PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 



165 



the crystal constants and paralleling capacitances which are usually involved. 

 They will be defined and their method of use and application in oscillators 

 will be pointed out, 



12.10 Solution by Differential Equations 



The most direct method of determining the oscillating conditions in a 

 circuit is to analyze the differential equation for the current in some particu- 

 lar branch of the circuit. The relations existing between the coefficients 

 determine whether the current builds up, dies out, or is maintained at a 

 constant value and frequency. Unfortunately the equations resulting from 

 the application of this method to the crystal oscillator circuit are quite 

 complicated. However, lower order differential equations result from the 



Fig. 12.4 — Equivalent circuit of oscillator with crystal connected between 

 grid and plate 



application of this method to similar electric oscillator circuits, and certain 

 qualitative information obtained from the latter is applicable to crystal 

 oscillators. Thus Heising's analysis of the Colpitts and Hartley circuits 

 gives much information directly applicable to the Pierce and Miller types 

 of crystal oscillators. From this the circuit conditions necessary for oscilla- 

 tions to exist and the effect of certain circuit variables upon the frequency 

 are ascertained. The more complex qualitative view is given by Terry^ 

 who shows the relations of the coefficients of linear differential equations 

 of the 2nd, 3rd, and 4th orders, and applies them to the analysis of three 

 common types of crystal oscillator circuits. The resulting equations, 

 together with certain qualitative information regarding their interpretation, 

 are repeated here. In making this analysis the grid current is disregarded 

 and the static tube characteristic is considered linear. 



