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BELL SYSTEM TECHNICAL JOURNAL 



small effect, while variations in the angle of rotation about the X-axis have 

 a relatively large effect on these quantities. 



Mason has shown^ how the magnitudes of the piezo-electric constants 

 and the moduli of compliance for any angle of rotation may be derived from 

 their magnitudes along the principal axes of quartz. Using these equations 

 and the magnitudes for the principal axes tabulated in a recent paper^ by 

 Mason, dn and 522 have been calculated as a function of the angle of rotation 

 of the plates about the X-axis. In turn, the frequency and inductance con- 

 stants have been calculated as a function of the angle of rotation, using the 

 relations shown in Fig. 14.3. Figure 14.4 is a plot of the frequency and in- 

 ductance constants as a function of the angle of rotation for angles between 

 about —70° and +70°. It shows how the inductance and resonant 



3600 

 3500 

 3400 

 3300 

 3200 

 3100 

 3000 

 2900 

 2800 

 2700 

 2600 

 2500 



280 

 260 



■^ I— o 

 220 i^;- 



I cc c 



200 I 2 ^ 



5 c/; I- 



ISO J;; 2 g 



160 



!40 2rO<o 



120 =) 



Q 



100 1 



80 



60 



Fig. 14 

 of their an 



-70° -60° -50° -40° -30° -20° -10° 10° 20° 30° 40" 50° 60° 70° 

 ANGLE OF ROTATION 



4. — Frequency and inductance constants of X-cut quartz plates as a function 

 s;le of rotation around the X-axis. 



frequency of these plates will change with deviations in the angle of rotation. 

 The frequency constant used is the product of the resonant frequency in 

 kilocycles and the length of the plate in millimeters. The inductance 

 constant used is the inductance per millimeter of thickness of a plate having 

 a width-to-length ratio equal to 0.1. 



The change of inductance and resonant frequency with deviations in 

 angle of rotation about the X-axis is of particular interest for the angles of 

 rotation most commonly used, that is for —18.5° and for +5°. The 

 calculations indicate that f i r an X-cut plate rotated —18.5°, a devia- 

 tion of ±1° in the angle of rotation will change the inductance by +1.2% 



2 "Electrical Wave Filters Employing Quartz Crystals as Elements", W. P. Mason. 

 B. S. T. J., July 1934, equations on page 451; also in Chapter VI. 



3 "Quartz Crystal Applications," VV. P. Mason, B. S. T. J., July 1943. 



