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Static dielectric constant assessment from capacitance over a wide range of electrode separations
Journal of Electrostatics 87 (2017) 19e25
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Static dielectric constant assessment from capacitance over a wide range of electrode separations Gokul Raj R, C.V. Krishnamurthy* Measurement and Modeling Lab, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 November 2016 Received in revised form 28 February 2017 Accepted 6 March 2017
Static dielectric constant extraction from two-electrode capacitance measurement over a wide range of electrode separations and dielectric constants involves careful assessment of fringe fields. Finite-element method has been employed to compute capacitance and quantify fringe fields for parallel electrode capacitor of (finite thickness, radii r, electrode separation d), with a homogeneous dielectric medium extending up to the geometric limits of the electrodes. Two distinct regimes, in the fringe field contributions are seen. A procedure to extract the static dielectric constant has been proposed for the first regime and a validation has been provided for the same. © 2017 Elsevier B.V. All rights reserved.
Keywords: Circular disc Capacitance Dielectric constant Fringing fields Parallel electrode capacitor
1. Introduction Static dielectric constant measurements are usually done by forming a simple capacitor with the medium sandwiched between two electrodes. Experimentally measurable quantity is capacitance and one has to use the geometrical parameters to extract dielectric constant information from the experiment. For large electrode area (A), small electrode separation (d) and negligible electrode thickness the approximate expression,
Cideal ¼
εr ε0 A ¼ εr C 0 d
(1)
derived analytically, neglecting fringing effects has been widely employed to determine εr the relative permittivity. Here, ε0 is the free space permittivity. C0 denotes the ideal capacitance for relative permittivity εr ¼ 1: Capacitance measurements for the parallel circular electrode configuration that deviate from the ideal capacitance are usually handled through stray capacitance and fringe field capacitance corrections. Attempts to deal with the capacitance due to fringe fields have been of two kinds.
1) Minimize fringe fields through use of guard ring. 2) Evaluate fringe field capacitance through the solution of Laplace's equation as a small correction term using analytical, semianalytical and numerical techniques. Table 1 provides a summary of various approaches, reported in the literature, to evaluate/measure capacitance for parallel circular electrodes arranged in various configurations as shown in Fig. 1. Kirchhoff [4,5] deduced an approximate formula to quantify fringe fields for parallel electrode capacitor (no dielectric medium) at small electrode separations. Ali Naini and Mark Green [7] tried to eliminate the logarithmic divergence of fringing field effect in Kirchoff's formula by incorporating thickness for circular electrodes. Experiments were performed, without any dielectric medium, for aspect ratio range 0:01 dr 0:5. The thicknesscorrection formula was found to work well for dr < 0:1 but failed above it. Carlson and Illman [10] provided a numerical evaluation of solutions to Love's equation and obtained capacitance for various aspect ratios 103 dr 10 with εr ¼ 1. They have also compared their numerical values of capacitance with previous investigators [6,8]. Wintle and Goad [9] employed semi-analytical methods for finding capacitances of circular disk capacitors having different dielectric
* Corresponding author. E-mail address: [email protected] (C.V. Krishnamurthy). http://dx.doi.org/10.1016/j.elstat.2017.03.001 0304-3886/© 2017 Elsevier B.V. All rights reserved.
constants
for
aspect
ratios
0:01 dr 10.
They
compared excess capacitance with Kirchhoff equation for dr < 0:1 with εr ¼ 1. A recent publication [1] reviews analytical and semi-
20
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
Table 1 A summary of analytical, semi-analytical and numerical approaches for capacitance assessment. S.No.
Aspect ratio d/r
Circular Electrodes 1. 0.007e0.2 2. 0.01e0.5 3. 0.001e10 4. 0.0001e1 5. 0.01e10 6. 0.01e10 7. 0.1e10 8. 0.001e10 9 0.00001e0.01
Relative permittivity (εr Þ
Plate thickness (h)
Approach
Ref.
Configurations
1, and 2.02 to 64.9 (liq.) 1 1 1 0.5e100 1 1,3 1 1
Not Specified 2.3 mm Negligible Negligible Negligible Negligible Negligible Negligible Negligible
Experimental Experimental Semi-Analytical Semi-Analytical Semi-Analytical Numerical (BEM) Numerical (MoM) Semi-Analytical Semi-Analytical
[3] [7] [10] [8] [9] [11] [12] [13] [14]
Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.
1(a) 1(a) 1(a) 1(a) 1(c) 1(a) 1(c) 1(a) 1(a)
Fig. 1. Different configurations of identical circular disk parallel electrode capacitor (a) same dielectric in every region (b) dielectric extending up to the geometric limit (c) dielectric extending beyond the geometric plate limits.
analytical approaches and presents an analytical treatment for the asymmetric electrode configuration. A procedure to calculate capacitance co-efficient at large aspect ratio of two bodies with different orientations has been reported recently [2]. It can be seen from Table 1 that the most extensively studied case, for the assessment of fringing effects, has been for the relative permittivity εr ¼ 1 pertaining to configuration 1(a). Wintle and Goad [9] study configuration 1(c) and tabulate the fringe field capacitance over a wide range of dielectric constants and aspect ratios. It is further seen that effects of electrode thickness are (a) neglected in analytical treatments due to difficulties in formulating the problem statement and (b) largely ignored in experiments since corrective measures are not clear, despite having to employ finite thickness electrodes. More importantly, there has been very little done to study the effect of fringing field capacitance pertaining to configuration 1(b) for εr > 1. Practically, schemes to determine relative permittivity, εr ; from capacitance measurements, using a symmetric electrode configuration (Fig. 1(b)), involving (i) electrodes with finite thicknesses, and (ii) medium to large aspect ratios, do not seem to have been reported. With the availability of powerful, yet economical, computational resources, numerical approaches, such as those based on FEM, can provide an assessment of simple as well as practical measurement scenarios. The work presented in what follows has been conceived to provide an insight on how fringe fields influence capacitance measurements using a symmetric electrode configuration in the presence of dielectric media and to arrive at a procedure to extract the relative permittivity from capacitance measurements. Section 1 outlines the FEM approach for the problem and describes the implementational details of the numerical aspects. Section 2 details the FEM-based calculations carried out for a wide range of aspect ratios and relative permittivities to assess the fringe field contributions to capacitance. The results are examined for trends that can be understood and quantified in a useful manner. Section 3 provides an alternative assessment based on total capacitance and establishes trends that can be quantified for practical extraction of relative permittivity. Section 4 presents validation with experimental data.
