A comparative study of supercapacitor capacitance characterization methods

Abstract

This paper compares two supercapacitor capacitance measurement methods: the method 1A of the IEC standard 62391-1 and a method utilizing the total charge stored in the supercapacitor. These two methods are applied to three supercapacitor samples with different rated capacitances from different manufacturers at various terminal voltages. Experimental results show that the capacitance determined using the IEC method decreases when the discharge current increases. Besides, the capacitance measured using the IEC method is lower than that estimated using the total charge method. Moreover, the ratio of the capacitance estimated using the total charge method to the one measured using the IEC method is greater when the supercapacitor terminal voltage is lower. These three observations on the capacitances measured using the two methods are explained by analyzing a five-branch RC ladder circuit model capturing multiple aspects of the supercapacitor physics: porous electrode structure, voltage dependence of capacitance, charge redistribution, and self-discharge.

1. Introduction

Energy storage is becoming an increasingly critical asset in many systems. Among various energy storage technologies, supercapacitors are advantageous in several aspects such as high power density and long cycle life. In fact, supercapacitor-based energy storage systems have been employed by a variety of applications including power grids [1–3], electric vehicles [4–7], and wireless sensor networks [8–10].

To exploit the supercapacitor technology, a comprehensive and in-depth understanding of its characteristics at the device level is crucial. Therefore, modeling and characterization of supercapacitors have been of great interest. Various equivalent circuit models [11,12] capturing different aspects of the supercapacitor physics such as voltage dependence of capacitance [13,14], charge redistribution [15,16], and self-discharge [17,18] have been proposed to reduce the complexity of the generic ladder circuit model. On the other hand, accurate estimation of the supercapacitor state of charge (SOC) is important at both the device and system levels. Although the supercapacitor terminal voltage is a natural indicator of its state, accurate SOC estimation is still challenging because the supercapacitor capacitance and equivalent series resistance (ESR) are affected by multiple factors such as its terminal voltage, operating temperature, and aging condition in a complex manner. Numerous frameworks [19–22] have been developed to identify supercapacitor parameters and estimate supercapacitor states.

Among various aspects of the supercapacitor physics, charge redistribution is of particular interest, which is a relaxation process originated from the porous structure of the electrodes. The physical mechanisms leading to this process have been revealed: the electrode pore sizes are nonuniform. Therefore, the ions in the electrolyte need extra time to penetrate the middle-size mesopores and small-size micropores compared to the large-size macropores [23–26]. The effects of charge redistribution on the supercapacitor terminal voltage behavior [27,28], supercapacitor energy delivery capability during a constant power discharge process [29,30], supercapacitor charge capacity associated with a constant current discharge process [31,32], and the applicability of Peukert’s law to supercapacitors with constant current [33–38] or constant power [39] loads have been extensively studied.

As one of the most important specifications of supercapacitors, capacitance is a key parameter in the design, control, and management of supercapacitor-based energy storage systems for both high power (e.g., power grids and electric vehicles) and low power (e.g., wireless sensor networks and biomedical devices) applications. For example, wireless sensor networks need the information of the available charge or energy stored in the energy storage system to schedule tasks such as reading the sensor measurements, processing the collected data, and transmitting the data packets. In fact, energy efficiency is a major concern in wireless sensor networks and a variety of power management strategies have been developed based on the state of the energy storage system [8–10], which are ultimately based on the information of the supercapacitor capacitance among others. To obtain an accurate estimate of the supercapacitor capacitance, effective approaches are needed. A preliminary work [40] revisits the supercapacitor capacitance measurement method 1A of the IEC standard 62391-1 [41] and reveals that the characterized capacitance decreases when the discharge current used in the experiment increases.

This paper develops [40] in three aspects and compares two supercapacitor capacitance measurement methods: the IEC 62391-1 method 1A and a method utilizing the total charge stored in the supercapacitor. First, the original five-branch RC ladder circuit model for supercapacitors analyzed in [40] is modified to take into account the voltage dependence of the supercapacitor capacitance. Second, the total charge stored in the supercapacitor and the corresponding capacitance determined using the total charge method are simulated while [40] only examines the capacitance estimated using the IEC method. Third, the physical mechanisms accounting for the differences between the capacitances estimated using the two methods are elaborated. It should be noted that this paper is not intended to evaluate all supercapacitor capacitance characterization methods. For instance, this paper does not examine the galvanostatic charge-discharge method that charges and discharges the supercapacitor using constant currents of the same magnitude between certain maximum and minimum voltages [42,43]. Rather, this paper focuses on the specific IEC method for the following consideration. The IEC standard categorizes supercapacitors into different classes based on their capacitances and internal resistances. For each category, the standard specifies multiple methods that may be used to characterize the supercapacitor capacitance and internal resistance. Compared to the galvanostatic charge-discharge method, the test procedures are well defined in the IEC standard. Therefore, the IEC methods can be readily implemented and evaluated. Moreover, while this paper only examines the capacitance measurement method 1A specified in the standard, more work is being conducted to evaluate other characterization methods for the supercapacitor capacitance and internal resistance, which may help reveal some of the potential issues with the standard.

The remainder of this paper is organized as follows. Section 2 examines the supercapacitor capacitance measurement method 1A of IEC 62391-1. Section 3 presents an alternative method utilizing the total charge stored in the supercapacitor. Section 4 compares the capacitances determined using these two methods. Section 5 explains the differences between the capacitances measured using these two methods. Section 6 concludes this paper.

