PERFORMANCE INDEX OF QUARTZ PLATES 233 



which is a plausible assumption when Ct > Co , and we assume that 

 — - — = We then, 



. _ ^?_ 



^'^ ^ ""' ■ (15.43) 



[C02 — 0) 

 C02 — OOlJ 



The relationship between Ri and Re does not involve i?i, and may be 

 expressed in terms of Co and Ci instead of frequency. Neglecting Rl to 

 determine the relationship between capacitance and frequency in the P.I. 

 meter as derived in Section 15.92, we find 



. [-/-a 





where, 



M 

 Pi 



(15.44) 



(15.45) 



(■ - f) 



If Pi » 2 then the expression between Re and Ri may be written 



R, = i?x (l + |j (15.46) 



The restriction that Ci > Co may be removed if the error between (15.42) 

 and (15.46) is taken into account. This error may be expressed as 



Per Cent Error in R. = 100 [^] [ i + Vf+VP f] ^^^'^^^ 



The resonant resistance, i^i, together with (15.46) provides a means of 

 checking the P.I. meter. The magnitude of Ri may be determined by the 

 substitution method and from this, the value of Re calculated. Fig. 15.5 

 represents the agreement between the P.I. meter and those expected from 

 resonant frequency measurements. 



The remaining crystal constants Li and Ci (Fig. 15.1) may be evaluated 

 from the measurement of wi , wo and Co . The resonant frequency, coi , 

 is defined as 



col = ^ (15.48) 



The anti-resonant frequency, C02 , is defined as 



