A Perturbative Method for Calculating the Impedance of Coils on Laminated Ferromagnetic Cores

Abstract

A new set of formulas for calculating the self and mutual impedances of coils on straight and closed laminated ferromagnetic cores of circular cross-section has been derived. The obtained formulas generalize the well-known formulas for impedances of coils on homogeneous ferromagnetic cores, for the case of laminated cores, and improve the previously known formulas for laminated cores. The obtained formulas are fully consistent with Maxwell's equations and, therefore, offer an excellent accuracy. The perturbation theory and the average field technique are used to solve Maxwell's equations inside and outside the core. The solution inside the core can also be used in the analysis of thermal effects occurring inside the laminated core.

1. Introduction

Analytic calculation of the self and mutual impedances for coils on ferromagnetic cores is not a new topic [1–7]. A rigorous analytic solution of Maxwell’s equations with the relevant boundary conditions is the natural basis for deriving accurate formulas for the coil impedances. In this approach, eddy currents induced into a laminated core and the frequency dependence of the impedance due to skin effect are properly represented. The mathematical complexity of the approach is related with the geometry and structure of the ferromagnetic core, so, rigorous analysis of a realistic case may become complicated. Nevertheless, the problem may be solved to any required degree of accuracy by using different approximate techniques and modern computational methods.

In the paper, an analytic study of electromagnetic field in a transformer with an infinitely long laminated ferromagnetic core of circular cross-section is presented. The formula for self and mutual impedance for filamentary turns is derived. Due to the laminated structure of the ferromagnetic core, some of the parameters involved in the Maxwell’s equations, such as magnetic permeability, permittivity, and electrical conductance, are discontinuous functions. As a result, the electromagnetic field becomes discontinuous, it loses the axial symmetry, and the use of numerous boundary conditions on the surface between each lamination and dielectric is required. The problem becomes too complicated.

In this paper, a new approach to the impedance calculation for transformers, which considers the laminated core as a whole, is presented. In order to simplify the problem, preserving a good accuracy, the use of the average electromagnetic field is proposed. In this approach, the problem becomes similar to the case of a uniform but anisotropic ferromagnetic core, where the magnetic permeability, permittivity, and electrical conductance are slightly different in the 𝑋 and 𝑌 directions. In turn, it means that there exists a small parameter which can be taken, for example, as the relation between the difference of the magnetic permeability in the 𝑌 and 𝑋 directions and the magnetic permeability of the uniform ferromagnetic core. Then, Maxwell’s equations can be solved by employing the perturbation theory [8, 9]. In this approach, every function involved in Maxwell’s equations is considered as a power series in the small parameter. The leading term of this series is the solution of the exactly solvable problem with the axial symmetry. The higher-order terms are the corrections that complement the leading term to the full solution of Maxwell’s equations. An approximate solution of the full problem is obtained by truncating the series, keeping only the first two terms of the power series. The higher orders become successively less important, and, therefore, they can be neglected.

2. Equations for the Average Electromagnetic Field

In Figure 1, a ferromagnetic laminated core with filamentary turns is presented. The ferromagnetic core is taken to be infinitely long; each lamination is of permeability 𝜇₂ and conductivity 𝜎₂. The laminations are separated by a dielectric of permittivity 𝜀₂. The medium around the core is of permittivity 𝜀₁ and permeability 𝜇₁. The core radius is 𝑅. An energizing coil of 𝑁 filamentary turns of radius 𝑟₁, placed at the coordinates 𝑧=𝑛𝜏, is considered. The ferromagnetic lamination width is 𝑎, and the distance between two near laminations is 𝑏.

Figure 1 

Filamentary turns on an infinite ferromagnetic core.

The permeability, permittivity, and electrical conductance can be modeled by introducing the function Δ(𝑥) presented in Figure 2, such that Δ(𝑥)=1 inside and Δ(𝑥)=0 outside the ferromagnetic lamination. Once the function Δ(𝑥) is defined, the electrical conductance in the whole space can be represented in the following form:𝜎𝑥,𝑦=𝜎₂𝜃(𝑅−𝑟)Δ(𝑥), (2.1) where 𝜃(𝑅−𝑟) is the Heaviside step function [10], such that 𝜃(𝑅−𝑟)=1 inside and 𝜃(𝑅−𝑟)=0 outside the core (the medium 1 is nonconducting).

Figure 2 

Function Δ(𝑥).

The permittivity in terms of the function Δ(𝑥) can be modeled as follows:𝜀𝑥,𝑦=𝜃(𝑅−𝑟)𝜀₂(1−Δ(𝑥))+𝜀₁Δ(𝑥)+𝜀₁𝜃(𝑟−𝑅), (2.2) and the permeability takes the form:𝜇𝑥,𝑦=𝜃(𝑅−𝑟)𝜇₁(1−Δ(𝑥))+𝜇₂Δ(𝑥)+𝜇₁𝜃(𝑟−𝑅). (2.3)

The mutual impedance between the energizing turn and a second filamentary turn of radius 𝑟, as well as the self impedance of a nonfilamentary turn of small cross-section, follows directly from the solution for the electric field intensity. We assume that the electromagnetic field is quasistationary, and all capacitive effects may be taken into account separately.

As it has been mentioned before, the problem has not cylindrical symmetry. Moreover, due to unhomogeneities of the ferromagnetic core, the structure of Maxwell’s equations becomes complicated. In order to avoid such difficulty, we consider the electromagnetic field averaged with respect to the 𝑋 direction over 𝑙=𝑎+𝑏. The procedure of transforming the exact electromagnetic fields to the average fields comprises having on hand, for example, a bump function 𝑝(𝜉) [9], that is, compactly supported smooth (in the sense of 𝑝(𝜉)∈𝐶^∞) function, such that(a)𝑝(𝜉)≥0 for 0≤𝜉≤𝑙;(b)𝑝(𝜉)=0 for 𝜉<0 and 𝜉>𝑙;(c)∫^+∞_−∞𝑝(𝜉)𝑑𝜉=1.

