On The Inverse Relaxation Approach To Supercapacitors Characterization

TL;DR

This paper introduces a new inverse relaxation method for supercapacitor characterization that simplifies testing, avoids fixed current sources, and can replace traditional impedance measurements, with potential for automation.

Contribution

A novel inverse relaxation technique for supercapacitors is developed, modeled, and experimentally validated, offering a simplified, automated alternative to existing characterization methods.

Findings

The method measures capacitance at different time scales without fixed current sources.

It distinguishes between 'easy' and 'hard' to access capacitance.

The technique can replace traditional impedance and IEC 62391 tests.

Abstract

A novel inverse relaxation technique for supercapacitor characterization is developed, modeled numerically, and experimentally tested on a number of commercial supercapacitors. It consists in shorting a supercapacitor for a short time $\tau$, then switching to the open circuit regime and measuring an initial rebound and long-time relaxation. The results obtained are: the ratio of "easy" and "hard" to access capacitance and the dependence $C(\tau)$, that determines what the capacitance the system responds at time-scale $\tau$; it can be viewed as an alternative to used by some manufacturers approach to characterize a supercapacitor by fixed capacitance and time-scale dependent internal resistance. Among the advantages of proposed technique is that it does not require a source of fixed current, what simplifies the setup and allows a high discharge current regime. The approach can be used as a replacement of low-frequency impedance measurements and the ones of IEC 62391 type, it can be effectively applied to characterization of supercapacitors and other relaxation type systems with porous internal structure. The technique can be completely automated by a microcontroller to measure, analyze, and output the results.

Equations

$$Q(\tau )$$

$$\Delta Q_{C_1}$$

$$C_1\cdot (U_2-U_1)$$

$$\eta =\frac{C_1}{C_2}$$

$$C_1$$

$$R_1$$

$$C(\tau )$$

$$Z_2(\omega )$$

$$Z(\omega )$$

$$C_{impedance}(\tau )$$

$$C_{\Sigma }$$

$$C(\tau )$$

$$\eta (\tau )$$

$$R_s$$

Full text

A novel inverse relaxation technique for supercapacitor characterization is developed, modeled numerically, and experimentally tested on a number of commercial supercapacitors. It consists in shorting a supercapacitor for a short time $\tau$, then switching to the open circuit regime and measuring an initial rebound and long–time relaxation. The results obtained are: the ratio of ‘‘easy’’ and ‘‘hard’’ to access capacitance and the dependence $C(\tau)$, that determines what the capacitance the system responds at time–scale $\tau$; it can be viewed as an alternative to used by some manufacturers approach to characterize a supercapacitor by fixed capacitance and time–scale dependent internal resistance. Among the advantages of proposed technique is that it does not require a source of fixed current, what simplifies the setup and allows a high discharge current regime. The approach can be used as a replacement of low–frequency impedance measurements and the ones of IEC 62391 type, it can be effectively applied to characterization of supercapacitors and other relaxation type systems with porous internal structure. The technique can be completely automated by a microcontroller to measure, analyze, and output the results.

Dedicated to the memory of S.L. Kulakov

I Introduction

A distributed internal RC structure is manifested in electrical measurements of supercapacitors. The distribution is caused by hierarchical porous structure of electrodes. The two most commonly used technologies for manufacturing carbon structures for supercapacitor electrodes are Carbide–derived carbon (CDC) and Activated carbon. CDC materials are derived from carbide precursors[1]. An initial crystal structure of the carbide is the primary factor affecting CDC porosity. Activated carbon is typically derived from a charcoal or biochar[2]. It’s structure is inherited from the starting material and has a surface area in excess of $2,000m^{2}/g$ [3]. See [4] for a review of carbon materials used in supercapacitor electrodes. All the technologies used for supercapacitor manufacturing lead to a complex, ‘‘self–assembled’’ type of internal structure. In applications the most interesting is not the internal structure of a device per se, but it’s manifestation in the electric properties.

While Li–ion systems are the most effective in energy storage applications[5], supercapacitors are the most effective in high–power applications[6]. For Li–ion batteries the two characteristics are typically provided by manufacturers: specific energy and specific power. For supercapacitors the other two characteristics are typically provided by manufacturers: capacitance and internal resistance. Standard methods of characterization create a substantial uncertainty, because a supercapacitor’s characteristics change during the discharge process.

