_applied_  
_sciences_  
_Article_  
**Fractance with Tunable Factor's Order for Microwave Circuit Applications**  
**[PERSON]**  
Faculty of Engineering, International Telematic University Uninettuno, 00186 Rome, Italy;  
[EMAIL_ADDRESS]  
**Featured Application: The paper intends to illustrate a methodology to design and implement fractance with tunable fractor's order, suitable for implementing fractional order circuits in a more flexible way. **  
There is an increasing number of studies in the literature to implement fractional order components by means of equivalent circuits based on integer order components. Such implementations aim to realize laboratory equipment that can exhibit a fractional behavior in a certain range of frequencies. One of the main limitations of the existing implementation is the fixed fractor's order. In practice, every time the experimenter wants to change fractor's order, it is necessary to calculate again the equivalent circuit and implement it again. In order to overcome this limitation, in this paper we propose a new implementation of a fractional order component that enables to tune the fractor's order. This is achieved by means of variable resistors and a proper control methodology. The methodology can be applied in microwave circuits, for instance for the implementation of filters: a low-pass one is discussed in this paper. fractional calculus; fractional order circuits; low pass filter; microwave circuit; tunable component  
**Citation:** [PERSON] Fractance with Tunable Factor's Order for Microwave Circuit Applications. _Appl. Sci._**2022**, 12, 10108. [[https://doi.org/10.3390/app121910108](https://doi.org/10.3390/app121910108)]([https://doi.org/10.3390/app121910108](https://doi.org/10.3390/app121910108)) Academic Editor: [PERSON]  
Received: 22 September 2021  
Accepted: 30 September 2022  
Published: 8 October 2022  
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. **Citation:** 2022 by the author. Liconse MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/](https://creativecommons.org/licenses/by/)]([https://creativecommons.org/licenses/by/](https://creativecommons.org/licenses/by/)) 4.0/). **1. Introduction**  
Fractional calculus is a branch of mathematics dealing with the extension of the concept of derivative and integral. The name is due to fact that the order of derivative is not just an integer value but may be fractional or even real. Namely, for a function \(f(t)\), in fractional calculus it is defined as the expression:  
\[\frac{\mathrm{d}^{\alpha}f}{\mathrm{d}t^{\alpha}}\text{ with }\alpha\in\mathbb{R} \tag{1}\]  
Fractional derivative concept was discussed for the first time about three centuries ago, but only in the last fifty years it has been studied in deep. Formally, the operator \({}_{a}\mathrm{D}_{t}^{\alpha}f(t)\) denotes the fractional derivative of a function \(f(t)\) defined in the interval \([a,b]\). Several representations of the fractional derivative have been proposed in the literature. The most recognized ones are the so-called Caputo derivative [1] and the Caputo-Weyl derivative [2; 3; 4]. The Caputo definition is more commonly adopted in time domain, while the Caputo-Weyl definition is usually adopted in frequency domain, where the Fourier transform of the fractional derivative operator exhibits the following property:  
\[\mathcal{F}(^{W}_{-\infty}\mathrm{D}_{t}^{\alpha}\ f(t))=(\mathrm{j}\omega)^{ \alpha}F(\omega) \tag{2}\]  
being \(F(\omega)\) the Fourier transform of \(f(t)\) and \({}^{W}_{-\infty}\mathrm{D}_{t}^{\alpha}\ f(t)\) the Caputo-Weyl derivative of \(f(t)\). This is the fractional extension of the derivative property of the Fourier transform.