

562 BELL SYSTEM TECHNICAL JOURNAL 



where w denotes o/o),. It follows that 

 2.V 1 



/' = /- 



^c Vl-1 



and tiiat the oscillations build-up to the proximate stead\--state in a 

 time interval* r = 2N/u>c'\/\ — 'W- after the voltage is applied. 



Case I, it will be observed, does not hold for a)=o since B"{o) =o. 

 The condition that Case I shall apph' is that 



^r8]V-(l-w=)»/'2 '"'" 



{\^-2-w-yi' 



shall be sul)stantiall\- j^realer than unity. Hence Case I applies only 

 when l/v/18xV<w< 1. This however, includes the important part 

 of the signalling frequency range in properly designed lines, provided 

 that they are long (iV>100). 



In the range of applied fn'quencies, therefore, corresponding to 



l/\/l8iV<w<l, the current reaches 50 per cent, of its ultimate 



27V 1 

 steady value in a time interval /. after the voltage is applied 



^c \/\ — W 



and its rate of building-u]) at this time is proportional to 



^4N V^ 



For the non-dissipative coil-loaded line B"{b))=o when oi=o, and 

 Case II applies. Consequently when w=o, the oscillations reach 1/3 



of the ultimate stead\- \aluo at time / = 2..W, at which time their 

 rate of building-up is propcirtioiKd to 



The forfgding formul.is ha\c bci-n ^hown to in- in good agreement 

 with cxptTiuHMiial results, and ha\e been applied to the design ol 

 loaded lines in the Bell System. 



Maiiiimatical Discussion 



The functions p <ind a of equations (2) and (3) can be tornml.itci 

 as the Fourier integrals 



* It will lie iinlcd lli.it this funniila hreaks down at w=a^ or «• = 1. 



