Mutual and Self-Inductance Variation in Misaligned Coupler of Inductive Power Transfer System: Mechanism, Influence, and Solutions

Abstract

This article focuses on the self- and mutual inductance variation of a coupler with ferrite in an Inductive Power Transfer (IPT) system. The mechanism of the variation was analyzed using the magnetic field splitting method, revealing that varying the reluctance leads to inductance variability. Additionally, the inductance variation trends were explored by Finite Element Analysis (FEA), based on which the input angle and voltage gain of an LCC-S type IPT system were calculated when coupler misalignment occurred. Then, an input voltage adjustment and frequency tracking compound control method was designed to neutralize the effect of inductance variability, which was validated by simulation. Finally, a prototype LCC-S type IPT system was fabricated. Experimental results reveal a 35.28% variation in self-inductance over the misalignment range, and the compound control managed to stabilize the output voltage and maintain the soft switching of the inverter with system efficiency remaining above 86% up to 94.27%. The proposed mechanism of inductance variation and compound control are instructive for solving the coupler misalignment problem in IPT systems.

1. Introduction

Inductive power transfer (IPT) technology is a relatively mature form of wireless power transfer, which has significant advantages in terms of safety, convenience, and economy compared with traditional contact power transmission. It has been heavily researched and applied in electric vehicle charging [1,2,3], intelligent household appliances [4], implantable devices [5,6], and railway transportation [7]. However, in practical usage scenarios, the inevitable misalignment of the couplers of IPT systems introduces serious problems, such as increased switching losses of the inverters and fluctuations of the system output, which severely affect the safety and stability of power transmission.

This effect is rooted in the inductance variation when the couplers are misaligned, which has already been noted. Hurley et al. [8] provide a general calculation method for the self-inductance and mutual inductance of coaxial circular coils. In reference [9], the planar rectangular coil was simplified into the superposition of many coaxial single rectangular coils, and the vector model of mutual inductance of coils was established by using the second-order vector bit. Stankiewicz and Choroszucho [10,11] comprehensively studied the influence of geometry and type of periodically arranged planar coils on the inductance and the efficiency of the system. These articles focus on coils of different shapes and numbers. However, in high-power applications, coils are usually combined with ferrite and a shielding layer to form couplers, resulting in different trends of inductance variability.

As for solving the misalignment problem, there are two main areas of academic effort: coupler design and system control. Aiming to reduce the coupler inductance variation while misaligned, coupler design focuses on the targeted design of coupler structure [12,13,14], material [15], and winding method [16,17]. Despite the possibility of fundamentally alleviating the variability, this approach suffers from the disadvantage of limited application of the after-design coupler and limited system performance improvement.

On the other hand, system anti-aliasing control aims to compensate for inductance variability by tuning controllable modules in an IPT system. A composite control method is designed in [18] based on the linear active disturbance rejection control and impedance matching technology, realizing constant voltage output and transmission efficiency improvement, whereas the additional Buck–Boost may cause energy loss. Li et al. [19] proposed an FM phase-shift keying communication technique for closed-loop maximum efficiency point tracking control, but the usage of a controllable rectifier increased the control difficulty. The pulse-frequency modulation conceived in [20] could suppress the switching losses while maintaining full-range soft-switching, thus improving the system efficiency, whereas the resulting increase of harmonic component must be taken into account. In addition, researchers such as Yang et al. [21] and Liu et al. [22] have also explored this problem. Taken together, all the aforementioned works provide excellent possibilities to solve the coupler misalignment problem in IPT systems, but little attention has been paid to the case of self-induced variation.

In this paper, we focus on the variation in mutual and self-inductance and its consequences when the couplers of the IPT system are misaligned. Based on the structure of the coupler in practice, a magnetic circuit model is applied in addition to the magnetic field splitting method to give a mechanical explanation for the variability, and a 3D FEA is performed in Section II to explore the trend of the variability. The expressions for the input impedance angle and voltage gain at off-resonance are derived in Section III, where the operational characteristics of the LCC-S topological IPT system with misaligned coils are analyzed. In addition, a control-based solution for the misalignment of the coils is designed. In Section IV, a prototype of the 1 kW IPT system is fabricated to validate the previous analysis and solution. Finally, Section V concludes the paper.

