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I've made a simple circuit design in EasyEDA, and I'm planning to implement it in a prototype pcb, like this one, just by soldering wires on the bottom side. I'm not sure how to proceed in route designing, but allowing the software to auto-generate the routes for me. In case of EasyEDA, it provides options for tracing on different layers, but as I'm not going to order a pre-soldered pcb (with copper traces on both sides of the board), the result of auto-routing is not quite optimized for wires. So, I wonder how to optimize route tracing when I just have the circuit design and a prototype pcb to make it work.

I'm open to correct the terminology used in this question since I don't have the proper background.

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You have more freedom when hand wiring on a proto board, because you can cross wires anywhere and as often as you want. Without knowing more about your project, the signals, the packaging, etc, here are some general thoughts.

Start with a schematic that is as clear and uncluttered as possible. Move the parts around to minimize the number of crossed signals. This can lead to a approximate parts placement map on the board that has short, direct signal runs. If the schematic shows an organized flow of signal paths from inputs to outputs, that will drive a similar flow on the board.

Things like octal registers have numbers assigned to I/O pins, but these are not rigidly defined bit values. Sometimes the out0 to out7 arrangement is the exact opposite of the in0 to in7 arrangement of whatever the outputs are driving, and flipping over the outputs now becomes a nice flat parallel run. Of course if you move the output bit positions around you have to move the corresponding inputs, and this might create more problems that it solves. You won't know until you are further into the design and layout, but it is something to keep in mind.

Similarly, when assigning individual gates in logic devices, think about their placement in the package. Often this doesn't become clear until you are in the wiring phase - you'll see that if you swap these two gates here, or swap a gate in this chip for an identical one in another chip, it reduces signal crossing and clutter. Then go back and edit the schematic to reflect the final wiring.

Since the power and ground connections to the chips presumably will not change during debugging, put all of them down first and run the signal wires over them. Don't forget the decoupling capacitors at each power pin.

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You will want to have:

  • shortest airwires to save the copper. This may be important, but next items are of more importance though;
  • easy hand soldering. The points of solder must be reachable by the iron, there should a place for wire and holding tweezers while soldering. Avoid stacking wires into single point. I am sure you will find more and better advice how to perform point-to-point wire mounting on the internet;
  • last but not least. After you solder everything, you should be able to "read" the connections so that you can easily visually check your work.

And finally. Quality of routing - P2P or PCB - heavily depends on the layout. Before starting putting components into the breadboard, you must think several times what would be the best layout to satisfy all three requirements above.

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