PERFORMANCE INDEX OF QUARTZ PLATES 



245 



From this expression we can again see the same form as (15.52) except that 

 the coefficients of a, jS,7 and 5 are modified by the capacitance, Ci , in parallel 

 with the equivalent crystal circuit. It has been previously explained that 

 the frequency of oscillation in the generalized oscillator is determined by the 

 anti-resonant frequency of the impedance, Zab , and that this impedance 

 represents the exact definition of P.I. P.I. therefore is represented in 

 magnitude hy — Pi in Fig. 15.10. 



Equation (15.75) enables us to write the expression for the input im- 

 pedance, Zi , for the P.I. meter directly and furthermore, it enables us to 

 compute the error produced by the adjustment of Cr to minimum impedance 

 rather than unity power factor as assumed in the derivation of (15.28). 

 This error obviously will be a function of M, Co, Ct and Rl- Writing the 

 impedance expression for Zi directly, requires a little modification in that 

 the crystal is shunted by two elements, Cx (the crystal socket capacitance) 

 and Rl (the holder loss) as well as having the combination in series with 



the capacitor, C3, where C3 = - — , — ~. Shunting the crystal with Cx 



Cs + Ck 



modifies (15.77) only in that Ct = Cx', from this we can use (15.75) directly 



in a form readily adaptable to the determination of our original coefficients, 



a", jS", 7" and 8". Adding a series capacitor, C3, and a shunt resistor, 



Rl, modifies (15.77) in a manner specified by (15.75). Here, 



■ C. 



a = T -\- jMi 



J 



T = Rl 



'■('4:)| 



C. 



7 = -M^^+i 



<-ci) 



s = 



1 



JC0C3 



(15.78) 



Substitute these constants given by (15.78) in (15.75) then a", /3", 7" and 

 5" become, 



MrXr 



= a -\- S 



G+".)=h-=('+l)-^'] 



/3 + ^ 5 + 



0-D 



= 7+^ = 



f)=[ 



M 



+ j I Mr + 



( 



+ j 



MCxX, 



Co 



T 



X, 



1 



+ ^Y.,(1 + ^^ 



n/J -^L^.J 



> (15.79) 



+ R.. 



. Co R 



;] 



('*'i)-\k-<{'*m 



