Electrical equivalent circuit modelling of solid state fractional capacitor

Accepted Manuscript Electrical equivalent circuit modelling of solid state fractional capacitor Dina A. John, Karabi Biswas PII: DOI: Reference:

S1434-8411(17)30342-4 http://dx.doi.org/10.1016/j.aeue.2017.05.008 AEUE 51879

To appear in:

International Journal of Electronics and Communications

Accepted Date:

5 May 2017

Please cite this article as: D.A. John, K. Biswas, Electrical equivalent circuit modelling of solid state fractional capacitor, International Journal of Electronics and Communications (2017), doi: http://dx.doi.org/10.1016/j.aeue. 2017.05.008

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Electrical equivalent circuit modelling of solid state fractional capacitor Dina A. Johna,∗, Karabi Biswasa,1 a

Department of EE, IIT Kharagpur, Kharagpur, West Bengal, India 721302.

Abstract The paper proposes an electrical equivalent circuit model (EECM) for a new type of solid state fractional capacitor. The fractional capacitor is fabricated by sandwiching the polymer-CNT nanocomposite in between the electrodes. The parameters of the EECM are extracted from MEISP software and correlation of these parameters with wt% of CNT is studied in detail. The error analysis of the fractional exponent (α) values from experimental and simulated data are also presented. Keywords: Fractional capacitor, Constant Phase (CP) zone, Electrical Impedance Spectroscopy (EIS), EECM

∗

Corresponding author. Email addresses: [email protected] (Dina A. John), [email protected] (Karabi Biswas) 1 Associate Professor at Department of EE.

Preprint submitted to Elsevier

May 6, 2017

1. Introduction The electrical impedance of a fractional order element (FOE) can be given as [1] Z(s) =

1 F sα

(1)

where F is fractance and α, the fractional exponent. The magnitude and 1 −απ phase of the impedance is given as |Z(jω)|=| F (ω) reα | and ∠Z(jω)= 2

spectively. Depending on the values of α, the component can be termed as 5

resistor (α=0), capacitor (α=1) or inductor (α=−1). If α takes up values in between 0 to ± 1, then the device is named as fractional capacitor/ inductor since α is a non-integer value. The significance of fractional order element is that it is governed by fractional order differential equation and is utilized for many applications like fractional order system modelling, filters, oscillators,

10

impedance matching etc. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. As our studies are focused on fractional capacitor, the question : How can we get a fractional capacitor? arises. To answer this, one has to dig the literature for realization of fractional capacitor. Hence, one finds that the fractional capacitors can be realized either by single component or multicomponent method. In

15

the case of multicomponent method [12, 13, 14, 15, 16, 17, 18, 19], the desired α can be emulated by R-C combination. The main challenge is that the designed values of R and C may not be commercially available, making the implementation really cumbersome. To avoid this, researchers have developed single component fractional capacitor [20, 21, 22, 23, 24, 25, 26]

2

20

which can replicate the infinite number of R-C network in between the two electrodes. But the main shortcomings include: commercialization not yet done [20, 21, 22, 24, 25, 26], spillage of liquid [22, 24, 25] and difficulties in identifying the parameters to control the value of α [20, 21, 24]. Keeping these points in mind, we have developed a solid state fractional capacitor

25

where the α value can be varied by changing the filler % (i.e. CNT) and has dimension close to commercially available capacitor. The description of the fractional capacitor for which the electrical modelling is realized has been presented in [27]. For electrical modelling, we require EIS data. By definition, EIS [28, 29] is

30

a method by which one characterizes the dynamics of the process happening in a device. Even though EIS data can be obtained by using potentiostat [30] or impedance analyzers [31, 32], yet the complexity of extracting the exact information of the process is complicated and it is a hefty procedure for the material scientists and electrochemists [33]. Different techniques are avail-

35

able for the analysis of EIS data and one such approach is EECM method. This motivated us to understand the phenomenon in the developed fractional capacitor by correlating the EIS data with the electrical model. There are many softwares available to extract electrical equivalent circuit from the EIS data. In this work, we have used MEISP software [34](Kumho Chemical

40

Laboratories) to get the EECM of the solid state fractional capacitor. It is user friendly, uses complex non-linear squares (CNLS)[29] algorithm to get good fit of the experimental data and require no prior knowledge. 3

The primary aim of this paper is to find the EECM of the fractional capacitor which represents the different phenomenon in it and the relation of 45

various parameters (like resistance, fractance and the fractional exponent) of EECM to % CNT. The paper consists of 5 sections. First section deals with introduction and the experimental details are described in section 2. Section 3 provides the proposed electrical equivalent model, section 4 elaborates the results and finally concluding the work in section 5.

