New cuts for width‐extensional mode quartz crystal resonators

Abstract

AbstractThis paper describes various characteristics for a width‐extensional mode quartz crystal resonator. A new cut is proposed for a width‐extensional mode quartz crystal resonator, which is designated the “KT‐cut,” and clarifies its frequency characteristics, frequency temperature behavior, and electrical characteristics.First, a frequency equation is derived under the conditions that width x0 is much larger than thickness z0 but much smaller than length y0 by solving partial differential equations for a new‐shape GT‐cut quartz crystal resonator already obtained with variational principle. This is because the modified shape for the present resonator is referred to as that of the new‐shape GT resonator.Next, from the obtained equation, relationships of resonant frequency versus width x0 and the first‐ and second‐order temperature coefficients α, β versus a cut angle θ are analyzed theoretically. As a result, a cut angle θ which gives α = 0 approximately is θ = 24.5° at the reference temperature of 20°C and β has a value of β = −0.60 × 10−8/°C2. This value is then compared with the measured values of β = −1.25 × 10−8/°C2 and the absolute value in the experiments is found to be about twice the calculated value. In addition, α and β are very dependent on a thickness‐to‐width ratio Rzx (thickness z0/width x0), and have a parabolic curve versus the ratio Rzx. Thus, α and β are also found to vary with Rzx.Finally, series resistance R1, and a quality factor Q are primarily examined in the frequency range of approximately 8.7 to 32 MHz. Consequently, e.g., when f = 16.3 MHz, a width‐extensional mode quartz crystal resonator is found to be obtained successfully with satisfactory values of R1 = 89.7 Ω and Q = 180, 000.