The CDI system doesn't need a cap "across the switch" because it is an electronic switch and doesn't erode over time like a mechanical one exposed to air does. Additinally, the CDI doesn't have to oscillate to work. It is a pulse modulator type system....and you only get one half cycle (one unidirectional pulse) because the solid state device doesn't (materially) conduct backwards like a set of points can do.
The capacitor, charged to the 300 (or whatever volts) is a bang waiting to happen. It stores the energy that goes to the coil to operate the plug.
This stored energy (measured in Joules....Watt-seconds)is defined by the equation 1/2 C x (V squared) where C is the value of the capacitor in farads and V is the value of the stored voltage in volts. Notice V is squared, so increasing the voltage twice gives you 4x the stored energy, hence, with a good dielectric, you can reduce the space required to do the job. That's why you charge the C to 300 volts, rather than ring it out at 12V.
The switch (transistor, FET, MOSFET, SCR) dumps the cap into the coil and the secondary voltage takes off like a rocket (because the coil is unloaded....no current flowing). This is how you get your 40,000 volts in 40 microseconds which 2 cycle outboard mfgrs. just love. Also, note the gap on the plug is now much wider, like .045 -.060 rather than .030-.035 of the old conventional systems.....course you need better insulated coils and spark plug wires now with the higher voltage.
When the voltage hits the firing point of the gap, it arcs over and clamps the secondary of the ignition coil to a very low voltage....say 50 volts and it stays at that voltage till the coil has released most of it's energy (it got from the cap) which becomes insufficient to maintain the arc across the plug gap and the plug shuts off resulting is some residual oscillations in the coil as the core eats what energy is left....not much.
And that's the way it is. Just so happens this is the principle that pulsed RADAR systems for aircraft and ships have operated on for many 10's of years.
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