An Anti-Misalignment Method for Inductive Power Transmission System Based on Working Mode Switching

Abstract

In Inductive Power Transfer (IPT) systems, coils misalignment can significantly alter the system’s output power, thereby compromising operational stability. To address this issue, this article proposes a mode switching approach. Firstly, let the IPT system operate under conditions of primary-side inductive detuning and secondary-side resonance. Subsequently, by modifying the inverter’s switching frequency and conduction scheme, it is possible to vary the system’s operating frequency and the degree of primary-side detuning, thereby producing two distinct output power curves. Therefore, the controller can dynamically change the switching frequency of the inverter and the degree of primary-side detuning based on coil misalignment, thereby mitigating the effects of variations in the coupling coefficient on the output power by switching the output power curve. Finally, the parameter selection step and open-loop control system are presented. An experimental prototype with an output power of 600 W was constructed to validate theoretical analysis. The experimental results indicate that, when the system load is set to 10.4 Ω and the coupling coefficient varies between 0.23 and 0.50, the output power fluctuation of the system is 4.5%, and the transmission efficiency is 91.7%. The experimental results demonstrate that the proposed working mode switching method significantly improves the system’s anti-misalignment characteristics.

1. Introduction

Inductive power transfer (IPT) technology has been applied in various applications, including smartphone charging, electric vehicle charging, and underwater device charging, thereby substantially improving convenience in daily life [1,2,3,4]. In electric bicycle charging systems, transmission coils are generally designed to be flat due to spatial limitations of electrical components. The primary coil is fixed to the ground, while the secondary coil is integrated within the vehicle. Variations in the vehicle’s parking position resulted in shifts in the location of the receiving coil, resulting in fluctuations in system output, which subsequently impacted the stable operation of the charging system [5,6]. To mitigate the effects of coil misalignment on system output, researchers have primarily focused on developing control strategies, optimizing coupling structures, and optimizing compensation topology.

A common control strategy is to add a DC converter to either the primary-side or secondary-side of the system [7,8]. This allows the system output to be maintained at a constant value by adjusting the converter’s duty cycle. However, adding a DC converter increases the overall system volume. Therefore, References [7,8,9,10,11] used phase-shift control and pulse density modulation, respectively, to regulate the fundamental voltage amplitude of the inverter output and achieve a constant output. However, due to the inherent limitations of these modulation methods, the system may fail to achieve zero voltage switching (ZVS) when coil misalignment is significant, which consequently reduces transmission efficiency [12]. Meanwhile, the use of closed-loop control techniques requires the tuning of control parameters according to the system circuit model. However, the high-order and nonlinear characteristics inherent in IPT systems complicate the parameter tuning process.

In the design of coupling structures, researchers have found that optimizing coil geometry, structure, or the number of turns enables the primary coil to generate a relatively uniform magnetic field. As a result, when coil misalignment occurs, the mutual inductance fluctuates within a limited range or remains constant [13]. Based on this principle, Reference [14] connected a reversely wound circular coil in series with a square coil and optimized the number of turns to ensure that the system’s mutual inductance remained largely stable during misalignment. References [15,16] proposed the double-D (DD), double-D quadrature (DDQ), and triple-polar (TP) coil structures through coil structure optimization, which also exhibit strong misalignment tolerance. However, the process of optimizing coil parameters is generally complex, requiring multiple iterations with simulation software to achieve satisfactory anti-misalignment performance. Once the system design specifications are altered, the entire optimization process must be repeated, indicating a lack of design flexibility.

