Tubular Conductors Research Articles

Formulas for the Inductance of Rectangular Tubular Conductors

THE formula for the inductance per unit length of two parallel conductors of rectangular cross section, obtained by using the formal expressions for the geometric mean distance of a rectangle to itself and the geometric mean distance between two rectangles with sides parallel, is exceedingly cumbersome. More than a score of logarithms and arc tangents must be evaluated when computing the inductance for a given conductor spacing and cross section. To circumvent such lengthy calculations when computing the reactance of like parallel rectangular strap conductors, Dwight1,2 has published curves, each plotted for a certain conductor spacing and cross section, from which the reactance can be obtained. Unless, however, the conductor spacing and the ratio of conductor thickness to breadth coincides with those values for which the curves are plotted, interpolation is necessary. Recently Roth3 has expressed the inductance of such lines in the form of a rapidly converging series, the parameters of which are the spacing and dimensions of the conductors. Usually, a few terms of this series suffice to yield a value of inductance sufficiently accurate for all design purposes. As regards rectangular tubular conductors, the only analytical literature is a paper by Dwight and Wang4 giving formulas for thin square tubular conductors. These are derived on the assumption that the conductor walls are so thin that the sides can be considered as but line segments. The geometric mean distances for such segments are then used to calculate the inductance. Such a procedure is equivalent to asserting that the current flows solely on the surface of the conductors. It follows, then, that the values of inductance computed on this basis will be less than the actual values. Consequently, while these formulas suffice for many calculations on square tubular conductors, if accuracy is a desideratum of design this minimization of thickness restricts the range of application of these formulas to very thin square tubular conductors. Recognizing this restriction and noting, further, that commercial conductors have rounded corners, Dwight and Wang give formulas, empirically deduced, from which corrections for thickness and for rounded corners can be calculated.

Formulas for the inductance of rectangular tubular conductors

Formulas for the inductance of rectangular tubular conductors

The Temperature Coefficient of Inductances for Use in a Valve Generator

This paper discusses the relative importance of skin effect on the temperature coefficient of inductance. It starts with a definition of inductance which includes explicitly the effect of magnetic field within the wire and analyzes inductance into an internal and external component. It is the internal inductance which has a temperature coefficient which can make the total coefficient greater than that of linear expansion. Its ability to affect the total coefficient obviously depends on the ratio of internal to total inductance. This ratio is assessed for various typical coils of round, solid wire, and found to be seldom less than about 5 per cent. It is also assessed for tubular and flat-strip conductors. The dependence of internal inductance on frequency is explored in general terms, and it is suggested that the product of high-frequency resistance and internal inductance tends to be constant for all conductors. An expression is derived for the ratio of the total temperature coefficient of a typical coil to that of linear expansion; it is shown that this ratio may well be four or five but that it need never exceed unity appreciably if the radius of the wire is chosen suitably with respect to the frequency. Some general rules are given for this choice. Finally three methods of constructing coils on a ceramic former are compared in relation to the temperature coefficient.

Discussion on “The alternating-current resistance of tubular conductors”

Discussion on “The alternating-current resistance of tubular conductors”

The alternating-current resistance of tubular conductors

Simple formulae are developed for the effective resistance to alternating currents of tubular conductors when used under the following conditions:- (a) Isolated conductor. (b) Single-phase system of 2 concentric conductors. (c) Single-phase system of 2 conductors of equal size laid parallel to each other. (d) Three-phase system of 3 conductors of equal size laid parallel to each other in a plane, with the middle conductor midway between the outer conductors. (e) Three-phase system of 3 conductors of equal size laid parallel to each other in triangular formation, with equal distances between conductors. In cases (a) and (b) rigid formulae are already available. The formulae developed in this paper are shown not to differ from the rigid formulae by more than a few parts in 1 000, while the reduction of labour in using the simple formulae is probably of the order of one hundredfold. In cases (c), (d), and (e), no rigid formulae are available, and the solutions put forward in this paper are the simplest and the most accurate which have yet been published. Experimental work has been undertaken to check the accuracy of the formulae and it has been found that the discrepancy between the experimental values and the calculated values is less than 3 per cent, provided that th'e ratio of diameter of conductor to spacing between axes of conductors is less than 0.85. This covers the great majority of cases occurring in practice. When the conductors are closer together than this, the errors in the formulae may be a little larger, but, even with the conductors almost touching, the greatest observed discrepancy between the experimental and calculated values of the ratio of a. c. resistance to d. c. resistance was only 6 per cent (11 per cent of the ratio of increase of resistance with a. c. to d. c. resistance).

