Conventional Synchronous Reference Frame Phase-Locked Loop is an Adaptive Complex Filter

Despite the wide acceptance and use of the conventional synchronous reference frame phase-locked loop (SRF-PLL), no transfer function describing its actual input-output relationship has been developed so far. Arguably, the absence of such transfer function has hampered the application of SRF-PLL as a filter or controller inside the closed-loop control systems. In this paper, the transfer function describing the actual input-output relationship of the conventional SRF-PLL is presented. Using this transfer function, it is shown that the conventional SRF-PLL is a first-order adaptive complex bandpass filter. The accuracy of this transfer function is confirmed through numerical results.