Fractional order filter with two fractional elements of dependant orders
Graphical Abstract
Highlights
► General fundamentals of the fractional-dependent-order filters are introduced. ► The stability analysis of the fractional-order filters is presented. ► The critical frequencies (max., half-power, and right-phase) are calculated generally. ► A general transformation between the types of these filters is introduced. ► The fractional-order KHN filter is chosen to validate the presented concepts.
Introduction
A system which is defined by fractional order differential equations is termed a fractional order system [1], [2], [3], [4], [5], [6], [7], [8]. The significant advantage of fractional order systems over their counterpart integer order systems is that they are characterized by infinite memory, whereas integer order systems are characterized by a finite memory. As a result, modeling real-world phenomena using fractional order calculus has received great attention in the last few decades [6]. Literatures show a large number of natural phenomena, which have been modeled by fractional order differential equations to produce more accurate results. Many applications based on fractional-order systems have been recently discussed in the fields of bioengineering [9], [10], [11], chaotic systems [12], agriculture [13], electromagnetic [14], Smith-chart [15], [16] and control theorems [17], [18]. In addition, many fundamentals in the conventional circuit theories and stability techniques have been generalized into the fractional-order domain [19], [20].
Although the realization of fractional-elements has not yet become commercial, but there are many excellent research papers that have been introduced during the last three decades such as in the topics of fractal behavior of a metal–insulator solution interface, dynamic processes such as mass diffusion and heat conduction [21] Also in the analog domain where such an operation can be called a fractance device [21], [22], [23], [24], [25], [26].
A method for modeling and simulation of fractional systems using state space representation is proposed in [1], [2], [3], [4]. Fractional order differentiators and integrators are used to compute the fractional order time derivatives and integrals of the given signal.
The Riemann–Liouville definition of a fractional derivative of order α is given by [1], [5]:another approximation of fractional order derivative based on Grunwald–Letnikov is given bywhere Δt is the integration step and . By applying the Laplace transform to (1) assuming zero initial conditions yields
The expression for the s-domain impedance of the fractance device is given bywhere ko is a constant and α is the fractional order. This impedance is considered a generalized impedance since it covers the conventional passive elements (R, L, and C) where the impedance of R, L and C are expressed as s0R, s1L, and (sC)−1 for the values of α equal to 0, 1, and −1 respectively.
One of the most important and popular analog blocks are continuous time filters, which are widely used functional blocks, from simple anti-aliasing filters preceding ADCs to high-spec channel-select filters in integrated RF transceivers [27], [28], [29], [30], [31], [32], [33], [34], [35]. Many recent papers have been introduced to generalize the analysis of the conventional-filters in the fractional-order domain; however all of them assumed the same fractional-orders [4], [5], [6].
In this work, we seek to generalize the design of classical second-order filters to the fractional-order domain using two different fractional orders α and β. Generally, the conventional and equal-order fractional filter designs are considered special cases from this study. The design of a fractional order filter with different orders increases the degree of freedom since the design parameters can be enhanced which in turn increases the design flexibility. Moreover, fractional order filters can be controlled to obtain the exact requirements of the delay and frequency responses in the time and frequency domain respectively, which is a critical issue for many applications like the phase locked loop (PLL) [16], [17], [18].
This paper is organized as follows; Section 2 presents a literature review for the previous fractional order filter design. Section 3 discusses the proposed design procedures for the fractional order filters using fractional elements of dependant orders. Frequency transformation techniques from the fractional low-pass to the fractional high-pass and band-pass filter are presented in Section 4. In Section 5, fractional order KHN filter design is introduced using the proposed procedure and simulations using ADS are also presented. Finally the conclusion is introduced in Section 6.
Section snippets
Fractional-order filter design
Fractional order filters are represented by general fractional-order differential equations, and considered as the generalized case of the integer-order filters. Fractional order filters were first proposed in [4], where procedures for designing all filters with a single fractional-element were introduced. These procedures introduced the fractional-order low-pass (FLPF), fractional-order high-pass (FHPF), fractional-order band-pass (FBPF), and fractional-order all-pass (FAPF) filters with their
Fractional-order filter with dependent orders
Since fractional-orders are dependent, let us assume β=kα where 0<α, kα<2. Then, under these conditions the transfer function will be
Therefore, the characteristic equation can be rewritten as follows:
As a special case when k=1, then α=β and (9) will be similar to that given in [5]. As the stability represents one of the most important parameters in the filter design, therefore the first
Fractional-order frequency transformation
Until now, only filters using the low-pass configuration have been examined. The low-pass transfer function can be rewritten here as follows:
In this section, the transformation from low-pass prototype into the other configurations: high-pass and band-pass will be discussed.
Circuit simulation
Now it is important to prove the reliability of the proposed algorithm. So, the aim of this section is to apply the proposed design procedure on a practical filter like KHN filter. To test the fractional order filter response, it is important first to model the fractional order element. A finite element approximation of the special case was reported in [25]. This finite element approximation relies on the possibility of emulating a fractional-order capacitor via semi infinite RC trees
Conclusion
In this work, we have generalized classical continuous time filter networks to be of fractional-order by using two fractional elements of dependant orders. The design procedure based on the relation between the fractional orders α, β is introduced. We have deduced the expressions for the pole frequencies, the right-phase frequencies, and the half-power frequencies. Also stability analysis of the fractional order filter is introduced. It is clear that more flexibility in shaping the filter
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