Consequently, researchers have improved the system’s anti-misalignment characteristics by optimizing the compensation topology. Reference [17] proposes a hybrid compensation topology that combines the inductor capacitor capacitor–inductor capacitor capacitor (LCC-LCC) and Series–Series (S-S) topologies in a primary-side series and secondary-side series configuration. By leveraging the opposing trends of output current and coupling coefficient between the two topologies, thereby making the system’s output current becomes insensitive to coil misalignment. Building upon this principle, references [18,19,20,21] have proposed various hybrid compensation topologies through the selection of different topologies or the adjustment of connection configurations. Although hybrid compensation topologies can improve the system’s anti-misalignment characteristics, they necessitate a four-coil structure and incorporate numerous compensation components, resulting in a substantial increase in the system’s volume and weight. Consequently, researchers have shifted their focus toward optimizing components within single compensation topologies to improve the system’s anti-offset performance. For example, in the S-S topology, reference [22] proposes a bilateral detuning method that adjusts the compensation capacitance to enable the system to operate with primary-side inductive detuning and secondary-side capacitive detuning. When the coupling coefficient varies between 0.08 and 0.2, the output power fluctuation remains below 12%. Reference [23] optimized the compensation capacitors in the Series–Series–Series (S-S-S) topology to achieve primary-side inductive detuning and secondary-side resonance. With coupling coefficients ranging from 0.092 to 0.143, output power fluctuations remained below 5%. To broaden the system’s coupling variation range, reference [24] proposes a reconfigurable compensation topology. By changing the connection mode of primary-side compensation components through AC switches, the system can switch between inductor capacitor capacitor–Series (LCC-S) and S-S topologies, thereby modifying the output power curve and extending the coupling variation range to 0.07–0.33. Reference [25] controls the rectifier to switch between full-bridge and half-bridge rectification modes. By changing the equivalent resistance of the rectifier, it achieves a change in the system’s output power curve, extending the coupling variation range to 0.1–0.4. However, both References [24,25] exhibit output power fluctuations exceeding 10%.

To address this problem, this paper proposes an open-loop control method based on working mode switching, which dynamically switches the system’s working mode to modify its output curve and can thereby reduce fluctuations in output power. In this method, the controller continuously measures the system’s output power in real time and adjusts the inverter’s switching frequency accordingly, while simultaneously modifying the switching states of the switches to change the detuning factor of the primary-side circuit. By coordinating changes both the switching frequency and the primary-side detuning factor, the system can transition to the appropriate output curve under varying misalignment conditions, effectively reducing output power fluctuations.

The structure of this article is organized as follows: Section 2 and Section 3 analyze the system’s output characteristics and anti-misalignment characteristics, respectively. Section 4 introduces the parameter design process and validates the theoretical analysis through experimental results. Finally, Section 5 presents the conclusion based on theoretical analysis.

2. Circuit Analysis

The circuit structure of the IPT system used in this article is illustrated in Figure 1. Here, V_in represents the DC power supply voltage. L₁ and L₂ represent the inductances of the primary and secondary coils, respectively. $M = k\sqrt{L_1{L}_2}$ represent the mutual inductance between the coils, where k is the coupling coefficient. L_X represents the compensation inductor. C₁, C₂, and C_X represent the compensation capacitors. C_o represents the filter capacitor, and R represents the resistive load. The inverter is composed of switches S₁ through S₄, while the uncontrolled rectifier consists of diodes D₁ through D₄. G₁₋G₄ represent the driving signals of the corresponding switches, respectively.

By adjusting the switching frequency of the inverter and conduction mode of switches S₁ through S₄, the system can operate in two distinct working modes. The output characteristics of these modes are analyzed below.

2.1. Working Mode I

When the driving signals of the inverter are as shown in Figure 2a, that is, switches S₂ and S₃ remain continuously conducting, while S₁ and S₄ conduct alternately at a frequency f₁ with a duty cycle of 0.5, the corresponding equivalent circuit can be established based on the fundamental harmonic approximation (FHA) method, as illustrated in Figure 2b. In this equivalent circuit, the rectifier circuit and resistive load can be equivalent to an AC resistance R_ac, where the relationship between R_ac and R is R_ac = (8R)/π².

According to Kirchhoff’s law, the voltage equation for the equivalent circuit is expressed as follows:

$$\left\{\begin{array}{l}\left[j{\omega }_1{L}_1+{\displaystyle \frac{1}{j{\omega }_1\left(C_1+C_2\right)}}\right]{\dot{I}}_{1,1}-j{\omega }_{1}M{\dot{I}}_{2,1}={\dot{V}}_{\mathrm{a}1}\\ -j{\omega }_{1}M{\dot{I}}_{1,1}+\left(R_{\mathrm{ac}}+j{\omega }_1{L}_2+j{\omega }_1{L}_{\mathrm{E}}+{\displaystyle \frac{1}{j{\omega }_1{C}_3}}\right){\dot{I}}_{2,1}=0\end{array}\right.$$

(1)

where ω₁ = 2πf₁, f₁ represents the switching frequency. According to the reference [26], under the action of frequency f₁, parallel branches L_X and C_X can be equivalent to an inductive element L_E. The relationship between L_E and L_X, C_X is as follows:

$$L_{\mathrm{E}}={\displaystyle \frac{L_{\mathrm{X}}}{1-{\omega }_1^2{L}_{\mathrm{X}}{C}_{\mathrm{X}}}}$$

(2)

