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Look at the online help for the basics of:
Drawing paths and assigning strokes to them.
Appearance Palette's Add Stroke and Add Fill commands. Understand that youi can apply more than one stroke and/or fill to a single path.
For wiring diagrams, use the ability to apply multiple strokes to a single path. Each stroke can have its own weight. For the diagonal stripes, use a PatternBrush, and make the PatternBrush one of the multiple strokes applied to a path. Store such combinations as Graphic Styles so they can be applied to other paths with one click.
The sample below shows a set of Pattern Brushes I use for wiring in isometric illustrations, but the principle is the same. The Brushes would work fine for flat schematics. Wires like 2 and 5 would not require using Brushes; they could be done by simply applying two strokes to a path in the Appearnace palette. (They are Brushes in my sample, in order to include the terminal ends.)
Wire 4 is an example of using a Brush to create a stripe. The wire has two strokes, with a different Brush applied to each. The top stroke has the stripe Brush applied; the bottom stroke has the simple shaded Brush (with trimmed insulation ends) applied. Again, the bottom Brush would not be necessary if it were not for my intent to include the trimmed ends. In a schematic, that wire could be done with a set of three strokes: A thick stroke at the bottom to create the outer edge color; a middle stroke somewhat narrower to create the middle color; and a top stroke with the spiral stripe Brush applied.
...it emerged that it was the better tool for the job...
Yes. Please don't use Photoshop for wiring diagrams. Such diagrams need to be resolution-independent vector artwork. Think of your downstream users. Wiring schematics often become large and complex, requiring large sheets with relatively small text. A nice PDF containing vector artwork can be zoomed indefinitely to trace a circuit with legible text; and the file size is usually much smaller than if it were a large raster image. (Tech illustrators hate it when technical drawings like schematics default to a dumbed-down raster format. Just makes it a royal pain to work with and modify.)
I may add that you should also look at the Symbols section of the manual. Many components can be definde as symbols, which makes for a leaner Illustrator file. Convert geometry you will need more than once into a symbol and place it as often as you need it.
Right, I've got it to work, and I've traced all the wires with the pen and used either one or two colored strokes for each.
Next I needed to delete the original wires in the underlying TIFF image, so I used the Blob Removal Tool. It seemed to work, but the path for this tool has been saved and gets in the way when I try to adjust the upper layers.
Is there a better way to remove sections of the image below?
Just delete the image. Use layers to organize your work. Put the TIFF on one layer, wires on another above that layer, and components on another layer. Text labels should get their own layers, too. Just hide he layer with the TIFF image or delete the layer it you are positive you don't need to see the image anymore. Of course, now you also have to delete whatever you drew with the Blob Brush.