2. Analysis of Inductance Variation

2.1. Structure of Coupler in IPT Systems

As shown in Figure 1, the common coupler applied in IPT systems consists of a transmitter and receiver, which can be in the same or different shapes for diverse application scenarios. Without a loss of generality, this paper focuses on couplers with a symmetrical structure for simplicity, and the others can be analyzed similarly.

In the coupler, the coil is made of wound wire and serves as the conductor of the current; the materials of the magnetizer layer have strong magnetic permeability, which can guide the magnetic field and enhance the coupling of transmitter and receiver; the outermost shielding layer, usually made of metal, uses the eddy current effect to reduce magnetic leakage; and the waterproof rubber ring and encapsulation shell support and seal the whole coupler.

2.2. Mechanism Analysis of Inductance Variation

The self-inductance of bare coils is simply correlated with its size and winding method, not affected by changes in relative positions towards other coils. However, the existence of the magnetizer layer and shielding layer alter the natural distribution of the magnetic field, enhancing the performance of the coupler and, simultaneously, resulting in the mutability of self-inductance.

$$\begin{array}{l}M=N^2\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$R_{\mathrm{MP}}$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_{\mathrm{MF}}$}\right.\right)\\ L=N^2\left(\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$R_{\mathrm{LP}}$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_{\mathrm{LF}}$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_{\mathrm{MP}}$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_{\mathrm{MF}}$}\right.\right)\end{array}$$

(1)

Taking the magnetizer layer and shielding layer into consideration, the expressions of mutual and self-inductance of the coupler in the IPT system are established with the magnetic field splitting method in [23] as (1), where M and L denote the mutual and self-inductance of the coupler, respectively, N is the number of turns of the coil, α and β are the equivalent turns, and R_xx denotes the reluctance of different magnetic flow paths in the magnetic field.

The sectional view of the magnetic flow paths of the coupler is shown in Figure 2, and the arrows indicate the distribution of the magnetic field direction. The full path is divided into several parts (e.g., Φ_LP, Φ_LF, Φ_MP, and Φ_MF) according to the coupling degree between the magnetic flow and coils. The boundaries of the divided paths have no intersection with the magnetic field lines, and the magnetic field flows in one direction in each path. Taking Φ_LP for an example in Figure 3, the green part of volume V is the space occupied by the path and the average length of the magnetic field line in that path is l_av. Thus, R_LP can be calculated as (2), where μ is the permeability of the medium, and R_LF, R_MP, and R_MF can be calculated in similar ways.

$$R_{\mathrm{LP}}=\raisebox{1ex}{$l_{av}^2$}\!\left/ \!\raisebox{-1ex}{$\mu V$}\right.$$

(2)

Comprehensive analysis of Figure 2 and (2) reveals that when the coupler is misaligned, the shape of space occupied by magnetic flow paths will change and thus l_av and V will vary, resulting in the variation in the reluctance of magnetic flow paths (R_LP, R_LF, R_MP, and R_MF) and the mutual and self-inductance of the coupler.

2.3. Inductance Variation Trend Analysis

To accurately research the variation in the mutual and self-inductance of the coupler with different misalignments, a 3D finite element simulation model was established as shown in Figure 4. The misalignment distance was from 0 to ±20 cm along the X-axis and 3 to 30 cm along the Z-axis, and the inductance of the coupler for a total of 104 relative positions were calculated. Due to the symmetrical structure, the self-inductance simulation results of the transmitter and receiving were nearly the same and are uniformly denoted by L, and the results of misalignment along Y-axis are omitted for simplicity. The simulation results are plotted in Figure 5 and Figure 6.

Figure 5a is the overall diagram of simulation results and reveals the variation trend of self-inductance, from which we can determine that the self-inductance of the coupler decreases generally as the distance between the transmitter and receiver increases (both along the X- and Z-axis), which is particularly obvious at relatively strong coupling. Specifically, in Figure 5b, the self-inductance decreases markedly when the coupler is misaligned along the Z-axis (i.e., further in the energy transfer direction). Additionally, the better aligned in the X-axis, the more significant the reduction is, which could even reach 23.37%. Figure 5c illustrates that the variation of self-inductance is relatively smaller when the coupler is misaligned along the X-axis (i.e., further in the horizontal direction).

