Fractional-order L_βC_α Low-Pass Filter Circuit

Abstract

This paper introduces the fundamentals of the conventional LC low-pass filter circuit in the fractional domain. First, we study the new fundamentals of fractional-order LC low-pass filter circuit including the pure real angular frequency, the pure imaginary angular frequency and the short circuit angular frequency. Moreover, sensitivity analysis of the impedance characteristics and phase characteristics of the LC low-pass filter circuit with respect to the system variables is studied in detail, which shows the greater flexibility of the fractional-order filter circuit in designs. Furthermore, from the filtering property perspective, we systematically investigate the effects of the system variables (LC, frequency f and fractional orders) on the amplitude-frequency characteristics and phase-frequency characteristics. In addition, the detailed analyses of the cut-off frequency and filter factor are presented. Numerical experimental results are presented to verify the theoretical results introduced in this paper.

1. Introduction

There has been more than four centuries since the advent of fractional calculus. However, the study of the fractional calculus has been made dramatic advances in recent four decades [1]. In view of the extra fractional-order variables, the application of fractional calculus has many advantages superior to that of traditional integer calculus, such as more flexibility, freedom, best fit, and optimization techniques and so on. Therefore, many scientists have attempted to broaden the scope of fundamentals and theorems from integer order systems into fractional ones in biomedical applications [2, 3], chaotic systems [4, 5], signal processing [6], control design [7], and more. More specifically, the applications of fractional calculus on the filter design have yielded much recent progress in theory [8-10], noise analysis [11] and stability analysis [12]. However, fewer researchers have concentrated on the LC low-pass filter circuit, which is an important basic filter circuit.

Fractional-order mathematical models developed for inductors and capacitors could describe the electrical characteristics more accurately. In other words, the actual inductors and capacitors are fractional-order in nature [13-16]. More specifically, the fractional-order capacitors are obtained by different designing ways [17-20]. Also, Machado and Galhano [21] pointed that the discretional fractional orders of inductors could be designed based on the skin effect. Motivated by the above analysis, circuit designers have to face new challenges on the new phenomenon and laws owing to applications of the fractional-order components.

Recently, some researchers have published some papers on the filter circuit [22-26]. While few researchers have concentrated on the fractional-order filter circuit [27, 28]. From these considerations, the fundamentals of fractional-order LC filter circuit are mainly studied. Because the fractional-order capacitors and inductors are introduced into the filter circuit, there are four variables: L, C plus two fractional orders α and β. Therefore, there should be new fundamentals and laws which cannot be obtained in the conventional filter circuit.

From the above analyses, the following advanced research contents can make our research attractive. First, the new fundamentals of the impedance characteristics and phase characteristics have been systematically investigated, which lays the groundwork for the design of the fractional-order filter circuit. Second, from the filter perspective, the amplitude-frequency characteristics, phase-frequency characteristics and cut-off frequency are studied in detail with respect to LC and fractional orders to show the better filter properties of the fractional-order LC filter circuit.

This paper is organized as follows: Section 2 simply analyzes the fundamentals of the conventional LC filter circuit. In Section 3, basic definitions of fractional capacitors and inductors are presented, and the new fundamentals of the fractional-order filter circuit are discussed, including the pure real angular frequency, the pure imaginary angular frequency, the short circuit angular frequency and the sensitivity analysis of the impedance and phase characteristics with respect to the system variables. In Section 4, as for the filtering characteristics, amplitude-frequency characteristics, phase-frequency characteristics and cut-off frequency are studied in detail. Section 5 concludes the paper.

 

2. Integer Order LC Filter Circuit

The diagram of integer order LC filter circuit is shown in Fig. 1(a). Thus, we can get its impedance as

Fig. 1.Diagram of the LC filter circuit.

Hence, its magnitude of the impedance can be expressed by

From Eq. (1), the impedance is pure imaginary, which means the LC filter circuit is power lossless. And there is a critical operating frequency which is called as resonance frequency. Therefore, when ω > ωc , the impedance will be inductive. Otherwise, it will be capacitive.

