Laminated E-I cores are widely used in electrical machines, especially in power transformers and inductors. They are built using E and I pieces, such as those of Figure 1. There are two different lapping methods used for assembling the E-I cores:

Figure 1 Example of E - I pieces used for assembling the laminated core

Figure 2 Lapping method of E-I laminated core using butt joint (left) and alternate-lap (right)

The use of alternate-lap joint stacking in a laminated E-I core helps to increase the mechanical strength and to reduce energy losses. Figure 3 shows a simplified model, used in this work, with two laminations, using the alternate-lap joint lamination method. The core of Figure 3 has been excited with a 200 turns coil surrounding the central column of the E stack, fed with a current 2 A (Figure 4).

Figure 3 Simplified model of a laminated E-I core, with 2 laminations, using the alternate-lap joint lapping method

Figure 4 3D model of the core of Figure 3

The type of stacking shown in Figure 3 determines the distribution of the magnetic flux in the core. If all the E-I laminations have the same configuration, using a butt joint stack where all the z-cross sections are equal, the distribution of the magnetic flux in each one of these laminations, neglecting the end effects, is the same.

If the butt joint lapping method is used, a 2D analytical model based on lumped reluctances, shown in Figure 5, or a 2D FEM model, shown in Figure 6, can be used to analyze the magnetic flux distribution in the laminated E - I core. In the analytical model, the constructive air-gaps in the joints between the E and I pieces are introduced as added reluctances, *R*_{g}

$$\begin{array}{}{\displaystyle {R}_{g}=\frac{1}{{\mu}_{0}}\frac{g}{{S}_{g}}}\end{array}$$(1)

where *g* is airgap length, *S*_{g} is the total cross section of the columns of the E pieces, and *μ*_{0} is the permeability of the vacuum. In the FEM model, these airgaps are directly introduced as a layer of air between the E and I pieces.

Figure 5 2D reluctance model of the core of Figure 3

Figure 6 2D FEM model of of the core of Figure 3

Both models assume that the E-I configuration is constant in the z-direction, that is, they assume that the laminated E-I core is a butt joint stack. In fact, it is not possible to simulate the 3D inter-laminations fluxes of the alternate-lap joint method using a 2D model. In this case, the flux has an alternative path of lower reluctance in parallel with each joint between an I piece and an E piece. So, instead of crossing the E-I junction airgap, the magnetic field lines jump to adjacent laminations through the insulation layer, in the z-direction, which implies that a 3D model is necessary to capture this non-uniform distribution of the magnetic flux. This alternative path has a reluctance given by
$$\begin{array}{}{\displaystyle {R}_{g}=\frac{1}{{\mu}_{0}}\frac{{\delta}_{s}}{{S}_{A}}}\end{array}$$(2)

where *δ*_{s} is the separation between two adjacent laminations, and *S*_{A} is the surface of contact between them (see Figure 7).

Figure 7 Surface of the contact area between adjacent laminations (*S*_{A}), and surface of the contact area at the E-I junction airgap (*S*_{g})

It can be seen in Figure 7 that the reluctance of the path that crosses the insulation between adjacent laminations is much lower than the path that crosses the E-I junction, because the area of this alternative path is much greater than the area of the E-I junction airgap. In fact, the flux preferably flows through the adjacent laminations, which have the largest contact surface, and does not flow through the air gap existing between the E and I pieces. This fact is illustrated in Figure 8, using a 3D reluctance model of the core of Figure 3, and in Figure 9, using a 3D FEM model of the same core.

Figure 8 3D reluctance model of the core of Figure 3

Figure 9 3D FEM model of of the core of Figure 3

From Figure 8 and Figure 9 it can be concluded that the assumption of uniformity of the flux distribution in the z-direction is no longer valid when using an alternate-lap joint stack, which interchanges the relative positions of the E-I pieces in consecutive laminations. In this case the magnetic flux is not uniformly distributed, as assumed in 2D simulations. Indeed, there is a inter-lamination flux in the z axis nearby the E-I joints, which highly reduces the magnetic flux density in the I pieces, as can be seen clearly in Figure 9. This results in the I pieces being practically unloaded, and the presence in the E pieces of highly saturated points.

## 2.1 Comparison of a 3D FEM model of the laminated E-I core using butt joint and alternate-lap joint lapping methods

To evaluate numerically the effect of the lapping method on the average value of the magnetic flux density in the laminated E-I core, a 3D FEM simulation has been made using the two cores displayed in 2, one with a butt joint stack (Figure 2, right), and the other one with an alternate-lap joint stack (Figure 2, left). From these simulations, the average value of the magnetic flux density in each core has been computed, and the results are presented in .

Table 1 Average value of the magnetic flux density in the laminated E-I core as a function of the lapping method.

That is, the change of the lapping method of the laminated E-I core from a butt joint to an alternate-lap joint method produces a reduction of the effective section of the magnetic core of about 44%, for the core used in this work. In Figure 10 the inter-lamination fluxes in the z-direction are clearly visible for the alternate-lap joint stack, and also their main effect: the unloading of the I piece in each lamination due to the jump of the main flux from one E-piece to the adjacent E-pieces, avoiding the crossing of the high reluctance E-I junction airgap.

Figure 10 3D FEM model of the laminated E-I core of Figure 3 with 8 laminations, using an alternate-lap joint lapping method, showing the inter-lamination zig-zag fluxes in the z-direction

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