Dual Resonant Frequency Inductive Power Transfer in an Underwater Tight Coupling System

Abstract

The application of wireless power transfer technology in the underwater environment proposes both opportunities and challenges to undersea power feeding. Due to the attenuation of electromagnetic waves in seawater, the distance between transmitter and receiver is always maintained at a minimum value, which results in tight coupling between the transmitter and receiver. The tight coupling condition provides a low impedance loop for high-order harmonic, so the component of the harmonic wave is thus significantly increased and cannot be ignored in the power transmission system. In order to fully utilize the harmonic energy, a fundamental-harmonic dual-channel system was proposed and studied in this paper. Compared with single-channel systems transmitting fundamental wave only, the dual-channel system has higher power transmission capability, while the loss in dual channel system can be less than that of a single-channel system after proper optimization. A 3 kW experiment platform is established to verify the effectiveness of theoretical analysis.

1. Introduction

The development of underwater robotics is an important field of research and a promising industrial application since it represents a sustainable solution to exploit ocean resources with a reasonable environmental impact [1]. The application of wireless power transfer technology to underwater devices [2,3,4] has potential technical and economic benefits, compared to the wet plug technology, which is complicated to implement and expensive to maintain(shown in Figure 1). Wireless power transfer facilitates electrical isolation between the transmitter and receiver and eliminates the compulsory physical contact from power feeder to consumer [5,6,7]. However, currently, the majority of research mainly focuses on wireless power transfer in the atmospheric environments [8,9], the application in the undersea environment raised new challenges to the development of the WPT(wireless power transfer) technology. On one hand, the conductivity of air at standard temperature and pressure is nearly zero, so the medium loss is not incorporated in the analysis of the theoretical model. The eddy current loss in seawater cannot be ignored, and it has a significant influence on the system efficiency at high operation frequency. On the other hand, wireless power transfer requires a stable alignment of the transmitter and receiver; however, the electromagnetic coupler in seawater is vulnerable to ocean current disturbance, additional mechanical fixing design is required.

Due to the attenuation of electromagnetic wave in seawater [10,11,12], the distance between transmitter and receiver is maintained at a minimum value, which results in tight coupling between the transmitter and receiver(the coupling coefficient is greater than 0.5), so the underwater inductive power transfer system is a strong coupling system [13,14,15,16,17]. The component of the harmonic wave is thus significantly increased [18,19,20] and cannot be ignored in the power transmission system. This paper proposes a dual resonant frequency inductive power transfer system, in which the fundamental and third harmonic waves are transferred simultaneously. The multi-frequency WPT for smart power distribution among multi-standard receivers has been investigated in [21]. A GaN (Gallium nitride)-based programmable pulse width modulation [22] and a concurrent transmitter consisting of two independent transmitters [23] were proposed and implemented to support the dual-mode WPT operation. A scheme that amplifies both the fundamental and third harmonic generated by a full-bridge inverter was proposed to implement the double-frequency WPT (DF-WPT) [24]. A multi-frequency superposition methodology [25] and a multi-frequency drive configuration [26] have been proposed to achieve high efficiency and flexible power distribution for multi-load WPT. In order to reduce the effect of electromagnetic interference, snubbers for power switch devices are necessary, and their effectiveness has been verified in previous literature [27].

In this paper, the compensation network on the primary side is designed with LC stepped structure featuring multiple resonant frequencies. The dual-channel wireless power transfer system is compared with the single-channel system in power losses, including winding loss and circuit loss. Theoretical analysis is validated through simulation and experimental results.

2. Dual Resonant Frequency Inductive Power Transfer

This section starts with the analysis in a mutual inductance model transmitting only a fundamental wave, and an additional circuit branch is added to build the fundamental-harmonic dual-path topology. As extra rectifier bridge and coils were used, the dual channel system transmitting both fundamental and harmonic wave was analyzed and compared with single-channel WPT system in system losses.

2.1. Dual Channel Inductive Power Transfer

The single-channel S-S (series to series) compensated wireless power transfer system is shown in Figure 2.

Through the Fourier decomposition of the Inverter output square wave voltage,

$$v_{AB}=\frac{4U_s}{\pi }{\displaystyle \sum _{m=1,3,5...}\frac{1}{m}}\sin (m\omega t)$$

(1)

w is the switching angle frequency.

