Quartz crystal resonator model measurement and sensitivity analysis

A quartz crystal resonator that can be modeled as a one- or two-port electrical circuit is discussed. Extraction of the equivalent electrical parameters can be accomplished using scattering parameters measured near the resonant frequency of the crystal. Due to measurement errors inherent in the network analyzer, the extracted parameters will vary from run to run. These measurement inaccuracies lead to greater errors in the electrical parameters as the magnitude of S₁₁ (one-port model) or S₁₂ (two-port model) approaches one. This work presents results of measured and simulated data and a first-order sensitivity theory. An expression for the error in the extracted series resistance is derived. Simulated and actual crystal data are compared to the theoretical values for the one-port model