1.1. Section 1: FEM implementation and results Finite element method is used to evaluate the capacitance of two parallel circular electrodes with variable spacing and dielectric filling by exploiting the axi-symmetry. The calculations are carried out for a wide range of aspect ratios (103 dr 10) and dielectric constants (1 εr 80). Finite element method (FEM) converts the differential equations to difference equations. The discretization of computational domain of interest has been done with triangular meshes. Given the boundary conditions, the potential at different nodes of the meshes are computed which will help in calculating capacitance eventually. Electrostatic module in COMSOL solves the Laplace's equation (V2 V ¼ 0) with zero charge boundary condition/ Perfectly insulating boundary conditions, in which the electric field ! b :D ¼ 0), where nb is the unit lines are tangential to the boundary ( n ! vector normal to the boundary and D represents the electric displacement field. The potential difference across the parallel electrodes is provided by specifying one terminal to be at 1 V and the other to be at 0 V. Convergence check on capacitance has been
Fig. 2. Schematic representation of parallel electrode capacitor employed in FEM simulation.
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
performed by employing extremely fine meshing. Fig. 2 represents a schematic representation of parallel electrode capacitor with a dielectric. The semi-circular domain surrounding the electrodes and the dielectric is taken to have a relative permittivity of unity and is meshed to deal with the non-uniform fringing fields. The whole analysis is based on the assumption that conductors are ideal and dielectric material is linear, and isotropic. Physics-based meshing (extremely fine) option in COMSOL [15] was employed in setting up the FEM model. All the calculations reported here have been carried out with a Intel (R) Core TM i3-3220 @3.30 GHz system having 16 GB RAM. Numerical computations based on FEM using COMSOL have been carried out for a wide range of aspect ratios 0:001 dr 10. The thickness of the circular electrode was taken to be 5 mm. Relative permittivity was set to unity to facilitate comparison with semi-analytical calculation reported by Carlson and Illmann [10]. Fig. 3 presents the comparison between capacitance computed using FEM (CFEM ) and capacitance evaluated semi-analytically (CSA ). The ideal capacitance (Cideal ) included in Fig. 3 serves as a useful reference. Following Carlson and Illman [10], a log-log plot of normalized capacitance with respect to the aspect ratio is shown in Fig. 3. The normalization is done with respect to the capacitance of two uncoupled circular disks in series (4ε0 r), where 12
ε0 ¼ 8:854188 10m F is the permittivity of free space. Fig. 3 indicates that CFEM is in very good agreement with CSA on the log-log scale. Non-ideal behavior, regarded as the deviation from Cideal, becomes significant (>1%) beyond an aspect ratio of 0.005 (corresponding to 2:3 in the log-scale). Recalling that Cideal is derived by assuming that the entire field is confined in the disclike volume (Vin) between the inner electrode surfaces facing each other, the deviation from ideal behavior is seen to give rise to positive capacitive contributions. These positive capacitive contributions (CFEM e Cideal) are due to the field that permeates the remaining volume (Vout) surrounding the electrodes e the fringefield capacitance. While the total capacitance varies over a wide range from about 100 pF to about 0.1 pF, it is found that CFEM agrees with the semi-analytical results up to two significant digits over the
21
treatment approaches the analytical treatment of idealized twodimensional electrodes, albeit slowly, it is more important to note that FEM calculations can assist measurement scenarios dealing with electrodes of finite thicknesses.
1.2. Section 2: examining Cfringe Fig. 3 also indicates a change in the trend shown by fringe-field capacitance (CFEM e Cideal) as the aspect ratio increases from dr < 1; to dr 1. It is worth pointing out that the total capacitance begins to level off beyond rd > 1, approaching that of two independent or uncoupled conductors as expected. To understand the role of dielectric media in fringe field behavior, the following discussion is presented for εr ¼ 1 and εr > 1 separately. 2.1) εr ¼ 1 (electrodes with finite thickness): It is well known [4] that the fringe field contributions can be expressed as a capacitance in parallel to C0 ,
Ctotal ¼ C0 þ Cfringe
(2)
According to Table 1 and the discussion preceding and following it, Cfringe for circular electrodes of equal radii has been handled largely through Kirchhoff approximation expressed, following [3,5], as
16pr 1 Cfringe ¼ ε0 r ln d
(3)
aspect ratio range 1 < dr 10, CFEM agrees with the semi-analytical results up to the first significant digit in the pico-farad scale. The agreement with semi-analytical results has been found, in the present study, to improve as the electrode thickness is reduced further. However, FEM calculations increasingly become memory intensive and time consuming. While it is good to see that FEM
Fig. 4 presents the variation of fringe capacitance with aspect ratio obtained from equation (3) and equation (29) in Ref. [9] along with semi-analytical and FEM calculations carried out for thin and thick electrodes. The FEM fringe capacitance calculations with thin electrodes (5 mm) is in fair agreement with semi-analytical [10] results over the whole range of aspect ratios. FEM calculations with 2 mm thick electrodes give a systematically higher value 1 pF for Cfringe at smaller aspect ratios and a lower value 0:1pF at larger aspect ratios. It is seen that the results from predictions based on equation (3) and equation (29) in Ref. [9], from semi-analytic calculations [10] and current thin-electrode FEM evaluations agree with each other till about d=r ¼ 0:1. Predictions from equation (3) break down beyond d=r > 0:1.
Fig. 3. Comparison between semi-analytical [Ref.[10]] and present numerical (FEM) results for symmetric electrode configuration.
