Driving Stepper motor using ULN2003a IC

For most simple applications, you needn't have two separate grounds. For heavier applications or where there is need for greater cleanliness of the power supply, it is generally considered a good practice to keep the grounds separate.

Being able to join two grounds together is usually possible (Though not always).

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In the case of batteries, any two batteries are isolated from each other for most practical purposes. Once you connect them (or the circuits they are part of) together, the point at which the connection is made specifies how the voltages are related to each other. For conventional applications, you could just call the connection point the combined circuit's main ground reference point, and connect the negative terminals of the batteries to it.

In the case of something connected to a wall power supply, such as with using a transformer, a similar logic can be applied. The transformers isolate the outputs from the input (wall power), and so the DC voltages derived by rectification are effectively isolated from each other. Again, you can connect the two negative points (the 'grounds') together to make a common reference point. Most commercial power supplies that feed off of wall power will output isolated DC voltage due to regulatory requirements. If the power supply also uses the third pin (Earth) on the wall socket, it may be using that to bias the output (fix it to a certain point relative to the Earth at your house / office / what have you), in which case some care must be taken to connect them together. If the power supplies produce single ended DC output (5V and GND, or 12V and GND), then you can pretty safely assume that if the supply is fixing the output voltage relative to Earth, it most likely will be connecting the output ground (0V) to the Earth. Make sure to make the connection on your circuit as well anyway, because the output may also be isolated.

The problem comes if you are using an unusual power supply combination. If you are using a power supply which produces 5V,5VGND; 12V,12VGND, and the two supplies are for reason not isolated and 5VGND is not connected to 12VGND, then you cannot connect them directly. Which means you will have to most likely treat the two grounds as truly separate.

Treating the two grounds as separate is also necessary if, say, you are driving high power motors which may interfere with your Atmega, or if you are also trying to make sensitive analog measurements, or for a variety of other reasons. If you do have to use separate grounds, though, then your Atmega outputs cannot be directly sent to the L293. The signals have to be referenced against the L293 ground (which we'd usually refer to as the power ground) and not the Atmega ground, and these two are not necessarily at the same voltage (the potential difference between the grounds cannot be guaranteed to be zero unless you actually connect them together). This 'translation' can be done using ICs such as optocouplers and other methods of galvanic isolation, which means that the signal is translated from one 'ground' to another without using any usual conducting connection.

Your 12V linear regulator is most likely running out of headroom, due to the fact that you have a fixed turns-ratio step-down transformer as its input source (which is of course gets rectified and smoothed). As the AC voltage coming in drops, so must the AC voltage coming out of it. Once the voltage gets too low, your linear regulators don't stand a chance.

You can either replace the whole power supply with a switcher as the other answer suggested, or use a switcher to generate a regulated 15VDC bus to feed the regulators (instead of the transformer and rectifier scheme). If your loads are noise sensitive, keeping the linear regulators may give better performance (at the cost of lower efficiency).