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The LEGO EV3 Hardware developer kit says that the first and second pin of output port provides a current of max of 700 mA with 1 A but the motors appendixes don't have the maximum current that can I (safely) supply to the motor.
What is the max current that the motors can safely drain?
It is important to consider not just the current itself but also the amount of time that the current is applied. Since the motors are not 100% efficient, there is energy lost to heat. If there is too much heat, the motor will be damaged.
So, while it is no problem to run the motors at maximum current by stalling the motor for a short amount of time (a few seconds), it is not OK to stall it for a long period of time. I accidentally did this once with an EV3 medium motor and I noticed the problem after a few minutes because I could smell a burning smell coming from the motor. I have also been at LEGO conventions where motors are running all day long. Even though they are not stalled, some of the motors eventually quit working and have to be replaced.
The motors themselves do have internal protection that is supposed to cut off power to prevent overheating problems due to too much power for too long (see R7 in both the large and small motor schematics in the hardware developer kit and datasheets). But based on my experience, it doesn't seem like a good idea to completely rely on this protection.
So unfortunately, I don't have an exact answer in amps. But if you are running the motor for short periods of time with plenty of time in-between to cool down, it should be OK no matter what the current is.
For reference, these are my best guess for the data sheets for R7.
The amperage drawn by any motor depends on the voltage supplied and the resistance (load) on the axle. Even if you provide the motor with more ampers than it could safely take, it will only consume as much as the voltage and load require.
As for the maximum value that each motor can draw when supplied with 9V, potentially unlimited amperage and given an unlimited load to move (stalled), see the comprehensive component characteristics comparison compilation by Philo. Here we can see that when given 9V, the EV3 large and medium motors will consume 1.8 A and 780 mA respectively when stalled, so those are the maximum values they can draw from any 9V power supply.
See also this Q&A over at Electronics.SE about motors, the relevant values and their relations.
