Frequency-Temperature Characteristics

      The frequency-temperature characteristic defines how the resonant frequency of the quartz crystal resonator varies in response to changes in temperature.  For both AT- and SC-cut resonators, it is found that the frequency shift due to the variation of temperature can be expressed in a cubic curve of the form:

            Df/f₀ = a₀(T – T₀) + b₀(T – T₀)² + c₀(T – T₀)³

       T is the temperature variable and T₀ is the inflection temperature, which is approximately 25°C for the AT-cut and 92°C for the SC-cut.  The coefficients a₀, b₀, and c₀ are the first -, second-, and third order temperature coefficients of frequency, and they are constants that depend on quartz properties and the angle of cut.  The above equation gives a family of curves as shown below for four values of the relative cut angle.  These curves show that a resonator can be designed to give a relatively small frequency variation over a broad temperature range.

                                AT-cut Frequency vs. Temperature Curves

                                SC-cut Frequency vs. Temperature Curves

          Both cuts are useful for temperature controlled (ovenized) applications.  In addition, since the AT-cut inflection temperature (T_o) is approximately 25°C, it is widely used for uncontrolled temperature applications such as filters and non-ovenized oscillators.  Both cuts vibrate in thickness modes that can be designed to have very low mounting losses and hence provide exceptional Q (up to 1.5 x 10⁶ at 10 MHz and 35 x 10³ at 300 MHz) and stability with time (example 1 x 10^-10/day at 10 MHz).