50

2. Experimental Details 2.1. Description of solid state fractional capacitor The developed fractional capacitor [27] consists of 3 plates (shown in Figure 1). The middle plate is of porous material with size less than the left and right plates. The nanocomposite is placed between left and the

55

middle plate as well as in between middle plate and the right plate. The nanocomposite used in our study comprises of conductive filler (i.e. Multiwalled Carbon Nanotubes (MWCNT)) dispersed in the epoxy resin and the procedure for its fabrication is given in [27]. When the fractional capacitor is excited by a sinusoidal voltage of varying frequency, the time taken by

60

the electrons to travel the distance in between the plates through CNTs determines the CP zone. But the CP zones are obtained for the cases when wt% of CNT > 0.75, since the percolation threshold2 was found to be in 2

Percolation threshold [35, 36] by definition, is the concentration of CNT which causes the formation of conductive pathways in the CNT-polymer nanocomposite.

4

between 0.5 - 0.75 % [27]. CNTs Right plate

Left plate Epoxy resin

Inter tubular spacing between CNT B Electrical connection

A Electrical connection

Middle plate Figure 1: Cross sectional view of solid state fractional capacitor

2.2. Measurements 65

The experimental data required for the electrical modelling is recorded with the help of Alpha-A impedance analyzer from Novocontrol Technologies. It is an impedance analyzer used for measuring the impedance when the frequency sweeps from 0.01 Hz to 20 MHz. The magnitude and phase angle of the impedance is captured by defining the frequency points in the WinDETA

70

software of the analyzer. Our study is limited to the frequency range 100 Hz - 20 MHz as the CP angle is obtained in this frequency range and research is going on to extend the frequency range to mHz region, thereby making the fractional capacitor to be employed for controller design applications.

5

2.3. Configuration Details 75

The configurations for which EECM is proposed are from [27] and the details are stated in Table 1. Table 1: Details of the configurations

Configuration Size of the middle plate Size of the left and right plate I

1 cm of 11 µm pore size fil- Copper plate of size - 1.5 cm2

ter paper II

1 cm of 20 µm pore size fil- Copper plate of size - 1.5 cm2

ter paper

3. Electrical equivalent circuit modelling of the solid state fractional capacitor The proposed EECM is shown in Figure 2. We have tried various elec80

trical models to fit the EIS data but the model presented here (in Figure 2) provided the minimum absolute error as well as minimum relative SD for the parameters extracted from MEISP software. Each configuration of fractional capacitor mentioned in Table 1 is modelled as two parallel combination of resistance (R) and constant phase element (X) connected in series. It is already

85

mentioned that the simulation of the electrical model is done using MEISP software package. The software takes frequency, real (ZR ) and imaginary (Zi ) part of impedance as the input to extract the final parameters of EECM

6

using CNLS algorithm. Initial values of the electrical components can be given manually or the software takes it automatically. It gives best results if 90

the relative standard deviation (SD) of each parameter is less than 20%. In Figure 2, R1 and R2 are the resistances, X1 and X2 represents CPEs.

Figure 2: Proposed electrical equivalent circuit model; Ref - Reference voltage and Gnd Ground

X1 has the parameters fractance, F1 and fractional exponent, α1 whereas X2 has the parameters F2 and α2 . Parallel combination of R1 -X1 represents the grain effect or due to CNT alone. Correspondingly, in the frequency re95

gion where R1 -X1 is dominant (>100 kHz), the movement of electrons inside the MWCNT defines the data points of the Nyquist plot. The parallel combination R2 -X2 is due to the motion of electrons across the grain boundary (i.e. in the interface between the CNT and polymer matrix) which is usually prominent in the low frequency region (up to 100 kHz). In the low frequency