Subscript m is defined as the circuit parameter in m-th harmonic.

The impedances on the primary and secondary sides are:

$$Z_{p\_m}=j(m\omega )L_p+\frac{1}{j(m\omega )C_p}$$

(2)

$$Z_{s\_m}=j(m\omega )L_s+\frac{1}{j(m\omega )C_s}+R_L$$

(3)

The reflected impedance of Z_{R_m} on the primary side is:

$$Z_{R\_m}=\frac{{[(m\omega )M]}^2}{Z_{s\_m}}$$

(4)

The total input impedance on the primary side is:

$$Z_{in\_m}=Z_{P\_m}+Z_{R\_m}$$

(5)

At resonant frequency f_r, the fundamental input impedance is:

$$Z_{in\_r}=\frac{{\omega }_r^2{M}^2}{R_L}$$

(6)

The quality factor of secondary coil Q_s is defined as:

$$Q_s=\frac{{\omega }_r{L}_s}{R_L}$$

(7)

The normalized input impedance Z_{inN_m} is:

$$Z_{inN\_m}=j\frac{m^2-1}{m{Q}_s}\frac{1}{k^2}+\frac{m^3}{m+j{Q}_s(m^2-1)}$$

(8)

The normalized input harmonic current is:

$$I_{PN\_m}=\frac{I_{s\_m}}{I_{s\_r}}=\frac{U_{s\_m}/\left|Z_{in\_m}\right|}{U_{s\_r}/\left|Z_{in\_r}\right|}=\frac{1}{m\left|Z_{in\_mN}\right|}=\frac{1}{m\left|j\frac{m^2-1}{m{Q}_s}\frac{1}{k^2}+\frac{m^3}{m+j{Q}_s(m^2-1)}\right|}$$

(9)

I_{s_m} is the total input current on the primary side, and I_{s_r} is the fundamental input current at resonant frequency f_r.

Under different quality factors, the variation of the third harmonic component with the coupling coefficient is shown in Figure 3. With the increase of the coupling coefficient, the third harmonic component increases first before decreasing. When the transmitter and receiver coils are under tight coupling condition, the component of third harmonic cannot be neglected.

In an underwater WPT system, the gap between the transmitter and receiver coil is tiny, which results in a high coupling coefficient, and the harmonic component is significantly increased.

In order to transfer the fundamental and harmonic wave simultaneously, an additional circuit is usually added to transfer the harmonic wave. A schematic diagram of the fundamental-harmonic dual-path topology is shown in Figure 4. The fundamental and harmonic compensation network is designed at its respective resonant frequency. The two networks share the same inverter bridge. However, this strategy requires two compensation networks and two couplers, which significantly increase the cost and volume of the system; meanwhile, the two outputs will affect the original input angle, and the inverter no longer works under the ZVS (zero voltage switching) condition.

As a result, in order to transfer the fundamental and harmonic wave simultaneously, the primary compensation network needs to have multiple resonant frequencies, as shown in Figure 5. Resistor–capacitor–diode clamp snubber was added to the inverter to reduce electromagnetic interference. Multiple resonant frequency network usually employs the LC stepped structure, and the combination of multiple resonant networks enables the system to have multiple resonant frequencies, due to the frequency-selective characteristic of a single LC network. X_e is a reactance unit used to control the phase. The secondary side employs multiple output circuits, to transfer the fundamental and third harmonic wave independently. As the proposed system topology features constant voltage output, the rectifier bridge outputs can be connected in series.

The additional passive compensation network is added on the receiver side (L_r1 and C_r1, L_r2 and C_r2) to eliminate the effects of cross-coupling. As Channel 1(L_s1 and C_s1) is designed to transfer the fundamental wave and Channel 2(L_s2 and C_s2) is designed to transfer only the third harmonic, L_r1 and C_r1 are designed to be resonant at the third harmonic frequency, and L_r2 and C_r2 are designed to be resonant at the fundamental frequency. For Channel 1, the impedance of the third harmonic is infinite, so the third harmonic can hardly go through Channel 1. Similarly, a fundamental wave can hardly go through Channel 2.