Fig. 4. Fringe field capacitance comparisons between various methods of evaluation with.εr ¼ 1.
aspect ratio range 0:001 dr 1 in the pico-farad scale. For the
22
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
2.2) εr > 1 (electrodes of finite thickness): We consider that fringe capacitance can be taken to be parallel to ideal capacitance even in the presence of a dielectric medium (Fig. 1(b)) and express the total capacitance as, 0
Ctotal ¼ εr C0 þ Cfringe
(2a)
Fig. 5 shows several interesting trends in the behavior of fringe capacitance. Firstly, over the range of dielectric constants considered, the fringe field contribution to the capacitance decreases sharply from about 3 pF at rd ¼ 0:001 to about 1 pF as dr/1 from below and then remains within a narrow range between 0.7 pF and 1.1 pF as dr/10. Two regions may be defined based on these distinct variations in the fringe capacitance e region I, where the fringe capacitance decreases sharply till about d/r < 1 and region II, where the fringe capacitance increases gently till about d/r < 10. Secondly, the fringe capacitance is seen to be largely insensitive to relative permittivity in the aspect ratio range 0:001 dr < 1. As this range of aspect ratios is widely employed for dielectric measurements, it is comforting to note that Equation (2a) can be used to extract relative permittivity from measured capacitance since Cfringe needs to be determined only once without any dielectric medium. Thirdly, the fringe capacitance for dr > 1 shows a nearly linear increase with aspect ratio for εr > 1: It may be noted that the slope for εr ¼ 80 is only slightly different from that for εr ¼ 20 indicating that as εr increases further, the slope may tend to a constant value. These trends can be seen from another perspective where the roles of fringe field capacitance (second term in equation (2a)) and the uniform field capacitance (first term in equation (2a)) are compared as shown in Fig. 6 for three relative dielectric permittivities. In regime I, as the electrodes are brought closer, there is an increase in both the uniform field due to charges on the inner electrode surfaces, and the fringe field due to charges near the edges. However, it can be seen from Fig. 6(a) that the first term in equation (2a) dominates for small dr and masks this increasing trend in fringe capacitance. In regime II, as the aspect ratio increases, the first term in equation (2a) gradually drops and becomes comparable to the fringe capacitance (second term in equation (2a)) and beyond a certain aspect ratio, the fringe capacitance dominates as can be seen in Fig. 6(a). With increasing aspect ratios, the charge distribution tends to be more uniformly distributed on the inner as well
as the outer surfaces, including the edges, leading to a significant drop in the contributions from the uniform field between the electrodes (ideal capacitance). Thus, for εr ¼ 1, the fringe capacitance becomes nearly independent of dr as dr /10. In other words, the “fringe field” loses its identity and becomes the field surrounding each of the conductors such that the electrodes can be said to be uncoupled at large aspect ratios. This can also be seen from Fig. 3 where the total capacitance for a coupled pair of electrodes approaches that of uncoupled pair of conductors. When εr > 1, Fig. 6 (b) and (c) show how the fringe field contributions are weakly dependent on the relative permittivity and how the ideal capacitance dominates in both the regimes. Given the configuration under study (dielectric medium extending only up to the edges of the electrodes), εr > 1 brings in an asymmetry e the medium does not permeate everywhere as in the case for εr ¼ 1: This asymmetry seen by each of the two electrodes e dielectric medium on one side and vacuum on the other side - exists at all separations considered. Hence, there is still a dependence of fringe field capacitance on dr for dr > 1. The fringe field behavior is bound to simplify if the dielectric medium between the plates and the surroundings are one and the same (as for measurements in air and fluids, for instance). The entire flux (uniform and non-uniform) passes through a single dielectric medium suggesting the following scaled expression:
h i Ctotal ¼ εr C0 þ Cfringe
(4)
It is clear from the above discussion that Cfringe does not show simple trends for the symmetric electrode configuration, particularly for εr > 1, and that quantification of Cfringe is essential for a precise assessment of relative permittivity over the range of dr and εr studied. For instance, a thick sample of low dielectric constant presents a combination where the uniform field capacitance becomes comparable to fringe field capacitance.
1.3. Section 3: examining Ctotal as a function of
d r
and εr
We re-examine the total capacitance and drop the idea of splitting it as uniform and non-uniform part. From an experimental point of view, total capacitance and geometric parameters are known and the dielectric constant is an unknown quantity which is of interest. Fig. 7 shows the variation of total capacitance in region I (see Fig. 3), normalized to C0 , as a function of aspect ratio for a wide range of relative permittivities ð1 εr 80Þ . It is striking to see the nearly parallel nature seen for the whole range of relative permittivities considered and suggests that a simple procedure can successfully permit the estimation of relative permittivity without having to deal with fringe field capacitance explicitly. The reasonableness of the relation εr ¼ CC0 for small aspect ratios and the nearly linear increase observed in
C C0
with the aspect ratio suggest the
form,
C ¼AþB C0
Fig. 5. Variation of fringe capacitance with aspect ratio for three relative permittivities.
m d r
(5)
The constant A is found to be a good approximation to the actual relative permittivity. The value of B and m are found to be reasonably constant with values 1.57 ± 0.03 and 0.828 ± 0.002 respectively over region I. It is proposed that the relative permittivity be estimated from a single measurement of total capacitance measured using electrodes of finite thickness for an aspect ratio that falls within region I by using the following expression
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
23
Fig. 6. Comparison of uniform field capacitance with fringe field capacitance for (a) εr ¼ 1; (b) εr ¼ 20; and (c) εr ¼ 80.
Fig. 7. Variation of normalized capacitance with normalized separations in region I for different relative permittivities.
εr ¼
0:828 C d 1:57 C0 r
(6)
The spread in the estimation of low dielectric constants is found to vary in the first decimal place. For media with higher dielectric constants, the spread is found to be in the second decimal place. Experiments have been carried out to validate the procedure to extract the dielectric constant from a single measurement using equation (6) for aspect ratios falling in region I.
