CIKCIIT TIIEOKY .IXH OrER.ITIOX.U. CIlXriAS 607 



iiUfrprt'tation of optTational im-tho;ls anil lead to an auxiliary formula 

 from which the rules of the Opirational Calculus are direclly de- 

 (luril)le. 



^B) The solution of problem (2) carries with it the solution of 

 problem ('.i) and also serves as a basis for the theory of alternating 

 currents. 



(C) The solution of problem (2) leails direclly to an extension of 

 circuit theory to the c.ise where the network contains variable ele- 

 ments: i.e., circuit elements which \-.iry with time and in which non- 

 linear relations obtain. 



Problem (2) is therefore the fundamental problem of circuit theory 

 ami the formula which we shall now derive may be termed the fimd.i- 

 mental formula of circuit theory. 



Consider a network in any branch of which, sa>- branch 1, a imit 

 e.m.f. is inserted at time t = o, the network having been pre\ iously in 

 equilibrium. By unit e.m.f. is meant an electromotive force which 

 has the value unity for all positive values of time (/^o). Let the 

 resultant current in any branch, say branch n, be denoted by Ani{t). 

 Aki [t) will be termed the indickil admittance of branch n with respect 

 to branch I — or, more fully, the transfer indicial admittance. 



The indicial admittance, aside from its direct ph\sical significance, 

 plays a fundamental role in the mathematical theory of electric cir- 

 cuits. In words, it may be defined as follows: The indicial admittance, 

 .•l„i(/), is equal to the ratio of the current in branch n, expressed as a 

 time function, to the magnitude of the steady e.m.f. suddenly inserted 

 at time l = o in branch I. It is evidently a function which is zero for 

 negative values of titne and approaches either zero or a steady value 

 (the d.c. admittance) for all actual dissipative systems, as / approaches 

 infinity. It may be noted that, aside from its mathematical determi- 

 nation, which will engage our attention later, it is an experimentally 

 determinable function. 



We note, in passing, an important property of the indicial admit- 

 tance A}k{l), which is deducible from the reciprocal theorem:- this 

 is that Ajk{l)=Akj{t). That is to say, the value of the transfer 

 indicial admittance is unchanged by an interchange of the driving 

 point and recei\ing point. It is therefore immaterial in the expression 

 A;k(t) whether the e.m.f. is inserted in branch j and the current 

 measured in branch k, or vice-versa. In general, unless we are con- 

 cerned with particular branches, the subscripts will be omitted and 

 we shall simply write •-!(/), it being understood that any two branches 



• Exceptions to this relation e.xist where the network contains sources of energy- 

 such as amplifiers. These need not engage our attention here. 



