Abstract
This review paper introduces real-valued two-terminal fully passive RC ladder structures of the so-called constant phase elements (CPEs). These lumped electronic circuits can be understood as two-terminal elements described by fractional-order (FO) dynamics, i.e., current–voltage relation described by non-integer-order integration or derivation. Since CPEs that behave almost ideally are still not available as off-the-shelf components, the correct behavior must be approximated in the frequency domain and is valid only in the predefined operational frequency interval. In this study, an audio frequency range starting with 20 Hz and ending with 20 kHz has been chosen. CPEs are designed and values tabularized for predefined phase shifts that are commonly used in practice. If constructed carefully, a maximum phase error less than 0.5° can be achieved. Several examples of direct utilization of designed CPEs in signal processing applications are provided.
Highlights
Fractional-order (FO) calculus is an ancient mathematical idea, it has still attracted significant interest from analog design engineers in the last two decades
An FO capacitor can be understood as a two-terminal element that forms a bridge between the resistor and conventional capacitor, which has a module frequency response of admittance linearly increasing with frequency slower than the capacitor, and has constant phase shift of current and voltage between 0◦ and 90◦
constant phase elements (CPEs) can be understood as a generalization of FO capacitors since the same abbreviation can be used for FO inductors, FO
Summary
Fractional-order (FO) calculus is an ancient mathematical idea, it has still attracted significant interest from analog design engineers in the last two decades. 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]. 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 This limitation is a source of the functional errors in many papers published in renowned journals. Concluding remarks and possible future research topics are provided
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