Resonance and Quality Factor of the $RL_{\alpha } C_{\alpha }$ Fractional Circuit

This paper introduces the general fundamentals of the fractional order RL_αC_α circuit, where α is the order of the elements. The generalized complex impedance of the RL_αC_α circuit can be purely real, imaginary, or short circuit. The stability analyses where two regions have been classified are introduced with many numerical examples. General maps for transient and frequency responses are investigated showing the damping parameters for each case. The resonance, 3 dB frequencies, bandwidth, and quality factor for all possible cases have been investigated with detailed analytical formulas. Numerical and PSpice simulations are provided using different examples to validate these concepts.