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At its simplest, think about how a flashlight works.
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A battery, a light bulb, and a switch are all in series.
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You close and open the switch (Arduino produces an output signal).
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Electrons leave the negative side of the battery, go through the closed switch, go through the light bulb, enter the positive side of the battery.
The bulb (motor driver) reacts to the electron flow, the bulb emits light (the motor turns). -
If the switch is open (the common GND wire is missing) the bulb does not emit light (the motor doesn’t rotate).
Master the above and you can solve many wiring problems.