Independence of the Unloaded $Q$ of a Planar Electromagnetic Resonator From Its Shape

A figure of merit in characterizing resonance performance is the unloaded quality factor
$Q_{0}$
. Classically, this figure is calculated for planar microwave resonators in the same manner it is calculated for any arbitrary structure, by finding the electromagnetic field equations of the resonant mode and substituting them into volume and surface integrals across the concerned resonating structure. It had been observed in the literature, however, that the calculation results for some planar cases would be “surprising” in that the fields' modal functions (which are governed by the structure's shape) had no bearing on the value of
$Q_{0}$
. It is shown in this paper, using simple electromagnetic analysis, that any planar resonator that does not exhibit considerable power loss to radiation will have its
$Q_{0}$
independent from its shape, and that becomes a characteristic feature of such planar resonators. The paper presents the model predicting the
$Q_{0}$
for such resonators, and it shows that
$Q_{0}$
only depends on the resonant frequency, on the substrate materials height, and on losses in the materials used to construct the resonating structure. Only when significant power is allowed to radiate out of the structure does the
$Q_{0}$
become dependent on shape (as shape usually controls radiation). While radiation or housing power loss calculations are often complex to perform, this theory also provides a simple and accurate method of estimating such losses. In addition to its theoretical importance, this result carries an important feature for practical design as it offers flexibility in choosing shapes for planar resonators, to achieve a required value of
$Q_{0}$
at a certain frequency (wavelength) and using a given set of materials.