The propagation of radio frequency currents along a wire of finite length

In the above paper it is pointed out that before we may use the usual wave equation for the propagation of current along a straight wire in free space it is necessary that (a) the inductance L and capacity C per unit length of the wire are independent of the current distribution and (b) that the product LC = 1/c2. This has been shown to be the case.The inductance and capacity of an element of a conductor are defined generally in the way suggested by Moullin and it is shown that the radiation resistance term, which is a function of the current distribution, is small compared with the corresponding terms involving inductance and capacity. The usual wave equation is derived from Maxwell's equations for the case of a tubular conductor in which the current is a function of the distance z along the conductor only. If the curl of the current is not equal to zero this will not represent the true state of affairs.It is pointed out in the earlier part of the paper that a complete solution of the problem, taking into account the radiation resistance, and the skin effect, involves the solution of an integral equation which is given.The field about a tubular current, with a current distribution J0 sin kz along it, is examined. It is found that, for a long wire, the electric force is perpendicular to the surface at every point since we neglect the contribution from the ends. The magnetic force is in circles about the axis as would be expected.A small component parallel to the current, and in phase with it, is introduced by the charges upon the ends of the tubular conductor, and this component determines the radiation resistance. From this it follows that the radiation resistance could be increased by placing a capacity on the ends—thus increasing the charges there.

Bessel Functions for A-C. Problems

Tables of Bessel functions of zero and higher orders are given, for use in problems of skin effect and proximity effect in conductors carrying alternating currents. Formulas for deriving numerical values of the functions are given, in the form of series for large and for small values of the argument. The series are complete with their general terms. The application of the tables to the calculation of the skin effect resistance ratio of an isolated tubular conductor is described and an example is given.

Bessel functions for A-C. problems

Tables of Bessel functions of zero and higher orders are given for use in problems of skin effect and proximity effect in conductors carrying alternating currents. Formulas for deriving numerical values of the functions are given in the form of series for large and for small values of the argument. The series are complete with their general terms. The application of the tables to the calculation of the skin effect resistance ratio of an isolated tubular conductor is described and an example is given.

Wave Propagation Over Parallel Tubular Conductors: The Alternating Current Resistance

On the basis of Maxwell's laws and the conditions of continuity of electric and magnetic forces at the surfaces of the conductor, the fundamental equations are established for the axial electric force and the tangential magnetic force in a non-magnetic tubular conductor with parallel return. The alternating current resistance per unit length is then derived as the mean dissipation per unit length divided by the mean square current. The general formula is expressed as the product of the alternating current resistance of the conductor with concentric return and a factor, termed the “proximity effect correction factor,” which formulates the effect of the proximity of the parallel return conductor. The auxiliary functions which appear in the general formula are each given by the product of the corresponding function for the case of a solid wire and a factor involving the variable inner boundary of the conductor. In general, the resistance may be calculated from this formula, using tables of Bessel functions. The most important practical cases, however, usually involve only the limiting forms of the Bessel functions. Special formulae of this kind are given for the case of relatively large conductors, with high impressed frequencies, and for thin tubes. A set of curves illustrates the application of the formulae.

A precise method of calculation of skin effect in isolated tubes

A precise method of calculation is describ d for determining the skin effect in isolated tubular conductors. This may be used where more accurate results are desired than are given by the curves and approximate methods of calculation previously published by the writer. The present calculation requires the use of certain numerical values of Bessel functions, a table for which is given. Asymptotic series for calculating them are given, which are complete with their general terms. An example is worked out, and the result checked up with the published curves.

Skin effect and proximity effect in tubular conductors

Skin effect and proximity effect in tubular conductors

Skin effect and proximity effect in tubular conductors

Skin effect and proximity effect in tubular conductors

Skin Effect and Proximity Effect in Tubular Conductors

The effective a-c. resistance of tubular conductors is required to be predetermined by designers, for radio installations, for large underground cables with non-magnetic cores, and for electric furnace circuits. (See Examples I to IV). For the above purpose, sets of curves are given in this paper. The skin effect ratio for isolated tubes is shown in Fig. 1. For stranded conductors, the resistance must be multiplied also by a ratio for the spirality effect, as is approximately indicated in Fig. 2. When the return conductor is near, a ratio for the proximity effect, as indicated in Fig. 3, is also to be used. A calculation for the proximity effect ratio for thin tubes is made, and the results are compared with tests in Fig. 3. Some of the requirements for future research work on skin effect are discussed. The conclusion is expressed that it seems scarcely worth while providing a non-magnetic core with a 2,000,000 cir. mil, 25-cycle cable in order to reduce the skin effect, but with the other cases considered, the tubular form seems very advantageous.