2. The method 1A of IEC standard 62391-1

To examine the supercapacitor capacitance measurement method 1A of IEC 62391-1, the three supercapacitor samples listed in Table 1 are tested using an automated Maccor Model 4304 tester at room temperature. The samples are made by different manufacturers. Their rated capacitances and voltages are denoted as C_N and U_R, respectively.

Table 1.

Supercapacitor samples.

Sample	1	2	3
Manufacturer	Eaton	AVX	Maxwell
Model	HV10302R7106R	SCCV60B107MRB	BCAP0350
CN (F)	10	100	350
UR (V)	2.7	2.7	2.7

The measurement procedure [41] for the IEC 62391-1 method 1A is summarized as follows while the supercapacitor terminal voltage profile during the experiment is illustrated in Fig. 1: (1) charge the supercapacitor using a constant current; (2) when the supercapacitor terminal voltage reaches the specified value, the charge process continues with a constant voltage; (3) after that, the supercapacitor is discharged by a constant current. In Fig. 1, U_R is the rated voltage, T_CV is the duration of the constant voltage charge process, which is set to be 30 min, and ∆U₃ is the voltage drop used in the internal resistance calculation, which is not considered in this paper. The capacitance measured using this method is calculated as

$$C=\frac{I_{cc}\times \left(t_2-t_1\right)}{U_1-U_2},$$ (1)

where I_cc is the discharge current, U₁ is the calculation start voltage, t₁ is the time at which the supercapacitor terminal voltage reaches U₁ from the start of the discharge process, U₂ is the calculation end voltage, and t₂ is the discharge time associated with U₂.

Fig. 1.

Voltage-time characteristics between supercapacitor terminals in capacitance and internal resistance measurement [41].

This measurement procedure is applied to the three supercapacitor samples. For example, Fig. 2 shows an experiment for sample 2, which is composed of three phases and the experiment conditions are set according to the standard. This supercapacitor is first charged from 0 V to the rated voltage of 2.7 V using a constant current source of 4.7 A. The constant charge current of 4.7 A uses the value suggested by the standard. Specifically, the charge current is set as U_R/38R_N to ensure that the supercapacitor is charged with an efficiency of 95%. For this supercapacitor, the rated voltage U_R is 2.7 V and the nominal internal resistance R_N is 15 mΩ [44]. Therefore, the charge current is approximately 4.7 A. Following the first phase, the supercapacitor is charged by a constant voltage source of 2.7 V for 30 min. Finally, a constant discharge current of 1 A is applied and the supercapacitor is discharged to 0.01 V. The capacitance determined from this experiment is denoted as C₃ because the discharge current of 1 A uses the suggested value for class 3 (power) supercapacitors: 4C_NU_R (units: mA), where C_N is the rated capacitance of 100 F and U_R is the rated voltage of 2.7 V. Note that the used value of 1 A is close to the suggested value of 1.08 A.

Fig. 2.

Experimental results using IEC method: measured voltage for supercapacitor sample 2 when initial voltage of discharge process is 2.7 V and discharge current is 1 A.

To calculate the capacitance using (1), the following procedure is adopted. According to the standard, the start and end voltages used to calculate the capacitance are set as U₁ = 0.8U_R = 2.16 V and U₂ = 0.4U_R = 1.08 V when U_R = 2.7 V, respectively. Considering the time and voltage resolutions of the supercapacitor tester, it is possible that none of the recorded voltages matches these two values exactly, which is actually the case. To examine if the voltage resolution significantly impacts the capacitance calculation, for both U₁ and U₂, the data point closest to the desired value together with another two closest neighboring data points are used. Specifically, the measurement with U_1C = 2.1604 V at t_1C = 53.74 s is the data point closest to the desired voltage of U₁ = 2.16 V and the subscript “C” means “closest”. The two closest neighboring data points are U_1H = 2.1613 V at t_1H = 53.64 s and U_1L = 2.1593 V at t_1L = 53.84 s, where the subscripts “H” and “L” represent the “higher” and “lower” voltages compared to the “closest” voltage, respectively. For U₂, the three data points are U_2C = 1.0803 V at t_2C = 168.64 s, U_2H = 1.0814 V at t_2H = 168.54 s, and U_2L = 1.0796 V at t_2L = 168.74 s. In total, nine data point combinations of U₁ and U₂ are generated and each gives an estimate of C₃, as shown in Table 2. Statistically, the variations of these estimates are minimal: the average is 106.40 F and the standard deviation is 0.013 F, which give a coefficient of variation of 0.0001. Therefore, it can be assumed that the capacitance calculated using the data points closest to the desired values is a good estimate of the actual value. The experiment shown in Fig. 2 is repeated three times and the average of the three C₃ estimates is 105.34 F, which is used as the final estimate of C₃, as listed in Table 3.

Table 2.

Measured C₃ for supercapacitor sample 2 using IEC method.

Data point combination no.	U1	U2	C3 (F)
1	U1C	U2C	106.38
2	U1C	U2H	106.39
3	U1C	U2L	106.40
4	U1H	U2C	106.38
5	U1H	U2H	106.40
6	U1H	U2L	106.41
7	U1L	U2C	106.39
8	U1L	U2H	106.41
9	U1L	U2L	106.42

Table 3.

Measured capacitances for supercapacitor sample 2.

