Carbon Black based capacitive Fractional Order Element towards a new electronic device

Int. J. Electron. Commun. (AEÜ) 84 (2018) 307–312

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Regular paper

Carbon Black based capacitive Fractional Order Element towards a new electronic device

T

⁎

A. Buscarinoa, R. Caponettoa, , G. Di Pasqualeb, L. Fortunaa, S. Graziania, A. Pollicinoc a

DIEEI, Universitá degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy DSC, Universitá degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy c DICAR, Universitá degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy b

A R T I C L E I N F O

A B S T R A C T

Keywords: Fractional order element Nano structured dielectrics Model identification Non integer order filters

In this paper, Fractional Order Elements (FOEs), fabricated by using a carbon black nano structured dielectrics, are presented. FOEs have been realized by varying different fabrication parameters such as the percentage of carbon black, the curing temperature, and the solvent type. Results on the experimental frequency characterization of one FOE device are given. The FOE has been, then, used for demonstrating the possibility of realizing a fractional order RC filter. The frequency analysis of the RC filter shows the coherence of the fractional order between the FOE and corresponding RC circuit.

1. Introduction

capacitors (C ) are electric passive circuit elements, whose impedance is given by:

Fractional calculus is a quite general approach that has been already proposed in a growing numbers of fields, due to its capability of modeling and controlling systems characterized by long term properties [1]. Typical applications can be found in automatic control area. In [2], fractional order PID controllers have been introduced. Their stability region have been investigated in [3]. The corresponding tuning procedure has been introduced in [4]. Applications, both in analog [5] and digital [6,7] controller implementation, have been proposed. Applications to system modeling can be found in the supercapacitor area [8], as well as in medical applications [9,10]. More specifically, in [9], the human respiratory system is emulated, using approximated fractional-order capacitors and inductors. In [10] the identification of a model for respiratory tree, by using an equivalent electrical model, is proposed. The relationship between chaos and fractional order systems has been investigated in [11], while their Lyapunov exponents characterization is introduced in [12]. Applications, related to electronic devices can be found in [13], where the numerical analysis of the fractional harmonic oscillator has been proposed. The Fractional Order Impedance (FOI) characterization has been studied in [14–16]. Modeling of non integer order electronic devices is needed for the devices proposed in this paper. Resistors (R) , inductors (L) and the

Z (s ) = Ks−α

⁎

where K is a gain, s represents the Laplace variable and α ∈ {−1,0,1} , for capacitor, resistor and inductance, respectively. The impedance is described in the frequency domain by substituting jω for s, where j is the imaginary unit and w is the angular frequency. The impedance then becomes:

Z (jw ) = K (jw )−α

(2)

K /w α

and The magnitude and phase of this impedance are |Z| = Arg (Z ) = −απ /2 , respectively. For α ∈ {−1,0,1} , the phase is π/2 (i.e., inductor), 0 (i.e., resistor), and −π/2 (i.e., capacitor), respectively. Actually, the value of α is not necessary an integer number. It can rather assume any value in  . In such a case, the systems are modeled using the fractional order approach, see [17,18]. According to (2) a fractional element exhibits a constant phase behavior and is often referred as a Constant Phase Element (CPE). The availability of CPEs will play a fundamental role in the possibility of realizing non integer order electronic circuits. CPE realization and applications can be found respectively [19–22]. Realization of Fractional Order Element (FOE), have been proposed in [23,24] where electrolytic process and Ionic Polymeric Metal Composite, respectively, are investigated in the perspective of the

Corresponding author. E-mail addresses: [email protected] (R. Caponetto), [email protected] (G. Di Pasquale), [email protected] (S. Graziani), [email protected] (A. Pollicino). https://doi.org/10.1016/j.aeue.2017.12.018 Received 3 October 2017; Accepted 15 December 2017 1434-8411/ © 2017 Published by Elsevier GmbH.

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Int. J. Electron. Commun. (AEÜ) 84 (2018) 307–312

