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spectroscopy. The impedance is determined by finding the ratio of the voltage to the currentand doesnt matter on which parameter amongst them is the input or the output. Thus any ofthe impedance spectroscopy measurement techniques can be used [7]-[12].

The results obtained from measurements are used to draw a Nyquist plot, which is a plot of the imaginary impedance (Z) versus the real impedance (Z ) as a function of

frequency. Oscilloscopes have been used to obtain plots in a case where the input and outputsignals are connected to the plates of an oscilloscope. The resulting plot is called a Lissajousfigure which takes the form of an ellipse. Further analysis is done based on which of thesignals is connected to the vertical plate and which is connected to the horizontal plate. Thismethod is called the Lissajous analysis. Bode plots are often used in some cases [7]-[12].

Nowadays, computers are used in measuring impedance by the application of FastFourier Transform (FFT) in which the setup consists of an embedded measurementinstrument connected to a computer workstation which has software installed for data

presentation and analysis. The embedded measurement system contains analogue to digitalconverters, instrumentation amplifiers and a data acquisition board which transfers data to thework station computer as well as process any command issued via software in order tocontrol the experimental setup [7]-[12].

MethodologyImpedance spectroscopy was first carried out on a commercial supercapacitor using

VersaSTAT 3 and its accompanying V3-studio software. The purpose was to model it firstsuch that any other supercapacitor could be modeled based on results obtained. A GS113capxxsupercapacitor was selected for the purpose. The supercapacitor is a thin prismatic

packaged supercapacitor suitable for application in portable devices such as mobile phone,camera, digital music players, PCMCIA cards, notebook PCs, location tracking devices andother automotive applications etc. Due to the type of packaging used capxxsupercapacitorsare well known for their low equivalent series resistance implying that they permit maximum

power delivery [13]. An equivalent circuit model was generated for the data obtained usingZsimpwin software.

Figure 1 : Picture of the Capxx prismatic supercapacitor.

The Nyquist plot from the impedance spectroscopy of the capxxsupercapacitor anddata obtained from equivalent circuit fitting is shown below:

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Figure 2 : Impedance spectra of capxxsupercapacitor with fitted equivalent circuit data.

Figure 2 above showed a legend composed of Z, Msd and Z, Calc. Z, Msd is themeasured impedance spectra, while Z, Calc is the data for the fitted equivalent circuit. It will

be noticed that the data was almost the same indicating how matched the measured data is tothe fitted equivalent circuit data.

The fitted circuit model resulted by matching the data obtained by measurementsagainst all the circuit models available in the software and selecting the best among them.The software Zsimpwin by iteration process computes the global minimum chi square ( )for a selected circuit and shows the optimal circuit parameters using the Down-hill SimplexMethod [14]. Since is the sum of squared errors divided by the variance [15], the circuit

that gave the least chi square was selected as the appropriate model. A kind of circuit codingsystem was used to indicate a circuit by symbols rather than by figures. In such codes,elements in series are represented by a concatenation of their symbol letters. Parallelcomponents are bracketed and concatenated with other series or parallel elements. In the casewhere the first element or first set of elements in series are in parallel with the remainingcircuit components, the whole expression is put into bracket. Table I below shows the list ofcircuit elements used for equivalent circuit fitting.

Table 1 : Circuit elements used for modelling impedance spectroscopy data.Description Symbol ParameterResistance R R

Capacitance C CInductance L LWarburg W Y o

Constant phase Element Q Y o, Tanhyperbole T Y o, BCothyperbole O Y o, B

Gerischer G Y o, B

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. The equivalent circuit model for the capxxsupercapacitor and its code is shown below:

Figure 3: Equivalent circuit model of capxxsupercapacitor.

Circuit code: R(CR)(CR)T

For easy identification, each of the circuit components was given a number designation. Numbers 0 to 5 were used as shown below:

R(CR)(CR)T0 12 34 5

Table 2 shows each parameter, its designated number and value.

Table 2: Equivalent circuit parameters of the capxxsupercapacitor

Six carbon electrode supercapacitors were fabricated, each of which was given a letterdesignation from A to F. Impedance spectroscopy test was performed on each supercapacitorand the results were fitted unto the equivalent circuit that matched the capxxsupercapacitor.

The results are shown below:

Number designation Component Parameter Value0 R( ) Solution resistance. 0.06601 C(F) Double layer capacitance. 0.0165

2 R( ) Solution resistance. 0.00533 C(F) Double layer capacitance. 2.36004 R( ) Solution resistance. 0.09365 T Y o(S) Admittance. 3.4180

B(s) Time constant. 0.3128

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Figure 4 : Impedance spectra of supercapacitor A with fitted equivalent circuit data.

Figure 5 : Impedance spectra of supercapacitor B with fitted equivalent circuit data.

Figure 6 : Impedance spectra of supercapacitor C with fitted equivalent circuit data.

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Figure 7: Impedance spectra of supercapacitor D with fitted equivalent circuit data.

Figure 8: Impedance spectra of supercapacitor E with fitted equivalent circuit data.

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Figure 9 : Impedance spectra of supercapacitor F with fitted equivalent circuit data.

The equivalent circuit parameters of each supercapacitor are shown in the table below:

Table 3: Values of equivalent circuit parameters of the fabricated supercapacitors

The chi-square ( ) value obtained from the equivalent circuit fitting of eachsupercapacitor is shown below:

Table 4 :Chi square values obtained from the equivalent circuit fitting of eachsupercapacitor

Supercapacitor capxx A B C D E F

Chi-square X 10 -3 2.339 3.705 3.880 2.833 8.979 1.210 1.081

N0. Component SupercapacitorA B C D E F

0 R( ) 0.5447 0.3843 0.5279 0.4263 0.4193 2.27901 C(F) 54.050 9.9480 12.210 3.8640 46.48 0.56682 R( ) 2.4900 5.8410 1.9370 7.3530 0.3327 2.87403 C(F) 0.0754 1496.0 113.90 604.60 4226 233.34 R( ) 1.1300 0.4664 0.3640 0.1747 0.0751 0.43875 T Y o(S) 0.1063 1.4270 0.6464 0.4104 0.5225 0.2372

B(s) 54.710 2.4580 2.9530 3.3700 1.5140 1.1880

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Discussion of ResultsFrom the equivalent circuit fitting it was seen that the supercapacitor could be

represented with a resistance in series with 2 time constants and a Tanhyperbole (T) element.The presence of the T element indicates that finite diffusion takes place in the supercapacitor.This is a sort of an impurity in an electrochemical double layer supercapacitor. But the level

of diffusion is not much because a T element describes a process with a limited number ofelectro-active species. The moment they are finished, they cant be replenished.

Resistors with designation numbers 0, 2 and 4 are solution resistances together withcontact resistance and separator resistance. By mentioning the term solution resistance, theelectrolyte resistance and charge transfer resistance are being referred to. This is because thefitting was done using software and one cant differentiate the contributions of eachresistance. Meanwhile a totality of there effects is shown by the resistors. The same alsoapplies to the double layer capacitors.

ConclusionsImpedance spectroscopy tests were carried out on seven supercapacitors, the result of

which was used in determining an equivalent circuit that models the behavior of asupercapacitor. Results showed that a supercapacitor can be represented by an equivalentcircuit consisting of a resistor in series with two time constants and a Tanhyperbole.

References

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A. Schneuwly, R. Gallay, Properties and applications of supercapacitors from the state-of-the- art to future trends , 2000, Proceeding PCIM, pp1-8,http://www.garmanage.com/atelier/root/public/Contacting/biblio.cache/PCIM2000.pdf , retrieved June 1, 2010.

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