Fig. 2 shows the relation between the impedance magnitude versus ω and LC. The large magnitude response will be obtained at very low frequency, or very high frequency and larger LC. Also, it’s clear from the figure that the minimal magnitude of the impedance exists at ωc , which is zero.

Fig. 2.The graphic model of the magnitude of the impedance versus LC and ω of the conventional LC filter circuit.

As for phase, it is a fixed value with the inductive impedance, or with capacitive impedance.

 

3. Fractional Order LβCα Filter Circuit

3.1 Basic definitions of fractional capacitor and inductor

At present, there are three frequently-used definitions of the fractional derivative, which are the Grunwald-Letnikov, the Riemann-Liouville and the Caputo definition [29]. Here, 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. The fractional derivative by Caputo is denoted as

where Γ(x) is the well-known Euler’s Gamma function and n −1 ≤ α ≤ n .

Under zero initial conditions, we apply the Laplace transform to the Caputo definition (3). Thus, one gets

Then, we use the relationship between the voltage and current of the fractional order capacitor and fractional order inductor. Consequently, the following expressions are given as

and

Similarly, under zero initial conditions, by applying the Laplace transform to Eq. (5) and Eq. (6), the impedances of the fractional order capacitor and inductor can be given as and , respectively

The diagram of fractional-order LβCα filter circuit is shown in Fig. 1(b). So, we get the impedance as

Furthermore, one gets

The equivalent impedance of the fractional-order LβCα filter circuit is presented by Eq. (7). Note that there are both the real and imaginary parts in the impedance. For the integer order LC filter circuit, the previous cases cannot be satisfied, which validates the deficiency of the integer order filter circuit. Additionally, both the real and imaginary parts in the impedance vary with the four parameters, fractional orders α and β, LC and ω.

Therefore, we will systematically analyze the new fundamentals of the fractional-order LβCα filter circuit in the following subsections.

3.2 Pure real angular frequency

The equivalent impedance of the fractional-order LβCα filter circuit can be pure real only at a certain frequency which we named pure real angular frequency [27]. In this case, the energy is converted into loss in the fractional-order filter circuit. From Eq. (7), if the imaginary part is zero, the pure real angular frequency can be given by

where α, β≠2.

Furthermore, the pure real impedance with the pure real angular frequency can be expressed by

Fig. 3 illustrates the effects of the fractional orders with different LC values on the pure real angular frequency. From Eq. (9), if α=β , the pure real angular frequency can be simplified as and the corresponding pure real impedance can be rewritten as which can also explain the results in Fig. 4. Specifically, the results in Fig. 3 agree with the expected outcome, in that ωpri is a fixed value 1 when LC=1. When LC<1, ωpri decreases with the increase of fractional orders α=β, and it also decreases as LC increases with fixed fractional orders α=β. In addition, ωpri has a very wide range varying with the fractional orders. However, when LC >1, it’s quite opposite as shown in Fig. 3(b).

Fig. 3.The relation between ωpri versus α=β for different LC values. (a) LC<1. (b) LC≥1.

Fig. 4.The relation between the pure real impedance versus α = β for different L and C values.

From Fig. 4, we can get that when α = β =1, the pure real impedance is zero, which is the integer order case. Interestingly, if α = β>1, we can get a negative resistor.

3.3 Pure imaginary angular frequency

Similarly, the pure imaginary impedance can also be obtained at a certain frequency which we named as pure imaginary angular frequency [25]. In this case, all the power is stored in the filter circuit without energy loss. From Eq. (7), the pure imaginary angular frequency can be given by

where α , β ≠ 1.

From Eq. (11), the pure imaginary impedance can be derived as

From Eq. (11), if α=β, pure imaginary angular frequency ωpii can be simplified as Since ωpii must be positive, the acceptable results of ωpii can be obtained if and only if If α ≠ β, the acceptable results of ωpii can be obtained in the following two cases. First, Y is positive when α ∈(0, 1) and β ∈(1, 2) or α ∈(1, 2) and β ∈(0, 1). Second case is without considering the positive or negative Y. As a result, we can get acceptable results of ωpii only at some critical values of the fractional orders.