2.1.1. Determination of Circuit Parameters

As to the LCL compensation network (shown in Figure 6a, L_A resonant with C_A ensures constant current output, and when L_A equals L_B, the input impedance is resistive). The LCL compensation network is a third-order system with a single resonant frequency. A dual resonant frequency system can be obtained by adding C_B after L_B, as shown in Figure 6b.

Define s_ω as negative frequency, $s_{\omega \_m}=j{\omega }_{\_m}$

The current gain of dual resonant frequency network G_iP is:

$$G_{iP}=\frac{I_P}{U_{AB}}=\frac{1}{s_{\omega }^4{L}_A{L}_B{C}_A+s_{\omega }\left(L_A+L_B\right)+\left[s_{\omega }^4{L}_A{L}_B{C}_A{C}_B+s_{\omega }^2\left(L_A{C}_A+L_A{C}_B+L_B{C}_B\right)+1\right]Z_o}$$

(10)

when $s_{\omega }^4{L}_A{L}_B{C}_A{C}_B+s_{\omega }^2\left(L_A{C}_A+L_A{C}_B+L_B{C}_B\right)+1=0$, G_iP is independent of load Z_o. When the system resonant at the fundamental and third harmonic frequency, G_iP is independent of load Z_o.

$$\frac{L_A{C}_A+L_A{C}_B+L_B{C}_B}{L_A{L}_B{C}_A{C}_B}=10{\omega }_{\_1}^2$$

(11)

$$\frac{1}{L_A{L}_B{C}_A{C}_B}=\frac{1}{9}{\omega }_{\_3}^4$$

(12)

The current gain is:

$$G_{iP}\left({\omega }_{\_1}\right)=G_{iP}\left({\omega }_{\_3}\right)=\frac{1}{s_{\omega }^3{L}_A{L}_B{C}_A+s_{\omega }\left(L_A+L_B\right)}$$

(13)

The current gains for fundamental and third harmonic are defined as G_{_1} and G_{_3}, respectively:

$$G_{\_1}=\frac{1}{L_A{L}_B{C}_A{\omega }_{\_1}^3-\left(L_A+L_B\right){\omega }_{\_1}}$$

(14)

$$G_{\_3}=\frac{1}{L_A{L}_B{C}_A{\omega }_{\_3}^3-\left(L_A+L_B\right){\omega }_{\_3}}$$

(15)

Solving the equations,

$$\{\begin{array}{c}\begin{array}{l}{L}_A=\frac{1}{3{\omega }_{\_1}\left(G_{\_3}-3G_{\_1}\right)}\\ \end{array}\\ L_B=\frac{{\left(G_{\_1}-3G_{\_3}\right)}^2}{8{\omega }_{\_1}{G}_{\_1}{G}_{\_3}\left(3G_{\_1}-G_{\_3}\right)}\end{array}$$

(16)

$$\{\begin{array}{c}{C}_A=\frac{{\left(3G_{\_1}-G_{\_3}\right)}^2}{8{\omega }_{\_1}\left(3G_{\_3}-G_{\_1}\right)}\\ \\ C_B=\frac{8G_{\_1}{G}_{\_3}}{3{\omega }_{\_1}\left(G_{\_1}-3G_{\_3}\right)}\end{array}$$

(17)

Therefore, given the current gains G_{_1} and G_{_3} and fundamental frequency, the values of L_A, L_B, C_A, and C_B can be determined.

2.1.2. System Output Characteristics

The constant voltage output condition is:

$$\{\begin{array}{c}{\omega }_1{L}_{s1}-\frac{1}{{\omega }_1{C}_{s1}}+\frac{{\omega }_1{L}_{r1}}{1-{\omega }_1^2{L}_{r1}{C}_{r1}}=0\\ {\omega }_3{L}_{s2}-\frac{1}{{\omega }_3{C}_{s2}}+\frac{{\omega }_3{L}_{r2}}{1-{\omega }_3^2{L}_{r2}{C}_{r2}}=0\end{array}$$

(18)