U (V)	C2 (F)	C3 (F)	C4 (F)	CT (F)	r2	r3	r4
2.7	108.13	105.34	102.54	111.30	1.029	1.057	1.085
2	102.55	98.80	94.33	106.61	1.040	1.079	1.130
1.35	97.35	92.79	86.80	103.60	1.064	1.116	1.194
0.7	92.24	86.85	76.73	99.52	1.079	1.146	1.297

Other than C₃, Table 3 also lists another two capacitances measured using the IEC method: C₂ and C₄. The capacitance C₂ is measured and calculated using the discharge currents for classes 1 (memory backup, 1C_N = 0.1 A) and 2 (energy storage, 0.4C_NU_R = 0.108 A) supercapacitors, which are approximately equal and therefore only one experiment is performed using 0.1 A. The capacitance C₄ is determined using the discharge current of 10 A, which is close to the suggested current for class 4 supercapacitors: instantaneous power, 40C_NU_R = 10.8 A. Considering the voltage dependence of the supercapacitor capacitance, experiments are also conducted to investigate the supercapacitor capacitances at different voltages. Therefore, the supercapacitor voltage is swept and three more voltages are examined in addition to the rated voltage of 2.7 V: 2, 1.35, and 0.7 V. The capacitances determined using the IEC method are also listed in Table 3.

3. A method using supercapacitor total charge

To characterize the supercapacitor capacitance, this section adopts an alternative method utilizing the total charge stored in the supercapacitor, which has been used to study the voltage dependence of the supercapacitor capacitance [14]. As elaborated in [31,32], the total charge stored in the supercapacitor cannot be fully extracted during one discharge process because of the porous electrode structure and the charge redistribution behavior. The total charge capacity can be estimated by applying multiple discharging-redistribution cycles to the supercapacitor. For example, Fig. 3(a) shows the measured voltage for supercapacitor sample 2 during a 10 A constant current discharge experiment. The supercapacitor is first conditioned by ten charging-redistribution-discharging cycles to minimize the effect of residual charge. It is then charged by a constant voltage source of 2.7 V for 3 h (this charge time is determined based on a preliminary study reported in [31]), which is designed to fully charge the supercapacitor. After that, a 10 A constant discharge current is applied and the supercapacitor is discharged to 0.01 V, as highlighted in Fig. 3(a). It should be noted that the supercapacitor is not fully discharged to 0 V to avoid overdischarge that might lead to safety incidents. Fig. 3(b) shows the supercapacitor voltage during this discharge process. Taking 2.7 V as the initial voltage and 0.01 V as the cutoff voltage, the charge delivered during this process is 252.4 C. Then the discharge current is disconnected and the supercapacitor experiences charge redistribution, which results in an increase in the terminal voltage. Once the terminal voltage increase rate is less than 0.01 V per 5 min, the charge redistribution process is considered complete and the discharge current is applied again. This discharging-redistribution cycle is repeated ten times to extract the charge stored in the supercapacitor to the maximum extent possible. The remaining charge still stored in the supercapacitor is minimal compared to that already extracted during the ten cycles [31]. Therefore, the total charge is estimated by adding up the charge delivered during the ten cycles, which is 293.7 C in this experiment.

Fig. 3.

Experimental results using total charge method: measured voltage for supercapacitor sample 2 when initial voltage of discharge process is 2.7 V and discharge current is 10 A. (a) Overview. (b) Constant current discharge process.

To estimate the total charge, the experiments and results reported in [31,32] are reused. Specifically, seven constant current discharge experiments using the following discharge currents are conducted: 10, 5, 1, 0.5, 0.1, 0.05, and 0.01 A. This set of experiments is repeated once. The average of the seven total charge measurements is calculated for both sets of experiments and the average of them is used as the estimate of the total charge, which is 299.41 C. The capacitance is then calculated as

$$C_T=\frac{Q_T}{U_B-U_E},$$ (2)

where Q_T is the total charge, U_B is the beginning voltage of the discharge process (i.e., 2.7 V), and U_E is the ending voltage of the discharge process (i.e., 0.01 V). This capacitance is denoted as C_T in Table 3, which is 111.30 F at 2.7 V. This method is applied to the other three voltages (i.e., 2, 1.35, and 0.7 V) and the results are also included in Table 3.

4. Comparisons of two methods

The capacitance characterization results for supercapacitor samples 1 and 3 are listed in Tables 4 and 5, respectively. The following observations can be made based on the results presented in Tables 3–5.

Table 4.

Measured capacitances for supercapacitor sample 1.

U (V)	C2 (F)	C3 (F)	C4 (F)	CT (F)	r2	r3	r4
2.7	11.40	10.74	10.02	11.85	1.039	1.103	1.182
2	10.36	9.65	8.88	10.94	1.057	1.135	1.233
1.35	9.57	8.80	8.14	10.34	1.081	1.176	1.271
0.7	8.68	7.91	7.33	9.97	1.149	1.261	1.360

Table 5.

Measured capacitances for supercapacitor sample 3.

U (V)	C2 (F)	C3 (F)	C4 (F)	CT (F)	r2	r3	r4
2.7	375.85	372.82	370.52	376.23	1.001	1.009	1.015
2	354.50	349.99	346.89	356.37	1.005	1.018	1.027
1.35	331.61	325.11	320.84	338.63	1.021	1.042	1.055
0.7	307.63	299.25	294.16	325.52	1.058	1.088	1.107

First, when the IEC method is used, the measured capacitance becomes smaller (C₂ > C₃ > C₄) as the discharge current increases for all the samples at all the voltages: 0.4C_NU_R for C₂, 4C_NU_R for C₃, and 40C_NU_R for C4. For sample 2 at 2.7 V, the capacitance decreases from 108.13 (C₂) to 105.34 (C₃) and finally to 102.54 F (C₄) when the discharge current increases from 0.1 to 1 and finally to 10 A.

