MANUFACTURING DEVIATIONS IN CRYSTAL UNITS 



277 



anti-resonant electrical network. The curve labeled (Xi -\- X2) shows the 

 efifect of the wire resonance on the response of the crystal plate. It may be 

 observed that the apparent resonance has been reduced by a small frequency 

 decrement. The amount of frequency shift and the increase in effective 

 resistance depend on the Q of the wire resonance, its frequency location 

 compared with the resonance of the crystal plate, the mass of the wire rela- 

 tive to that of the quartz plate, and the distance from the node to the point 

 at which the wire is actually fastened to the plate. 



The slope of the frequency-reactance characteristic corresponding to the 

 mechanical resonance of the quartz plate is very steep and the efifect of the 

 wire resonance will be noticed only when an anti-resonant frequency of the 

 wire is close to the resonant frequency of the plate. The changes in resonant 

 frequency and effective resistance due to wire resonance have been measured 

 for some filter crystal units and the measurements are tabulated in Table II. 



Table II 

 Effect of Wire Vibrations on the Resistance of a Quartz Crystal Plate 



Crystal Type 



-f-5° X-Cut 



+5° X-Cut 



-18° X-Cut 



-18° X-Cut 



5th Harmonic 



GT 



Mode of Vibration 

 for Crystal 



Flexural 

 Longitudinal 

 Longitudinal 

 Longitudinal 



Longitudinal 



Resonant 

 Frequency 



12 kc 

 164 kc 

 335 kc 

 552 kc 



164 kc 



Crystal 

 Mass in 

 Grams 



.51 

 .142 

 .075 

 .068 



.98 



Maximum 

 Increase in 

 Resistance 



250% 



640% 



360% 



1100% 



370% 



(N) Specified Dimension. 

 (M) Measured Dimension. 



The relation between the length of a wire and the frequencies at which it 

 will resonate in flexural modes is expressed by the following equation: 



I = m 



vLf 



where v is the velocity of sound in the wire 



d is the diameter of the wire 



/ is the length of the wire 



/ is the frequency of wire resonance in cycles per second 



w is a number that depends on the manner in which the ends of the 

 wire can move. 

 At a particular frequency and for wire of a particular material and diam- 

 eter there is a series of critical wire lengths which must be avoided. The 

 critical lengths are these which cause the wire to present a high impedance 

 to the motion of the plate. This high impedance may be considered, from 

 the electrical point of view, as corresponding to an anti-resonance of the wire. 



