Analog Realization of a Fractional Control Element - Revisited

Attempts up to now to develop an analog realization of a fractional order control (FOC) element have focused on constructing an approximation of the FOC using ideal resistor/capacitor networks.[1] In fact, an ideal capacitor cannot exist. An ideal dielectric in a capacitor having an impedance of the form 1/(jωC) would violate causality. It seems sensible to look for dielectric materials exhibiting the more realistic fractional behavior 1/(jωC), where α ≈ 0.5. Such a component would display “fractance” attributes and could be termed a “fractor,” as opposed to a resistor or capacitor. In this case, a single component would do the job of an entire network of “ideal” components. In hybrid, analog plus digital, devices, the job of calculating the fractional order element could be offloaded from a digital approximation[2] to an external analog element. This would be of particular interest in small hybrid microprocessor based systems where the processor memory is severely restricted due to cost constraints on parts.