# Capacitance Estimation for Electrical Capacitance Tomography Sensors Using Digital Processing of Time-Domain Voltage Response to Single-Pulse Excitation

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Prototype Sensor Board

#### 2.2. Equivalent Circuit Representation of the Sensor

#### 2.3. Capacitance Estimation

#### 2.4. Forward Model

## 3. Results and Discussion

#### 3.1. Forward Simulation

#### 3.2. Estimated Capacitances Using the Algorithm

#### 3.3. Sensitivity to Noise

#### 3.4. Effect of Integration Time Window

#### 3.5. Effect of Denoising Filter

#### 3.6. Effect of Stray Capacitances

#### 3.7. Results with Measured Data

#### 3.8. Comparison with Existing ECT Systems

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A block diagram of the prototype ECT sensor. A material is placed near the sensor, a single voltage pulse is applied to each electrode, and the response voltages are measured at all the electrodes. The measured voltages are digitally processed to obtain the capacitances, which are in turn used to create an image representing the permittivity distribution of the material.

**Figure 2.**Photograph of the experimental setup with the sensor board of curved geometry connected to the signal generator and oscilloscope is shown on the left. The view of the board from below, showing the frame, the ground plate, and the electronics is shown on the right.

**Figure 3.**Equivalent circuit representing a sensor with three electrodes. The capacitances ${C}_{1,2},{C}_{2,3},$ and ${C}_{1,3}$ are the inter-electrode capacitances. Additionally, ${C}_{TG,i}$ and ${C}_{KG,i}$ for $i=1,2,3$ are influenced by the unknown stray capacitance in the system.

**Figure 4.**The 2D geometry simulated by the FEM solver. The sensor PCB has a relative permittivity of 4 and is supported by the frame with a relative permittivity of 3. A cylindrical material with a relative permittivity of 2 is shown on top of the sensor, and the background is air with a relative permittivity of 1.

**Figure 5.**The voltages on the K-node electrodes when the pulse input is applied to electrode ${E}_{1}$.

**Figure 6.**Plot of true and estimated capacitances for three different material permittivities. The true capacitances were computed with the FEM solver and were then used to obtain the voltage data for the capacitance estimation algorithm. The x-axis represents all 105 unique electrode pairs 1-2, 1-3, …, 2-3, 2-4, …, 14-15, some of which are not explicitly labeled to avoid clutter.

**Figure 7.**Plot of relative errors in estimated capacitances for three different material permittivities. The true capacitances were computed with the FEM solver and were then used to obtain the voltage data for the capacitance estimation algorithm. The x-axis represents all 105 unique electrode pairs 1-2, 1-3, …, 2-3, 2-4, …, 14-15, some of which are not explicitly labeled to avoid clutter.

**Figure 8.**The coefficient of variation of the estimated capacitances for different SNR levels. The results are shown for four different SNR levels and the material permittivity set to that of air. The x-axis represents all 105 unique electrode pairs 1-2, 1-3, …, 2-3, 2-4, …, 14-15, some of which are not explicitly labeled to avoid clutter.

**Figure 9.**The relative errors in estimated capacitances for different SNR levels. The results are shown for four different SNR levels and the material permittivity set to that of air. The x-axis represents all 105 unique electrode pairs 1-2, 1-3, …, 2-3, 2-4, …, 14-15, some of which are not explicitly labeled to avoid clutter.

**Figure 10.**Convergence of the maximum relative capacitance errors, ${\eta}_{C,max}$, with the SNR levels for different material permittivities.

**Figure 11.**Plot of maximum relative capacitance error, ${\eta}_{C,max}$, vs. SNR for different integration time windows specified by ${T}_{w2}$ with ${T}_{w1}=0$. The results were obtained with the material permittivity ${\epsilon}_{r,material}=1$.

**Figure 12.**Plot of maximum relative capacitance error, ${\eta}_{C,max}$, vs. the length of the Savitzky–Golay filter, for different SNR levels. The results were obtained with the material permittivity ${\epsilon}_{r,material}=1$ and an integration time window defined by ${T}_{w2}=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$s. The case in which the filter is not applied is denoted by ${L}_{SGF}=0$.

**Figure 13.**Plot of maximum relative capacitance error, ${\eta}_{C,max}$, vs. K-node to ground capacitance for different SNR levels. The capacitances, ${C}_{KG,i}$, are taken as the same value, ${C}_{KG}$, for each electrode, $i=1,2,\cdots 15$, and are varied from 25 pF to 200 pF. The results were obtained with the material permittivity ${\epsilon}_{r,material}=1$ and an integration time window defined by ${T}_{w2}=5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$s.

**Figure 14.**Plot of capacitances obtained from measured data with cylindrical phantoms for two different materials: oil and water. The results are compared against the value obtained from the FEM simulation using the oil and water permittivity values of 3 and 80, respectively. The x-axis represents all 105 unique electrode pairs 1-2, 1-3, …, 2-3, 2-4, …, 14-15, some of which are not explicitly labeled to avoid clutter.

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**MDPI and ACS Style**

Kalarickel Ramakrishnan, P.; Westwood, T.; Magalhães Gouveia, T.; Taani, M.; de Jager, K.; Murdoch, K.; Orlov, A.A.; Ozhgibesov, M.S.; Propodalina, T.V.; Wojtowicz, N.
Capacitance Estimation for Electrical Capacitance Tomography Sensors Using Digital Processing of Time-Domain Voltage Response to Single-Pulse Excitation. *Electronics* **2023**, *12*, 3242.
https://doi.org/10.3390/electronics12153242

**AMA Style**

Kalarickel Ramakrishnan P, Westwood T, Magalhães Gouveia T, Taani M, de Jager K, Murdoch K, Orlov AA, Ozhgibesov MS, Propodalina TV, Wojtowicz N.
Capacitance Estimation for Electrical Capacitance Tomography Sensors Using Digital Processing of Time-Domain Voltage Response to Single-Pulse Excitation. *Electronics*. 2023; 12(15):3242.
https://doi.org/10.3390/electronics12153242

**Chicago/Turabian Style**

Kalarickel Ramakrishnan, Praveen, Timothy Westwood, Tomé Magalhães Gouveia, Mahdi Taani, Kylie de Jager, Kenny Murdoch, Andrey A. Orlov, Mikhail S. Ozhgibesov, Tatiana V. Propodalina, and Natalia Wojtowicz.
2023. "Capacitance Estimation for Electrical Capacitance Tomography Sensors Using Digital Processing of Time-Domain Voltage Response to Single-Pulse Excitation" *Electronics* 12, no. 15: 3242.
https://doi.org/10.3390/electronics12153242