Optimized design of a low-resistance electrical conductor for the multimegahertz range

Abstract

We propose a design for a conductive wire composed of several mutually insulated coaxial conducting shells. With the help of numerical optimization, it is possible to obtain electrical resistances significantly lower than those of a heavy-gauge copper wire or litz wire in the 2-20 MHz range. Moreover, much of the reduction in resistance can be achieved for just a few shells; in contrast, litz wire would need to contain ∼ 10 4 strands to perform comparably in this frequency range.In this letter, we show that a structure of concentric cylindrical conducting shells can be designed to have much lower electrical resistance for ∼ 10 MHz frequencies than heavy gauge wire or available litz wires. At such frequencies, resistance is dominated by skin-depth effects, which prevent the current from being uniformly distributed over the cross section; this is typically combatted by breaking the wire into a braid of many thin insulated wires (litz wire 1 ), but the ∼ 10 µm skin depth at these frequencies makes traditional litz wire impractical (∼ 10 4 µm-scale strands). In contrast, we show that as few as 10 coaxial shells can improve resistance by more than a factor of 3 compared to solid wire, and thin concentric shells can be fabricated by a variety of processes (such as electroplating, electrodeposition, or even a fiber-drawing process 2-4 ). Good conductors at these frequencies are increasingly important, e.g. to make low-loss resonators for wireless power transfer 5,6 , or for other applications (e.g. RFID) operating at ISM (Industrial, Scientific, and Medical 7 ) frequencies (e.g. 6.78 and 13.56 MHz). We derive an analytical expression for the impedance matrix of both litz wire and nested cylindrical conductors starting from the quasistatic Maxwell equations; in particular, a key factor turns out to be the proximity losses 8 induced by one conductor in another conductor via magnetic fields. Using this result combined with numerical optimization, we are able to quickly optimize all of the shell thicknesses to minimize the resistance with a given frequency and number of shells.For a cylindrically symmetrical system of nested conductors oriented along the z direction, we first see below that Maxwell's equations reduce to a Helmholtz equation in each annular layer. In the quasistatic limit of low frequency, we show that this further simplifies into a scalar Helmholtz equation for E z alone, which can be solved in terms of Bessel functions, the coefficients of which are determined by the boundary conditions at each interface: continuity of E z and of H φ ∼ ∂E z /∂r. Once the solution for E z , and thus the current density σE z (for conductivity σ) and the magnetic fields (from Ampere's law), are obtained, the impedance matrix can be derived from energy considerations. Of course, a real wire is not perfectly cylindrical because of bending and other perturbations, but these effects can typically be neglected (e.g., if the bending radius is much larger than the wire radius).We start by analytically solving Maxwell's equations in...