Is it okay to directly connect together the grounds of the logic supply and the motor supply when using a Pololu style stepper driver?
If yes/no, why so?
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(This is more of an electronics.stackexchange.com question btw).
It is not only ok, it is absolutely necessary. The systems won't operate correctly unless all the gnds are tied together. Since this is a stepper motor, you don't have to worry about noise getting back onto the power supply lines for the logic supply, since the stepper board has a decoupling capacitor(s) to keep stabilize the power to the motors and the VCC of the logic supply will be unaffected by the GND of the other supply that its tied to.
That depends on how much noise you have on your motor power supply ground. You definitely want the 100 µF capacitor to have a good high frequency response. Motors turning on and off can be noisy, and that noise can cause false clock signals in your logic circuitry if you tie the grounds together.
I could not find a schematic of the Pololu A4988 stepper motor driver, but I did find a photo of the board that includes the ground planes for both the motor and logic. On the reverse side of the board, they were not connected. They should not be connected on the driver cards.
Instead, all the motor ground lines should be brought together at a point, and the Motor supply ground connected to that point. The V-motor should also be run radially to each stepper driver and connected to the V-motor supply.
There should be a single connection point between the motor ground and logic ground. No motor current should flow through any logic ground wire or circuit board trace. The motors are switching high voltages and high currents. If these currents pass through a logic ground line, they will introduce voltage spikes into the logic which can result in unexpected operation.
The 100 μF capacitor in the schematic is good for supplying low frequency energy needs, but it should be paralleled with a 1 μF capacitor and a 0.01 μF capacitor placed as close to the radial feed points as is practical. A larger capacitor tends to have higher leakage inductance, which limits the high-frequency response. A range of capacitors in parallel will perform better.
Is it okay to directly connect together the grounds of the logic supply and the motor supply when using a pololu style stepper driver?
Most stepper drivers don't have separate ground pins for the logic and motor supplies. The two ground pins on the A4988 board you've shown are electrically connected on the board. The question is moot.