# Equivalent Circuit Models: An Effective Tool to Simulate Electric/Dielectric Properties of Ores—An Example Using Granite

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sample

#### 2.2. Experimental

^{10}Ω·m [20]). Thus, when measuring the electric properties of granite, the surface leakage current should be avoided. The sample was placed in a resistivity chamber (Model 24, ADCMT, Saitama, Japan) where the leakage current of the sample was eliminated by a guard electrode of the chamber. The diameter of the upper electrode of this chamber was 6 mm. An ultra-high resistance meter (5451, ADCMT) was used to measure the I–V properties of the sample. In order to measure the dielectric properties of the sample, we used an LCR meter (ZM2376, NF, Yokohama, Japan) with the same resistivity chamber. The capacitance C and dielectric loss tanδ were measured with the range of frequency from 1 Hz to 1 MHz with 500 measurement points of frequency.

#### 2.3. Simulation Methods

## 3. Results and Discussion

#### 3.1. Electrical Properties

^{−11}S/m and its standard deviation was 9.0 × 10

^{−12}S/m. Thus, the value of resistivity (1/σ) of our sample was 1.0 × 10

^{10}Ω·m, and it was similar to the value found in the literature (10

^{10}Ω·m) [20].

#### 3.2. Dielectric Properties of Granite

_{r}ε

_{0}) is the dielectric constant, L is the thickness of the sample (1.1 mm) and S is the surface area of the electrode (28.3 × 10

^{−6}m

^{2}). Ε

_{r}is the relative dielectric constant/permittivity of granite (i.e., 4–7 at 100 MHz) [26,27] and ε

_{0}is the dielectric constant of vacuum/air (8.85 × 10

^{−12}F/m). From our measurements, the average value of ε

_{r}was found to be 10.8 at the range of frequency from 1000 to 10,000 Hz (10 kHz). The value was not exactly the same as the literature values (i.e., 4–7 at 100 MHz) [26,27]. However, it should be noted that the frequency used in the above reference was 100 MHz, and in general, the capacitance C and the relative dielectric constant ε

_{r}decreased with frequency. Using the calculated values of ε, the ε-f properties of our granite sample were plotted in a double logarithmic graph (Figure 6). The resistance R is expressed by the following equation, which was used to calculate the general resistance value [25]:

^{−6}m

^{2}). The problem of an electromagnetic wave in dielectric materials (conductivity σ ≠ 0) can be understood by solving Maxwell’s equations for electromagnetics [25]. We will not discuss the details here, but by solving Maxwell’s equations for dielectric materials, the following relationship can be found [25]:

^{−12}F (393 Hz), 8.44 × 10

^{−13}F (21 kHz), 1.59 × 10

^{−12}F (435 kHz), 4.37 × 10

^{−13}F (761 kHz)—Figure 5a; tanδ:0.109 (18.2 kHz), 0.479 (33.4 kHz), 0.0682 (533 kHz)—Figure 5b; ε: 6.54 × 10

^{−11}F/m (393 Hz), 3.28 × 10

^{−11}F/m (21 kHz), 6.59 × 10

^{−11}F/m (435 kHz), 1.70 × 10

^{−11}F/m (761 kHz)—Figure 6) at different frequencies in the measured data of the sample. From Equations (2), (4), and (5), it is clear that the values of capacitance C and dielectric constant/permittivity ε have a negative correlation with the angular frequency (ω = 2πf)/frequency f. The slope of log ε at the range of frequency 1 to 1000 Hz was found −1.12 (Figure 6). The theoretical value of this slope is −1 (i.e., Equation (5)) and, thus, our measurement results of the dielectric properties of granite show the same trends as the theoretical analysis of dielectric materials. However, after 1000 Hz of frequency, the values of ε started to possess peak values, which indicate the dielectric relaxation in the granite, and as Maxwell’s equations on electromagnetics cannot explain the dielectric relaxation of composite materials, our simulation works would not be able to simulate some of the measured data obtained in that frequency range.

#### 3.3. Electrical Properties of a Void (Pore in Mineral/Rock)

^{6}V/m). The dielectric-breakdown voltage of air depends on the air pressure, temperature, etc., and in general, it is known at 3.0 × 10

^{6}V/m (3.0 kV for 1 cm of gap length), thus, our result of air breakdown was very similar to other reports [13,20]. From our measurement results, it is clear that the conductivity of a void (pore) is a voltage-dependent parameter.