The techniques currently used for characterization of a supercapacitor’s electric properties can be classified as:

• •

Low–current impedance AC spectroscopy is a frequency domain technique. High frequency range $10^{-3}\div 10^{6}$Hz allows an information of porous structures to be obtained. However a low-current measurement regime, interpretation difficulties, and equipment complexity limit the technique applicability.

• •

Cyclic Voltammetry is a time–domain technique, where the voltage is swept between lower and upper limits at a fixed scan rate. The voltage scan rate is the slope of the $U(t)$; current evolution is measured as a function of the voltage. This technique is quite common for electrochemical materials study, it is less convenient for supercapacitors characterization, where a high current regime is often required.

• •

Constant current charge/discharge regime is the most used time–domain technique for supercapacitors characterization. A measurement starts by switching to a given constant charging current. The initial instantaneous voltage jump determines the capacitor series resistance. The current is switched off at time when the capacitor has reached the maximal voltage, the voltage instantaneously drops due current interruption via the series resistance. Then similar processes take place in the discharge regime. The value of the current is defined in [7] IEC 62391-{1,2} standards. The value of the internal resistance is determined from the potential jumps. The value of the capacitance is determined from the time necessary to charge/discharge the capacitor at given current.

Multiple extensions of the discharge techniques[8, 9, 10] have been recently proposed, the most noticeable are: total charge measurement and the difference of supercapacitors characteristics in constant power and constant current discharge regimes[6].

In this paper a novel technique of supercapacitor characterization is developed. The technique has all the measurements performed in time–domain, possibly at high current. The results obtained are:

1. the ratio $\eta$ of ‘‘easy’’ and ‘‘hard’’ to access capacitance and 2. the dependence $C(\tau)$, that determines what the capacitance the system responds at time–scale $\tau$, varied in at least three orders range. The technique was microcontroller–automated to measure, analyze, and output the results; the measurement of $\eta$ does not require total change measurement, but the measurement of $C(\tau)$ does require one, it is implemented using a microcontroller with ADC. There are two distinguished features of the technique: no fixed–current source requirement and the measurements are performed at various time–scales $\tau$, what introduces a parameter $\tau$ similar to inverse frequency in impedance spectroscopy technique. The approach can be effectively used as a replacement of low–frequency impedance measurements to determine the internal resistance and capacitance.

II Inverse Relaxation Model

Supercapacitor equivalent circuit has multiple internal $RC$s, what corresponds to it’s hierarchical internal structure[11, 4, 12, 13]. The dynamics of such a system is rather complex, it is exhibited, for example, in multi–exponent evolution of $U(t)$ relaxation (see experimental Fig. 5 below) and in deviation from a rectangle in a cyclic voltammetry (CV) plot. For an application in electronics the most convenient characteristic is: how a supercapacitor behaves at a given time–scale $\tau$, how much $Q$ it can be charged/discharged during time–interval $\tau$. In terms of electric properties supercapacitor’s electrodes internal porous structure can be considered as electric capacitance of two kind: ‘‘easy’’ (accessible at low $\tau$) and ‘‘hard’’ (accessible only at high $\tau$), Fig. 1. Actual distribution of the internal $RC$ can be of various forms, because the internal structure manifests itself in the distribution of $RC$. This approach is more objective compared to impedance technique (which is a low amplitude technique), since it characterizes the discharge as a whole. When a supercapacitor is in the stationary state, all the capacitors in Fig. 1 model have equal potential, it is equal to the one on the electrodes, and there is no internal current. When a supercapacitor is in a non–stationary state then internal charge redistribution takes place, it can be directly observed through the dynamics of electrodes potential.

Consider a measurement technique: the system is charged to some initial potential $U_{0}$, then it is short–circuited for a short interval time (lower than the supercapacitor’s internal $RC$) to create a non–stationary state, after that it is switched to the open circuit regime and $U(t)$ is recorded to observe internal relaxation. The $U(t)$ dependence is:

• •

First, from the initial potential $U_{0}$ to almost zero (shorting to create a non–stationary state). Instead of shorting, a connection to a low–resistance circuit (we denote it $R_{s}$, a typical value is about $10-50m\Omega$) can be used, Fig. 1, in this case the potential is non–zero, the potential at the moment right before switching to the open circuit regime is denoted as $U_{s}$.

