Vectorial measurement of the 2nd harmonic response of an FBAR resonator

We have measured the phase and magnitude (“vector”) response of the 2^nd harmonic emissions of a thin film bulk acoustic (FBAR) resonator. The phase measurement is the difference in phase between the emitted 2^nd harmonic waveform and that of the incident waveform. Although the magnitude has been measured previously by various investigators [1], [2], we are not aware of a phase measurement prior to this work. The phase measurement presents a challenge because it requires an elaborate measurement setup with a sophisticated calibration procedure. Using Agilent´s non-linear vector network analyzer (NVNA) we have measured the standard S₁₁ response of a resonator as well as the vector response of its 2^nd harmonic emissions. We have observed that: (1) phase vs. frequency plots of the 2^nd harmonic have the same shape as the phase of S₁₁ and (2) the magnitude vs. frequency plots of the 2^nd harmonic are proportional to the group delay of S₁₁.