MUTUAL IMPEDANCES OF PARALLEL WIRES 521 



form the second pair or secondary. We have 



O1O3 = O2O4 = 2c 

 and 



O1O2 = OxOi = ^[2c. 



The notation for the dimensions and coordinate systems is shown in 

 Fig. 4. The theoretical values of mutual inductance are calculated 

 from the geometry and electrical constants of this system by means of 

 the formulas which will be derived herein, while the measured values 

 are those obtained for this system by R. N. Hunter and R. P. Booth. '^ 



Numerical Results 



A close agreement between the values of mutual inductance com- 

 puted on the basis of the approximate formulas derived below and 

 the experimental results is shown by the curves in Figs. 5 and 6. 

 In fact, for No. 18 gauge wires, in which case the proximity effect is 

 comparatively small, the computed and measured values are indis- 

 tinguishable in Figs. 6A and 6B. Evidently the error introduced by 

 the fact that actually the line is comparatively short while theoretically 

 we assume it of doubly infinite length, is inappreciable. The drawings 

 give relative values of the real and imaginary components of the com- 

 plex mutual inductance M = Ma + iMb, for 74 inch lengths of wires 

 with vertical and horizontal interaxial spacing of 0.14 inch over a 

 frequency range of 1 to 1000 kilocycles per second. (The value 

 0.565 X 10~'^ emu. is assumed for the conductivity of the wires and 

 unit permeability for both wires and dielectric.) The solid curves 

 represent computed values and the dotted curves measured values. 

 The values shown are the ratios of Ma and Mh to the value of Ma at 

 1 kilocycle. In Figs. 5A and 6A the frequency scale is linear, while in 

 Figs. 5B and 6B it is logarithmic. The computed curves of Fig. 5 

 (obtained from formula (13) below) assume a pair of No. 10 A.W.G. 

 wires (0.102 inch in diameter) as the primary and a filamentary 

 secondary. Actually the secondary was a pair of No. 28 A.W.G. wires. 

 In the two cases shown in Fig. 6, computed from formula (14) below, 

 both pairs of conductors are of the same size; namely, No. 10 and No. 

 18 A.W.G. wires, respectively (the latter being actually 0.0410 inch in 

 diameter). 



It will be observed that we have the relation 



M = Zm/icO, 



Z,n denoting the mutual impedance, oj/lir the frequency and i the 



