Energy storage and loss in fractional-order circuit elements

The efficiency of a general fractional-order circuit element as an energy storage device is analysed. Simple expressions are derived for the proportions of energy that may be transferred into and then recovered from a fractional-order element by either constant-current or constant-voltage charging and discharging. For a half-order element, it is shown that the efficiency of the charging phase of the cycle is equal to the efficiency of the discharging phase. The results demonstrate the duality of the fractional capacitive and inductive elements, in that the efficiency of one under constant-current cycling is the same as the efficiency of the other under constant-voltage cycling, and vice-versa.