Research on Maximum Efficiency Tracking in Wireless Power Transfer Systems Based on Seven-Level Inverter

Abstract

To address the issues of low fundamental content in the output voltage of high-frequency inverters within wireless power transfer (WPT) systems and efficiency degradation caused by coupling coefficients and load variations, this paper proposes a novel seven-level inverter topology and a closed-loop PI control strategy based on current amplitude ratio. First, the influence of LCC-S WPT system parameters on current and efficiency is analyzed. Subsequently, by comparing fundamental content in inverter output voltage across different level structures, a seven-level configuration is selected. A novel seven-level inverter topology with fewer switches and lower voltage stress is proposed, and its efficiency enhancement advantage is validated through optimized switch turn-on angles. Finally, a closed-loop PI control strategy based on current amplitude ratio is adopted. By merely acquiring coil currents and calculating their amplitude ratio, the duty cycle of the Buck-Boost circuit is adjusted to optimize current amplitude, achieving maximum efficiency tracking for the system. Experimental results demonstrate that system efficiency approaches theoretical calculations during coil spacing variations. When the load varies between 5 Ω and 105 Ω, system efficiency remains around 91.4%, with maximum efficiency point tracking error maintained at approximately 2%. This validates the system’s reliability and the effectiveness of the control strategy.

1. Introduction

Wireless Power Transfer (WPT) technology enables the transmission of electrical energy from a power source to an electrical load without physical connectors, offering significant advantages in terms of flexibility, reliability, and safety. Over the past decade, WPT has evolved from a conceptual technology into a practical solution, witnessing widespread adoption across various sectors. In consumer electronics, it has become a standard feature for charging smartphones, smartwatches, and wireless earbuds. More critically, WPT is actively being investigated and deployed in high-power applications such as electric vehicles (EVs), where it facilitates convenient static and dynamic charging, eliminating range anxiety and the inconvenience of plugging in. Furthermore, its inherent galvanic isolation and ability to operate in harsh environments make it indispensable for powering underwater equipment, medical implants, and industrial automation systems [1,2,3,4,5]. As WPT systems are increasingly integrated into complex and dynamic scenarios, such as the energy internet for distributed energy interconnection, the research frontier is shifting towards enhancing system intelligence and adaptive capabilities to ensure robust and efficient operation under varying conditions.

A primary challenge in WPT systems is maintaining high power transfer efficiency, which is highly susceptible to variations in operating conditions. The transmission efficiency is critically influenced by factors such as the transmission distance, lateral and angular misalignment between the transmitter and receiver coils (which affects the coupling coefficient), and fluctuations in the load resistance. Additionally, environmental interference can further degrade performance. Traditionally, the front end of many WPT systems employs a full-bridge inverter. While simple to implement, this topology produces a square wave output voltage with a low fundamental frequency component and high harmonic distortion, which inherently limits the achievable efficiency. When the coupling mechanism shifts or the load changes, the system’s operating point deviates from its optimally designed condition, leading to a further decline in efficiency [6].

To address these efficiency challenges, extensive research has been conducted globally, focusing on both inverter topology optimization and transmission efficiency enhancement. Early efforts in inverter topology explored clamped Class-E and push-pull Class-E inverters [7,8]. While these topologies offered some efficiency improvements, their optimization was often limited, and they struggled to systematically reduce reactive power losses, hindering sustained high-efficiency operation. Multilevel inverters have emerged as a superior alternative because they produce output voltage waveforms that more closely approximate a sine wave, resulting in lower total harmonic distortion (THD) and higher efficiency. For instance, flying-capacitor and cascaded H-bridge seven-level inverters have been proposed [9,10,11]. Although these designs increase the fundamental component of the output voltage, they often require complex control strategies for capacitor voltage balancing. More recently, a switched-capacitor seven-level inverter topology was introduced, featuring self-balancing capacitor voltages [12]. However, this topology suffers from relatively high total standing voltage (TSV) per unit on its switches, which can increase component stress and cost.

Alongside innovations in topology, significant efforts have been dedicated to enhancing transmission efficiency, primarily through impedance matching to achieve maximum efficiency tracking (MET). Impedance matching methods can be broadly classified as passive or active. Passive impedance matching utilizes networks of inductors and capacitors, such as T-type matching networks [13,14]. While these provide strong equivalent impedance adjustment and design flexibility, they suffer from complex control requirements, significant switching losses, and a large system footprint. Conversely, active impedance matching optimizes efficiency by modulating the pulse width of active rectifiers [15] or the duty cycle of DC-DC converters [16]. This approach effectively expands the impedance matching range, ensuring precise matching even under load variations. However, when the coupling coefficient varies due to changes in coil spacing, these methods alone prove insufficient for real-time tracking of the maximum efficiency point [17].