#### 3.4. Equivalent Circuit Models for Minerals

_{b}is Boltzmann’s constant (8.618 × 10

^{−5}eV/K). The constants A and B for each mineral in the granite, and the relative permittivity ε

_{r}used to calculate the capacitance are shown in Table 1. In this study, we used an absolute temperature of 293.15 K, which is considered as room temperature (i.e., 20 °C). As shown in Figure 2b, two resistive elements were placed vertically in one cube, and the value of one resistive element was R/2. The resistance values R of the minerals were calculated using Equation (3). In this study, the capacitance of a mineral was calculated from Equation (2). Since the value of C is the value of the entire mineral in the cube of Figure 2b, the value of C for the mineral element was 2C. The values of L and S in Equations (2) and (3) were L = 1.0 × 10

^{−4}m and S = 1.0 × 10

^{−8}m

^{2}because the minerals were arranged in 0.1 mm cubes, as shown in Figure 2b.

#### 3.5. Equivalent Circuit Model of Void (Pore in Mineral/Rock)

^{−12}F/m). As Equation (2) is applicable for any dielectric materials that include a void (pore), this equation was used to calculate the void capacitance that was 0.885 × 10

^{−15}F and was used in our simulation work. As we described before, the conductivity of air depends on the input voltage, thus, we used a voltage-dependent resistance (VDR) system which can fit with the measurement data of the I–V properties of air. The equivalent circuit was simulated with the LTspice circuit simulator. The I–V properties of air (void) were simulated for the input voltage ranging from 1 to 350 V. The I–E characteristics of the simulation data of a void (pore/air) were prepared and plotted in a double logarithmic graph. The equivalent circuit model of the void and the I–E characteristics of the void are shown in Figure 8a,b, respectively. The electrical properties of a void (see Figure 8) indicates that before the dielectric breakdown of the air, the electric current through a void can be negligible as the value of the electric current is very low (i.e., 10

^{−12}A at 1 to 350 V) when comparing to that of near the dielectric-breakdown area (i.e., 10

^{−3}A at 680 V) of a void. We aimed to make an equivalent circuit model for the void, which can represent the I–V properties in a way that could fit the dielectric-breakdown region of the void. Figure 8b compared the I–E properties of the void obtained via measurement and simulation. From the simulation results, the electric current was 20 pA for the input voltage of 10 V (i.e., 100,000 V/m), and it remained less than 10

^{−9}A even at 150 V (i.e., 1,500,000 V/m). The values of the electric current were 0.1 × 10

^{−9}at 50 V (i.e., 500,000 V/m) and 0.21 × 10

^{−9}A at 100 V (1,000,000 V/m)). The electric current increased gradually with the increase in input voltage, as we found it was 15.7 × 10

^{−9}A at 200 V (i.e., 2,000,000 V/m) and 0.172 × 10

^{−6}A at 300 V (i.e., 3,000,000 V/m). Finally, the current reached 1 mA in between the input voltage of 340 V (3,400,000 V/m) and 350 V (i.e., 35,000,000 V/m). The dielectric-breakdown voltage was found in our measurement at 680 V/0.2 mm (i.e., 340 V/0.1 mm), and, thus, we confirmed the reproduction of the I–V characteristics of a void (air) in our simulation work. It is to be noted that, while measuring the dielectric breakdown of the void, the system was very unstable to measure, but from the simulation works, the details of the electric current were calculated.

#### 3.6. Simulation Results of Granite Sample

^{12}Ω·cm (i.e., 100 pS/m in the value of conductivity) [21] which is the same as our measured and simulation values. Thus, our simulation results are considered reliable. However, a difference in the conductivity properties of granite was found between the measurement results and simulation works. As shown in Figure 11a, it is clear that the simulation values are almost constant regardless of the input voltages, while the measured values increase with the input voltage. This difference might occur from the Schottky effect observed under a HV applied voltage on dielectric materials [13]. For an electric field E at temperature T, the emission of electrons/ions can be expressed by the following equation:

^{6}A/m

^{2}K

^{2}), T is the absolute temperature in K, φ is the activation energy (i.e., similar to a work function in metal), e is the elementary charge quantity, and the others were described before. The details will not be discussed here, however, there is a linear relationship between the natural logarithm of electric current I and electric field E

^{1/2}(Equations (8)–(10)).