According to the reference [24], when the secondary side of the system operates in a detuned state, the transmission efficiency of the system decreases. Therefore, the secondary compensation element must satisfy the following relationship:

$$j{\omega }_1{L}_2+j{\omega }_1{L}_{\mathrm{E}}+{\displaystyle \frac{1}{j{\omega }_1{C}_3}}=0$$

(3)

According to Equations (1)–(3), the current phasors ${\dot{I}}_{1,1}$ and ${\dot{I}}_{2,1}$ flowing through coils L₁ and L₂ can be calculated as follows:

$$\left\{\begin{array}{l}{\dot{I}}_{1,1}={\displaystyle \frac{R_{\mathrm{ac}}{\dot{V}}_{\mathrm{a}1}}{{\omega }_1^2{M}^2+j{\alpha }_1{\omega }_1{L}_1{R}_{\mathrm{ac}}}}\\ {\dot{I}}_{2,1}={\displaystyle \frac{j{\omega }_{1}M{\dot{V}}_{\mathrm{a}1}}{{\omega }_1^2{M}^2+j{\alpha }_1{\omega }_1{L}_1{R}_{\mathrm{ac}}}}\end{array}\right.$$

(4)

where α₁ represents the degree of detuning of the primary-side loop in working mode I, and the relationship between α₁ and the system parameters is given by

$${\alpha }_1=1-{\displaystyle \frac{1}{{\omega }_1^2{L}_1\left(C_1+C_2\right)}}$$

(5)

Furthermore, the output impedance Z_in1 and the input impedance angle θ₁ of the system can be calculated from Equation (4) as follows:

$$\left\{\begin{array}{l}{Z}_{\mathrm{in}1}={\displaystyle \frac{{\dot{V}}_{\mathrm{a}1}}{{\dot{I}}_{1,1}}}={\displaystyle \frac{k^2{\omega }_1^2{L}_1{L}_2}{R_{\mathrm{ac}}}}+j{\alpha }_1{\omega }_1{L}_1\\ {\theta }_1={\displaystyle \frac{\mathrm{Re}\left(Z_{\mathrm{in}1}\right)}{\mathrm{Im}\left(Z_{\mathrm{in}1}\right)}}=\arctan \left({\displaystyle \frac{{\alpha }_1{R}_{\mathrm{ac}}}{k^2{\omega }_1{L}_2}}\right)\end{array}\right.$$

(6)

Finally, according to Equation (4), the system’s output power can be calculated as follows:

$$P_1={\left|{\dot{I}}_{2,1}\right|}^2{R}_{\mathrm{ac}}={\displaystyle \frac{k^2{V}_{\mathrm{a}1}^2{\omega }_1^2{L}_1{L}_2{R}_{\mathrm{ac}}}{{\left(k^2{\omega }_1^2{L}_1{L}_2\right)}^2+{\left({\alpha }_1{\omega }_1{L}_1{R}_{\mathrm{ac}}\right)}^2}}$$

(7)

where V_a1 represents the fundamental effective value of the inverter output voltage v_a1. According to the reference [27], the relationship between V_a1 and the input voltage V_in is as follows:

$$V_{\mathrm{a}1}={\displaystyle \frac{\sqrt{2}}{\pi }}{V}_{\mathrm{in}}$$

(8)

2.2. Working Mode II

When the drive signal of the inverter is as shown in Figure 3a, that is, switch S₁ is continuously turned on, S₂ is turned off, while S₃ and S₄ are alternately turned on with a frequency of f₂ and a duty cycle of 0.5, the corresponding equivalent circuit can be established based on the FHA method, as illustrated in Figure 3b.

Similarly, according to Kirchhoff’s law, the voltage equation for the equivalent circuit can be expressed as follows:

$$\left\{\begin{array}{l}\left(j{\omega }_2{L}_1+{\displaystyle \frac{1}{j{\omega }_2{C}_2}}\right){\dot{I}}_{1,2}-j{\omega }_{2}M{\dot{I}}_{2,2}={\dot{V}}_{\mathrm{a}2}\\ -j{\omega }_{2}M{\dot{I}}_{1,2}+\left(R_{\mathrm{ac}}+j{\omega }_2{L}_2+{\displaystyle \frac{1}{j{\omega }_2{C}_{\mathrm{E}}}}+{\displaystyle \frac{1}{j{\omega }_2{C}_3}}\right){\dot{I}}_{2,2}=0\end{array}\right.$$