1.4. Section 4: experimental validation with air and Teflon as media A series of experiments have been carried out with various spacing between two identical circular electrodes arranged in parallel. The electrode diameter is 40 mm and electrode thickness is 2 mm with air as the medium. The experiments were performed with Alpha-A impedance analyzer from Novo control Technologies
[16]. The frequency sweep is from 1 Hz to 40 MHz and experiments were performed at ambient temperature (25 deg C) and a relative humidity of 45%. Experimental data in the frequency regime from 1 Hz to 10 MHz were found to be free from dispersion and the values at 1 kHz have been used for comparison with electrostatic simulations. Table 2 presents a comparison of total capacitance obtained from thick electrode experimental data with various techniques. Experiments have also been performed with Teflon as the dielectric medium. Teflon of diameter 40 mm with various thicknesses 2 mm, 3 mm, 4 mm, 5 mm, 7 mm, and 12 mm were considered. Repeated measurements with air as well as Teflon show that the capacitance values can be determined within a narrow dispersion of ± 0:1pF. Fig. 8(a,b) presents a comparison between simulations and experiments. The experimental capacitance results obtained for the Teflon samples fall in the range 0:1 < dr < 0:35 and are found to be systematically less than that deduced from simulations as depicted in Fig. 8(a). The lowering of capacitance is believed to be due to the distribution of air pockets [4] at the interface between the smooth electrodes and the rough surfaces of the sample. The finite element simulations were carried out using a simple model where uniform air films with variable thicknesses were considered to lie between top and bottom electrode-sample interfaces. The FEM calculations indicate that the trend in capacitance closely follows the experimental trend. Experiments were then carried out with Aluminum foils pressed between the top and bottom electrode-sample interfaces. Results are shown in Fig. 8(b). Applying conducting paint over the sample was an option but was not considered due to possible physical/chemical contamination of the sample. Table 3 compares the FEM results and experiments performed with and without Aluminum foil for different sample thicknesses. The mean sample thicknesses were used in FEM simulations. Simulations with thickness that includes the scatter were also performed and found to lead to insignificant contributions to the
Table 2 Experimental verification of thick plate FEM numerical results with air as dielectric. S.No
d/r
1. 2. 3. 4. 5.
0.05 0.1 0.15 0.2 0.3
Total Capacitance (pF) Kirchoff
Carlson and Illman [10]
Thin Plate FEM
Thick Plate FEM
Thick Plate (2 mm)Experiment
12.2 6.4 4.5 3.5 2.5
12.2 6.5 4.6 3.6 2.6
12.2 6.5 4.6 3.6 2.6
12.7 6.9 4.9 3.9 2.9
12.8 ± 0.1 6.9 ± 0.1 5.0 ± 0.1 3.9 ± 0.1 2.8 ± 0.1
24
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
Fig. 8. Measured Capacitance of Teflon of different thicknesses along with simulation(a) bare samples (b)with Al foil.
Table 3 Comparison of simulated capacitance with experiment and validation of dielectric constant extraction procedure. Teflon thickness (mm)
Simulated Capacitance (F)
Experimental Capacitance (F) at 1 kHz
Experimental Capacitance (F) at 1 kHz with aluminum foil
C= C0
0:828 εr ¼ CC0 1:57
d r
(True value for Teflon ¼ 2.1) 2.056 3.074 4.094 5.372 7.013
± ± ± ± ±
0.004 0.004 0.004 0.053 0.023
1.27E-11 8.8E-12 6.8E-12 5.4E-12 4.4E-12
1.05E-11 7.4E-12 5.7E-12 4.3E-12 3.5E-12
1.29E-11 8.7E-12 6.6E-12 5.2E-12 4.4E-12
2.4 2.5 2.4 2.5 2.8
2.1 2.1 2.0 2.0 2.1
± ± ± ± ±
0.1 0.1 0.1 0.1 0.1
capacitance. Table 3 describes the procedure, corresponding to region I, to extract the dielectric constant, given the measured capacitance and sample thickness. It may be noted that the values in the fifth column ought to have given the true value of εr (sixth column entries).
practical requirements that goes beyond what is possible with currently available analytical, and semi-analytical solutions for. εr 1.
2. Conclusions
One of us (Mr.Gokul Raj R) would like to express his sincere gratitude to Mr.Satyanarayana Raju, for his assistance in carrying out the experiments and for related discussions.
FEM calculations of capacitance with thin circular electrodes of equal diameters have helped in reproducing semi-analytical results for a wide range of electrode separations. For small aspect ratios, the fringe field contributions estimated from FEM calculations are in good agreement with that predicted by Kirchoff's formula for relative permittivity εr ¼ 1: FEM calculations for electrodes with finite thickness, used routinely in experiments, have been found to agree reasonably well with measurements for relative permittivity εr ¼ 1: These calculations have been extended to cover a range of dielectric constants representing a large number of commonly used materials taken to be linear, homogeneous and isotropic. Qualitatively, analysis of the trends in the total capacitance indicates that (a) fringing fields will be less significant when the dielectric constant is high and electrode separation is not large, and (b) fringing fields become significant when the dielectric constant is low and electrode separation is large. Quantitatively, two regimes have been identified based on dr values. A functional relationship between the total capacitance, the aspect ratio and the dielectric constant is established in the first regime. This functional relationship allows the dielectric constant to be estimated from a single measurement of capacitance for any aspect ratio within regime I. The range of applicability of the function is also indicated. The suggested procedure is believed to cover a significant range of
Acknowledgements
References [1] G. Paffuti, E. Cataldo, A. Di Lieto, F. Maccarrone, Circular plate capacitor with different discs, Proc.R. Soc. A 472 (2016), http://dx.doi.org/10.1098/ rspa.2016.0574, 20160574. [2] G. Paffuti, F. Maccarrone, Capacitance and potential co-efficients at large distances, J. Electrost. 83 (2016) 22e27. [3] Ion. C. Halalay, Versatile cell for complex permittivity measurements on liquids, Rev. Sci. Instrum. 70 (1999) 3432e3434. [4] A.H. Scott, H.L. Curtis, Edge correction in the determination of dielectric constant, J. Res. Natl. Bur. Stds. 22 (1939) 747e775. [5] G. Kirchhoff, Zur theorie des kondensators, Monatsb. Acad. Wiss. Berl. (1877) 731e734. [6] I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory, Wiley, New York, 1996, pp. 230e246. [7] Ali Naini, Mark Green, Fringing fields in a parallel plate capacitor, Am. J. Phys. 45 (1977) 877e879. [8] J. Wintle, S. Kurylowicz, Edge corrections for strip and disc capacitors, IEEE Trans. Instrum. Meas. IM-34 (1) (MAR. 1985). [9] H.J. Wintle, D.G.W. Goad, Capacitance corrections for disc electrodes on sheet dielectrics, J. Phys. D. Appl. Phys. 22 (11) (1989) 1620e1626. [10] G.T. Carlson, B.L. Illman, The Circular disk parallel plate capacitor, Am. J. Phys. 62 (12) (Dec. 1994) 1099. [11] Hitoshi Nishiyama, Mitsunobu Nakamura, Capacitance of circular disk capacitors, IEEE Trans. comp., Hyb., Man. Tech. 16 (3) (May 1993). [12] C.Y. Kim, A numerical solution for the round disk capacitor by using annular patch subdomains, Prog. Electromag. Res. B 8 (2008) 179e194.
