QUARTZ CRYSTAL PLATES 529 



temperature can be explained. In addition, the dimensional ratios or 

 tuning points which yield zero temperature coefficients for a given 

 temperature can be predicted if the coupling is known. 



Referring again to Fig. 6, suppose the two coupled circuits have 

 temperature coefficients of opposite sign, circuit No. 1 being positive 

 and circuit No. 2 negative, for co2 less than wi say at the point A, co' 

 has a positive and co" a negative temperature coefficient. For a 

 value of C02 greater than co] say at B, co' now has a negative and co" a 

 positive temperature coefficient, co' and co" having interchanged roles. 

 Somewhere between therefore, both co' and co" must have had a zero 

 temperature coefficient. Returning to equation (4), if this expression 

 for CO be differentiated with respect to the temperature, regarding k, 

 the coupling as constant, and the result placed equal to zero, the 

 condition that co is independent of temperature ^* is obtained as 

 follows : 



coi2co2^(w - n) , . 



where 



m = — -TTfT = temperature coefficient of circuit No. 1, 

 coi di 



n = -jTp = temperature coefficient of circuit No. 2 ; 



C02 dl 



now let Q = n/m then equation (6) becomes 



1 -Q 



U = C02" 



~<ir 



(7) 



solving equation (7) for CO2/CO1 replacing co" by its value from equa- 

 tion (4) 



co2\^_^^(l - <2)'+i± 



coi/ 2(2 



which when k is small becomes 



2(2 



' , ^'^(1 - QY 

 ^ Q 



-y=i±Mi_^. (8) 



coi/ V(2 



This equation gives the tuning points, or the values of C02 at which 

 the angular frequencies of the coupled system, co' and co", will have 



^^ Dr. F. B. Llewellyn of the Bell Telephone Laboratories is responsible for this 

 analysis. 