(9)

where ω₂ = 2πf₂, f₂ represents the switching frequency, and the relationship between the switching frequencies f₁ and f₂ is f₁ < f₂. Under the action of frequency f₂, parallel branches L_X and C_X can be equivalent to a capacitor element C_E. The relationship between C_E and L_X, C_X is as follows:

$$C_{\mathrm{E}}=C_{\mathrm{X}}-{\displaystyle \frac{1}{{\omega }_2^2{L}_{\mathrm{X}}}}$$

(10)

When the operating frequency varies, the secondary side must continue to operate in a resonant state, and the compensating element should satisfy the following relationship:

$$j{\omega }_2{L}_2+{\displaystyle \frac{1}{j{\omega }_2{C}_{\mathrm{E}}}}+{\displaystyle \frac{1}{j{\omega }_2{C}_3}}=0$$

(11)

Similarly, according to Equations (9)–(11), the current phasors ${\dot{I}}_{1,2}$ and ${\dot{I}}_{2,2}$ flowing through coils L₁ and L₂ can be calculated as follows:

$$\left\{\begin{array}{l}{\dot{I}}_{1,2}={\displaystyle \frac{R_{\mathrm{ac}}{\dot{V}}_{\mathrm{a}2}}{{\omega }_2^2{M}^2+j{\alpha }_2{\omega }_2{L}_1{R}_{\mathrm{ac}}}}\\ {\dot{I}}_{2,2}={\displaystyle \frac{j{\omega }_{2}M{\dot{V}}_{\mathrm{a}2}}{{\omega }_2^2{M}^2+j{\alpha }_2{\omega }_2{L}_1{R}_{\mathrm{ac}}}}\end{array}\right.$$

(12)

where α₂ represents the degree of detuning of the primary-side loop in working mode II, the relationship between α₂ and the system parameters is given by:

$${\alpha }_2=1-{\displaystyle \frac{1}{{\omega }_2^2{L}_1{C}_2}}$$

(13)

According to Equation (12), the input impedance Z_in2, input impedance angle θ₂, and the output power P₂ of the system in working mode II can be calculated as follows:

$$\left\{\begin{array}{l}{Z}_{\mathrm{in}2}={\displaystyle \frac{{\dot{V}}_{\mathrm{a}2}}{{\dot{I}}_{1,2}}}={\displaystyle \frac{k^2{\omega }_2^2{L}_1{L}_2}{R_{\mathrm{ac}}}}+j{\alpha }_2{\omega }_2{L}_1\\ {\theta }_2={\displaystyle \frac{\mathrm{Re}\left(Z_{\mathrm{in}2}\right)}{\mathrm{Im}\left(Z_{\mathrm{in}2}\right)}}=\arctan \left({\displaystyle \frac{{\alpha }_2{R}_{\mathrm{ac}}}{k^2{\omega }_2{L}_2}}\right)\\ P_2={\left|{\dot{I}}_{2,2}\right|}^2{R}_{\mathrm{ac}}={\displaystyle \frac{k^2{V}_{\mathrm{a}2}^2{\omega }_2^2{L}_1{L}_2{R}_{\mathrm{ac}}}{{\left(k^2{\omega }_2^2{L}_1{L}_2\right)}^2+{\left({\alpha }_2{\omega }_2{L}_1{R}_{\mathrm{ac}}\right)}^2}}\end{array}\right.$$

(14)

where V_a2 represents the fundamental effective value of the inverter output voltage v_a2. The relationship between V_a2 and the input voltage V_in is as follows:

$$V_{\mathrm{a}2}={\displaystyle \frac{\sqrt{2}}{\pi }}{V}_{\mathrm{in}}$$

(15)

3. Analysis of Anti-Misalignment Characteristics

According to Equations (7) and (14), once the system parameters are determined, the output powers P₁ and P₂ can be expressed as functions of the coupling coefficient k. The relationship between output power and the coupling coefficient is shown in Figure 4. As shown, the output power initially increases and subsequently decreases with increasing coupling coefficient.