100

region, the distance travelled by electrons via tunneling after the introduction of delay caused by the middle plate, contributes the data points in the Nyquist plot. In general, the diameter of R1 -X1 semicircle is smaller than the diameter 7

of R2 -X2 semicircle. For configurations I and II, electrical modelling is done 105

with % CNT varied from 0.75% onwards. The total impedance of the circuit in Figure 2 is given as below:

Z(jω) =

R1 R2 + α 1 1 + R1 F1 (jω) 1 + R2 F2 (jω)α2

(2)

And rearranging equation 2, we get the real and imaginary part of the impedance as:

ZR (jω) =

R1 (1 + R1 F1 ω α1 cos( α21 π )) + 1 + 2R1 F1 ω α1 cos( α21 π ) + (R1 F1 ω α1 )2 R2 (1 + R2 F2 ω α2 cos( α22 π )) 1 + 2R2 F2 ω α2 cos( α22 π ) + (R2 F2 ω α2 )2

R1 F1 ω α1 sin( α21 π ) − 1 + 2R1 F1 ω α1 cos( α21 π ) + (R1 F1 ω α1 )2 R2 F2 ω α2 sin( α22 π ) 1 + 2R2 F2 ω α2 cos( α22 π ) + (R2 F2 ω α2 )2

(3)

Zi (jω) = −

(4)

4. Results and Discussions In this section, the EECM parameters obtained from MEISP software for the configurations mentioned in Table 1 as well as the relation of the parameters with % CNT are elaborated. 110

1. For Configuration I As we need to find the α corresponding to CP zone, the results are plotted for the variation of real (ZR ) and imaginary (Zi ) part of impedance 8

with frequency instead of Nyquist plot. The experimental (Z-Exp) and simulated (Z-Sim) data of ZR and Zi are shown in Figure 3 and 4. And from these figures, α corresponding to the CP zone is obtained from the slope of experimental imaginary impedance (Zi -Exp) plot; Whereas, αc is determined from simulated imaginary impedance (Zi -Sim) plot and both the fractional exponents are illustrated in Table 2. Variation of ZR and Zi with frequency

Variation of ZR and Zi with frequency for 0.75% of CNT

107

for 2 % of CNT

106

ZR-2-Sim

ZR-0.75-Sim

106

ZR-2-Exp

ZR-0.75-Exp

Zi-2-Sim

5

10

Zi-0.75-Sim

Zi-2-Exp

Zi-0.75-Exp

105 i 4

10

Zi-0.75-Sim Zi-0.75-Exp

103

ZR and Zi

104

ZR-0.75-Exp

R

Z and Z

115

Zi-2-Exp

103 Zi-2-Sim

ZR-2-Exp

102

102

ZR-0.75-Sim

ZR-2-Sim 1

1

10 102

104

106

108

10 102

104

106

Frequency (Hz)

Frequency (Hz)

Figure 3: Comparison of real and imaginary part of impedance with frequency for simulated and experimental data (0.75 and 2% of CNT)(ZR -real part of impedance and Zi imaginary part of impedance)

Table 2: Comparison of α from the simulation and experimental data of configuration I Frequency range 100 kHz - 20 MHz 2 MHz - 20 MHz *

% of CNT 0.75 2 2.5

α

αc

|Ea |*

0.931 0.904 0.885

0.910 0.834 0.814

0.02 0.07 0.07

Absolute Error |Ea |=|αc − α|

9

108

Variation of Z R and Z i with frequency for 2.5% of CNT

106

Z R-2.5-Sim Z R-2.5-Exp

105

Z i-2.5-Sim

ZR and Z i

Z i-2.5-Exp

104 Zi-2.5-Sim Z i-2.5-Exp

103

Z R-2.5-Exp

102 Z R-2.5-Sim

101 2 10

103

104

105

106

107

108

Frequency (Hz)

Figure 4: Comparison of real and imaginary part of impedance with frequency for simulated and experimental data (2.5% of CNT)

The details of the electrical parameters obtained from EECM simulation of configuration I are given in Table 3. Whereas the plot of R’s and F’s with % of CNT is demonstrated in Figures 5 and 6. 108

