Modeling and design of three-phase systems using complex transfer functions

The use of complex vectors in the modeling, analysis and design of three-phase AC power systems is gaining wider use. Largely complex vectors have been used with coordinate transformations, signal flow diagrams and, in a few cases, with transfer functions, root loci and Bode plots. This paper formally and comprehensively extends complex vector representation to the standard set of control system tools including transfer functions, Laplace transforms, Nyquist and Bode plots, and stability criterion. The simple and insightful approach is developed and demonstrated in terms of the well known principles of control theory. A design example is given for a three phase AC voltage regulator with linear and nonlinear unbalanced loading. Direct design and analysis of a complex controller with complex gains are performed based on the results of this paper. Simulation results further illustrate the complex regulator performance. Lastly experimental results validate the developments and confirm the simulation results.