Transformer: 2D vs 3D modeling

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When modeling an electromechanical device, a designer needs to keep in mind that a 2D problem set introduces an error in a field distribution in comparison with a 3D problem solution. However, a solving process of 2D problems is multiple times faster than of 3D. So, what is a range of the error specifically for a transformer design application?
An example of transformer is shown on the pictures. It has a rectangular iron core with 15 turns of primary coil (all dimensions are in millimeters).

 3D view Front view Side view

A 2D model considers only a part of a coil with a current direction “in” and “out” a plane and neglecting end effects of “front” and “end” coil parts with currents flowing along the plane. To estimate the error let’s assume a linear iron core with a constant relative permeability μr = 1000. Also, to maximize the effect of end parts the coil turn has a square shape (length is equal to width).
For analyzing this problem of magnetostatics the QuickField software is used. This robust and powerful package is extremely user-friendly and you obtain results in few minutes. At first, the solution for the 2D front view of the transformer is shown on a figure below (model depth is 10 mm). All magnetic flux is concentrated inside the iron core that has low reluctance. Total magnetic flux is Ψmain = -77.7 μWb and average flux density in the core is Bmain = 0.778 T.

At second, let’s analyze a 2D model of a side view, which considers front and end parts of a coil. The model has the same depth of 10 mm and the same excitation current. Flux lines close in surrounding air, as shown on a figure below. The reluctance of magnetic path now is high, consequently, magnetic flux is Ψside = -1.37 μWb and average flux density is only is Bside = 0.014 T.

This example clearly shows that even when coil end parts have the same dimensions and conduct the same current with main sides of the coil, the error in flux density stays below 2% (Bside / [Bside + Bmain] = 1.77%).

It worth to mention that the software used for simulations easily allows to incorporate non-linear iron permeability and you can practice it on your own.  Here you can find model files to investigate them in a student version of QuickField software.