The two receiver circuits are decoupled under the condition:

$$\{\begin{array}{c}1-{\omega }_3^2{L}_{r1}{C}_{r1}=0\\ 1-{\omega }_1^2{L}_{r2}{C}_{r2}=0\end{array}$$

(19)

The AC voltage gain is:

$$\{\begin{array}{c}{G}_{v1}=\left|\frac{U_{o1}}{U_{i1}}\right|=\left|\frac{{\omega }_1{M}_{PS1}{I}_{P1}}{U_{i1}}\right|={\omega }_1{M}_{PS1}\left|G_{iP1}\right|\\ G_{v2}=\left|\frac{U_{o2}}{U_{i3}}\right|=\left|\frac{{\omega }_3{M}_{PS2}{I}_{P3}}{U_{i3}}\right|={\omega }_3{M}_{PS2}\left|G_{iP3}\right|\end{array}$$

(20)

As the rectifier bridge outputs connect in parallel,

$$U_o=\frac{\pi }{4}(U_{o1}+U_{o2})$$

(21)

The power in each receiver circuit is distributed according to the ratio of voltage distribution,

$$\{\begin{array}{c}{P}_{o1}=\frac{U_{o1}}{U_o}·\frac{U_o^2}{RL}=\frac{U_{o1}{U}_o}{R_L}\\ P_{o2}=\frac{U_{o2}}{U_o}·\frac{U_o^2}{RL}=\frac{U_{o2}{U}_o}{R_L}\end{array}$$

(22)

If the rectifier bridge loss is neglected, the power flows in the rectifier equals the power flows out,

$$\{\begin{array}{c}{U}_{o1}{I}_{s1}=P_{o1}=\frac{U_{o1}{U}_o}{R_L}=I_o{I}_{o1}{R}_L\\ U_{o2}{I}_{s2}=P_{o2}=\frac{U_{o2}{U}_o}{R_L}=I_o{I}_{o2}{R}_L\end{array}$$

(23)

Therefore, the equivalent resistances with the rectification of two receiver circuits are:

$$\{\begin{array}{c}{R}_{E1}=\frac{U_{o1}}{I_{s1}}=\frac{16}{{\pi }^2}{R}_L\frac{\left|M_{PS1}{G}_{iP1}\right|}{\left|M_{PS1}{G}_{iP1}\right|+\left|M_{PS2}{G}_{iP3}\right|}\\ R_{E2}=\frac{U_{o2}}{I_{s2}}=\frac{16}{{\pi }^2}{R}_L\frac{\left|M_{PS2}{G}_{iP3}\right|}{\left|M_{PS1}{G}_{iP1}\right|+\left|M_{PS2}{G}_{iP3}\right|}\end{array}$$

(24)

2.2. Comparison of Dual and Single Channel Systems

The comparison of dual- and single-channel systems is carried out in terms of inverter circuit conduction loss and winding loss in a loosely coupled transformer.

The total power output of the system is P_o, the power output of receiver circuit 1 is P_o1, and the power output of receiver circuit 1 is P_o2.

To define power factor λ, the equivalent load resistance of the single-channel system is:

$$\{\begin{array}{c}\lambda =\frac{P_{o1}}{P_o}=\frac{\left|M_{PS1}{G}_{iP1}\right|}{\left|M_{PS1}{G}_{iP1}\right|+\left|M_{PS2}{G}_{iP3}\right|}\\ 1-\lambda =\frac{P_{o2}}{P_o}=\frac{\left|M_{PS2}{G}_{iP3}\right|}{\left|M_{PS1}{G}_{iP1}\right|+\left|M_{PS2}{G}_{iP3}\right|}\end{array}$$

(25)

Assuming the current gain of the single-channel system is G_iP, the total power output in the single-channel system is:

$$R_E=\frac{8R_L}{{\pi }^2}$$

(26)

$$P_o=\frac{{\left|{\omega }_1{M}_{PS1}{U}_{i1}{G}_{iP}\right|}^2}{R_E}$$

(27)

while the power outputs in the dual-channel system are:

$$\{\begin{array}{c}{P}_{o1}=\frac{{\left|{\omega }_1{M}_{PS1}{U}_{i1}{G}_{iP1}\right|}^2}{R_{E1}}\\ P_{o2}=\frac{{\left|{\omega }_3{M}_{PS2}{U}_{i3}{G}_{iP3}\right|}^2}{R_{E2}}\end{array}$$

