Modified Modeling Method of Quartz Crystal Resonator Frequency–Temperature Characteristic With Considering Thermal Hysteresis

The frequency-temperature (${f}$ -${T}$ ) characteristic of quartz crystal resonators is an important topic closely related to the frequency-deviation compensation in the design of cost-effective oscillators. Traditional studies depict the ${f}$ -${T}$ characteristic as a polynomial function (usually a cubic function). However, this omits the thermal hysteresis phenomenon and cannot provide a very accurate frequency compensation. To handle this issue, this article is to propose two modified ${f}$ -${T}$ characteristic modeling methods with considering the thermal hysteresis. First, to reflect the thermal hysteresis property of quartz crystal resonators, a two-directional ${f}$ -${T}$ model is designed by introducing two individual submodels for describing the increasing and decreasing temperature stages, respectively. Furthermore, to integrate the submodels and provide the more accurate frequency-deviation estimation, a holistic ${f}$ -${T}$ characteristic model based on double-hidden layer extreme learning machine (DHL-ELM) is presented. Different from the basic ELM model, two hidden layers, including one deterministic nonlinear mapping and one random nonlinear activation layer, are constructed for a better description of ${f}$ -${T}$ characteristic. To validate our studies, an experiment system is applied to obtain the testing data of the real crystal resonators, and the applications demonstrate the effectiveness of the proposed methods.