It looks like the business end - the burner controls - are the same as on mine.
I would suggest a couple of steps to determine where the fault lies:
1) First, disconnect power at the breaker panel to the range. Try a new burner coil in the left front spot. If you don't have a spare, you can use the known good right rear one, just swap them out.
2) If swapping the burners doesn't fix the problem, the fault is either bad wiring or bad control. I suspect the left front control tends to go out before the others because it tends to get the brunt of spills and boil overs.
3) Getting at the control is no fun. You have to pull all the knobs, remove a bunch of screws, gently separate the control panel L-shaped face from the guts underneath and without breaking any wires and such. I found some wire connectors were so shot that I had to replace them (spade type connectors). If the wiring looks ok, or you found bad wiring and fixed it, and the coil still doesn't work, then you'll probably need a new control. These are available online, just plug in the number you find on your old control into Google.
4) If you do get a new control, you'll need to shorten the already short screws holding it to the face panel. Otherwise they'll bottom out on the moving parts inside the replacement control and ruin it. And yes, a number of the screws on my range were so corroded/rusted that I had to replace them with ones I got from the hardware store. After I cut them down to proper size (which is about 1/4" long!).
5) In my opinion while the Compact 30 line looks cool and works well enough, it was not made with easy serviceability in mind. The coils are relatively easy to replace, but the controls are a bitch. Such is the cost of a compact design that tries to squeeze old school parts into a too small space. Perhaps Frigidaire was thinking this was an entry level range and when anything more than the elements broke, the customer would be ready to junk it and move up to a big range that would be easier to service. Sort of ignores the fact that these are drop-in units, made for built-in cabinetry, and it's no so easy to convert a built-in setup to a free standing setup. OK, end of rant.