(28)

Defining δ = M_PS1/M_PS2, solving (17)–(20), and normalizing current gain G_iP1, G_iP3 and equivalent resistance R_E1 and R_E2:

$$\{\begin{array}{c}{G}_{iP1\_N}=\frac{G_{iP1}}{G_{iP}}=\sqrt{2}\lambda ,G_{iP3\_N}=\frac{G_{iP3}}{G_{iP}}=\sqrt{2}\delta (1-\lambda )\\ R_{E1\_N}=\frac{R_{E1}}{R_E}=2\lambda ,R_{E2\_N}=\frac{R_{E2}}{R_E}=2(1-\lambda )\end{array}$$

(29)

2.2.1. Rectifier Circuit Conduction Loss

Assume there is no loss in passive elements, the output power from the inverter equals the input power to the rectifier, and the input current on the primary side is as follows:

Single channel:

$$I_{in}=\frac{{\left|{\omega }_1{M}_{PS1}{G}_{ip}\right|}^2}{R_E}{U}_{in}$$

(30)

Dual channel:

$$\{\begin{array}{c}{I}_{in1}=\frac{{\left|{\omega }_1{M}_{PS1}{G}_{iP1}\right|}^2}{R_{E1}}·U_{in1}\\ I_{in3}=\frac{{\left|{\omega }_3{M}_{PS2}{G}_{iP3}\right|}^2}{R_{E2}}·U_{in3}\end{array}$$

(31)

The normalized input current expression is:

$$I_{in\_N}=\frac{\sqrt{I_{in1}^2+I_{in3}^2}}{I_{in}}=\sqrt{{\lambda }^2+9{(1-\lambda )}^2}$$

(32)

The normalized input current curve of the dual-channel system is plotted in Figure 7. When the power factor λ is greater than 0.8, the input current in the dual-channel system is smaller than the single-channel system, and when λ = 0.9, the normalized current achieves its minimum value of 0.9. As the input current into the resonant network is the break-over current in the inverter circuit. As a result, to maintain a low conduction loss, the dual-channel system should be designed to work in the shaded region shown in Figure 7.

2.2.2. Winding Loss on Loosely Coupled Transformer

The fundamental and third harmonic current on the primary side is:

$$\{\begin{array}{c}{i}_{P1}=U_{in1}\left|G_{iP1}\right|\sin ({\omega }_{1}t+{\varphi }_1)\\ i_{P3}=U_{in3}\left|G_{iP3}\right|\sin (3{\omega }_{3}t+{\varphi }_3)\end{array}$$

(33)

in which, φ₁ is the phase difference between i_p1 and u_in1, and φ₃ is the phase difference between i_p3 and u_in3. As i_p1 lags behind u_in1 by 90°, i_p3 leads u_in3 by 90°, so i_p1 and i_p3 are in-phase. The primary current can be written as:

$$i_p=i_{p1}+i_{p3}=U_{in1}\left|G_{iP1}\right|\sin ({\omega }_{1}t)+U_{in3}\left|G_{iP3}\right|\sin (3{\omega }_{3}t)$$

(34)

Through the derivation with w_1t, when

$${\omega }_{1}t=\arcsin (\sqrt{(\left|G_{iP1}\right|+\left|G_{iP3}\right|)/4\left|G_{iP3}\right|})$$

(35)

i_P achieves its maximum value,

$$I_{p\_\max }=\frac{U_{in1}(\left|G_{iP1}\right|+\left|G_{iP3}\right|)}{3}·\sqrt{1+\left|\frac{G_{iP1}}{G_{iP3}}\right|}$$

(36)

Through the derivation with |G_iP3|, when |G_iP3| = |G_iP1|/2, i_{P_max} achieves its minimum value,

$$\min (i_{p\_\max })=\frac{\sqrt{3}}{2}{U}_{in1}\left|G_{ip1}\right|$$

(37)

Therefore, the primary current amplitude of the dual-channel system can be reduced to $\sqrt{3}/2$ of the single-channel system, which facilitates the reduction of maximum flux density, so as to reduce EMI.

