An Improved Formula for Estimating Stored Energy in a BAW Resonator by its Measured S11 Parameters

Abstract

In 2008, a formula was proposed to estimate the Time Averaged Stored Energy (TASE) in a bulk acoustic wave (BAW) resonator using only the measurement of its S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> parameters and the incident power [1]. When the formula was applied to a BAW resonator, it provided an estimate of TASE across the entire resonant frequency range, but gave erroneous estimates of TASE at frequencies where spurious modes were present. In this work, we will derive an improved TASE formula that resolves this issue by accurately estimating the TASE in both Main-mode and Spurious-mode regions. The improved formula is derived by analyzing an extended Butterworth-Van Dyke (eBVD) model comprising several spurious branches. It is assumed that, at any given frequency, the overall admittance is dominated by one particular branch of the multibranch circuit. It follows that the estimated TASE of the eBVD circuit is the product of the incident power and |S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> /dω|. This improved estimate for TASE has been validated using a physically realistic 2D FEM resonator model, consisting of lateral modes, that compares the improved estimate (from S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> ) to that in which the stored energy was integrated through the FEM's mesh elements. Since quality factor (Q) employs the TASE formula, an improved Q formula is also proposed. This improved Q formula has been shown to provide a superior estimate of a resonator's Q factor in the lateral mode region, and we propose that it should replace the existing Q formula currently in wide use.