Iron core coils are widely used for reactors, filters, and transformers [1–12]. In many applications, a magnetic shield is used to isolate the coil fields from the external environment. Hence, the analysis of an iron core coil with a magnetic shield has practical importance. If iron core coils are designed for power reactors or power filters, the manufacturing cost must be considered; a hollow iron core, reducing both the cost and equipment weight, could be a reasonable choice for the coil design (Fig. 1). Conventionally, such a complicated structure should be analyzed with numerical methods, for example, the finite element method (FEM). However, an analytical model may be more convenient for the initial design steps due to the explicit relationship among the parameters of the coil structure. For air core coils with a magnetic shield, classical methods can be employed [13]. However, for iron core coils, a method termed TREE (truncated region eigenfunction expansion) was pioneered by Hannakam and his colleagues [14–21], developed by several researchers [22–31], and used to establish analytical models [32]. Note that in [32], the permeability of the magnetic shield is assumed to be infinite, and the TREE analysis is simplified considerably. In contrast, the permeability of both the core and the shield are both assumed to be finite in the current study.

In conventional TREE, the Newton–Raphson method or contour integral method [25, 33, 34] are routinely used to obtain the eigenvalues. However, the Newton–Raphson method becomes cumbersome when dealing with the regions composed of multiple subregions, as it requires a complicated nonlinear equation to be solved. For the TREE analysis of the problem shown in Fig. 1, many Bessel functions are included to establish the nonlinear equation, and the computational efficiency may be significantly reduced. The contour integral method may also become tedious for the same reason; that is, an integrand consisting of many Bessel functions must be used in the quadrature. Recently, a technique of eigenvalue computation based on the Sturm–Liouville equation was proposed in [35–37]. Under this novel approach, the efficiency of eigenvalue locating can be essentially improved, but symbolic eigenfunctions are still adopted, which are inconvenient for the subsequent calculation of TREE, especially when there are multiple subregions. More recently, a novel approach has been proposed, using the Sturm–Liouville theory and one-dimensional (1D)-FEM discretization to obtain eigenvalues and eigenfunctions simultaneously [38, 39]. This novel approach enables the tedious symbolic eigenfunctions for the multiple subregions to be completely avoided. Consequently, in this study, the technique in [38, 39] is employed, and a performant analytical model will be obtained.

Section II presents an analytical model for coaxial coils with a hollow iron core and a magnetic shield, based on the enhanced TREE of [38, 39]. In Section III, the algorithm is verified by comparisons among the data of TREE, FEM simulation, and measurement. After the numerical verification, the paper finishes with conclusions.

Two coaxial coils, denoted coil 1 and coil 2, respectively, are shielded by a magnetic shield of permeability μ_r1μ₀. The shield has openings on the top and bottom faces, which help the ventilation and assembly process. A hollow iron core of permeability μ_rμ₀ is encircled by the coaxial coils. A perfect electric boundary (PEB) is introduced at r=a to discretize the eigenvalues of this boundary value problem (BVP). The geometry is shown in Fig. 2, in which the whole domain is divided into seven regions along the z-axis.

The vector potential satisfies Laplace or Poisson equations in the relevant regions, namely:

and

where J(r,z) is the current density. Note that we only consider the field of coil 1 (assuming coil 2 is absent) at this stage. Due to the axisymmetry, the Laplacian in the cylindrical coordinates is:

The vector potentials of the seven regions shown in Fig. 2 can be expressed by the separation of variables:

where J₁(r), F₂(r), F₃(r), and F₄(r) are the column vectors with the elements J₁(p_1,ir), F₂(p_2,ir), F₃(p_3,ir), and F₄(p_4,ir), respectively, which are the radial eigenfunctions of the relevant regions. The superscript “T” denotes the matrix transpose. J_n(x) is the Bessel function of the first kind of order n, and p_1,i are the positive roots of J₁(p_1,ia)=0. The other eigenvalues p_2,i to p_4,i are obtained from the Sturm–Liouville equations for k = 2, 3, 4:

In addition, P₁–P₄ are diagonal matrices of the eigenvalues p_1,i to p_4,i, and C₁–C₆ and D₂–D₇ are column vectors of undetermined coefficients. Eq. (5) can be solved by a recently developed method using 1D-FEM [38, 39]. In addition, Ψ(z) is the source function obtained from the Poisson equation:

where

with J₁ denoting the current density over the cross-section of coil 1. The interface condition of Bz and Hr at z = h₂, h₁, b₄, b₁, − h1, and −h2 provide the following equations for the coefficients C₁–C₆ and D₂–D₇:

where

The orthonormality

where I is the identity matrix, and the diagonal matrix S with the elements

were used to derive (8a)–(8l). With the eigenfunctions obtained from (5) by 1D-FEM, the orthonormality of Eq. (10) was proved in [39], which has simplified the TREE analysis considerably. According to [39], Eq. (5) can be transformed to:

where the stiffness and mass matrices are

and

respectively. ϕ_m and ϕ_n are 1D-FEM basis functions composed of cubic Lagrange polynomials. With the solution of (12), the eigenvalues p_k,i and (discretized) eigenfunctions can be obtained simultaneously. The continuous eigenfunctions can be obtained from the interpolation of the Ui′ normalized from U_i [39].

