Table 531 449 



THE CALCULATION OF THE HIGH-FREQUENCY RESISTANCE 

 OF CONDUCTORS 



(By Dr. F. W. Grover, Consulting Physicist, Bur. Standards, 1931.) 



The resistance of a conductor to high frequency alternating currents is not the same as 

 it offers to direct or low frequency currents. The linkages of flux with the inner portions 

 of the conductor are more numerous than with the outer portions. That is, the reactances 

 of the inner filaments are greater than those of the outer filaments. Consequently, the 

 current density decreases from the outside toward the center of the conductor. 



This tendency of the current to crowd toward the outer portions of the cross section 

 becomes more pronounced the higher the frequency, and at very high frequencies the 

 current density is sensibly zero everywhere except in the surface layer of the conductor. 

 This phenomenon is called the " skin* effect." It causes an increase in the effective resis- 

 tance of the conductor over its resistance to a direct current. 



What is of interest in the calculation of the high frequency resistance is the resistance 

 ratio, the quotient of the resistance at the given frequency by the direct current resistance. 

 The resistance ratio depends upon the distribution of current density in the cross section, 

 and this is a function of the frequency and the shape of the cross section. In general, 



however, the resistance ratio is a function of the parameter— ^5— , in which / is the fre- 



-ATO 



quency, and R« is the direct current resistance per unit length. In what follows Ro will be 

 taken as the direct current resistance per 1000 ft. of conductor. 



The distribution of current in the cross section is affected by a neighboring conductor 

 carrying high frequency currents. This proximity effect finds an explanation in that the 

 value of the mutual inductance of any filament A of one conductor on a filament B of the 

 other conductor depends upon the positions of A and B in their respective cross sections. 

 The proximity effect may be very appreciable for conductors nearly in contact ; falling off 

 rapidly as their distance is increased, it is negligible for moderate ratios of distance apart 

 to cross sectional dimensions. In such cases the resistance is sensibly the same as for an 

 isolated conductor. 



Beside the spacing factor of the conductors, the proximity effect depends upon the 

 frequency, and in lesser degree upon the shape of the cross sections. Quantitatively, the 

 proximity effect may be expressed by the proximity factor, which is the quotient of actual 

 resistance of the conductor by the resistance which it would have if removed to a great 

 distance from the disturbing conductor, both values of resistance being referred to the 

 same frequency. 

 That is, if 



Ro = the direct current resistance 



Ri = the resistance of the conductor when isolated, frequency / 

 Ri = the resistance in the presence of the disturbing conductor 

 at frequency / 



7? R 



then the proximity factor is P =-5^, and the resistance ratio -=^, in the presence of the 



K\ Ko 



disturbing conductor, is obtained from the resistance ratio -=- when isolated by the rela- 



Ko 



7? 7? 



tion -~- =■ P ~ . Resistance ratio may be obtained in any case if the resistance ratio 



Ro Ro 



when isolated is known, together with the value of the proximity factor. 



Formulas for the high-frequency resistance ratio have been developed in only a few 

 simple (but important) cases, and even then very complicated formulas result. For prac- 

 tical work tables are necessary for simplifying the calculations. The following tables cover 

 the most important cases. 



Formulas have been derived for the high-frequency resistance ratio of single-layer coils 

 wound with round wire. Generally, these differ from one another and from measured 

 values, because, simplifying assumptions are made which are not sufficiently realized in 

 practice. No tables of values for coils such as are met in practical radio work are available. 

 As a rough guide, the high-frequency resistance ratio for a single-layer coil is often from 

 two to five times as great as the resistance ratio of the same wire stretched out straight 

 and carrying current of the given frequency. The experimental work available indicates 

 that this factor due to the coiling of the wire, that is, the total proximity effect of the 

 turns of the coil, is largely dependent upon the frequency and the ratio of wire diameter 

 to pitch of winding, and in lesser degree to the ratio of length to diameter. 



Smithsonian Tables 