Let 𝖿(𝑥,𝑦,𝑧) be one of the components of the electromagnetic field. Then, the averaged component ⟨𝖿(𝑥,𝑦,𝑧)⟩ is obtained as the following integral:𝖿𝑥,𝑦,𝑧=^+∞_−∞𝖿𝑥+𝜉,𝑦,𝑧𝑝(𝜉)𝑑𝜉. (2.4)

As a result, the averaged function ⟨𝖿(𝑥,𝑦,𝑧)⟩ has derivatives of all orders𝑑^𝑛𝑑𝑥^𝑛𝖿𝑥,𝑦,𝑧=(−1)^𝑛+∞_−∞𝖿𝑑𝑥+𝜉,𝑦,𝑧^𝑛𝑑𝜉^𝑘𝑝(𝜉)𝑑𝜉. (2.5)

The averaged Maxwell’s equations in the quasistationary approximation together with the continuity equation take the form⟨𝐇⟩=⟨⟩+⟨𝐣⟩,⟨𝐄⟩𝜇∇×𝜎𝐄∇×=−𝜕𝐇,⟨⟩=𝜌,𝜕𝜌𝜕𝑡∇⋅𝜀𝐄∇⋅𝜇𝐇=0,⟨⟩𝜕𝑡+∇⋅𝜎𝐄=0, (2.6) where the averaged current density ⟨𝐣⟩ in the energizing coil is modeled by using the Dirac delta-function [10] as follows:⟨𝐣⟩=𝐼−𝑦,𝑥,0^𝜙𝑟𝛿𝑟−𝑟₁^𝑁_𝑛=1𝛿(𝑧−𝑛𝜏), (2.7) where 𝑟=𝑥²+𝑦². The energizing filamentary turn carries a sinusoidal current 𝐼_𝜙(𝑡)=𝐼_𝜙𝑒^𝑖𝜔𝑡.

The system of (2.6) is not closed due to the presence of the correlation functions ⟨𝜎𝐄⟩, ⟨𝜀𝐄⟩, and ⟨𝜇𝐇⟩. In the medium 1, the parameters 𝜎=0, 𝜀=𝜀₁, 𝜇=𝜇₁ are constants. Then, these correlation functions take the following simple form: ⟨𝜎𝐄⟩=0, ⟨𝜀𝐄⟩=𝜀₁⟨𝐄⟩, ⟨𝜇𝐇⟩=𝜇₁⟨𝐇⟩. In the medium 2, the correlation functions are to be estimated.

Let us approximate the bump function 𝑝(𝜉) by the following discontinuous function:𝑝(𝜉)=1𝑙for0≤𝜉≤𝑙,0otherwise. (2.8)

Then, we obtain𝜎𝐸_𝑥=1𝑙^𝑙₀𝜎𝑥+𝜉,𝑦𝐸_𝑥𝑥+𝜉,𝑦𝑑𝜉=𝑎𝜎₂𝑙𝐸_𝑥𝑎, (2.9) where ⟨𝐸_𝑥⟩_𝑎 is the average electric field intensity inside the ferromagnetic lamination, which is given by the equation𝐸_𝑥𝑎=1𝑎^𝑎₀𝐸_𝑥𝑥+𝜉,𝑦𝑑𝜉. (2.10)

Similarly, the average electric field intensity in the dielectric medium between the laminations is𝐸_𝑥𝑏=1𝑏_𝑎^𝑎+𝑏𝐸_𝑥𝑥+𝜉,𝑦𝑑𝜉. (2.11)

Applying the boundary condition for the average components ⟨𝐸_𝑥⟩_𝑎 and ⟨𝐸_𝑥⟩_𝑏, we can approximately write 𝜀₁𝐸_𝑥𝑎=𝜀₂𝐸_𝑥𝑏. (2.12)

On the other hand, the full average electric field intensity has the following form:𝐸_𝑥=𝑎𝑙𝐸_𝑥𝑎+𝑏𝑙𝐸_𝑥𝑏. (2.13)

Equations (2.12) and (2.13) form a system of equations which may be elementary solved with respect to the average ⟨𝐸_𝑥⟩_𝑎. Substituting ⟨𝐸_𝑥⟩_𝑎 into (2.9), we finally obtain approximately𝜎𝐸_𝑥=𝜀₂𝑎𝜀₂𝑎+𝜀₁𝑏𝜎₂𝐸_𝑥=𝜎₂𝐸_𝑥+𝑏𝑙𝑎𝜀₂−𝜀₁𝜀₂𝑎+𝜀₁𝑏𝜎₂𝐸_𝑥, (2.14) where ⟨𝜎₂⟩=𝑎𝜎₂/𝑙. The rest of the correlation functions can be calculated similarly by using the appropriate boundary conditions between a lamination and the dielectric. As a result, we obtain, that after averaging, the medium 2 becomes anisotropic, and the permeability, permittivity, and electrical conductance of the laminated ferromagnetic core acquire tensor properties, that is, ⟨𝜎𝐄⟩=𝝈⋅⟨𝐄⟩, ⟨𝜀𝐄⟩=𝜺⋅⟨𝐄⟩, and ⟨𝜇𝐄⟩=𝝁⋅⟨𝐄⟩, where𝜀𝝈=²𝑎𝜀₂𝑎+𝜀₁𝑏𝜎₂0𝜎00₂000𝜎₂𝜀,𝜺=¹𝜀₂𝑙𝜀₂𝑎+𝜀₁𝑏0𝜀00₂000𝜀₂,𝜇𝝁=¹𝜇₂𝑙𝜇₁𝑎+𝜇₂𝑏0𝜇00₂000𝜇₂. (2.15)

Finally, for both mediums, the tensors 𝝈, 𝜺, and 𝝁 can be represented in the forms⟨𝜎⟩𝑏𝝈=𝐈+𝑙𝜒_𝜎⟨𝜀⟩𝑏𝐐,𝜺=𝐈+𝑙𝜒_𝜀𝜇𝑏𝐐,𝝁=𝐈+𝑙𝜒_𝜇𝐐, (2.16) where (a)for medium 1, ⟨𝜎⟩=0, ⟨𝜀⟩=𝜀₁, ⟨𝜇⟩=𝜇₁ and 𝜒_𝜎=𝜒_𝜀=𝜒_𝜇=0;(b)for medium 2,⟨𝜎⟩=𝜎₂,⟨𝜀⟩=𝜀₂=𝜀₁𝑎+𝜀₂𝑏𝑙,𝜇=𝜇₂=𝜇₂𝑎+𝜇₁𝑏𝑙,𝜒_𝜎=𝑎𝜀₂−𝜀₁𝜀₂𝑎+𝜀₁𝑏𝜎₂,𝜒_𝜀𝑎𝜀=−¹−𝜀₂²𝜀₂𝑎+𝜀₁𝑏,𝜒_𝜇𝑎𝜇=−¹−𝜇₂²𝜇₁𝑎+𝜇₂𝑏. (2.17)

𝐈 is the identity matrix, and the tensor 𝐐 has the following form: 𝐐=100000000. (2.18)

