PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 177 



equation method and illustrated in Fig. 12.6 by the G-C curves. As the 

 plate reactance A'l is made small the frequency increases and approaches a 

 limiting value but does not quite reach it. This limiting value is the fre- 

 quency at which A's = 0. The dotted G-C curve shows that Reg tends to 

 lower the frequency and determines how close the limiting frequency is 

 approached. The plate circuit resistance R2 (component of Zi), if con- 

 sidered, would have a similar effect as shown by the experimental curves 

 12.9. The grid resistance Rg (component of Z2) has an opposite effect as 

 shown in Figure 12.10 because increasing Rg is equivalent to decreasing 

 the efTective resistance Reg . 



The effect of the various constants of the crystal and circuit upon the 

 oscillating frequency may be obtained from (12.24) upon substitution of 

 these constants for the reactances and resistance Reg . The equation is put 

 in a more convenient form for this purpose by Koga.^^ Equation (12.21) 

 is written, 



^'-''^'^ 23(1 + rJ, + j^ ' ° ('"« 



It is assumed that the current in the grid branch is small compared to the 

 plate current. This reduces the equation to 



^ + F3 + Z3(i t^izi) = " C^-^o) 



The admittance expression for the crystal is 

 ^1 - 3 



1 1 



uiLi — ^ 



coCi CO (Co + C\) \ / C4 Y . C0C4 



RX + coLi - ^ _ ^ T V<^« + ^V ■^"'Co + C4 



;Ci CO (Co + C4)J 



(12.27) 



Note that Koga considers the air gap capacitance C4 as a separate factor 

 but it may be included in the other constants of the crystal in which case 

 the equivalent circuit is as shown in Fig. 12.3. With the crystal con- 

 nected between grid and cathode the various circuit admittances are: 



1 1 



1 1 J_ 



Z2 Z,c JK-g 



1 



77 = ycoCa 



