So I've designed this CPU with purely logical components (basic logic gates), now how would I go about converting it into a circuit? I know that logical gates can be implemented using transistors, but it seems pretty cumbersome to do the equivalent conversion for every gate (there are a couple hundred). In addition to this, adding the required resistors and other passive components due to physical constraints seems a bit cumbersome. Is there a secret to how this is generally done?
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For practical purposes of getting it working on your desk, you probably want an FPGA.
In order to make sure it works, you should simulate it first with a program like Modelsim. This enables you to iron out the bugs before buying any hardware.
Texas Instruments, et al. have almost unbelievably extensive stables of digital logic functions available which can be assembled into pretty much anything you want.
While the above answers are correct, in the late 70's through the early 90's sometimes simple CPU's were constructed from TTL logic. While you may or may not want to get into such a project, it is interesting concept. Because the CPU is spread out and hand wired you can probe around and see it in its entirety rather than just as a small black box.
Here's one that I found that is along the lines of what I've seen in the past:
Here's another that I think would be a mess to debug but still fun to work on:
Good luck with whatever you choose.