To achieve maximum efficiency tracking under varying coupling coefficients, many studies combine active impedance matching with coupling coefficient identification. Various identification techniques have been explored, including estimating mutual inductance from the total harmonic distortion of the transmitter current [18] and extracting fundamental and harmonic components from transmitter-side measurements [19]. However, these methods often rely on complex formulas and demonstrate limited accuracy. Another approach employs Pulse Density Modulation (PDM) and utilizes Fast Fourier Transform (FFT) analysis of interharmonics to calculate the coupling coefficient [20]. While this method requires no additional hardware, it is computationally intensive due to its dependence on FFT and least-squares approximation algorithms.

Building upon the identified gaps in the literature, this paper introduces a novel seven-level inverter topology and a simplified MET control strategy for WPT systems. The main contributions of this article are listed as follows:

(1)

A novel seven-level inverter: A new seven-level inverter topology is proposed that enhances the fundamental component of the inverter’s output voltage while utilizing a reduced number of components compared to existing multilevel designs. This topology achieves a favorable balance between high performance and low complexity.

(2)

Optimized switching strategy: The optimal conduction angles for the switching devices in the proposed seven-level inverter are analytically derived, enabling efficient circuit regulation and maximizing the fundamental output voltage.

(3)

Simplified maximum efficiency tracking: A novel MET strategy based on a closed-loop PI controller regulating the current amplitude ratio between the receiver and transmitter coils is introduced. This method eliminates the need for complex and error-prone estimation of the coupling coefficient, requiring only the measurement of coil current magnitudes, thereby simplifying implementation and enhancing tracking accuracy.

The remainder of this paper is structured as follows. Section 2 introduces the overall architecture of the WPT system and derives the transmission efficiency formula in detail. Section 3 analyzes the fundamental wave content ratio of inverters under different level configurations, selects a seven-level inverter, and thoroughly examines its operating principles with theoretical validation. Section 4 establishes a mathematical relationship model between current amplitude ratio and system efficiency, validating the theory through simulation data. Section 5 introduces the experimental platform and relevant parameter settings, verifying the feasibility and effectiveness of the proposed method. Section 6 concludes the paper.

2. Analysis of the WPT Principle and Transmission Characteristics

The block diagram of the LCC-S type WPT system is shown in Figure 1. This system consists of a DC input voltage source U, a full-bridge inverter, an LCC-S type compensation network, a rectifier bridge, and a load R_L. The LCC-S compensation network comprises primary-side compensation inductors L_Q, compensation capacitors C_Q and C_P, and secondary-side compensation capacitor C_S. L_P and L_S represent the self-inductance of the transmitter coil and receiver coil, respectively. R_P and R_S denote the internal resistance of the transmitter coil and receiver coil, respectively. M is the mutual inductance between the coils.

According to the fundamental wave analysis method, the equivalent mutual inductance model of the LCC-S type WPT system is illustrated in Figure 2. In the figure, U_in and I_in represent the output voltage and output current of the inverter, respectively; I_P and I_S denote the currents of the transmitter-side coil and receiver-side coil, respectively; and R_eq is the equivalent resistance at the receiver end.

According to Kirchhoff’s voltage law, the loop formula is

$$\left\{\begin{array}{l}{U}_{\mathrm{in}}=I_{\mathrm{in}}\mathrm{j}\omega L_{\mathrm{Q}}+{\displaystyle \frac{(I_{\mathrm{in}}-I_{\mathrm{P}})}{\mathrm{j}\omega C_{\mathrm{Q}}}}\\ U_{\mathrm{in}}=I_{\mathrm{in}}\mathrm{j}\omega L_{\mathrm{Q}}+I_{\mathrm{S}}\mathrm{j}\omega M+I_{\mathrm{P}}(\mathrm{j}\omega L_{\mathrm{P}}+{\displaystyle \frac{1}{\mathrm{j}\omega C_{\mathrm{P}}}}+R_{\mathrm{P}})\\ -I_{\mathrm{S}}{R}_{\mathrm{eq}}=I_{\mathrm{S}}(\mathrm{j}\omega L_{\mathrm{S}}+{\displaystyle \frac{1}{\mathrm{j}\omega C_{\mathrm{S}}}}+R_{\mathrm{S}})+I_{\mathrm{P}}\mathrm{j}\omega M\end{array}\right.$$

(1)

When the system is in a resonant state, the following relationship holds

$$\left\{\begin{array}{l}{\displaystyle \frac{1}{\omega C_{\mathrm{Q}}}}=\omega L_{\mathrm{Q}}\\ {\displaystyle \frac{1}{\omega C_{\mathrm{S}}}}=\omega L_{\mathrm{S}}\\ {\displaystyle \frac{1}{\omega C_{\mathrm{P}}}}=\omega L_{\mathrm{P}}-\omega L_{\mathrm{Q}}\end{array}\right.$$

(2)

