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BELL SYSTEM TECHNICAL JOURNAL 



(1) are plotted as abscissas and the reactance components as ordinates. 

 To every value of the impedance, then, there is a corresponding 

 point, and to the values of the impedance over a range of variation 

 of some element, or over a frequency range, there corresponds a locus, 

 in the complex plane. This locus may be labelled at suitable points 

 with the corresponding \alue of the variable. So labelled, it com- 

 bines into one the curves which are usually plotted to show separately 

 the variation of the reactance and resistance components or to show 

 separately the variation of absolute value and angle. 



The use of the complex plane is not new: it is the basis of most 

 of the vector diagrams for electrical machinery. The character- 

 istics of both smooth and loaded transmission lines have also been 

 displayed by its means. Its application to electrical networks, how- 

 ever, is not common, and it is a subsidiary purpose of this paper 

 to illustrate the fact that the properties of certain networks, which 

 have complicated characteristics if exhibited in the usual way, may 

 be shown quite simply in the complex plane. This simplicity, com- 

 bined with generality, is attained by application of theorems con- 

 cerning functions of a complex variable which are immediately avail- 

 able. 



THii FuND.\Mii;NrAL Hyu.vrioNs 

 The impedance measured in branch 1 of any network is 



S = R-\-iX = ^ 



(1) 



where A is the discriminant of the network, (.itiier in terms of branches 

 or w independent meshes. - 



Assigning the reactances Zs and Za to meshes 2 and 3 



Rn Rn Ri3 ■ Rm 



Ri\ JR22 + Z2 R23 R'ln 



Rn Rsi Ri3-\-^i Ran 



A = 



R.n 



Rn-2 



R„, 



Run 



(2) 



where Rj^ is the resistance in mesh j and R,k{ = Rkj) that common 

 to meshes j and k. 



»Ste: (;. A. Cam[il)ell. Transactions of thu A. I. K. K., M), 1911, pages 873-909, 

 for a complete discussion of the stilution of networks by means of determinants. 