Second, the capacitance measured using the total charge method is greater than those determined using the IEC method: C_T > C₂ > C₃ > C₄. For sample 2 at 2.7 V, C_T is 111.30 F, which is greater than C₂ = 108.13 F (the largest value among C₂ − C₄). To quantify the difference, the ratio of C_T to C_i (i = 2 − 4) can be defined as

$$r_i=\frac{C_T}{C_i}.$$ (3)

For instance, r₂ is 1.029 for sample 2 at 2.7 V. The ratios for all the samples at all the voltages are calculated and listed in Tables 3–5. It can be observed that the ratios are greater than unity and r₂ < r₃ < r₄ because C_T > C₂ > C₃ > C₄.

Finally, for the same supercapacitor sample, the capacitance ratio associated with a particular discharge current increases when the voltage decreases. For sample 2, r₂ increases from 1.029 at 2.7 V all the way to 1.079 at 0.7 V. Similar patterns apply to r₃ and r₄. Although the magnitudes of the ratios vary among the three samples, similar patterns can be observed.

In Tables 3–5, the capacitance estimated using the total charge method is based on the average of the total charge measurements associated with various discharge currents. This is because the average is a better estimate of the total charge compared to the individual measurements corresponding to different discharge currents, as elaborated in [31,32]. Specifically, when the discharge current is relatively high, the supercapacitor terminal voltage change due to ESR is significant and the total charge estimation error tends to be noticeable. On the other hand, when the discharge current is sufficiently low, the discharge process is extended and the supercapacitor self-discharge process results in a significant energy loss, which also introduces a considerable total charge estimation error.

To verify if the observations above apply to the capacitance calculated using the individual total charge measurement, Table 6 lists the results for sample 2. The capacitance C_Ti is calculated using the total charge Q_Ti associated with a particular discharge current:

$$C_{Ti}=\frac{Q_{Ti}}{U_B-U_E},$$ (4)

where the index i = 2 – 4 denotes the discharge current used in the IEC method, as elaborated in Section 2. For example, when the voltage is 2.7 V, C_T4 is 108.62 F, which is based on the total charge corresponding to the 10 A discharge current: Q_T4 = 292.20 C. For comparison, C_T in Table 3 is 111.30 F, which is based on the average total charge: Q_T = 299.41 C. Similarly, C_T2 and C_T3 are based on the total charge measurement when the discharge current is 0.1 and 1 A, respectively.

Table 6.

Measured capacitances for supercapacitor sample 2 using individual total charge results.

U (V)	CT2 (F)	CT3 (F)	CT4 (F)	rT2	rT3	rT4
2.7	111.71	112.05	108.62	1.033	1.064	1.059*
2	107.08	107.41	103.17	1.044	1.087	1.094
1.35	104.71	104.87	98.36	1.076	1.130	1.133
0.7	101.79	101.86	89.86	1.104	1.173	1.171*

Utilizing the capacitance C_i estimated using the IEC method in Table 3 and the capacitance C_Ti calculated using the individual total charge in Table 6, the ratio of them is determined as

$$r_{Ti}=\frac{C_{Ti}}{C_i}.$$ (5)

The results are listed in Table 6. Clearly, the third observation strictly applies: the capacitance ratio for a specific discharge current increases when the voltage decreases. For instance, when the discharge current is 10 A, r_T4 increases from 1.059 to 1.171 when the voltage decreases from 2.7 to 0.7 V. The second observation applies in general. For all the four voltages, the capacitance ratios are greater than unity. Moreover, for 2 and 1.35 V, the capacitance ratio strictly increases when the discharge current increases: r_T2 < r_T3 < r_T4 (e.g., for 2 V, the capacitance ratio increases from 1.044 to 1.087 and finally to 1.094). On the other hand, for 2.7 and 0.7 V, while the capacitance ratio still strictly increases from r_T2 to r_T3, it shows minor decrease from r_T3 to r_T4. For instance, when the voltage is 2.7 V, the capacitance ratio first increases from 1.033 to 1.064 and then decreases to 1.059, as indicated by the * sign. A similar pattern applies when the voltage is 0.7 V and an * is therefore associated with 1.171. This is because when the discharge current is 10 A, the total charge measurement Q_T4 tends to underestimate the actual value and the corresponding capacitance C_T4 is therefore also an underestimate, as further illustrated using the simulation results in Section 5.2. In fact, Table 6 clearly shows that while C_T2 and C_T3 are approximately equal, C_T4 is noticeably smaller. For samples 1 and 3, the results are similar. In summary, although the capacitances determined using the individual total charge measurements show a certain level of variation, the observations on the capacitance estimated using the average total charge apply as well.