Define,

$$Q=\frac{{\omega }_1{M}_{PS1}}{R_E},Q_1=\frac{{\omega }_1{M}_{PS1}}{R_{E1}},Q_2=\frac{{\omega }_3{M}_{PS2}}{R_{E2}}$$

(38)

The currents in the secondary coil single system and dual-channel system are as follows:

Single-channel:

$$I_s=Q\left|G_{iP}\right|U_{in1}$$

(39)

Dual-channel:

$$\{\begin{array}{c}{I}_{s1}=Q_1\left|G_{iP1}\right|U_{in1}\\ I_{s2}=Q_2\left|G_{iP3}\right|U_{in3}\end{array}$$

(40)

As shown in Figure 8, R_p, R_s1, R_s2 are the equivalent resistance of each coil. The winding loss in single- and dual-channel systems when considering the coil resistance is

$$\{\begin{array}{c}{P}_{Cu\_s}={\left|U_{in1}{G}_{iP}\right|}^2{R}_{p1}+{\left|Q{G}_{ip}{U}_{in1}\right|}^2{R}_{s1}\\ \begin{array}{l}{P}_{Cu\_D}={\left|U_{in1}{G}_{iP1}\right|}^2{R}_{p1}+{\left|U_{in3}{G}_{iP3}\right|}^2{R}_{p3}\\ +{\left|Q_1{G}_{ip1}{U}_{in1}\right|}^2{R}_{s1}+{\left|Q_2{G}_{ip3}{U}_{in3}\right|}^2{R}_{s2}\end{array}\end{array}$$

(41)

Based on the winding loss in the single-channel system, the winding loss in the dual-channel system is normalized:

$$\begin{array}{l}{P}_{Cu\_D\_N}=\frac{P_{Cu\_D}}{P_{Cu\_S}}\\ =\frac{18{\lambda }^2{R}_{p1}+2{[(1-\lambda )\delta ]}^2{R}_{p3}+4.5Q^2(R_{s1}+R_{s3})}{9(R_{p1}+Q^2{R}_{s1})}\end{array}$$

(42)

According to the measured values R_p1 = 130 mΩ, R_p3 = 890 mΩ, R_s1 = 120 mΩ, R_s3 = 115 mΩ, δ = 1.88, the normalized winding loss is plotted in Figure 9 (the trend of the normalized winding loss curve is subject to change according to the variation of internal resistances of coils). In the region below P_{Cu_D_N} = 1, the winding loss of the dual-channel system is less than that in the single-channel system.

3. Experiment Verification

A 3 kW underwater wireless power transfer (UWPT) platform is established to verify the effectiveness of theoretical analysis. The schematic diagram of the proposed UWPT coupling structure is shown in Figure 10. The side column is designed with an incision angle to increase the contact area and to reduce the magnetic reluctance. Meanwhile, the proposed coupler also has the self-latching function once the transmitter and receiver get to the aligned position, which facilitates its resistance to ocean current disturbance (design and optimization of the coupling structure can be found in [28]). When the underwater vehicle lands on the base station, the receiver is plugged into the transmitter; thus in the charging process, the transmitter and receiver would remain in the fixed position. DMR95 material is utilized as the ferrite core. The gap between transmitter and receiver side varies between 5 and 20 mm, as the waterproof coating is applied on the coupler’s surface; the gap of 5 mm is the thickness of the waterproof coating, and the effects of seawater as transmission medium are reduced to a minimum. The couplers are shown in Figure 10. The experimental platform is shown in Figure 11. The experimental setup is shown in

Table 1. Component third harmonic as a variation of coupling coefficient.

k	0.52	0.61	0.71	0.82
IP_3/IP_1 (%)	6.72	10.64	22.39	48.56

.

The influence of the strong coupling condition on system performance was first investigated on a single channel system. When the load resistance is 50 Ω, the inverter output voltage and current waveform are shown and compared in Figure 12. The component of the fundamental wave and third harmonic is then calculated and demonstrated in Table 1. As the coupling coefficient increases from 0.52 to 0.82, the inverter current waveform distortion worsens and the component of third harmonic increases from 6.72% to 48.56%, which means the component of harmonic increases as the increase of coupling coefficient, which agrees with the theoretical analysis.