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realization of fractional order capacitors. Nanocomposite based fractional capacitor have been proposed in [25] and their packaging study is given in [26]. The focus of this paper is on the possibility of exploiting nanostructured materials, in particular Carbon Black (CB ) based composites, for realizing non integer order devices. More specifically, CB composites are exploited for realizing the dielectric of capacitors that can be modeled as FOEs. By using such components, a new class of filters is proposed, with frequency characteristics that can not be easily obtained by using traditional components. The realization of analog fractional order systems, in the form of fractional order filters, represents the counterpart of their digital implementation [7]. At present, fractional order transfer functions are essentially digitally implemented. The fractional order and the corner frequency approximations bring to the implementation of high order digital filters. In the paper the possibility of using the proposed FOEs for realizing RC filter is implemented in hardware. This allows for directly using an analog device, without any approximation. 2. Carbon Black as dielectric for new FOE electronic devices It is widely known in the literature that, by adding a conductive filler to an insulating matrix (generally a polymeric matrix), both the electrical and mechanical properties of the composite can change dramatically. Such a phenomenon has been widely exploited for realizing polymer conducting composites, with application, e.g., such as thermistors, deformation sensors [27], pressure sensors [28], tactile sensors, and gas sensors [29]. Here, the possibility of using CB, as the filler of a polymeric matrix, for realizing nano composites with dielectrics properties is investigated. Sylgard184 has been used for realizing the polymeric matrix of the capacitor dielectric. It was purchased from DowCorning as a two part liquid elastomer kit. Part A (consisting in the vinyl-terminated PDMS prepolymer), and Part B (the crosslinking curing agent, consisting in a mixture of methylhydrosiloxane copolymer chains with a Pt catalyst and an inhibitor). CB (acetylene, 100% compressed, 99.9%, specific area 75 m2 /g , bulk density 170–230 g/L, average particle size 0.042 μm ) was purchased from AlfaAesar and used as received. The described CB based composite material is the dielectric of the capacitances investigated in the following of the paper. The resulting structure of the CB-Fractional Order Element (CB-FOE) is given in Fig. 1. The CB-FOE samples have been prepared by mixing the PDMS and the crosslinking agent in a weight ratio of 1:10 in a Teflon crucible. The mixture was mixed for 10 min. CB has been added for achieving the desired concentration. The mixture was stirred for further 10 min for enhancing the dispersion of the CB. Curing at different temperatures were carried out, taking into account both the manufacturer recommended curing time and the heat propagation through the mold. This results in a stabilization time, required for the temperature of the curing PDMS approaching the desired curing temperature. The mixture was used for realizing the capacitors dielectric by pouring the viscous mixture into the device. The mixture was allowed to crosslink at room temperature, or in an oven preheated to the desired temperature, for 48 h. More specifically, the obtained dielectrics have been used to realize cylindrical capacitors, whose geometry is shown in Fig. 2. Capacitances considered in the following had copper-based shell with hight h = 8 cm, internal diameter a = 0.6 cm and external diameter b = 1.2 cm. The curing time was 53 min at 100 °C,38 min at 125 °C, and 28 min at 150 °C. The CB percentage, the curing temperature and the solvent type, have been fixed as reported in Table 1. Experiments reported in the following will refer only to the capacitors C −100,C −150 and C −200 , all realized with a CB percentage of 8%. This choice comes from the idea of characterizing the capacitors with

Fig. 1. A scheme of the CB-FOE structure and corresponding electrical approximation.

respect to the curing temperature parameter. As a first step, the frequency behavior of the capacitive devices has been investigated. The experimental surveys were performed by using a network analyzer Agilent Technologies E5071B. Moreover, the devices were connected to the measuring instrument, by using adequate connectors, in such a way to avoid any high-frequency parasitic effect. A picture of a FOE device under test, along with the used connector is shown in Fig. 3. As an example, the results of the frequency analysis of the FOE C125 is shown in Fig. 4. As it can be noticed, in the frequency range 105–109 Hz, the phase assumes approximatively a constant value equal to −75°. Such a phase value corresponds to α = 0.85. The slope of the magnitude diagram is equal to 17.14 dB/dec , which is in good agreement with expected value, i.e. α∗20 dB = 0.85∗20 dB = 17 dB/dec . 3. RC filters by CB-FOE The CB-FOE element, described so far can be valuable resource for the hardware implementation of non integer order filters. In the following, experimental results obtained by the investigation of non integer order low-pass filters. fabricated by using the CB-FOE capacitors, are reported. When a low pass filter is considered, the transfer function assumes the following form:

Z (s ) =

K (1 + τs )α

(3)

where τ is the pole of the system and, as reported in Section 1, α ∈  . The fractional nature of (3) reflects in its Bode diagram. More specifically, both the asymptotic slope of the magnitude and the phase, are multiplied by α . If ω → ∞,|Z| becomes −20α log10 (ωτ ) . On a semidB logarithmic plane, a line with slope −20α dec (instead of −20 dB/dec as for first order system), will be observed. The expression of Arg (Z ) shows that α modulates the scale of the phase law [17]. In fact, it can be easily seen that, for ω → ∞, the phase angle approaches −απ/2 , instead of −π/2 , as it occurs for a first order low-pass filter. 308

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Fig. 2. Sample of the CB-FOE under investigation.

