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Fractional Resonance-Based Filters
We propose the use of a fractional order capacitor and fractional order inductor with orders , , respectively, in a fractional series circuit to realize fractional-step lowpass, highpass, bandpass, and bandreject filters. MATLAB simulations of lowpass and highpass responses having orders of , 1.5, and 1.9 and bandpass and bandreject responses having orders of 1.5 and 1.9 are given as examples. PSPICE simulations of 1.1, 1.5, and 1.9 order lowpass and 1.0 and 1.4 order bandreject filters using approximated fractional order capacitors and fractional order inductors verify the implementations.
Fractional calculus, the branch of mathematics concerning differentiations and integrations to noninteger order, has been steadily migrating from the theoretical realms of mathematicians into many applied and interdisciplinary branches of engineering . These concepts have been imported into many broad fields of signal processing having many diverse applications, which include electromagnetics , wave propagation in human cancellous bone , state-of-charge estimation in batteries , thermal systems , and more. From the import of these concepts into electronics for analog signal processing has emerged the field of fractional order filters. This import into filter design has yielded much recent progress in theory [6–9], noise analysis , stability analysis , and implementation [12–14]. These filter circuits have all been designed using the fractional Laplacian operator, , because the algebraic design of transfer functions are much simpler than solving the difficult time domain representations of fractional derivatives. A fractional derivative of order is given by the Caputo derivative  as where is the gamma function and . We use the Caputo definition of a fractional derivative over other approaches because the initial conditions for this definition take the same form as the more familiar integer-order differential equations. Applying the Laplace transform to the fractional derivative of (1) with lower terminal yields where is also referred to as the fractional Laplacian operator. With zero initial conditions, (2) can be simplified to Therefore it becomes possible to define a general fractance device with impedance proportional to , where the traditional circuit elements are special cases of the general device when the order is , , and for a capacitor, resistor, and inductor, respectively. The expressions of the voltage across a traditional capacitor and inductor are defined by integer order integration and differentiation, respectively, of the current through them. We can expand these devices to the fractional domain using integrations and differentiations of non-integer order. Then the time domain expressions for the voltage across the fractional order capacitor and fractional order inductor become where , are the fractional orders of the capacitor and inductor, respectively, is the current through the devices, is the capacitance with units , is the inductance with units , and [s] is a unit of time not to be mistaken with the Laplacian operator. Note that we will refer to the units of these devices as [F] and [H] for simplicity.
By applying the Laplace transform to (4), with zero initial conditions, the impedances of these fractional order elements are given as and for the fractional order capacitor and fractional order inductor, respectively. Using these fractional elements in circuits increases the range of responses that can be realized, expanding them from the narrow integer subset to the more general fractional domain. While these devices are not yet commercially available, recent research regarding their manufacture and production shows very promising results [17, 18]. Therefore, it is becoming increasingly important to develop the theory behind using these fractional elements so that when they are available their unique characteristics can be fully taken advantage of.
While a thorough stability analysis of the fractional circuit has been presented in , the full range of filter responses possible with this topology have not. In this paper we examine the responses possible using a fractional order capacitor and fractional order inductor with orders of , in a series circuit to realize fractional step filters. With this topology, fractional lowpass, highpass, bandpass, and bandreject filters of order are realized. MATLAB simulations of lowpass and highpass responses having orders of , , and and bandpass and bandreject having orders of and are presented. PSPICE simulations of , , and order lowpass and and order bandreject filters are presented using approximations of both fractional order capacitors and fractional order inductors to verify the circuit and its implementation.
2. Fractional Responses
The traditional RLC circuit uses standard capacitors and inductors with which only 2nd order filter responses can be realized. We can further generalize this filter to the fractional domain by introducing fractional orders for both frequency-dependent elements. This approach of replacing traditional components with fractional components has previously been investigated for fractional order capacitors in the Sallen-Key filter, Kerwin-Huelsman-Newcomb biquad , and Tow-Thomas biquad . The addition of the two fractional parameters allows the circuit to realize any order . With this modification fractional lowpass, highpass, bandpass, and bandreject filter responses requiring only rearrangement of the series components are realizable. The topologies to realize these four fractional order filter responses are shown in Figure 1.