Figure 6a shows the global trend of mutual inductance variation and illustrates that the change of mutual inductance is more conspicuous than that of self-inductance and is positively correlated with coupler misalignment. Hence, the reduction in both Figure 6b,c is much larger than the parallels in Figure 5.

When combined, the variation in both the mutual and self-inductance of the misaligned coupler is non-negligible, and its corresponding influence on the IPT system will be studied in the next section.

3. Influenced System Characteristics and Targeted Solution

Due to its advantages in the flexibility of parameter design, stability of output, and mutual inductance independence of the resonance condition, an LCC-S compensation topology has been attracting increasing attention and is adopted in this paper. For an LCC-S topology IPT system, the soft-switching state of the inverter and the output voltage of the system are two of the most decisive and impressionable operating characteristics. Thus, accounting for the immediate correlation with the aforementioned characteristics, the system input impedance angle θ_in and the voltage gain G_v deserve comprehensive study, especially in coupler misalignment conditions.

3.1. Modeling of LCC-S Topology IPT System

Figure 7 shows the structure of a typical LCC-S topology IPT system, where U_DC is the DC input voltage and R_L is the load resistance. The harmonic components in the AC square wave voltage output by the inverter are filtered out after passing through the compensation network, so the system analysis can be performed based on the fundamental wave equivalent circuit in Figure 8.

In the figure, L_f, C_f, C₁, and C₂ are the component inductance and capacitor of the compensation network; L₁ and R₁ are the self-inductance and AC equivalent internal resistance of the transmitter; and L₂ and R₂ are that of the receiver. M is the mutual inductance between the transmitter and receiver, U_in is the effective value of the fundamental voltage output by the inverter, and its magnitude can be obtained from (3) when the switching tubes of the inverter are complementarily turned on.

$$U_{\mathrm{in}}=\frac{2\sqrt{2}}{\mathsf{\pi }}{U}_{\mathrm{DC}}$$

(3)

R_eq is the equivalent resistance value of the load resistance converted to the circuit by the rectifier. When the rectifier is complementarily turned on, its size can be obtained from (4).

$$R_{\mathrm{eq}}=\frac{8}{{\mathsf{\pi }}^2}{R}_{\mathrm{L}}$$

(4)

Given the complexity of the circuit topology in Figure 8, the T-circuit shown in Figure 9 is used in this paper to simplify the analysis process.

In the figure, Z₁, Z₂, and Z₃ are the impedance on the three bridge arms of the T-circuit; Z_L is the load impedance; U_in and I_in are the port voltage and input current of the input port; and I_out and U_out are the port voltage and output current of the load port.

$$\{\begin{array}{l}{U}_{\mathrm{in}}=I_{\mathrm{in}}{Z}_1+I_{\mathrm{out}}{Z}_2+U_{\mathrm{out}}\\ U_{\mathrm{in}}=I_{\mathrm{in}}{Z}_1+\left(I_{\mathrm{in}}-I_{\mathrm{out}}\right)Z_3\\ U_{\mathrm{out}}=I_{\mathrm{out}}{Z}_{\mathrm{L}}\end{array}$$

(5)

By arranging Equation (5), we can obtain

$$\{\begin{array}{l}{Z}_{\mathrm{in}}=\frac{U_{\mathrm{in}}}{I_{\mathrm{in}}}=Z_1+\frac{\left(Z_2+Z_{\mathrm{L}}\right)Z_3}{Z_2+Z_3+Z_{\mathrm{L}}}\\ G_{\mathrm{v}}=\frac{U_{\mathrm{out}}}{U_{\mathrm{in}}}=\frac{Z_3}{\left(Z_1{Z}_2+Z_1{Z}_3+Z_2{Z}_3\right)/Z_{\mathrm{L}}+Z_1+Z_3}\end{array}$$

(6)

Equation (6) is the expression of the input impedance Z_in and voltage gain G_v of the T-circuit. For the convenience of derivation, the components are classified as in Figure 8; that is, let

$$X_1=\omega L_{\mathrm{f}},X_2=\omega L_1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C_1$}\right.,X_3=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C_{\mathrm{f}}$}\right.,X_4=\omega L_2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C_2$}\right.$$

