164 BELL SYSTEM TECHNICAL JOURNAL 



upon the oscillator as consisting only of inductances, capacitances, and 

 resistances, along with the vacuum tube. The crystal is replaced by the 

 proper circuital elements arranged as in Fig. 12.3. This circuit or equivalent 

 of the crystal is that of a series resonant circuit having capacitance parallel- 

 ing it. The circuit will show both phenomena of series resonance and 

 parallel resonance, the two frequencies being very close together. By 

 making suitable measurements on a crystal, the magnitudes of the in- 

 ductance, resistance, and the two capacitances can be determined. It is 

 usually found that the series inductance is computed as hundreds or thou- 

 sands of henries, and the series capacitance is a small fraction of a micro- 

 microfarad. The magnitudes of the inductance and capacitance are beyond 

 what it is possible to construct in the usual forms of building inductances 

 and capacitances. This accounts for its superior frequency control 

 properties. 



Although reducing the crystal to an equivalent electrical circuit provides 

 one notable step in understanding the performance of the crystal oscillator, 

 it does not readily lead to a full understanding. The electric oscillator in 

 itself is not fully and completely analyzed in all its ramifications, although 

 it has been under study for over 25 years. These studies have been mathe- 

 matical and experimental in character, but in all cases it appears there have 

 been approximations of some kind, made because the variable impedance 

 characteristics both of the plate circuit and the grid circuit of the tubes 

 did not lend themselves readily to a rigorous analysis. The earlier investi- 

 gations assumed a linear relation between grid voltage and plate current 

 and assumed constant plate impedance. Later investigations brought in 

 further elements and further variables, the different investigators attacking 

 the problem in different ways and attempting to prove different points. 

 By this means a large number of factors in oscillators have been ascertained 

 to a first degree of approximation so that a qualitative review of the per- 

 formance of the'electric oscillator is very well known. It is the quantitative 

 view upon the first order magnitude which is still difficult or uncertain. 

 This is particularly true of the crystal oscillator because of the slightly 

 different circuit. 



It is proposed, therefore, in this paper to cover briefly a number of the 

 studies on crystal oscillators so as to point out the different modes of attack 

 and the different behavior points in the oscillators which the various investi- 

 gators have studied. After covering these points, there will be discussed 

 the frequency control properties of the crystal and the frequency stability of 

 crystal oscillators. The performance of the crystal in the oscillator with 

 respect to activity is then treated. There will be introduced two new yard- 

 sticks for measuring or indicating crystal quality, one called "figure of 

 merit" and the other called "performance index." These are related to 



