Harmonic State-Space Model of Second-Order Generalized Integrator Phase-Locked Loop

Single-Phase Phase-Locked Loop (PLL) are widely used for integrating single-phase power converters. Over the years, many models for single-phase PLLs have been developed. One such PLL is Second-Order Generalized Integrator (SOGI)-PLL, which is a quadrature signal generator-based PLL. The modeling of these kinds PLLs is usually done using traditional Linear Time-Invariant (LTI) approach, which only considers fundamental frequency in the system, ignoring other higher frequencies. In this paper, Linear Time-Periodic (LTP) framework is utilized for modeling SOGI-PLL to obtain a more accurate picture of the system. First, from the nonlinear model of the PLL, a LTP model is derived. Next, using the harmonic balance method, a LTI harmonic state-space model is obtained. The two models obtained are then simulated and validated with the nonlinear model in MATLAB/Simulink. In addition to time-domain simulation, frequency-domain responses from the LTI model and the nonlinear model are compared for validation. While the frequency-domain response of the LTI model can be obtained by plotting the Bode plots of an input-output transfer function, the frequency-domain response of the nonlinear model is obtained using the harmonic injection method.