Then, after averaging, the system of Maxwell’s equations (2.6) takes the following form:⟨𝐇⟩⟨𝐄⟩+⟨𝐣⟩,⟨𝐄⟩𝜕⟨𝐇⟩∇×=𝝈⋅∇×=−𝝁⋅,∇(⟨𝐄=𝜌,∇(⟨𝐇𝜕𝜌𝜕𝑡𝜺⋅⟩)𝝁⋅⟩)=0,(⟨𝐄𝜕𝑡+∇𝝈⋅⟩)=0. (2.19)

3. The Zero-Order Perturbation Theory

The perturbation theory is an asymptotic method of solving differential equations [8, 9]. This method is commonly used in quantum mechanics [11, 12]. In order to develop the perturbation theory for the system of (2.19), the relation 𝑏/𝑙≪1 is taken as a small parameter. Then, the average electromagnetic field components should be expanded in a power series in (𝑏/𝑙) as follows:⟨𝐄⟩=⟨𝐄⟩₀+^∞_𝑘=1𝑏𝑙^𝑘⟨𝐄⟩_𝑘,⟨𝐇⟩=⟨𝐇⟩₀+^∞_𝑘=1𝑏𝑙^𝑘⟨𝐇⟩_𝑘, (3.1) where ⟨𝐄⟩₀ and ⟨𝐇⟩₀ are the leading terms of the series (3.1), ⟨𝐄⟩_𝑘 and ⟨𝐇⟩_𝑘 are the 𝑘-order terms of the perturbation theory. In this paper, we were limited only to the first order of the perturbation theory.

Substituting the expansions (3.1) into the system of (2.19), we obtain for the zero-order perturbation theory in the frequency domain∇×⟨𝐇⟩₀=⟨𝜎⟩⟨𝐄⟩₀+⟨𝐣⟩,∇×⟨𝐄⟩₀=−𝑗𝜔𝜇⟨𝐇⟩₀∇⟨𝜀𝐄⟩,⟩⟨₀=𝜌₀,∇𝜇⟨𝐇⟩₀𝜌=0,𝑗𝜔₀+∇⟨𝜎⟩⟨𝐄⟩₀=0. (3.2)(3.3)(3.4)

Equations (3.2)–(3.4) are similar to the Maxwell’s equations for the case of the homogeneous ferromagnetic core with the core permeability ⟨𝜇₂⟩, conductivity ⟨𝜎₂⟩ and permittivity ⟨𝜀₂⟩. This problem carries axial symmetry, which makes the use of the cylindrical coordinates appropriate. Maxwell’s equations for coils on homogeneous ferromagnetic cores are solved in [1]. The complete solution of the system of (3.2)–(3.4) for the electric field intensity can be represented in the form𝐸_𝜙0=𝐸_𝜙10𝜃(𝑟−𝑅)+𝐸_𝜙20𝜃(𝑅−𝑟), (3.5) where ⟨𝐸_𝜙⟩₀ is the Fourier-transform of the electric field intensity ⟨𝐸_𝜙⟩₀ with respect to the 𝑧-coordinate𝐸_𝜙(𝑟,𝑘_𝑧)₀=^+∞_−∞𝐸_𝜙(𝑡,𝑟,𝑧)₀𝑒^{−𝑗𝑘𝑧𝑧}𝑑𝑧. (3.6)

The electric field intensities outside and inside the core, that is, ⟨𝐸_𝜙1⟩₀ and ⟨𝐸_𝜙2⟩₀, are given by the following equations:𝐸_𝜙10=−𝑗𝜔𝜇₁𝑟₁𝜃𝑟₁−𝑟𝐼₁𝑘_𝑧𝑟𝐾₁𝑘_𝑧𝑟₁+𝜃𝑟−𝑟₁𝐼₁𝑘_𝑧𝑟₁𝐾₁𝑘_𝑧𝑟+𝐾₁𝑘_𝑧𝑟𝐾₁𝑘_𝑧𝑟₁×𝐼₀𝑘_𝑧𝑅𝐾₀𝑘_𝑧𝑅1−𝜇₁/𝜇₂𝛼𝐼₀(𝛼𝑅)𝐼₁𝑘_𝑧𝑅/𝑘_𝑧𝐼₁(𝛼𝑅)𝐼₀𝑘_𝑧𝑅1+𝜇₁/𝜇₂𝛼𝐼₀(𝛼𝑅)𝐾₁𝑘_𝑧𝑅/𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀𝑘_𝑧𝑅×𝐼_𝜙^𝑁_𝑛=1𝑒^{−𝑗𝑘𝑧𝑛𝜏},𝐸_𝜙20=−𝑗𝜔𝜇₁𝜇₂𝑘_𝑧𝑟₁𝐾₁𝑘_𝑧𝑟₁𝐼₁()×𝐼𝛼𝑟₀𝑘_𝑧𝑅𝐾₁𝑘_𝑧𝑅+𝐼₁𝑘_𝑧𝑅𝐾₀𝑘_𝑧𝑅𝜇₁𝛼𝐼₀(𝛼𝑅)𝐾₁𝑘_𝑧𝑅+𝜇₂𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀𝑘_𝑧𝑅𝐼_𝜙^𝑁_𝑛=1𝑒^{−𝑗𝑘𝑧𝑛𝜏}, (3.7)(3.8) where 𝛼²=𝑘_𝑧²+𝑗𝜔⟨𝜇₂⟩⟨𝜎₂⟩, and 𝐼₀, 𝐼₁, 𝐾₀ and 𝐾₁ are the well-known Bessel functions [13]. The difference between the solutions (3.5), (3.7), and (3.8) and the results obtained in [1] is that (3.5), (3.7), and (3.8) contain the averaged core parameters ⟨𝜇₂⟩ and ⟨𝜎₂⟩, which are smaller than the ferromagnetic parameters 𝜇₂ and 𝜎₂.

The solutions (3.5), (3.7), and (3.8) are going to be used in the solution of equations for the first-order perturbation theory.