## 4. Conclusions

- (1)
- The calculated electrical conductivity of the granite model and the actual granite were close to each other (measurement: 53.5 pS/m, simulation: 36.2 pS/m), and the standard deviation was very small in the simulation results (i.e., 2.77 pS/m).
- (2)
- The Schottky effect was observed in the I–V properties of granite, and it emphasized the necessity to consider the dielectric constant in order to simulate the I–V properties.
- (3)
- Comparing the simulation values of C and tanδ of the granite model and the measurement data, the dielectric relaxation phenomenon was observed in both the simulation and measurement data for the tanδ and frequency (f) relationship, and their values were close to each other. However, our simulation works did not imitate the relaxation phenomenon for the C–f relationship. For the higher frequencies (i.e., larger than 1 kHz), the simulation results showed a larger dielectric constant ε (i.e., 10
^{−9}F/m), while the measured values were around 10^{−10}F/m.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Granite sample. (

**a**) Optical microscopic image. (

**b**) Micro-X-ray fluorescence spectrometer image identifying quartz (red), plagioclase (green), K-feldspar (blue), and biotite (pink).

**Figure 2.**Concept of equivalent circuit model. (

**a**) Granite model and (

**b**) equivalent circuit model for a small, divided cube of granite.

**Figure 3.**Example of mineral distribution in the simulated granite model. (

**a**) Example of a granite 3D model, (

**b**) XY plane of the granite model shown in (

**a**).

**Figure 4.**J–E relationship of granite sample (measured data) with error bars showing the standard deviation values.

**Figure 5.**Measured dielectric properties of granite with error bars showing the standard deviation values. (

**a**) C–f relationship, (

**b**) tanδ–f relationship.

**Figure 6.**ε-f properties of granite calculated from the measured results of dielectric properties with error bars showing the standard deviation values.

**Figure 8.**Equivalent circuit model and electrical properties of a void (pore) in minerals. (

**a**) Equivalent circuit and (

**b**) I–E properties of a void.

**Figure 9.**Average of simulation results of J–E for granite with error bars showing the standard deviation values.

**Figure 10.**Simulation results and analysis (

**a**) simulation results of conductivity of granite, (

**b**) Schottky effect on granite at room temperature.

**Figure 11.**Simulation results of dielectric properties with error bars showing the standard deviation values. (

**a**) C–f relationship, (

**b**) tanδ–f relationship.

**Figure 12.**Comparison of dielectric constant with frequency in between the measurement and simulation results.

Mineral | Log(A) [log(s/m)] | B [eV] | ε_{r} |
---|---|---|---|

Quartz | 6.3 | 0.82 | 6.53 |

Plagioclase | 0.041 | 0.85 | 6.91 |

K-Feldspar | 0.11 | 0.85 | 6.2 |

Biotite | −13.8 | 0 | 9.28 |

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**MDPI and ACS Style**

Fukushima, K.; Kabir, M.; Kanda, K.; Obara, N.; Fukuyama, M.; Otsuki, A.
Equivalent Circuit Models: An Effective Tool to Simulate Electric/Dielectric Properties of Ores—An Example Using Granite. *Materials* **2022**, *15*, 4549.
https://doi.org/10.3390/ma15134549

**AMA Style**

Fukushima K, Kabir M, Kanda K, Obara N, Fukuyama M, Otsuki A.
Equivalent Circuit Models: An Effective Tool to Simulate Electric/Dielectric Properties of Ores—An Example Using Granite. *Materials*. 2022; 15(13):4549.
https://doi.org/10.3390/ma15134549

**Chicago/Turabian Style**

Fukushima, Kyosuke, Mahmudul Kabir, Kensuke Kanda, Naoko Obara, Mayuko Fukuyama, and Akira Otsuki.
2022. "Equivalent Circuit Models: An Effective Tool to Simulate Electric/Dielectric Properties of Ores—An Example Using Granite" *Materials* 15, no. 13: 4549.
https://doi.org/10.3390/ma15134549