Combining Formulas (1) and (2), the output current I_in of the inverter, the transmitter-side coil current I_P, and the receiver-side coil current I_S are given by

$$\left\{\begin{array}{l}{I}_{\mathrm{in}}={\displaystyle \frac{U_{\mathrm{in}}\left[{\omega }^2{M}^2+R_{\mathrm{P}}(R_{\mathrm{eq}}+R_{\mathrm{S}})\right]}{{\omega }^2{L}_Q^2(R_{\mathrm{eq}}+R_{\mathrm{S}})}}\\ I_{\mathrm{P}}={\displaystyle \frac{U_{\mathrm{in}}}{\mathrm{j}\omega L_{\mathrm{Q}}}}\\ I_{\mathrm{S}}=-{\displaystyle \frac{U_{\mathrm{in}}M}{L_{\mathrm{Q}}(R_{\mathrm{eq}}+R_{\mathrm{S}})}}\end{array}\right.$$

(3)

The input power P_in and output power P_out of the system are

$$\left\{\begin{array}{l}{P}_{\mathrm{in}} =\left|U_{\mathrm{in}}\right|\left|I_{\mathrm{in}}\right|={\displaystyle \frac{U_{\mathrm{in}}^2\left[{\omega }^2{M}^2+R_{\mathrm{P}}(R_{\mathrm{eq}}+R_{\mathrm{S}})\right]}{{\omega }^2{L}_{\mathrm{Q}}^2(R_{\mathrm{eq}}+R_{\mathrm{S}})}}\\ P_{\mathrm{out}}={\left|I_{\mathrm{S}}\right|}^2{R}_{\mathrm{eq}}={\displaystyle \frac{U_{\mathrm{in}}^2{M}^2{R}_{\mathrm{eq}}}{L_{\mathrm{Q}}^2{(R_{\mathrm{eq}}+R_{\mathrm{S}})}^2}}\end{array}\right.$$

(4)

The efficiency of the fundamental wave output from the inverter to the load is then obtained as

$${\eta }_L={\displaystyle \frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}}={\displaystyle \frac{{\omega }^2{M}^2{R}_{\mathrm{eq}}}{\left[{\omega }^2{M}^2(R_{\mathrm{S}}+R_{\mathrm{eq}})+R_{\mathrm{P}}{(R_{\mathrm{S}}+R_{\mathrm{eq}})}^2\right]}}$$

(5)

In the WPT system, the full-bridge inverter produces a square wave voltage with relatively high harmonic content. Additionally, energy passing through the compensation network experiences power losses, which reduce system efficiency. Therefore, increasing the fundamental component ratio in the inverter’s output voltage is an effective method to enhance the overall transmission efficiency of the WPT system.

3. Design of a Novel Seven-Level Inverter

To increase the fundamental wave content ratio in the inverter output waveform and reduce topological complexity, this can be achieved by optimizing the switch conduction angle and appropriately selecting the number of levels.

3.1. Optimization of Switch Conduction Angle and Selection of Optimal Number of Levels for Inverter

Taking the output voltage of a three-level inverter as an example for analysis, its schematic diagram is shown in Figure 3.

If the DC input voltage to the inverter is U, its output voltage can be expressed as

$$U_{\mathrm{in}}=\left\{\begin{array}{l} U,\omega t\in \left[{\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\theta }{2}},{\displaystyle \frac{\pi }{2}}+{\displaystyle \frac{\theta }{2}}\right]\\ -U,\omega t\in \left[{\displaystyle \frac{3\pi }{2}}-{\displaystyle \frac{\theta }{2}},{\displaystyle \frac{3\pi }{2}}+{\displaystyle \frac{\theta }{2}}\right]\\ 0, \mathrm{others}\end{array}\right.$$

(6)

where θ represents the conduction angle of the switch, where θ ∈ [0, 2π].

According to Fourier decomposition, the output voltage expression is

$$U_{\mathrm{in}}={\displaystyle \frac{4U}{\pi }}{\displaystyle \sum _{k=1,3,5}^{\infty }\frac{{\left(-1\right)}^{\frac{k-1}{2}}\sin (\frac{k}{2}\theta )}{k}\sin (k\omega t)}$$

(7)

The effective value of its fundamental component is

$$U_{1\mathrm{m}}={\displaystyle \frac{2\sqrt{2}U}{\pi }}\sin {\displaystyle \frac{\theta }{2}}$$

(8)

Since the total width of the interval where U_in takes values of ±U within one cycle is 2θ, the effective value of the inverter output voltage is

$$U_{\mathrm{in}\left(\mathrm{m}\right)}=\sqrt{{\displaystyle \frac{2}{T}}{\displaystyle {\int }_0^{\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}U{(t)}^{2}dt}}=U\sqrt{{\displaystyle \frac{\theta }{\pi }}}$$

(9)

The fundamental wave content ratio is

$$\eta ={\displaystyle \frac{U_{1\mathrm{m}}}{U_{\mathrm{in}\left(\mathrm{m}\right)}}}={\displaystyle \frac{2\sqrt{2}}{\sqrt{\pi \theta }}}\sin {\displaystyle \frac{\theta }{2}}$$

(10)

Through iterative calculation, it is determined that the maximum fundamental wave content ratio of the system is η_max = 96.06% when the conduction angle θ = 133.6°.