• •

Then, after switching to the open circuit regime, the potential jumps from $U_{s}$ to $U_{1}$. There is a similar current–interruption technique used in fuel cell measurements [14], page 64, the immediate rise voltage $V=IR_{i}$ is an equivalent of $U_{1}-U_{s}$.

• •

A slow final rise from $U_{1}$ to $U_{2}$, Fig. 1. The $U(t)$ relaxation from $U_{1}$ to $U_{2}$ may be of a single of multiple exponent type, this depends on the supercapacitor’s internal structure. For two–$RC$ model a pure linear dependence is observed, For three–$RC$ supercapacitor model there are two exponents in $U(t)$ evolution, one can clearly observe a deviation from a linear dependence in Fig. 3a below.

• •

While the $U(t)$ measurement can be performed using traditional equipment a progress in microcontrollers (e.g. the STM32F103C8T6 ARM which costs below $5 and has a 72Mhz CPU with 12–bit analog–to–digital converter (ADC)) allows the data to be easily recorded and stored. A microcontroller allows the total charge passed on the shorting stage to be calculated by direct integration:

[TABLE]

‘‘Right rectangle’’ integration rule is used to simplify microcontroller implementation, it is more that adequate for a typical sampling frequency $10^{5}/sec$.

Before we consider a more realistic model, let us demonstrate how the ratio of easy and hard to access capacitance can be found with the inverse relaxation technique for a two–$RC$ model. In this case the separation on ‘‘easy’’ and ‘‘hard’’ to access capacitance is trivial: $C_{1}$ is easy to access, $C_{2}$ is hard to access. In two–$RC$ model an internal charge redistribution between $C_{1}$ and $C_{2}$ is:

[TABLE]

Important, that the ratio (4) of ‘‘easy’’and ‘‘hard’’ capacitance does not depend on shorting time and on specific values of $R_{1}$ and $R_{2}$. In two capacitors model the values $U_{0}$, $U_{1}$, and $U_{2}$ can be obtained analytically, but we are going to present a numerical solution with the goal to study a more complex model later on. In Fig. 2a the dependence of $\eta$ on $\tau$ is presented for two different two–$RC$ capacitor models with the same $\eta$. One can clearly see that the ratio (4) does not differ from the exact value $C_{1}/C_{2}=0.4$ when shorting time $\tau$ changes in two orders of magnitude range. A deviation from the constant arises only when shorting time $\tau$ becomes comparable to the supercapacitor’s internal $RC$. For a small $\tau$ charge redistribution inside a supercapacitor leads to $\eta$ independence on $\tau$ in a wide interval. When one starts to increase the $\tau$ — an initial charge redistribution becomes more prolonged and the deviation of $\eta$ from a constant can be observed. The independence of $\eta$ on $\tau$ allows us to consider the ratio $\eta(\tau\to 0)$ as an immanent characteristic of the system, it is a characteristic that separates easy (accessible at low $\tau$) and hard (accessible only at high $\tau$) to access capacitance.

The $\eta$ is obtained only from the measurement of the potential (requires no charge measurement), and, while useful for structural characterization, lacks the information about absolute values. To obtain these the total charge $Q$ passed on shorting stage is required, this requires a microcontroller to calculate (1). Once the $Q$ is obtained, absolute values of $R_{1}$ and $C_{1}$ are:

[TABLE]

The (5) follows from $\tau\ll RC$; the (6) follows immediately from current balance: $U_{s}/R_{s}=\left[U_{1}-U_{s}\right]/R_{1}$. Now consider an increase of $\tau$ to the values above $RC$. Then $R_{1}$ (6) stays the same, it does not depend on $\tau$ at all. The $C_{1}$ (5) increases with $\tau$, in the limit of large $\tau$ it becomes equal to total capacitance $C_{\Sigma}=C_{1}+C_{2}+\dots$. Introduce

[TABLE]

that determines what the capacitance the system responds at time–scale $\tau$. A large $\tau$ corresponds to full discharge, a small $\tau$ corresponds only to a discharge of some porous branches (partial discharge). In the Fig. 2b the dependence of $C(\tau)$ is presented for two–$RC$ models. As expected $C(\tau=0)=C_{1}$ and $C(\tau=\infty)=C_{1}+C_{2}+\dots$. A similar to $C(\tau)$ concept can be conceived from impedance consideration. For two–$RC$ model the impedance $Z(\omega)$ is:

