# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 1 Introduction
FO feedback control of DC motors is the subject of [56]. The properties of common structures of the integrated voltage regulator were studied in [57]. Therein, a passive RC approximant located in a closed feedback loop significantly improved the performance of a designed regulator, as documented by circuit simulations. FO elements can find useful applications in DC-DC converters, as recognized by authors of [58]. This paper illustrates a promising method for analysis of FO DC-DC converters demonstrated on boost converters working in continuous conduction mode. The same operational regime, but a general class of DC-DC converters (buck, boost, and buck-boost), is assumed in the case of the investigations presented in [59]. Therein, the authors include an FO inductor in the standard topology of converters. This is motivated by non-idealities in large commercial inductors, i.e., the necessity to obtain a better model and more accurate results. To this end, the modeling of real physical phenomena, where descriptive information is usually available as measured data sequences, by using FO differential equations can be observed in many other research fields. FO wireless power transmission systems are shown in paper [60]. Wideband CPEs need to be utilized while studying chaotic dynamics with the lowest possible order [61] if new FO systems with robust strange attractors are to be discovered [62], or if FO circuit elements should be working in wideband applications such as FO memristors in generators of chaotic waveforms based on Wien bridge oscillator topology [63]. Shortcomings, which have been recognized in many papers focused on the utilization of FO elements in analog systems, are based on frequency limitations of CPE approximants. In other words, CPEs keep a predefined phase shift only in the finite frequency bandwidth (with a top and bottom boundary), and this should be respected by circuit designers and developers of applications. Unfortunately, this limitation is a source of the functional errors in many papers published in renowned journals. For example, many \"fractional-order\" filters proposed in the literature can be hardly utilized in real situations, since correct operation is restricted to two frequency decades of processed signals. This is especially obvious in the case of FO frequency filters. Some authors adopt FO sub-circuits that work well only within one, two [64], or, at most, three decades [65; 66] with significant phase corrugation. Generators of the harmonic signals are not demanding applications with respect to CPEs unless these are supposed to be tunable in a wide frequency range. Thus, the CPEs utilized in [67] cannot be transferred to other systems that oscillate at a different frequency. This paper is organized as follows: Section 2 describes the different possibilities for lumped circuit realization of CPEs. Individual types of circuits are equivalent in the sense that any phase shift between \(0^{\circ}\) and \(90^{\circ}\) can be realized. Section 3 details fully passive RC implementations of CPEs, including numerical values of the circuit components and error analysis, calculated in Mathcad. Section 4 of this work discusses a few practical applications of audio CPEs; introduced two-port phase shifters can perform non-asymptotical phase shifting, i.e., connect two arbitrary points on the phase frequency response. Furthermore, a bass-corrector and a subsonic voltage-mode two-port with arbitrary constant phase shift is designed. Section 5 is devoted to discussion; several problems with CPEs are revealed and solutions are suggested. Finally, concluding remarks and possible future research topics are provided.