By extending the inverter level structure through the principle of phase superposition, when the number of output voltage levels is m = 2n + 1, the effective value of the fundamental component of the output voltage is

$$U_{1\mathrm{m}}=\left\{\begin{array}{l} \hspace{1em} {\displaystyle \frac{2\sqrt{2}U}{\pi }}\hspace{1em}\hspace{1em} ,\mathrm{level} \mathrm{number} \mathrm{is} 2\\ {\displaystyle \frac{2\sqrt{2}U}{\pi }}\left[{\displaystyle \sum _{i=1}^n\sin \frac{{\theta }_i}{2}}\right],\mathrm{level} \mathrm{number} \mathrm{is} 2\mathrm{n}+1\end{array}\right.$$

(11)

The effective value of the inverter output voltage is

$$U_{\mathrm{in}\left(\mathrm{m}\right)}=\left\{\begin{array}{c} \hspace{1em} U \hspace{1em}\hspace{1em} ,\mathrm{level} \mathrm{number} \mathrm{is} 2\\ U\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(2i-1){\theta }_i}}{\pi }}},\mathrm{level} \mathrm{number} \mathrm{is} 2\mathrm{n}+1\end{array}\right.$$

(12)

The fundamental component ratio of the output voltage is

$${\eta }_i=\left\{\begin{array}{l}\hspace{1em} \hspace{1em}{\displaystyle \frac{2\sqrt{2}}{\pi }}\hspace{1em}\hspace{1em} ,\mathrm{level} \mathrm{number} \mathrm{is} 2\\ {\displaystyle \frac{2\sqrt{2}{\displaystyle \sum _{i=1}^n\sin \frac{{\theta }_i}{2}}}{\sqrt{\pi \times {\displaystyle \sum _{i=1}^n(2i-1){\theta }_i}}}},\mathrm{level} \mathrm{number} \mathrm{is} 2\mathrm{n}+1\end{array}\right.$$

(13)

By deriving from Formula (13), the optimal conduction angles and the fundamental wave content ratio of the inverter under different level numbers are obtained. Taking level numbers 2, 3, 5, 7, and 9 as examples, the results are shown in

Table 1. Optimal conduction angles and fundamental wave percentages at different voltage levels.

Level Number	θ1	θ2	θ3	θ4	Fundamental Wave Proportion
2	/	/	/	/	90.01%
3	133.6°	/	/	/	96.06%
5	154.2°	96.3°	/	/	98.68%
7	162.0°	124.2°	79.2°	/	99.34%
9	171.5°	135.6°	97.8°	67.5°	99.61%

.

As shown in Table 1, the fundamental wave content proportion of multilevel inverters increases with the number of levels. When the multilevel inverter outputs seven levels or more levels, the fundamental wave content proportion approaches saturation. Furthermore, as the number of levels increases, the inverter topology becomes more complex. Therefore, this paper sets the multilevel count to seven levels.

3.2. Structure and Advantages of a Novel Seven-Level Inverter

Compared to conventional multilevel inverters, the switched-capacitor multilevel inverter structure features lower voltage stress on switches, reduced number of required switches, simpler modulation methods, balanced capacitor voltages, and high voltage gain.

Based on the fundamental building blocks of switched-capacitor multilevel inverters, this paper proposes a novel seven-level inverter topology as shown in Figure 4. In Figure 4, resistor R₁ serves to limit the high surge charging current of capacitors C₁ and C₂, whose rated voltage is U.

To validate the advantages of the proposed topology, a similar topology was selected for comparison, as shown in

Table 2. Comparative analysis of different inverter topologies.

References	NL	NDC	NC	NSW	ND	TSVpu	K
[10]	7	1	2	8	/	12.0	3
[11]	7	2	4	8	2	5.66	1
[12]	7	1	2	10	1	6.0	3
Proposed topology	7	1	2	7	3	5.33	3

N_L number of levels; N_DC number of DC sources; N_C number of capacitors; N_SW number of switches; N_D number of diodes; TSV_pu total standing voltage per-unit; K voltage gain.

, the topology proposed in [10] does not require diodes and has a similar number of switches to the present work, but it exhibits the highest total stress per unit voltage for the switches. The topology proposed in [11] has a similar total stress per unit voltage for the switches as the present work, but it requires two external power supplies, the highest number of switching capacitors, and only one voltage gain. The topology proposed in [12] uses fewer diodes but has the highest number of switches, while also exhibits relatively high total stress per unit voltage for its switches. Therefore, the topology proposed in this paper offers advantages in reducing both the total stress per unit voltage of the switches and the number of switches required.