5. Effects of supercapacitor physics on capacitance characterization methods

5.1. Supercapacitor model

To explain the three observations on the capacitances characterized using the two methods, the five-branch RC ladder circuit model shown in Fig. 4(a) is analyzed, which is a modified version of the original model shown in Fig. 4(b). As elaborated in [31,32], the original model is conceived to illustrate the impact of three aspects of the supercapacitor physics on its charge capacity: porous electrode structure, charge redistribution, and self-discharge. The original model includes five RC branches (R₁ through C₅) to capture the distributed nature of the supercapacitor capacitance and resistance, which is a result of the porous electrode structure and also the origin of the charge redistribution process. The parallel leakage resistor R₆ is used to represent the self-discharge process. The supercapacitor terminal voltage is denoted as V_T, which is a measurable parameter. In fact, V_T equals the voltage across the first RC branch composed of R₁ and C₁. When a source or load is applied to the supercapacitor terminals, the capacitor of each RC branch is accessed through a series connection of all the resistors from the supercapacitor terminals to the branch in question. The time constant of each RC branch can be written as

$${\tau }_i=C_i\sum _{k=1}^i{R}_k,$$ (6)

and the porous electrode theory gives that

$${\tau }_1<{\tau }_2<\cdots <{\tau }_n.$$ (7)

Fig. 4.

Two five-branch RC ladder circuit models for 100 F supercapacitors. (a) Modified model with voltage-dependent C₁. (b) Original model with constant C₁.

As shown in Fig. 4(b), all the component values are constant in the original model. To investigate the relationship between the delivered charge and the discharge current in [31,32], the component values of the five RC branches are tuned to generate time constants that can be used to characterize the supercapacitor behavior on various time scales: τ₁ = 1.05, τ₂ = 10, τ₃ = 100, τ₄ = 1000, and τ₅ = 10000 s. The total capacitance of the five branch capacitors is 100 F. The C₁ capacitance is 70% of the total capacitance because the first branch is the main branch. The capacitances are 16, 8, 4, and 2 F for the remaining four branches with a scale factor of 0.5 based on the fact that a slower branch makes a smaller contribution to the total capacitance. As for the resistors, the first branch resistance R₁ uses the typical ESR value specified in the supercapacitor sample 2 datasheet [44]. The other four branch resistances are calculated based on the conrresponding time constant and capacitance using (6). The value of the parallel leakage resistor R₆ is estimated based on the rated voltage and the leakage current specified in the sample 2 datasheet [44].

It should be noted that the model shown in Fig. 4(b) is constructed to characterize the supercapacitor behavior over a wide time span (on the order of 1 to 10,000 s) because a wide range of discharge current (10–0.0025 A) is swept in [31,32] for two considerations. First, from the theoretical analysis perspective, the effects of supercapacitor physics such as charge redistribution and self-discharge on the delivered charge pattern may vary depending on the discharge current magnitude. As a matter of fact, the effects of self-discharge are negligible when the discharge current is relatively high. For sufficiently low discharge currents, self-discharge results in a significant energy loss, as elaborated in [32]. Second, from the practical application perspective, understanding the supercapacitor behavior over a wide range of discharge current is critical to improve the system energy efficiency. In fact, supercapacitors have been used in both high power applications (e.g., power grids and electric vehicles) and low power applications (e.g., wireless sensor networks and biomedical devices). Even for the same application, multiple power levels may exist. For instance, the current consumptions of transceivers, microcontroller units, and sensors vary significantly in wireless sensor networks. Therefore, the model shown in Fig. 4(b) is conceived to include multiple RC branches with extended time constants. While the models in Fig. 4 include five RC branches to study the supercapacitor behavior in [31,32] and this paper, the number of RC branches can be increased or decreased to examine the supercapacitor characteristics for other applications. Clearly, as more RC branches are incorporated, the model will be more complicated, which will improve the model accuracy and increase the computational complexity at the same time.

The modified model shown in Fig. 4(a) incorporates a variable C₁ to take into account the voltage dependence of the supercapacitor capacitance [13,14], which is defined as follows in LTspice:

$$Q={43}^{\ast }x+{10}^{\ast }{x}^{\ast }x,$$ (8)

where Q is the charge stored in this capacitor and x is the voltage across it: x = V₁. The equivalent capacitance of this voltage-dependent capacitor [13] is then determined as

$$C_{1eq}=43+10\times V_1,$$ (9)

which gives C_1eq = 70 F at V₁ = 2.7 V. Like other components in the model, the values in (8) are assumed with certain arbitrariness to qualitatively characterize the voltage dependence of the supercapacitor capacitance. Specifically, the coefficient of 10 is first selected to represent the voltage-dependent component of the capacitance C_1eq. At V₁ = 2.7 V, the voltage-dependent component is 10 × 2.7 = 27 F, as indicated in (9). To hold the same amount of charge as the constant C1 shown in Fig. 4(b), the constant component of C_1eq should be 70 − 27 = 43 F. The other component values remain unchanged in the modified model.

5.2. Simulation results

The modified model shown in Fig. 4(a) is implemented and simulated using a widely adopted SPICE tool: LTspice. The capacitance is estimated using the two methods. For the IEC method, Fig. 5 shows the supercapacitor terminal voltage during a simulation: the supercapacitor initial voltage is 2.7 V and the discharge current is 10 A. In this simulation, the initial voltages of the five branch capacitors are set to be 2.7 V and a discharge current of 10 A is applied to bring the supercapacitor terminal voltage down to 0.01 V. As elaborated in Section 2, two data points during the discharge process are selected according to the IEC standard: 2.16 V at 3.7360 s and 1.08 V at 13.1322 s. The capacitance calculated using (1) is then 87.00 F.

Fig. 5.

Simulation results using IEC method: terminal voltage when initial voltage of discharge process is 2.7 V and discharge current is 10 A.