Variation of R's with CNT

10-8

R1

Variation of F's with CNT F1 F2

107

-1)

106

F1 and F2(F.sec(

)

R2

R1 and R2

120

105 0.5

1

1.5

2

2.5

% CNT (wt %)

10-9

10-10 0.5

1

1.5

2

% CNT (wt %)

Figure 5: Variation of the electrical parameters (R and F) with % CNT for configuration I

10

2.5

Table 3: Circuit Parameters of Configuration I % CNT 0.75

Parameter F2 (F.secα−1 ) α2 R2 (Ω) R1 (Ω) F1 (F.secα−1 ) α1 F2 (F.secα−1 ) α2 R2 (Ω) R1 (Ω) F1 (F.secα−1 ) α1 F2 (F.secα−1 ) α2 R2 (Ω) R1 (Ω) F1 (F.secα−1 ) α1

2

2.5

Value 1.368 × 10−10 0.912 1.508 × 107 1.161 × 106 1.351 × 10−10 0.909 3.413 × 10−10 0.844 2.168 × 105 6.476 × 105 2.794 × 10−9 0.802 5.400 × 10−10 0.827 1.260 × 105 4.856 × 105 3.07 × 10−9 0.788

Relative SD 5.627 × 10−6 1.008 × 10−6 1.814 × 10−6 7.758 × 10−6 4.310 × 10−6 8.896 × 10−7 7.485 × 10−6 1.003 × 10−6 6.92 × 10−6 4.740 × 10−6 1.777 × 10−5 3.36 × 10−6 1.083 × 10−5 1.486 × 10−6 1.07 × 10−5 5.057 × 10−6 1.945 × 10−5 3.671 × 10−6

1 0.95

c

0.85

2

0.8

1

0.9

0.75

1

c 2

0.7 0.65 0.6

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

% CNT (wt %)

Figure 6: Variation of the electrical parameters(α) with % CNT for configuration I

The inferences from the parameters (for the cases 0.75, 2 and 2.5% )

11

2.6

obtained through MEISP software are given below: a. The value of resistances R1 and R2 decreases with increase in wt% of 125

CNT. b. The values of fractional exponents α1 and α2 decreases with increase in wt% of CNT. In Figure 6, the value of αc (calculated from the imaginary part of impedance (Zi -Sim)) for the CP frequency zone is also shown. α1 is a parameter in the frequency region where R1 -X1 is

130

dominant and α2 is obtained from the frequency region where R2 -X2 is effective. But αc is prevailing only in the CP frequency zone (details in Table 2). c. The value of fractances F1 and F2 increases with increase in wt% of CNT.

135

In general, the distance between the CNTs get reduced with increase in CNT loading, which is explicitly shown by the decrease in resistances as well as the increase in fractances. 2. For Configuration II The experimental and simulated data of ZR and Zi are shown in Figures 7

140

and 8 and Table 4 presents α and αc obtained for the CP zone calculated from Zi -Exp and Zi -Sim. The details of the parameters obtained from simulation are given in Table 5. The plot of R’s and F’s with % CNT loading is depicted in Figures 9 and 10.

12

Variation of ZR and Zi with frequency for 0.75% of CNT

108

Zi-0.75-Sim

ZR-2-Exp

105

Zi-2-Sim

Zi-0.75-Exp

Zi-2-Exp

ZR and Zi

ZR-0.75-Exp

i

ZR-2-Sim

ZR-2-Exp

ZR-0.75-Exp Zi-0.75-Sim

106

for 2% of CNT

106

ZR-0.75-Sim

Zi-0.75-Exp

4

10

R

Z and Z

Variation of ZR and Zi with frequency

102

104 Zi-2-Sim Zi-2-Exp

103

102 ZR-2-Sim

ZR-0.75-Sim 0

1

10 102

104

106

10 102

108

104

Frequency (Hz)

106

108

Frequency (Hz)

Figure 7: Comparison of real and imaginary part of impedance with frequency for simulated and experimental data (0.75 and 2% of CNT) Variation of Z R and Z i with frequency for 2.5% of CNT

