THE m'ti.nixc.vi' of sixusom.ii. cruRfLvrs ?59 



In an iMi>;itu>tTin>; stmh' of the Iniilding-up process wi- arc i)rin- 

 ( ipalK concerned with the vmclope of the oscillations, which, hy (l{J, 

 is proportional to 



The problem is therefore to determine the functions p and a and to 

 examine the effect of the applied fre(|iienc\' a; 27r and the character- 

 istics of the circuit on their rate of buiidinn-up and iimdi' of appro.ich 

 to their ultimate stcad\- \alues. 



Two pro(>ositions will now he slated which cover the building-up 

 process in the practically important cases. Since the line is assumetl 

 to bo approximateh' equalized, as rcgartls the absolute value of the 

 received current in the neighborhood of the applied frequency to 27r, 

 the building-up process depends only on the total phase angle B{w). 

 The successi\e derivatives of the phase angle with respect to w will 

 be denoted by B'{ij>), B"{u), B"'(u), B" (oj), etc. 



Case I. B"{u>) iO and y/B"{w)2\ large C()m|)ared with \/B"'{w)/ZV. 



The envelope of the oscillations in response to an c.ni.f. K cos ut ap- 

 plied at time t=o, is proportional to 



lV{l+Pr + <r'- (4) 



'ichere 



P=C(.x=)-|-5(.v=). (5) 



a=C{x^)-Six^), (6) 



.= -^^£a= /' . (7) 



and C(.v), 5(.v) are Fressel's Integrals to argument x. 



The envelope therefore reaches 50 per cent, of its ultimate steady value 

 at time t = T = B'(oi) and its rate of building-up is inversely propor- 

 tional to y/B'^w). 



I 



The curve n( Fig. 1 is a plot of the en\-clope function vd + p)'^^' 



to the argument .v and is therefore applicable to all l>pes of loafling 

 and lengths of line, subject to the restrictions noted abo\e. 



Case II. B"{(ji) =0: B"'(ui)=^0 and y/B"'{w)/'i\ large compared with 



