# Capacitive-Coupling Impedance Spectroscopy Using a Non-Sinusoidal Oscillator and Discrete-Time Fourier Transform: An Introductory Study

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- By coupling electrodes capacitively to the measured object and by incorporating the resulting couplings into an oscillation circuit, an alternating current is applicable inside the object covered with a thin insulating layer.
- (2)
- By measuring the amplitude and phase of the object’s current and those of the object’s potential difference resulting from oscillation, even with unknown coupling capacitance, the impedance of the object is measurable.
- (3)
- By estimating the impedance of the measured object from the amplitude and phase spectrum obtained from the waveform of a few oscillation cycles, the temporal resolution of IS is improved.
- (4)
- By making the oscillation waveform a non-sinusoidal wave, the fundamental frequency of oscillation and its higher harmonic waves are usable for the analysis. In this manner, the operation to switch frequency of a sinusoidal wave becomes unnecessary.

## 2. Approach of Capacitive-Coupling IS

#### 2.1. Non-Sinusoidal Oscillator Circuit with Capacitive Couplings

#### 2.2. Determination of Unknown Capacitance and Resistance in Series Connection

#### 2.3. Experimental Method

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic model of electrodes capacitively coupled to a conductive material via a thin insulator: (

**a**) cross-sectional diagram; (

**b**) equivalent circuit.

**Figure 2.**Circuit diagrams illustrating the measurement of impedance shown in Figure 1: (

**a**) with a sinusoidal voltage source; and (

**b**) in a non-sinusoidal oscillator.

**Figure 3.**Example of the oscillation waveforms of ${v}_{12}\left(t\right)$ and $v1\left(t\right)$ measured in the circuit of Figure 2b. The ${R}_{\mathrm{X}}$ and ${C}_{\mathrm{X}}$ were set to 4.0 kΩ and 10 nF, respectively. The segment of time with a colored background corresponds to two cycles of the oscillation and was used for the subsequent discrete Fourier transform (DFT) analysis.

**Figure 4.**Frequency spectra of: (

**a**) amplitude $\left|{\dot{V}}_{12}\right|$; (

**b**) phase ${\theta}_{\mathrm{V}12}$; (

**c**) amplitude $\left|{\dot{V}}_{1}\right|$; and (

**d**) phase ${\theta}_{\mathrm{V}1}$ at $\left(2m-1\right){f}_{0}$ ($m=1,2,3,\cdots ,20$ ) Hz. The spectra were obtained using DFT from the two-cycle segment of ${v}_{12}\left(t\right)$ and ${v}_{1}\left(t\right)$ in Figure 3. The ${R}_{\mathrm{X}}$ and ${C}_{\mathrm{X}}$ were set to 4.0 kΩ and 10 nF, respectively.

**Figure 5.**Frequency spectra of: (

**a**) absolute impedance $\left|{\dot{Z}}_{\mathrm{A}}\right|$; (

**b**) phase ${\theta}_{\mathrm{ZA}}$; (

**c**) absolute impedance $\left|{\dot{Z}}_{\mathrm{X}}\right|$; and (

**d**) phase ${\theta}_{\mathrm{ZX}}$ at $\left(2m-1\right){f}_{0}$ ($m=1,2,3,\cdots ,20$ ) Hz.

**Figure 6.**Three-dimensional perspective plots of: (

**a**) impedance ${\dot{Z}}_{\mathrm{X}}$; and (

**b**) admittance ${\dot{Y}}_{\mathrm{X}}$. Three-dimensional DFT data is projected onto each plane. The solid and dashed lines are theoretical curves of ${\dot{Z}}_{\mathrm{X}}$ and ${\dot{Y}}_{\mathrm{X}}$.

**Figure 7.**Frequency spectra of: (

**a**) real part $\mathrm{Re}({\dot{Z}}_{\mathrm{X}})$; and (

**b**) imaginary part $\mathrm{Im}({\dot{Z}}_{\mathrm{X}})$ of impedance ${\dot{Z}}_{\mathrm{X}}$. The DFT data were fitted to Equations (22) and (23) for determining ${R}_{\mathrm{X}}$ and ${C}_{\mathrm{X}}$ (dashed lines).

**Figure 8.**${R}_{\mathrm{X}}$ and ${C}_{\mathrm{X}}$ estimated using oscillation waveforms and DFT: (

**a**) estimated ${R}_{\mathrm{X}}$; (

**b**) absolute error of estimated ${R}_{\mathrm{X}}$; (

**c**) estimated ${C}_{\mathrm{X}}$; (

**d**) relative error of estimated ${C}_{\mathrm{X}}$.

**Figure 9.**Equivalent circuit modeling of ${\dot{Z}}_{\mathrm{A}}$: (

**a**) Cole–Cole plot for ${C}_{\mathrm{X}}=$ 0.10 nF and ${R}_{\mathrm{X}}=$ 0 Ω; (

**b**) equivalent circuit with a stray resistance ${R}_{\mathrm{AS}}$ and stray inductance ${L}_{\mathrm{A}}$. The symbol $n$ represents the harmonic number of ${\dot{Z}}_{\mathrm{A}}\left(n{f}_{0}\right)$. We used pyZwx software to fit the DFT data to the equivalent circuit model [59].

**Figure 10.**${R}_{\mathrm{X}}$ and ${C}_{\mathrm{X}}$ estimated using optimized ${\dot{Z}}_{\mathrm{A}}$: (

**a**) estimated ${R}_{\mathrm{X}}$; (

**b**) absolute error of estimated ${R}_{\mathrm{X}}$; (

**c**) relative error of estimated ${C}_{\mathrm{X}}$.

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**MDPI and ACS Style**

Yamaguchi, T.; Ueno, A.
Capacitive-Coupling Impedance Spectroscopy Using a Non-Sinusoidal Oscillator and Discrete-Time Fourier Transform: An Introductory Study. *Sensors* **2020**, *20*, 6392.
https://doi.org/10.3390/s20216392

**AMA Style**

Yamaguchi T, Ueno A.
Capacitive-Coupling Impedance Spectroscopy Using a Non-Sinusoidal Oscillator and Discrete-Time Fourier Transform: An Introductory Study. *Sensors*. 2020; 20(21):6392.
https://doi.org/10.3390/s20216392

**Chicago/Turabian Style**

Yamaguchi, Tomiharu, and Akinori Ueno.
2020. "Capacitive-Coupling Impedance Spectroscopy Using a Non-Sinusoidal Oscillator and Discrete-Time Fourier Transform: An Introductory Study" *Sensors* 20, no. 21: 6392.
https://doi.org/10.3390/s20216392