In the marine environment, the inductive power transfer (IPT) technology offers many benefits such as increased flexibility and complete rejection of the winding problem. However, for underwater vehicles, the impact of seawater flow vibration will affect the stability of the power transmission. In this paper, a symmetrical double coil structure named 1×1×1 system is proposed to reduce the eddy current loss (ECL) and enhance the power transmission stability of the system under the impact of seawater flow vibration. Firstly, based on the discretization of the electric fields excited by each turn of two transmitting coils, the novel calculation method proposed is that the synthetic electric field at each discrete point is the sum of the electric field vector. Then the synthetic electric field and induced voltage of the 1×1×1 system are simulated by the finite element simulation software COMSOL to verify the calculated results. Finally, experiments are conducted to illustrate that the 1×1×1 system has good ability to resist Z-axial interference and reduce the impact of the seawater flow vibration.

## I. INTRODUCTION

Inductive power transfer (IPT) technology has been applied in many fields, and the influences of coil structure on the energy transmission system have gotten more attention. On the one hand, many researchers improved the transmission efficiency and distance by changing the coil structure. F. Jolani et al. used the manufacturing transmitting and receiving coils with the plane printed coil and printing circuit to reduce the resistance of the printed coil resonator and increased its quality factor.^{1} Zhang W showed that the coupling coefficient of the coil is mainly determined by the coil structure and size, which is independent of the number of turns.^{2} On the other hand, the influences of the number of coils on the system performance are studied. The IPT system with relay coils is proposed by Sang-Cheol Moon et al, and the enhancement of mutual inductance increases the system transmission efficiency.^{3} Y. Zhang et al. proposed that even number of resonators in IPT system will produce constant current characteristics, and the odd number of resonators will produce constant voltage characteristics.^{4}

The remarkable difference between WPT systems in seawater and air is that the electric field passing through seawater causes eddy current loss (ECL), which inevitably impairs the transmission efficiency of the WPT systems. A single circular coil is proposed to calculate the eddy current loss and an IPT prototype is designed and built to analyze and measure the eddy current loss. The test shows that the power transmission efficiency is about 85% with 5 millimeters gap distance.^{5} In order to get more accurate analytical solutions, the eddy current loss calculation model of spiral winding is presented.^{6} Although the IPT system in seawater using four-coil structure can store more energy and enhance output power, the loss increases because of the long distance.^{7}

This article proposes a new three-coil structure utilizing two transmitter coils placed symmetrically adjacent to each side of the receiver coil, which can reduce the ECL generated by the two transmitter coils in Section II. Section III establishes the method to calculate the synthetic electric field of two transmitter coils considering the distance between the gap of the coils. The results were verified by finite element simulation and the experiment. Finally, the advantages of the three-coil structure to resist the seawater flow vibration are analyzed in Section IV.

## II. ANALYSIS OF EDDY CURRENT LOSS IN SEAWATER

In this section, the electric fields of a 1 × 1 system and a 1×1×1 system are analyzed, as depicted in FIG. 1, and the eddy current losses in the two systems are compared. The eddy current losses in the 1 × 1 system and the 1×1×1 system are compared under the condition that *M*_{13} equals *M*_{23}, where *M*_{13} and *M*_{23} are the mutual inductances between Coil_{1} and Coil_{3} and between Coil_{2} and Coil_{3}, respectively. In FIG. 1(a), the center coordinates of the transmitting coil, Coil_{1}, and the receiving coil, Coil_{3,} are (0, 0, -*L*_{13}) and (0, 0, 0), respectively. In the 1×1×1 configuration, there is another transmitting coil, Coil_{2}, shown in FIG. 1(b), centered at (0, 0, *L*_{13}).

In FIG. 1(a), *E*_{si-j} represents the electric field excited by Coil_{i} in Zone *j*. In FIG. 1(b), *E*_{mi-j} represents the electric field excited by Coil_{i} in Zone *j*. To compare the eddy current losses of the 1×1 and the 1×1×1 systems when the power transferred to the receiver side of each system is equal, *I*_{m3} is set to be equivalent to *I*_{s3}, and the following expressions is obtained:

From (1), the reduction of the eddy current loss in the 1×1×1 system is equal to half of the eddy current loss caused by the transmitting coil in the 1×1 system.

## III. THE SYNTHETIC ELECTRIC FIELD OF TWO TRANSMITTER COILS

Due to the seawater flow impact, the distance between the primary coil and secondary coil will be changed in the 1 × 1 system, then it will change the mutual inductance between Coil_{1} and Coil_{3}, and influence the stability of the power transmission. The 1 × 1 × 1 system ensures that the system has a strong resistance against the axial displacement. When the axial displacement occurs, the transmission power of the system is almost unchanged. In order to keep the stability of electric field in the certain range, a synthetic electric field model is built in FIG. 2(a), where *r*_{p}=172mm, *r*_{s}=147mm.

