198 BELL SYSTEM TECHNICAL JOURNAL 



copper current and the sheath current are nearly in phase, but with 

 increasing frequency, the copper current lags behind the sheath current, 

 until at high fretjuencies the two currents approach a queidrature 

 jihase relation. 



It may be said that at high frequencies the current in the loading 

 material is practically all "wattless" current, in the sense that it 

 contributes very little to the energy delivered to any receiving device 

 connected to the line, but it dissipates energy, of course. At 10 

 kilocycles, for the 19-gauge loaded wire, the current carried by the 

 magnetic sheath contributes only 2 per cent of the useful current 

 (see Fig. 3) ; yet 75 per cent of the energy loss occurs in the sheath 

 (see Fig. 2). 



The difference in phase between the component currents in wire 

 and sheath is explained by the consideration that the reactance of a 

 given filament of current is proportional to the magnetic flux external 

 to it. In the copper, therefore, the elementary current paths have a 

 small resistance, but a large reactance, due to the fact that nearly all 

 the magnetic flux is in the loading material. Near the outer surface 

 of the loading material, on the other hand, the current paths have 

 less internal reactance, but the resistance is large. 



This brings the discussion to Fig. 4, which shows how the amplitude 

 and phase of the current varies over the cross-section of the bi-metallic 

 conductor for direct current and for 2 and 10 kilocycle alternating 

 currents. For the 19-gauge loaded wire, illustrated, the "skin effect" 

 in the copper is seen to be very small, the alternating current dis- 

 tribution being practically uniform, as for direct current. At the 

 boundary between the copper and the magnetic material, the current 

 amplitude suffers a discontinuity, but the phase is continuous. The 

 discontinuity in the current amplitude conforms to the law that the 

 component of electric force along the conductor must be continuous 

 at a boundary, which requires that the ratio between the current 

 amplitudes on the two sides of the boundary must equal the ratio of 

 the conductivities of the two materials. The current density dis- 

 tribution over the cross-section of the magnetic sheath is uniform for 

 direct current, of course, but for alternating currents, the density 

 increases and the phase advances abruptly toward the outer surface 

 of the sheath. 



APPENDIX A 



The impedances ' which appear in equation (1) in the bod\- of the 

 paper as the coefficients of the currents are given b\-: 



^ See abo\'e noted paper (reference i) tor ihc development of tliese forniulas. 



