# Design of Magnetic Concrete for Inductive Power Transfer System in Rail Applications
## 3 Materials and Methods
### Electromagnetic-Thermal Simulation in IPT Systems  
A coupled electromagnetic-thermal simulation model was developed to evaluate the heat generation and temperature distribution within the IPT system under 85 kHz operation. The electromagnetic analysis is based on [PERSON]'s equations in the frequency domain, which describes how time-varying electric and magnetic fields propagate and interact with materials. The primary field quantities considered are the magnetic vector potential \(A\) and the electric scalar potential \(V\). The governing equation for \(A\) in a lossy medium is derived from [PERSON]'s law and [PERSON]'s law, given by the following:  
\[\
abla\times\mathbf{H}=\mathbf{J_{e}}+\mathbf{J_{i}},\ \
abla\times\mathbf{E}=-\mathbf{j}\omega\mathbf{B} \tag{3}\]  
where \(\mathbf{H}\) is the magnetic field intensity, \(\mathbf{E}\) is the electric field intensity, \(\mathbf{B}\) is the magnetic flux density, \(\mathbf{J_{e}}\) is the excitation current density, and \(\mathbf{J_{i}}\) is the induced current density. We used the following constitutive relations:  
\[\mathbf{B}=\mathbf{\mu}\mathbf{H},\ \mathbf{J_{i}}=\sigma\mathbf{E} \tag{4}\]  
where \(\mathbf{\mu}=\mu_{0}\mathbf{\mu_{r}}\) is the absolute magnetic permeability and \(\sigma\) is the electrical conductivity. \(\mathbf{A}\) is defined as follows:  
\[\mathbf{B}=\
abla\times\mathbf{A} \tag{5}\]  
Substituting into [PERSON]'s law gives the following:  
\[\mathbf{E}=-\mathbf{j}\omega\mathbf{A}-\
abla\mathbf{V} \tag{6}\]  
Substituting Equations (4) and (6) into [PERSON]'s law results in the following:  
\[\
abla\times(\frac{1}{\mathbf{\mu}}\
abla\times\mathbf{A})+\mathbf{j}\omega\sigma\mathbf{A}+ \sigma\
abla\mathbf{V}=\mathbf{J_{e}} \tag{7}\]  
In time-harmonic eddy current simulations, the scalar potential gradient \(\
abla\mathbf{V}\) is often neglected due to the use of the coulomb gauge or the assumption of low capacitive effects. This simplifies the formulation to the following:  
\[\
abla\times(\frac{1}{\mathbf{\mu}}\
abla\times\mathbf{A})+\mathbf{j}\omega\sigma\mathbf{A}= \mathbf{J_{e}} \tag{8}\]  
After solving for \(\mathbf{A}\), the magnetic flux density \(\mathbf{B}\) and electric field \(\mathbf{E}\) are calculated using Equations (5) and (6), respectively. The induced current density \(\mathbf{J_{i}}\) is then obtained from Equation (4), which defines the relationship \(\mathbf{J_{i}}=\sigma\mathbf{E}\). The volumetric Ohmic power loss \(\mathbf{P_{ohmic}}\), which primarily occurs in conductive regions such as Litz wire, is calculated by the following:  
\[\mathbf{P_{ohmic}}=\frac{1}{2}\frac{\left|\mathbf{J_{i}}\right|^{2}}{\sigma} \tag{9}\]  
Core losses in ferrite materials are modeled using an empirical Steinmetz-type expression:  
\[\mathbf{P_{core\_loss}}=\mathbf{C_{m}}\mathbf{f}^{e}\mathbf{B}_{m}^{y} \tag{10}\]  
where \(\mathbf{P_{core\_loss}}\) is the volumetric core loss, \(\mathbf{C_{m}}\) is a material-specific coefficient, \(\mathbf{B_{m}}\) is the peak magnetic flux density, and \(\mathbf{x}\) and \(\mathbf{y}\) are empirical exponents. This model is applied only to ferrite cores, where \(\mathbf{C_{m}}=3.2\times 10^{-3}\), \(\mathbf{x}=1.46\), and \(\mathbf{y}=2.75\)[62]. The thermal analysis is performed by solving the steady-state heat conduction equation:  
\[\
abla\cdot(\mathbf{k}\
abla\mathbf{\Gamma})+\mathbf{P}_{\mathbf{total}}=0 \tag{11}\]  
where \(\mathbf{k}\) is the thermal conductivity, \(\mathbf{T}\) is the temperature, and \(\mathbf{P}_{\mathbf{total}}\) is the total volumetric heat generation applied as input to the thermal solver. The simulation workflow, depicted in Figure 7, begins with defining the initial parameters, including material properties, excitation current, and initial temperature settings. The process consists of several key steps. First, system parameters are established--specifically, material properties such as thermal conductivity and magnetic permeability, along with the excitation current based on the system's operating conditions.