3.3. Modulation Method for Seven-Level Inverter

By controlling the turn-on timing of each switch, combined with capacitor charging/discharging and diode freewheeling, the seven-level inverter can achieve outputs at different voltage levels.

Using the seven-level optimal turn-on angle derived in Table 1, the turn-on timing diagram for the switch is obtained, as shown in Figure 5. Calculations indicate that the turn-on sequences for transistors S₁–S₃ are α_S1 = 124.2°, α_S2 = 79.2° and α_S3 = 27.9°, respectively.

Based on the switching combinations of the power switches,

Table 3. The output voltage, switch states, and capacitor status of the seven-level inverter.

Working Mode	Output Level	S1	S2	S3	Q1	Q2	Q3	Q4	C1/C2
I	3U	1	1	0	1	0	0	1	D
II	2U	1	0	0	1	0	0	1	D
III	U	0	0	1	1	0	0	1	C
IV	0	0	0	1	1	1	0	0	C
V	0	0	0	1	1	1	0	0	C
VI	−U	0	0	1	0	1	1	0	C
VII	−2U	1	0	0	0	1	1	0	D
VIII	−3U	1	1	0	0	1	1	0	D

‘1’ and ‘0’ represent the ‘on’ and ‘off’ states of the switches, respectively; ‘C’ and ‘D’ represent the ‘charging’ and ‘discharging’ states of the capacitors, respectively.

shows the output voltage, switch states, and capacitor states of the seven-level inverter during one cycle. Figure 6 illustrates the operating mode of the seven-level inverter during the positive half-cycle.

Mode I: In Figure 6a, switches Q₁ and Q₄ are turned on, and capacitors C1 and C2 discharge in series, providing voltage together with the power supply U. At this time, the seven-level inverter output a voltage level of 3U.

Mode II: In Figure 6b, switches Q₁ and Q₄ are turned on, and capacitors C1 and C2 discharge in parallel, providing voltage together with the power supply U. At this time, the seven-level inverter output a voltage level of 2U.

Mode III: In Figure 6c, switches Q₁ and Q₄ are turned on, and capacitors C1 and C2 continue charging. The power supply U provides the voltage, and the seven-level inverter output a voltage level of U.

Mode IV: In Figure 6d, switches Q₁ and Q₂ are turned on, and capacitors C₁ and C₂ begin charging. At this time, the seven-level inverter output a zero voltage level.

Similarly, by controlling the switches Q₂ and Q₃ along with S₁–S₃, the seven-level inverter can generate a three-level trapezoidal wave output during the negative half-cycle.

The novel seven-level inverter topology proposed in this paper requires fewer switching state transitions during level switching compared to traditional multilevel inverter topologies, thereby helping to reduce switching losses. Furthermore, through capacitor charging and discharging, this topology can achieve up to three times the voltage gain.

Within a modulation cycle, the capacitor operating states are illustrated in Figure 7, with the continuous discharge intervals for capacitors C₁ and C₂ are [α_S3, π − α_S3] and [π + α_S3, 2π − α_S3], respectively.

To suppress voltage ripple, an appropriate capacitance value must be selected. The inverter’s output current i_in can be simplified as

$$i_{\mathrm{in}}=I_{\mathrm{m}}\sin (\omega t+\phi )$$

(14)

where I_m represents the output current magnitude, ω denotes the angular frequency of the output voltage, and φ is the impedance angle.

Since the seven-level inverter WPT system operates at high frequencies, it is assumed that the output current remains constant over one cycle. Therefore, the currents i_c1 and i_c2 through capacitors C₁ and C₂ can be expressed as

$$i_{\mathrm{c}1}=i_{\mathrm{c}2}=i_{\mathrm{in}}=I_{\mathrm{m}}\sin (\omega t+\phi )$$

(15)

When capacitors C₁ and C₂ are in parallel, the discharge capacity is

$$∆Q_1={\displaystyle \frac{1}{2}}{\displaystyle {\int }_{{\alpha }_{\mathrm{S}3}}^{\frac{\pi -{\alpha }_{\mathrm{S}2}}{2}}{i}_{\mathrm{in}}d\omega t}$$

(16)

where ΔQ₁ represents the respective discharge quantities of capacitors C₁ and C₂.

When capacitors C₁ and C₂ are in series, the discharge capacity is

$$∆Q_2={\displaystyle {\int }_{\frac{\pi -{\alpha }_{\mathrm{S}2}}{2}}^{\frac{\pi +{\alpha }_{\mathrm{S}2}}{2}}{i}_{\mathrm{in}}d\omega t}$$

(17)

where ΔQ₂ represents the respective discharge quantities of capacitors C₁ and C₂.

Within half a cycle, the total discharge of capacitors C₁ and C₂ is

$$∆Q_{\mathrm{T}}=2\times ∆Q_1+∆Q_2$$

(18)

The optimal capacitance value is expressed as

$$C_{\mathrm{opt}}\ge ∆Q_{\mathrm{T}}/\beta \cdot V_{\mathrm{ci}}$$

(19)

where β represents the percentage of ripple voltage, and V_ci denotes the rated voltage of capacitors C₁ and C₂.

