22 BELL SYSTEM TECHNICAL JOURNAL 



force sin (pt-\-Q) acting during the time interval T, to that calculated 

 on the assumption of a steady state in this interval. 



Calculations of these formulas are of particular interest and im- 

 portance in multiplex carrier telephone and telegraph systems where 

 they furnish a measure of the interference between channels operating 

 at different frequencies. 



In order to exhibit clearly the significance of the formulas without 

 detailed computation, consider an ideal selective circuit, for which 

 in the range wi^co^co 2 ,|Z(ico)| = Z T (a constant) and everywhere else 

 \Z(iw)\=Z s (a constant, very large compared with Z T ). Under 

 these assumptions, formulas (8) and (9) become approximately, for 

 the case when £>co 2 , 



2Z\ "*" 2tt (p - cu 2 ) (p - co:) Z\ K ' 



and 



\ 7Ti Z T {p — U2) (P — 0)i)J 



These formulas admit of some quite interesting deductions which 

 are applicable to band filters in general. 



(1) The energy absorbed in excess of that calculated in the steady 

 state basis is 



1 C02 — COi 1 



2ir (p — C02) (p — coi) Zj~ 



This is independent of the duration of the applied force and of the 

 degree to which the filter discriminates against steady state currents 

 outside the frequency range coi^co^co 2 . It is proportional to the 

 band width and inversely to the product {p — uiz) (p — ui). It follows 

 therefore that no amount of selectivity will appreciably reduce the energy 

 absorbed from a sinusoidal force of finite duration outside the transmission 

 range of the filter, below the value given above. 



(2) The fractional excess of energy absorbed is given by 



1 /Z s \ 2 co 2 — COl 



TtT\ZtJ (p — U2)(p — U\)' 



This decreases with the duration of the applied force but increases as 

 the square of the selectivity (Z S /Z T ) of the filter. Hence for forces of 

 short duration the energy absorbed may be very large compared 

 with that calculated on the steady state basis. 



