# A Multi-Domain Model for Variable Gap Iron-Cored Wireless Power Transmission System
## 1 Introduction  
Wireless power technologies represent a growing industry and research field due to the increasing employment of battery-powered devices. As described in [1, 2, 3], the main technologies for wireless power devices in the near field are inductive power transfer (IPT) and capacitive power transfer (CPT). Studies [4, 5, 6] describe how these types of applications are already heavily used in the automotive and telecommunication field, generally on systems that are capable of disposing of high amounts of energy on the primary side, thus, compensating for possible efficiency losses during coupled communication [7, 8]. In systems where the amount of available energy on the transmitter is significantly contained, or in applications where the gap presents a continuous chaotic fluctuation that drastically affects the transmission performances--such as in tiny mechatronic application, implanted medical devices or, more generally, in small portable electronic devices--the optimization of the coupling factor is decisive. In the literature, the current standard approach sees an equivalent circuit-based analysis; this simplification is acceptable since NFC devices have a light coupling between the two air-cored coils [9, 10]. Some further effort can be found in automotive applications where the necessity of transferring energy at frequencies higher than tens of MHz [4], and the necessity to embed the devices in an external medium [11, 12], push the analysis towards the integration of finite element analysis (FEA) models [13]. Instead, this paper aims to introduce a co-simulation methodology and define a different characterization procedure for the analysis and evaluation of the dynamic and energetic performances of cored-coil IPT devices subjected to gap fluctuations. Figure 1 presents an explicative schematization of the device taken under analysis. The apparatus can essentially be described as a ferrite core coupled inductor used in high-frequency applications, in which the unwanted chaotic variation of the air gap, due to the geometric application characteristics of the assembly schematized here with Body 1 and Body 2, greatly influences the energy transmission performances. From a mathematical point of view, the main objective is to provide a general and reliable multi-physics model based onto the co-simulation of a Spice-based ECM analysis and ESRF Radia-based 3 DFVM analysis, placing particular emphasis on the intrinsic sensitivity with respect to variables that cannot be directly controlled, such as the variations in transmission efficiency induced by undesirable changes in the coupling factor. This behavior pushes towards the introduction of a co-simulation architecture capable of integrating variations imposed in the electrical and magnetic domains, resulting from a modification of the mechanical and geometric properties of the device, and introducing in the numerical model an analytical-based characterization of the static and dynamic phenomena, such as: self/mutual inductance, coupling factor, stray capacitive behavior, and parasitic core and copper losses related to high-frequency phenomena. Furthermore, this work provides a detailed outline of a smart and effective experimental methodology for the estimation of the previously reported parameters and the validation of numerical models, giving particular importance to the reconstruction of the transfer performances in the time domain and frequency domain as a result of an imposed and controlled variation of the air gap through a dedicated actuation and measurement apparatus. The paper is structured as follows: Section 2 evaluates the mutual induction mathematical models, focusing on the description of the static parameters and the method by which they were obtained. Section 3 highlights the state-of-the-art mathematical models available in the literature for the modeling and analysis of dynamic coupling aspects such as high-frequency AC core and copper losses.