Mutual Impedances of Parallel Wires * 



By RAY S. HOYT and SALLIE PERO MEAD 



This is a theoretical paper relating to circuits of straight parallel wires 

 traversed by alternating currents under such conditions (frequency of the 

 alternating currents, diameter and spacing of the wires) that the resulting 

 non-uniformity of the current distribution is sufficient to play an important 

 part in determining the mutual and self impedances. The paper deals 

 primarily with the mutual impedances; but incidentally the self impedances 

 are dealt with almost as fully, except that no numerical calculations are 

 made for them. 



Part I is mainly a discussion of the physical nature of the mutual and self 

 impedances in the generalized manner necessitated by the non-uniformity 

 of the current distribution. It deals with wires which are short enough com- 

 pared with the wave-length so that the complicating effects of propagation 

 are negligible and so that the current in each wire can be regarded as an ag- 

 gregate of filamentary currents. 



Part II establishes, by recourse to electromagnetic wave theory, calcula- 

 tion formulas for the mutual and self impedances per unit length of a pair 

 of long straight parallel transmission circuits forming a square array. 

 Values of the mutual impedance are calculated over a frequency-range of 

 1 to 1000 kilocycles per second, for three cases of the circuits, and are com- 

 pared with measured values. 



Introduction 

 ^ I ^HE concept of the mutual impedance per unit length between two 

 -*- straight parallel filamentary conductors is well understood by 

 engineers, and its calculation formula is simple. This mutual im- 

 pedance is a pure reactance (directly proportional to the frequency), 

 the induced electromotive force being in phase quadrature with the 

 inducing current. 



In the case of open-wire circuits, even when operating with carrier 

 currents of very high frequency, the mutual impedance can be calcu- 

 lated with high accuracy by regarding the wires as filamentary. 



For cable circuits, however, the foregoing statement is not true, 

 because of the close juxtaposition of the wires. In such circuits the 

 wires may be termed "thick," meaning that their diameter is ap- 

 preciable compared with their interaxial separation. Depending in a 

 complicated manner on the conductivity, permeability, diameter, and 

 interaxial separation of such wires, the frequency may easily be so 

 high as to render the filamentary formulas for the mutual impedance 

 of even straight wires quite inaccurate and unreliable. In such cases 

 it is necessary to consider the current distribution over the cross-section 



* The two parts of this paper are distinct, though complementary. Part I 

 was written by Ray S. Hoyt, Part II by Sallie Pero Mead. 



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