Is it safe to charge iPhone while it is powered off? When I am in a rush I often plug my phone into it's charger and then turn it off (if I do it vise-versa it will turn on when I plug it in).  I have found it charges a estimated 1.5x faster then while charging turned in. I have never had a problem with this, although, since I am in a rush, I only have it plugged in for a little amount of time. 

Could I damage the battery or some other component if I have it charging, turned off, for an extended period of time?
Short answer

This is perfectly fine to do, although the benefit of doing so is negligible.

The reason an iDevice will typically switch on when connected to a charging source is for usability reasons (i.e. due to User Experience considerations, not due to any hardware limitations). Likewise, you'll find in the case of a totally dead battery that plugging an iDevice into a charging source will not allow the user to switch on the device until the battery has been charged to about 5%.

Long answer

Your impression that it charges a estimated 1.5x faster then while charging turned on is probably due to the different stages of charging rather than the fact your iPhone was on/off during the period. 

To clarify this somewhat, your iPhone uses a process known as trickle charging once your iPhone battery has reached 80% of its charge capacity. That is, the first 80% of a battery's capacity is charged at a much faster rater than the last 20%. This process increases the lifespan of your battery, not to mention lets you get going again faster in between charging sessions. 

So, if you're plugging your phone in for short spurts every now and then in between uses, and thereby it's still switched on, it's much more likely to be at 80% or above or that it reaches that point sooner rather than later. And, once it does, trickle charging kicks in and that explains the impression that it's charging slower while your iPhone is switched on.

Another factor that affects battery performance and charging times is temperature. If your iPhone is switched on in the car and streaming Apple Music while also using Maps to navigate you to a destination it will charge a lot slower than it would otherwise, not only because of how much the phone is being actively used, but also because the temperature levels will rise under those conditions. So, for these reasons, if your phone is switched off while charging, then it's temperature level isn't likely to be an issue and, of course, it's not actively being used either.

However, based on my own testing, having an iPhone switched off while charging only reduces charge time by between 1% to 2%, so it really is negligible. You'd get close to the same benefit by putting your iPhone into Airplane Mode and switching off Wi-Fi, which may be more convenient than switching off the phone altogether.

If you're keen, the following may be of interest to you: 


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