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Resonators and Q measurements at fundamentals/overtones

Question asked by rogerfreeman on Dec 7, 2011
Latest reply on Dec 18, 2011 by rogerfreeman

This is a bit long, sorry; I wanted to be as complete as I can be so as to best armour readers to come up with helpful suggestions. I hope!

Question: how do you measure the Quality Factor of a resonator of some sort, when the response of that resonator at some overtone is sufficiently small to be less than 3dB above the local noise floor?  I have an old 8753C which can't "zoom in" on a Smith chart, and the |logmag| measurements diminish in amplitude such that the 3dB points can't be seen. I might have access to a PNA-L too if I can borrow it but I need toassume I'm limited to the 8753 only.

Background: If I wanted to use a scalar analyser  in some form or other, maybe a spectrum analyser with tracking generator, I could sweep across the fundamental frequency of a resonator , measure the centre frequency and the 3dB bandwidth, and calculate Q from the resultant plot of (say) response vs. frequency.  The response could be a function of the base input impedance, and the resonator in question in my particular case is an air-cored single layer solenoid as if it were an electrically small normal-mode helical antenna. The response could be an electric field probe close to but not detuning this resonator - signal magnitude increases both at resonance and at overtone re-resonances in that case.

But if I wanted to use a VNA - and I do - I could look at the response on a Smith chart. When a resonator is driven at a resonant frequency, its quality factor is a function of the  energy lost per cycle of RF power supplied to it compared with the energy stored per cycle - a standard definition of "Q". At resonance, the resonator’s complex input impedance appears as a real value only; the imaginary components of the complex impedance cancel one another out as the magnetic storage energy and the electric storage energy are equal but opposite.   Above or below the resonant frequency the impedance will again appear complex; at the point where the resistive part of the impedance is equal in magnitude to the complex part then 

R =      ± jX

This shows that, at the frequency where the resonator’s inductive or capacitive reactance is equal in magnitude to the resistive part of the resonator’s impedance, half of the RF power supplied to the
resonator is being dissipated and half is stored as a circulating field exchange between the electric and magnetic vector fields.   The complex impedance of the resonator can now be measured by a
VNA (vector network analyser) and displayed as a Smith chart. Measuring the real input impedance Z at resonance and then find the frequencies where the +j and –j contributions are equal will allow
the 3dB points to be found for the fundamental and for the reresonances which appear at various overtones.

But - if those resonances are low amplitude such that I can't "see" 3dB changes but only (say) 1 or 1.5dB changes, how do I relate the bandwidth of the response to the equivalent Q?

This also applies if I am looking at the match into the base of the resonator - Q can be seen from rate of change of match at resonance but if the response is weak at higher overtones - how do I relate (say) 1dB bandwidth to3dB bandwidth in order to derive Q?