10 



BELL SYSTEM TECHNICAL JOURNAL 



The formulas for the indicial admittances of all the band pass 

 filters are approximate, as stated above, and are deduced on the as- 

 sumption that the band width is narrow. Practically, however, as 

 regards the essential deductions drawn from them, they are not so 

 restricted but are applicable to the case of relatively wide bands. 

 (See Appendix I.) 



There are, of course, an infinite variety of band pass filters; the 

 ones investigated in the present paper are, however, representative 

 and the conclusions drawn from a study of them are, in their 

 general aspects, applicable to all types. 



3. Band Pass Wave-Filter, Type L\C\L 2 C 2 , Fig. 3. 



w 

 A n (t) = — r J 2 „(y) sin X 



where x = u m t; y = wt/2; and the filter elements are given by 



Li = 2k/w, L 2 = wk/2w z m , 



C l = w/2ku' i m , C 2 = 2/wk. 



iU ..2C, 



o Ifflft^ 



(3a) 



Fig. 3 



This is the "constant k" type of filter and, as will be noted, the ele- 

 ments are so proportioned that LiCi=L 2 C 2 = l/io 2 m , and Li/Cj = 

 L 2 /d = k\ 



From the properties of Bessel functions discussed in Appendix II, 

 it follows that A n (t) is very small until y^2n. For values of y>2n, 

 the character of the function is clearly exhibited by the following 

 approximate formulas, although these are not sufficiently accurate 

 for the purposes of precise computation. 



w l — o~ 



A„(t)= — r- h 2n * — cos (q 2n y—Q 2n ) sin x (3b) 



u m R \ ^y 



w i o 



= k — ;*!»!- [sin (x-q 2n y + Q 2n )+s\n (x+q 2 „ y-Q 2 „)] (3c) 



Z<Ji m K \ Txy 



