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When a motor, connected to a battery that has a constant voltage, spins without a load it's speed is higher than with load. I'm told that because of back emfs the current is very small when there's no load because of the higher speed. And so when there is a load the back emf is less as the motor spins slower, and so the current is higher.

What is a back emf and what is the relationship between that and the speed of the motor (well the coils inside the motor)?

Is the output energy of the motor constant, whether it has a load or not? (ignoring friction and electrical resistance) because I don't understand how the current can be higher when the motor is (or seems to be) doing more work due to the load?

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In a DC motor, when the armature rotates its coils cut the magnetic and induce a voltage in the coils. This voltage is of opposite polarity of the voltage that is powering the motor (the battery) and is called the back emf. It is modeled as a voltage source that is proportional to the speed of the motor times a constant. The faster the motor is rotating the higher the back emf.

The power of the motor is not constant and neither is the energy. The power into a motor is just the voltage measured at the lead wires times the current in the lead wires (P=V*I). The output power of the motor is just the speed of the motor times the torque. The power will be zero when there is zero torque and it will be zero when there is zero speed. In between those two points, the power will increase, peak, and then return to zero. In general, on a DC motor, speed is proportional to voltage and torque is proportional to current. As your load increases and you need more torque, the motor will draw more current. If you need to speed your motor up or slow it down, you need to raise or lower your voltage.

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