# Research on Three-Phase Wireless Power Transfer System
## 3 Experiment and Analysis  
### Comsol Simulation  
At present, there are many types of coupling mechanisms in the wireless power transfer system. By analyzing the coil structure in the existing literature, it is concluded that the coupling coefficient and transmission effect of the three-coil structure are better. In order to verify the correctness and feasibility of the three-phase coil, we will perform COMSOL Multiphysics 5.6 simulation verification on the transmission effects of different coil structures. Due to the magnetic effect of the current, the energized coil will generate a magnetic field in space. This phenomenon can be described by [PERSON]'s equations, including the  
Figure 3: Topology of three-phase wireless power transfer system. [PERSON] loop law and [PERSON]'s law of electromagnetic induction in [PERSON]'s equations. The [PERSON]'s loop law describes the magnetic field generated by the current passing through the conductor, while [PERSON]'s law of electromagnetic induction shows that a change in the magnetic field will induce an electric field. These equations describe the interaction of electromagnetic fields, which are very important for understanding and simulating the magnetic field characteristics of energized coils. \[\left\{ \begin{array}{l}\
abla\times E=-\frac{\partial B}{\partial t}\\ \
abla\times H=J\\ B=\mu H,J=\sigma E\end{array} \right. \tag{5}\]  
where \(B\) is the magnetic induction intensity of any point in space; \(H\) is the magnetic field intensity; \(J\) is the current density; \(J=0\) in the air; \(J=J/A\) in the coil (\(I\), \(A\) are the coil current and the cross-sectional area of the wire, respectively); \(E\) is the induced electromotive force; \(\mu\) is the permeability of the material; and \(\sigma\) is the conductivity of the material. The magnetic diffusion equation between coils can be derived from [PERSON]'s equation:  
\[\frac{\
abla\times\
abla\times B}{\mu\sigma}+\frac{\partial B}{\partial t}=0 \tag{6}\]  
According to the formula, \(\
abla\times\
abla\times B=\
abla\cdot(\
abla\cdot B)-\
abla^{2}B\) and \(\
abla\cdot B=0\) can obtain the magnetic density \(B\) diffusion equation:  
\[-\frac{1}{\mu\sigma}\
abla^{2}B+\frac{\partial B}{\partial t}=0 \tag{7}\]  
The loss model is added to calculate the transmission efficiency. The main loss comes from the resistance heat expression of the coil. When the magnetic diffusion equation is applied to the coupling coil, the three dimensions of \(x\), \(y\), and \(z\) can be calculated separately:  
\[\left\{ \begin{array}{l}\frac{1}{\mu\sigma}(\frac{\partial^{2}B_{y}}{\partial y ^{2}}+\frac{\partial^{2}B_{z}}{\partial z^{2}})=\frac{\partial B_{x}}{ \partial t}\\ \frac{1}{\mu\sigma}(\frac{\partial^{2}B_{y}}{\partial x^{2}}+\frac{\partial^ {2}B_{y}}{\partial y^{2}})=\frac{\partial B_{y}}{\partial t}\\ \frac{1}{\mu\sigma}(\frac{\partial^{2}B_{z}}{\partial x^{2}}+\frac{\partial^ {2}B_{z}}{\partial y^{2}})=\frac{\partial B_{z}}{\partial t}\end{array} \right. \tag{8}\]  
where \(\sigma\) is the conductivity of the material, \(B_{x}\) is the \(x\)-axis component of the magnetic flux density in the coil space, \(B_{y}\) is the \(y\)-axis component of the magnetic flux density, and \(B_{z}\) is the \(z\)-axis component of the magnetic flux density. The magnetic flux density of the space can be calculated by the above formula. \[Q=\frac{1}{T}\int\limits_{Tend^{-T}}^{Tend}J\cdot Edt \tag{9}\]  
In order to simplify the finite element analysis, the Litz coil is simplified into a multistrand lumped coil in the COMSOL simulation. This method can effectively simplify the details while retaining the main electromagnetic characteristics. This simplified method is particularly useful in the design of coupling mechanisms because it can reduce the computational cost while maintaining accuracy. The specific parameters of the COMSOL simulation comparison are shown in Table 1. The distance D is the distance between the transmitting coil and the receiving coil.