TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 9 

 where 



u c = 2tt times critical frequency below which the filter attenuates, 

 Ci = l/2« e fe, k =VL 2 /C u 



L 2 = k/2cc c , w c = \/2VL2~Ci- 



2C 4 „C| 



u 



Fig. 2 



The symbol D~ m denotes multiple integrations, repeated m times and 



»»(«) = /oW - ^ /:(*) + (m) ( ^ ~ 1} Ux) + • . + (-l)-/.,(*). 



A large amount of time and effort have been devoted to an attempt 

 to reduce this and other forms of solution (see Appendix I) to a form 

 in which its properties would be exhibited by direct inspection, but 

 without success. Numerical computations and curves must, there- 

 fore, be largely relied upon in the study of the high pass filter in the 

 transient state. 



For sufficiently large values of x (x>4n 2 ) the ultimate behavior of 

 the filter is shown by the asymptotic formula 



A n {t) «.(-!)" \ ^- cos (x-tt/4). 



(2b) 



Band Pass Wave- Filters. 



In all the band pass types of filters discussed below the transmission 



band lies in the frequency range between wi/27r and co 2 /27r so that the 



band width is (a>2 — o)i)/2ir. We shall write v coi«2 = w m and a>2 — o>i=Z0. 



For each type the filter elements are determined by the parameters 



o> m , w and a third parameter 7 k which may be so chosen as to fix 



the magnitude of the impedance of the filter. 



7 The parameter k is equal to the characteristic impedance, both mid-series and 

 mid-shunt, at mid-frequency of the confluent band, "constant k" type of wave- 

 filter. See Theory and Design of Uniform and Composite Electric Wave-Filters, this 

 Journal, Jan., 1923. 



