Realizing Fractional Capacitor for the Proposed Oscillator Designing

The work on realizing the fractional order capacitors is going on since decades [7-9], however, still they are not available on a commercial scale. Various approximation techniques are being considered to depict the fractional characteristics in a circuit. For a fractional capacitor, the impedance can be represented as:
\begin{equation} Z\left(\text{jw}\right)=\frac{1}{C{(jw)}^{\alpha}}\nonumber \\ \end{equation}\begin{equation} Z\left(s=jw\right)=\frac{1}{Cs^{\alpha}}\nonumber \\ \end{equation}
This expression has a constant phase which equals to (–απ/2) and is dependent upon α, the fractional order. Therefore a close optimization network is required to be designed that closely mimics this behavior. Fractional order capacitors are approximated by using a geometrically progressive RC ladder structure. These structures are used to implement tank circuits with low phase shift and are used to create fractional order oscillators. The LC circuits built using these capacitors are tested using AC sweeps in order to create Bode plots and verify fractional order reactance [32]. The only drawback is that the huge network comprising of R and C of different values tends to increase the economic cost and space occupied by these ladder circuits.