## 1 Introduction  
Fractional-order (FO) systems, i.e., systems of non-integer order described by non-integer order differential and integral equations, are able to provide more arbitrary properties and characteristics in comparison with their integer-order counterparts [1, 2]. Thus, FO systems are employed in many branches, such as signal processing, modeling various real-world phenomena, or system control [3]. Analog electrical circuits can also be designed with fractional order, and for their implementation, elements with FO impedance are frequently employed. These elements are also often referred to as fractional-order elements (FOEs) [4] or factors [5]. The impedance of FOE can be expressed in the \(s\)-domain by the following non-integer power-law dependence  
\[Z(s)=\frac{1}{s^{\alpha}F}, \tag{1}\]  
where the real number \(\alpha\) is fractional exponent or order of FOE and the positive constant \(F\) represents fractance having the unit \(\Omega^{-1}\)-sec\({}^{\alpha}\) or F-sec\({}^{\alpha-1}\); whereas, here \"F\" is farad. After the substitution \(s=\mathrm{j}\omega\), the relation (1) can be written in the magnitude-phase format in the frequency domain:  
\[Z(\mathrm{j}\omega)=\frac{1}{\omega^{\alpha}F}\angle-\frac{\alpha\pi}{2}. \tag{2}\]  
The roll-off of the impedance magnitude is \(-20\alpha\) dB per decade of frequency, and the phase remains constant regardless of frequency. If \(\alpha>0\), the FOE has a capacitive character and is labeled as FO capacitor or capacitive factor. Its impedance magnitude decreases with increasing frequency, and the phase angle is negative. If \(\alpha<0\), the FOEis called as an FO inductor or inductive factor, its impedance magnitude increases with increasing frequency, and the phase angle is positive. The values of \(\alpha\) are considered \(-1<\alpha<1\) here. The special cases with \(\alpha=0\), \(1\), or \(-1\) denote resistor, classic capacitor, and classic inductor, respectively. Currently, the research of FO capacitors is particularly conducted as surveyed, e.g., in Reference [4], though it is also necessary to study implementations of FO inductors, as their importance is growing in various applications. FO inductors are used, for example, in parallel and series FO LC resonant circuits [6; 7; 8] and FO frequency filters [9; 10], where they enable achieving high-quality factor. Furthermore, FO inductors are part of electrical equivalent models of various systems, e.g., in medicine, they are used for modeling mechanical impedance of respiratory systems [11; 12]. A design example of an FO inductor for this application is presented in this paper. The FO inductors were also found to very accurately describe the real behavior of coils with ferromagnetic cores, eddy currents, and hysteresis losses [13]. The employment of FO inductors in wireless power transmission systems has been investigated recently, leading to higher power efficiency and lower switching frequency than conventional methods [14]. FO inductors can be approximated by passive ladder circuits consisting of resistors and standard inductors [15; 16; 17]. However, the implementation of inductors by conventional passive coils is currently avoided, due to their drawbacks, such as bulky dimensions, complicated production, incompatibility with integrated technologies, low-quality factor, and other parasitic electrical properties. A class of active FO inductance simulators studied, e.g., in Reference [18], consists of an FO differentiator or integrator with a voltage/current converter connected in a feedback loop. Here the FO differentiator or integrator is emulated by multiple-feedback structure, such as follow-the-leader feedback (FLF). Their advantage is the use of integer-order elements only; however, the circuit complexity grows when an accurate operation in a wide frequency band is required. Since FO capacitor as a solid-state discrete device is expected to be obtainable soon, another type of active FO inductance simulator is becoming attractive.