(7)

then there are

$$\begin{array}{l}{Z}_1=\mathrm{j}\omega L_{\mathrm{f}}=\mathrm{j}{X}_1\\ Z_2=\mathrm{j}\omega L_1+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{j}\omega C_1$}\right.=\mathrm{j}(\omega L_1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C_1$}\right.)=\mathrm{j}{X}_2\\ Z_3=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{j}\omega C_{\mathrm{f}}$}\right.=-\mathrm{j}{X}_3\\ Z_{\mathrm{s}}=R_2+R_{\mathrm{eq}}+\mathrm{j}\omega L_2+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{j}\omega C_2$}\right.=R_2+R_{\mathrm{eq}}+\mathrm{j}(\omega L_2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C_2$}\right.)=R_2+R_{\mathrm{eq}}+\mathrm{j}{X}_4\end{array}$$

(8)

Introduce reflected impedance Z_ref

$$Z_{\mathrm{ref}}=\frac{{(\omega M)}^2}{Z_{\mathrm{s}}}=\frac{{\omega }^2{M}^2}{R_2+R_{\mathrm{eq}}+\mathrm{j}{X}_4}$$

(9)

Then, the load impedance of the T-circuit can be expressed as

$$Z_{\mathrm{L}}=R_1+Z_{\mathrm{ref}}=R_1+\frac{{\omega }^2{M}^2}{R_2+R_{\mathrm{eq}}+\mathrm{j}{X}_4}$$

(10)

Substituting Formulas (8)–(10) into Formula (6), we can obtain

$$Z_{\mathrm{in}}=Z_1+\frac{(Z_2+Z_L)Z_3}{Z_2+Z_3+Z_{\mathrm{L}}}=\mathrm{j}{X}_1+\frac{X_2{X}_3-\mathrm{j}{X}_3(R_1+\frac{{\omega }^2{M}^2}{R_2+R_{\mathrm{eq}}+\mathrm{j}{X}_4})}{\mathrm{j}(X_2-X_3)+R_1+\frac{{\omega }^2{M}^2}{R_2+R_{\mathrm{eq}}+\mathrm{j}{X}_4}}$$

(11)

Let

$$\Delta =\frac{{\omega }^2{M}^2}{{(R_2+R_{\mathrm{eq}})}^2+X_4{}^2},O=R_2+R_{\mathrm{eq}}$$

(12)

Continue to organize (10), and finally the expression of input impedance angle θ_in can be reached as

$$\begin{array}{l}{Z}_{\mathrm{in}}=\frac{X_3{}^2(R_1+\Delta O)}{{(R_1+\Delta O)}^2+{(X_2-X_3-\Delta X_4)}^2}+\mathrm{j}\{X_1-\frac{X_3[{(R_1+\Delta O)}^2+(X_2-\Delta X_4)(X_2-\Delta X_4-X_4)]}{{(R_1+\Delta O)}^2+{(X_2-X_3-\Delta X_4)}^2}\}\\ {\theta }_{in}=\arctan \frac{\mathrm{Im}\left(Z_{\mathrm{in}}\right)}{\mathrm{Re}\left(Z_{\mathrm{in}}\right)}\end{array}$$

(13)

Similarly, the expression of the voltage gain G_v under the same condition can be deduced as follows

$$G_{\mathrm{v}}=\frac{1}{A^2+B^2}\left|\left(AC-BD\right)-\mathrm{j}\left(AD+BC\right)\right|$$

(14)

where

$$\begin{array}{l}A=\left(X_1{X}_3+X_2{X}_3-X_1{X}_2\right)\left(R_2+R_{\mathrm{eq}}\right)-R_1\left(X_1-X_3\right)X_4\\ B=X_4\left(X_1{X}_3+X_2{X}_3-X_1{X}_2\right)+R_1\left(X_1-X_3\right)\left(R_2+R_{\mathrm{eq}}\right)\\ +\left(X_1-X_3\right){\omega }^2{M}^2\\ C=R_1{X}_3{X}_4\\ D=R_1\left(R_2+R_{\mathrm{eq}}\right)X_3+X_3{\omega }^2{M}^2\end{array}$$

(15)

3.2. System Operation Characteristics with Varied Inductance

Equations (13) and (14) illustrate that the input impedance angle θ_in and voltage gain G_v of the LCC-S topology IPT system are susceptible to coupler misalignment and inductance variation. To observe the influenced system operation characteristics intuitively, the coupler inductance at the 104 relative positions mentioned in Section II and the other necessary parameters listed in

Table 1. Necessary parameters for calculation.

