266 BELL SYSTEM TECHNICAL JOURNAL 



14.22 Deviations in Linear Dimensions 



In the case of X-cut plates, the length dimension is used to control the 

 location of their resonant frequency. The length is lapped to its final dimen- 

 sion after all other processes have been completed, so that this dimension 

 will be such as to compensate for the effect of any other deviations that may 

 have occurred. The sensitivity of this adjustment depends on the mode of 

 vibration. For plates vibrating in their extensional mode, the resonant 

 frequency is inversely proportional to the length, as shown in Fig. 14.3, 

 while for plates vibrating in their fiexural mode, the resonant frequency is 

 inversely proportional to the square of the length. The amount of the ad- 

 justment required depends on the magnitude of the frequency errors that 

 may have been introduced due to deviations in the width or in the angular 

 orientation of the plates or due to still other causes. The magnitude of 

 such frequency errors, in turn, depends to a considerable degree on the angle 

 of rotation of the plates. For example, it was shown in Section 14.21 that 

 a deviation in the angle of rotation of a +5° plate changes its resonant fre- 

 quency more than ten times as much as a similar deviation in a —18.5° 

 plate. It must be noted that the adjustment of length compensates for 

 frequency errors only and that errors in inductance or temperature coefficient 

 may be increased by such adjustment. 



Deviations in the thickness dimension principally affect the impedance 

 level of the plates. As shown by Fig. 14.3, the inductance is directly and 

 the capacity inversely proportional to the thickness. In the case of GT-cut 

 plates the thickness dimension is important also because it controls the 

 location of the most prominent unwanted resonances, which arise from 

 vibrations in thickness flexure. However, plates are designed to avoid 

 critical thicknesses and small deviations from the nominal thickness will 

 not usually result in plates having unwanted resonances. 



Deviations in the width dimension affect the equivalent electrical charac- 

 teristics appreciably. The effect of deviations in width on the frequency 

 of X-cut plates vibrating in their extensional mode can be deduced from 

 the curves of Fig. 14.5. The curves show that this effect is more pro- 

 nounced for larger values of the width-to-length ratio where coupling with 

 the width extensional mode becomes appreciable. For a — 18.5° plate with 

 a width-to-length ratio of 0.8 an increase of 1% in the width dimension 

 will decrease the frequency by about .04%. For a -1-5° plate with a 

 width-to-length ratio of 0.4 an increase of 1% in the width dimension 

 will also decrease the frequency by about .04%, but for a ratio of 0.6 the 

 decrease in frequency will amount to 0.13%. 



Similar information is available for crystals vibrating in width flexure 

 from the measurements published by Harrison^ and the calculations pub- 



^ "Piezo-Electric Resonance and Oscillatorv Phenomena with Fiexural Vibration in 

 Quartz Plates", J. R. Harrison, Proc. L R. £.,'Dec. 1927. 