By setting the derivative of power P₁ equal to zero, the coupling coefficient k_1max can be determined as follows:

$${\displaystyle \frac{d{P}_1}{dk}}=0\Rightarrow k_{1\max }={\displaystyle \frac{\sqrt{{\alpha }_1{R}_{\mathrm{ac}}}}{\sqrt{{\omega }_1{L}_2}}}$$

(16)

By substituting Equation (16) into Equation (7), the maximum output power, denoted as P_1max, is calculated as follows:

$$P_{1\max }={\displaystyle \frac{V_{\mathrm{in}}^2}{{\pi }^2{\alpha }_1{\omega }_1{L}_1}}$$

(17)

Similarly, according to Equation (14), the coupling coefficient k_2max and the maximum output power P_2max are calculated as follows:

$$\left\{\begin{array}{l}{k}_{2\max }={\displaystyle \frac{\sqrt{{\alpha }_2{R}_{\mathrm{ac}}}}{\sqrt{{\omega }_2{L}_2}}}\\ P_{2\max }={\displaystyle \frac{V_{\mathrm{in}}^2}{{\pi }^2{\alpha }_2{\omega }_2{L}_1}}\end{array}\right.$$

(18)

As demonstrated in Equations (16) and (17), an increase in the detuning factor α₁ (α₂) results in an increase in the coupling coefficient k_1max (k_2max), while the maximum output power P_1max (P_2max) decreases. Therefore, the primary-side detuning factor should be appropriately selected based on the design specifications. According to references [28,29], to minimize the fluctuation of the output power, the maximum power under the two working modes is set to be equal, i.e., P_1max = P_2max. By substituting this constraint into Equations (17) and (18), the relationship among the parameters α₁, α₂, ω₁ and ω₂ can be obtained as follows:

$${\alpha }_1{\omega }_1={\alpha }_2{\omega }_2$$

(19)

According to Equations (16), (18) and (19), it can be inferred that the coupling coefficient k_1max is greater than k_2max. Consequently, the position of the power curve is shown in Figure 5.

Subsequently, the intersection of the power curves P₁ and P₂ is analyzed. According to Equations (7) and (14), the coupling coefficient k_C and the output power P_C are calculated as follows:

$$\left\{\begin{array}{l}{k}_{\mathrm{C}}={\displaystyle \frac{\sqrt{{\alpha }_1{R}_{\mathrm{ac}}}}{\sqrt{{\omega }_2{L}_2}}}\\ P_{\mathrm{C}}={\displaystyle \frac{2{\omega }_2{V}_{\mathrm{in}}^2}{{\pi }^2{\alpha }_1{L}_1\left({\omega }_1^2+{\omega }_2^2\right)}}\end{array}\right.$$

(20)

Furthermore, by substituting Equation (20) into Equations (7) and (14), the coupling coefficients k_min and k_max can be calculated as follows:

$$\left\{\begin{array}{l} k_{\min }={\displaystyle \frac{{\omega }_1\sqrt{{\alpha }_1{R}_{\mathrm{ac}}}}{{\omega }_2\sqrt{{\omega }_2{L}_2}}}\\ k_{\max }={\displaystyle \frac{\sqrt{{\alpha }_1{\omega }_2{R}_{\mathrm{ac}}}}{{\omega }_1\sqrt{L_2}}}\end{array}\right.$$

(21)

As shown in Figure 5, the system operates in working mode II when the coupling coefficient varies between k_min and k_C. Conversely, when the coupling coefficient ranges from k_C to k_max, the system operates in working mode I. This method can expand the permissible range of coupling coefficient variations. Under these conditions, the output power fluctuation ΔP of the system is expressed as follows:

$$\Delta P={\displaystyle \frac{P_{1\max }-P_{\mathrm{C}}}{P_{1\max }+P_{\mathrm{C}}}}={\displaystyle \frac{P_{2\max }-P_{\mathrm{C}}}{P_{2\max }+P_{\mathrm{C}}}}={\displaystyle \frac{{\left({\omega }_2-{\omega }_1\right)}^2}{{\left({\omega }_2+{\omega }_1\right)}^2}}$$

(22)

Finally, based on Equation (21), the allowable range of coupling variation, Δk, for the system can be calculated as follows:

$$\Delta k=k_{\max }-k_{\min }={\displaystyle \frac{\sqrt{{\alpha }_1{R}_{\mathrm{ac}}}\left({\omega }_2^2-{\omega }_1^2\right)}{{\omega }_1{\omega }_2\sqrt{{\omega }_2{L}_2}}}$$

(23)

4. Theoretical Verification

In this section, a physical platform is constructed to validate theoretical analysis. The system’s input voltage is set to 330 V, the rated output power P_ref is 600 W, and the power fluctuation ΔP is 4%. The operating frequency range of the system is 100–200 kHz. The system design is detailed as follows.