Parameter (Unit)	Value	Parameter (Unit)	Value
f (kHz)	85.00	RL (Ω)	25.00
Cf (nF)	98.76	C1 (nF)	29.81
C2 (nF)	22.90	Lf (uH)	35.50
R1 (Ω)	0.45	R2 (Ω)	0.35

are substituted into a calculation program in MATLAB R2019a, and the calculation results are plotted in Figure 10 and Figure 11. The figures have been interpolated for legibility.

Figure 10 shows the calculation results of θ_in with different misalignment of the coupler. It is known that θ_in will vary between less than, equal to, and greater than 0 as misalignment varies. In particular, when the coupling is relatively tight (i.e., the distance along the Z-axis is short) and the coupler is greatly misaligned along the X-axis, θ_in will decrease below zero, which means that the input impedance is capacitive. The capacitive input impedance is highly unfavorable and can easily cause device damage and system shutdown. On the other side, when the coupler is misaligned along the X-axis with a long distance in the Z-axis, θ_in will increase above zero, even approaching 90°, and the input impedance will become inductive. At that time, the switching stress of the device and switching loss of the inverter will greatly increase, resulting in a higher risk of malfunction and depressed energy transmission efficiency.

Figure 11 illustrates the variation of G_v in the same condition as Figure 10. Analysis of Figure 11 reveals that the voltage gain of the LCC-S topology IPT system is positively correlated with the coupling strength between the transmitter and receiver. Therefore, if the input voltage of the system remains unchanged, the output voltage will fluctuate with the coupler shifting along the X- or Z-axis.

In summary, under the circumstance of coupler misalignment and inductance variation, θ_in and G_v of the LCC-S topology IPT system will change drastically, which seriously affects the soft-switching state of the inverter and system output voltage U_out, endangering the safety of the system and stability of energy transmission. Thus, necessary measures should be taken to solve the problem.

3.3. A Control-Based Solution

The previous analysis expounds that the soft-switching state of the inverter and U_out are affected by coupler misalignment due to the variation of mutual- and self-inductance. This problem is solved from two aspects in this section and a corresponding compound control method is designed.

$$U_{\mathrm{out}}=U_{\mathrm{in}}\cdot G_{\mathrm{v}}$$

(16)

Firstly, the output voltage stabilization control is designed. We rewrote Equation (6) into (16), by analysis of which U_in and G_v were found to be directly related to U_out. Equation (14) reveals that G_v has a complicated relationship with the inductance of the coupler, the parameters of compensation components, the operating frequency of the system, and the equivalent resistance of the load. Among these factors, the inductance of the coupler is variable, caused by unavoidable coupler misalignment; the adjustment of the equivalent resistance of load faces difficulties in adjustable range and transfer efficiency decline [24]; the parameters of the compensation components are usually fixed once designed for the problem of volume, cost, and accuracy in adjusting them [25,26,27]; and the operating frequency is not the ideal instrument of voltage regulation for its influence on other operating characteristics, such as θ_in and transfer efficiency. Relatively, adjusting U_in is an effective way to compensate for the variation of inductance and stabilize U_out for its decisive influence on the output. With this in mind, an output voltage stabilization method based on adjusting input voltage was designed and is shown in Figure 12.

Figure 12 is the diagram of the output voltage stabilization method. It shows that the working flow of the method is as follows: U_out collected by the voltage sensor is sent to the receiving processor in real-time and transmitted to the transmitting side through the WiFi modules. Then, the transmitting processor adjusts U_in to the desired level after logical judgment through a built-in PI controller.

As for maintaining the soft-switching state of the inverter, θ_in should be kept within a slightly inductive range, which was set as 10 to 15° based on practical experience in this paper. Equation (13) shows that θ_in is similarly related to the inductance of the coupler, the parameters of compensation components, the operating frequency of the system, and the equivalent resistance of the load. Additionally, the other parameters cannot be used in the regulation of θ_in for the aforementioned reasons, except for the operating frequency of the system, f. Easily available by controlling the switches of the inverter, varying f can change θ_in effectively without adding additional structures to the system. Then, a soft-switching maintaining method based on frequency regulation is designed and shown in Figure 13.

