A battery experiment

Abstract

Equations derived from electrical circuit theory form the basis of a laboratory technique to characterize the components of the electrical equivalent circuit of rechargeable batteries. The experiment is a useful laboratory complement to lectures on energy courses for electrical engineering students. Results for lead acid batteries are presented in this study.

Keywords

Teaching battery circuit battery characterization experiment

Introduction

The paper concerns an experiment to characterize the electrical properties of a rechargeable battery. The author developed this experiment for use in his course on renewable electric energy sources; a course that is offered to electrical engineering students. The purpose of the experiment was to determine the electrical properties of a D size 2 V lead-acid cell. The properties of interest were the internal resistance, the double layer capacitance, and the resistance of the plate–electrolyte contact. The values of these parameters constitute the components of the Thevenin model of the battery. The most basic circuit of the battery is a voltage source in series with the internal resistance.¹ The electrochemical processes of rechargeable batteries^2,3 lead to the more precise equivalent circuit of the Thevenin model, comprising the open circuit battery voltage in series with a resistor (representing the internal resistance of the cell) in series with the parallel RC combination of a resistor (representing the resistance of the plate–electrolyte contact) and a capacitor (representing the electrode double layer capacitance).^4–6 The Thevenin model is used here to represent the battery under conditions of constant state of charge (SOC) and constant temperature. This model was used in the present experiment.

The battery of the present work was the Cyclon 2 V sealed lead-acid cell. (Although henceforth referred to as a battery it is in fact a single cell where the 12 V car battery is made up of a battery of six nominally 2 V such cells.) Its size, ability to be used in any position, reliability, and good performance make the Cyclon a good choice for portable power applications. An electrical transient method was used to determine the circuit components of the Thevenin model of rechargeable batteries.⁷ The results of the application of this characterization technique to the lead-acid battery are given here for the first time. The characterization method was based on driving the cell into the discharge-rest-charge-rest sequence while plotting the temporal variation of the cell voltage during each of these four transient phases. Then, a set of seven cell voltages, which were read off of these plots, were used to calculate the resistances and capacitor for each phase.

Theory

The equivalent circuit of the battery appears to the left of terminals a and b in Figure 1 where $v (t)$ is the battery voltage. It is a series combination of the open circuit battery voltage $V oc$ , the series resistor, and the equivalent of the parallel combination of a resistor and a capacitor. The switch is closed for the discharge phase and open for the subsequent rest period. The capacitor value is C₁ and the series resistor is R for both the discharge and the subsequent rest period, while R₁ and R₂ are the resistors in parallel with the capacitor in the discharge and rest periods, respectively. R_ex is the external resistor.

Figure 2 shows the battery circuit during the charge phase and the subsequent rest period. V_g is the charging voltage. The same battery model applies in the charge phase as in the discharge one, but now with different values for the components in the parallel circuit. The capacitor value is C₂ for both the charge and the subsequent rest period, while R₃ and R₄ are the resistors in parallel with the capacitor in the charge and rest periods, respectively. The series resistor R stays the same as in the discharge-rest case. Since V_g > V_oc the voltage source V_g acts as a generator. The current i(t) flows into terminal a and ultimately into the positive terminal of V_oc as a charging current to the cell. The source V_oc now acts as the energy absorber.

Transients are involved during the charging and discharging of batteries. Abrupt changes are recorded in the voltage across the terminals of the battery as the switch is closed or opened at the start of these transients. These boundary value changes in voltage as well as voltage levels in the intervening transients are used here to calculate the circuit parameters. The temporal variation of the voltage across the battery terminals v(t) during the discharge-rest-charge-rest sequence appears in Figure 3. The variation in v(t) gives equations^7,8 that correspond to the voltage levels V₁ to V₇ that are shown in Figure 3 where V₁ is V_oc. The equations that were used to calculate the equivalent circuit parameters of the lead-acid battery are functions of these voltage levels and are

R = R ex (\frac{V oc}{V 2} - 1)

(1)

R 1 = R (\frac{V oc}{V 4 - V 3} - 1) - R ex

(2)

R 2 = \frac{R 1 V oc}{(R + R 1 + R ex) C 1 (\frac{dv}{dt}) r 1}

(3)

C 1 = - \frac{R ex V oc}{(R + R ex) 2 (\frac{dv}{dt}) d}

(4)

R 3 = R (\frac{V g - V oc}{V 6 - V 7} - 1) - R ex

(5)

R 4 = - \frac{R 3 (V g - V oc)}{(R + R 3 + R ex) C 2 (\frac{dv}{dt}) r 2}

(6)