In summary, the fundamental efficiency of a seven-level inverter can be expressed as

$${\eta }_m={\displaystyle \frac{P_{\mathrm{m}}}{P_{\mathrm{s}}}}={\displaystyle \frac{\frac{1}{T}{\displaystyle {\int }_0^T{u}_{\mathrm{m}}{i}_{\mathrm{m}}dt}}{\frac{1}{T}{\displaystyle {\int }_0^T{u}_{\mathrm{in}}{i}_{\mathrm{in}}dt}}}$$

(20)

where P_m and P_s represent the fundamental power and input power of the seven-level inverter, respectively, while u_m and i_m denote the fundamental output voltage and current. Since the system operates at high frequencies, it can be assumed that i_m = i_in. Therefore, the efficiency of the seven-level inverter can be expressed using the fundamental power fraction.

When the inverter outputs a seven-level trapezoidal wave, combining Formulas (5) and (13) yields the efficiency η₇ of the WPT system from the DC power supply input to the load is

$$\begin{array}{l}{\eta }_7={\eta }_{\mathrm{L}}{\eta }_3={\displaystyle \frac{{\omega }^2{M}^2{R}_{\mathrm{eq}}}{{\omega }^2{M}^2(R_{\mathrm{eq}}+R_{\mathrm{S}})+R_{\mathrm{P}}{(R_{\mathrm{eq}}+R_{\mathrm{S}})}^2}}\times {\displaystyle \frac{2\sqrt{2}(\sin \frac{{\theta }_1}{2}+\sin \frac{{\theta }_2}{2}+\sin \frac{{\theta }_3}{2})}{\sqrt{\pi ({\theta }_1+3{\theta }_2+5{\theta }_3)}}}\\ \end{array}$$

(21)

where η₃ represents the fundamental wave content ratio of the seven-level inverter output.

When the inverter outputs a two-phase square wave, the efficiency η₁ of the WPT system from the DC power source input to the load is

$${\eta }_1={\eta }_L{\eta }_0={\displaystyle \frac{2\sqrt{2}{\omega }^2{M}^2{R}_{\mathrm{eq}}}{\pi \left[{\omega }^2{M}^2(R_{\mathrm{S}}+R_{\mathrm{eq}})+R_{\mathrm{P}}{(R_{\mathrm{S}}+R_{\mathrm{eq}})}^2\right]}}$$

(22)

where η₀ represents the proportion of fundamental wave content in the two-level output.

Comparing Formulas (21) and (22), the overall transmission efficiency ratio of the WPT system is

$${\displaystyle \frac{{\eta }_7}{{\eta }_1}}>1$$

(23)

The above formula demonstrates that the novel seven-level inverter proposed in this paper exhibits significantly higher efficiency than that of a full-bridge inverter.

4. Maximum Efficiency Tracking Based on Current Amplitude Ratio with Closed-Loop PI Control

When the coupling coefficient or load changes deviate from the maximum efficiency point, the efficiency of the WPT system significantly decreases. To address this, this paper employs a closed-loop PI control strategy based on the current amplitude ratio to ensure maximum efficiency tracking when the system deviates from the maximum efficiency point.

4.1. Maximum Efficiency Based on Current Amplitude Ratio

To further enhance the efficiency of the WPT system, the transmission efficiency η₇ must be increased to its theoretical maximum value, defined as

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\eta }_7}{\partial R_{\mathrm{eq}}}}=0\\ {\displaystyle \frac{{\partial }^2{\eta }_7}{\partial {R_{\mathrm{eq}}}^2}}<0\end{array}\right.$$

(24)

From Formulas (21) and (24), the optimal equivalent load value for the system at maximum transmission efficiency can be determined as

$$R_{\mathrm{eq}.\mathrm{opt}}=\sqrt{R_{\mathrm{S}}^2+{\displaystyle \frac{{\omega }^2{k}^2{L}_{\mathrm{P}}{L}_{\mathrm{S}}{R}_{\mathrm{S}}}{R_{\mathrm{P}}}}}$$

(25)

From Formula (25), it is evident that when the system operates at the point of maximum transfer efficiency, R_eq.opt is positively correlated with k. However, obtaining the value of k in real time requires simultaneous measurement of primary and secondary side voltages and currents, increasing sampling complexity. Therefore, this paper employs a closed-loop PI controller based on the current amplitude ratio to achieve maximum efficiency tracking.