2.1. Fractional Lowpass Filter (FLPF)
The circuit shown in Figure 1(a) can be used to realize a lowpass filter response with a transfer function given by This transfer function realizes an FLPF response with DC gain of , high frequency gain of zero, and fractional attenuation of dB/decade in the stopband. With the magnitude of (5) given as The MATLAB simulated magnitude responses of (5) with fractional orders of , , and with , Ω, H, and F are illustrated in Figure 2, respectively. From the magnitude responses of Figure 2 we see stopband attenuations and frequencies of , , and dB/decade and , , and rad/s for the , , and order FLPFs, respectively. This confirms the decreasing fractional step of the stopband attenuation as the order, , increases.
The half power frequency, , can be found by numerically solving the following equation: for . Solving (7) for fixed values of when is varied from to in steps of and yields the values given in Figure 3. These values show a general trend that as both and increase, the half power frequency increases. It should be noted that all subsequent MATLAB simulations are performed for fixed values of when is varied in steps of from to with all components fixed with values of , H, and F.
2.2. Fractional Highpass Filter (FHPF)
The circuit shown in Figure 1(b) can be used to realize a highpass filter response with a transfer function given by with DC gain of zero, high frequency gain of 1, and fractional attenuation of dB/decade in the stopband. The magnitude of (8) is given as The MATLAB simulated magnitude response of (8) with fractional orders of , , and when , H, and F are illustrated in Figure 4. From the magnitude responses of Figure 4 we see stopband attenuations and frequencies of of , , and dB/decade and , , and rad/s for the , , and order FHPFs, respectively.
The half power frequency, , can be found by numerically solving the equation for . The values calculated with fixed and varied are shown in Figure 5. These values show a general trend that as both and increase, the half power frequency decreases.
2.3. Fractional Bandpass Filter (FBPF)
The circuit shown in Figure 1(c) can be used to realize a bandpass filter response with a transfer function given by This transfer function realizes an asymmetric FBPF response with DC and high frequency gains of zero and fractional attenuations of and dB/decade for frequencies lower and higher, respectively, than the maxima frequency (). With the magnitude of (10) given as The MATLAB simulated magnitude response of (10) with fractional orders of , and when , , H, and F are illustrated in Figure 6. From the simulated magnitude responses, we see that the low frequency stopband has attenuations of and dB/decade while the high frequency stopband maintains an attenuation of dB/decade. The stopband attenuations closely match those predicted and confirm that the low and high frequency stopband attenuations are independent of each other (which is unique for fractional-order bandpass filters), with low frequency stopband attenuations dependent only on the order of the fractional capacitor, , and the high frequency stopband only on the order of the fractional inductor, .
The maxima frequency can be found by numerically solving the following equation: for . Solving (12) for varying and fixed values of yields the maxima frequencies given in Figure 7(a). The quality factor, , can be found by numerically solving for its two real roots, and , and then evaluating . Solving these equations numerically for varying and fixed values of yields the quality factors given in Figure 7(b). While there is no clear trend for , shows an increase with increasing order for fixed when , H, and F. The maxima frequency (), maximum magnitude (), half power frequencies, (), and quality factors () of the FBPF responses shown in Figure 6, solved numerically as described previously, are given in Table 1.
It is possible to increase the quality factor of these circuits by decreasing the value of for fixed order and values of and . The quality factors of the FBPF when , H, and F for and varied from to in steps of are given in Figure 8. From this figure, we can see that the increasing is more pronounced with higher orders.