106

Z R-2.5-Sim Z R-2.5-Exp

5

10

Z i-2.5-Sim

ZR and Z i

Z i-2.5-Sim

104 Z R-2.5-Exp Z R-2.5-Sim

103 Z i-2.5-Sim

Z i-2.5-Exp

2

10

1

10 2 10

3

10

4

5

10

10

6

10

7

10

Frequency (Hz)

Figure 8: Comparison of real and imaginary part of impedance with frequency for simulated and experimental data (2.5% of CNT) Figure 9: Variation of the electrical parameters (R and F) with % CNT for configuration II

13

8

10

Table 4: Details of the α of configuration II Frequency range 100 kHz - 20 MHz 2 MHz - 20 MHz 3 kHz - 38 kHz

*

% of CNT 0.75 2 2.5 0.75 2 2.5

α

αc

|Ea |*

0.951 0.856 0.809 0.899 0.405 0.335

0.938 0.794 0.753 0.919 0.451 0.373

0.01 0.06 0.05 0.02 0.04 0.03

Absolute Error |Ea |=|αc − α|

Table 5: Circuit Parameters Configuration II % CNT 0.75

2

2.5

Parameter F2 (F.secα−1 ) α2 R2 (Ω) R1 (Ω) F1 (F.secα−1 ) α1 F2 (F.secα−1 ) α2 R2 (Ω) R1 (Ω) F1 (F.secα−1 ) α1 F2 ((F.secα−1 ) α2 R2 (Ω) R1 (Ω) F1 (F.secα−1 ) α1

Value 4.556 × 10−11 0.943 7.798 × 107 2.998 × 106 9.849 × 10−11 0.929 1.228 × 10−8 0.726 1.831 × 105 1.591 × 104 8.452 × 10−10 0.844 3.229 × 10−8 0.683 1.320 × 105 1.450 × 104 1.663 × 10−9 0.800

14

Relative SD 3.532 × 10−6 6.479 × 10−7 1.417 × 10−6 8.380 × 10−6 3.581 × 10−6 9.701 × 10−7 2.054 × 10−5 3.332 × 10−6 4.363 × 10−6 1.236 × 10−5 2.806 × 10−5 2.686 × 10−6 3.058 × 10−5 5.438 × 10−6 7.500 × 10−6 1.474 × 10−5 3.257 × 10−5 3.283 × 10−6

1 1

c2

0.8

2

0.4 1

1

2

c1

0.6

c2

2

0.2

c1 c2

0 0.6

0.8

c1

(100 kHz - 20 MHz) (3 kHz - 38 kHz)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

% CNT (wt %)

Figure 10: Variation of the electrical parameters(α) with % CNT for configuration II

145

The inferences from the parameters of the EECM are given below: a. R1 and R2 decreases with increase in % CNT. This means that the tunnelling distance between the CNTs has decreased enhancing the electrons to travel more in between the plates. b. α1 and α2 (fractional exponent) decreases with increase in % CNT.

150

In Figure 10, the values of αc1 and αc2 (calculated from the imaginary part of impedance(Zi -Sim)) for the CP frequency zones are also shown. αc1 exists in the CP frequency zone 100 kHz - 20 MHz, whereas αc2 occur in the CP frequency range 3 kHz - 38 kHz. α1 is a parameter in the frequency region where R1 -X1 is dominant and α2 is obtained from

155

the frequency region where R2 -X2 is effective. c. F1 and F2 increases with increase in % CNT. The fractance has increased as the number of CNT-CNT pairs with polymer in between has increased. 15

2.6

As seen from Table 3 and 5, the value of relative SD for the electrical 160

parameters are less than 0.2, hence justifying that the model which we have assumed gives good fit for the EIS data in the frequency range 100 Hz 20 MHz. It is difficult to retrieve the frequency region in which the effect due to CNT or the electron travel across the interface is dominant from the Nyquist plot obtained through MEISP. Evaluating configuration I, the

165

values of EECM parameters are not changing much for R2 -X2 and R1 -X1 for a specific % of CNT (Table 3) which establishes the fact that both R2 -X2 and R1 -X1 is imparting the CP zone. Again from Table 4, there are two CP zones attained for configuration II and in the CP zone (3 kHz - 38 kHz), the effect due to R2 -X2 is more prominent as the electron gets more time to travel and