4. The First-Order Perturbation

In the first-order perturbation theory, we obtain from the system of (2.19) in the frequency domain ∇×⟨𝐇⟩₁=⟨𝜎⟩⟨𝐄⟩₁+𝜒_𝜎𝐐⋅⟨𝐄⟩₀,∇×⟨𝐄⟩₁=−𝑗𝜔𝜇⟨𝐇⟩₁+𝜒_𝜇𝐐⋅⟨𝐇⟩₀,∇⟨𝜀⟩⟨𝐄⟩₁+𝜒_𝜀𝐐⋅⟨𝐄⟩₀=𝜌₁,∇𝜇⟨𝐇⟩₁+𝜒_𝜇𝐐⋅⟨𝐇⟩₀𝜌=0,𝑗𝜔₁+∇⟨𝜎⟩⟨𝐄⟩₁+𝜒_𝜎𝐐⋅⟨𝐄⟩₀=0. (4.1)

Equations (4.1) are equivalent to Maxwell’s equations for a polarized medium with the electric current density 𝐉, magnetization 𝐌, and the polarization density 𝐏, given by the following equations:𝐉=𝜒_𝜎𝐐⋅⟨𝐄⟩₀=𝜒_𝜎𝐞_𝑥𝐸_𝑥0,𝐌=𝜒_𝜇𝐐⋅⟨𝐇⟩₀=𝜒_𝜇𝐞_𝑥𝐻_𝑥0,𝐏=𝜒_𝜀𝐐⋅⟨𝐄⟩₀=𝜒_𝜀𝐞_𝑥𝐸_𝑥0, (4.2) where 𝐞_𝑥 is the unit vector codirectional with the 𝑋 axis. These vector fields are calculated based on the solutions for the zero-order perturbation theory, and, therefore, in the first-order perturbation theory, they are considered as given functions. The contribution of the vectors 𝐉 and 𝐏 to the fields ⟨𝐄⟩₁ and ⟨𝐇⟩₁ can be taken into account in a future research.

From the definition of the coefficients 𝜒_𝜎 and 𝜒_𝜀 given above it follows, that in the case 𝜀₁=𝜀₂, these two constants are zero, that is, 𝜒_𝜎=𝜒_𝜀=0. As a result, the vector fields 𝐉 and 𝐏 become zeros, and therefore, the fields ⟨𝐄⟩₁ and ⟨𝐇⟩₁ are caused only by the magnetization 𝐌. Further, only this case, in which the permittivities 𝜀₁ and 𝜀₂ are very close or equal, will be analyzed.

Substituting 𝐉=𝐏=0 into (4.1), we obtain∇×⟨𝐇⟩₁=⟨𝜎⟩⟨𝐄⟩₁,∇×⟨𝐄⟩₁=−𝑗𝜔𝜇⟨𝐇⟩₁,∇⟨𝜀𝐄⟩+𝐌⟩⟨₁=𝜌₁,∇𝜇⟨𝐇⟩₁𝜌+𝐌=0,𝑗𝜔₁+∇⟨𝜎⟩⟨𝐄⟩₁=0, (4.3) where ⟨𝜀₂⟩=𝜀₁≡𝜀₀ is the vacuum permittivity. From (4.3) it follows that ⟨𝜌⟩₁=0. Then, excluding the field ⟨𝐇⟩₁ from (4.3), we can obtain the equations for the electric field intensity ⟨𝐄⟩₁ for mediums 1 and 2, respectively,Δ𝐄₁₁Δ𝐄=0,₂₁−𝑗𝜔𝜇₂𝜎₂𝐄₂₁=𝑗𝜔∇×𝐌. (4.4)

Equation (4.4) is to be solved with the evident boundary condition at the core surface𝐸_{𝜙11𝑟=𝑅}=𝐸_{𝜙21𝑟=𝑅},𝐻_{𝑧11𝑟=𝑅}=𝐻_{𝑧21𝑟=𝑅}. (4.5)

The calculation of mutual impedance between the inducing coil and the filamentary turn, which is the aim of this paper, requires of the angular component ⟨𝐸_𝜙1(𝑟,𝑧,𝜙)⟩ of the intensity ⟨𝐄⟩, integrated over the turn of the radius 𝑟 [1], that is,𝑍=−∫₀^2𝜋𝐸_𝜙1𝑟,𝑧,𝜙𝑟𝑑𝜙𝐼_𝜙=𝑍₀+𝑏𝑙𝑍₁+⋯, (4.6) where 𝑍₀ is the mutual impedance in the zero-order approximation, which can be represented in the following form [1]:𝑍₀=𝑗𝜔𝜇₁𝑟₁𝑟^+∞_−∞𝑑𝑘_𝑧𝜃𝑟₁−𝑟𝐼₁𝑘_𝑧𝑟𝐾₁𝑘_𝑧𝑟₁+𝜃𝑟−𝑟₁𝐼₁𝑘_𝑧𝑟₁𝐾₁𝑘_𝑧𝑟+𝐾₁𝑘_𝑧𝑟𝐾₁𝑘_𝑧𝑟₁×𝐼₀𝑘_𝑧𝑅𝐾₀𝑘_𝑧𝑅1−𝜇₁/𝜇₂𝛼𝐼₀(𝛼𝑅)𝐼₁𝑘_𝑧𝑅/𝑘_𝑧𝐼₁(𝛼𝑅)𝐼₀𝑘_𝑧𝑅1+𝜇₁/𝜇₂𝛼𝐼₀(𝛼𝑅)𝐾₁𝑘_𝑧𝑅/𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀𝑘_𝑧𝑅×^𝑁_𝑛=1𝑒^{𝑗𝑘𝑧(𝑧−𝑛𝜏)} (4.7) and 𝑍₁ is the first-order perturbation of the whole mutual impedance. Let us denote the component ⟨𝐸_𝜙1(𝑟,𝑧,𝜙)⟩₁ of the electric field intensity ⟨𝐄⟩₁, averaged over all the directions 𝜙, as ⟨⟨𝐸_𝜙1(𝑟,𝑧)⟩⟩, where𝐸_𝜙1(=1𝑟,𝑧)2𝜋₀^2𝜋𝐸_𝜙1(𝑟,𝑧,𝜙)₁𝑑𝜙. (4.8)

Then, the first-order perturbation term 𝑍₁ of the full mutual impedance can be calculated as follows:𝑍₁𝐸=−2𝜋𝑟_𝜙1(𝑟,𝑧)𝐼_𝜙. (4.9)

Let us apply the averaging procedure (4.8) to (4.4). The advantage of the use of the averaging (4.8) is that the equations for ⟨⟨𝐸_𝜙⟩⟩ acquire axial symmetry, that is, this averaged component loses the dependence on the angle 𝜙. Then, after averaging procedure (4.8) and (4.4) can be written in the cylindrical coordinates as follows:(a)for medium 1, 1𝑟𝜕𝑟𝜕𝐸𝜕𝑟^𝜙1+𝜕𝜕𝑟²𝐸_𝜙1𝜕𝑧²−𝐸_𝜙1𝑟²=0; (4.10)(b)for medium 2, 1𝑟𝜕𝑟𝜕𝐸𝜕𝑟^𝜙2+𝜕𝜕𝑟²𝐸_𝜙2𝜕𝑧²−𝐸_𝜙2𝑟²−𝑗𝜔𝜇₂𝜎₂𝐸_𝜙21=𝑗𝜔2𝜋₀^2𝜋𝐞_𝜙(∇×𝐌)𝑑𝜙, (4.11) where the right-hand side of (4.11) can be easily calculated as follows: 12𝜋₀^2𝜋𝐞_𝜙()𝜒∇×𝐌𝑑𝜙=^𝜇2𝜋₀^2𝜋𝜕𝐻_𝑟20𝜕𝑧cos²𝜒𝜙𝑑𝜙=^𝜇2𝜕𝐻_𝑟20𝜕𝑧. (4.12)