Based on the single-channel UWPT system, an additional set of coils were added on the receiver side, which shares the same ferrite core with the original coil. The schematic diagram of the fundamental-harmonic dual-path wireless power transfer system is shown in Figure 5. DC power source provides power into the inverter before feeding into the compensation network together with the LC resonant tank. The two receiver coils pick up the power individually, and through the compensation network and rectifier on the receiver side, energy is provided to the load. The output of two receiver networks connects in series. The key parameters of the proposed dual-channel system are shown in

Table 2. Parameters of the dual-channel system.

Parameters	LA/μH	LB/μH	Lp/μH	Ls1/μH	Ls2/μH	CA/nF
Value	106.1	79.6	254.87	252.99	49.01	31.8
Parameters	CB/nF	Cs1/nF	Cs2/nF	Lr1/μH	Lr2/μH	Cr1/nF
Value	42.4	36.8	26.3	20	50	56.3
Parameters	Cr2/nF	f/kHz	Mps1/μH	Mps2/μH	Ms12/μH	RL/Ω
Value	202.6	50	200.08	106.41	92.34	50

. As few inductance elements were added for dual-frequency resonant network, and the high work frequency can effectively reduce the inductance volume, independent inductance was utilized.

The key waveform of dual-channel UWPT system is shown in Figure 13. The current in the primary coil is composed of fundamental wave and third harmonic, which proves the dual-frequency resonant characteristic of the transmitter network. The current in Channel 1(L_s1 and C_s1) only has the fundamental wave, and the current in Channel 2(L_s2 and C_s2) only has the third harmonic, which proves the frequency-selective characteristic of the receiver compensation network.

The frequency sweeping curve of system voltage gain is plotted in Figure 14. The crossing point in Channel 1 is right on 50 kHz and the voltage gain is 1.15. Meanwhile, the crossing point in Channel 2 is right on 49.8 kHz and the voltage gain is 0.45. The resonant frequency in Channel 2 is 49.8 × 3 = 149.4 kHz. The third harmonic of the inverter output is the input signal into Channel 2.

The maximum power transfer curves of the single- and dual-channel systems are compared in Figure 15, and the dual-channel system’s DC-DC efficiency is plotted in Figure 16 and compared with that of the single-channel system (both theoretically and experimentally). For the dual-channel system, the output power equals the output voltage multiplied by output current ($P_O=U_O{I}_O$). The DC-DC efficiency in this paper is defined as the ratio of output power to input power ($\eta =\left|U_o{I}_o/U_{in}{I}_{in}\right|\times 100\%$). The dual-channel system possesses higher energy output capability, with higher transmission efficiency. As the load resistance increases, the output power increases first before decreasing. When the output power is 2.69 kW, the system efficiency reaches its maximum value, 92.3%. Utilizing the experimental parameters in equations to calculated total output power and efficiency at specific load conditions R_L = 30, 40, 50, 60 Ω, the theoretical values are plotted and compared with experimental results in Figure 15 and Figure 16.

4. Conclusions

For the underwater WPT system, the distance between transmitter and receiver is very small, which results in tight coupling between the transmitter and receiver. The system output characteristics under the tight coupling condition were analyzed based on the calculation of input impedance. The tight coupling condition provided a low impedance loop for high-order harmonic, which significantly increased the component of harmonics in UWPT. In order to fully utilize the harmonic energy, a fundamental-harmonic dual-channel system was proposed and studied in this paper. An LC stepped structure was applied on the primary side, and the combination of multiple resonant networks enabled the system to have multiple resonant frequencies. An additional LC compensation network was added on the secondary side to decouple the two receiver coils. The power transmission capability of the dual-channel system surpassed that of the single-channel system. The dual-channel system was compared with single-channel system in loss and efficiency. As to the rectifier conduction loss and winding loss, the loss in the dual-channel system was less than that of the single-channel system when the power factor was within a certain range. Theoretical analysis was then validated on a 3 kW underwater wireless power transfer platform. Maximum DC-DC efficiency of 92.3% was achieved. The dual-channel system has a better output characteristic compared with the single-channel system in underwater applications.