C0.\5TANT FREQUENCY OSCILLATORS 

 Equations (1) may be rewritten in determinant form as follows: 

 (/-p + Zi + Z5) + (Zi + Z„.) - (Z„. + tir„) 



71 



+ (Zi + Z„.) 



Zo 



(Z, + Z J 



- (Z2 + Z J 

 (r, + Z2 + Z4) 



= 0. 



(3) 



This determinant form of (1) follows immediately from reducing (1) 

 to three equations. 



Me 



•Z4 



-wvw 



MWVv- 



(^1 



Z6' 



Z| 



vw\^ 



Z3 



Fig. 1 — Equivalent circuit network of Hartley or Colpitts-type oscillator. 



In accordance with the theory of the operation of oscillators, dis- 

 cussed in the appendix, both the conditions necessary for oscillation to 

 exist and the frequency of oscillation may be found from (3). That is: 



(r, + Zi + Z5)Zo(r, + Z2 + Z4) + (Z: + Z„0(Z2 + ZJC/zr, + 2ZJ 

 = ZoZ,„(/.r, + Z„) + (Zi + Z^nr, + Zo + Z4) + (Z2 + Z^Y 



(rp + Zi + Z5). (4) 



The next step is to express each of the generalized Z's in the equiva- 

 lent form of (R + iX) where i stands for the imaginary quantity, 

 A — 1, and both R and X are real, representing, respectively, resist- 

 ance and reactance. A great simplification results when it is recalled 

 that the circuits external to the vacuum tube are assumed to have 

 very little resistance, and that practically all of the losses in the net- 

 work are caused by the tube resistances, r^ and fp, so that these two 

 are the only resistances which need be retained in the analysis. With 

 this understanding, (4) becomes: 



[_rp^i{Xi + X,)yXo[_r,+i{X2+X,):\-{X,+Xj{Xo + X,n)((Jir, + 2iXm) 

 = -XoX„,(Mr,-f 7X.) - {X,-{-X„,y-lr„-hi{X2 + X,)2 

 - (T2-f XJ2[,^+,-(Xi+X5)]. (5) 



