510 BELL SYSTEM TECHNICAL JOURNAL 



III 



Representation of Impressed Field by Equivalent 

 Electromotive Forces 



In the present section we shall start anew with the problem dealt 

 with in Section II, and attack it by a synthetic method, as distinguished 

 from the analytical method employed there. While the results so 

 derived are all deducible from the analytical theory and formulas of 

 Section II, the synthetic or physical mode of attack has important 

 advantages in engineering applications, in giving a physical picture 

 of the phenomena and an intuitive grasp of the problem. In many 

 cases it enables us to deduce results very simply, when the physical 

 picture is well in mind, whereas the purely analytical solution may be 

 laborious. 



The essence of this synthetic method consists in replacing the 

 known electric field impressed on the physical system by a set of 

 equivalent electromotive forces; the current at any point in the 

 system can then be calculated when the transfer admittances between 

 that point and the points where the electromotive forces are situated 

 are known or calculable (as is often the case in practical applications). 

 For, considering any linear system containing any number m of elec- 

 tromotive forces inserted at any points 1, • • • m, it is known, from the 

 principle of superposition, that the current Ih at any point A is a linear 

 function of all the electromotive forces, that is, 



m 



h= T. AnkEu. (40) 



The coefficient Auk is called the 'transfer admittance' from k to h, 

 because Ahk is equal to the ratio of h to Ek when all of the electro- 

 motive forces except Ek are zero. If the system contains any uni- 

 lateral element (such as a one-way amplifier, for instance), Ahk is 

 not in general equal to Akh- 



Fundamental Set of Equivalent Electromotive Forces: 

 General Formulation 

 Consider any system of parallel wires situated in an arbitrary 

 impressed field, with any number of localized admittance bridges 

 between wires or between wires and ground. (Evidently, distributed 

 bridged admittance can be analyzed into infinitesimal elements, and 

 these can be regarded as localized.) The cross-sectional dimensions 

 of the wires are assumed to be small enough so that the axial (longi- 

 tudinal) impressed electric force is sensibly constant over each cross- 

 section. 



I 



