I'.l.l'.CIRIC CIRCirr IIII'.OKV 37S 



inj^ till' conwrgencc of the s )liili()n nci'd not prrwiil ihf use of ihe 

 inelliod in a great nuiiu' ])r()l)k'ms wIutc |)h\sical considerations 

 furnish a safe guide. For example this method oi solution, when 

 applied to the i)r()blem of the induction generator, discussed above, 

 leads to the usual simplified engineering theory of the induction 

 generator and motor, besides exhibiting effects which the usual treat- 

 ment either ignores or fails to recognize. 



Non-Linear Circuits 



In the previous examples discussed, the variations of the variable 

 circuit elements are assumed to be specified time functions, which is 

 the same thing as postulating that these variations are controlled by 

 ignored forces which do not explicitly appear in the statement and 

 equations of the problem. We distinguish another type of variable 

 circuit element, where the variation is not an explicit time function 

 but rather a function of the current (and its derivatives) which is 

 flowing through the circuit. For example, the inductance of an iron- 

 core coil varies with the current strength as a consequence of mag- 

 netic saturation. The equation of a circuit which contains such a 

 variable element (provided it is a single valued function) may be 

 written down in operational notation 



Z/+0(/)=E(O, 

 or 



Z/=^£(O-0[/(/)]. (311) 



In this equation Z is, of course, to be taken as the impedance of 

 the invariable part of the circuit, the indicial admittance of which 

 is denoted by the usual symbol A{t). 



Equation (311) may be interpreted as the equation of the current 

 /(/) in a circuit of invariable impedance Z when subjected to an 

 applied e.m.f. £(/) — 0[/(/)]; consequently, by aid of our fundamental 

 formula, /(/) is given by 



I{t)=^J^A{t-y)E{y)dy~j^£A{t-y)4>{I{y)\)dy. 



The first integral is simply the current in the invariable circuit of 

 impedance Z in response to the applied e.m.f. E{t); denoting this 

 by /(/), we have 



I{t)=Io{t)-jJ^A{t-y)4>{l{y)\dy. 

 This is Q. Junctional integral equation, the solution of which is gotten 