2.4. Fractional Bandreject Filter (FBRF)
The circuit shown in Figure 1(d) is able to realize a bandreject filter response with a transfer function given by This transfer function realizes an asymmetric FBRF response with DC and high frequency gains of . With the magnitude of (14) given as The minima frequency, , can be found by numerically solving for . Solving this equation when , H, and F for fixed values of when is varied from to yields the values given in Figure 9(a), while , can be found by numerically solving for its two real roots, and , and then evaluating . The quality factor calculated for FBRFs while is varied for fixed is given in Figure 9(b), respectively, again, with the general trend showing that increases with increasing order.
The MATLAB simulated magnitude responses of (14) with fractional orders of and when , , H, and F are illustrated in Figure 10. From this figure, we can clearly observe the asymmetric nature of this FBRF, with the attenuation of the low and high frequency stop bands dependent on the element orders of and , respectively. The minima frequency (), minimum magnitude (), half power frequencies (), and quality factors () of these responses are given in Table 2. It is possible to increase the quality factor of the FBRF by decreasing the value of for fixed order and values of and . The quality factors of the FBRF when , H, and F for varied from to and varied from to in steps of are given in Figure 11. From this figure, we can see that like the FBPF the increasing is more pronounced with higher orders. It should be noted, though, that it is not possible to calculate a quality factor for FBRFs with both low order and resistance. The minima peak of these filters increases with decreasing order and resistance and when the minima increases above it is not possible to calculate the quality factor using the previous definition.
3. Circuit Simulation Results
Although there is currently much progress regarding the realization of fractional order capacitors [17, 18], there are no commercial devices using these processes available to implement these circuits. As well, even though supercapacitors have been shown to exhibit fractional impedances [19, 20], their high capacitance and limited order () limit their usefulness in signal processing circuits. Until commercial devices with the desired characteristics become available, integer order approximations must be used to realize fractional circuits. There are many methods to create an approximation of that include continued fraction expansions (CFEs) as well as rational approximation methods . These methods present a large array of approximations with the accuracy and approximated frequency band increasing as the order of the approximation increases. Here, a CFE method  was selected to model the fractional order capacitors for PSPICE simulations. Collecting eight terms of the CFE yields a 4th order approximation of the fractional capacitor that can be physically realized using the RC ladder network in Figure 12.
Now, while an RC ladder can be used to approximate a fractional order capacitor, this same topology cannot realize a fractional order inductor as it would require negative component values. However, we can use a fractional order capacitor as a component in a general impedance converter circuit (GIC), shown in Figure 13, which is used to simulate a grounded impedance . Figure 13 realizes the impedance simulating a fractional inductor of order with inductance .
Using both the approximated fractional order capacitor and fractional order inductor, we can realize the FBRF shown in Figure 1(d) with orders and when . The realized circuit is given in Figure 14 with the RC ladder shown as the inset for the fractional order capacitor and the GIC circuit as the inset for the fractional order inductor. The , , and values required to realize these circuits after applying a magnitude scaling of 1000 and frequency scaling to 1 kHz are given in Table 3. The component values required for the 4th order approximation of the fractional capacitances with values of 12.6 and 0.382 μF and orders of 0.5 and 0.9, respectively, using the RC ladder network in Figure 12, shifted to a center frequency of 1 kHz, are given in Table 4. The magnitude and phase of the ideal (solid line) and 4th order approximated (dashed) fractional order capacitor with capacitance μF and order , shifted to a center frequency of 1 kHz, are presented in Figure 15. From this figure we observe that the approximation is very good over almost 4 decades, from 200 Hz to 70 kHz, for the magnitude and almost 2 decades, from 200 Hz to 6 kHz, for the phase. In these regions, the deviation of the approximation from ideal does not exceed 1.23 dB and 0.23∘ for the magnitude and phase, respectively. The simulated fractional order inductors of and 0.382 H can be realized using fractional capacitances of 12.6 and 0.382 μF, respectively, when Ω.