The magnetic field in the zero-order perturbation theory in virtue of (3.3) can be obtained from the solution for medium 2 (3.8) in the form𝐻_𝑟201=−𝑗𝜔𝜇₂𝜕𝐸_𝜙20𝜕𝑧. (4.13)

Then, (4.10) and (4.11) become 𝜕²𝐸_𝜙1𝜕𝑟²+1𝑟𝜕𝐸_𝜙1−𝑘𝜕𝑟^2𝑧+1𝑟²𝐸_𝜙1𝜕=0,₂𝐸_𝜙2𝜕𝑟²+1𝑟𝜕𝐸_𝜙2−𝛼𝜕𝑟²+1𝑟²𝐸_𝜙2=𝜒_𝜇𝑘²_𝑧2𝜇₂𝐸_𝜙20, (4.14)(4.15) where the Fourier transform ⟨⟨𝐸_𝜙⟩⟩ of the field ⟨⟨𝐸_𝜙⟩⟩ with respect to the 𝑧-coordinate is taken. Equations (4.14) and (4.15) are to be complemented with the averaged boundary conditions at the core surface𝐸_{𝜙1𝑟=𝑅}=𝐸_{𝜙2𝑟=𝑅},𝐻_{𝑧1𝑟=𝑅}=𝐻_{𝑧2𝑟=𝑅}. (4.16)

The solution of (4.14), taking into account the limit requirement ⟨⟨𝐸_𝜙1⟩⟩→0 as 𝑟→∞, takes the following form:𝐸_𝜙1=𝐶₁𝐾₁𝑘_𝑧𝑟. (4.17)

Equation (4.15) can be solved by means of the method of variation of constants. The solution of (4.15), after applying the condition ⟨⟨𝐸_𝜙2⟩⟩→0 as 𝑟→0, is the following:𝐸_𝜙2=𝐶₂𝐼₁(𝛼𝑟)+𝑓𝑘_𝑧×(𝛼𝑟)²𝐼₁²(𝛼𝑟)−𝐼₀(𝛼𝑟)𝐼₂(𝛼𝑟)𝐼₁(𝛼𝑟)+2𝑗𝜋𝐾₁(𝛼𝑟)−2𝛼²𝐼₁(𝛼𝑟)^𝑟_𝑅𝐼₁𝛼𝜂𝐼₁𝛼𝜂+2𝑗𝜋𝐾₁𝛼𝜂𝜂𝑑𝜂, (4.18) where 𝑓𝑘_𝑧=−𝜋𝜒_𝜇𝜔𝜇₁𝑟₁𝑘³_𝑧8𝛼²𝐾₁𝑘_𝑧𝑟₁×𝐼₀𝑘_𝑧𝑅𝐾₁𝑘_𝑧𝑅+𝐼₁𝑘_𝑧𝑅𝐾₀𝑘_𝑧𝑅𝜇₁𝛼𝐼₀(𝛼𝑅)𝐾₁𝑘_𝑧𝑅+𝜇₂𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀𝑘_𝑧𝑅𝐼_𝜙^𝑁_𝑛=1𝑒^{−𝑗𝑘𝑧𝑛𝜏}. (4.19)

The integral in the right-hand side of (4.18) can be calculated giving as a result the Meijer 𝐺-function [13]^𝑟_𝑅𝐼₁𝛼𝜂𝐼₁𝛼𝜂+2𝑗𝜋𝐾₁=1𝛼𝜂𝜂𝑑𝜂2𝜂²𝐼²₁(𝛼𝜂)−𝐼₀(𝛼𝜂)𝐼₂+𝑗(𝛼𝜂)𝜋^3/2𝛼²𝐺²²₂₄^1,3/2_1,2,0,0𝛼𝜂^2𝑟_𝑅. (4.20)

The constants 𝐶₁, 𝐶₂ are obtained from the boundary conditions (4.16). Substituting (4.17) and (4.18) into (4.16), we get𝐶₁=−𝑗𝜔𝜒_𝜇𝜇²₁𝑘²_𝑧𝑟₁4𝐾₁𝑘_𝑧𝑟₁×𝐼²₁(𝛼𝑅)−𝐼₀(𝛼𝑅)𝐼₂(𝛼𝑅)𝜇₁𝛼𝐼₀(𝛼𝑅)𝐾₁(𝑘_𝑧𝑅)+𝜇₂𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀(𝑘_𝑧𝑅)²𝐼_𝜙^𝑁_𝑛=1𝑒^{−𝑗𝑘𝑧𝑛𝜏} (4.21) and𝐶₂=𝐾₁𝑘_𝑧𝑅𝐼₁()𝐶𝛼𝑅¹−𝑓𝑘_𝑧(𝛼𝑅)²𝐼₁()−𝐼𝛼𝑅⁰(𝛼𝑅)𝐼₂(𝛼𝑅)𝐼₁()×𝐼𝛼𝑅₁(𝛼𝑅)+2𝑗𝐾₁(𝛼𝑅)𝜋. (4.22)

Substituting constant 𝐶₁ into (4.17) and then, the inverse Fourier transform of the field ⟨⟨𝐸_𝜙1⟩⟩ into (4.9), we obtain as a result the correction for the mutual impedance 𝑍₁=𝑗𝜔𝜒_𝜇𝜇₁²𝑟₁𝑟4^+∞_−∞𝑑𝑘_𝑧𝑘_𝑧²𝐾₁𝑘_𝑧𝑟₁𝐾₁𝑘_𝑧𝑟×𝐼₁²(𝛼𝑅)−𝐼₀(𝛼𝑅)𝐼₂(𝛼𝑅)𝜇₁𝛼𝐼₀(𝛼𝑅)𝐾₁(𝑘_𝑧𝑅)+𝜇₂𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀(𝑘_𝑧𝑅)^2𝑁𝑛=1𝑒^{𝑗𝑘𝑧(𝑧−𝑛𝜏)}. (4.23)

Finally, the whole mutual impedance of coils on laminated ferromagnetic core is obtained by substituting (4.7) and (4.23) into (4.6).

Note that both terms 𝑍₀ and 𝑍₁ take into account the laminated structure of the core.

Formulas (4.6), (4.7), and (4.23) generalize the results of [1] in the sense that the result for the mutual impedance obtained in [1] is a particular case of the formulas obtained in the present paper.