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25 [13] M. Norgren, An Alternative Integral Equation for Calculating the Surface Charge in a Parallel Disk Capacitor. 2010 URSI International Symposium on Electromagnetic Theory, 2010, Aug [Online]. Available: http://ieeexplore.ieee. org/xpls/abs_all.jsp?arnumber¼5637176&tag¼1. [14] M. Norgren, B.L.G. Jonsson, The capacitance of the circular parallel plate capacitor obtained by solving the love integral equation using an analytic
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expansion of the kernel, Pro. Electromag. Res. 97 (2009) 357e372. [15] COMSOL Multiphysics® v. 5.2a. www.comsol.com. COMSOL AB, (Stockholm, Sweden). [16] Alpha-A High Resolution Dielectric, Conductivity, Impedance and Gain Phase Modular Measurement System, Issue 7/2012 Rev. 3.3 by Novocontrol Technologies GmbH & Co. KG.
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Static dielectric constant assessment from capacitance over a wide range of electrode separations Gokul Raj R, C.V. Krishnamurthy* Measurement and Modeling Lab, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 November 2016 Received in revised form 28 February 2017 Accepted 6 March 2017
Static dielectric constant extraction from two-electrode capacitance measurement over a wide range of electrode separations and dielectric constants involves careful assessment of fringe fields. Finite-element method has been employed to compute capacitance and quantify fringe fields for parallel electrode capacitor of (finite thickness, radii r, electrode separation d), with a homogeneous dielectric medium extending up to the geometric limits of the electrodes. Two distinct regimes, in the fringe field contributions are seen. A procedure to extract the static dielectric constant has been proposed for the first regime and a validation has been provided for the same. © 2017 Elsevier B.V. All rights reserved.
Keywords: Circular disc Capacitance Dielectric constant Fringing fields Parallel electrode capacitor
1. Introduction Static dielectric constant measurements are usually done by forming a simple capacitor with the medium sandwiched between two electrodes. Experimentally measurable quantity is capacitance and one has to use the geometrical parameters to extract dielectric constant information from the experiment. For large electrode area (A), small electrode separation (d) and negligible electrode thickness the approximate expression,
Cideal ¼
εr ε0 A ¼ εr C 0 d
(1)
derived analytically, neglecting fringing effects has been widely employed to determine εr the relative permittivity. Here, ε0 is the free space permittivity. C0 denotes the ideal capacitance for relative permittivity εr ¼ 1: Capacitance measurements for the parallel circular electrode configuration that deviate from the ideal capacitance are usually handled through stray capacitance and fringe field capacitance corrections. Attempts to deal with the capacitance due to fringe fields have been of two kinds.
1) Minimize fringe fields through use of guard ring. 2) Evaluate fringe field capacitance through the solution of Laplace's equation as a small correction term using analytical, semianalytical and numerical techniques. Table 1 provides a summary of various approaches, reported in the literature, to evaluate/measure capacitance for parallel circular electrodes arranged in various configurations as shown in Fig. 1. Kirchhoff [4,5] deduced an approximate formula to quantify fringe fields for parallel electrode capacitor (no dielectric medium) at small electrode separations. Ali Naini and Mark Green [7] tried to eliminate the logarithmic divergence of fringing field effect in Kirchoff's formula by incorporating thickness for circular electrodes. Experiments were performed, without any dielectric medium, for aspect ratio range 0:01 dr 0:5. The thicknesscorrection formula was found to work well for dr < 0:1 but failed above it. Carlson and Illman [10] provided a numerical evaluation of solutions to Love's equation and obtained capacitance for various aspect ratios 103 dr 10 with εr ¼ 1. They have also compared their numerical values of capacitance with previous investigators [6,8]. Wintle and Goad [9] employed semi-analytical methods for finding capacitances of circular disk capacitors having different dielectric
* Corresponding author. E-mail address: [email protected] (C.V. Krishnamurthy). http://dx.doi.org/10.1016/j.elstat.2017.03.001 0304-3886/© 2017 Elsevier B.V. All rights reserved.
constants
for
aspect
ratios
0:01 dr 10.
They
compared excess capacitance with Kirchhoff equation for dr < 0:1 with εr ¼ 1. A recent publication [1] reviews analytical and semi-
20
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
Table 1 A summary of analytical, semi-analytical and numerical approaches for capacitance assessment. S.No.
Aspect ratio d/r
Circular Electrodes 1. 0.007e0.2 2. 0.01e0.5 3. 0.001e10 4. 0.0001e1 5. 0.01e10 6. 0.01e10 7. 0.1e10 8. 0.001e10 9 0.00001e0.01
Relative permittivity (εr Þ
Plate thickness (h)
Approach
Ref.
Configurations
1, and 2.02 to 64.9 (liq.) 1 1 1 0.5e100 1 1,3 1 1
Not Specified 2.3 mm Negligible Negligible Negligible Negligible Negligible Negligible Negligible
Experimental Experimental Semi-Analytical Semi-Analytical Semi-Analytical Numerical (BEM) Numerical (MoM) Semi-Analytical Semi-Analytical
[3] [7] [10] [8] [9] [11] [12] [13] [14]
Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.
