Types of measurements

A super-capacitor, often referred to an electrical double layer capacitor, has similar characteristics to a normal capacitor, but offers significantly higher capacitance values per unit volume. According to Table 5.2, the super-capacitors have a superior cycle efficiency which reaches around 97%-98% and more than 50,000 cycle life. Since the

model of super-capacitors is introduced for fast lifetime calculation, it is inconvenient to use a complex super-capacitor’s model. In this work, for achieving fast simulation, a simple super-capacitor model, which is just taking into account the super-capacitor parameters from the datasheet, is developed. The equivalent circuit of the super-capacitor is shown in Figure 5.9. From the figure, a series resistor can be determined by measuring the potential difference between the two terminals at the beginning when charging the super-capacitor with a large constant charging current.

is very close to the equivalent resistance value provided by the super-capacitor datasheet (Zhang and Yang, 2011). Thus, it is assumed that is equal to the equivalent resistance of the super-capacitor. Since the charging frequency is quite low in energy harvesting systems, the series inductor is neglected in this work. The parallel capacitor value can be determined by the capacitance of the super-capacitor. As the super-capacitor has a high self-discharge behaviour, a high parallel resistor is used to model a leakage resistor. In order to identify the value of , the self-discharge process has been conducted in the laboratory.

Figure ‎5.9 Proposed electrical equivalent circuit of super-capacitor

Normally, the capacitor model description can be divided into the charging mode and discharging mode, as illustrated in Figure 5.9. The relationship for the electrical charges of the super-capacitor and the terminal voltage of the super-capacitor

is

(5.21)

where is the capacitance of the super-capacitor. From the current side of view, the charges of the capacitor can be expressed as:

∫ (5.22)

where refers to the initial charges on the capacitor before charging/discharging and is electrical current charging the super-capacitor at time t, and is electrical charge difference during a time interval . As shown in Table 5.2, a capacitor is not an ideal electrical circuit, the charge and discharge efficiency of a super-capacitor , is considered. According to the super-capacitor’s characteristics, is around 97%-98%. Based on Figure 5.9, the charging current is approximately equal to the output current of the power conversion circuit times charge and discharge efficiency . By considering the leakage current of the super-capacitor . The relationship between and can be expressed as:

The discharging current from the super-capacitor is:

where is the current consumed by the whole energy harvesting system at the time t. Equation 5.22 can be rewritten as:

∫ (5.25)

∫ (5.26) where refers to the initial voltage of the super-capacitor. Generally, to characterize the model with different I-V inputs, the model description can be extended to have a constant voltage charging/discharging phase.

(1) Charging/discharging using a constant voltage

By using a constant voltage to charge a super-capacitor, the previous equations can be rewritten by using Kinhowff’s voltage laws to analyse Figure 5.9. The charging Equation 5.28, the equation can be rewritten as:

∫ By using the Laplace transform, Equation 5.29 can be expressed as

where S is the notation of Laplace transfer. By writing the equation as the expression of , Equation 5.31 can be obtained.

, then the express becomes:

By integrating Equation 5.33 into Equation 5.25, the equation is rewritten as:

[ ]

The charging time period , which is the time that the supercapacitor is charged from

to , can be expressed as:

By using Laplace transform, Equation 5.37 can be rewritten as:

The discharging time period needed to discharge the super-capacitor to a certain voltage can be given by:

(2) Charging and discharging with a constant current

Similar to using a constant voltage to charge/discharge a super-capacitor, it can also be charged or discharged by using a constant current . Then Equations 5.23, 5.24, 5.25, and 5.26 can be rewritten as:

The duration or it takes by using a constant current to charge or discharge the super-capacitor to a certain voltage can be calculated by Equations 5.46 and 5.47, respectively.

(3) Model evaluation

In order to evaluate the model, the leakage resistance of the super-capacitor under various states should be determined. As stated in (Guan and Liao, 2008), charge on the super-capacitor drops by around 35% per month due to the leakage. . The self-discharge resistance of the super-capacitor can be estimated as (Guan and Liao, 2008):

[

]

where is super-capacitor’s voltage decreased by self-discharging during a time duration . and can be determined by experiments. For the evaluation of the model, a 2.3V 22F super-capacitor from Panasonic (Panasonic 22F super-capacitor, 2008), was used in this work. The specifications of the super-capacitor are listed in Table 5.3.

Table ‎5.3 Specifications of Panasonic Super-capacitor (Panasonic 22F super-capacitor, 2008)

Parameter Specification Unit

Capacitance 22 F

Internal resistance at 1KHz 0.1 Ω

Maximum operating voltage 2.5 V

In order to determine the self-discharging resistance of the super-capacitor, experiments have been conducted in the laboratory with constant environmental factors (25 and 30% humidity). Initially, the super-capacitor was fully charged by using a DC power supply and the self-discharge process has been taking place for two hours after removing the power source. In order to reduce random measurement errors, the experiment has been repeated ten times by using four 22F super-capacitors. The average value gained from these experiments and the testing results are shown in Figure 5.10. By integrating the parameters into Equation 5.48, the self-discharging resistor can be estimated. By comparing the simulation results with the experimental results, the error of the self-discharging model is around 2.7%.

Figure ‎5.10 Self-discharging effect of 22F 2.5V super-capacitor The accuracies of the proposed model during the charging and the discharging processes have been evaluated by using experiments. Firstly, the model was validated with a 2.V constant charging voltage and the result is depicted in Figure 5.11(a). The corresponding experimental test, which used a 2.5V battery as the constant voltage

source to directly charge the super-capacitor, has been carried out in the laboratory.

The accuracy of the proposed super-capacitor model is 96.1% by comparing the simulation result with the experimental result. Figure 5.11 (b) shows the model, in which the super-capacitor is charged by a constant current source. In order to see the difference, 10mA and 5mA input currents have been simulated, respectively. Initially, the voltage level of the super-capacitor is set to be 1.3V. By using these two current sources to charge the super-capacitor with the same time interval (3200s), respectively, the final voltages of the super-capacitor in these two cases are 2.49V and 1.9V, respectively. The corresponding experiments have been tested and the final voltages of the super-capacitor in these two experiments are 2.39V and 1.79V, respectively. By comparing the simulation results and the experimental results, a 6% error could be found in the worst case scenario.

Figure ‎5.11 (a) 2.5v constant charging voltage (b) 10mA and 5mA constant charging current

The discharging processes were simulated and tested with constant discharge voltages and constant discharge currents, respectively. Figure 5.12 shows that fully charged super-capacitors were discharged by 1.1V and 1.2V batteries. The simulation results show that the time periods for discharging the capacitors from 2.5V to 1.5V are 2.5s and 3.2s, respectively. The corresponding test results are 3s and 3.5s, which indicate the error of the model is around 20%. The increase of the model error is caused by a measurement error, which is normally caused by the equipment and those running the tests. The discharging model was evaluated by constant current sources, for which 5mA and 10mA were selected. The simulation results show that the capacitor needs 2680s and 1448s to discharge from 2.5V to 1.8V respectively. In order

to evaluate the accuracy of the proposed model, experimental tests have been held.

The comparison results show that the capacitor needs 2200s and 1309S to discharge from 2.5V to 1.8V by using these two current sources and the error of the model is around 10%.

Figure ‎5.12 (a) constant discharge voltages (b) constant discharging current