as it is said in your booklet, the controller is a DC to AC inverter.
The current in the motor is AC, the current from the battery is DC.
The efficiency of the motor is principally a function whose main variable is the RPM. How can overvolting a 36V controller make the motor more efficient? Would you plug a 12V car inverter into an 18V battery instead of buying the right inverter?
Unfortunately, you are comparing apples and oranges there
Ok, so you asked, why would anyone want to over volt their bike unless they want to do illegal speeds?
I've been playing with the simulator and hopefully, if I've done this right, it should explain why.
Our ebike motor simulator allows you to easily simulate the different performance characteristics of different ebike setups - with a wide selection of hub motors modeled, and the ability to add custom batteries and controllers and set a wide variety of vehicle parameters you'll be able to see...
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This compares a 36V battery and a 59V battery on the same motor.
The controller for 36V is 20A, which is what you would recommend over a standard 15A controller for having more oomph for hill climbing.
The second controller is 59V at 15A, so essentially a 15A "36V" controller over volted to 59V.
When climbing a 4.5% incline, which isn't incredibly steep, the 36V powered system can't even maintain 20kph at 100% throttle.
However, the 59V powered system has no trouble maintaining 25kph at 80% throttle (which is effectively what the limiter will do, throttle the output).
It's using more Watts, but it's well within the tolerable range for these motors and it's likely that the hill will be climbed faster, so putting less stress on the motor as RPMs are higher and it is put under pressure for a shorter time.
In addition, because 100% throttle isn't necessary, efficiency is higher for the 59V system, by about 2%.
Imagine if you are going into a strong headwind, it would have a similar effect. The 36V powered system would not be able to maintain 25kph.
Most controllers labelled as "36V" these days are perfectly capable of running at 48V, or even higher. Also, just swapping to 48V is a modification that doesn't need any shunt soldering or changing other system components, so long as the user is aware of the LVC difference.
For a someone who is a complete novice about this sort of thing, it may not be recommended, but to users who have some knowledge and are well aware of what may or may not happen, I think over volting is a sensible option.
If I've not used the right parameters somewhere, please let me know, but from my experience, this does seem to back up what happens in the real world.