In FIG. 2(a), the nearest distance between the Coil_{1} and Coil_{2} are *L*_{12}. The turns of Coil_{1} and Coil_{2} closest to the original point are the first turn, named the Coil_{1.1} and the Coil_{2.1} respectively, and the turn of the Coil_{1} and the Coil_{2} which keeps the gap *d*_{1} away from the original point, are expressed as Coil_{1.n1}, *n*1=1…*N*_{1} and Coil_{2.n2}, *n*2=1… *N*_{2}. *N*_{1,} *N*_{2} are the turns of the Coil_{1} and Coil_{2}. The displacement area of the receiver coil is demonstrated as 2Δ*S*, which is divided into 2*k*+1 points with the minimum *d* gap. P (0, *r*_{s}, *z*) is the point on the Coil_{3}, and the electric field at P point generated by the Coil_{1.n1} and Coil_{2.n2} can be expressed as *E*_{1.n1}(*z*), *E*_{2.n2}(*z*).

Since it is difficult to determine the primitive function of the integrand $f\lambda =\lambda /u\u22c5J1\lambda rpJ1\lambda rse\u2212u|(n1\u22121)d+L12/2+z|$, the numerical integration method is used to acquire an approximation. In the numerical integration method, the integration interval is divided into parts, and the sum of their areas is the approximation of the integral of the function *f*(*λ*).

When *I*_{m1}= *I*_{m2}=1A, *f*=300kHz, *d*_{1}=6.8mm, *d*=2mm, *L*_{12}=28mm, *N*_{1}=*N*_{2}=4, the electric field waveform diagram excited by Coil_{1.N1} and Coil_{2.N2} is obtained by using Matlab as FIG. 2(b). At 2*k*+1 points, *E*_{j.nj}(*z*) is discretized as *E*_{j.nj}(*id*), *i*=-*k*…0…*k*. *j*=1,2 represents the electric field excited by Coil_{1.n1} and Coil_{2.n2}.

So the synthetic electric field at each point of 2*k*+1 points is the sum of the electric field vector generated by the Coil_{1} and Coil_{2}, which is expressed as follows:

*E*(*id*) represents the synthetic electric field at *z*=*id* point. Therefore, the inductive voltage of receiving Coil_{3} is indicated as follows:

*N*_{3} is the turns of the Coil_{3}. *E* (*md*) represents the synthetic electric field at the position (*z*=*md*) of the left-most turn of Coil_{3}.

## IV. CALCULATIONS, SIMULATIONS AND EXPERIMENTS

In this section, in all calculations, simulations and experiments, the fundamental parameters are *N*_{1}=*N*_{2}=11, *N*_{3}=5, and the impedance of the load *R*_{L}=30Ω, *f* =300kHz, *d*_{1}=6.8mm, *d*=2mm, σ=3.38S/m. In FIG. 2, under the condition that the power transferred to the receiving side of each system is equal, *I*_{s1}= 2 A in the 1×1system and *I*_{m1}= *I*_{m2}=1A in the 1×1×1 system.

In the displacement area from -10*d* to 10*d*, the synthetic electric fields of the 1×1 system and the 1×1×1 system are calculated using (3) and simulated utilizing the finite element simulation software COMSOL. The calculated and simulated synthetic electric field of different systems is shown in FIG. 3(a). In the 1×1×1 system, the synthetic electric field is calculated and simulated with different *L*_{12} in FIG. 3(b). Besides, the induced voltage using (4) is calculated and simulated from -10*d* to 10*d* at different *L*_{12} as shown in FIG. 4(a), which is compared with the experimental results when *L*_{12} is 28mm in FIG. 4(b).

According to FIG. 3, the calculated results are in good agreement with the simulation results, which shows that the calculation method is rational. Apparently, the synthetic electric field of the 1×1×1 system in the displacement area is steadier than that of the 1×1 system.

In FIG. 4(a), the change of induced voltage is within 5% in the displacement area, illustrating that the 1×1×1 system has good ability to resist the Z-axial interference, so the 1×1×1 system can reduce the impact of the seawater flow vibration. Besides, *L*_{12} has influence on the stability of the synthetic electric field and can be optimized in the actual application. When *L*_{12} is 28mm, the calculated, simulated and the experimental induced voltages in the receiver coil are showed in FIG. 4(b). The results are basically the same, which shows that the method proposed in this paper is reasonable.

## V. CONCLUSION

This paper has investigated a new three-coil structure named 1×1×1 system. Based on the discretization of the electric fields excited by each turn of two transmitter coils, the novel calculation method is that the synthetic electric field at each discrete point is the sum of the electric field vector. The induced voltage of receiver coil in the displacement area of the 1×1×1 system can keep almost invariable, which demonstrates that the 1×1×1 system can resist the impact of the seawater flow vibration.

## ACKNOWLEDGMENTS

This work was supported by the Natural Science Basic Research Plan in Shaanxi Province of China under Grant 2018JM5033 and the China Scholarship Council under Grant 201806295003.