As for the pure imaginary impedance shown in Eq. (12), it must be pure imaginary. If α = β, the corresponding pure imaginary impedance can be simplified as However, there is no acceptable result because of the existence of ( −1)α and which is illustrated in Fig. 5. Specifically, the imaginary part of X is always negative. On the other hand, if α ≠ β, there are also two cases shown in Fig. 6. First, from Fig. 6, some acceptable results of the pure imaginary impedance exist when the imaginary part of X is 0 and the real part of X is not zero. In this case, the pure imaginary impedance could be inductive or capacitive. Second, acceptable results of the pure imaginary impedance can be obtained when . However, it is impossible for 0<α, β<2.

Fig. 5.The imaginary of X versus 1/4α when α=β with different LC values.

Fig. 6.The graphic model of the imaginary and real part of X versus the fractional orders with LC = 10−3 : (a) Imaginary part of X when α ∈(0, 1) and β ∈(1, 2); (b) Real part of X when α ∈(0, 1) and β ∈(1, 2); (c) Imaginary part of X when α ∈(1, 2) and β ∈(0, 1); (d) Real part of X when α ∈(1, 2) and β ∈(0, 1).

3.4 Short circuit angular frequency

If Eq. (10) is zero, the real and imaginary part of the impedance of the fractional-order filter circuit will be zero, which means there are no power loss and storing energy. In this case, the fractional-order filter circuit can oscillate freely without any source, which is impossible in the integer order filter circuit. Therefore, the frequency of oscillation (also called as short circuit angular frequency) [27] can be obtained as ωsci = ωpri , and the condition of oscillation can be expressed by

With 0< α, β< 2, α + β = 2 can be derived, where α ≠ β ≠ 1 .

3.5 Impedance characteristics

3.5.1 Effect of LC

From Eq. (8), the magnitude of the impedance can be amplified by because the values of LC are very small in practice. And the magnitude of the impedance will increase rapidly with the increase of LC.

Also, the critical magnitude of the impedance can be obtained when Therefore, the critical LC can be expressed by

where 1<α+β<3.

Hence, the critical magnitude of the impedance is derived as

Fig. 7(a) shows the values of (LC)cr versus α + β for different ω values. Clearly, the critical LC decreases as α + β increases. In addition, the critical LC also decreases with the increase of ω with fixed fractional orders. The relation between the critical magnitude of the impedance at critical LC versus α for different β values is presented in Fig. 7(b), where some local minimum exist when α + β =2.

Fig. 7.(a) The relation between the critical LC versus α+β for different ω values. (b) The critical magnitude of the impedance at critical LC versus α for different β values when ω=100.

3.5.2 Effect of fractional orders

From Eq. (8), the effects of fractional orders on the magnitude of the impedance are systematically discussed in the following contents.

It’s clear from the Fig. 8(a) that the critical maximum values exist at different fractional orders and its location changes as ω changes. The critical minimum values will be discussed in detail later in Fig. 10. The graphic model of the magnitude of the impedance versus fractional orders of the fractional-order filter circuit is presented in Fig. 8(b), where the larger effects of the fractional orders on the magnitude of the impedance can be obtained with the larger β or smaller α.

Fig. 8.(a) The relation between magnitude of the impedance versus the fractional orders for different ω values when LC = 10−3 . (b) The graphic model of the magnitude of the impedance versus fractional orders of the fractional-order filter circuit when ω=100 and LC = 10−3 .

Fig. 10.(a)The critical value of α=β versus ω for different LC values. (b) The critical value of magnitude of impedance versus ω for different LC values.

The critical magnitude of the impedance versus α and β can be easily obtained when and which are presented in Eqs. (16) and (17), respectively.

