The Measurement of the Performance Index of Quartz Plates 



By C. W. HARRISON 



15.00 Introduction 



THE theory of the general behavior of crystals in oscillator circuits has 

 been described by I. E. Fair\ In Fair's paper as well as in others^, it 

 has been pointed out that in the neighborhood of the operating frequency a 

 crystal is equivalent to the circuit shown in Fig. 15. lA. The crystal 

 possesses two resonant frequencies, a series resonant frequency determined 

 by the efifective inductance, Li , and effective capacitance, Ci , and an anti- 

 resonant frequency determined by these same elements plus the paralleling 

 capacitance, Co . This paralleling capacitance is the static capacitance 

 between electrodes of the crystal and any capacitance connected thereto by 

 the crystal holder and lead wires within the holder. The dotted resistor, 

 Rl , shunting the equivalent crystal circuit represents the effective shunt 

 loss of the holder. In the ideal case and in many practical instances this 

 loss is negligible. 



It is rather difl&cult to express the circuital merit of a crystal quantitatively 

 in a single term such as has been found useful for inductances and capacitances. 

 It is customary to express the circuital merit of these two elements in the 

 form of the ratio of reactance to resistance. That is, for an inductance 



(15.1) 

 a: 



and for a capacitance 



(15.2) 



For filter purposes, the () of a crystal involving only the inductance, Zi, and 

 resistance, Ri , of Fig. 15. lA is adequate to express its usefulness in certain 

 parts of a filter network, but for oscillator purposes it is insufficient. At 

 frequencies other than the series resonant frequency the paralleling capacitor 

 Co together with the associated shunt loss of the holder enters into the 

 determination of a crystal's performance. The term Q therefore is not com- 

 pletely indicative of the crystal performance. There has been devised, as 

 pointed out in Fair's paper\ a term called "figure of merit" for a crystal 



^I. E. Fair, "Piezoelectric Crystals in Oscillator Circuits," this issue of the B. S.T.J. 

 -K. S. Van Dyke, "The Electrical Network of a Piezo-Electric Resonator", Physical 

 Review, Vol. 25, pp. 895, 1925. 



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