[TABLE]

Then, formally considering a ‘‘capacitance’’ as inverse proportional to imaginary part of $\omega Z(\omega)$ and time–scale as $\tau=1/\omega$ one can introduce:

[TABLE]

This definition treats impedance reactive component as caused by some ‘‘effective capacitance’’, this is reasonable while there is no inductances[15] in the system. The $C_{impedance}(\tau)$ behaves similarly to $C(\tau)$, e.g. it has the same asymptotic $C_{impedance}(\tau=0)=C_{1}$ and $C_{impedance}(\tau=\infty)=C_{1}+C_{2}+\dots$. It is presented in Fig. 2b to compare with $C(\tau)$. One can see a similar dependence. The $C_{impedance}(\tau)$ from (9) can be used for an arbitrary distributed $RC$ system, not only for the two–$RC$ impedance $Z(\omega)$ from (8), see Fig. 3b below for a three–$RC$ system example. The major advantage of the $C(\tau)$ over the $C_{impedance}(\tau)$ is that it can be measured purely in time–domain.

If the inverse relaxation potential $U_{2}$ is used in (7) instead of the $U_{1}$ obtain the total capacitance

[TABLE]

which is independent on the value of shorting time $\tau$, but it is often difficult to measure $U_{2}$ experimentally at low $\tau$ because of charge leaks. There is no such a difficulty to measure the $C(\tau)$ from (7); for this reason it is convenient to express the $\eta(\tau)$ from (4) via the $C(\tau)$:

[TABLE]

The expression (12) is mathematically identical to (4) but it is not sensitive to a presence of charge leaks in experimental measurements if the charge $Q$ in $C(\tau)$ from (7) is calculated using numerical integration (1).

The two–capacitors inverse relaxation model provides a single exponent behavior of $U(t)$ in the open circuit regime with $R_{2}C_{1}C_{2}/(C_{1}+C_{2})$ exponent time. In a system with a number of porous branches the behavior of $U(t)$ is more complex. In the Fig. 3 a three–$RC$ supercapacitor model is presented. The system has multi–exponents in $U(t)$ evolution, stable $\eta$ for $\tau\ll RC$ in two orders or magnitude range, and $C(\tau)$ asymptotic $C(\tau=0)=C_{1}$ and $C(\tau=\infty)=C_{1}+C_{2}+C_{3}+\dots$.

The $C(\tau)$ shows what the capacitance the system responds at time–scale $\tau$. It is measured from $U(t)$ sampling with subsequent integration (1). An important advantage of $C(\tau)$ is that it requires only $\tau$–long $U(t)$ measurement: during shorting period and the potential immediately after switching to open circuit regime, this makes (7) non–susceptible to charge leaks (no $U_{2}$ measurement is required). The $C(\tau)$ has a very clear practical meaning: If partial discharge takes time $\tau$, what is the ratio of charge/potential change for the $\tau$. This makes the inverse relaxation technique a well suitable tool for characterization of supercapacitors and other relaxation type systems with porous structure.

III The Experimental Measurement Of Supercapacitors

The circuit in Fig. 1 provides an implementation of the inverse relaxation measurement technique. It consists of two computer–controlled switches ‘‘Charge’’ and ‘‘Short’’ (they can be either MOSFET transistors or fast mechanical relays); the output potential $U(t)$ is measured by ADC port of a computer. If a controller has more than a single ADC then it is convenient also to record the potential $U^{*}(t)$ directly from $R_{s}$, this allows to increase measurement precision and we can avoid (13) calibration of the $R_{s}$, which is required when a single potential $U(t)$ is recorded, in which case the resistance of ‘‘Short’’ switch is combined with the $R_{s}$.

Measured potentials, before connecting to ADC ports, must be passed through an operational amplifier with MOSFET input, e.g. AD823, to decrease parasitic discharge and, especially for two–potential measurement setup, we can set operational amplifier to a constant amplification to bring small potential on shorting stage to the range of maximal ADC precision. This setup allows to overcome most of the difficulties[16] of charge measurement at low $U$. For example, the $R_{s}$ can be chosen such a small value that maximal value of $U^{*}(t)$ Fig. 1, will be about $10mV$. An operational amplifier AD823 can be used to bring it to a standard ADC max level $3.3V$ with the gain set to just $330$ (a very stable regime) and $U(t)$ numerical integration (1) then gives an accurate estimation of the total charge passed. Formally, the inverse relaxation potential $U_{2}$ gives the total charge as in Eq. (10). While one might think this allows to avoid a microcontroller–implemented numerical integration (1), our experimental measurements show that numerical integration gives a much more accurate total charge since it is not sensitive to charge leaks.