The self impedance of a coil of 𝑁 turns of small cross section can be obtained from (4.6), (4.7), and (4.23) by taking 𝑟 and 𝑧 successively to coincide with the point on the surface of each turn of the coil and then summing the results for each turn of the coil. The self impedance of a single turn is a particular case which can be obtained by substituting 𝑁=1 into the formula for self impedance of a coil.

5. Special Cases

5.1. Closed Cores

An approximate formula for the mutual impedance for turns on a closed toroidal core can be obtained by placing the finite-length core between two infinite plates of perfect magnetic materials, that is, infinite permeability and conductance [1], with respective boundary conditions. In order to provide a reasonable accuracy to this approximation, the core is to be sufficiently long (𝜏≫𝑅), and the turns should be taken close to the core surface. In practice, these assumptions are equivalent to the case of an infinitely long core with an infinite number of energizing turns distributed periodically along the core surface [1]. The formulas (4.6), (4.7), and (4.23) are still applicable in this case, provided that the number of turns 𝑁 becomes infinite, 𝜏 is the closed core length, and the Fourier transform is to be replaced by a Fourier series, due to the periodical structure of the problem. Formally, the transition from the Fourier transform to the Fourier series in (4.7) and (4.23) can be fulfilled by substituting [1]^+∞_𝑛=−∞exp−𝑗𝑘_𝑧𝑛𝜏=2𝜋𝜏^{+∞𝑘=−∞}𝛿𝑘_𝑧−2𝜋𝑘𝜏. (5.1)

As a result, the formula for the complete mutual impedance takes the form𝑍=𝑗𝜔𝐿_𝐴+𝑍_𝐶0+𝑏𝑙𝑍₁+⋯, (5.2) where𝐿_𝐴=𝜇₁𝑟₁𝑟2𝜋𝜏^{+∞𝑘=−∞}𝜃𝑟₁−𝑟𝐼₁𝛽_𝑘𝑟𝐾₁𝛽_𝑘𝑟₁+𝜃𝑟−𝑟₁𝐼₁𝛽_𝑘𝑟₁𝐾₁𝛽_𝑘𝑟𝑒^{𝑗𝛽𝑘𝑧} (5.3) is the inductance in air [1] and𝑍_𝐶0=𝑗𝜔𝜇₁𝑟₁𝑟2𝜋𝜏^{+∞𝑘=−∞}𝐾₁𝛽_𝑘𝑟𝐾₁𝛽_𝑘𝑟₁×𝐼₀𝛽_𝑘𝑅𝐾₀𝛽_𝑘𝑅1−𝜇₁/𝜇₂𝛼_𝑘𝐼₀𝛼_𝑘𝑅𝐼₁𝛽_𝑘𝑅/𝑘_𝑧𝐼₁𝛼_𝑘𝑅𝐼₀𝛽_𝑘𝑅1+𝜇₁/𝜇₂𝛼𝐼₀𝛼_𝑘𝑅𝐾₁𝛽_𝑘𝑅/𝑘_𝑧𝐼₁𝛼_𝑘𝑅𝐾₀𝛽_𝑘𝑅𝑒^{𝑗𝛽𝑘𝑧},𝑍₁=𝑗𝜔𝜒_𝜇𝜇₁²𝑟₁𝑟42𝜋𝜏^{+∞𝑘=−∞}𝛽_𝑘²𝐾₁𝛽_𝑘𝑟₁𝐾₁𝛽_𝑘𝑟×𝐼₁²𝛼_𝑘𝑅−𝐼₀𝛼_𝑘𝑅𝐼₂𝛼_𝑘𝑅𝜇₁𝛼𝐼₀(𝛼_𝑘𝑅)𝐾₁(𝛽_𝑘𝑅)+𝜇₂𝑘_𝑧𝐼₁(𝛼_𝑘𝑅)𝐾₀(𝛽_𝑘𝑅)²𝑒^{𝑗𝛽𝑘𝑧}, (5.4) and 𝛽_𝑘=2𝜋𝑘/𝜏 and 𝛼_𝑘²=𝛽_𝑘²+𝑗𝜔⟨𝜇₂⟩⟨𝜎₂⟩.

In Figures 3 and 4, contours of constant 𝐿_𝑚=Im𝑍(𝜔)/𝜔 are presented for different frequencies and as a function of 𝑧 and 𝑟. For illustrative purposes, the following parameters values have been taken: 𝜎=1.667×10⁶ S/m, 𝜇₁=𝜇₀, 𝜇₂=4045𝜇₀, 𝑅=0.1 m, 𝑟₁=0.11 m, 𝜏=1.4 m, 𝑏=106 μm, and 𝑎=248 μm. Figure 3 represents contours of constant 𝐿_𝑚(𝜔) at the frequency 1 MHz. Figure 4 illustrates contours of constant inductive part of the mutual impedance at 𝜔=2𝜋60 rad/s. The mutual inductance 𝐿_𝑚(𝜔) increases from the dark parts to the light regions. Figures 3 and 4, respectively, illustrate the effects of different ranges of flux penetration into the core. The behavior of the resistive part 𝑅_𝑚=Re𝑍(𝜔) of the mutual impedance is somewhat similar, and therefore, the respective figures have been omitted.

Figure 3 

Contours of constant 𝐿𝑚 at 1 MHz.

Figure 4 

Contours of constant 𝐿𝑚 at 60 Hz.

In Figures 5, 6, 7, and 8, the behavior of the correction term 𝑍₁ at different frequencies is presented. At 60 Hz, the resistive part of the correction impedance 𝑍₁ brings a negative contribution to the complete mutual impedance near the energizing filamentary turn, whereas at 1 MHz it is positive.

Figure 5 

Contours of constant Re𝑍1 at 1 MHz.

Figure 6 

Contours of constant Im𝑍1/𝜔 at 1 MHz.

Figure 7 

Contours of constant Re𝑍1 at 60 Hz.

Figure 8 

Contours of constant Im𝑍1/𝜔 at 60 Hz.

From Figures 3 and 4 and [1], we can see that the behavior of the mutual impedance of turns on homogeneous and laminated ferromagnetic cores is somewhat similar. In order to compare the mutual impedance of coils on a laminated ferromagnetic core 𝐿_𝑚 and on a homogeneous ferromagnetic core 𝐿_𝑚ℎ, and to estimate the maximum contribution of the laminated core to the mutual impedance, let us plot the relative increment of the mutual impedance and resistance. The core parameters are the same as those in Figures 3–8.

