82 BELL SYSTEM TECHNICAL JOURNAL 



the operating frequency without affecting the currents and voltages 

 in the remaining parts of the circuit. 



It is well known that the frequency of such a piezo-electric oscillator 

 is less affected by changes in battery voltage than is the frequency 

 of the ordinary, nonstabilized electric oscillator. However, the bat- 

 tery voltage does influence the frequency of the piezo-electric oscillator 

 to an extent which is undesirable for certain accurate types of work. 

 It therefore becomes useful to apply stabilization to the piezo-electric 

 oscillator. It will be shown that the stabilization may be accomplished 

 by adjusting the size of the output tuning condenser to such a value 

 that the impedance of the output circuit bears a certain critical rela- 

 tion to the impedance of the crystal, while at the same time, the circuit 

 as a whole fulfills the conditions necessary for the existence of oscil- 

 lations. 



The same kind of stabilization is, of course, applicable to an elec- 

 tric oscillator having analogous relations between the input and output 

 impedances. Thus, it is often possible to stabilize the Hartley oscil- 

 lator by moving the connection between the filament and coil to differ- 

 ent positions on the coil, until that one which gives the proper ratio 

 of input to output impedances has been found. In the case of the 

 Hartley and Colpitts oscillators, however, it is more often preferable 

 to stabilize by the special circuit arrangements illustrated in Figs. 1 

 to 7, while, on the other hand, the tuned-grid, tuned-plate type of 

 circuit lends itself readily to stabilization by adjustment of the output 

 circuit. 



Numerical expressions for the proper impedance relations may be 

 obtained by noting that the circuit of Fig. li may be generalized into 

 the circuit of Fig. 1 by regarding Z4 and Z5 as zero, while Z2 comprises 

 the whole input network which may consist of various arrangements 

 of coils, condensers, grid leaks, and the like, and, in a similar fashion, 

 Z] comprises the whole output network. The mathematical analysis 

 given in connection with Fig. 1 may therefore be adapted to fit Fig. 14 

 immediately, and in place of (6) and (7) we have the two expressions: 



Xo(r,Xi -f r^Xaj + ixVgX.X^ = r^AV + r,X{\ (11) 



Xoir,r, - X1X2) = - XrX,(Xr + X^). (12) 



The requirements of (11) are: 



''X2 



uXo + Xq — Xi 



Xo — An 



(13) 