When working in a setup of single–potential recording the ‘‘on’’–resistance of the ‘‘Short’’ switch is combined with the $R_{s}$. We can consider some ‘‘effective’’ $R_{s}$ to enter (6) and the ‘‘Short’’ switch to be ideal. To obtain the value of an ‘‘effective’’ $R_{s}$ correctly one can either:

• •

Do a calibration to total charge:

[TABLE]

• •

Disconnect SC, put ‘‘Short’’ and ‘‘Charge’’ switches to ‘‘on’’ state, and connect the $U_{0}$ terminal to a fixed current source, typically about a few ampere. Measured potential $U$ determines the value of effective $R_{s}$, this is a variant of four–terminal sensing technique.

In the setup used by the authors an ‘‘effective’’ $R_{s}$ was $0.020\Omega$.

We tested four commercial supercapacitors: AVX-SCCS20B505PRBLE, Eaton-HV1020-2R7505-R, IC-505DCN2R7Q, and Nesscap-ESHSR-0005C0-002R7; all $5F$ with $2.7V$ max. In Fig. 4 internal $R_{i}C$ time is presented as a function of nominal capacitance for supercapacitors of the same series according to manufacturer datasheets. The $R_{i}C$ depends mostly on the technology used and increases slowly with the capacitance. Supercapacitors have a developed internal structure, which manifests itself in multi–exponent $U(t)$ dependence on inverse relaxation stage, see Fig. 5, which illustrates that inverse relaxation is typically not a single–exponent type of behavior. The relaxation at small time is faster than at large time. An ultimate situation of such a behavior is presented in Fig. 3a for a model system. The deviation $\log(U_{2}-U(t))$ from a linear law is related to a distributed internal $RC$. Fitting of a multi–exponent $U(t)$ relaxation is a common field of study[13]. A deviation from a single exponent (linear dependence in $\log$ axis) can be used as a source of information about supercapacitor’s internal structure. However, such an approach is more susceptible to measurement errors111 There is a much more advanced Radon–Nikodym technique[17] that can be applied to obtain relaxation rate distribution as matrix spectrum for relaxation type of data such as in Fig. 5. The distribution of the eigenvalues (using the Lebesgue quadrature[18] weight as eigenvalue weight) is an estimator of the distribution of relaxation rates observed in the measurement; Radon–Nikodym approach is much less susceptible to measurement errors compared to inverse Laplace transform type of analysis. See [19] for application example to Li–Ion degradation rate estimation.

and has interpretation difficulty.

Let us start discussing our experimental results with internal resistance $R_{1}$, Eq. (6), it is presented in Fig. 6. Measured $R_{1}$ does not depend on shorting time at all, the value corresponds to minimal possible internal resistance. Some manufacturers present the internal resistance for different $\tau$ (e.g. [20] presents internal resistance for $0.01$sec and $5$sec). In their setup (which assumes constant capacitance and variable resistance) larger $\tau$ corresponds to current propagation to supercapacitor deep pores, what involves a contribution from $R_{2}$. In our setup we have a constant $R(\tau)=R_{1}=const$ and instead consider an ‘‘effective’’ capacitance as a function of $\tau$, this is how the $C(\tau)$ is defined in Eq. (7). It has a very clear meaning: the ratio of charge/potential change for the $\tau$; this corresponds to a typical SC setup: discharge as much as you can in a given time $\tau$.

In Fig. 7a $\eta$ as a function of shorting time is presented. Only three potentials $U_{0}$, $U_{1}$, and $U_{2}$ have to be measured; no measurement of exponentially small values is required: the $U_{0}-U_{1}$ and $U_{2}-U_{1}$ are not small for actual $\tau$ values used in the experiment. The potentials $U_{0}$, $U_{1}$, and asymptotic $U_{2}$ are measured directly. The only difficulty that may arise if a supercapacitor has a parasitic charge leak, both internal and through $U(t)$ measurement circuit; this affects $\eta$ (4) and $C_{\Sigma}$ (10), but does not affect the $C(\tau)$ (7). A good heuristic for $U_{2}$ in case of a substantial self–discharge is the maximal value of $U(t)$ on inverse relaxation stage. A maximal $\tau$, for which a plateau can still be observed, is the value below characteristic scale of inverse relaxation. For a typical supercapacitor a characteristic scale of the inverse relaxation can be estimated as a multiple of internal $R_{i}C$ (available from Fig. 4 at $C=5F$ for the supercapacitors we use). This scale is different from $R_{i}C$ (typically several times greater), but about the same order of magnitude.

