Quasi-Analytical Calculation of Frequency-Dependent Resistance of Rectangular Conductors Considering the Edge Effect

Abstract

This paper presents an accurate quasi-analytical approximation of frequency-dependent ac resistance of single rectangular conductors. In this work, first, a two-dimensional analytical ac resistance of rectangular conductors is derived. Unlike circular conductors, where current density distributes evenly in each layer of the conductor’s cross-section, the edge effect is involved for rectangular conductors. Due to the edge effect, one cannot define an accurate boundary condition for solving the two-dimensional partial differential equation of magnetic field or current density of rectangular conductors. Hence, the calculated two-dimensional analytical current density result is not accurate and is modified and fitted on FEM simulation, taking the conductor’s thickness into account using the least-square problem to improve its accuracy. Unlike numerical approaches, the proposed method yields an easy-to-use formula applicable to industrial applications in different fields. Contrary to the one-dimensional approach, which is only valid for very thin rectangular conductors, this method takes edge effect into account and can be used for any thickness (from square to very thin rectangular conductors). The proposed method can be used in applications where an accurate ac resistance of rectangular conductors over a wide frequency range is required, such as white-box modeling of power transformers and interpreting its frequency response analysis (FRA), and calculating the resistance of electric machine winding, busbars, and printed circuit board traces.

On the Fabrication of High-Performance Additively Manufactured Copper Winding Using Laser Powder Bed Fusion

“…The conductor’s cross-section geometry and shape considerably affect the current density distribution. As depicted in Figure 16 e, the skin effect reduces the effective area of the square conductor by inducing “current crowding” toward the corners [ 38 ]. Accordingly, the R ac of the solid and printed square coil is relatively higher, as found in Figure 16 c. where is the skin depth, is the angular frequency, is the absolute magnetic permeability of the conductor.…”

Skin and Proximity Effect Calculation of a System of Rectangular Conductors Using the Proper Generalized Decomposition Technique

“…Skin effect is a phenomenon in which ac current distributes with more concentration near the surface of a conductor due to the interaction between the internal magnetic field and the electric current of the conductor. When it comes to rectangular conductors, the edge effect should also be considered [20,59]. PGD is a suitable technique for rectangular conductors since mesh selection is easy and straightforward in Cartesian coordinates.…”

“…A z (x, y) goes to zero at infinity. To solve (16), the Dirichlet boundary condition (i.e., A z (x, y) = 0) at a far enough boundary is employed (i.e., truncating the outer boundary at a distance much larger than a conductor size) [20,29,44]. In this section, ( 16) is solved at each frequency using the PGD approach and approximated as…”

“…There are many approximations for the calculation of the ac resistance of single rectangular conductors, such as one-and two-dimensional calculations [12,13], the circular cross-section formula [13] used for rectangular conductors [14][15][16], and the Ametani formula [17][18][19]. The shortcomings of these approaches have been discussed in [20], and a quasi-analytical approach has been developed for the calculation of ac resistance of single rectangular conductors taking skin and edge effects into account [20]. However, the proximity effect is also important in a system of multiple conductors, but the approach presented in [20] cannot be used for multiple conductors or proximity effect calculations.…”

“…The shortcomings of these approaches have been discussed in [20], and a quasi-analytical approach has been developed for the calculation of ac resistance of single rectangular conductors taking skin and edge effects into account [20]. However, the proximity effect is also important in a system of multiple conductors, but the approach presented in [20] cannot be used for multiple conductors or proximity effect calculations. The main obstacle in the analytical calculation of the ac resistance of rectangular conductors is that it is difficult to define accurate boundary conditions.…”