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An intuitive model is proposed in this paper to describe the electrical behavior of certain ultracapacitors. The model is based on a simple expression that can be fully characterized by five real numbers. In this paper, the measured impedances of three ultracapacitors as a function of frequency are compared to model results. There is good agreement between the model and measurements. Results presented in a previous study are also reviewed and the paper demonstrates that those results are also consistent with the newly described model.

Smart grids and energy systems always need energy storage. Ultracapacitors represent an alternative to batteries for storing electrical energy and can help to compensate for the limited power density of batteries [

Often, ultracapacitors are modeled with simple

Practical ultracapacitor models are more complicated. Conway [

Equivalent circuit developed by Gualous et al.

Zubieta and Bonert [

Equivalent circuit developed by Zubieta and Bonert.

However, the pores of activated carbon in an ultracapacitor have a complex branch pore structure for which any impedance analysis model should account. Furthermore, the parameters in the above models, especially the voltage dependent capacitance, can be difficult to quantify.

Buller et al. [

Equivalent circuit developed by Buller et al.

Qu and Shi [

Equivalent circuit developed by Qu and Shi.

Itagaki et al. [

This paper proposes a new electrical model for ultracapacitors that is also based on a fractal interpretation of their structure, but does not make any assumption about the size or shape of the pores. The new model is relatively simple and can be fully described by five parameters.

Ultracapacitors are comprised of two highly porous activated carbon electrodes, which take surface area and charge separation distance to an extreme. The surface areas can be greater than 21,500 square feet per gram and the separation between the charged surfaces is reduced to distances on the order of nanometers [

Consider the sample pore structure illustrated in Figure

Pore model of the ultracapacitor electrodes.

If we start by analyzing the capacitance of the structure without any teeth, the total capacitance is

The capacitance,

Capacitance scheme of porosity.

A circuit model for the post-only ultracapacitor is provided in Figure

The post-only model.

Now, consider the structure of the intermeshed teeth between posts. In fractal geometry, each tooth would be a scaled down version of the post to which it was attached. If we assume that each post has

The capacitance,

From (

Equivalent circuit for a capacitor with one set of posts and teeth.

Employing (

Electrical model of a fractal electrode structure.

Although the example in Figure

As the different number of time between charging and discharging, the deterioration of ultracapacitors is also not the same. Few electrolytes will be decomposed to form the free insoluble product particles, which increase the resistance. Therefore, a resistor is paralleled in the model in Figure

Considering its ohmic resistance characteristics and ohmic capacitive characteristics, a resistor and a capacitor are in series; thus, the model in Figure

Ultracapacitor model described by only 5 real numbers.

Although the model in Figure

Complex impedance ladder network.

The equivalent circuit of ladder network in Figure

In this case,

Electrochemical impedance spectroscopy (EIS) is one of the most frequently used analytical tools for the characterization of ultracapacitors. This method was used to determine input resistance and reactance of the ultracapacitor as a function of frequency at a given excitation voltage.

Two ultracapacitors with nominal values of 3.3 F and 2 F were measured at room temperature. Potentiostat cycling tests were performed in the capacitor designing lab of St. Jude Medical’s Cardiac Rhythm Manufacture, the Liberty, S.C., with a Gamry potentiostat (Reference 3000). The impedance was measured by applying a sinusoidal 5 mV excitation superimposed on a 1.2 V DC bias to the ultracapacitor and measuring the magnitude and phase of the current. The frequency range was from 100 mHz to 1 kHz.

In this study, a Panasonic 3.3 F/2.3 V (EEC-HW0D335) ultracapacitor which is a new one and a Taiyo Yuden 2 F/2.3 V (PAS1016LR2R3205) which had been charged and discharged for many times were measured. Figures

The Comparison between measured numbers and calculated numbers in Nyquist plot.

Nyquist plot of a 3.3 F ultracapacitor (values for the model calculation:

Nyquist plot of a 2 F ultracapacitor (values for the model calculation:

The values of

Complex-plane representation of a 400 F ultracapacitor (values for the model calculation:

A new model for describing the electrical behavior of ultracapacitors is introduced based on an intuitive representation of the electrode pores as a fractal structure. The new model has an infinite number of elements but is fully represented by only five real numbers. A closed-form expression for the input impedance was derived making it relatively easy to fit measured results to the model.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Fundamental Research Funds for the Central Universities (no. 2014JBZ017).