In Figure 9, the |𝑅_𝑚−𝑅_𝑚ℎ|/𝑅_𝑚ℎ curve versus the frequency is shown at 𝑟=0.1112 m and 𝑧=0.7 m. A maximum difference (up to 66%) under the conditions mentioned above occurs at the limit 𝜔→0.

Figure 9 

|𝑅𝑚−𝑅𝑚ℎ|/𝑅𝑚ℎ curve versus frequency.

In Figure 10, the (𝐿_𝑚−𝐿_𝑚ℎ)/𝐿_𝑚ℎ curve at 𝑟=0.1112 m and 𝑧=0.7 m is shown. The maximum contribution (17.5%) of the laminated structure of the core, for the given conditions, occurs at approximately 16.5 kHz.

Figure 10 

(𝐿𝑚−𝐿𝑚ℎ)/𝐿𝑚ℎ curve versus frequency.

5.2. Coils on an Infinite Core

The mutual and self impedance for coils on an infinite core can be obtained under the same assumptions that are given in [1]. We consider two coils of the radius 𝑟₁ and 𝑟₂, widths 𝑤₁ and 𝑤₂, thicknesses ℎ₁ and ℎ₂, and the turn numbers 𝑁₁ and 𝑁₂. The distance between two coils is 𝑧. Then, the mutual impedance caused by the laminated ferromagnetic core is the following:𝑍_𝐶=𝑍_𝐶0+𝑏𝑙𝑍₁=2𝑁₁𝑁₂ℎ₁ℎ₂𝑤₁𝑤₂×^∞₀𝑃₁𝑘_𝑧𝑟₁+ℎ₁,𝑘_𝑧𝑟₁𝑃₁𝑘_𝑧𝑟₂+ℎ₂,𝑘_𝑧𝑟₂×𝑄₁𝑘_𝑧𝑤₂,𝑘_𝑧𝑤₁𝐺𝑘_𝑧cos𝑘_𝑧𝑧𝑑𝑘_𝑧, (5.5) where 𝐺𝑘_𝑧=𝑗𝜔𝜇₁𝐼₀𝑘_𝑧𝑅𝐾₀𝑘_𝑧𝑅1−𝜇₁/𝜇₂𝛼𝐼₀(𝛼𝑅)𝐼₁𝑘_𝑧𝑅/𝑘_𝑧𝐼₁(𝛼𝑅)𝐼₀𝑘_𝑧𝑅1+𝜇₁/𝜇₂𝛼𝐼₀(𝛼𝑅)𝐾₁𝑘_𝑧𝑅/𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀𝑘_𝑧𝑅+𝑏𝑙𝜒_𝜇𝜇₁𝑘_𝑧²4𝐼₁²(𝛼𝑅)−𝐼₀(𝛼𝑅)𝐼₂(𝛼𝑅)𝜇₁𝛼𝐼₀(𝛼𝑅)𝐾₁(𝑘_𝑧𝑅)+𝜇₂𝑘_𝑧𝐼₁(𝛼𝑅)𝐾₀(𝑘_𝑧𝑅)²,𝑃₁=1𝑥,𝑦𝑘²_𝑧𝑝₁(𝑥)−𝑝₂𝑦,𝑄₁=4𝑥,𝑦𝑘²_𝑧𝑥sin2𝑦sin2,𝑝₁(𝑥)=𝜋𝑥2𝐾₁(𝑥)𝐿₀(𝑥)+𝐾₀(𝑥)𝐿₁(𝑥), (5.6) where 𝐿_𝜈(𝑥) is the modified Struve function [13]𝐿_𝜈(𝑥)=^∞_𝑚=0𝑥2^𝜈+2𝑚+11Γ(𝑚+3/2)Γ(𝜈+𝑚+3/2). (5.7)

In the case of the self impedance calculations of a coil on an infinite ferromagnetic laminated core we should put: 𝑟₁=𝑟₂, 𝑤₁=𝑤₂, ℎ₁=ℎ₂ and 𝑁₁=𝑁₂.

5.3. Coils on a Closed Core

For the case of coils on a closed core, we can fulfill the same considerations that have been done in the previous section. Then, the formula for the mutual impedance for two coils on a ferromagnetic laminated toroidal core takes the form𝑍=𝑗𝜔𝐿_𝐴+𝑍_𝐶, (5.8) where𝑍_𝐶=𝜋𝜏𝑁₁𝑁₂4𝐶+ℎ₁ℎ₂𝑤₁𝑤₂^∞𝑘=1𝑄₁𝛽_𝑘𝑤₂,𝛽_𝑘𝑤₁×𝑃₁𝛽_𝑘𝑟₁+ℎ₁,𝛽_𝑘𝑟₁𝑃₁𝛽_𝑘𝑟₂+ℎ₂,𝛽_𝑘𝑟₂𝐺𝛽_𝑘cos𝛽_𝑘𝑧,𝐶=𝑗𝜔𝑅²2𝜇₂𝐼₁𝑗𝜔𝜇₂𝜎₂𝑅𝑗𝜔𝜇₂𝜎₂𝑅𝐼₀𝑗𝜔𝜇₂𝜎₂𝑅−𝜇₁. (5.9)(5.10)

In Figures 11 and 12, the theoretical curves and experimental points for the resistive (Figure 11) and inductive (Figure 12) components of self impedance in the frequency range 10² Hz to 1 MHz for test coil on an outer limb of the 25 kVA core are presented. The experimental data and the core and coil parameters were taken from [3]: 𝜎=97.0874 Ω⁻¹, 𝑎=0.27 μm, 𝑏=0.02 μm, 𝜇₁=𝜇₀, 𝜇₂=843𝜇₁, 𝜏=0.976 m, 𝑅=0.047 m, and the test core has 12 turns of heavy-gauge wire, and so forth. The inductance in air is calculated following the recommendations given in [3]. Both theoretical curves show a good agreement between measurements and calculations, which confirms once again the idea expressed in [3] about the validity of treating the laminated core as solid.

Figure 11 

Resistive component of self impedance (test coil on an outer limb of the 25 kVA core). Dashed line corresponds to the solid core and the continuous line represents the laminated core.

Figure 12 

Inductive component of self impedance (test coil on an outer limb of the 25 kVA core). Dashed line corresponds to the solid core and the continuous line represents the laminated core.

Also, in [3] it has been expressed that in spite of a good accuracy of obtained formulas, the prediction of transient behavior of transformer under general conditions requires some more accurate considerations. In particular, the consideration of the differential mutual impedance Δ𝑍(𝜔)=𝑍₁₁(𝜔)−𝑍₁₂(𝜔)=Δ𝑅(𝜔)+𝑗𝜔Δ𝐿(𝜔) is required. Any substantial errors obtained in the calculated quantities Δ𝑅(𝜔) and Δ𝐿(𝜔) for the solid core (dotted lines in Figures 13 and 14) may result in significant errors in transient oscillations. Therefore, it is important to keep these errors small.

