Determining the boundary line between normal versus excessive self-discharge on lithium ion cells
2020-07-15 | 12 min read
Testing self-discharge of lithium ion cells has been discussed extensively here through several posts for good reason; it is a critical parameter that all cells need to be tested for during manufacturing. All cells have some self-discharge. This is normal and expected. However, excessive self-discharge is an indicator of underlying defects that could lead to failure later in life. In this posting I will elaborate on determining where the boundary line should lie between what is normal versus what is excessive self-discharge in lithium ion cells. For reference, “Keysight Solutions for Measuring Self-Discharge of Lithium Ion Cells Achieves Revolutionary Reduction in Test Time” (click on title to access) is a worthwhile post to review to learn more about cell self-discharge and its testing.
There are two main methods of measuring self-discharge. First is the traditional delta OCV method. The loss of the cell’s open circuit voltage (OCV), measured typically over weeks of time, indirectly indicates the amount of self-discharge in a cell. Second is the potentiostatic method. Here the cell is held at a fixed potential with a stable external source. After settling on the order of an hour or so for the cell and external source to equilibrate, the current being furnished by the external source equals the cell’s self-discharge current. Either method, properly done, will yield the same relative distribution of results for self-discharge on a given group of cells. For reference, another posting here “Shortening Lithium Ion Cell Manufacturing Time: A Comparative Study of Two Methods of Making Self-Discharge Measurements” (click on title to access) provides further insights on how these two methods compare and what factors determine their respective test times.
So, how does one determine the boundary line where normal self-discharge ends, and excessive self-discharge starts? This is not a straightforward thing to answer. It involves several things, including:
- How external factors impact self-discharge and its measurement, to what degree, and how to control or compensate for them.
- Establishing a history though characterizing self-discharge of the cells over many lots and time.
- Applying statistical methods to establish a boundary line between what is normal versus excessive self-discharge and making further refinements over time based on the cumulative history of results.
The principles used to determine where the boundary line between normal versus excessive self-discharge should lie are the same regardless of the method being used to measure the self-discharge.
A significant consideration and major challenge when testing self-discharge and determining what is normal versus excessive is that self-discharge is not fixed. It can vary greatly, dependent on several external factors, including:
- How long have the cells rested after being discharged or charged.
- The % SoC of the cells.
- The temperature of the cells when tested.
- The consistency of how the measurement methodology is applied each time.
- The accuracy and stability of the test equipment.
These external factors can change the measured values by an amount greater than the self-discharge itself! Therefore, they need to be kept the same across all the cells within a lot to get meaningful results. Similarly, they need to be kept the same or minor differences properly compensated for from lot to lot, in order to determine where the boundary line should lie between normal versus excessive self-discharge over the long term. Often this may not be much more than using the average, or mean, of each lot as the reference base line to work from, assuming any external influences are affecting all cells equally within a lot, which is often the case.
Once the measurement methodology and external influencing factors are addressed, it then comes down to characterizing the distribution of the self-discharge values for the population of cells, to know what is within the range of normal versus not. One statistical technique most often used is to analyze the data based on a symmetrical bell-shaped normal probability distribution, as many things in nature fall into a normal distribution. Because of the wide use of the normal probability distribution, its function is built into most popular software applications. It is defined by the following mathematical expression:
The mean, µ, is the center of the distribution and the standard deviation, σ, is a parameter characterizing the distribution’s width, where +/-1σ is 68.26% of the population. +/-3σ limits are commonly used as a boundary lines as they include most all the expected main population. The +3σ limit serves well as a starting point for an upper boundary line, beyond which can be considered excessive self-discharge. This can be refined over time and testing numerous lots of cells, as more experience and history of the cells’ self-discharge characteristics is gained.
An example of setting a boundary line based on a normal probability distribution is shown in Figure 1. This is for self-discharge of a group of 16 cells using a Keysight BT2152B Self-Discharge Analyzer to make potentiostatic method self-discharge measurements on them.
Figure 1: Setting an upper boundary line using normal distribution statistical analysis
First the mean and standard deviation values were calculated for setting the boundary line for self-discharge on this group of cells. Three of the cells were clearly bad outliers, having much greater self-discharge than the main population. This being the case, they were omitted from calculations for determining the mean and standard deviation values for the normal population. A +3σ limit was then used to set the upper boundary line for normal versus cells with excessive self-discharge. 99.87% of the main population falls below the +3σ limit. Again, as experience and results are accumulated over numerous lots of cells and time, this can provide additional insight for refining where the boundary should lie, in combination with using statistical methods.
Self-discharge for a cell is characterized to be a value, regardless of whether the delta OCV method or potentiostatic method is used. Note however, that the graph in Figure 1 for the potentiostatic method measurement self-discharge for the cells is represented as a trace, exponentially rising from zero and then levelling off. This is due to the settling time nature of the potentiostatic method measurement. The levelled off area beyond about 1.5 to 2 hours represents the final settled values for each of the cells. However, the determination between normal versus excessive self-discharge can be made earlier from the potentiostatic method measurement, if desired, before the measurement is fully settled. Here, it can be seen the determination can be made between normal versus high discharge within about 30 minutes. For reference, a 10 minute rolling average was used here to reduce noise in the self-discharge current measurement dataset. Additional denoising techniques exist that can be used on the self-discharge dataset to further reduce noise, potentially improving the determination between normal versus excessive self-discharge in less time. Another posting here, “Denoising algorithms enhance self-discharge current measurements on Li-Ion cells” (click on title to access) goes into much greater detail about denoising techniques.
While statistics based on a normal distribution is most commonly used for setting a boundary line, as already described, other statistical techniques exist. Another statistical technique that can be employed is referred to as quartiles. Applying quartiles is more of a procedural process. Here the data is broken into four equal groups by count, in ascending value order. These four groups are bounded by the five values at the ends and in-between the four groups. These four groups and five values are often represented in what is referred to as box and whisker plot, illustrated in Figure 2.
Figure 2: Quartiles box and whisker plot
The interquartile range (IQR) is the difference between the third quartile and first quartile values. This is analogous to the standard deviation value of a normal distribution for quantifying the width of the distribution. Upper and lower fences are then calculated to set the boundaries for the distribution. Statistical analysis using quartiles is supposed to be more robust when the distribution deviates significantly from a normal distribution bell curve.
Quartiles analysis was performed on the self-discharge current measurement dataset in Figure 1, to establish the upper fence to use as the boundary line, as shown in Figure 3. In this example it was reasonably consistent with the +3σ boundary line for the normal distribution analysis. It exhibited a bit more deviation, which is attributed to greater variance introduced by having relatively small lot of just 13 cells defining the normal population.
Figure 3: Setting an upper boundary line using quartiles analysisIn closing, self-discharge is an important parameter that lithium ion cells must be screened for in manufacturing. However, determining where the boundary line is between what is normal versus excessive self-discharge is not a straightforward thing to answer. Demonstrated here is that by characterizing the distribution of the self-discharge values of the cells by either of two different statistical means provides a good and objective way to determine where the boundary line should be set for defining what is normal versus excessive self-discharge. This is regardless of what measurement method is used for determining the self-discharge, as properly done, they should yield comparable results. As experience and results are accumulated over numerous lots of cells and time, this can provide additional insight for refining where the boundary line should lie, if needed, in combination with using a statistical method like those detailed here.