CIKCUIT THF.ORY .IXl> >)/7:A'. / //O.V.-//. C.II.Cn.CS 701 



The Inle^rul Equation for the Indiciat Admittance 



So far \vi> have tacitly assunu-d that the iiulicial admit taiicc is 

 known. As a matter of fact its tieterminalion constitutes the essentia! 

 |)art of our prolilem. It is, in fact, the llea\isi(le ()rol)Iem, and its 

 in\estij;ation, to which we now proceed, will U'.id us directly to the 

 ( )perational Calculus. 



Hea\iside's method in iiueslij^atinj; this problem was intuitive and 

 "exiH-rimental". We, however, shall establish a \er\- ijeneral integral 

 eijuation from whii'h we shall directlx di'duce his nuthods and e.\- 

 tensions thereof. 



I.ct us suppose that an e.m.f. e'", where /> is either positi\e real 

 (luantity or complex with real part positive, is suddenly impressed 

 on the network at time t = o. It follows from the foregtjing theory 

 ih.it the resultant current /(/) will be made up of two parts, (I) a 

 forced exponential part which \-aries with time as e*", and (2) a com- 

 plementary part which we shall denote by y{t). The exponential or 

 "forced" component is simply e'''/Z{p). where Zip) is functionally of 

 the same form as the usual symbolic or complex impedance Z(ioi). 

 It is gotten from the ditTerential equations of the problem, as explained 

 in a preceding section, b>- replacing d" dl" by p", cancelling out the 

 common factor e*", and solving the resulting algebraic equation. The 

 complementary or characteristic component, denoted by y(t), depends 

 on the constants and connections of the network, and on the value of 

 p. It does not, howe\er, contain the factor e'" and it dies away for 

 sufficiently large \alue of /, in all actual dissipative systems. Thus 



Now ri'turn to formula (20-a) and replace E{t) by e^ . We get 



/(/)=^eN f'A(T)e~t^dT 

 dt Jo 



which can be written as 



j' j el" f A{T)e-!^dT-et"f .4(r)e-'>'^/r j- . 

 Carrying out the indicated difTercntiation this becomes 



J(t)=pei"f A{T)e-^\lT-pct" I' A(T)e-''^dT+A{t). (27) 



