I am doing some requirements analysis work on backup power systems for GPON ONTs. As part of this work, I need to perform an evaluation of a number of batteries for charge capacity and energy. As part of this evaluation, I decided to write a Mathcad worksheet to perform the analysis. When I write an analysis routine, I usually test it using some data that has already been acquired and processed. In this case, I grabbed some data from the web and used this data to test my worksheet. If you wish to look at the actual worksheet, you can download it here.
I will be evaluating some lithium ion batteries for charge capacity and energy as a function of the battery's discharge termination voltage (i.e. battery voltage at which discharge current is terminated). The same approach can be used for any battery.
Reference Battery Data
I found some excellent data on this web site, which I will replicate here. The data is for a single-cell, lithium ion (LiCoO2) battery (type 18650). There is a lot of excellent data on the web about this type of battery.
Figure 1 shows data for this battery in the same format as I often receive from our lab when testing batteries. Basically, you use a current source to apply a fixed load on the battery. You then measure the variation in battery voltage versus time.
In addition to the raw test data, this web site also contains graphs of energy (Watt-hours) and charge capacity (amp-hours). Thus, I can compare my worksheet's results with those of the web site as check on my routine's correctness.
Gather Battery Data
Using Figure 1 and Dagra, I converted the graph into numeric form. Figure 2 shows how this data appears in my Mathcad worksheet. Normally, I receive this data from the lab in the form of an Excel spreadsheet. Because I am looking for test data prior to doing my own tests, grabbing some test data from an exisiting graph is a good way to evaluate my Mathcad worksheet.
The arrays A, B, C, D, E, F each contain two columns of data (time and battery voltage) that corresponds to discharge currents 0.2 A, 0.5 A, 1.0 A, 2.0 A, 3.0 A, and 5.0 A, respectively.
Compute Battery Energy
The energy extracted from a battery as we draw current from it is given by Equation 1, which assumes the discharge begins with a battery charged to 4.2 V. As we draw energy from the battery, its terminal voltage decreases. Equation 1 will be used to generate a plot of energy drawn versus battery voltage.
- E(v) is the energy drawn from the battery as the terminal voltage has dropped to v [Watt-hours].
- vBattery(t) is the battery voltage [V].
- iBattery(t) is the current being drawn from the battery [A].
- τ is the time at which vBattery=v [hours].
Figure 3 shows the Mathcad formulas I used to compute the battery energy and a plot of the data generated from this analysis.
Compute Battery Capacity
The calculation of the battery capacity is a bit simpler than the calculation of battery energy. The formula I used is shown in Equation 2.
- Q(v) is the charge drawn from the battery as the terminal voltage has dropped to v [Ampere-Hours].
Figures 4 shows how the charge capacity calculation was setup. Note that while Equation 2 contains an integral, the use of a constant current source load means that we just need to compute the product of the current and time.
Figure 5 contains that graph of the charge capacity calculation results.
My worksheet generates very similar results to those posted on the reference web site. The only differences that I see between Figures 6 and 7 and the results my worksheet is generating are related to errors in grabbing the data off of the images -- there is always some error in that process.
At this point, I feel comfortable that my Mathcad routine is running properly and I can use this routine to process data being taken in my lab.