Electromagnetic field modeling in composite materials such as carbon fiber reinforced polymers (CFRPs) or multifilamentary high-temperature superconductive (HTS) tapes, is in many cases a challenging task. Numerical modeling is then often necessary [1, 2], where the conventional numerical tools have difficulties to provide accurate solutions in reasonable computing times due to multiscale dimensions, nonlinearities and high anisotropies. In these cases, specific modeling approaches meeting a better compromise between accuracy and computing time are needed.
In previous works, we have developed an electromagnetic model, based on an integro-differential formulation, for a rapid eddy current computation in carbon fiber reinforced composite polymers for nondestructive testing (NDT) applications . In this work, this model is extended to model eddy current losses in multifilamentary HTS composite tapes submitted to external sinusoidal time varying magnetic fields. Indeed, both materials have similarities in their geometrical structures, constituted of filaments embedded in a matrix. In the case of CFRPs, the filaments are carbon fibers embedded in a polymer matrix resulting in poor conductivities and high anisotropies, whereas in the HTS tapes, the filaments are made of a nonlinear conductive material (HTS) and the matrix is made of a high conductive material. A numerical scheme is proposed to take into account the nonlinearity of the constitutive law, relating the electrical current density to the electrical field, in the frequency domain rather than in the time domain as it is commonly done. This ensures the stability and the rapidity of convergence of the numerical scheme. The proposed approach is validated by comparison to theoretical and experimental results. For loss computation, it is particularly shown that the nonlinearity of the HTS tapes can be avoided by considering a high anisotropy factor as reported in .
The modeled system and the electromagnetic formulation are presented in the next sections. Numerical results and validation are given in the last section.
2 The modeled system
The modeled system (Figure 1) consists of a multifilamentary composite sample of length Lt, width Wt and thickness Tt, constituted of filaments embedded in a matrix. The sample is submitted to an external sinusoidal time varying magnetic flux density (B⃗a). The whole system is characterized by the vacuum magnetic permeability (μ0). To avoid the modeling at the filament scale, a homogenized nonlinear electrical conductivity tensor is defined for the sample, given by (1), where σ// and σT denote the conductivities parallel and transverse to the filaments [2, 3, 4].
For the CFRPs, the matrix is a non-conductive material and the transverse conductivity depends essentially on the volume fraction of the carbon filaments. In the case of HTS tapes, the transverse conductivity depends on both the filaments volume fraction (ηs) and shape, the matrix conductivity (σm), the interface between the filaments and matrix, and the source filed orientation [4, 5]. The longitudinal conductivity depends mainly on the filaments volume fraction as given in (2), where, E// and J// are respectively the components of the electric field and current density parallel to the filament orientation.
The CFRPs are characterized by linear electromagnetic properties, whereas, HTS tapes are characterized by a nonlinear conductivity tensor. In this work, the nonlinearity is reported only on σ//. The power law is used to express the nonlinear constitutive law between the components of the electric field and current density parallel to the filament direction, given by (3), where Ec is a characteristic electric field usually set to 1μV/cm, and Jc is the critical value of the electric current density. The power exponent (n) characterizes the steepness of the transition shape from the superconducting to the normal state.
The critical value of the electric current density depends in its turn on the magnetic flux density (B⃗). The Kim’s law is used to take into account this dependence, as given in (4), where Jc0 is the critical current density at zero magnetic field, and B0 is a characteristic magnetic flux density :
3 Model formulation and implementation
Starting from the electrical current conservation, we derive the latter as the curl of the electric vector potential (T):
Considering that the magnetic field varies sinusoidally with time, from the Maxwell-Faraday law in the frequency domain, with the angular frequency ω, we have:
We separate the magnetic flux density into its source and reaction terms, evaluated by integral equations (7) involving the source (J⃗S) and the eddy currents. In (7) B⃗a is an externally applied magnetic flux density satisfying ∇⃗ .B⃗a = 0.
In the following, to compact the expressions of the formulas, the matrix writing is adopted, where C̿, R̿ and M̿ are respectively the curl, the resistivity and the integral matrices. Equation (7) can thus be rewritten as follows:
The integral part of the formulation allows a direct interaction between the system elements, and thus only the active parts of the system are discretized. The finite difference method is used to discretize the curl operator. The matrix equation (9) is solved iteratively which allows to take into account the nonlinearity of the resistivity matrix as shown in the flowchart described in Figure 2. We start by solving (9) by considering only the source term of the magnetic flux density, then we evaluate the eddy currents, after that we evaluate the reaction term of the magnetic flux density; at this step, we actualize the value of the critical current density, then we actualize the conductivity tensor and solve again the equation (9) with the actualized values of the magnetic flux density and the conductivity tensor. The operation is repeated until convergence. An iterative method (conjugate gradient method) is used to solve the matrix equation. As the source term is divergence free, no gage condition is needed as shown by Z. Ren in .
