Modelling of negative equivalent magnetic reluctance structure and its application in weak-coupling wireless power transmission

In weak-coupling wireless power transmission, increasing operating frequency, and incorporating metamaterials, resonance structures or ferrite cores have been explored as effective solutions to enhance power efficiency. However, these solutions present significant challenges that need to be addressed. The increased operating frequency boosts ferrite core losses when it exceeds the working frequency range of the material. Existing metamaterial-based solutions present challenges in terms of requiring additional space for slab installation, resulting in increased overall size. In addition, limitations are faced in using Snell’s law for explaining the effects of metamaterial-based solutions outside the transmission path, where the magnetic field can not be reflected or refracted. To address these issues, in this work, the concept of a negative equivalent magnetic reluctance structure is proposed and the metamaterial theory is extended with the proposed magnetic reluctance modelling method. Especially, the negative equivalent magnetic reluctance structure is effectively employed in the weak-coupling wireless power transfer system. The proposed negative equivalent magnetic reluctance structure is verified by the stacked negative equivalent magnetic reluctance structure-based transformer experiments and two-coil mutual inductance experiments. Besides, the transmission gain, power experiments and loss analysis experiments verify the effectiveness of the proposed structure in the weak-coupling wireless power transfer system.


Supplementary Note 2. Parameters and configuration of the NEMR structure-based WPT system
The schematic configuration of the proposed NEMR structure and WPT system is given in Figure S1.The system consists of a receiver coil, transmitter coil, and two coil-embedded NEMR structures with a transfer distance, d.The geometric parameters of the transmitter and receiver coils, as well as the NEMR structure, are given in Table S2.The diameter l of the substrate FR4 plate is 150 mm.The input power of the transmitter coil is set as 1 W and the load in the receiver coil is selected as 50 ohm.Considering the load of those four systems [ i) system without NEME structure, ii) system with NEMR structure in the transmitter coil, iii) system with NEMR structure in the receiver coil, iv) system with NEMR structure in both the transmitter and receiver coil], and the input power is constant, the higher current in the receiver coil indicates a higher receiver power and efficiency.The magnetic field intensity is directly proportional to the current  in a conductor (receiver) based on Ampère's circuital law, described as follows.
Considering that the input power of the transmitter coil is constant, by comparing the magnetic field strength H around the receiver coil, the current I in the receiver coil can be obtained and the efficiency comparison between WPT systems can be found roughly.
As the given results in the manuscript, the NEMR structures increase the magnetic field intensity H around the receiver coil to a different extent, which is directly connected to the power transfer efficiency of the WPT system.
Supplementary Note 4. The detailed analysis of the transformer with core of stacked NEMR structures.
The voltage proportion between the primary coil and the secondary coil is determined by the self-inductance, mutual inductance, and current in coils, given in Eq. ( S2).
where   ,  ,   ,   ,   ,   ,   ,   are the voltage, self-inductance, current, and resistance of the primary side and secondary side, respectively;  is the mutual inductance between the two coils.
In the no-load operating condition, the secondary current   is considered as zero.The relationship between input voltage   and output voltage   can be found as follows.
In this transformer, as the primary coil and secondary coil share the same magnetic core and magnetic flux path, the magnetic reluctance of the primary inductance and secondary inductance are considered to be almost the same.Besides, the number of turns of the abovementioned coils are the same.Hence, based on the definition of the mutual inductance given in Eq. ( S4), the mutual inductance is considered to be k times the inductance of the primary coil.
The coupling coefficient k is determined by the magnetic core and its value is between 0 to 1.
The total magnetic reluctance of the NEMR structure-based transformer is defined as   , consisting of the magnetic reluctance of the proposed structure   and that of air   .
Based on the definition of mutual inductance and inductance   =  2 /  , Eq. ( S3) can be rewritten as.
Based on Eq. ( S5), as the primary resistance   is greater than zero and the coupling coefficient  is less than 1, only if the magnetic reluctance   is negative, the proportion between the output voltage   and primary voltage   could be larger than 1.

Supplementary Note 5. The analysis of mutual inductance on the efficiency of the proposed WPT system
The voltage proportion between the impact of mutual inductance on the efficiency of the WPT system is concluded as follows.The generalized equivalent circuit of the WPT system with the series-series (SS) compensation network is shown in Figure S2.In Figure S2,   and   are the AC power supply and the load of the WPT system, respectively.  and   ,   and   ,   and   are the internal resistance, compensation capacitor, and coil self-inductance of the primary and secondary sides, respectively.
The Kirchhoff voltage balance equation for the SS topology WPT system in Figure 3 can be expressed as follows.
where   and   are the currents of the primary and secondary coils of the WPT system, respectively.
Based on Equation S6, the input power   and the output power   of the WPT system can be found as follows.
Equivalent circuit of the WPT system with series-series network.
where  is the phase difference between the primary voltage and current.
Under the rated operating condition, the frequency  equals the resonant frequency   of the system, which should meet the requirements of   = ) zero.Under this condition, the value of the phase difference  is near zero and cos() is equal to 1. Based on Equations.S6, and S7, the efficiency  of the WPT system is expressed as follows.
As shown in Equation ( S8), the mutual inductance has a great impact on the efficiency of the WPT system.As for the generalized kHz WPT system with short transfer distance (smaller than a quarter of coil diameter) and large coil size, the mutual inductance  between the primary and secondary coil is larger enough (about mH) and the coupling coefficient is always larger than 0.15.In this condition, the internal resistance of the primary side   and that of the secondary side   would have a slight impact on the efficiency.However, as for the weak coupling WPT system, the mutual inductance  and coupling coefficient are small.Therefore, the efficiency of the WPT system is highly dependent on the resistors   and   .
where   and   are the leakage flux linkage and main flux linkage, respectively; while   and   are the leakage flux and mutual flux, respectively.
As for the proposed design, based on the magnetic circuit in Figure .S4, as well as Equation S13 and S14, Equation S14 can be rewritten as.
where   is the magnetic reluctance of the magnetic core determined by the material, which where  0 and   is the vacuum and relative permeability of the core material, respectively.  and   are the magnetic reluctance of ferrite material and NEMR structure, respectively.Considering the relative permeability   of air is equal to 1, the permeability of air is equal to   0  .
As shown in Equation S15 and S16, the mutual inductance of the two-coil system is inversely proportional to   .Hence, if   is lower than   , the effect of NEMR structure on mutual inductance enhancement is better than that of ferrite material theoretically.

Figure S1 .
Figure S1.The configuration of the proposed NEMR structure-based WPT system.

Figure S4 .
Figure S4.The equivalent magnetic circuit of the proposed design.

Table S1 .
General Comparison of the Proposed NEMR Structure and State-of-art Works For Weak-

Table S2 .
Geometric parameters of the coils and NEMR structure