The cigar lighter does NOT swith off when the ignition switch is turned to START. It is connected to an ALWAYS HOT branch circuit.
What does happen is that when the starter motor cranks, its pull of 150 A (+/-+) causes the battery voltage to sag, often below the threshold needed to keep attached accessory devices operational. The devices power themselves off due to insufficient power.
Adding the large value cap provides an additional capacity that will keep the voltage from sagging as much.
Big Al, either your battery is getting weak, your starter is pulling too much current, or your battery cables are degrading and increasing in resistance.
Steve