When the initial voltage of the discharge process is 2.7 V, the capacitances estimated using the IEC method are plotted in Fig. 6. In addition to the three currents (i.e., 10, 1, and 0.1 A) examined in Section 2, more currents are swept. In particular, the currents between 0.05 and 0.5 A are more densely swept. It can be observed that the capacitance decreases from 102.52 to 87.00 F when the discharge current increases from 0.05 to 10 A, which is consistent with the experimental results listed in Tables 3–5. It should be noted that the capacitances associated with the currents between 0.05 and 0.2 A exceed the actual value of 100 F (e.g., 102.52 F resulted from 0.05 A and 100.05 F associated with 0.2 A), which suggests that the IEC method might need to be reconsidered for certain applications.

Fig. 6.

Simulation results: capacitances estimated using two methods when initial voltage of discharge process is 2.7 V.

One possible explanation for the overestimated capacitance is the straight line approximation used in the IEC method: the supercapacitor capacitance is assumed to be constant and the voltage dependence of the capacitance is not taken into account. With this constant capacitance assumption, the capacitance calculated using (1) may lead to erroneous results. For illustration, the impact of the voltage dependence of the capacitance is investigated. Referring to Fig. 4(a) and (9), the general form of the voltage-dependent capacitance can be written as

$$C_{1gen}=a+b\times V_1,$$ (10)

where the coefficients a and b must satisfy that for V₁ = 2.7 V, C_1gen = 70 F. As illustrated in Section 5.1, after b is selected, a can be calculated accordingly. As another example, for b = 4, a is determined as 70 − 4 × 2.7 = 59.2. The coefficient b can be used to represent the voltage dependence of the capacitance: a larger b means that the dependence is stronger. To study the effects of this dependence on the IEC method, b is swept from 12 to 0 with a step of 2. Note that Fig. 4(a) corresponds to the case with b = 10. Moreover, Fig. 4(b) corresponds to the case with b = 0: the voltage dependence of the capacitance is removed and the capacitance is constant. For each case of b, the IEC method is applied and the capacitances associated with different discharge currents are plotted in Fig. 7. Note that the curve for b = 10 reuses the IEC results in Fig. 6. It can be observed that when b is relatively large (i.e., 6–12), which means that the voltage dependence of the capacitance is relatively strong, the capacitance associated with 0.05 A overestimates the actual value of 100 F. For instance, the capacitance corresponding to 0.05 A is 103.48 F for b = 12. This observation also applies to 0.1 A when b varies from 8 to 12. On the other hand, when b is relatively small (i.e., 0–4), even the capacitance associated with 0.05 A does not exceed the actual value of 100 F. In particular, for b = 0 representing the constant capacitance, all the capacitance estimates including 97.34 F associated with 0.05 A are less than 100 F. To better understand the origin of the capacitance overestimation results shown in Fig. 7, a further study needs to be conducted to investigate the impact of the straight line approximation used in the IEC method on the capacitance calculation formula.

Fig. 9.

Simulation results using IEC method: capacitances when initial voltage of discharge process varies.

Fig. 7.

Simulation results: capacitances estimated using IEC method when initial voltage of discharge process is 2.7 V. Parameter b represents voltage dependence of capacitance.

To estimate the capacitance using the total charge method, Fig. 8 plots the simulated supercapacitor terminal voltage when the initial voltage of the discharge process is 2.7 V. Note that the following simulation results use the model shown in Fig. 4(a) with b = 10. As elaborated in Section 3, the total charge stored in the supercapacitor can be estimated by summing the charge delivered during multiple discharge processes. In fact, ten discharging-redistribution cycles are simulated in Fig. 8. In Fig. 8(a), the discharge current is 10 A. The supercapacitor is first discharged to 0.01 V in 20.4247 s and then experiences charge redistribution during the following 1800 s. As in the experiment shown in Fig. 3, the discharge termination voltage is set as 0.01 V to ensure that the simulation and experimental setups are similar. After that, the discharge current is applied again and the supercapacitor is discharged to 0.01 V again in 3.1947 s. The charge redistribution time is 600 s in this cycle and in the remaining eight cycles. The total charge delivered during the ten cycles is 254.24 C and the capacitance calculated using (4) is 94.51 F. Similarly, Fig. 8(b) shows the simulation results when the discharge current is 0.1 A. Note that the supercapacitor is also discharged to 0.01 V to ensure that the simulation setups are the same when the discharge current varies. For this simulation, the total charge is 265.40 C and the capacitance is 98.66 F.

Fig. 8.

Simulation results using total charge method: terminal voltage when initial voltage of discharge process is 2.7 V. (a) Discharge current is 10 A. (b) Discharge current is 0.1 A.

The capacitances calculated using the total charge method are also plotted in Fig. 6. Clearly, for the discharge current of 10–0.5 A, the capacitances estimated using the total charge method are greater than those determined using the IEC method, which is consistent with the experimental results. When the discharge current is 0.4 A, the two capacitances are close: 98.53 F from the IEC method and 98.65 F from the total charge method. For 0.3 A, the IEC method results in a larger capacitance: 99.24 versus 98.66 F. For the discharge currents less than or equal to 0.2 A, the IEC method results in capacitances greater than 100 F (e.g., 100.05 F corresponding to 0.2 A), which can be excluded as erroneous data points.

