Oscillation-Hijsteresis in Simple Triode Generators. 181 



equating the coefficients of terms involving cos wt and simvt 

 separately to zero, we have 



«*=«,■ (4) 



7 -j /^t 



and 77 "^ T I %>( a Sln w ^ sm lci • ^ = 0- • ■ 0>) 



From (4) we see that to the particular degree o£ approxi- 

 mation we are considering, the frequency of the free 

 vibrations depends only on the inductance and capacity of 

 the oscillatory circuit, while from (5) we see at once that 

 the stationary values (e. g., <z s ) of a are given by the real 

 roots of 



if* . 



J-Jo 



sinwt) &in lot .dt = 0. . . . (6) 



In considering the stability of the oscillations we may 

 imagine the amplitude to be altered slightly from* its 

 stationary value by some external means, and investigate 

 whether the amplitude will eventually return to or depart 

 further from its original value. Thus, if we consider a 

 change of amplitude from a s to a s + oa s , we have from (5) 



1 f T 

 Tml %{ ( a s+ & a s) sin w t} si n w t • dt 



dha s 



i f T 



- — 1 % (« s sin wt) sin wt .dt = 0, 



dSa 8 &a s i T . 



■ \a s sin iclj sin wo . t*f> — 



iha s Ba s i T , . . . , 2 , n 



-7- 1 + -7|t 1 % (a s sm wrt) sin- u;£ .dt = i) 

 do 1 Jo 



Now for any amplitude a. ? to be stable, dt must be 



negative ; that is to say, we must have 



1 f T 



Tpl % '(a s sinu^)sin 2 i^.^>0. ... (7) 



J- Jo 



Now by partial differentiation, (6) can further be written 



1 f * 



T 1 %' ( tts s * n M *) * cos ' 2 "^ ' ^ = 0' • • (6 a) 



with the aid of which we are able to write (7) more simply 

 as 



^r X , («..rin«t>ft>0 (8) 



In the case of a triode, the oscillation characteristic 



