In Concentric geometry windings, there are many reasons for simplify the problem.

For reactance (impedance) calculations, however, values can be estimated reasonably close to test values by considering only the window cross section. A high level of accuracy of 3-D calculations may not be necessary since the tolerance on reactance values is generally in the range of ±7.5% or ±10%. For uniformly distributed ampere-turns along LV and HV windings (having equal heights), the leakage field is predominantly axial, except at the winding ends, where there is fringing (since the leakage flux finds a shorter path to return via yoke or limb). The typical leakage field pattern shown in figure 1 (a) can be replaced by parallel flux lines of equal length (height) as shown in figure 2 (a).

The equivalent height (Heq) is obtained by dividing winding height (Hw) by the Rogowski factor KR (<1.0).

The leakage magnetomotive (mmf) distribution across the cross section of windings is of trapezoidal form as shown in figure 2 (b). The mmf at any point depends on the ampere-turns enclosed by a flux contour at that point; it increases linearly with the ampere-turns from a value of zero at the inside diameter of LV winding to the maximum value of one per-unit (total ampere-turns of LV or HV winding) at the outside diameter. In the gap (Tg) between LV and HV windings, since flux contour at any point encloses full LV (or HV) ampere-turns, the mmf is of constant value. The mmf starts reducing linearly from the maximum value at the inside diameter of the HV winding and approaches zero at its outside diameter.

The core is assumed to have infinite permeability requiring no magnetizing mmf, and hence the primary and secondary mmfs exactly balance each other. The flux density distribution is of the same form as that of the mmf distribution. Since the core is assumed to have zero reluctance, no mmf is expended in the return path through it for any contour of flux. Hence, for a closed contour of flux at a distance x from the inside diameter of LV winding, it can be written that

For deriving the formula for reactance, let us derive a general expression for the flux linkages of a flux tube having radial depth R and height Heq. The ampere-turns enclosed by a flux contour at the inside diameter (ID) and outside diameter (OD) of this flux tube are a(NI) and b(NI) respectively as shown in figure 3, where NI are the rated ampere-turns. The general formulation is useful when a winding is split radially into a number of sections separated by gaps. The r.m.s. value of flux density at a distance x from the ID of this flux tube can now be inferred from equation 3 as

But about asymmetrical/ non-uniform winding configurations, the Finite Element Method (FEM) is the most commonly used numerical method for reactance calculation of non-standard winding configurations and asymmetrical/ non-uniform ampere-turn distributions, which cannot be easily and accurately handled by the classical method. The FEM analysis can be more accurate than the analytical methods. User-friendly commercial FEM software packages are now available. Two-dimensional FEM analysis can be integrated into routine design calculations. The main advantage of FEM is that any complex geometry can be analyzed since the FEM formulation depends only on the class of problem and is independent of its geometry. It can also take into account material discontinuities easily. The FEM formulation makes use of the fact that Poisson’s partial-differential equation is satisfied when total magnetic energy function is a minimum. The problem geometry is divided into small elements. Within each element, the flux density is assumed constant so that the magnetic vector potential varies linearly within each element. For better accuracy, the vector potential is assumed to vary as a polynomial of a degree higher than one. The elements are generally of triangular or tetrahedral shape. Windings are modeled as rectangular blocks. If ampere-turn distribution is not uniform (different ampere-turn densities), the windings are divided into suitable sections so that the ampere-turn distribution in each section is uniform.

Author : Hamid - From: Iran