Fig. 9 shows the relations between the critical α (β) and the critical value of magnitude of impedance versus the operating frequency for different β (α) values. From Fig. 9(a), the critical α gradually decreases with the increase of frequency, and the critical α also decreases as β increases. However, it’s quite opposite for the critical value of magnitude of impedance shown in Fig. 9(b). It’s clear from the Fig. 9(c) that the local maximal critical β exists at different frequency for different α. Note that there is more than one critical β in the high frequency when α<1. With the increase of frequency in Fig. 9(d), the critical magnitude of impedance rapidly decreases, which means the critical magnitude of impedance can become zero at very high frequency.

Fig. 9.(a) The critical value of α versus ω for different β when LC = 10−3. (b) The critical value of magnitude of impedance versus ω for different β when LC = 10−3 . (c) The critical value of β versus ω for different α when LC = 10−3 . (d) The critical value of magnitude of impedance versus ω for different α when LC = 10−3 .

Also, the critical magnitude of the impedance versus α and β can be obtained when and α = β, which is given by

The relations between the critical value of α=β and magnitude of impedance versus ω for different LC values are presented in Fig. 10(a). As expected, the critical fractional orders decrease with the increase of frequency, which can also explain the phenomena happened in Fig. 8(a). From Fig. 10, the three minimal points can be obtained in Fig. 8(a), which are (0.77, 533.8), (0.71, 817.3) and (0.68, 1018), respectively.

3.6 Phase characteristics

From Eq. (7), the phase response of fractional order LβCα filter circuit is given by

3.6.1 Effect of LC

From Eq. (19), the relation between the phases of the impedance of the fractional-order LC filter circuit versus LC for different fractional orders is presented in Fig. 11.

Fig. 11.The phase of the fractional-order filter circuit versus LC for different fractional orders when ω=100.

Comparing the figures in Fig. 11, when α=β<1, with the increase of LC, the phase is gradually increases. In addition, the rate of increase is becoming larger with the increase of fractional orders. However, it’s quite opposite when α=β>1.

When α = β =1, phase could be infinite because the real part of the impedance is 0 in the conventional case.

3.6.2 Effect of the fractional orders

From Eq. (19), the relation between the phase of the impedance of the fractional-order filter circuit versus α =β is illustrated in Fig. 12.

Fig. 12.Phase of the impedance versus the fractional orders α = β when LC=0.001. (a) 0<α =β<1. (b) 1<α =β<2.

Fig. 12 shows the relation between the phases of the impedance versus the fractional orders for the different ω values. Clearly, from Fig. 12(a), the phase decreases as the fractional orders increase at the low frequency, while there is a critical phase at the high frequency, which will be illustrated in detail. In addition, the effect of the frequency on the phase is minimal when the fractional orders are very small. However, there is no critical phase at the high frequency as shown in Fig. 12(b), and the minimal effect of the phase on impedance can be obtained with the large fractional orders.

Based on above analyses, the critical phase of the impedance versus α = β can be easily obtained when Therefore, the relations between the critical fractional orders and phase of the impedance can be presented in Fig. 13.

Fig. 13.(a) The critical fractional orders versus the frequency with different LC values. (b) The critical phase of the impedance versus the frequency with different LC values.

With increase of the frequency and LC in Fig. 13(a), the critical fractional orders decrease. However, the interrupted phenomena happens for all curves when α = β =1, which is the conventional case. From Fig. 13(b), more than one critical phase of the impedance exists at the low frequency. Furthermore, the minimal phase of the impedance happened in Fig. 12(a) for ω=100 can be proved by Fig. 13. Specifically, αcp = βcp =0.56 and corresponding phase of the impedance is −0.85.

Fig. 14 shows the effects of fractional orders on the magnitude of phase. The results of the magnitude of the phase agree with the expected outcome, in that the larger magnitude of the phase can be obtained when α, β→1 shown in Fig. 14(a), (c) and (d). In addition, note that the local critical magnitudes of the phase exist in Fig. 14(b).

