phase four-wire system. From the vector m
odels, two digital controllers, which employ two different reference frames
, are proposed. Also, the use of the same gains for both controllers simpli
fies the equations of the controller in the stationary reference frame. As
an example, current control in a PWM-VSC system is presented.