Figure 13 

Resistive component of differential impedance between two test coils (coils on outer limb of the 25 kVA core at separation of 0.2 m). Dotted line corresponds to the solid core, the continuous line represents the laminated core with one parameter calculated 𝜒𝜇 and the dashed line is for two calculated parameters 𝜒𝜇 and 𝜒𝜖.

Figure 14 

Inductive component of differential impedance between two test coils (coils on outer limb of the 25 kVA core at separation of 0.2 m). Dotted line corresponds to the solid core, the continuous line represents the laminated core with one parameter calculated 𝜒𝜇 and the dashed line is for two calculated parameters 𝜒𝜇 and 𝜒𝜖.

To improve these results, it was proposed in [3] to enhance the core resistivity. This proposal seems reasonable because this is that really occurs inside the core. Indeed, the averaged conductivity, that is, presented in (5.9), is smaller than the conductivity of lamination, that is, ⟨𝜎₂⟩=𝑎𝜎₂/𝑙<𝜎₂, and therefore, the core resistivity increases. Nevertheless, on the contrary, a simple numerical analysis shows that the reduction of the averaged conductivity results in a larger error between the theoretical and experimental results. To corroborate this, it is enough to substitute 𝜒_𝜇=0 in (5.6) and plot the differential resistivity and inductance curves. Therefore, a simple change of the core resistivity cannot improve the agreement between the theoretical and experimental results.

On the other hand, taking into account the first-order perturbative term, that is, assuming 𝜒_𝜇≠0, improves the theoretical results only a little. A reasonable explanation of this fact can lie in how the correlations ⟨𝜎𝐄⟩, ⟨𝜀𝐄⟩, and ⟨𝜇𝐇⟩ have been calculated. The correct calculation of these correlations implies having on hand the solutions for 𝐄 and 𝐇 before solving Maxwell’s equations, which is impossible. Then, there are two possible ways to deal with these terms. Both methods consider the parameters 𝜒_𝜎, 𝜒_𝜀, and 𝜒_𝜇 as unknown constants. In the first one, these parameters should be recalculated after solving the exact Maxwell’s equations. The second way is to calculate these constants by comparing the theoretical and the experimental results. However, the first way is not feasible in this particular case because the technique proposed in this paper has the purpose of avoiding the necessity of calculating the exact fields 𝐄 and 𝐇. Then, the second program should be carried out.

In Figures 13 and 14, the theoretical and experimental behavior of the differential resistivity and inductance, respectively, is presented. The dotted line shows the theoretical behavior of the differential impedance in the case of the solid core [3]. The continuous line represents the theoretical differential impedance for the laminated core with the parameter 𝜒_𝜇 calculated to obtain a good agreement with the experimental differential inductance. In this case, to obtain the parameter 𝜒_𝜇, one experimental point as a reference is enough. However, to obtain the best result, the least squares method can be used. The obtained result is 𝜒_𝜇=32.06 H, and the theoretical result obtained with this parameter value shows an excellent agreement with the experimental data in the case of the differential inductance, and a better approximation for the differential resistance.

In turn, in this method, the rest of parameters, that is, 𝜒_𝜀 and 𝜒_𝜎 cannot be taken as zero, and therefore, the impedance terms corresponding to the vectors 𝐉 and 𝐏, cannot be neglected. Figures 13 and 14 show the differential resistivity and inductance calculated (the dashed line), taking into account the terms 𝐉 and 𝐏 in the first-order perturbation theory (the corresponding contribution of these terms to the impedance is calculated similarly as the contribution corresponding to the term 𝐌). The parameters 𝜒_𝜇=33.24 H, 𝜒_𝜀=−1.30282×10⁻⁹ F/m, and 𝜒_𝜎=4764.1 Ω⁻¹ are calculated to obtain a better correlation with the experimental data by using the least squares method. We can see that, involving the vectors 𝐉 and 𝐏 into the differential impedance calculation, we obtain a better approximation of the theoretical differential resistivity to the measured data, especially in the high frequency range.

In [3], it was proposed to adjust the theoretical results with measurements by introducing an additional empiric term in the formula for the differential impedance. On the contrary, the formulas obtained in this paper are theoretical and completely consistent with Maxwell’s equations. Therefore, it seems more logical and consistent to compare our results with the experimental data and the respective theoretical results for the solid core [3], which has been carried out in this section.

6. Conclusions

New formulas for calculating the self and mutual impedances of coils on laminated ferromagnetic cores of circular cross-section have been derived. These formulas generalize the previously established formulas for the homogeneous ferromagnetic cores [1], for the case of the laminated cores. In the limit 𝑏→0 and 𝑎→𝑙, the obtained formulas are transformed into the well-known formulas for the self and mutual impedances for coils on homogeneous ferromagnetic cores published in [1].

The obtained result for the frequency-dependent self and mutual impedances for coils on laminated core is compared with the experimental and theoretical results for coils on a practical transformer core, published in [3]. The presented numerical results show a very good correspondence of the new formulas to the measured data. These new formulas do not require introducing the additional empiric terms in order to obtain a better correspondence to the experimental points; all the new terms, involved in the impedance formula, are fully consistent with Maxwell’s equations. Moreover, the accuracy, reached in the new formulas, is not the limit; it can be improved by a more thin analysis of the correlation terms ⟨𝜎𝐄⟩, ⟨𝜀𝐄⟩, and ⟨𝜇𝐇⟩. Thus, all the stated above can be considered as a method for further theoretical research.

As the proposed method implies the strict solving of Maxwell’s equations, it can be used not only for impedance calculation, but also for the electromagnetic fields analysis inside the core. For example, substituting (4.22) into (4.18), we can obtain the averaged electric field intensity inside the core, which, in turn, can be used in the analysis of the laminated core heating. In turn, it calls for numerical evaluation of different special functions such as the Meijer 𝐺-function [13]. However, with available computers and established numerical techniques, this is no longer a problem [1]. The programming of the Meijer 𝐺-function is not a great obstacle; this function can be calculated with any required degree of accuracy, which is not particularly time consuming.

Further theoretical work could be done to establish the exact formulas for the toroidal cores and cores of different geometries. Also, it is of interest the theoretical analysis of the thermal effects occurring inside the laminated core during the transformer operation. The proposed methods, taken as a principle for solving this problem, make possible such type of analysis.

Acknowledgments

This paper was supported by the Dirección General de Educación Superior Tecnológica and the Consejo Nacional de Ciencia y Tecnología-Sistema Nacional de Investigadores of México.