4 Numerical examples and validation
In this section a numerical example is considered for validation purpose. We consider an HTS multifilamentary untwisted tape submitted to an external magnetic field oriented normal to its large surface (width). The results of the eddy current losses are compared to coupling losses (Qc) given by the Campbell’s model , expressed as J/m3/cycle in (10), involving a time constant of the magnetic flux penetration (τ) and a geometrical factor (χ) related to the demagnetizing factor .
The system specifications are given in Table 1. For the comparison, we use in (10) both theoretical and experimental values of the parameters (τ, χ). The experimental ones are given by D. Zola et al., obtained through measurements for the specified geometry .
Figure 3 shows a comparison between the analytical and numerical results of the eddy current losses. The comparison is performed in the limits of the validity of the analytical model (feeble fields) [8, 9]. The results present constant discrepancy over the entire interval of calculation between the numerical and analytical values, which is of 2.8% when theoretical values of the parameters (τ, χ) are used and about 11% when the measured ones are used. This discrepancy is due to errors in both models. Indeed, in the analytical model, the tape is considered isotropic and the dependence of the critical current on the magnetic field is not taken into account. The magnetic field is also assumed to be homogeneously distributed on the tape, which is not really satisfied as it can be shown in Figure 4 through the eddy current repartition in the tape reflecting a spatial variation of the magnetic field. Moreover, the measured parameters used in the analytical model have been determined experimentally with a certain precision. In the other hand, the numerical model is also based on some assumptions, in particular, the use of an homogenized conductivity tensor, and present a certain numerical error. In particular, the transverse conductivity is calculated simply by increasing the conductivity of the matrix by the volume fraction of the superconductor, i.e., σT = (1 + ηs)σm, and may thus be overestimated.
In our investigations, we were also interested in verifying a result presented by G.B.J Mulder et al. announcing that the behavior of a multifilamentary HTS tape become similar to that of an anisotropic material as the anisotropy increases , with less assumptions by considering the magnetic reaction of the tape, which was ignored in the work presented by G.B.J Mulder et al. At this end, we have considered a linear σ// in (1). Figure 5 represents the normalized variation of the eddy current losses with respect to the anisotropy factor (ratio between σ// and σT). It is interesting to notice that the eddy current losses become constant beyond an anisotropy factor of about 104 which confirms the results reported by G.B.J Mulder et al.
Figure 6 represents the variation of the eddy current losses according to the applied magnetic flux density for three cases: linear anisotropic conductivity (σ// = 104 σT) and nonlinear anisotropic conductivity (σ//(J) ≠ σT) with and without taking account of the variation of Jc(B). An agreement is shown between the results corresponding to the constant value of the longitudinal conductivity and those corresponding to the case where it depends only on the electrical current density. The results corresponding to the case where the longitudinal conductivity depends on both the current and magnetic flux densities show a little discrepancy with the others results, presenting a greater power loss for the high values of the applied magnetic flux density, due to the decrease of the value of the longitudinal conductivity. For feeble values of the applied magnetic flux density, one can thus neglect the variations of σ//(J) and Jc(B), considering only an anisotropic linear conductivity (σ// = 104 σT).
In this work, we have extended a model developed in a previous work for eddy current modeling in carbon fiber reinforced polymers (CFRPs) to model eddy current losses in multifilamentary high-temperature superconductive (HTS) tapes using the geometrical and physical similarities between these two materials constituted of filaments embedded in a matrix, characterized by multiscale dimensions, nonlinearities and high anisotropies.
Considering sinusoidal time varying magnetic fields, the problem is solved iteratively, taking account of the nonlinearity of the constitutive laws, in the frequency domain rather than in the time domain as it is commonly done, resulting in a considerable gain in computation time. Moreover, it was shown that, for low values of the applied magnetic flux densities, one can consider the HTS tape as a high anisotropic linear conductor.
The model is tested on a simple geometry for validation purpose, but it is intended to be used for complex geometries. In future works, it will be interesting to test new numerical schemes for the discretization of the formulation, such as the degenerated hexahedral Whitney elementsin a finite element scheme as presented in .
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Published Online: 2018-04-18