MATHEMATICAL MODELING OF THREE-PHASE CLARKE AND PARK TRANSFORM SYSTEMS APPLIED TO INVERTER CONTROL

Abstract

This chapter addresses the mathematical modeling of three-phase electrical systems through the application of the Clarke and Park Transforms, essential tools for power electronics and the integral analysis of electrical networks. The direct control of voltage and current in natural coordinates abc presents high mathematical complexity due to the intrinsic coupling of the phases and the time-varying behavior of sinusoidal quantities. To overcome this limitation, the Clarke Transform is applied to project the three phases into a two-dimensional orthogonal stationary reference frame (αβ), reducing the dimensionality of the system and facilitating modulation and pre-filtering steps. Sequentially, the Park Transform maps these variables into a rotating synchronous reference (dq), following the natural angular frequency of the electrical network and converting alternating signals into continuous numerical values. This transformation allows the elegant separation between the real and imaginary parts of the resulting vector and enables the use of classic linear controllers, such as Proportional-Integral (PI), in inverter control loops and synchronism algorithms (PLL). It is concluded that the benefits of these spatial representations transcend the design of static converters, consolidating themselves as an indispensable analytical basis for dynamic stability studies, fault modeling, digital protection and energy quality monitoring in power systems.