Define the ratio of the amplitude of the receiving coil current I_S to the transmitting coil current I_P as ρ. When the WPT system operates in a resonant state, Formula (3) expresses ρ as

$$\rho =\left|{\displaystyle \frac{I_{\mathrm{S}}}{I_{\mathrm{P}}}}\right|={\displaystyle \frac{\omega M}{R_{\mathrm{eq}}+R_{\mathrm{S}}}}$$

(26)

Combining Formulas (21) and (26) yields the relationship between the efficiency η₇ and the current amplitude ratio ρ:

$${\eta }_7={\displaystyle \frac{2\sqrt{2}(\omega M\rho -R_{\mathrm{S}}{\rho }^2)(\sin \frac{{\theta }_1}{2}+\sin \frac{{\theta }_2}{2}+\sin \frac{{\theta }_3}{2})}{(\omega M\rho +R_{\mathrm{P}})\sqrt{\pi ({\theta }_1+3{\theta }_2+5{\theta }_3)}}}$$

(27)

In high-frequency wireless power transfer systems, since $R_{\mathrm{S}}/\omega M\ll 1$, combining Formulas (25) and (26) yields the optimal current amplitude ratio ρ_opt as

$${\rho }_{\mathrm{opt}}\approx \sqrt{R_{\mathrm{P}}/R_{\mathrm{S}}}$$

(28)

According to Formula (27), the relationship between efficiency η₇ and current amplitude ratio ρ for different k values is shown in Figure 8.

As shown in Figure 8, the system exhibits a maximum efficiency point at different values of k, and it can reach the maximum efficiency point when ρ is close to ρ_opt. Formula (28) indicates that ρ_opt is independent of k. Since only the current amplitude needs to be detected without requiring phase information, this method simplifies the current sampling circuit while achieving maximum efficiency tracking.

4.2. Design of Closed-Loop PI Control Structure Based on Maximum Efficiency Tracking

Based on Formulas (3) and (29), when the other parameters are known, the current I_S in the receiving coil can be modified by adjusting the equivalent resistance R_eq, thereby indirectly regulating the parameter ρ. To enable continuous adjustment of R_eq over a wide range, the system employs a buck-boost circuit. After its integration, the relationship between the equivalent resistance R_eq at the WPT receiving end and the load resistance R_L is as follows:

$$R_{\mathrm{eq}}={\displaystyle \frac{8R_{\mathrm{L}}}{{\pi }^2}}{({\displaystyle \frac{1-D}{D}})}^2$$

(29)

Combining Formulas (26) and (29) yields the current amplitude ratio ρ:

$$\rho ={\displaystyle \frac{\omega M}{\frac{8R_{\mathrm{L}}}{{\pi }^2}{(\frac{1-D}{D})}^2+R_{\mathrm{S}}}}$$

(30)

According to Formula (30), the relationship between the current amplitude ratio ρ and the duty cycle D for different k values is shown in Figure 9.

As shown in Figure 9, ρ increases monotonically with increasing D. Therefore, a PI controller can be employed to regulate, thereby adjusting ρ to ρ_opt.

The WPT system structure incorporating the buck-boost circuit is shown in Figure 10.

In summary, the current amplitude ratio PI control block diagram shown in Figure 11 can be obtained.

5. Experimental Verification

5.1. Experimental Setup

To validate the novel seven-level inverter structure and the effectiveness and correctness of the maximum efficiency tracking method proposed in this paper, an experimental setup was constructed as shown in Figure 12.

In Figure 12, a RIGOL DP832A is used to provide a DC power supply, a Chroma 11050−5M is used to measure the self-inductance of the coil, and a Tektronix MSO2024B oscilloscope is used to record the experimental waveforms.

The basic parameters of the experimental setup are shown in

Table 4. Experimental setup parameters.

Parameters	Values
Input voltage U	20 V
Frequency f	100 kHz
Self-inductance LP, LS	142.26 μH, 142.16 μH
Compensation inductor LQ, Ldc	22 μH, 200 μH
Compensation capacitor CQ, CP, CS	115.14 nF, 21.06 nF, 17.82 nF
Coil resistance RP, RS	0.4 Ω, 0.49 Ω
Storage capacitor C1, C2, Cdc	680 μF, 680 μF, 100 μF

.

5.2. Experimental Results

To verify that the transmission efficiency of the seven-level inverter WPT system outperforms that of the conventional inverter WPT system, the load resistance was set to 20 Ω under open-loop conditions. Experimental measurements of the transmission efficiency for both WPT systems at different coil spacings were obtained and compared with theoretical values. The resulting efficiency comparison curves for the two WPT systems are shown in Figure 13.

As shown in Figure 13, compared to the conventional inverter WPT system, the novel seven-level inverter WPT system demonstrates significant efficiency advantages. When the coil spacing is within 12 cm (indicating a high coupling coefficient), its transmission efficiency approaches the theoretical maximum value. As the coil spacing varies, the overall transmission efficiency of the system increases by an average of 7.4%, showing an increasing trend with greater spacing.

At this point, when only different inverters were used while all other experimental conditions remained consistent, the inverter output voltage and current waveforms for the two WPT systems are shown in Figure 14a,b. Compared to the full-bridge inverter WPT system, the new seven-level inverter WPT system produces an output voltage that more closely approximates a sine wave and a sinusoidal output current, thereby enabling more efficient operation of the WPT system.