1(a) 1(a) 1(a) 1(a) 1(c) 1(a) 1(c) 1(a) 1(a)
Fig. 1. Different configurations of identical circular disk parallel electrode capacitor (a) same dielectric in every region (b) dielectric extending up to the geometric limit (c) dielectric extending beyond the geometric plate limits.
analytical approaches and presents an analytical treatment for the asymmetric electrode configuration. A procedure to calculate capacitance co-efficient at large aspect ratio of two bodies with different orientations has been reported recently [2]. It can be seen from Table 1 that the most extensively studied case, for the assessment of fringing effects, has been for the relative permittivity εr ¼ 1 pertaining to configuration 1(a). Wintle and Goad [9] study configuration 1(c) and tabulate the fringe field capacitance over a wide range of dielectric constants and aspect ratios. It is further seen that effects of electrode thickness are (a) neglected in analytical treatments due to difficulties in formulating the problem statement and (b) largely ignored in experiments since corrective measures are not clear, despite having to employ finite thickness electrodes. More importantly, there has been very little done to study the effect of fringing field capacitance pertaining to configuration 1(b) for εr > 1. Practically, schemes to determine relative permittivity, εr ; from capacitance measurements, using a symmetric electrode configuration (Fig. 1(b)), involving (i) electrodes with finite thicknesses, and (ii) medium to large aspect ratios, do not seem to have been reported. With the availability of powerful, yet economical, computational resources, numerical approaches, such as those based on FEM, can provide an assessment of simple as well as practical measurement scenarios. The work presented in what follows has been conceived to provide an insight on how fringe fields influence capacitance measurements using a symmetric electrode configuration in the presence of dielectric media and to arrive at a procedure to extract the relative permittivity from capacitance measurements. Section 1 outlines the FEM approach for the problem and describes the implementational details of the numerical aspects. Section 2 details the FEM-based calculations carried out for a wide range of aspect ratios and relative permittivities to assess the fringe field contributions to capacitance. The results are examined for trends that can be understood and quantified in a useful manner. Section 3 provides an alternative assessment based on total capacitance and establishes trends that can be quantified for practical extraction of relative permittivity. Section 4 presents validation with experimental data.
1.1. Section 1: FEM implementation and results Finite element method is used to evaluate the capacitance of two parallel circular electrodes with variable spacing and dielectric filling by exploiting the axi-symmetry. The calculations are carried out for a wide range of aspect ratios (103 dr 10) and dielectric constants (1 εr 80). Finite element method (FEM) converts the differential equations to difference equations. The discretization of computational domain of interest has been done with triangular meshes. Given the boundary conditions, the potential at different nodes of the meshes are computed which will help in calculating capacitance eventually. Electrostatic module in COMSOL solves the Laplace's equation (V2 V ¼ 0) with zero charge boundary condition/ Perfectly insulating boundary conditions, in which the electric field ! b :D ¼ 0), where nb is the unit lines are tangential to the boundary ( n ! vector normal to the boundary and D represents the electric displacement field. The potential difference across the parallel electrodes is provided by specifying one terminal to be at 1 V and the other to be at 0 V. Convergence check on capacitance has been
Fig. 2. Schematic representation of parallel electrode capacitor employed in FEM simulation.
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
performed by employing extremely fine meshing. Fig. 2 represents a schematic representation of parallel electrode capacitor with a dielectric. The semi-circular domain surrounding the electrodes and the dielectric is taken to have a relative permittivity of unity and is meshed to deal with the non-uniform fringing fields. The whole analysis is based on the assumption that conductors are ideal and dielectric material is linear, and isotropic. Physics-based meshing (extremely fine) option in COMSOL [15] was employed in setting up the FEM model. All the calculations reported here have been carried out with a Intel (R) Core TM i3-3220 @3.30 GHz system having 16 GB RAM. Numerical computations based on FEM using COMSOL have been carried out for a wide range of aspect ratios 0:001 dr 10. The thickness of the circular electrode was taken to be 5 mm. Relative permittivity was set to unity to facilitate comparison with semi-analytical calculation reported by Carlson and Illmann [10]. Fig. 3 presents the comparison between capacitance computed using FEM (CFEM ) and capacitance evaluated semi-analytically (CSA ). The ideal capacitance (Cideal ) included in Fig. 3 serves as a useful reference. Following Carlson and Illman [10], a log-log plot of normalized capacitance with respect to the aspect ratio is shown in Fig. 3. The normalization is done with respect to the capacitance of two uncoupled circular disks in series (4ε0 r), where 12
ε0 ¼ 8:854188 10m F is the permittivity of free space. Fig. 3 indicates that CFEM is in very good agreement with CSA on the log-log scale. Non-ideal behavior, regarded as the deviation from Cideal, becomes significant (>1%) beyond an aspect ratio of 0.005 (corresponding to 2:3 in the log-scale). Recalling that Cideal is derived by assuming that the entire field is confined in the disclike volume (Vin) between the inner electrode surfaces facing each other, the deviation from ideal behavior is seen to give rise to positive capacitive contributions. These positive capacitive contributions (CFEM e Cideal) are due to the field that permeates the remaining volume (Vout) surrounding the electrodes e the fringefield capacitance. While the total capacitance varies over a wide range from about 100 pF to about 0.1 pF, it is found that CFEM agrees with the semi-analytical results up to two significant digits over the
21
treatment approaches the analytical treatment of idealized twodimensional electrodes, albeit slowly, it is more important to note that FEM calculations can assist measurement scenarios dealing with electrodes of finite thicknesses.
1.2. Section 2: examining Cfringe Fig. 3 also indicates a change in the trend shown by fringe-field capacitance (CFEM e Cideal) as the aspect ratio increases from dr < 1; to dr 1. It is worth pointing out that the total capacitance begins to level off beyond rd > 1, approaching that of two independent or uncoupled conductors as expected. To understand the role of dielectric media in fringe field behavior, the following discussion is presented for εr ¼ 1 and εr > 1 separately. 2.1) εr ¼ 1 (electrodes with finite thickness): It is well known [4] that the fringe field contributions can be expressed as a capacitance in parallel to C0 ,
Ctotal ¼ C0 þ Cfringe
(2)
According to Table 1 and the discussion preceding and following it, Cfringe for circular electrodes of equal radii has been handled largely through Kirchhoff approximation expressed, following [3,5], as
16pr 1 Cfringe ¼ ε0 r ln d
(3)
aspect ratio range 1 < dr 10, CFEM agrees with the semi-analytical results up to the first significant digit in the pico-farad scale. The agreement with semi-analytical results has been found, in the present study, to improve as the electrode thickness is reduced further. However, FEM calculations increasingly become memory intensive and time consuming. While it is good to see that FEM
Fig. 4 presents the variation of fringe capacitance with aspect ratio obtained from equation (3) and equation (29) in Ref. [9] along with semi-analytical and FEM calculations carried out for thin and thick electrodes. The FEM fringe capacitance calculations with thin electrodes (5 mm) is in fair agreement with semi-analytical [10] results over the whole range of aspect ratios. FEM calculations with 2 mm thick electrodes give a systematically higher value 1 pF for Cfringe at smaller aspect ratios and a lower value 0:1pF at larger aspect ratios. It is seen that the results from predictions based on equation (3) and equation (29) in Ref. [9], from semi-analytic calculations [10] and current thin-electrode FEM evaluations agree with each other till about d=r ¼ 0:1. Predictions from equation (3) break down beyond d=r > 0:1.
