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What kind of device can measure a coil's self resonance frequency?
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graybeard:
I would use a series through measurement on a network analyzer looking for the dip in S21.

Another method would to be to measure the coil at two different frequencies with a Q meter and calculate the self resonance.   

A third method would to find the resonance by loosely coupling it to the coil on a grid dip meter and tuning to find the resonance. 

A fourth method would to be to connect the coil to two different small capacitors and hit it with a current pulse (through a large resistor connected to a pulse generator) and monitor the frequency of the ringing with a scope.  Don't forget to account for the capacitance of the scope probe.  You can then calculate it from the two measurements. 
G0HZU:

--- Quote ---I would use a series through measurement on a network analyzer looking for the dip in S21.
--- End quote ---

For many coil types I think this can give a different result compared to a shunt S11 measurement. A lot depends on the way the coil is wound.

I'd expect a Tesla coil to be a large solenoid with lots of turns. Generally speaking, a solenoid is best modelled as a distributed structure, a bit like a complicated transmission line. This means that the classic model of an inductor in parallel with a capacitor will only be valid up to (say) a third of the parallel resonance frequency of the solenoid coil when measured using an S11 measurement with a VNA. Beyond this the simple/classic lumped LC model will show errors.

The parallel resonance frequency using a shunt S11 measurement using a VNA often doesn't coincide with any null seen in a series S21 measurement of the same solenoid. Often the null seen in S21 will be very rounded and the centre of the null will be slightly higher in frequency than the parallel resonance frequency. Depending on the way the coil is wound the difference can become fairly significant. This is because the solenoid is way more complicated than it first appears. It can often be modelled fairly well using a transmission line based model but the model has to be quite elaborate in order to be fairly accurate. A simple transmission line model won't be good enough.
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