STABILIZED FEEDBACK OSCILLATORS 463 



and the value of /Xj8, when tan ^ is zero, by 



The angle <p may be either zero or 180 degrees when tan <p is zero, but 

 which it is may be determined by the sign of m/3o- If this is positive, 

 the phase angle is zero, the phase angle of /8 being then 180 degrees. 



For the simplified case of the pure reactance coupling network the 

 expression for ^t/3 in terms of the image parameters takes different forms 

 depending upon whether the frequency lies in a transmission band or 

 in an attenuating band. At frequencies within a transmission band 

 the image impedances are resistive and the transfer constant is a 

 pure imaginary quantity indicating a phase shift without attenuation. 

 Denoting this phase shift by xp, equation (6) becomes 



Q ^ - aR2^K,K2 



''^ (RiKi + R2KO cos lA + j(KiK2 + R1R2) sin i/' ' 



(11) 



from which 



{K1K2 -\- R1R2) . . 



^^" ^ ^ iR,K2 + R.KO ^^" ^- (^^) 



At the cut-ofif frequencies, equations (11) and (12) become indetermi- 

 nate. At frequencies in the attenuation bands, the transfer constant 

 6 takes the form 



e = A+j'^, (13) 



where A denotes the attenuation and n is an integer the value of 

 which may be different for the different transmission bands of a com- 

 plex network. The image impedances are pure imaginaries, but their 

 product is real and may be either positive or negative. Simplified 

 forms of equation (6) may be written down for any particular case. 



Frequency Stabilization 



From equation (9), which gives the value of the phase angle of /i/8, 

 it is at once evident that an essential condition for zero phase angle is 

 that 



D - RiRoDn, 22 - 0. (14) 



Since the quantities D and D\i, 22 are functions of frequency, equation 

 (14) determines the frequency or frequencies at which the phase shift 

 is zero and hence determines the oscillation frequency. The equation 



