Frequency and amplitude modulations in crystal resonators due to transient thermal effects

We show through a theoretical analysis that when a crystal resonator is subjected to a transient temperature change, in addition to the known frequency shift or modulation, there always exists an accompanying amplitude modulation. The theory for small fields superposed on finite initial fields is employed and reduced to an ordinary differential equation with a time-dependent coefficient. An approximate analytical solution is obtained. The solution shows that the frequency modulation is a first-order effect, and the amplitude modulation is a second-order effect. The results are useful to the understanding and design optimization of crystal resonators. (c) 2014 AIP Publishing LLC.