Articles techniques et présentations

An important attribute for numerical simulations is to provide accurate quantified solution values. Thus it is useful to find complex problems with analytic solutions to confirm the numerical results obtained from software. The well-known proximity effect causes a redistribution of currents and added resistance in conductors induced by currents in nearby parallel conductors, and hence provides a valuable opportunity to achieve such a validation. Here a new analytic solution for the proximity effect between parallel round wires is obtained from a series of integral equations and yields quantified analytic solution values for the redistribution of currents in the wires as well as the amount of added resistance per unit length induced in each wire. The added resistance is expressed as a fractional increase over that for skin-effect resistance on a single isolated wire.

The proximity effect calculation is a severe challenge for numerical software since it involves fields that vary rapidly with distance, currents that vary radially and azimuthally and requires accurate post-processing to obtain net resistance values for conductors with highly non-uniform currents.

The proximity effect was calculated using the AC/DC Module of COMSOL Multiphysics^® simulation software. Wire arrangements from 2 to over 20 parallel wires were selected and many different wire-to-wire separations values from 10% to over 300% of the wire diameter were used. Because the analytic solution is for parallel wires the 2D planar solver was employed in these calculations. Furthermore the analytic solution assumes the current is confined to the wire surface, a strong “skin-effect” condition, and so for copper conductors a frequency of 100 MHz was used to cause a very thin current skin depth of just 6.3 µm with wires of 500 times larger diameter (3.175 mm).

An example magnetic field solution by COMSOL® for 3 parallel 1/8^th inch wires with wire-to-wire separation of ½ wire diameter is depicted in Figure 1. An extremely fine mesh was employed along with greater mesh density both just inside and just outside the wire surface to achieve good accuracy. The ratio of corresponding wire resistance values Rp/Ro for theory and by COMSOL Multiphysics^® are shown in the Table 1 and exhibit less than 0.4% deviations, where Rp is the resistance with proximity effect and Ro that for skin effect resistance of a single wire. The corresponding surface current densities are also depicted.

The solutions with COMSOL Multiphysics^® were found to be in excellent agreement with the analytic values for corresponding wire configurations, results differed by less than 0.5%. Both surface current distributions and the resultant net resistance values were accurately quantified by the software in comparison to the analytic solutions for the complete range of test conditions. The added resistance was consistently different for each wire and for each arrangement of number and wire spacing. This wide range of conditions and solution results provided further opportunity to exercise the ability of the software to match analytic solutions.

  Table 1   Proximity Effect Resistance Comparison: 3-Parallel Wires

                                       (Rp/Ro)-ext        (Rp/Ro)-int         (Rp/Ro)-ave (=total/3)

               Theory:              0.4986               0.0393                  0.3455

               COMSOL®:       0.4968               0.0391                  0.3443