TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 15 



and their instantaneous frequency (measured by intervals between 

 zeros) 



2 



Vi - (2«M/) 2 . 



The oscillations are therefore ultimately of cut-off or critical frequency 

 w c /2ir in all sections, but this frequency is approached more and 

 more slowly as the number of filter sections is increased. 8 



The indicial admittances of the band pass filter, type L1C1L2C2, are 

 shown in Figs. 11, 12 and 13 for the initial, the 3rd and the 5th sec- 

 tions. These curves show not the actual oscillations but their envelopes. 

 That is to say the curves must be multiplied by sin w m t to give the 

 actual oscillations. The "mid-frequency" o} m /2ir may therefore be 

 regarded as the "carrier frequency" which is modulated by the rela- 

 tively low frequency oscillations shown in the curves. 



Comparison of the formulas for the indicial admittances of the 

 band filters of type L1C1C2 and L1C1L2 with that of type L1C1L2C2 

 shows that these curves are applicable to the two former types pro- 

 vided the number of sections is doubled and the phase of the oscilla- 

 tions of frequency w m /27r is correctly modified. 



Referring to Figs. 11, 12, 13 it will be observed that the oscillations 

 are small until time t = 4n/w; consequently they are transmitted with 

 an apparent velocity of propagation roughly equal to w/4 = 1/2 V L1C2 9 

 sections per second. 



After time / = 4w/w, the low frequency oscillations shown in the 

 curves are of increasing frequency and diminishing amplitude, their 

 envelope being roughly equal to 



— I JL. 



The actual oscillations are analyzable into two frequencies 



i(»*+|>Jl-(4»M)°) and l(^-^ 1-iin/wtY) 



so that the ultimate oscillations are of the two critical frequencies 



7r {u m -\-w/2) and 7r (u m -w/2). 

 Zir Z7r 



8 For curves showing the indicial admittance of the low pass filter when n is very 

 large, the reader is referred to Transient Oscillations, Trans. A. I. E. E., 1919. 



9 For types L\C\Ci and LiGL 2 the velocity in sections per second is double this. 

 This corresponds to the fact that two sections of these types are approximately 

 equivalent, as regards their selectivity, to one section of type L1C1L2C2. 