Table 1 List of the realized CB-FOE. Name

CB %

Curing temperature

Capacitance value

C1-R C2-R C-100 C-125 C-150 C-S1 C-S2

1% 2% 8% 8% 8% 5% Xiline 5% CHCl3

Room Room 100°C 125°C 150°C Room Room

≈ 25 pF ≈ 30 pF ≈ 76.6 pF ≈ 207 pF ≈ 150 pF ≈ 137 pF ≈ 50 pF

Fig. 4. Frequency analysis of FOE C-125. m1, m2, and m3 pointers have been used for estimating the module slope and the phase lag, respectively.

Magnitude [dB]

Bode plot

Fig. 3. Device under test.

0

α=1.3 α=1 α=0.7

−50

−100 −4 10

−2

10

0

10

2

10

4

10

Frequency

In Fig. 5, examples of the Bode diagram obtained for three first order low-pass filter are given. More specifically, the value of the pole is the same (τ = 1 s ), while α assumes the values [0.7,1,1.3] respectively. Figs. 6 and 7, shown the comparison of the Bode diagram of a non integer low-pass filter, realized by using the FOE described in the following of the paper, with two low-pass RC filters realized by using commercial components. The plots outline the non integer order nature of the fabricated devices. The transfer functions of the first order lowpass filters are:

G1 (s ) =

1 (1 + 105s )

Phase [deg]

0 −50 −100 −150 −4 10

−2

10

0

10

2

10

4

10

Frequency Fig. 5. Bode diagram of fractional system F (s ) = 1/(s + 1)α with α = 1 (-), α = 1.3 (∗), α = 0.7 (+).

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G2 (s ) = and

1 (1 + 0.87∗105s )

for C1 = 130 pF and C2 = 150 pF respectively, and R = 76.8 kΩ . 309

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Int. J. Electron. Commun. (AEÜ) 84 (2018) 307–312

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0

Magnitude [dB]

Bode plots 130 pF

Magnitude [dB]

−5 −10

FOE

−15

0 −10 −20 −30 2 10

3

10

150 pF

6

10

5

0

Phase [deg]

10

5

10

Frequency

−20 −25 4 10

4

10

6

10

Frequency

−50

−100 2 10

3

10

Fig. 6. Magnitude comparison of the realized and commercial, order one, capacitors.

4

10

5

10

6

10

Frequency Fig. 9. Bode data fitting for the CB-FOE@100° .

red 150pF − green FOE − blue 130pF 0 −10

were fixed in such a way to experimentally investigate 100 frequency values per decade. Both the input Vin and output Vout signals were acquired by using a National Instrument NI / USB−6251 board. In order to study the behavior of the CB-FOE based RC filters, two different approaches have been applied. The first one consists in fitting the filter Bode diagram of transfer function (3) to the corresponding experimental Bode diagram, estimated from the acquired data. The unknown parameters in (3), K ,α , and τ , were determined by using a standard genetic algorithm [30], with the following parameters: population size = 2000, generations number = 100, generation gap 0.9 and number of bit per variable equal to 10. The cost function, that has been minimized, takes into account both the absolute mean square errors on the magnitude and phase estimation. The obtained results can be observed in Figs. 9–11 for the filters realized by using the CB-FOEs, C −100,C −125 and C −150 (for adopted symbols refer to Table 1), respectively. The estimated values of the parameters, K ,τ , and the value of α are given in Table 2. It is possible to observe that the interpolation is effective only for C −100 and C −125, while for the C −150 CB-FOE based filter, the parameters of the Bode model obtained for (3) does not allow for a satisfactory fitting of experimental data. In the second approach, the asymptotic phase values was used for determining the values of the α parameters. For each CB-FOE based RC filter the mean valus of the last 50 points, of the estimated phase Bode diagram, were considered. The values of α has been established as the mean values of the phase lag, in degrees, relative to the value 90°. The slope of the magnitude was estimated multiplying α by 20 dB/dec . Note that in this second method the value of α has been estimated by using only the Bode diagram

130pF

Phase [deg]

−20 −30 −40 −50

FOE 150pF

−60 −70 −80 −90 −100 3 10

4

10

5

10

6

10

7

10

Frequency Fig. 7. Phase comparison of the realized and commercial, order one, capacitors.

In fact, a perusal of Figs. 6 and 7, shows that the commercial capacitors of capacitance C = 150 pF and C = 130 pF , respectively, have the classical Bode trend. The FOE under investigation (indicated as FOE in the figure) has a non integer order Bode diagram, characterized by an asymptotic behavior of magnitude and phase lag which differ, from −20 dB/dec and −π/2, respectively. Graphs reported in Figs. 6 and 7 are part of a deeper investigation performed on non integer FOE-based filters, realized by using the CBbased FOEs. The capacitors were connected in series configuration with a resistor of 12 kΩ. Such a resistance value was chosen for fixing the pole position of the low-pass filters at frequencies laying in the frequency range investigated for the FOE characterization. In such a way, it was possible analyzing both the low frequency behavior of the filter, where the FOE corresponds to an open circuit, see Fig. 8, and the high-frequency filter behavior, which are affected by the FOE dynamics. A sinusoidal input voltage Vin was produced by using a waveform generator Agilent 33220A. The sinusoidal signal was applied through an operational amplifier, ST TL082CP , in buffer configuration, as shown in Fig. 8. The frequency range from 102 to 107 Hz was investigated and the signal amplitude was fixed a to V = 3.8. Moreover, frequencies values

