4-54
(ppm)
5
3
M
DT, 0C
50
40
30
20
10
0
-10
-20
-30
-50
-100
-80
-40
-20
-0
20
40
60
80
                            AT-cut
Reference angle-of-cut (q) is about
8 minutes higher for the overtone modes.
(for the overtone modes of the SC-cut,
the reference q-angle-of-cut is about 30
minutes higher)
1
-60
-40
¥
Effects of Harmonics on f vs. T
   Shown above are the frequency vs. temperature characteristics of a resonator excited on the fundamental mode, third overtone, and fifth and higher overtones.  The f vs. T of the fundamental mode is different from that of the same resonator’s third and higher overtones.  When excited on the third overtone, the f vs. T is only slightly different from that of the 5th overtone (because the higher the overtone, the less the effects of piezoelectricity, as is explained in the Ballato reference).  The rotation of the f vs. T with overtone is as if the angle of cut had been lowered.  For example, for an AT-cut, the third overtone’s f vs. T is the same as that of a fundamental mode resonator the angle of cut of which is eight minutes lower.
   When the f vs. T characteristics are described by polynomials, it is found that the change between the fundamental and 3rd and higher overtones are due almost entirely to a change in the first order temperature coefficient, i.e., the linear terms of the polynomials.  This fact is exploited in the microcomputer compensated crystal oscillator (MCXO - see the MCXO discussions in chapter 2).  In the MCXO, the fundamental (f1) and third overtone (f3) frequencies are excited simultaneously (“dual mode” excitation) and a beat frequency f is generated such that f = 3f1 - f3  (or = f1 - f3/3). The f is then a monotonic and nearly linear function of temperature.  The resonator can, thereby, become its own thermometer (“self-temperature sensing”).  The f senses the resonator’s temperature exactly where the resonator is vibrating, thereby eliminating temperature gradient effects, and, because an SC-cut is used, thermal transient effects are also eliminated.


“Quartz Resonator Handbook - Manufacturing Guide for AT-Type Units,” edited by R. E. Bennett, prepared for the US Dep’t of the Army, pp. 77-103, 1960, AD-274031.

A. Ballato, and T. Lukaszek, "Higher Order Temperature Coefficients of Frequency of Mass-Loaded Piezoelectric Crystal   Plates," Proc. 29th Ann. Symp. on Frequency Control, pp. 10-25, 1975, AD-A017466.  Proc. copies available from NTIS.