The matrices Ψ of (7) and T₁–T₄ are calculated by the Clenshaw–Curtis quadrature. In principle, the eigenvalues and eigenfunctions can also be obtained by the conventional TREE; however, the eigenfunctions would become extremely clumsy (consisting of many Bessel functions).

In (10), (13), and (14), the permeability functions are:

Accordingly, the matrix equation can be obtained from (8a)–(8l):

The submatrices A₁₁–A₆₆ and E₂–E₅ are provided in Appendix. Consequently, with the solution of (16), the self-inductance of the coil 1 can be found:

where N₁ is the number of turns, r₁, r₂ are inner and outer radii, and b₂ and b₃ are axial positions of coil 1.

For coil 2 with the number of turns N₂, inner and outer radii r₃, r₄, and axial position b₅ and b₆, the self-inductance L₂ can be obtained by the replacement b₂→b₅, N₁→N₂, b₃→b₆, r₁→r₃, and r₂→r₄ in (17).

In addition, the mutual inductance between coil 1 and coil 2 can be obtained in the same manner:

where

and

Eq. (19) is evaluated by the Clenshaw–Curtis quadrature, and Eq. (20) can be solved analytically. The results depend on the relative position of the two coils.

In this section, the TREE method is validated by FEM simulation and experiment. The TREE algorithm is coded in Wolfram Mathematica, and the FEM simulation is carried out by COMSOL Multiphysics. The experiment arrangement is shown in Fig. 3, where the N4L Power Analyzer PPA500 is used for the measurement of the voltages and currents.

The same type of silicon steel is used for the iron core and shield, with the permeability μ_r = μ_r1 = 17,000. A lamination structure is used for the iron core and shield to diminish the eddy currents as much as possible. The experimental layout is shown in Fig. 4. The geometric parameters of the coils, the hollow iron core, and the shield are listed in Table 1.

The self-inductance of coil 1 and coil 2 is shown in Fig. 5. The initial position of coil 1 is at d₁ = b₂ + h₁ = 4 cm. Coil 2 has the same initial position of d₂ = b₅ + h₁ = 4 cm. In the experiment, d₁ and d₂ span from 4 cm to 11 cm with a step of 0.5 cm. The TREE results are plotted with solid lines, and the FEM and measured data are denoted by circles and triangles for comparisons.

The mutual inductance is plotted in Fig. 6, in which coil 1 is fixed at positions d₁ = 7 cm, 8 cm, and 9 cm, while the position of coil 2 varies from d₂ = 4 cm to 11 cm. The TREE results are illustrated by solid lines, circles indicate FEM data, and triangles indicate measurements. The data of d₁ = 7 cm, 8 cm, and 9 cm are shown in blue, red, and green, respectively.

Good agreements were achieved in all cases between the results of TREE, FEM, and the experiment. The relative errors are within 0.56%. The calculations were carried out on a laptop with a 2.3-GHz processor (Intel Core i9-9800K) and 32 GB RAM. Other computation configurations of the enhanced TREE are provided in Table 2.

Using the method described in [38, 39], the eigenvalues and eigenfunctions of (5) can be calculated within 1 second, and the overall computation time is within 1.5 seconds, making a notable contrast with the time taken by eigenvalue evaluation under the Newton–Raphson (65.3 seconds) or contour integral method (245 seconds). For the FEM simulation, the axisymmetric module is used, and the execution time is about 6–7 seconds with a mesh of about 13,000 triangular elements.

Moreover, using the method described in [38, 39], the complicated combinations of the Bessel functions are avoided, and the eigenfunctions of Regions 2–6 can be conveniently handled in a unified manner.

An additional investigation is carried out of the impact of the radial thickness of the iron core on the coil mutual inductance. In the calculation, coil 1 and coil 2 are set at six fixed positions. Fig. 7 shows that about 29% (volumetric ratio) core material (corresponding to the experimental configuration of a₂ – a₁ = 1.5 cm, indicated by the red circles) can provide 83% mutual inductance of a full iron core. On the other hand, the mutual inductance of the present configuration is around seven times larger than that of air core coils.

The magnetic induction decreases dramatically when the magnetic shield is used. The comparison is shown in Fig. 8, where the ratio of B₂ to B₁ is calculated by the TREE equations shown above. B₂ and B₁ represent the magnetic induction without and with the magnetic shield, respectively. In the calculation, a source current of 10 A is applied to coil 1. The blue and red lines show the paths of the top and side (Fig. 9), along which the calculations are carried out. Moreover, d₃ in Fig. 8 refers to the distance between the evaluated and initial points, which are marked by the circles. The positions of the initial points are r = 0 cm and z = 17.1 cm, r = 19.6 cm and z = −17 cm for the blue and red lines, respectively.

An analytical model of the coaxial coils with a hollow iron core and magnetic shield is presented in this work, based on the enhanced TREE. The formulas of self- and mutual inductance are provided, considering the impact of the radial thickness of the iron core and the openings on the magnetic shield. Numerical eigenfunctions were used to simplify the solving process. The theoretical results were verified by FEM simulation and experimental data, confirming the accuracy and efficiency of the proposed method. The results show considerable potential for saving the core materials, and the attenuation of the magnetic field is also revealed.