PERFORMANCE INDEX OF QUARTZ PLATES 



227 



The above equation involves only the original approximation that maxi- 

 mum current indicates resonance. If R,i and C,i are selected such that 

 R.i < < X a and if .4 ^ equals unity, then 



PI 



Cp eu 



A 



- Ra Ca 



(15.28) 



(c, - c.y 



From this expression it can be seen that the PI measurement is inde- 

 pendent of calibration of both the Cp and Ci vacuum tube voltmeters, pro- 

 vided that the same voltmeter scale factors are used for the "operate" and 

 "calibrate" conditions. The absolute calibration then depends on the 

 magnitude of A, Ra , Ca , C/, , Ct and Cx . The "multiply-by" factor that 



1? r' C 



is to apj)ear on the C,. dial is determined by the magnitude of -—^ -— . 



{Ct — Cx) 



Accurate evaluation of this quantity by capacitance and resistance meas- 

 urements is a little difficult since the denominator represents the square of 

 the difference of two small capacitances. When Ct is large, the evaluation of 

 this factor is helped considerably. This "multiply-by" factor may be 

 experimentally determined by a voltage measuring means which permits 

 an evaluation of this factor to a higher degree of accuracy. Substituting 

 (15.11) in (15.28) we have 



PI = 



A 



Ra Ca 



{• * !)■ 



(15.29) 



Ck \ Cs 



Now by shorting the crystal socket terminals (Fig. 15. 3A) and applying a 

 voltage ei at the ei generator terminals of external origin (the crystal oscil- 

 lator circuit itself may be used if self-excitation is provided), the current ii 

 through the capacitors Ck and Cs is given as 



eicoCsCk _ eiojCk 



" :) 



ti = 



Cs + C, 



1 + 



c. 



(15.30) 



Now the voltage, eo , across the series capacitor, C/. , is 



62 = 



wCfc 



(15.31) 



The ratio of Ci/co may be expressed as given in (15.32) when (15.31) is sub- 

 stituted in (15.30) 



Ck\ 



Ci \ Cs 



If (15.32) is substituted in (15.29), we find 



RaCa 



PI = 



( 



V 

 6pc 6i _] 



-^ \A 



Ck \_ei_ 



Ci 



(15.32) 



(15.33) 



