**5. Conclusions**

In view of the increase of coil ESR caused by substrate, based on theoretical modeling, simulation calculation, and experimental measurement, a systematic study on the variation rule of coil ESR with a single-layer substrate and double-layer substrate is carried out in this paper. According to the results, when the size of single-layer substrate is much larger than the coil size, the material with the smallest conductivity should be selected as the substrate. The thickness should be increased in case of large conductivity of the substrate, with an optimal value whereas. When the size of single-layer substrate is close to the diameter of the coil outer ring, coil ESR is proportional to the area of ferrite. However, with the increase of the area of aluminum plate, coil ESR increases first and then decreases. At the constant volume, the closer the side length of ferrite substrate is to the diameter of coil outer ring, the larger coil ESR is. Coil ESR value with aluminum plate reaches the minimum at the thickness of 5 mm. For the double-layer substrate, when its size is far larger than the coil size, coil ESR is only subject to the thickness of ferrite substrate in the first layer, rather than that of aluminum plate in the second layer. When the size of substrate is close to the diameter of coil outer ring, coil ESR increases first and then decreases with the increase of area and finally tends to be constant. The peak value appears when the size is close to the diameter of coil inner ring. At the constant volume, coil ESR increases with the increase of the thickness of double-layer substrate. It is believed that the foresaid rules shall assist researchers' e fficient selection of the thickness, area and volume of substrate, thus reducing the coil loss and realizing the optimal design of WPT system. Although this paper summarizes the influence of substrate with di fferent size on the coil ESR, the variation law of the coil ESR needs further study when the excitation frequency of the coil, the distance between the substrate and the coil, and the shape of the substrate change.

**Author Contributions:** Conceptualization, J.L.; methodology, J.L., R.W., and M.Z.; software, J.L.; investigation, M.Z.; writing—original draft preparation, J.L.; writing—review and editing, L.T. and X.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Key Research and Development Project, gran<sup>t</sup> number 2018YFB0106300.

**Conflicts of Interest:** The authors declare no conflict of interest.