Fig. 14.The graphic model of the phase of the impedance versus fractional orders of the fractional-order filter circuit when ω=100 and LC=10−3. (a) α ∈(0,1) , β ∈(0,1) . (b) α ∈(0,1) , β ∈(1,2) . (c) α ∈(1,2) , β ∈(0,1) . (d) α ∈(1,2) , β ∈(1,2) .

Accordingly, new fundamentals of the fractional-order LC filter circuit can be obtained, including the critical factional orders, LC, the impedance and phase. Conequently, we can get the suitable electrical characteristics of the filter circuit by adjusting the two extra parameters and LC, which shows the greater flexibility of the fractional-order filter circuit on circuit design.

 

4. Filtering Characteristics

4.1 Amplitude-frequency characteristics

From Fig. 1 (c), the voltage gain function is presented as

Therefore, the expression of frequency characteristics is easily given according to the relationship between the frequency characteristics and the voltage gain function as

Furthermore, the amplitude of Eq. (21) can be expressed as

The amplitude-frequency characteristic is an important reflection of the filtering properties. In this subsection, we will systematically analyze the amplitude-frequency characteristics of the fractional-order LβCα filter circuit varying with the fractional orders, α+β and LC.

4.1.1 Effect of fractional orders

Fig. 15 illustrates the relation between the magnitude responses of Eq. (22) versus f for different fractional orders α+β values. From Fig. 15, the following new amplitude-frequency characteristics of the fractional-order LβCα filter circuit can be obtained.

Fig. 15.Amplitude-frequency characteristics of the fractional order LβCα filter circuit versus different fractional orders α+β values when LC=10−3.

Ripple output: the ripple output of the fractional order LβCα filter circuit is becoming larger with the increase of the fractional orders α+β.

Passband gain: the fractional order LβCα filter circuit will get larger passband gain as α+β increases in the passing band of the fractional-order filter circuit, which means the preferable effect of the passband gain can be acquired with smaller fractional orders.

Cut-off frequency: the cut-off frequency of the fractional order LβCα filter circuit decreases with the increase of the fractional orders just when LC=10−3, which will be proved later

Bandwidth: the bandwidth of the fractional-order filter circuit attenuations as α + β increase, which is also illustrated in Table 1.

Table 1.The relation between the filter factor versus the fractional orders when LC=10−3.

Damping coefficient and Quality factor: power loss of the inductance is less at a fixed frequency with the increase of α+β due to the smaller damping coefficient. However, it’s quite opposite for the quality factor because of the reciprocal relation between damping coefficient and quality factor.

With the increase of the fractional orders shown in Table 1, the filter factor is decreasing, which indicate that the filter circuit can get higher frequency resolution and selectivity.

Cut-off frequency, alias half power frequency, is also an important reflection of the filtering properties. In order to study the effects of cut-off frequency on the filtering properties better, Fig. 16 is presented.

Fig. 16.Cut-off frequency of the fractional order LβCα filter circuit versus the fractional orders α+β for different LC values.

As shown in Fig. 16, as expected, these results show a general trend that as α+β increase, the cut-off frequency rapidly decreases.

4.1.2 Effect of LC

Fig. 17 shows the effect of LC on the magnitude response with fixed fractional orders. Also, the new amplitude-frequency characteristics can be described as follows.

Fig. 17.Amplitude-frequency characteristics of the fractional order LβCα filter circuit with different LC values when α+β =1.5.

Ripple output: the ripple output of the fractional order LβCα filter circuit is unaffected by the change of LC with fixed α +β.

Passband gain: the passband gain is unaffected by LC when α +β is fixed.

Cut-off frequency: the cut-off frequency of the fractional-order filter circuit decreases with the increase of LC with fixed α +β, which will be proved in Fig. 18.

Fig. 18.Cut-off frequency of the fractional order LβCα filter circuit versus LC for different fractional orders.

Bandwidth: the bandwidth of the fractional-order filter circuit attenuations as LC increases, which is also illustrated in Table 2.

