MUTUAL IMPEDANCES OF PARALLEL WIRES 517 



may be present, nevertheless the forms of these expressions do not. 

 Thus, so far as the forms of the expressions are concerned, the two 

 wires A and B need not be alone but may be any two of a system of 

 parallel wires A, ■ • -, B, • • , D carrying arbitrary currents Ia, • • -, 

 Ib, ■ • ■ , Id respectively; still further, A and B may even be any two of 

 the parallel longitudinal parts of which any wire may arbitrarily be 

 regarded as composed. 



Equations (8) and (7) respectively show that the mutual and self 

 impedances of "thick" wires have the following significance: 



The mutual impedance between two wires is equal to the sum of the 

 weighted mutual impedances from every filament in one wire to every 

 filament in the other, the weighting factor of any filamentary mutual im- 

 pedance being the product of the corresponding two relative filamentary 

 currents (the Ts)}'^ 



The self impedance of a wire is equal to the sum of the weighted mutual 

 impedances from every filament to every other filament, including the 

 weighted mutual impedance from every filament to itself, the weighting 

 factor of any filamentary mutual impedance being the product of the 

 corresponding two relative filamentary currents (the J's)}^ 



Or, more briefly, the self impedance of a wire is equal to the sum of the 

 weighted mutual impedances from every filameyit to every other filament 

 and to itself. 



Several matters of interest regarding the "thick" wires A and B 

 (Fig.l) will next be discussed, mainly from the physical viewpoint 

 corresponding to equations (7) to (10). 



Reciprocity of the Two Mutual Impedances 



Since Zp^ and Z^p are unquestionably equal, because they relate to 

 filaments, comparison of formulas (8) and (10) shows that the mutual 

 impedances Zab and Zba between the wires A and B are equal. The 

 same conclusion follows also from the first italicised paragraph above, 

 which is based on formulas (8) and (10). 



Complex Nature of the Mutual and Self Impedances 



Although every mutual impedance between different filaments is a 

 pure reactance which is directly proportional to the frequency, never- 

 theless the mutual impedance Zab between the wires A and B has in 



1" In other words, the mutual impedance of two wires is equal to the sum of the 

 weighted mutual impedances between all of the various filaments taken in pairs 

 each pair consisting of one filament from each wire. 



'1 In other words, the self impedance of a wire is equal to twice the sum of the 

 weighted mutual impedances between all of the various filaments taken in pairs, 

 plus the sum of the weighted self impedances of the filaments. (The weighting 

 factor of the self impedance of any filament is evidently the sc|uare of its relative 

 filamentary current.) 



