In this paper, the enhanced phase-locked loop (EPLL) is modified to achieve a linear time invariant (LTI) and a pseudolinear (PL) EPLL called the LTI-EPLL and the PL-EPLL, respectively. The modification is based on using the estimated amplitude to make the EPLL operation independent from the input signal magnitude. The LTI-EPLL is input-output LTI, and its input-output relationship is represented by a transfer function, a representation that has not been possible for any other type of existing PLL structures. Having a transfer function representation is very useful for design and analysis purposes. The PL-EPLL introduces frequency adaptivity into the LTI-EPLL and is no longer LTI, but its performance is still independent from the input signal magnitude. The transfer function approach also facilitates the development and design of various EPLL extensions to estimate and reject the dc component and the harmonics as well as the extension with controlled attenuation of wideband noises. These topics are discussed in this paper. The presentation is focused on single-phase EPLL, but extension to three-phase PLLs is also briefly presented.