Design and Performance Study of Dynamic Fractors in Any of the Four Quadrants

Abstract

A fractor is a simple fractional-order system. Its transfer function is $1/Fs^{\alpha }$; the coefficient, F, is called the fractance, and $\alpha$ is called the exponent of the fractor. This paper presents how a fractor can be realized, using RC ladder circuit, meeting the predefined specifications on both F and $\alpha$. Besides, commonly reported fractors have $\alpha$ between 0 and 1. So, their constant phase angles (CPA) are always restricted between $0^{\circ }$ and $-90^{\circ }$. This work has employed GIC topology to realize fractors from any of the four quadrants, which means fractors with $\alpha$ between $-$2 and +2. Hence, one can achieve any desired CPA between $+180^{\circ }$ and $-180^{\circ }$. The paper also exhibits how these GIC parameters can be used to tune the fractance of emulated fractors in real time, thus realizing dynamic fractors. In this work, a number of fractors are developed as per proposed technique, their impedance characteristics are studied, and fractance values are tuned experimentally.

Appendix

1.1 A Discussion on Fractance Unit

This paper presents $\mho /$s$^\alpha$ as the unit of fractance. The ambiguity may arise from the fact that fractance unit is dependent on the exponent value. As the exponent changes, the unit changes! For example, in Table I, we see that the fractor $F_{0.3}$ has fractance value 13.3 $\upmu \mho /$s$^{0.3}$ and the fractor $F_{0.4}$ has fractance value 7.56 $\upmu \mho /$s$^{0.4}$. Both the elements are fractors, but they have different units for their fractance values. This is explained as follows: The word ‘fractor’ does not indicate a single type of element rather a set of elements. Every fractor who has different $\alpha$ is actually different element of this set named ‘fractor’. That means, though both $F_{0.3}$ and $F_{0.4}$ are categorized as fractor, but they are different elements because they have different $\alpha$ value. So, it is logical that they have different units. (just like resistor and inductor, both are categorized as CPE, yet they are different element, hence, has different units, $\Omega$ and $\Omega$s (H), respectively)

According to some recent works, fractor’s impedance can be presented as,

$$\begin{aligned} Z_F(j\omega ) = \dfrac{Q}{(j \omega /\omega _N)^{\alpha }}, \end{aligned}$$

(24)

where $\omega _N$ is a normalizing frequency, Q is the coefficient, and $\alpha$ is the exponent. The main advantage of this representation is that the unit of Q is Ohm only which is easier to interpret. However, such realization bring a third variable, $\omega _N$ which is yet to be defined properly. For the realized fractors, presented in this paper, if we chose $\omega _N$ as 1 Hz, then the value of 1 / Q will be same as the fractance F. The only difference is that the unit of 1 / Q will be $\mho$ instead of $\mho /$s$^{\alpha }$.