Unloaded Quality Factor of Transmission Line Resonators With Capacitors
A. V. Zakharov, Mykhailo Ilchenko
Abstract
This article contains analytical expressions allowing us to calculate unloaded quality factor Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> of transmission line resonators with one or several capacitors. These expressions include the Q -factors of the transmission line Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> and capacitor Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> and the sensitivity S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> of the main resonance frequency ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> of the resonator to small capacitance changing. The sensitivity function S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> (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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> . At Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> > Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> , the value of Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> of quarter-wave resonator with capacitor decreases with increasing frequency. The established expression for calculate the unloaded quality factor Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> of resonators with one capacitor allows us to solve the inverse problem and to determine approximately the capacitor quality factor Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> by measured frequency responses of tunable bandpass filters. Measured results are presented.