Study on Calculation Methods of AC Loss for a HTS Magnet with Iron Core

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The great advantage of HTS coils in magnet is that it can provide large excitation in a limited space. However, under high level excitation condition, especially in the case of fast adjusting process, AC losses will occur and lead to reduction of thermal stability [1], [2]. Therefore, it is necessary to calculate AC loss of HTS coils quickly and accurately. The homogenization method based on H formulation basically meets the calculation requirements of AC loss for a thousand-turn magnet with simple structure [3]. However, for magnets with non-linear ferromagnetic materials such as HTS controllable reactors [4], [5], the non-linear saturation in ferromagnetic domains makes it difficult to calculate AC loss rapidly and accurately with homogenization methods. Three simplified calculation methods of AC loss for HTS magnets have been proposed in this paper. In order to verify the validity of the simplified algorithm quickly and effectively, a two-dimensional axisymmetric model is adopted. The key to reducing the nonlinearity is to quickly calculate the magnetic permeability distribution in the core region. Three methods have been proposed to simplify the calculation. The first method is A+H formulation method. In the A formulation, set the same resistivity through the current flowing area, apply a uniform current density excitation and set BH magnetic properties in the core. In the H formulation, set the nonlinear resistivity decided by E-J characteristic through the current flowing area [6], apply a total current constraint and the magnetic permeability of the core region is from the real-time calculation of the A formulation results. The coupling between the two formulations does not occur during the solution, but the permeability of the core in the H formulation is provided by the calculation of the A formulation. The second method is magnetic permeability transfer of the core region method. Core area is divided into different regions. The magnetic permeability in all the core regions is solved in the A formulation, which is made into a data table. Then the magnetic permeability is applied in the H formulation model as a known item. The third method is A formulation coupling with H formulation method (A&H formulation). A formulation and H formulation share the same model. The PDE module (the control formulation is the H formulation) only contains the superconducting domain and part of the air domain around the superconducting domain. All the domains are contained in the magnetic field module (the control formulation is the A formulation). The core domain is described by the BH curve. The HTS coils are excited by the uniform current density. The section boundary of air domain is shared with the PDE module to transfer the magnetic field strength to the PDE module. The example model is a small iron-containing superconducting magnet, which is modeled with a homogenization method. Considering that the H-formulation method is widely used and supported by a large number of experiments [7], [8], the present example uses the H-formulation results as the benchmark in the error analysis of each method. Fig.1 shows the calculation time with different methods. In the linear discrete model, A+H formulation solves two physical formulations for the whole domain, which has the highest degree of freedom and a 13.5% increase in the number of degrees of freedom compared with other linear discrete models. The A-formulation coupling with H-formulation adopts quadratic discrete and has the highest degree of freedom, which also has the longest solution time. Fig.2 shows the average loss of each method. The average loss of each model with linear discreteness is not much different from that of the H-formulation model and the maximum deviation is 0.527%. The error of the third method using the quadratic discrete is larger, ranging from 11.59% to 20.47%. Both of them are larger than the H-formulation calculation results. In summary, the first method is the first choice when we calculate AC Loss for a HTS Magnet with Iron Core. If the degree of freedom is too large in the model, we can choose the second method. If the AC loss is still difficult to calculate, we need to choose the third method at the expense of a little accuracy. This paper presents three simplified calculation methods of AC loss for HTS magnets, which make it possible to calculate the AC loss of the core-containing superconducting magnet rapidly and accurately. The first two methods have high precision, but they are not suitable for the large-scale model. The third method can calculate the large-scale model quickly, but there is a small amount of error.