C 2 = \frac{R ex (V g - V oc)}{(R + R ex) 2 (\frac{dv}{dt}) c}

(7)

where the respective gradients at the start of the discharge phase and at the start of the subsequent rest period are

(\frac{dv}{dt}) d

and

(\frac{dv}{dt}) r 1

, while the respective gradients at the start of the charge phase and at the start of the subsequent rest period are given by

(\frac{dv}{dt}) c

and

(\frac{dv}{dt}) r 2

Apparatus and experimental method

The Cyclon battery was examined. This is a sealed battery with a nominal voltage of 2 V. It has a 10 h capacity of 2.5 Ah. It is a D size cell. The discharge-rest-charge-rest curve of a fully charged battery was first obtained. The battery was then discharged, at the 10 h rate of 250 mA, to 70% SOC and then to 40% SOC, and the discharge-rest-charge-rest curve was recorded at each SOC. Each curve was a plot of the battery voltage v(t) as a function of time and was recorded with the battery at room temperature. Figures 4 and 5 show the circuit diagram and apparatus, respectively, for measuring the discharge-rest-charge-rest curve. The battery is at A in Figure 5. A single pole double throw switch with center off was used to switch between phases. This switch and the external resistor are on the protoboard at B in Figure 5 and the power supply providing the charging voltage is at C. A National Instruments USB-6009 data acquisition board sampling at 1000 samples per second was used to acquire the data with 14-bit voltage resolution. The DAQ board is to the left of the protoboard. The resulting temporal variation in the battery voltage was recorded by connecting the data acquisition probes between terminals a and b in Figure 4. The data were displayed on the computer at D in Figure 5. Figure 3 shows the discharge-rest-charge-rest curve of a battery at 100% SOC.

The charging voltage V_g during the charge phase was 3.75 V. The external load resistor $R ex$ was 10 Ω. The open circuit cell voltage V_oc was 2.14, 2.05, and 1.98 V at 100%, 70% and 40% SOC, respectively.

The battery displayed large abrupt voltage changes at the beginning of each phase in its discharge-rest-charge-rest curve. These changes in v(t) were then followed by transients where the cell voltage recovered to a steady state value within seconds.

Electrical parameters could be calculated to within ±0.02. The component values of the equivalent circuit of three cells are given in Table 1. The results show that the internal resistance of the cell R is approximately constant as SOC varies. The resistance of the plate–electrolyte contact for all phases, R₁, R₂, R₃, and R₄, increase as SOC increases. The double layer capacitance values C₁ and C₂ decrease as SOC increases. Since capacitance is inversely proportional to distance between separated charges the results mean that the width of the medium between charges increases as SOC increases. This also explains the observed increase in plate–electrolyte contact resistance with the increase in SOC.

Table 1.

Electrical circuit parameters of three Cyclon lead-acid batteries at states of charge of 40%, 70%, and 100%.

SOC	40%	70%	100%
(i)
R	0.29 Ω	0.26 Ω	0.26 Ω
R ₁	0.02 Ω	0.03 Ω	0.07 Ω
R ₂	0.04 Ω	0.08 Ω	0.11 Ω
C ₁	5.48 F	2.46 F	1.69 F
R ₃	0.02 Ω	0.02 Ω	0.32 Ω
R ₄	0.01 Ω	0.03 Ω	0.28 Ω
C ₂	7.50 F	3.13 F	1.68 F
(ii)
R	0.24 Ω	0.28 Ω	0.29 Ω
R ₁	0.03 Ω	0.04 Ω	0.08 Ω
R ₂	0.07 Ω	0.08 Ω	0.11 Ω
C ₁	4.61 F	1.57 F	1.57 F
R ₃	0.02 Ω	0.05 Ω	0.27 Ω
R ₄	0.01 Ω	0.04 Ω	0.24 Ω
C ₂	10.53 F	1.90 F	1.38 F
(iii)
R	0.36 Ω	0.37 Ω	0.30 Ω
R ₁	0.02 Ω	0.05 Ω	0.10 Ω
R ₂	0.04 Ω	0.05 Ω	0.18 Ω
C ₁	4.02 F	2.42 F	1.78 F
R ₃	0.03 Ω	0.07 Ω	0.43 Ω
R ₄	0.05 Ω	0.05 Ω	0.31 Ω
C ₂	3.80 F	2.62 F	2.30 F

Conclusions

A battery characterization experiment was found to be a valuable addition to a course on renewable energy sources. It gives students an experience in building a useful switching circuit and to acquire data to display on the computer. The results provide realistic component values for the equivalent circuit of rechargeable batteries that can be used in the design and analysis of battery driven electronic instruments.

Footnotes

Conflict of interest

None declared.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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