Quote of the Day
Successful ... politicians are insecure and intimidated men. They advance politically only as they placate, appease, bribe, seduce, bamboozle or otherwise manage to manipulate the demanding and threatening elements in their constituencies.
— Walter Lippmann
I frequently am asked to comment on data that other engineers send me. This morning I received some test data obtained from an engineer measuring the backup time of an Uninterruptible Power Source (UPS) containing multiple lithium-ion (Li-Ion) batteries. The engineer was disappointed with the backup time provided by this UPS and wanted to know if his test results were reasonable considering the battery capacity of the UPS. While there were numerous circuit parameters measured during this testing, the critical information was the battery voltage versus time.
The engineer's routine calculations showed that the UPS should provide 4 hours of backup time, but he measured 2.7 hours. While batteries are listed with a nominal charge capacity, their actual capacity varies strongly with the load presented to the battery. His testing was performed at room temperature and with a load of 0.8 A.
In this post, I will show that his test results are reasonable when when the effects of the load current are taken into account. I will perform the calculations three ways:
- Method 1: Simple capacity/load modeling (ignoring current load on capacity).
- Method 2: Using a graphical curve of capacity versus load.
- Method 3: Using numerical methods for interpolating digitized capacity graphs.
- C-rate is the charge/discharge current normalized to the battery capacity. A charge/discharge rate of one C for one hour draws a charge equal to the battery capacity. For example, the 1C discharge rate for a 2.2 A-hour battery is 2.2 A.
- Cutoff Voltage (VCutoff)
- The battery voltage at which the UPS terminates the discharge of the battery.
- The available charge in the battery, which is a function of the load current. A battery's charge capacity is generally specified at some low current draw. For example, the capacity of lead-acid batteries is specified at a 20 hour (C/20 or 0.05C) rate.
Here are the key facts about this UPS:
- It contains 4 series-connected, Li-Ion, 18650 cylindrical batteries (Figure 1).
- These batteries have a nominal cell voltage of 3.7 V.
- I assume that each 18650 battery is rated to have a nominal 2800 mA-hour charge capacity. The actual rating is 2700 mA-hour (minimum) and 2900 mA-hour (typical). I will average the two values for this analysis.
- The UPS load is a device that requires an input voltage between 10 V and 16 V.
- The voltage range requirement is met by connecting the batteries in series.
- The engineer modeled the load using 800 mA of constant current draw.
- Most UPS hardware stops discharging the battery at VCutoff, which for Li-Ion batteries is typically near 3.3. V. In this case, the UPS cutoff was ~2.9 V.
Measured Battery Voltage Versus Time for Constant Load Current
Figure 2 shows the data that was emailed to me. I have literally seen hundreds of these test plots. This UPS I include a calculation on Figure 3 that shows that the charged cell voltage for the batteries in this 4 cell-string pack is 3.92 V. This is below the manufacturers 4.2 V charge voltage in their specification. The cutoff voltage is 2.87 V.
This data was measured by instrumenting the battery series inside the UPS. We will need the initial cell voltage and the final cell voltage in order to estimate the charge drawn from the battery.
Method 1 will not use this data, but Method 2 and 3 will.
Method 1: Simple Calculation Ignoring Load Current Impact on Capacity
Figure 3 shows how to estimate the backup time provided by this UPS assuming nominal battery characteristics.
The key problem with this analysis is that it assumes that the battery capacity does not depend on the load. I model the effect of load current on battery capacity any time the load currents exceed 0.1C.
Method 2: Graphical Analysis
Figure 4 shows how to compute the expected backup time using the battery's capacity versus load chart. All the calculations are shown on the graph and I obtain a backup time estimate of 2.8 hours.
The calculations shown on the graph can be described as follows:
- Us the initial cell voltage to determine how much charge is lost because the UPS did not fully charge the battery.
- Use the final cell voltage to determine how much charge is available from a fully charged battery.
- Determine the difference between the final and initial charge, which reflects the charge available for backup energy.
- Divide the available charge (in mA-hours) by the load current, which give the backup time.
Method 3 : Numerical Analysis
Manufacturer's Capacity Rating Versus Load
Figure 5 shows the typical discharge specification for Li-Ion battery from Panasonic with a nominal rating of 2900 mA-hour (2700 mA-hour minimum). As you can see in Figure 4, the typical capacity is measured when the battery has a minimal load (0.2C). One unusual aspect of this chart is that you also get full capacity with a high load current (2C). I only rarely see this characteristic on a capacity chart.
I digitized this data using Dagra and pasted it into Mathcad. Figure 6 shows the digitized the data and a routine to interpolate the data. I will assume that batteries actual capacity is 2800 mA-hour, the mean of the minimum and maximum.
Mathcad Model of Battery Capacity Data
Given the discharge data shown in Figure 7, I can determine the effective capacity of the UPS battery at a 0.8 A load. Figure 6 shows the my interpolated results for the manufacturer's data and my interpolation for a 0.8 A load. The graph shows that the effective battery capacity is ~2779 mA-hour with an initial charge voltage of 4.2 V. I also show that 554 mA of charge is missing if the battery is initially charged to only 3.92 V, which is this case. In the next section, I will show how to algebraically obtain these results.
Estimated Discharge Time Assuming that Capacity is Load-Dependent
In Figure 8, I use 2-dimensional interpolation to compute the battery capacity assuming VCutoff = 3.3 V and ILoad = 0.8 A.
My calculated discharge time of 2.8 hours roughly agrees with the measured discharge time of 2.7 hours.
My estimate for the battery operating time is 2.8 hours with an 800 mA load. We actually measured 2.7 hours, so the estimate is ~4%. This is error is within reason for batteries – they are subject to individual variation.