Skin effect in tubular and flat conductors

A method is presented for calculating the skin effect resistance ratio of a tube, which is a form of conductor to be recommended for high-frequency work. A formula is also developed by means of which the asymptote to the curve of the ratio R'/R may be drawn, and thus the magnitude of the skin effect at extremely high frequencies may be obtained. The values of the ratio R'/R for tubes of various thicknesses, are plotted in a set of curves (Fig. 3) which may be used for the solution of practical problems. A similar method is described for the calculation of skin effect in a thin strap. Although the calculations cannot be carried out for as high frequencies as the calculation for tube, it indicates a method of coordinating the test results for straps, which have been published. A set of empirical curves for straps is given in Fig. 7, from which approximate values of R'/R for any case may be read.

Skin Effect in Tubular and Flat Conductors

A method is presented for calculating the skin effect resistance ratio of a tube, which is a form of conductor to be recommended for high-frequency work. A formula is also developed by means of which the asymptote to the curve of the ratio R'/R may be drawn, and thus the magnitude of the skin effect at extremely high frequencies may be obtained. The values of the ratio R'/R for tubes of various thicknesses, are plotted in a set of curves (Fig. 3) which may be used for the solution of practical problems. A similar method is described for the calculation of skin effect in a thin strap. Although the calculations cannot be carried out for as high frequencies as the calculation for tube, it indicates a method of coordinating the test results for straps, which have been published. A set of empirical curves for straps is given in Fig. 7, from which approximate values of R'/R for any case may be read.

Effects of self-induction in iron cylinder

If a solid cylindrical conductor be divided into imaginary concentric tubular conductors, the ordinary self and mutual induction theory shows that when the conductor is subjected to an alternating potential difference, the interior shells carry electric currents of smaller density than the exterior ones, and the currents suffer a phase displacement greater the nearer the centre of the cylinder. The theory shows that the permeability and conductivity of the material play an important part, the effects above mentioned being increased in magnitude with increasing permeability and conductivity. When, however, the material of the conductor has variable permeability the problem becomes more complicated, and it is the object of this paper to examine more closely what goes on in an iron cylinder when electric currents are reversed in it, and maintained steady after reversal. A second part of this research will deal with alternating currents of varying frequency and wave-form. The cylinder employed is of mild steel and has a diameter and length each equal to 10 inches (25.4 cm.). It is provided with holes drilled in a plane containing its axis of figure in such a manner that exploring coils can be threaded to inclose certain portions of that plane. The exploring coils are three in number. They are each 2 inches wide in a direction parallel with the axis of figure and midway between the ends of the Cylinder. Their depths in a radial direction are 1, 2, and 2 inches, and their average radii are 0.5, 2 and 4 inches respectively. These coils are referred to as Coils Nos. 1, 2, and 3, No. 1 being near the centre of the cylinder. The cylinder has been already described, but for the purpose of the present research has had a hole ¼-inch diameter drilled through it coinciding with the axis of figure. At each end of the cylinder are two massive gunmetal projections, which in the present research serve to conduct the electric current into the cylinder. The conductors attached to these projections are connected to a reversing switch so constructed that at its mid position it short-circuits the circuit of the cylinder and its conductors, which were arranged in the form of a circle about 6 feet diameter. The electric current was supplied by storage cells through an adjustable resistance and the shunt of an ampère-meter. The potential difference of the cells and the adjustable resistance were such that on reversing the current in the cylinder its value in the main circuit remained constant. In fact, the reversal of the main current was practically instantaneous. The epoch of reversal was noted on a seconds clock and at two-second intervals after reversal the deflections of the dead-beat galvanometers in circuit with the exploring coils were noted. The deflections have been reduced to volts per turn per square centimetre of the coils from which they were obtained, and plotted in terms of the time. Figs. 1, 2, and 3 give the results obtained from Coils Nos. 1, 2, and 3 respectively, and each curve is numbered to correspond with the total amperes reversed when it was observed. The curves have been integrated in order to find the maximum average value of the induction density B for the respective coils. The average values are set out in Table I.