# Numerical Simulation-Based Investigation of the Limits of Different Quasistatic Models  
[PERSON]  
1 Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697-1-2 EP, 59000 Lille, France; [EMAIL_ADDRESS] (M.T.); [EMAIL_ADDRESS] (T.H.); [EMAIL_ADDRESS] (Y.L.M.); [EMAIL_ADDRESS] (F.S[PERSON])  
[PERSON]  
1 Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697-1-2 EP[PERSON] 59000 Lille, France; [EMAIL_ADDRESS] (M.T.); [EMAIL_ADDRESS] (T.H.); [EMAIL_ADDRESS] (Y.L.M.); [EMAIL_ADDRESS] (F.S[PERSON])  
[PERSON]  
1 Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697-1-2 EP, 59000 Lille, France; [EMAIL_ADDRESS] (M.T.); [EMAIL_ADDRESS] (T.H.); [EMAIL_ADDRESS] (Y.L.M.); [EMAIL_ADDRESS] (F.S.)  
[PERSON]  
1 Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697-1-2 EP, 59000 Lille, France; [EMAIL_ADDRESS] (M.T.); [EMAIL_ADDRESS] (T.H.); [EMAIL_ADDRESS] (Y.L.M.); [EMAIL_ADDRESS] (F.S.)  
[PERSON]  
1 Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697-1-2 EP, 59000 Lille, France; [EMAIL_ADDRESS] (M.T.); [EMAIL_ADDRESS] (T.H.); [EMAIL_ADDRESS] (Y.L.M.); [EMAIL_ADDRESS] (F.S.)  
[PERSON]  
2 EDF R&D, ERMES, 7 Boulevard Caspard Monge, 91120 Palaiseau, France; [EMAIL_ADDRESS]  
on the distribution of the current density in the winding, the coupled resistive-inductive effects have to be taken into account, and so the magneto-quasistatic (MQS) [1] model is generally used.  
Moreover, if all these coupled effects are characterized at the same time, the full Maxwell model [2; 6] can be used, even it is more time-consuming and has an instability issue at low frequencies, as reported in [6; 7]. On the other hand, if the modeling is only considered in the intermediate frequency range, the [PERSON] model can be a good choice [8; 9; 10; 11; 12], which includes all the capacitive, resistive, and inductive effects, but neglects the radiation one.  
In the literature, the classification or the theoretical limitations of the different quasistatic models are mainly based on theoretical considerations, with several assumptions for the computational domain. For example, only the conductive domain is considered, not the case with multi-domains. The existing reference [13] showed a comparison of the results obtained with different quasistatic models in the time domain, when a simple axisymmetric test model, represented by a parallel plate capacitor, is used. The comparison results were illustrated for the electric field.  
The aim of our work is to investigate the limits of the above-mentioned quasistatic models in the general case, particularly for the computational domain, including both conductive and non-conductive domains, which has not been well addressed in the literature. We are interested in the frequency domain due to the impedances' computation.  
To achieve this objective, in this work, we first validated the Darwin model with an industrial example by comparing our numerical results and measurement results. To the best of the author's knowledge, the comparison of the Darwin model with real measurement results has not been reported in the literature. Secondly, with an academic example, we compared all the quasistatic models in the frequency domain, by regarding different numerical aspects, namely, the current density, the electric field, the magnetic field, and the impedance.  
The paper is organized as follows: the full-wave Maxwell system is briefly introduced in Section 2. In Section 3, the EOS, MQS, and Darwin models are presented. In Section 4, two different examples, an industrial application and an academic electromagnetic device are provided for the numerical parts. Finally, the conclusion is given in Section 5.