PERFORMANCE INDEX OF QUARTZ PLATES 



251 



From this expression, as previously explained, (see Section 15.81), the 

 following values for a and rj may be determined. 



M Co 



2 Co + Ca (15.96) 



ri = 1 



The anti-resonant impedance of Zab is represented by — P4 in Fig. 15.10. 

 The left hand term of (15.95) for the Pi operating point becomes 



ZABO^eiCo + Ci) = a -i- y/a^ — 1 



Substituting this value in (15.95) and solving for Z, we find 



(15.97) 



= -/mT 



jM Ks 



Co 



L Co + c, 



] 



(15.98) 



where 



1 1 / 1 



A3 - - + 2 y 1 - -2 



If Kz is expanded and all except the first two terms are neglected, Z 

 may be expressed as 



Z = -j 



M 



1 + 



+ 



(-c~:)" 



M 



(15.99) 



The next step is to obtain a similar expression to (15.99) only for the 

 minimum frequency impedance of Z, in Fig. 15.3 with p disconnected. For 



1 



this application S = r~^ and T = x . Substituting these values, as well as 

 jwLt 



those in (15.59), in (15.75), we have 



(15.100) 



j+jz 



From this equation, values for a and 77 may be determined as described in 

 Section 15.81. 



Mt 



Co + Ct 

 y] = — ^ T 



Ct 



(15.101) 



