[General Topics] waterplanet:
I'm interested in how I can predict the future battery life of a smartphone after a certain number of days of usage. This is somewhat important in the debate of user removable vs sealed in batteries. Or when deciding if you want to buy a phone with a sealed in battery, or when you are going to need battery service. I did not really find any such method online, so I made the following calculations myself. Please point out mistakes or errors if you see any. I have used lower case letters for variables, and upper case for constants.

Edit: Formula for calculating future battery life (including calendar loss)

I wanted to create a new formula that takes into account that batteries not only degrade when they perform work (w), but also as they age. The first I posted only took cycle loss into account. So i set up this expression that describes how the battery capacity (c) is reduced over time as the battery ages simultaneously as they lose capacity while continuously working at a constant rate:

By dividing both sides of the equation with the rate of doing work, power (P), that is assumed constant, we get this expression:

Then doing some substituting b = c/P, B=C/P and t = w/P:

By doing some manipulation of this expression and calculating the geometric serie:

This is equivalent with:

Where:

"Degraded Battery Life" (b) is the expected lifetime of a full charge in the end of the period. The unit is usually hours (h) or days (d), and the model is valid as long as b is 0 or larger.

"Initial Battery Life" (B) is the Battery Life before the degrading has happened, unit is hours (h), and valid values is everything from 0 and up. Please note that battery life in these calculations is the overall battery life, including use during both night and day. When people talk about battery life in everyday speech, they often mean battery life during the day when the user is awake and actively using the phone.

"Time" (t) is the time period the battery has been in use. Usually measured in years, and valid from 0 and up.

"Longevity rate" (L) is fraction of battery capacity remaining after a certain time period. It's assumed to be constant. The unit is usually ^-y (y = year), for instance 0.9^-y if 90% of its capacity remains after one year. Valid values are from 0.3^-year and below 1^-year. Suggested value: It's hard to find any useful information about this, but I guess it could be about 0.9^-y.

"Degradation to work ratio" (D) is the amount of degraded capacity relative to the work performed by the battery (D = -dc/dw). Similar to L, this is also assumed to be constant. This parameter does not have a unit. If the battery is described as being able to retain a fraction R of its original capacity after N cycles, D can be calculated as: D=1-R^-N. Suggested value: For common smartphone batteries they usually retain 80% of its initial capacity after 500 cycles. This means that D is approximately 0.000444 (Or 0.0444%)

Z is a dimensionless variable approaching 0, but is different from 0. Suggested value Z = 0.0001

LaTex code for the formulas:

c=\lim_{n\to\infty}(CL^t-\sum_{k=1}^{n}D\frac{w}{n}L^{t(\frac{n-k}{n})})

(\frac{c}{P})=\lim_{n\to\infty}((\frac{C}{P})L^t-\sum_{k=1}^{n}D\frac{(\frac{w}{P})}{n}L^{t(\frac{n-k}{n})})

b=\lim_{n\to\infty}(BL^t-\sum_{k=1}^{n}D\frac{t}{n}L^{t(\frac{n-k}{n})})

b=\lim_{n\to\infty}(BL^t-\frac{L^{t}-1}{n(L^{t/n}-1)}Dt)

b=\lim_{Z\to0}(BL^t-\frac{L^{t}-1}{L^{Zt}-1}DZt)