In Fig. 6, the capacitances characterized using the total charge method are approximately equal for the discharge current range of 1–0.05 A while the values are smaller for 10 and 5 A, which is consistent with the experimental results listed in Table 6. This is because a relatively high discharge current results in a significantly high residual voltage due to the presence of ESR, which introduces a relatively large error when it comes to estimating the total charge [31]. In fact, when the discharge current is 10 A, the voltage change due to ESR is 0.15 V, which is a major contributor to the supercapacitor terminal voltage, as shown in Fig. 8(a). Since 0.15 V is a significantly high voltage (i.e., 5.6% of the initial voltage of 2.7 V), the supercapacitor is not fully discharged after the ten discharging-redistribution cycles and the total charge estimated using the 10 A current underestimates the real value. On the other hand, when the discharge current is 0.1 A, the voltage change due to ESR is 0.0015 V and it can be assumed that the supercapacitor is fully discharged at the end of the ten cycles. Therefore, the total charge can be more accurately estimated, as shown in Fig. 8(b). Following the analysis in Section 4, the ratio of the capacitances estimated using the inidividual total charge and the IEC method increases from 1.006 to 1.086 when the discharge current increases from 0.5 to 10 A, which is also consistent with the experimental results listed in Table 6.

In addition to the rated voltage of 2.7 V, another three voltages are examined: 2, 1.35, and 0.7 V. Fig. 9 shows the capacitances estimated using the IEC method. For all the four voltages, the capacitance increases when the discharge current decreases. For a particular discharge current, the capacitance increases when the initial voltage of the discharge process increases, which reflects the voltage dependence of the supercapacitor capacitance defined in (9). For instance, the capacitance associated with the 10 A discharge current increases from 55.17 to 87.00 F when the initial voltage of the discharge process increases from 0.7 to 2.7 V.

For all the four initial voltages of the discharge process, the capacitance ratio is calculated using (3) while C_T is the capacitance determined using the total charge associated with the 0.1 A current. The ratios are plotted in Fig. 10. It is clear that for a specific voltage, the ratio increases when the discharge current increases, which means that the capacitance calculated using the IEC method decreases. For instance, the ratio increases from 0.973 to 1.134 when the discharge current increases from 0.1 to 10 A for 2.7 V, which is consistent with the first observation on the experimental results elaborated in Section 4. All the ratios are greater than unity except when the discharge current is 0.1 A for 2.7 and 2 V. Specifically, for 2.7 V, the IEC method results in a capacitance greater than 100 F and should be excluded, as elaborated in Fig. 6. For 2 V, the IEC method results in 92.32 F and the capacitance calculated using the total charge method is 91.63 F. The difference between these two values is minimal. Therefore, the second observation revealed in Section 4 is also verified. Moreover, for the same discharge current, the ratio increases when the voltage decreases. For instance, when the discharge current is 10 A, the ratio increases from 1.134 to 1.423 when the voltage decreases from 2.7 to 0.7 V, which is consistent with the third observation analyzed in Section 4. In summary, the simulation results are consistent with the experimental results, which suggests that the physical mechanisms leading to these observations can be explained using the supercapacitor model shown in Fig. 4(a).

Fig. 10.

Simulation results: ratios of capacitances estimated using two methods when initial voltage of discharge process varies.

5.3. Supercapacitor physics

The effects of the supercapacitor physics on the capacitances characterized using the two methods are elaborated in this section. To explain the first observation that the capacitance determined using the IEC method decreases when the discharge current increases, Fig. 11 shows the simulated supercapacitor terminal and branch capacitor voltages when the initial voltage of the discharge process is 2.7 V. The supercapacitor terminal voltage V_T is used as the condition to terminate the discharge process. The simulation results show that the same discharge termination condition of V_T = 0.01 V results in different branch capacitor voltages (V₁ − V₅) when the discharge current varies. While the initial voltages of the branch capacitors are the same, the final voltages at the end of the discharge process are lower when a lower discharge current is applied. Specifically, when the discharge current is 0.1 A, Fig. 11(b) shows that V1 − V₅ is 0.0115, 0.0347, 0.2142, 1.2221, and 2.2842 V, respectively. For the 10 A current, Fig. 11(a) shows that V₁ − V₅ is 0.1600, 1.3512, 2.6000, 2.6994, and 2.6972 V, respectively. Therefore, the branch capacitors are more deeply discharged by the 0.1 A current, which means that a larger fraction of the total charge is extracted during the discharge process and a larger capacitance estimate is obtained.

Fig. 11.

Simulation results using IEC method: terminal and branch capacitor voltages when initial voltage of discharge process is 2.7 V. (a) Discharge current is 10 A. (b) Discharge current is 0.1 A.

As shown in Fig. 11, the branch capacitors are less deeply discharged when the discharge current is higher because the discharge time is shorter: 20.4246 s for 10 A versus 2571.3179 s for 0.1 A. For an ideal capacitor with a zero ESR, the discharge time is inversely proportional to the discharge current. For supercapacitors, the discharge process is accelerated and the discharge time is shortened when the discharge current is higher for three reasons. First, a larger ESR voltage drop is resulted at the beginning of the discharge process when the discharge current is higher. As shown in Fig. 11(a), the voltage drop due to ESR is 0.15 V when the discharge current is 10 A, which is a significant portion of the supercapacitor terminal voltage drop during the discharge process: 5.6% (0.15 versus 2.69 V, note that the discharge termination voltage is 0.01 V). On the other hand, when the discharge current is 0.1 A, the voltage drop due to ESR is only 0.0015 V, which is negligible compared to the terminal voltage drop of 2.69 V.

