TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 21 



It thus requires an infinite time when the applied frequency is equal 

 to the critical frequency while infinite applied frequencies build up 

 instantly. 



When the applied frequency is outside the transmission band, the 

 current subsides to its steady value, the time required being pro- 

 portional to the ratio n/oi c and decreasing as the applied frequency is 

 decreased. 



The fact that the initial value of the current is of the same order 

 of magnitude as that of steady state currents in the transmission 

 range is an outstanding feature of the process and reflects the failure 

 of the selective properties of this type of filter in the transient state. 



V. The Energy Absorbed from Transient Applied Forces 



In only a relatively few cases is the solution for the transient current, 

 in response to suddenly applied forces, reducible to a manageable 

 form, which admits of interpretation or of computation without 

 prohibitive labor. Fortunately, however, it is usually possible to 

 calculate the energy absorbed by a receiving element in a selective 

 network from suddenly applied forces of finite time duration and 

 such a calculation throws a great deal of light on the general proper- 

 ties of selective circuits in the transient state. The calculation is 

 based on formulas VI to IX of Section II. 



A particularly important example is the energy absorbed from the 

 force sin (pt+Q), applied at time / = and removed at time t—T. 

 If the energy is averaged with respect to the phase angle 0, we get 17 



Jo 2tt./o \Z(iw)\ 2 \ (u-p) 2 («>+Pr ) 



If p/2ir is in the neighborhood of the frequency which the network 

 is designed to select, this becomes approximately 



r^-ir^^mr (8) 



In the steady state the time integral of the square of the current in 

 response to the e.m.f. sin (pt+Q) during the time interval T is simply 

 T/2\Z(ip)\ 2 . The expression 



\Z(ip)\ 2 r*l- cos(u-p)T do, 



tT Jo (u-PY \Z(io>)\ 2 K J 



is therefore the relative amount of energy actually absorbed from the 



17 Here Z(iu) is the steady-state transfer impedance and the integral measures 

 the energy absorbed by a unit resistance in the receiving branch. 



