The main reason that skinny coils are difficult to get to work is the very low coupling they provide. Any coil, where the primary resonates exactly at the same frequency of the secondary resonance, will eventually transfer all primary energy to the secondary. If the frequencies are different, not all primary energy will be transferred. Low coupling coils are very sensitive to these frequency offsets, so they have to be precisely tuned. This problem is aggravated by the slow transfer of energy to the secondary by weakly coupled coils. The arc will have time to break out and detune the coil by its capacitance. So later during the burst it becomes off tune. By tuning you have the choice of the coil either being in tune initially or later in the burst. A compromise is hard to find for low coupling. Bill Wysocks coil, Kurt wrote about, is of the magnifier type. There is no magnetic coupling to the skinny secondary. What I've written above does not apply in that case. Skinny coils can perform well then. Udo David Thomson wrote:
...Not having done any calculations on the eBay coil, my experience in working with the geometries of coils reminds me of my own failures. I, too, burned out a couple transformers on tall solenoid coils. Tesla ultimately isolated the power circuit to the primary by making the spark gap totally independent of the transformer frequency, such that he could generate pulses at whatever rate he chose. I don't know how he did it, but I recall him discussing this either in his Inventions book, or the Colorado Springs Notes. The key to his eventual success was the shortest pulse at highest energy and the right rate...
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