268 BELL SYSTEM TECHNICAL JOVRN.IL 



ence, and their applications to the static problem and the protection 

 afforded by selective networks against static will be discussed. 



The analysis takes its start with certain general formulas gi\en by 

 the writer in a recent paper', which may be stated as follows: 



Suppose that a selective network is subjected to an impressed 

 force (0- We shall suppose that this force exists only in the time 

 interval, or epoch, o^t^T, during which it is everywhere finite and 

 has only a finite number of discontinuities and a finite number of 

 maxima and minima. It is then representable b\' the Fourier Integral 



4,{t) = \;-,rf i/(ai)|-COsM + e(a))lf/a) (1) 



1/(0,) \- = \f <i>(t) COS co/f//T+ r /* <p{n sin ojtdt'X. (2) 



where 



Now let this force (/) be apiilied lo the network in tiie drrcing branch 

 and let the resulting current in the receiving branch be denoted by 

 I (t). Let Z {i w) denote the steady-state transfer impedance of the 

 network at frequency w/2ir: that is the ratio of e.m.f. in driving 

 branch to current in receiving branch. Further let z (i u) and cos 

 a (a,) denote the corresponding impedance and power factor of the 

 receiving branch. It ma\' then i)e shown that 



f\nr>Vdi = V.fl^,d. (3) 



and that ilie total energy IF absorbed by the reccix'iiig i)rancli is 

 giV'Cn by 



W=l/w r 1^1^ I z{io,) 1 cos a{co) ■ da,. (4) 



To apply the formulas given above to the problem of random 

 interference, consider a time interval, or epoch, sa\- from l = o to t=T, 

 during which the network is subjected to a disturbance made up of a 

 large numljer of unrelated elementary disturbances or forces, <t>i {(), 

 <t>t (t) ... 4>,, it). 



If we write 



*(/)=<^,(/)-t-</,2(0+ . . . +<i>nU), 



then i)y (1), <!>(/) can be represented as 



*(/) 



= 1/V f ' I F(a)) 1 • cos M + 0(co)h/a 

 •/(I 



' Transient Oscillations in KIcctric \\a\o I^'iltcrs, Carson and Zohcl, Hell Syslen 

 Technical Journal, July, 1923. 