In Fig. 7b measured $C(\tau)$ dependence is presented. This is the most informative chart of the inverse relaxation technique. For all four supercapacitors ($5~{}{\rm F}$ nominal) low $\tau$ discharge is equivalent to a discharge of about $2F$ ideal capacitor, then the $C(\tau)$ increases with $\tau$. At high $\tau$ the discharge is equivalent to a $5F$ nominal capacitor. The $C(\tau)$ at high $\tau$ being equal to nominal capacitance demonstrates correct operation of ADC and numerical integration (1). A selection of insufficiently small $R_{s}$ may introduce an error to the value of $\tau$ at which $C(\tau)$ starts to increase. Typically the internal relaxation resistance is much greater than the internal resistance (i.e. $R_{2}\gg R_{1}$ for a simple two–$RC$ model in Fig. 1). For $R_{s}$ a good choice is lower (or even about) than the internal resistance $R_{i}$ (for a simple two–$RC$ model it is $R_{s}\lesssim R_{1}$). If $R_{s}$ is chosen greater than the internal relaxation resistance ($R_{2}$ in two–$RC$ model) then the plot $C(\tau)$ will be shifted to larger $\tau$, thus a ‘‘too high $R_{s}$’’ measurement setup will underestimate the qualtity of a supercapacitor.

III.1 Impedance characteristics of the supercapacitors

The biggest advantage of impedance spectroscopy is that it can capture a wide range (many orders) of frequencies222The biggest advantage of impedance spectroscopy is that impedance function is a ratio of two polynomials, thus it can be measured/interpolated/approximated with a high degree of accuracy for the measurements in a wide range (over $9$ orders, typically $10^{-3}\div 10^{6}$Hz) of frequency responses. However, in time–domain, where exponentially small values need to be measured, a much smaller range of time–scales are accessible (less than $2$ orders, often just a single order), hence in standard mathematical techniques, such as inverse Laplace transform, any type of noise/discretization/measurement error/window effect have a huge impact on exponentially small Laplace transform contributions[17]. The disadvantages of the technique are: measurement equipment complexity, impedance interpretation difficulty, and typically a low current linear regime, thus non–linear effects are problematic to study[21]. Most manufacturers provide equivalent ESR at fixed frequency $1000Hz$ in the datasheets, which is typically several times lower than the one at DC. In this section we apply impedance technique to obtain DC characteristics of supercapacitors. The goal is to compare impedance approach with inverse relaxation.

A common impedance analysis method is Nyquist plot. In Fig. 8 the Nyquist plot $Z^{\prime}Z^{\prime\prime}$ is presented along with ZView fitting by two–$RC$ model for AVX-SCCS20B505PRBLE and Eaton-HV1020-2R7505-R supercapacitors. The impedance measurements have been performed in a frequency range $10^{-3}\div 10^{5}$Hz. In this range Nyquist plot has a complex behavior caused by a complex internal structure of the device. In supercapacitor applications the frequencies of practical interest are the ones below $30\div 50$Hz. For simple models (such as in Fig. 1) it would be rather naïve trying to fit many orders of frequency range by a simple circuit of several $RC$ chains. For these reasons we limit the frequency range by $10^{-3}\div 30$Hz. A simple one–$RC$ model has a vertical asymptotic behavior at low frequencies. Two–$RC$ chains give some slope at low frequencies, observed in Fig. 8. In ZView, a model with two PCE elements (one with small exponent, a second one is close to $1$, almost the capacitance) allows to obtain a very good fit of the impedance curve in the entire $10^{-3}\div 10^{5}$Hz frequency range. The PCE element by itself can be modeled as a sequence of $RC$ elements[22], thus the value and exponent of PCE describe the supercapacitor’s internal structure. However, a limited range of practically interesting frequencies along with interpretation difficulties makes this approach not very appealing.

