Alternating current distribution in cylindrical conductors

The fundamental equations of Maxwell for the scalar and vector potentials 4> and A are simplified by an approximation which is valid from the lowest to the highest (radio) frequencies.This approximation is due to the difference in the order of magnitude of the electrical conductivity of dielectrics and conductors.The case of N conducting cylindrical groups surrounded by any number of dielectrics is discussed and forms of solution obtained for 4> and A which are proportional to e-vt-n-representing a simple possible type of propagation along the cylinders.These lead to general formulas for the coefficients of leakage, capacity, resistance, and inductance (all of which are functions of the frequency in the general case) as well as expressions for the attenuation and phase velocity.The mean energy relations are also expressed in terms of these coefficients.Application is made to the case of two circular cylinders of different conductivity, permeability, and radius, surrounded by a homogeneous, slightly conductive dielectric.Asymptotic formulas for the alternating current resistance R and inductance L of the line at high frequencies are obtained, which, together with the exact expressions for the coefficients of leakage and capacity, lead to highfrequency expressions for the attenuation and phase velocity.These are all functions of the frequency due to the fact that the current distribution is not uniform in the conductors.CONTENTS Page I. Introduction-Statement of problem 278 II.Fundamental equations 278 1. Type of waves-Thepropagation constant y 278 2. General approximations 279 3. Differential equations and boundary conductions for and A. 281 III.The complex scalar potential <£ 284 1. Uniqueness of a solution 284 2. Existence and uniqueness of a generalized Green's function G 286 3.The existence of and its integral representation 289 4. Properties and physical interpretation of the partial potentials D 291 5.The complex coefficients of leakage and capacity 294 IV.The complex vector potential A 295 1. Existence and properties of a magnetic flux function MJ 295 2. Integral equations for A and E 297 3. Existence and uniqueness of a solution of (36) 298 4. Other methods of solving the integral equation (36) 301 (a) By normal functions 301 (b) Method of successive integration at low frequencies 303 (c) Method of harmonic analysis 304 (d) Mixed method 304 5. Form and properties of the solution 305 6. Coefficients of resistance and inductance 311 277 278 Scientific Papers of the Bureau of Standards ivoi.wPage V.The propagation constant 7. Attenuation and phase velocity 313 1. Determinantal equation for 7 313 2. Determination of constants to fit terminal conditions 316 3. Special cases 316 (a) Case of two conductors 316 (b) Uniform proximity effect 317 VI.Energy relations and mean values 318 VII.Examples and applications 321 1. Case of two conductors with circular symmetry Snote] A. C. Distribution in Cylindrical Conductors 279 Be lpt~' ** where p is 2t times the frequency, and 7 is the complex propagation constant which may be written 7 = 6 + % p T (1) The real quantities b and V represent the attenuation constant and phase velocity, respectively, of this type of wave.There will be a finite number of types of waves or values of 7 possible for a given system, and we shall first consider the case where only one type is present.It will be more convenient to deal with the complex scalar and vector potentials 4> and A, respectively, and to derive E and B from them by the relations E=-V(j>-ipA} B = curl A = l xH\ (2) The electromagnetic cgs system of units will be used throughout with the exception of the dielectric constant Tc, which will be taken in electrostatic cgs units.The electrical conductivity being X, and the ratio of the units being c = 3 X 10 10 , the Maxwell equations ipJc^ curl B = Att/j.( X curl E= -ipB Aire ) E Div# = Div E=0\ (3) require that v*.+»u.-f|V 2 A y + ?i 2 A y = dx dy dx V 2 A z +h 2 A z = -^=-y X d 2 d 2 Where V 2 ^+^-2 l 2 = y 2 - dA, Airip/j(} ipk\ Aire 2 ) X dA _.A , 7 2 " dx dy yA z + h 2 4> %p (4) Conversely any pair and A which satisfy these equations (4) and give continuity to the tangential components of E and H will give the correct field vectors.2. GENERAL APPROXIMATIONS It will be assumed that 7 is a small quantity of the order of -(first c order).In all materials Tc is of the order of unity, and hence the complex conductivity a, defined by ->+£(5)will be very approximately equal to X in conductors; that is, a finite magnitude (for copper X = 0.0006).