PERFOEI'IANCE IXDEX OF QUARTZ PLATES 



241 



Now substituting the values given in (15.59) in (15.60) and (15.61) to deter- 

 mine the coordinates of the center of the circle and the radius respectively, 

 we find 



Mr 



J-r 



(15.62) 



As might be expected, the radius of the circle is equal to the real component 

 of the coordinates of the center of. the circle. This indicates that Fig. 15.10 



NEAREST ORIGIN 



Fig. 15.10 — Circle diagram for crystal circuit combinations. 



graphically represents the circle diagram where 77 = 7. The impedance, 

 Zc , is represented for a given frequency, as a vector from the origin to 

 the corresponding point on the perimeter of the circle. From Fig. 15.10 

 the following crystal properties may be deduced. 



1. The anti-resonant impedance designated as — P4 is simply cr + 

 s/c^ — if, and when evaluated equals Ri{M^ — 1) = RiM^. 



2. The resonant resistance, — P3 given by a- — -s/o-^ — ij- is equal to Ri . 



3. The maximum positive reactance between coi and C02 is represented by 

 the distance from P5 to Pe • It is given as cr — r; which equals 



2 



X 



.,). 



