PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 191 



frequency of the oscillating loop, it is apparent that this effective reactance 

 will alter the frequency. However, this reactance is small when the im- 

 pedance of the circuit is low at the harmonic frequencies and is zero when 

 the external circuit is a pure resistance. 



12.50 Efficiency and Power Output of Oscillators 



In many applications of erystal oscillators the efficiency and power output 

 are important factors. These are not treated here but reference is made to 

 the work of Heising which covers this aspect for various electric oscillator 

 circuits. Much of the analysis is directly applicable to crystal oscillators. 



12.60 Frequency Stability of Crystal Oscillators 



The equations for frequency show that the frequency is governed some- 

 what by the amplification factor, the grid resistance and internal plate 

 resistance of the vacuum tube. Since these factors are functions of voltages 

 applied to the tube and amplitude of oscillation, they cannot be considered 

 fixed. If the frequency change resulting from these variables is great, the 

 frequency stability is said to be low, and if very little frequency change 

 takes place the frequency is determined principally by the circuit constants 

 and the frequency stability is said to be high. 



Llewellyn^ shows how it is possible to compensate for the change in plate 

 resistance by the proper value of circuit elements. This was done by deter- 

 mining the relations necessary for Rp to be eliminated from the frequency 

 equation. It is sometimes helpful in designing very stable oscillators for 

 frequency standards to select circuit elements which will reduce the effect 

 of plate voltage changes on the frequency. It is more the purpose of this 

 section, however, to show Llewellyn's derivation of the equations for fre- 

 quency stability which have not heretofore been published and from them 

 point out the characteristic of crystals which enable them to stabilize 

 oscillators. 



12.61 The Frequency Stability Equation 

 The steady state oscillating condition is 



Mj8 = 1 (12.51) 



In general /3 is a function of the frequency, the amplitude of oscillations, 

 and of some independent variable V. This independent variable is the one 

 for which it is desired to stabilize the frequency. It may be the potential 

 applied to the tube, or it may be a capacitance located somewhere in the 

 circuit. (3 depends upon these three variables thus: 



M/3 =/(/>, a, F) (12.52) 