4.1. Parameter Design

Firstly, determine the system’s operating frequency. Equation (23) indicates that the angular frequency ω₁ is inversely proportional to the allowable range of coupling variation Δk. Therefore, to maximize Δk, the switching frequency f₁ of operating mode I was set to 100 kHz. Subsequently, according to Equation (22), the switching frequency f₂ of operating mode II was calculated to be 150 kHz. Subsequently, a coupling mechanism is constructed, with the coil structure and parameters are shown in Figure 6a. The coil is constructed using Litz wire with a diameter of 2 mm. The relationship between the coupling coefficient and the misalignment distance (in either the x or y direction) as well as the transmission distance, is obtained through simulation and shown in Figure 6b. Owing to the coil’s symmetrical structure, the variations in coupling along the x and y directions are effectively equivalent.

To ensure the safe operation of the system, let P_1max = P_2max = P_ref, and substitute this into Equation (10) to obtain the values of the coefficients α₁ and α₂, which are given by:

$$\left\{\begin{array}{l}{\alpha }_1={\displaystyle \frac{V_{\mathrm{in}}^2}{{\pi }^2{\omega }_1{L}_1{P}_{\mathrm{ref}}}}\\ {\alpha }_2={\displaystyle \frac{V_{\mathrm{in}}^2}{{\pi }^2{\omega }_2{L}_1{P}_{\mathrm{ref}}}}\end{array}\right.$$

(24)

Subsequently, substitute the coefficients α₁ and α₂ into Equations (5) and (13) to calculate the capacitances C₁ and C₂ as follows:

$$\left\{\begin{array}{l}{C}_1={\displaystyle \frac{{\omega }_2^2\left(1-{\alpha }_2\right)-{\omega }_1^2\left(1-{\alpha }_1\right)}{\left(1-{\alpha }_1\right)\left(1-{\alpha }_2\right){\omega }_1^2{\omega }_2^2{L}_1}}\\ C_2={\displaystyle \frac{1}{\left(1-{\alpha }_2\right){\omega }_2^2{L}_1}}\end{array}\right.$$

(25)

Finally, to ensure that the secondary side of the system operates in a resonant state, the values of the compensation elements L_X, C_X, and C₃ are determined based on Equations (3) and (11) as follows:

$$\left\{\begin{array}{l}{L}_{\mathrm{X}}=p{L}_2\\ C_{\mathrm{X}}={\displaystyle \frac{{\omega }_1^2+{\omega }_2^2\pm \sqrt{{\omega }_1^4+{\omega }_2^4-\left(4p+2\right){\omega }_1^2{\omega }_2^2}}{2{\omega }_1^2{\omega }_2^2{L}_{\mathrm{X}}}}\\ C_3={\displaystyle \frac{1-{\omega }_1^2{L}_{\mathrm{X}}{C}_{\mathrm{X}}}{{\omega }_1^2\left(L_2+L_X\right)-{\omega }_1^4{L}_2{L}_{\mathrm{X}}{C}_{\mathrm{X}}}}\end{array}\right.$$

(26)

where p is a variable. To ensure that capacitor C_X has a positive solution, variable p must satisfy the inequality ${\omega }_1^4+{\omega }_2^4-\left(4p+2\right){\omega }_1^2{\omega }_2^2>0$. According to the above parameter design procedures, the system parameters are determined and presented in

Table 1. The system parameters.

Parameter	Value	Parameter	Value
L1	47.2 μH	R1	422 mΩ
L2	49.7 μH	R2	437 mΩ
LX	4.9 μH	RLX	34 mΩ
C1	100.7 nF	C2	40.7 nF
C3	41.9 nF	CX	275.9 nF

.

4.2. Control System Design

Figure 4 shows that the switching between the two working modes is determined by the coupling coefficient k_C. However, real-time calculation of the coupling coefficient requires sampling the high-frequency current flowing through the coil, which requires a high sampling rate from the sampling circuit. To simplify the design of the sampling circuit, this article determines the switching time based on the output power and designs an open-loop control system, as shown in Figure 7. Firstly, the control system is initialized. The parameter P_C is defined, and the system is operated in working mode I, i.e., the driver signal of the inverter as shown in Figure 2. Subsequently, the output voltage V_o and current I_o are measured, and the output power P is calculated. Based on the relationship between P and P_C, the system’s working mode was adjusted using the following method:

(1)

When P is greater than or equal to P_C, the system continues to operate in its current mode.

(2)

When P is less than P_C, the working mode of the system must first be determined. If the system is operating in working mode I, it should be switched to working mode II, i.e., the driver signal of the inverter as shown in Figure 2. Conversely, if the system is operating in working mode II, it should be switched to working mode I, i.e., the driver signal of the inverter as shown in Figure 2.

