Eddy-Current Losses in a Conducting Plate Due to a Collection of Bus Bars Carrying Currents of Different Magnitudes and Phases
Robert M. Del Vecchio, Member, IEEE
Abstract—Losses in bus bars carrying ac current and passing near the tank wall in transformers can contribute significantly to stray losses and can produce hightemperatures on the tank wall. Usually these losses are mitigated by moving the bus bars farther away from the wall or by placing a shield of high conductivity and low permeability on the tank wall near the bus bars. Although these losses in the wall or shield can be calculated by modern finite-element codes, an analytical method would obtain these losses quickly during the design phase. We presentsuch a method here. The method works for an arbitrary collection of bus bars carrying currents from different phases. Applications to other situations where long cables or bus bars carrying ac currents and running parallel to a conducting plate are also possible. Index Terms—Bus-bar losses, eddy-current losses, plate losses, shield losses, transformer stray losses, transformer tank losses.
I.INTRODUCTION US BARS or cables carrying high ac currents must often pass near the tank wall in transformers. These walls are generally made of magnetic steel of relatively high permeability and conductivity so that eddy-current losses are easily generated in them. When these losses are high, several options are possible for reducing them. The simplest choice is to move the bus bars farther from thetank wall. If this is not possible, a plate or shield of high conductivity and low permeability can be placed on the tank wall next to the bus bars. Another option is to place bus bars from different phases, from a balanced three-phase network, near each other so that their magnetic fields can partially cancel each other out. These options have been explored in  using a finite-elementapproach. Analytic methods can not only help one to quickly assess the effect on losses of repositioning the bus bars or using shields of different materials, but also make it possible to incorporate such loss calculations in other computer programs which calculate, for example, the plate temperature rise resulting from these losses. Previous work in this area focused on plate losses due to a currentsheet  and on plate losses due to a delta function current . We extend these results by considering an arbitrary collection of bus bars as approximated by a collection of delta function currents. The bus bars can carry currents of differing magnitudes and phases and these properties are inherited by their associated delta function current approximations. Different
Manuscript received April 10,2002; revised October 11, 2002. The author is with the Waukesha Electric Systems, Milpitas, CA 95035 USA (e-mail: Bob.DelVecchio@ WaukeshaElectric.spx.com). Digital Object Identifier 10.1109/TMAG.2002.806353
bus-bar shapes can be approximated by positioning the delta function currents (current filaments) strategically. For example, a rectangular shape can be approximated by a centralcurrent filament and one at each corner of the rectangle for a total of five filaments, each one carrying 1/5 of the total current. Other shapes, such as circular, can be approximated by a central filament and others along the circumference. Additional filaments can be positioned for greater accuracy. It should be noted that we consider the currents in the bus bars to be uniformly distributed. Thus, wedo not take into account the current redistribution in solid bus bars, which would result from the ac currents within the bus bar or any of its neighbors or from the eddy currents in the plate. This would certainly affect the losses in the bus bars. Strictly speaking then, the bus bars are assumed to be made of current filaments which are transposed to produce a uniform current distribution....