Using the component values in Tables 3 and 4, the approximated FBRF, shown in Figure 14, was simulated in PSPICE using MC op amps to realize responses of order and when and and . The PSPICE simulated magnitude responses (dashed lines) compared to the ideal responses (solid lines) are shown in Figure 16. We can clearly see from the magnitude response of both the MATLAB and PSPICE simulated responses that this filter does realize a fractional band reject response and that the simulation shows good agreement with the ideal response observing both symmetric and asymmetric characteristics for the and order filters, respectively. Note, though, that the PSPICE simulations deviate from the MATLAB simulated transfer function at low and high frequencies, which is a result of using the 4th order approximation of the fractional order capacitors. We highlight that this filter presents two characteristics not possible using integer order filters: (a) bandreject responses have not been possible from integer order filters with order less than and (b) asymmetric responses with independent control of stopband attenuation have never been presented.
3.1. Fractional Bruton Transformation
The Bruton transformation applies an impedance transformation to each element of a passive ladder circuit to create an active circuit realization using the concept of frequency-dependent negative resistance that does not require the use of inductors . This method can be expanded to the fractional-order domain to remove the fractional order inductor from the circuit. Where the integer order transformation scales each element by , the fractional Bruton transformation scales each element by . Applying this scaling to a resistor transforms it to a fractional order capacitor, a fractional order inductor to a resistor, and a fractional order capacitor to a new fractional element of order . This new fractional element can be a resistor, capacitor, or frequency-dependent negative resistor (FDNR) when , , and , respectively. Note that we will use the units of when referring to this fractional element in order to remain consistent with the FDNR, whose symbol we are also using to represent this element. The FLPF of Figure 1(a) after applying the fractional Bruton transformation is shown in Figure 17. The transfer function of this transformed circuit is given by where , , and for the transfer function to be equivalent to (5).
This new fractional element can be realized using a GIC with two fractional order capacitors, this topology is shown in Figure 18 and has an impedance simulating a fractional element with impedance . A GIC has also previously been employed to realize a fractional order capacitor of μF and in . Using the topology of Figure 17 to simulate the FLPFs of Figure 2, magnitude scaled by and frequency shifted to kHz, requires μF, , and n, n, p for the , , and order filters, respectively, when . The components required to simulate the impedance of the fractional element using the GIC topology are given in Table 5, with the values of the fractional order capacitor approximated with the RC ladder of Figure 12 given in Table 4.
The approximated FLPF, shown in Figure 19, was simulated in PSPICE using MC op amps to realize a order filter when . The PSPICE simulated magnitude responses (dashed lines) compared to the ideal responses (solid lines) are shown in Figure 20. Note that a M resistor has been added to bypass the capacitor and provide a DC path to the noninverting input terminal of the upper op amp in the fractional element realization for its bias current .
The PSPICE simulated magnitude responses of the FLPFs show very good agreement with the MATLAB simulated ideal response. Verifying the fractional Bruton transformed FLPFs as well as the GIC realizations of a fractional element of order using approximated fractional order capacitors. The deviations above kHz can be attributed to the approximations of the fractional order capacitors and nonidealities of the op amps used to realize the fractional order inductors.
We have proposed modifying the traditional series RLC circuit to use a fractional order capacitor and fractional order inductor to realize a fractional circuit that is capable of realizing fractional lowpass, highpass, bandpass, and bandreject filter responses of order requiring only modification of the element arrangement. This topology can realize bandpass or bandreject responses with order less than which are not possible using an integer order circuit. In addition, these proposed bandpass and bandreject filters show asymmetric bandpass characteristics with independent control of the stopband attenuations through manipulation of the fractional elements orders, which is not easily accomplished using integer order filters. We have also shown how to realize a new fractional element of order and a fractional order inductor using fractional order capacitors in a GIC circuit. PSPICE simulations of FBRFs and FLPFs verify the fractional characteristics of these circuits as well as the proposed realizations using approximated fractional elements.
T. J. Freeborn would like to acknowledge Canada's National Sciences and Engineering Research Council (NSERC), Alberta Innovates-Technology Futures, and Alberta Advanced Education and Technology for their financial support of this research through their graduate student scholarships.
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