Fig. 3. Comparison between semi-analytical [Ref.[10]] and present numerical (FEM) results for symmetric electrode configuration.
Fig. 4. Fringe field capacitance comparisons between various methods of evaluation with.εr ¼ 1.
aspect ratio range 0:001 dr 1 in the pico-farad scale. For the
22
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
2.2) εr > 1 (electrodes of finite thickness): We consider that fringe capacitance can be taken to be parallel to ideal capacitance even in the presence of a dielectric medium (Fig. 1(b)) and express the total capacitance as, 0
Ctotal ¼ εr C0 þ Cfringe
(2a)
Fig. 5 shows several interesting trends in the behavior of fringe capacitance. Firstly, over the range of dielectric constants considered, the fringe field contribution to the capacitance decreases sharply from about 3 pF at rd ¼ 0:001 to about 1 pF as dr/1 from below and then remains within a narrow range between 0.7 pF and 1.1 pF as dr/10. Two regions may be defined based on these distinct variations in the fringe capacitance e region I, where the fringe capacitance decreases sharply till about d/r < 1 and region II, where the fringe capacitance increases gently till about d/r < 10. Secondly, the fringe capacitance is seen to be largely insensitive to relative permittivity in the aspect ratio range 0:001 dr < 1. As this range of aspect ratios is widely employed for dielectric measurements, it is comforting to note that Equation (2a) can be used to extract relative permittivity from measured capacitance since Cfringe needs to be determined only once without any dielectric medium. Thirdly, the fringe capacitance for dr > 1 shows a nearly linear increase with aspect ratio for εr > 1: It may be noted that the slope for εr ¼ 80 is only slightly different from that for εr ¼ 20 indicating that as εr increases further, the slope may tend to a constant value. These trends can be seen from another perspective where the roles of fringe field capacitance (second term in equation (2a)) and the uniform field capacitance (first term in equation (2a)) are compared as shown in Fig. 6 for three relative dielectric permittivities. In regime I, as the electrodes are brought closer, there is an increase in both the uniform field due to charges on the inner electrode surfaces, and the fringe field due to charges near the edges. However, it can be seen from Fig. 6(a) that the first term in equation (2a) dominates for small dr and masks this increasing trend in fringe capacitance. In regime II, as the aspect ratio increases, the first term in equation (2a) gradually drops and becomes comparable to the fringe capacitance (second term in equation (2a)) and beyond a certain aspect ratio, the fringe capacitance dominates as can be seen in Fig. 6(a). With increasing aspect ratios, the charge distribution tends to be more uniformly distributed on the inner as well
as the outer surfaces, including the edges, leading to a significant drop in the contributions from the uniform field between the electrodes (ideal capacitance). Thus, for εr ¼ 1, the fringe capacitance becomes nearly independent of dr as dr /10. In other words, the “fringe field” loses its identity and becomes the field surrounding each of the conductors such that the electrodes can be said to be uncoupled at large aspect ratios. This can also be seen from Fig. 3 where the total capacitance for a coupled pair of electrodes approaches that of uncoupled pair of conductors. When εr > 1, Fig. 6 (b) and (c) show how the fringe field contributions are weakly dependent on the relative permittivity and how the ideal capacitance dominates in both the regimes. Given the configuration under study (dielectric medium extending only up to the edges of the electrodes), εr > 1 brings in an asymmetry e the medium does not permeate everywhere as in the case for εr ¼ 1: This asymmetry seen by each of the two electrodes e dielectric medium on one side and vacuum on the other side - exists at all separations considered. Hence, there is still a dependence of fringe field capacitance on dr for dr > 1. The fringe field behavior is bound to simplify if the dielectric medium between the plates and the surroundings are one and the same (as for measurements in air and fluids, for instance). The entire flux (uniform and non-uniform) passes through a single dielectric medium suggesting the following scaled expression:
h i Ctotal ¼ εr C0 þ Cfringe
(4)
It is clear from the above discussion that Cfringe does not show simple trends for the symmetric electrode configuration, particularly for εr > 1, and that quantification of Cfringe is essential for a precise assessment of relative permittivity over the range of dr and εr studied. For instance, a thick sample of low dielectric constant presents a combination where the uniform field capacitance becomes comparable to fringe field capacitance.
1.3. Section 3: examining Ctotal as a function of
d r
and εr
We re-examine the total capacitance and drop the idea of splitting it as uniform and non-uniform part. From an experimental point of view, total capacitance and geometric parameters are known and the dielectric constant is an unknown quantity which is of interest. Fig. 7 shows the variation of total capacitance in region I (see Fig. 3), normalized to C0 , as a function of aspect ratio for a wide range of relative permittivities ð1 εr 80Þ . It is striking to see the nearly parallel nature seen for the whole range of relative permittivities considered and suggests that a simple procedure can successfully permit the estimation of relative permittivity without having to deal with fringe field capacitance explicitly. The reasonableness of the relation εr ¼ CC0 for small aspect ratios and the nearly linear increase observed in
C C0
with the aspect ratio suggest the
form,
C ¼AþB C0
Fig. 5. Variation of fringe capacitance with aspect ratio for three relative permittivities.
m d r
(5)
The constant A is found to be a good approximation to the actual relative permittivity. The value of B and m are found to be reasonably constant with values 1.57 ± 0.03 and 0.828 ± 0.002 respectively over region I. It is proposed that the relative permittivity be estimated from a single measurement of total capacitance measured using electrodes of finite thickness for an aspect ratio that falls within region I by using the following expression
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
23
Fig. 6. Comparison of uniform field capacitance with fringe field capacitance for (a) εr ¼ 1; (b) εr ¼ 20; and (c) εr ¼ 80.
Fig. 7. Variation of normalized capacitance with normalized separations in region I for different relative permittivities.