170

is aided by the increased pore size of the middle plate. Considering the CP zone (100 kHz - 20 MHz), the effect due to R1 -X1 is more and is assisted by the increased pore size; additionally there is a reduction in CP zone for 2% and 2.5% of both the configurations (Table 2 and 4) due to more number of interconnected CNTs. A wider notion regarding the effect of pore size on

175

CP zone can be obtained by further experimentation with middle plate of different pore sizes and this is considered as future work. Another point that needs to be mentioned is that though the model fits the ZR and Zi part of impedance, the fractional exponent of the EECM (in Table 3 and 5) is not matching with the fractional exponent corresponding to

180

the CP zone obtained from Zi plot (in Table 2 and 4). This is the limitation of the software as MEISP uses Nyquist plot to extract the parameters. But 16

we can find the actual fractional exponent value of the CP zone from the slope of Zi plot and it is shown in Table 2 and 4. From the tables for CP zone (in Table 2 and 4), maximum absolute error of calculated fractional 185

exponent(αc ) w.r.t the actual fractional exponent (α) is less than 0.08; hence, we have defined the absolute error limit to be 0.08 as a criteria for validating the model from MEISP. That is, if Ea is less than 0.08 (in Table 2 and 4), then the model that we have assumed is good in terms of α matching with αc of the CP zone otherwise, we have to discard the model. A drawback of

190

MEISP is that if the initial assumptions for the parameters of EECM are not close to the exact value, then EECM simulation may not give good fit of the data.

5. Conclusions In this paper, we have tried to obtain an EECM which illustrates the 195

various phenomenon inside the solid state fractional capacitor. The parameters of the EECM are extracted using MEISP software. It also includes preliminary study on how the R’s, F’s and α vary with % of CNT in the polymer. Hence, one finds that, with increase in % CNT, the resistance and α values decrease whereas, the fractance (F) increases which substantiate

200

the fact of decrease in distance between the CNTs as well as enhancing the number of micro-capacitors in between the plates for higher loading of CNT filler. From the model, we have found that there are two phenomenon in the developed solid state fractional capacitor. One is due to the movement of 17

electrons across CNT-epoxy resin interface and the other is due to the move205

ment of electrons in the CNT itself which is a localized phenomenon. We also observed that the actual α value corresponding to the CP zone cannot be obtained from the model but it can be calculated from the plot of imaginary part of the impedance. As part of future work, extensive experimentation is required to understand the effect of pore size of middle plate on the CP zone

210

as well as to improve the bandwidth of the developed fractional capacitor.

Acknowledgements The authors would like to thank SGBSI Grant of IITKGP as well as DSTFIST (No:SR/FST/ETI/ET/-374/2014) project for providing the financial support in carrying out the research.

215

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http://www.keysight.com/en/pd-715495-pn-E4980A/

precision-lcr-meter-20-hz-to-2-mhz?cc=IN&lc=eng. [33] J. Ross MacDonald, Journal of Physics Condensed Matter 24 (2012). [34] Details of MEISP, accessed on January, 2017. URL: http:// impedance0.tripod.com/. 285

[35] Y. W. Mai, Z. Z. Yu, Polymer nanocomposites, Woodhead publishing Ltd., 2006. [36] C.-W. Nan, Y. Shen, J. Ma, Annual Review of Materials Research 40 (2010) 131–151.

Authors’ Biographies 290

Dina A. John received her B.Tech in Instrumentation and Control Engineering from NSS College of Engineering, Palakkad and M.E degree in Instrumentation Engineering from MIT Chennai. She is currently working as Research scholar in Department of Electrical Engineering, IIT Kharag-

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pur. Her research interests include development of electrochemical, solid state sensors and the associated electronic circuits. Dr. Karabi Biswas has received her B. Tech and PhD. degree from I. I. T. Kharagpur in 1992 and 2007 respectively. She did her M. Tech from Jadavpur University in 2000. She 22

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is presently Associate Professor in the department of Electrical Engineering, I.I.T. Kharagpur. Before that she served Jadavpur University as a faculty member in Instrumentation and Electronics Engineering. Her research interests include Fractor development and study of fractional order systems. She also works on Sensor

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development and Instrument system design.

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