Figure 13 is the diagram of the output voltage stabilization method. It shows that the working flow of the method is: The current sensor measures the output current waveform of the inverter in real time and sends it to the zero-cross detection module, whose output flips accordingly. In this way, the zero-crossing moment of the inverter output current is learned by the transmitting processor and then compared with that of the inverter voltage (decided by the driving program). The resulting time difference Δt, combined with the present f, is converted into the present θ_in by Equation (17). After that, the perturbation–observation method is applied and f is regulated by the newly produced driving signal until θ_in turns into the given range.

$${\theta }_{\mathrm{in}}=\frac{\Delta t}{T}·360°=\Delta t·f·360°$$

(17)

Combining the foregoing two methods of output voltage stabilization and soft-switching maintenance, a compound control method is achieved and a model is assembled in Simulink as in Figure 14 to preliminarily verify its effectiveness. The whole simulation is executed in two steps to verify the two methods, respectively, and during each simulation, the coupler inductance is set as that at the rated position (aligned in the X-axis, 6 cm away in the Z-axis) initially and switched to misaligned positions later. Without loss of generality, 9 cm away on the X-axis and 12 cm away on the Z-axis are taken as examples.

In the first simulation, the rated value of U_out and the initial value of U_in are set at 240 V and 120 V, respectively; the control module begins to work at 0.1 s and the inductance of the coupler is altered at 0.2 s. The waveform of U_in and U_out during the simulation are plotted in Figure 15a,b severally. In Figure 14, U_out decreases sharply for the sudden change of inductance at 0.2 s and U_in increases accordingly due to the effect of the control module, turning U_out to the rated value rapidly, which verifies the effectiveness output voltage stabilization method.

In the second simulation, the given range of θ_in is set as 10–15° as mentioned and the inductance is altered at 0.15 s. The waveform of θ_in during the simulation is plotted in Figure 16a,b severally. In Figure 15, θ_in immediately varies the inductance changes at 0.2 s and the control module takes effect at once, making it out to maintain θ_in within the given range.

4. Experiments and Discussions

4.1. Experimental Setup

According to the foregoing analysis, a prototype IPT system with parameters listed in

Table 2. Parameters of the prototype.

Parameter (Unit)	Value	Parameter (Unit)	Value
UDC (V)	120	Uout_DC (V)	240
f (kHz)	85.00	RL (Ω)	60.00
L1 (uH)	153.71	L2 (uH)	154.27
Cf (nF)	105.21	C1 (nF)	26.884
C2 (nF)	23.103	Lf (uH)	35.597
R1 (Ω)	0.45	R2 (Ω)	0.35

was fabricated as shown in Figure 17. The start–stop of the system was controlled by a program installed in the upper computer and the three-axis gantry contributed to coil misalignment by moving the coupler along three directions. The coupler had the same structure as shown in Figure 1, and the processors consist of DSP chips TMS320F28335, as well as other peripheral components. The sensors DVL 1000 and LT 100-S from LEM were adopted to collect voltage and current, respectively. Additionally, SiC MOSFETs C2M0080120D from Cree were used in the inverter, and the SiC Schottky diodes IDW30G65C5 from Infineon were used in the rectifier.

In particularly, the capacitance of C₂ is a little smaller than its resonance value for the ZVS of the inverter in the rated situation. With a power analyzer (ZLG PA5000H, Guangzhou Zhiyuan Electronics Co., LTD, Guangzhou, China) and an oscilloscope (ZLG ZDS3024 PLUS, Guangzhou Zhiyuan Electronics Co., LTD, Guangzhou, China), as well as a digital bridge used to record data, two experiments are performed and elucidated below.

4.2. Inductance Variation

In this offline experiment, initially, the transmitter and receiver are closely placed and then adjusted to 20 cm away in the Z-axis with a step of 1 cm. Meanwhile, the self- and mutual inductance of the coupler are measured and plotted in Figure 18.

In Figure 18, the whole range can be divided into two parts, the variational self-inductance region and the stable self-inductance region, based on the stability of self-inductance with a distance of 9 cm, coupling coefficient k = 0.34 as a boundary.

