Synthetic electric field calculation method of three-coil structure to resist the seawater flow vibration

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.

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 M₁₃ equals M₂₃, where M₁₃ and M₂₃ are the mutual inductances between Coil₁ and Coil₃ and between Coil₂ and Coil₃, respectively. In FIG. 1(a), the center coordinates of the transmitting coil, Coil₁, and the receiving coil, Coil_3, are (0, 0, -L₁₃) and (0, 0, 0), respectively. In the 1×1×1 configuration, there is another transmitting coil, Coil₂, shown in FIG. 1(b), centered at (0, 0, L₁₃).

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.

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 Coil₁ and Coil₃, 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 Coil₁ and Coil₂ are L₁₂. The turns of Coil₁ and Coil₂ 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 Coil₁ and the Coil₂ which keeps the gap d₁ away from the original point, are expressed as Coil_1.n1, n1=1…N₁ and Coil_2.n2, n2=1… N₂. N_1, N₂ are the turns of the Coil₁ and Coil₂. The displacement area of the receiver coil is demonstrated as 2ΔS, which is divided into 2k+1 points with the minimum d gap. P (0, r_s, z) is the point on the Coil₃, 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 $fλ=λ/u⋅J1λrpJ1λrse−u|(n1−1)d+L12/2+z|$⁠, 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, d₁=6.8mm, d=2mm, L₁₂=28mm, N₁=N₂=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 2k+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 2k+1 points is the sum of the electric field vector generated by the Coil₁ and Coil₂, which is expressed as follows:

E(id) represents the synthetic electric field at z=id point. Therefore, the inductive voltage of receiving Coil₃ is indicated as follows:

N₃ is the turns of the Coil₃. E (md) represents the synthetic electric field at the position (z=md) of the left-most turn of Coil₃.

In this section, in all calculations, simulations and experiments, the fundamental parameters are N₁=N₂=11, N₃=5, and the impedance of the load R_L=30Ω, f =300kHz, d₁=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 -10d to 10d, 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 L₁₂ in FIG. 3(b). Besides, the induced voltage using (4) is calculated and simulated from -10d to 10d at different L₁₂ as shown in FIG. 4(a), which is compared with the experimental results when L₁₂ 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, L₁₂ has influence on the stability of the synthetic electric field and can be optimized in the actual application. When L₁₂ 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.

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.

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.