# Analytical Solution and Experimental Validation of the Electromagnetic Field in an IPT System
## 1 Introduction  
Inductive power transfer (IPT) technology is a new type of energy transfer method by which energy is transferred from source to load wirelessly, according to the electromagnetic induction theory. In recent years, IPT has developed rapidly and been used for electric vehicles [1; 2], biomedical implantable devices [3; 4] and wireless charging in intelligent mobile phones [5]. An adaptive four-coil structure was used to improve the power transfer efficiency of the whole system [3]. A square coil was utilized to maximize the power received by the load and the resonant frequency was adjusted to maximize the coil's quality factor [4]. A mobile phone shell and the coil in the secondary side were used as the resonance bodies, which proposed a novel way to provide electrical energy for a mobile phone [5]. There are many applications for IPT systems, but the power transfer mechanism in between the coupled coils in an IPT system is indistinct. Therefore, research on the mechanism of energy transfer of the coil coupler is very important for IPT technology. Currently, research methods for IPT systems can mainly be classified into two categories: The coupled-mode theory and the equivalent circuit method. These have been proven to be equivalent through analysis with a 4-coil IPT system [6]. In 2007, the team led by [PERSON] in MIT (Massachusetts Institute of Technology) elaborated on the analysis and modelling method of the system from the perspective of the coupled-mode theory, which laid a theoretical foundation for the further development and popularization of IPT technology [7]. Coupled-mode theory is used to study the coupling principle of power transmission. The power transfer process from one resonant body to another can be expressed as a first-order partial differential equation according to it. This equation provides an accurate and effective modelling method for IPT systems, as well asan analytical tool to understand the power transfer mechanism in strongly coupled systems. In IPT systems, when the two resonant coils are close, strong coupling exists between them. Under these conditions, with the help of coupled-mode theory, it was found that the energy loss in the primary coil was much slower than the power transferred from the primary coil to the secondary coil [8], thus, the energy transmission was quite efficient. From the perspective of circuit theory and practical applications, the mutual inductance model was used to obtain the relationship between system transfer efficiency and parameters. The mutual inductance model can also be utilized to analyze the power transfer capability as well [9; 10]. Although the mutual inductance model is mature, its accuracy depends on the accurate acquisition of the coil parameters. In high-frequency cases, the coil impedance characteristic varies greatly with the operating frequency. Coupled-mode theory has been widely adopted in recent years, but in the analysis process, the state quantities are employed instead of electrical parameters. In addition, the coupled-mode theory model is relatively abstract, and it is difficult to build the mathematical model with lumped circuit parameters. By using the Biot-Savart law, the analytical solution of the electromagnetic field in the IPT system was calculated through the magnetic vector potential. However, the derivation process of the analytical solution of the electric field and magnetic field in IPT systems and the experimental verification were not introduced [11]. In [12], the derivation of the magnetic field, based on the analysis of a current-carrying conductor, is accurately calculated and a good approximation has been done when the far-field is calculated. The mutual inductance and self-inductance of two coaxial coils in a multilayer media have been introduced, and the frequency limitations of magnetic substrates are also established [13].