(3)

Upon completion of the system’s mode switching, the output voltage and current are measured to calculate the output power P. If P exceeds the P_C, the current operating mode is retained; otherwise, the system stops operation.

4.3. Simulation Verification

To validate the accuracy of the above theoretical analysis, a simulation platform was constructed based on the parameters listed in Table 1 for validation. When the resistive load R was set to 10.4 Ω, the output power and transmission efficiency were measured through simulation, with the results depicted in Figure 8. It was observed that the system operates in working mode II when the coupling coefficient k ranges from 0.23 to 0.34, and transitions to working mode I as the coupling coefficient varies between 0.34 and 0.5. The output power reaches its maximum value of 600 W at k = 0.27 or k = 0.41, and its minimum value of 554 W at k = 0.23 or k = 0.5.

Two cases with coupling coefficients k of 0.23 and 0.41 were selected, and their corresponding simulation waveforms were recorded as shown in the figure. In Figure 9, v_a1 and v_a2 represent the output voltages of the inverter; I1 represents the current flowing through coil L₁; and V₂ and i₂ correspond to the input voltage and current of the rectifier, respectively. In working mode II, the frequency of the inverter output voltage v_a1 is 150 kHz, whereas in working mode I, the frequency of the inverter output voltage v_a2 is 100 kHz. Furthermore, when the coupling coefficient k equals 0.23, the output power is 551 W with an input impedance angle of 54.7°; when k equals 0.41, the output power increases to 600 W with an input impedance angle of 45.3°. The simulation results align with the calculated values derived from Equations (16)–(21), thereby validating the theoretical analysis.

To verify the robustness of the system, the resistance load R was decreased to 7.4 Ω, and the output power, transmission efficiency, and input impedance angle were measured using the same method. The results are shown in Figure 10. When the coupling coefficient ranges from 0.19 to 0.29, the system operates in working mode II; when the coupling coefficient ranges from 0.29 to 0.43, the system operates in working mode I. The output power reaches a maximum value of 603 W at k = 0.24 or k = 0.37, and a minimum value of 557 W at k = 0.19 or k = 0.43. Based on the simulation results shown in Figure 8, it can be concluded that a decrease in load resistance does not affect the system’s rated output power P_ref or the power fluctuation range, but it does reduce the allow range of coupling coefficient variation. This finding aligns with Equations (22) and (23) and confirms the validity of the theoretical analysis.

Similarly, two cases with coupling coefficients k of 0.24 and 0.43 were selected, and their corresponding simulation waveforms were recorded as shown in Figure 11. As depicted, when the coupling coefficient k is 0.24, the output power is 600 W and the input impedance angle is 44.4°; when k is 0.43, the output power decreases to 554 W and the input impedance angle is 34.4°.

4.4. Experimental Verification

To validate the accuracy of the simulation, an experimental platform was constructed using the parameters listed in Table 1, as shown in Figure 12. Among them, MOSFETs IRF260NPBF were selected for switches S₁ through S₃, diodes BYW29-200G were selected for diodes D₁ through D₄, and DC power supply DPA30010U3 is used to provide power to the system. The Tektronix oscilloscope is used to capture experimental waveforms.

When the system load R is 10.4 Ω, the output power and transmission efficiency were experimentally measured, as shown in Figure 13. Within the coupling coefficient range of 0.23 to 0.34, the system operates in working mode II, with output power varying between 554 W and 607 W and achieving a maximum transmission efficiency of 91.7%. As the coupling coefficient increases from 0.34 to 0.5, the system transitions to working mode I, with the power variation ranges for working modes I and II being essentially equivalent. The calculated power fluctuation ΔP of the system is 4.5%. Due to errors in the components of the experimental prototype, discrepancies exist between the actual output power P and its fluctuations ΔP compared to the simulation results.

Figure 14a presents the dynamic experimental waveform as the coupling coefficient increases from 0.23 to 0.41. Figure 14b,c present the steady-state waveforms corresponding to coupling coefficients of 0.23 and 0.41, respectively. In these figures, v_a1 and v_a2 represent the output voltages of the inverter; i₁ represents the current flowing through coil L₁; and v₂ and i₂ represent the input voltage and current of the rectifier. As illustrated, when the coupling coefficient is 0.23, the frequency of the v_a1 is 150 kHz, indicating that the system operates in mode II. Conversely, when the coupling coefficient is 0.41, the frequency of the v_a2 decreases to 100 kHz, indicating the system operates in mode I. In both working modes, the current i₁ lags the voltages v_a1 or v_a2, indicating that the system’s input impedance is inductive. Furthermore, the current i₂ and voltage v₂ are in phase, signifying that the secondary side is in a resonant state. The experimental waveform confirmed the accuracy of the simulated waveform presented in Figure 9.