εr ¼
0:828 C d 1:57 C0 r
(6)
The spread in the estimation of low dielectric constants is found to vary in the first decimal place. For media with higher dielectric constants, the spread is found to be in the second decimal place. Experiments have been carried out to validate the procedure to extract the dielectric constant from a single measurement using equation (6) for aspect ratios falling in region I.
1.4. Section 4: experimental validation with air and Teflon as media A series of experiments have been carried out with various spacing between two identical circular electrodes arranged in parallel. The electrode diameter is 40 mm and electrode thickness is 2 mm with air as the medium. The experiments were performed with Alpha-A impedance analyzer from Novo control Technologies
[16]. The frequency sweep is from 1 Hz to 40 MHz and experiments were performed at ambient temperature (25 deg C) and a relative humidity of 45%. Experimental data in the frequency regime from 1 Hz to 10 MHz were found to be free from dispersion and the values at 1 kHz have been used for comparison with electrostatic simulations. Table 2 presents a comparison of total capacitance obtained from thick electrode experimental data with various techniques. Experiments have also been performed with Teflon as the dielectric medium. Teflon of diameter 40 mm with various thicknesses 2 mm, 3 mm, 4 mm, 5 mm, 7 mm, and 12 mm were considered. Repeated measurements with air as well as Teflon show that the capacitance values can be determined within a narrow dispersion of ± 0:1pF. Fig. 8(a,b) presents a comparison between simulations and experiments. The experimental capacitance results obtained for the Teflon samples fall in the range 0:1 < dr < 0:35 and are found to be systematically less than that deduced from simulations as depicted in Fig. 8(a). The lowering of capacitance is believed to be due to the distribution of air pockets [4] at the interface between the smooth electrodes and the rough surfaces of the sample. The finite element simulations were carried out using a simple model where uniform air films with variable thicknesses were considered to lie between top and bottom electrode-sample interfaces. The FEM calculations indicate that the trend in capacitance closely follows the experimental trend. Experiments were then carried out with Aluminum foils pressed between the top and bottom electrode-sample interfaces. Results are shown in Fig. 8(b). Applying conducting paint over the sample was an option but was not considered due to possible physical/chemical contamination of the sample. Table 3 compares the FEM results and experiments performed with and without Aluminum foil for different sample thicknesses. The mean sample thicknesses were used in FEM simulations. Simulations with thickness that includes the scatter were also performed and found to lead to insignificant contributions to the
Table 2 Experimental verification of thick plate FEM numerical results with air as dielectric. S.No
d/r
1. 2. 3. 4. 5.
0.05 0.1 0.15 0.2 0.3
Total Capacitance (pF) Kirchoff
Carlson and Illman [10]
Thin Plate FEM
Thick Plate FEM
Thick Plate (2 mm)Experiment
12.2 6.4 4.5 3.5 2.5
12.2 6.5 4.6 3.6 2.6
12.2 6.5 4.6 3.6 2.6
12.7 6.9 4.9 3.9 2.9
12.8 ± 0.1 6.9 ± 0.1 5.0 ± 0.1 3.9 ± 0.1 2.8 ± 0.1
24
G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25
Fig. 8. Measured Capacitance of Teflon of different thicknesses along with simulation(a) bare samples (b)with Al foil.
Table 3 Comparison of simulated capacitance with experiment and validation of dielectric constant extraction procedure. Teflon thickness (mm)
Simulated Capacitance (F)
Experimental Capacitance (F) at 1 kHz
Experimental Capacitance (F) at 1 kHz with aluminum foil
C= C0
0:828 εr ¼ CC0 1:57
d r
(True value for Teflon ¼ 2.1) 2.056 3.074 4.094 5.372 7.013
± ± ± ± ±
0.004 0.004 0.004 0.053 0.023
1.27E-11 8.8E-12 6.8E-12 5.4E-12 4.4E-12
1.05E-11 7.4E-12 5.7E-12 4.3E-12 3.5E-12
1.29E-11 8.7E-12 6.6E-12 5.2E-12 4.4E-12
2.4 2.5 2.4 2.5 2.8
2.1 2.1 2.0 2.0 2.1
± ± ± ± ±
0.1 0.1 0.1 0.1 0.1
capacitance. Table 3 describes the procedure, corresponding to region I, to extract the dielectric constant, given the measured capacitance and sample thickness. It may be noted that the values in the fifth column ought to have given the true value of εr (sixth column entries).
practical requirements that goes beyond what is possible with currently available analytical, and semi-analytical solutions for. εr 1.
2. Conclusions
One of us (Mr.Gokul Raj R) would like to express his sincere gratitude to Mr.Satyanarayana Raju, for his assistance in carrying out the experiments and for related discussions.
FEM calculations of capacitance with thin circular electrodes of equal diameters have helped in reproducing semi-analytical results for a wide range of electrode separations. For small aspect ratios, the fringe field contributions estimated from FEM calculations are in good agreement with that predicted by Kirchoff's formula for relative permittivity εr ¼ 1: FEM calculations for electrodes with finite thickness, used routinely in experiments, have been found to agree reasonably well with measurements for relative permittivity εr ¼ 1: These calculations have been extended to cover a range of dielectric constants representing a large number of commonly used materials taken to be linear, homogeneous and isotropic. Qualitatively, analysis of the trends in the total capacitance indicates that (a) fringing fields will be less significant when the dielectric constant is high and electrode separation is not large, and (b) fringing fields become significant when the dielectric constant is low and electrode separation is large. Quantitatively, two regimes have been identified based on dr values. A functional relationship between the total capacitance, the aspect ratio and the dielectric constant is established in the first regime. This functional relationship allows the dielectric constant to be estimated from a single measurement of capacitance for any aspect ratio within regime I. The range of applicability of the function is also indicated. The suggested procedure is believed to cover a significant range of
Acknowledgements
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G.R. R, C.V. Krishnamurthy / Journal of Electrostatics 87 (2017) 19e25 [13] M. Norgren, An Alternative Integral Equation for Calculating the Surface Charge in a Parallel Disk Capacitor. 2010 URSI International Symposium on Electromagnetic Theory, 2010, Aug [Online]. Available: http://ieeexplore.ieee. org/xpls/abs_all.jsp?arnumber¼5637176&tag¼1. [14] M. Norgren, B.L.G. Jonsson, The capacitance of the circular parallel plate capacitor obtained by solving the love integral equation using an analytic
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