To verify whether the seven-level inverter WPT system can achieve maximum efficiency tracking under different load resistance values, experiments were conducted with a coil spacing of 12 cm and ρ_opt set to 0.9. The experimental results were used to plot the system transmission efficiency versus load relationship diagram shown in Figure 15.

As shown in Figure 15, when the WPT system operates without current-amplitude-ratio-based PI control, the transmission efficiency initially increases but then declines as the load R_L continues to rise. Upon incorporating current-amplitude-ratio-based PI control, the system’s transmission efficiency fluctuates minimally and approaches the theoretical calculated value by adjusting the duty cycle of the Buck-Boost circuit. For R_L values of 35 Ω, 55 Ω, and 75 Ω, respectively, the overall efficiency of the system with this control increased by 3.52%, 6.69%, and 9.46%, respectively, compared to the system without this control.

At this point, the output voltage and current waveforms of the seven-level inverter are shown in Figure 16a,b. According to Formula (3), when the system is in steady state under the maximum efficiency tracking control strategy, the resistance values of R_eq corresponding to different loads converge to the same level. Therefore, the magnitude of the inverter output current I_in remains essentially consistent, demonstrating the effectiveness of the maximum efficiency tracking control strategy under varying loads. When R_L is 10 Ω and 30 Ω, the magnitudes of the inverter output current I_in are 5.2 A and 5.1 A, respectively.

To verify whether the seven-level inverter WPT system can achieve maximum efficiency tracking at different coil spacings, experiments were conducted with R_L set to 15 Ω and ρ_opt set to 0.9. The experimental results were used to plot the relationship between system transmission efficiency and coil spacing, as shown in Figure 17.

As shown in Figure 17, when the WPT system lacks current-amplitude-ratio-based PI control, the transmission efficiency noticeably declines as the coil spacing increases, gradually deviating from the theoretical calculation values. When this control is incorporated, the transmission efficiency still decreases but remains close to the theoretical calculation value. At coil spacings of 16 cm, 20 cm, and 24 cm, the overall efficiency of the system with this control increased by 1.2%, 4.6%, and 11.3%, respectively, compared to the system without this control.

At this point, the inverter output voltage and current waveforms are shown in Figure 18a,b. According to Formula (3), when R_L is the same, the inverter output current I_in is only positively correlated with k, resulting in different magnitudes of the inverter output current I_in. When the coil spacing is 10 cm and 16 cm, respectively, the magnitudes of the inverter output current I_in are 6.6 A and 3.4 A.

To verify the impact of ρ_opt values on the system’s maximum efficiency tracking, three sets of different coil spacings were set for comparison. When R_L is 15 Ω, the relationship between system transmission efficiency and the optimal current amplitude ratio was plotted as shown in Figure 19.

As shown in Figure 19, the transmission efficiency of the system under three different coil spacing configurations exhibits a trend of first increasing and then decreasing as the value of ρ_opt increases. In all cases, the maximum efficiency point is achieved near ρ_opt = 0.9, thereby enabling optimal efficiency tracking for the system.

As shown in Figure 15 and Figure 17, when the system was tested under varying load and coil spacing conditions, the deviation between the efficiency tracked by the algorithm and the theoretical maximum efficiency was approximately 3%. This deviation primarily stems from losses inherent to the switches and coils themselves. It also demonstrates that the proposed method can effectively approximate the theoretical optimum value.

The comparison results between the maximum efficiency tracking method proposed in this paper and existing literature methods are shown in

Table 5. Comparison with the existing literature methods.

Reference	Fundamental Proportion Optimization	Coupling Coefficient Estimation	Maximum Efficiency Tracking	Efficiency
This paper	Yes	No need	Yes	91.4%
[6]	No	Need	Yes	88%
[16]	No	Need	Yes	89%
[17]	No	Need	No	86.2%

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6. Conclusions

This paper proposes a novel seven-level inverter topology for wireless power transfer (WPT) systems. By employing a maximum efficiency tracking strategy based on current amplitude ratio closed-loop PI control, it achieves maximum efficiency tracking under varying coupling coefficient and load. First, by comparing the conduction angle of the inverter’s switches and the fundamental wave content ratio across different levels, a seven-level inverter is selected to enhance overall system efficiency. Second, the relationship between current amplitude ratio and system transmission efficiency is derived. Analysis indicates that system transmission efficiency peaks at the optimal current amplitude ratio, which depends solely on coil resistance and is independent of the coupling coefficient. Finally, a buck-boost circuit is introduced at the receiver side, employing closed-loop PI control based on current amplitude ratio to dynamically regulate this ratio. Experiments demonstrate that system efficiency remains stable at approximately 91.4% during load variations. When coil spacing changes (i.e., coupling coefficient variations), system efficiency approaches theoretical calculations, achieving maximum efficiency tracking.