To verify the effect of the system load changes on the system’s anti-misalignment performance, the system load R was reduced to 7.4 Ω. Subsequently, the output power and transmission efficiency were measured using an oscilloscope, as shown in Figure 15. When the coupling coefficient ranges from 0.19 to 0.3, the system operates in working Mode II; conversely, when the output of the coupling system varies between 0.3 and 0.43, the system operates in working Mode I. In both working modes, the system’s output power fluctuates between a minimum of 560 W and a maximum of 610 W, while the transmission efficiency ranges from 83.7% to 90.1%.

Figure 16a presents the dynamic experimental waveform of the system as the coupling coefficient increases from 0.24 to 0.43. Figure 16b,c present the steady-state waveforms of the system corresponding to coupling coefficients of 0.23 and 0.43, respectively. The experimental waveform confirmed the accuracy of the simulated waveform presented in Figure 11.

4.5. Discussion

Table 2. Comparison between the proposed method and other literature.

Proposed In	Ref. [22]	Ref. [23]	Ref. [24]	Ref. [25]	This Work
Number of coils	2	3	2	2	2
Number of inductors	0	0	1	0	1
Number of capacitors	2	3	4	2	4
Operating frequency	200 kHz	−	400 kHz	250 kHz	100 or 150 kHz
Coupling variation	0.08–0.24	0.09–0.14	0.07–0.4	0.1–0.4	0.23–0.50
Output power	70 W	3.3 kW	163 W	400 W	600 W
Power fluctuation	12%	4.9%	11.2%	9.7%	4.5%
Efficiency	77.5–91.4%	90.1–94.4%	80.5–93.5%	87.5–95.6%	83.7–91.7%
Complexity control	Simple	Simple	Difficult	Medium	Medium

presents a comparison between the method proposed in this article and those reported in the existing reference. The studies presented in references [1,2] achieved relatively stable output by optimizing the parameters of compensation components, offering the advantage of not requiring any control strategy. However, the allowed range of system coupling variations remains relatively limited. In contrast, references [22,23] use a topology reconfiguration method that dynamically alters the system topology to modify the output characteristic curve, thereby accommodating a broader range of coupling changes. The working switching method proposed in this article attains output curve switching objectives comparable to those of topology reconfiguration techniques by concurrently adjusting the system operating frequency and the degree of primary-side detuning. Notably, compared to the methods described in references [24,25], the proposed method significantly reduces output power fluctuations. Furthermore, in comparison to conventional variable frequency control methods [30,31], the proposed method requires only open-loop control without necessitating control parameter tuning, thereby reducing the system’s control complexity. However, this method necessitates that the secondary side maintain resonance at two distinct frequencies, thereby increasing the number of required compensation components.

The typical wireless charging systems for electric vehicles typically operate within a range of 500 W to 30 kW, with switching frequencies between 79 kHz and 300 kHz [32,33,34]. Higher output power corresponds to faster charging speeds. The method proposed in this paper delivers an output power of 600 W, which satisfies the basic requirements for wireless charging of electric vehicles, although it remains limited by relatively low power.

5. Conclusions

This article proposes a working mode switching method that reduces the effects of coil misalignment on output power. The specific contributions are as follows:

(1)

The output characteristics and anti-misalignment characteristics of the IPT system operating inductance detuning and secondary-side resonance were analyzed. At the same time, parameter selection processes were given based on theoretical analysis.

(2)

According to the variations in output power, the controller adjusts the switching frequency and conduction mode of the inverter to modify the system output curve. This approach not only reduces the impact of coil misalignment on output power but also simplifies system control.

(3)

A simulation platform and an experimental prototype with an output power of 600 W were constructed. The simulation and experimental results indicate that as the coupling coefficient increases from 0.23 to 0.5, the system’s output power reaches 600 W, with an output power fluctuation of 4.4%. The theoretical analysis has been validated through these simulations and experiments.

Due to limitations in the power supply, the output power of the experimental prototype is relatively low and is currently suitable only for charging electric bicycles. Future research will focus on developing anti-misalignment techniques for high-power charging applications.