After 60 years: A new formula for computing quality factor is warranted

A new formulation for computing the unloaded quality factor (Q) of a resonator or of any terminated electromagnetic or piezoelectric cavity from its measured S parameters is proposed. This formulation, based on the work of Dicke, and Bode, computes Q at a continuum of frequencies from below the resonant frequency (fs) to above the anti-resonant frequency (fp). Sixty years ago Dicke and Beringer derived a pair of formulations for Q in terms of the measured impedance (Z) and admittance (Y). The new formulation has two important advantages over Dicke's formulations: (1) Even for resonators that are free of spurious resonances - whose S-parameters are near ideal -- Dicke's Z and Y based formulations yield erroneous Q values over a range of frequencies (called ldquodead zonesrdquo) in the vicinity of the frequencies at which the functions Z and Y are singular (at fp and fs respectively). Since the S parameters of such an ideal resonator do not have any singularities, the new Q formulation provides reliable Q values over the full range of frequencies. (2) For non-ideal resonators exhibiting spurious lateral-mode resonances in certain frequency ranges, both the new and Dicke formulations report Q values that are clearly erroneous since they oscillate from positive to negative. Dicke does not explain why his formulation breaks down in such regions nor have we found such an explanation in the literature. In contrast to this, this work explains why the S parameter based Q formula is invalidated over regions of frequency space in which ln(mag(S)) fluctuates wildly. We show that in such regions that pre-conditioning of the S-parameter data with a Gaussian window average is useful in yielding a meaningful value for Q. We propose that the S parameter based Q formulation, being devoid of ldquodead zonesrdquo should be used as a standard by which piezoelectric resonators fabricated in different technologies e.g. FBAR, SAW, BAW be compared.