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Today, I ran my e-bike out of battery for the first time. Interestingly enough, the bike didn't actually turn off, but automatically moved the mode from "Eco" (light pedal assist) to "Off" (no pedal assist). The screen stayed on, telling me my MPH, etc, so clearly, the bike wasn't completely out of battery.

I continued to pedal with the screen on until I got home. I then plugged the bike into its charger.

Is this bad for the bike? I can't find information anywhere about this... even in the bikes manual, it says nothing about it.

I called the company that made the bike, and they seemed to be uncertain, but ultimately said it wasn't a big deal... they actually suggested it might be good for the bike to do this?

Is running an e-bike out of battery bad for the bike?

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    You did it right. What's bad is if you keep turning it off and back on again trying to squeeze out another 30 seconds of eco and really push the battery all the way to the bottom. – Affe Jun 28 '19 at 22:24
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Most rechargeable batteries have a certain voltage threshold below which they will get permanently damaged. Li-ion cells should never drop below 3.0V (slightly higher at 3.2V for LiPo). Smart batteries have integrated systems monitoring remaining voltage and will have a way to at least tell you they're getting too low.

In your case, it seems the system has a way to shut off electric assist but keep low-energy functions going for some time. I'm surprised the manufacturer could not give you a definitive answer, but I would expect the battery to be fine.

If you can get the specifications of your battery pack, you can divide the total voltage by the nominal voltage of one cell. For li-ion, that's 22.2V in a 6S pack so 6 cells of 3.7V in series. Multiply 3.0V by 6 and you get 18.0V. So if your E-bike has a display with remaining voltage and you're still above 20V, you shouldn't have a problem. If the system's smart, it will cut off the assist well before it gets to a critical point anyway.

Just make sure to recharge the batteries and not let them low when in storage.

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    Additional data point for older e-bikes, SLA batteries should never be discharged below 85% of nominal. So for a 12V battery that would be "discharged" at 10.5V , and 24V-> 21V etc. Continuing to power your motor at that discharge level will absolutely kill the batteries, Source personal experience. – Criggie Jul 1 '19 at 2:21
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The bike will certainly not let you run the battery down past safe levels, however if you avoid letting it drop below 10% or at least 5%, your battery will last a bit longer.

The biggest danger is if you don't use your bike for 6 months or a year and don't charge the battery in the meantime. The cells can self-discharge below the safe level, and you may not be able to use it again. This happened to one bike I bought used. The owner bought it and basically never used it, and then it wouldn't work. I ended up having to buy a new battery.

Some eBikes (particularly older ones) will be extremely hard to ride with the power off, while most newer ones are not too difficult.

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