Magnitude [dB]

Bode plots 0

−20

−40 2 10

3

10

4

10

5

10

6

10

Frequency Phase [deg]

0

−50

−100 2 10

3

10

4

10

5

10

Frequency Fig. 10. Bode data fitting for the CB-FOE@125° .

Fig. 8. Conditioning circuit for data acquisition.

310

6

10

Int. J. Electron. Commun. (AEÜ) 84 (2018) 307–312

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Bode plots

Magnitude [dB]

Bode plots

Magnitude [dB]

0 −10 −20 −30 2 10

3

10

4

5

10

10

6

10

0 −10 −20 −30 2 10

3

10

−50 −100 2 10

3

10

4

10

5

10

6

5

10

10

Frequency

0

4

5

10

10

0

Phase [deg]

Phase [deg]

Frequency

6

−50

10

−100 2 10

Frequency

3

10

4

10

6

10

Frequency

Fig. 11. Bode data fitting for the CB-FOE@150° .

Fig. 13. Data fitting for the CB-FOE@125° by using the asymptotic phase value. The solid line shows the slope of the diagram. Table 2 Determination of the α value by using the Bode diagram.

C-100 C-125 C-150

α

τ

0.98 0.938 0.91

≃ 0.75 ≃ 0.84 ≃ 0.7

0.00003 0.00006 0.00008

Magnitude [dB]

Bode plots K

0 −10 −20 −30 2 10

3

10

4

5

10

6

10

10

Frequency Phase [deg]

phase, while the corresponding Bode diagram amplitude was used to check the adequacy of the obtained high frequency asymptotic slope. The pole position linked to the τ value was not taken into account, since it does not depend from on α value. The results obtained with the described procedure are plotted in Figs. 12–14 and summarized in Table 3. It is worth noticing that the results reported for FOE C-125 characterization are fully confirmed by the behavior observed for the corresponding low-pass filter. More generally, the results obtained for the filters that use different CB-based FOE elements allows to foresee a dependence of the order α from the curing temperature and, therefore, the possibility of controlling, by design, the behavior of non integer filters. In fact, in Table 3 it is possible to note that α increases from 0.77 to 0.86 via 0.83 when the curing temperature raises from 100° to 150°, with a corresponding influence on the Bode diagram of the low-pass filter.

0 −20 −40 −60 −80 −100 2 10

3

10

4

5

10

6

10

10

Frequency Fig. 14. Data fitting for the CB-FOE@150° by using the asymptotic phase value. The solid line shows the slope of the diagram.

Table 3 Determination of the α value by using the asymptotic phase value (90°) .

C-100 C-125 C-150

Curing temperature (°C)

Asymptotic value (°)

α

Magnitude slope (dB/dec)

100 125 150

≃ −70 ≃ −75 ≃ −78

≃ 0.77 ≃ 0.83 ≃ 0.86

≃ 15.5 ≃ 16.6 ≃ 17.2

Magnitude [dB]

Bode plots 0

4. Conclusions

−10

The fabrication of fractional order capacitors, called here FOEs, with desired specification has been demonstrated in the paper. Both the realization of CB nano-structured composites FOEs and their application in fractional low pass filters has been reported. The frequency analysis shows that these filters possess a non integer order behavior. It has been shown that the manufacturing process plays a crucial role in determining the FOE characteristics. A dependance between the curing temperature and the fractional order of the FOEs have been demonstrated. The final goal of the proposed research consists in realizing process tunable analog devices as basic elements for implementing fractional order filters and circuits, without any need of using digital approximation procedures, nor complex hardware realization. This will require scouting of new composite materials. More specifically, graphene based FOEs, G-FOEs, will be considered in order to individuate the influence of this filler on the fractional order value of the G-FOEs.

−20 −30 2 10

3

10

4

10

5

10

6

10

Phase [deg]

Frequency 0

−50

−100 2 10

3

10

4

10

5

10

6

10

Frequency Fig. 12. Data fitting for the CB-FOE@100° by using the asymptotical phase value. The solid line shows the slope of the diagram.

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Particular attention will be given to the possibility of reducing the size of the electronic devices, providing, also, stable in time FOEs. The availability of better FOEs will allow for utilizing them to realize non integer order filters, including non integer order controllers and non integer order PIDs.

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