STABILIZED FEEDBACK OSCILLATORS 471 



DissiPATivE Feedback Networks 



The foregoing sections deal with non-dissipative feedback networks, 

 but the general ideas set forth are applicable to certain types, at least, 

 of dissipative networks. For such networks the transfer constant 5 is a 

 complex quantity and may be represented by 



^ = A+j^p, (34) 



where A denotes the attenuation and rp the phase constant. When the 

 phase constant is equal to 180 degrees the hyperbolic functions of the 

 transfer constant take the values 



sinh 6 = — sinh A, 



cosh 6 = — cosh A, (35) 

 and 



tanh 6 = tanh A . 



The image impedances will generally be complex, but, if they can be 

 made to become purely resistive at the frequency for which the phase 

 constant is 180 degrees, the value of fx^ at that frequency then becomes 



^'^^ (piP2 + RxR^) sinh A + (i?iP2 + i?2Pi) cosh A ' ^ ^ 



in which pi and p2 denote the resistive values of Ki and K2. Since this 

 is necessarily a positive real quantity the phase angle of fxjS is zero and 

 remains zero independently of variations of Ri and R2. Oscillations 

 occurring under this condition are therefore theoretically stable. 



Two Tube Oscillators 



The stabilization of the single-tube oscillator depends on the circum- 

 stances that the tube itself produces a constant phase shift of 180 

 degrees and that feedback networks can be devised to produce a phase 

 shift of this value which is independent of the terminal resistances. 

 Phase shifts of 90 degrees which are independent of the termination 

 can also be provided by means of reactive networks and this property 

 may likewise be made use of in the design of stabilized oscillators. For 

 this purpose it is necessary to have an amplifier which will give a uni- 

 form phase shift of 90 degrees over a fairly wide range of frequencies in 

 the neighborhood of the oscillation frequency. A suitable amplifier 

 may consist of two vacuum tubes coupled in tandem by a simple shunt 

 inductance or a simple shunt capacitance. The second tube should be 