Second, the effects of the charge redistribution process on the supercapacitor terminal voltage drop deceleration are weakened when the discharge current is higher. As elaborated in [31,32], due to the differences between the branch capacitor voltages, charge is unidirectionally redistributed from slow branches to fast branches during the discharge process, which decelerates the drop in the supercapacitor terminal voltage. When the discharge current is higher, the charge redistribution process is shortened and the discharge time extension effect associated with this process is therefore less significant.

Third, the energy loss is more significant when the discharge current is higher. Fig. 12 plots the resistor power when the initial voltage of the discharge process is 2.7 V. For instance, the power dissipated by R₁ is denoted as P₁:

$$P_1=I_1^2{R}_1,$$ (11)

where I₁ is the current through R₁. The energy loss during the discharge process is estimated by calculating the area under the power curve. The total energy loss of the six resistors is approximately 50 and 11 J when the discharge current is 10 and 0.1 A, respectively. A breakdown of the energy loss shows that R₁ is the major contributor in Fig. 12(a) with 30.6 J while R₄ dominates in Fig. 12(b) with 4.3 J. Furthermore, the energy loss can be partitioned into two groups: one from charge redistribution with R₁ − R₅ and the other from self-discharge with R₆. For 10 A, charge redistribution contributes 49.985 J and self-discharge contributes 0.015 J. For 0.1 A, the contributions from charge redistribution and self-discharge are 9.37 and 1.63 J, respectively. These results are consistent with the fact that the energy loss is mainly contributed by charge redistribution in the short term and the impact of self-discharge is significant in the long term [45].

Fig. 12.

Simulation results using IEC method: resistor power when initial voltage of discharge process is 2.7 V. (a) Discharge current is 10 A. (b) Discharge current is 0.1 A.

The second observation states that the capacitance determined using the total charge method is greater than those estimated using the IEC method. This is because during an IEC experiment, the supercapacitor is only partially discharged and the capacitance is therefore an underestimate of the actual value. On the other hand, when multiple discharge processes are employed, the supercapacitor is more deeply discharged and the capacitance calculated using the total charge method is greater. For example, Fig. 13 plots the supercapacitor terminal and branch capacitor voltages simulated using the total charge method when the initial voltage of the discharge process is 2.7 V and the discharge current is 10 A. The final voltages of the branch capacitors are 0.1601, 0.1653, 0.1684, 0.2376, and 1.2501 V, respectively. Compared to the results using the IEC method potted in Fig. 11(a), the branch capacitors are more deeply discharged and more charge is extracted. Therefore, the capacitance estimated using the total charge method is greater.

Fig. 13.

Simulation results using total charge method: terminal and branch capacitor voltages when initial voltage of discharge process is 2.7 V and discharge current is 10 A.

Finally, the third observation states that the ratio of the capacitances estimated using the two methods is larger when the initial voltage of the discharge process is lower. This is because less charge is stored in the supercapacitor and the discharge time is shorter at a lower voltage. For an ideal capacitor with a constant capacitance, the stored charge is proportional to the voltage. For supercapacitors, even less charge is stored at a lower voltage because of the voltage dependence of the supercapacitor capacitance: the capacitance is lower at a lower voltage. For example, Fig. 14 shows the supercapacitor terminal and branch capacitor voltages simulated using the IEC method when the initial voltage of the discharge process is 0.7 V and the discharge current is 10 A. The discharge time is 2.9004 s. Referring to Fig. 11(a), the discharge time is 20.4246 s when the initial voltage of the discharge process is 2.7 V and the discharge current is 10 A. The discharge time ratio is 14.2% (2.9004 versus 20.4246 s), which is lower than the supercapacitor terminal voltage drop ratio of 25.7% (0.69 versus 2.69 V). This is because the discharge process is accelerated and the discharge time is shortened when the initial voltage of the discharge process is lower. Specifically, the voltage drop due to ESR (0.15 V) is an even more significant portion of the supercapacitor terminal voltage drop of 0.69 V: 21.7%. This ratio is 5.6% when the initial voltage of the discharge process is 2.7 V. In the meantime, the effects of the charge redistribution process on the supercapacitor terminal voltage drop deceleration are weakened when the initial voltage of the discharge process is lower. Therefore, the supercapacitor is even less deeply discharged. Consequently, the capacitance estimated using the IEC method is even smaller and the ratio of the capacitances estimated using the two methods is even larger.

Fig. 14.

Simulation results using IEC method: terminal and branch capacitor voltages when initial voltage of discharge process is 0.7 V and discharge current is 10 A.

6. Conclusion

This paper comparatively studies two supercapacitor capacitance characterization methods: the IEC 62391-1 method 1A and a method based on the total charge stored in the supercapacitor. When the IEC method is used, a higher discharge current results in a smaller measured capacitance. In the meantime, the capacitance measured using the IEC method is an underestimate of the actual value. Moreover, the ratio of the capacitances estimated using these two methods is greater at a lower terminal voltage. The physical mechanisms accounting for these observations are illustrated by simulating a five-branch RC ladder circuit model that captures multiple aspects of the supercapacitor physics: porous electrode structure, voltage dependence of capacitance, charge redistribution, and self-discharge. Simulation results show that the supercapacitor is only partially discharged during an IEC experiment. When a lower discharge current is applied, the supercapacitor is more deeply discharged and the estimated capacitance is therefore greater, which is consistent with the experimental results.