Fractional-Order Inductor: Design, Simulation, and Implementation

Abstract

Fractional calculus has tremendous potential in modeling the evolution of complex systems including those with memory. Indeed, fractional-order models are more accurate in approximating non-locally distributed dynamics with short- or long-term memory effects. However, the realization of fractional systems is often hindered by the lack of robust fractional-order energy storage devices, particularly fractional-order inductors (FOIs). Inherent eddy currents, hysteresis losses, the lack of suitable materials, and a systematic design procedure are among the challenges of FOI synthesis. In this work, a straightforward and robust approach realizing FOIs with a coaxial structure is proposed. This approach relies on the fact that the wave impedance of the transverse electromagnetic (TEM) mode on the coaxial structure scales with $(\mathit {j}\omega)^{0.5}$ , where $\mathit {j}=\sqrt {(-1)}$ and $\omega $ is the angular frequency when the filling material is highly conductive. Indeed, experimental characterization of the realized device shows that it has a half-order inductive response (corresponding to 45° phase angle) that is stable in the frequency range 18 MHz – 1 GHz with a phase angle deviation not exceeding 5°. Furthermore, the effects of the device geometry and the permeability, the permittivity, the conductivity of the filling material on device response are investigated.

Keywords

• Coaxial cable
• coil
• electromagnetic fields
• fractional-order circuits
• fractional-order inductor
• fractional-order resonator