Table 2.The relation between the filter factor versus LC when α+β =1.5.

Damping coefficient and Quality factor: power loss of the inductance remains unchanged at fixed α +β with the increase of LC due to the unchanged damping coefficient. In other words, LC doesn’t affect the damping coefficient. Also, the quality factor is unchanged.

Also, the relation between the filter factor versus the LC is presented in Table 2.

From Table 2, the effect of LC on the filter factor is minimal.

To describe the effects of LC on the cut-off frequency in detail, Fig. 18 is presented below.

Fig. 18 shows the relation between the cut-off frequency versus LC for different fractional orders. Clearly, the cut-off frequency rapidly decreases as LC increases. And it’s worth noting that the cut-off frequency may not exist for the fixed LC when α+β =2, which validates the deficiency of the integer-order filter circuit in designs.

4.2 Phase-frequency characteristics

The phase-frequency characteristic is also an important indicator of the filtering properties, and the analysis of it is the main part of the subsection. Furthermore, the relationships between the phase-frequency characteristics of the fractional-order filter circuit and fractional orders α+β, LC are analyzed in the following contents.

4.2.1 Effect of LC

From Eq. (21), the effects of LC on the phase response are presented in Fig. 19.

Fig. 19.(a) The phase response of the fractional-order filter circuit versus frequency for different LC when α + β =1.5; (b) The phase response of the fractional-order filter circuit versus frequency for different LC when α + β=2; (c) The graphic model of the phase response of the fractional-order filter circuit versus different LC when f =100 and α + β =1.5; (d ) The graphic model of the phase response of the fractional-order filter circuit versus different LC when f =100 and α + β =2.

From Fig. 19(a), with increase of operating frequency, the lagging phase angle of the fractional-order filter circuit decreases to a fixed value with fixed fractional order α+β. Moreover, the filter circuit can obtain the same minimal and maximal lagging phase angle in the high frequency and very low frequency for different values. However, it’s quite different in the conventional case as shown in Fig. 19(b), in that the phase angle is zero at very low frequency, and it’s π at high frequency. Such results can be explained when the real part of Eq. (21) tends to zero at very low frequency, while it’s negative at high frequency. From Fig. 19(c) and (d), we see that L and C values have the same effects on the phase response, which agrees with theoretical analysis.

4.2.2 Effect of α+β

Also, the effects of α+β on the phase response are presented in Fig. 20.

Fig. 20.(a) The phase response of the fractional-order filter circuit versus frequency for different α +β when LC=10−3. (b) The graphic model of the phase response of the fractional-order filter circuit versus fractional orders when f =100 and LC=10−3.

From Fig. 20(a), as for the fractional-order filter circuit, the lagging phase angle decreases in the high frequency as fractional orders increase, while it’s a fixed value π for the conventional case. And the fractional order β plays a leading role in the phase-frequency characteristics as shown in Fig. 20(b).

 

5. Conclusions and Discussions

This paper systematically studies the fundamentals of using fractional-order capacitors and fractional-order inductors instead of integer-order elements, and creatively advances the LC filter circuit. From the point of view of the circuit, the new fundamentals of the fractional-order filter circuit are discussed in detail, including the pure real angular frequency, the pure imaginary angular frequency, the short circuit angular frequency and the sensitivity analysis of the impedance and phase characteristics, where many interesting phenomena are presented, and it shows a broad view of the fractional-order filter circuit since two extra parameters are introduced. Furthermore, we systematically study the amplitude-frequency characteristics and phase-frequency characteristics of the fractional-order LC filter circuit to show the greater flexibility of the fractional-order filter circuit.

In this paper, we make an important theoretical contribution to study the fractional-order LC low-pass filter circuit. Many new fundamentals are obtained by the results of the rigorous mathematical analyses and numerical experiments. In the future, we will devote ourselves to the physical experiments of the fractional-order filter circuit with the new fundamentals in the paper. Additionally, we will also try our best to the applications of the fractional-order LC low-pass filter circuit.