Unloaded Quality Factor of Transmission Line Resonators With Capacitors

This article contains analytical expressions allowing us to calculate unloaded quality factor Q
_u
of transmission line resonators with one or several capacitors. These expressions include the Q -factors of the transmission line Q
_l
and capacitor Q
_c
and the sensitivity S
_c
of the main resonance frequency ω
₀
of the resonator to small capacitance changing. The sensitivity function S
_c
(ω) is a circuit function, which similar to the input and transmission functions. Analysis based on these formulas extended understanding of the character of the resonator unloaded quality Q
_u
behavior under frequency variation. It is shown that, under certain conditions, connecting additional varactors to the resonator does not increase its unloaded quality factor Q
_u
. At Q
_c
> Q
_l
, the value of Q
_u
of quarter-wave resonator with capacitor decreases with increasing frequency. The established expression for calculate the unloaded quality factor Q
_u
of resonators with one capacitor allows us to solve the inverse problem and to determine approximately the capacitor quality factor Q
_c
by measured frequency responses of tunable bandpass filters. Measured results are presented.