A very important feature of a supercapacitor, not observed in a regular capacitor, is that the impedance curve depends on DC potential applied. When the DC potential changes from $0V$ to $2.5V$ the impedance curve shifts to the right (the supercapacitor’s internal resistance increases) and the Nyquist plot changes substantially. The dependence of the capacitance on the applied potential is a known effect. It can be caused by the density of state changes[23], double layer structure changes[24, 25, 26, 27], or redox–active electrolyte processes[28, 29] of both reversible (Faraday’s capacitance) and irreversible (electrochemical decomposition) types.

The data, obtained from $10^{-3}\div 30$Hz impedance fitting differ quite substantially from the results of previous section. While the value of $R_{1}$ and total capacitance are similar, the $\eta$ differ substantially, also it typically decreases with DC potential increase: changes from $2.82$ to $1.06$ for AVX-SCCS20B505PRBLE and from $7.76$ to $1.95$ for Eaton-HV1020-2R7505-R, the same behavior was observed in the other supercapacitors we measured.

These measurements make us to conclude that an approach of ‘‘stretching’’ small signal impedance technique down to DC range is not a good idea. The inverse relaxation has important advantages of being close to ‘‘natural’’ fast discharge regime of a supercapacitor deployment and the measurement technique itself is much simpler than the impedance technique.

IV Discussion

In this work a novel technique for supercapacitors characterization is developed, modeled numerically, and experimentally tested on a number of commercial supercapacitors. The technique does not have any exponentially small value to measure, while, in the same time, all the measurements are performed not in the frequency domain, but in the time domain; the measurement is technically feasible in at least three orders of time–scales: $10^{-2}\div 10^{1}$sec. Besides of the simplicity of the technique (no fitting is required), the most important feature of the inverse relaxation approach is it’s simple automation. Microcontroller–operated two switches and a single ADC can obtain the dependence of an ‘‘effective’’ capacitance on time–scale $C(\tau)$, Fig. 7b. The approach can be considered as an alternative to commonly used[20] consideration of a constant capacitance and time–scale dependent internal resistance. Among the advantages of our technique is that it does not require a source of fixed current, what simplify the setup and allows a very high discharge current regime. As the limitations of the techniques we would note: only about three order of available $\tau$ range (it is difficult to measure at shorting time $\tau\lesssim 10ms$) compared with about nine orders $10^{-3}\div 10^{6}$Hz of available frequency range in impedance spectroscopy, the necessity to use a microcontroller for numerical integration (1), and a risk to destroy a supercapacitor on shorting stage by a high discharge current.

Modeling supercapacitors internal structure in electronic circuit software is a common field of study[30, 31, 32, 33, 13]. In [34] a voltage rebound effect, shorting and then switching to open circuit was also modeled. However, only in our early work[12] the ratio $\eta(\tau\to 0)$ of ‘‘easy’’ and ‘‘hard’’ to access capacitance was introduced. Similar pulse–response characteristics of Li–ion batteries have been studied in [35] with an emphasis on time–scale.

The idea to use pulsed load for a battery (primary or secondary) or fuel cell is now widespread. For example GSM standard specifies $0.575ms$ transmission burst within a $4.6ms$ period (1/8 duty factor), thus DC–DC chips like [36], that are especially designed for pulsed load, have been used in all modern devices. S.L.Kulakov pioneered an application of pulsed load to metal–air power sources and then, about a decade later, brought to our attention the pulsed load technique in [37]. Developed in this paper pulsed technique for supercapacitors characterization is dedicated to his memory.

Appendix A Software Modeling

The systems have been modeled in Ngspice circuit simulator. The circuit have been created in gschem program of gEDA project. To run the simulator download[38] the file RCcircuit.zip and decompress it. To test the simulator execute

ngspice Farades_y_with_variables.sch.autogen.net.cir

Because original gschem+ngspice do not have a convenient parameterisation, a perl script run_auto.pl have been developed. To run the simulator with $\tau=0.2$sec execute:

perl -w run_auto.pl Farades_y_with_variables.sch 0.2

The script takes the Farades_y_with_variables.sch which corresponds to three–$RC$ system in Fig. 3 and substitute shorting time $\tau=0.2$. One can modify it accordingly, and run ngspice. The result is saved to n0_output.txt.