In the variational self-inductance region, the transmitter and receiver are closely coupled and both the self- and mutual inductance vary sharply. Specifically, the self-inductance dropped by 35.28% and mutual inductance by 74.4%. The strong link between self-inductance and the resonance of the IPT system means that a great variation of inductance could lead to serious detuning of the IPT system, failure of inverter soft-switching, reduction of transfer efficiency, and the enormous risk of system breakdown. Therefore, the influence of mutual- and self-inductance should be considered simultaneously in relatively moving coupler applications, especially in strongly coupled cases.

In the stable self-inductance region, the self-inductance tends to be stable, yet mutual-inductance still varies with the transmission distance. Due to the direct effect on the output of the IPT system, the variation of mutual inductance could result in the instability of output and deterioration of power-supply quality.

4.3. The Compound Control

In this experiment, the relative position of transmitter and receiver was changed between 0–10 cm away on the X-axis and 6–16 cm away on the Z-axis after power-on, and the output waveform, as well as energy transfer efficiency η of the IPT system, were recorded to analyze the compound control method. As a contrast, an uncontrolled case with the same parameters was conducted as well.

Taken as an example, the screen captures of instruments in 10 cm away misalignment along the X-axis experiment are shown in Figure 18. It can be seen from Figure 19a that there are obvious enlargements of Δt and decline in U_{out_DC} in this case. Additionally, the waveform suggests the severe switching stress of the inverter. Then, with the intervention of the control system in Figure 19b, those problems are well solved and η shows an increase to some extent.

The overall experimental results are plotted in Figure 20 and Figure 21. Noteworthily, the unknowable after-adjusting f produced by the program leads to the unavailability of final θ_in; hence, the achievement of soft-switching is reflected by the enhancement of η.

The ferrite core shown in Figure 20 shows the results of the X-axis misalignment experiment and reveals that, if not controlled, the output voltage of the system could drop by 31.44% during the moving range of the coupler. Through control, the output voltage was maintained at the rated value and η achieved a 1.28% increase at most.

The results of the Z-axis misalignment experiment are shown in Figure 21, which illustrates that the control performs better in dealing with Z-axis direction misalignment. The 69.1% decline of the output voltage was erased and η was improved by 4.28% at most. In addition, the transfer efficiency of the controlled system remained above 86%.

4.4. Discussion

The experimental results in Section 4.3 verify the effectiveness of the compound control system, whereas several challenges remain to be overcome. On the one hand, the current PI associated with the perturbation–observation method has large room for improvement in terms of dynamic performance and control limit. On the other hand, the impact of circuit parasitic parameters and harmonic component on the system performance is unknown. Therefore, future work will focus on investigating the effects of parasitic parameters and harmonic components in the circuit and attempt to adopt more advanced control algorithms to improve the system performance.

5. Conclusions

This paper focuses on the variation in the mutual- and self-inductance of the coupler and its corresponding influence on the IPT system. Furthermore, a targeted control-based solution is discussed. Specific contributions include:

• For the first time, the principle of inductance variability in misaligned couplers has been explained mechanically using the magnetic field splitting method, which reveals that the variability arises from the variation of the magnetic resistance, which is determined by the shape of the flux tube in the magnetic circuit. This is particularly remarkable in the case of tightly coupled coils.
• Considering the variation of self-inductance, the expressions of θ_in and G_v for the LCC-S topology IPT system at non-resonance are derived, and the operating characteristics of the system are analyzed. The corresponding calculations show that the variation gives rise to an increase in the switching loss or even the destruction of the inverter and, if neglected, to an alteration of the output voltage.
• A control-based solution is adopted and a 1 kW prototype system is fabricated. The experimental results show that within ±10 cm misalignment along the X-axis and 6–16 cm distance on the Z-axis, the output voltage deviation can be regulated within ±1.2% of the rated value, and θ_in is kept in a weakly inductive region simultaneously, maintaining the soft-switching state of the inverter, which could improve the energy transfer efficiency up to 4.28%. The closed-loop energy transfer efficiency of this prototype is maintained above 86%, up to 94.27%.

This paper reveals the mechanism and influence of the inductance of misaligned coupler in an IPT system and provides a targeted effective solution, which is beneficial to the design of IPT systems with unfixed applications, such as electric vehicle charging and wireless power supply of shipment. The methods used in this paper are also of reference for similar studies.