Dynamic Frequency–Temperature Characteristic Modeling for Quartz Crystal Resonator Based on Improved Echo State Network

Quartz crystal resonators are the key component of various kinds of electronic systems because they provide the reference frequency source of the system running clocks. However, the frequency stability is often affected by the temperature. Therefore, the frequency–temperature (
$f-T$
) characteristic modeling has been an important research topic in the frequency control field. The classic
$f-T$
modeling method omits the system dynamics and may lead to a large frequency compensation error in the case of rapid temperature changing. To deal with this issue, this article proposes a dynamic
$f-T$
modeling method based on improved echo state network (ESN), called residual scaled ESN (RSESN). In the proposed method, the residual modeling framework is designed for purposes of good physical understandability and high prediction precision. This framework uses the static polynomial
$f-T$
model to depict the approximated data relationship and applies the complicated network model to compensate the detailed dynamic error. To estimate the dynamic errors, one effective dynamic modeling tool, ESN, is introduced to build the dynamic compensation model for
$f-T$
characteristic of quartz crystal resonators. For a better fitting performance, the ESN activation limitations are analyzed and the scaled echo states are constructed in the improved ESN model. The modeling and testing results on the real experiment data show that the proposed method can capture the dynamic information effectively and provide better frequency deviation predictions.