Applying critical-points method in the presence of phase shift due to feed line

The critical-points method is adopted for measuring unloaded Q-factor of microwave resonators in the presence of phase shift caused by the feed line. The result is calculated from four frequencies of three points in the resonator’s impedance trace. In fact, the resonator’s impedance trace rotates in Smith Chart by the phase shift. If Q-factor were gotten directly from the measured impedance including feed line rather than the equivalent impedance of the resonator without feed line, the performance of measurement will be impaired. To de-embed the phase shift, objective function was introduced to find the proper rotation angle caused by the feed line instead of calibration using extra measurement. Another advantage of the proposed method lies in the fact that no special attention is needed to distinguish magnetic coupling and electric coupling. The effectiveness of the proposed method was demonstrated by one set of simulation data and two measurement examples, namely, a low Q dielectric resonator and a high Q hollow cylindrical cavity.