If I have to use Arduino to program a microcontroller for my project, can I burn the program separately to a controller that I can mount on my PCB or do I have to use the UNO R3 printed circuit board for the project?
I don't need a new UNO R3 card for every single project, do I?
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As to your question, the answer is yes. You can plug a DIP packaged micro into a Uno, program it using the IDE then unplug it and use it on a PCB
You can burn the program to the chip without the board...
But you need to add some components (crystal, caps) and you might need another chip to load the program...
Why make this more difficult than needed?
I’m a complete novice, and I was able to design my own custom PCB that contained the ATMEGA-328P chip (and a few basic supporting components and programming connection). The is certainly manageable.
Alternatively, with the right Arduino, you can program the Arduino (as normal) and then remove the ATMEGA DIP-chip and socket it into your board (again, with a few very basic supporting components).
There are many reasons why someone would want to move beyond the Arduino development board, once they are past the development stage.
Maybe the user wants a smaller footprint, or maybe their project needs a batch of boards (much less expensive to socket a single chip vs an entire Arduino)
Oh Sorry - but thank you for your help and your answer.
From op's question it was not clear whether he would like to use a standard board to program a chip and then transfer that chip to a non standard board or that he would like to program a chip without a board...
The first is well possible (as already ecmxplained above). The latter is pretty complicated and not worth the effort (personal opinion).