# A Multi-Channel Fast Impedance Spectroscopy Instrument Developed for Quality Assurance of Super-Capacitors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Performance Evaluation of the Supercapacitor Using Electrochemical Impedance Spectroscopy (EIS) Tests

#### 2.1. Electrochemical Impedance Spectroscopy

#### 2.2. Equivalent Circuit Model of the Supercapacitor

_{S}, a series resistance R

_{S}and a pore impedance Z

_{PORE}as shown in Figure 1. Here, Z

_{PORE}represents the non-ideal porous electrode by a ladder circuit in which an electrolyte resistance Re connected in parallel with a constant phase element (CPE) are connected in parallel as expressed by Equation (3) [18,19]. Figure 2 shows the Nyquist impedance representation of the equivalent circuit model of the supercapacitor.

#### 2.3. Power Loss by the Equivalent Series Resistance

_{S}+ R

_{e}/3. Also, as the SOC increases, the equivalent series resistance in the low-frequency region increases. The loss due to the resistance component can be obtained by Equation (4) [11].

_{e}is the electrode current, I

_{w}current flowing through the CPE component and R

_{w}is the resistance component of CPE. Therefore, the higher the electrode resistance R

_{e}and CPE resistance component at HF ESR, the higher the power loss [19].

#### 2.4. Evaluation of the Capacitance

#### 2.5. Evaluation of the Self-Discharge by Using ‘d’ Parameter of the Pore Impedance

_{C}the constant current and H the step function [10]. To investigate the self-discharge characteristics of the super-capacitor according to the change in ‘d’ value, the response of charge/self-discharge current of 100 mA is shown in Figure 5a.

_{C}) is applied to the supercapacitor and the current is cut off after 140 s. The voltage response of the supercapacitor is calculated by using the equivalent circuit model of the supercapacitor as shown in Equation (9) [10]. As shown in Figure 5b, the supercapacitor is charged up to its rated voltage 2.7 V when d = 1. However, it reaches 2.141 V and 1.693 V at 140 s when d = 0.95 and d = 0.9, respectively, which means that it takes more time to reach 2.7 V when d is less than 1. It is also observed that the self-discharge becomes severe with a smaller value of ‘d’. As ‘d’ approaches 1, it exhibits pure capacitor characteristics and results in less charge time and more self-discharge.

## 3. Proposed Multi-Sine Multi-Channel EIS Instrument

#### 3.1. Perturbation and Power Control

#### 3.2. Voltage Measurement and Acquisition

#### 3.3. Current Measuremnet and Acquisition

## 4. Software for the Proposed Instrument

#### 4.1. Graphical User Interface

#### 4.2. Implementing Lock-In Amplifier

_{in}is the input signal, ${\mathrm{V}}_{\mathrm{signal}}$ is the peak amplitude of V

_{in}, ${\mathsf{\omega}}_{\mathrm{signal}}$ is the frequency in radian of V

_{in}and ${\mathsf{\theta}}_{\mathrm{signal}}$ is the phase of V

_{in}. Equations (11) and (12) are the sine and cosine reference signals.

_{X}and Out

_{Y}as in Equation (13) and Equation (14).

#### 4.3. Complex Non-Linear Least Square Curve Fitting

_{j,}the value of Φ1 can be calculated, and then ∆θ is calculated. The updated θ is consequently used to calculate the new Φ. This calculation will be in iterative procedure until Φ converges to a certain limit, 10

^{−6}in our case. Figure 14 presents the flow chart for the CNLS procedure.

## 5. Discussions and Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Nyquist impedance plot of a supercapacitor (2.7 V, 10 F) at 0% SOC (state-of-charge) and the curve fitting result.

**Figure 3.**(

**a**) Nyquist impedance plots of a supercapacitor (2.7 V 10 F) at 0%, 60% and 100% SOC (

**b**) Parameters variation at each SOC.

**Figure 5.**(

**a**) Simulation of self-discharge with different “d” parameter up to Vrated = 2.7 V (Ic = 100 mA, C = 10 F, equivalent series resistance (ESR) = 0.065 mΩ) (

**b**) Simulation results of self-discharge with different ‘d’ parameters (Ic = 100 mA, C = 10 F, ESR = 0.065 mΩ).

**Figure 7.**(

**a**) Picture of the developed multi-sine/multi-channel electrochemical impedance spectroscopy (EIS) system. (

**b**) Multichannel differential data acquisition circuit.

**Figure 9.**Combined multi-sine method. (

**a**) Lowest frequency 0.1 Hz and 0.3 Hz imposed on 0.1 Hz (

**b**) 0.3 Hz, 0.6 Hz, 0.9 Hz super imposed on 0.1 Hz. (

**c**) All the frequencies including high frequency components superimposed on 0.1 Hz.

**Figure 15.**Comparison of EIS results by WEIS500 and the developed instrument for three supercapacitors A, B, C (C = 10 F, Vrated = 2.7 V).

**Table 1.**Comparisons of the parameters of samples A, B and C obtained by the proposed method and WEIS500 workstation.

Method/Parameters | Ls (nH) | Rs (Ω) | Re (Ω) | Q | d |
---|---|---|---|---|---|

Conventional | 230 | 0.0228 | 0.0485 | 6.7 | 0.984 |

Combined Mode | 200 | 0.0227 | 0.0483 | 6.72 | 0.981 |

Conventional | 232 | 0.0225 | 0.0481 | 6.74 | 0.984 |

Combined Mode | 216 | 0.0228 | 0.0478 | 6.7 | 0.986 |

Conventional | 195 | 0.0222 | 0.0475 | 6.63 | 0.984 |

Combined Mode | 189 | 0.0224 | 0.0476 | 6.6 | 0.983 |

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**MDPI and ACS Style**

Farooq, F.; Khan, A.; Lee, S.J.; Mahad Nadeem, M.; Choi, W.
A Multi-Channel Fast Impedance Spectroscopy Instrument Developed for Quality Assurance of Super-Capacitors. *Energies* **2021**, *14*, 1139.
https://doi.org/10.3390/en14041139

**AMA Style**

Farooq F, Khan A, Lee SJ, Mahad Nadeem M, Choi W.
A Multi-Channel Fast Impedance Spectroscopy Instrument Developed for Quality Assurance of Super-Capacitors. *Energies*. 2021; 14(4):1139.
https://doi.org/10.3390/en14041139

**Chicago/Turabian Style**

Farooq, Farhan, Asad Khan, Seung June Lee, Mohammad Mahad Nadeem, and Woojin Choi.
2021. "A Multi-Channel Fast Impedance Spectroscopy Instrument Developed for Quality Assurance of Super-Capacitors" *Energies* 14, no. 4: 1139.
https://doi.org/10.3390/en14041139