A Six-Switch Mode Decoupled Wireless Power Transfer System with Dynamic Parameter Self-Adaption

Abstract

For the fully resonant wireless power transfer (WPT) system, the high coupling of the converter and the resonant network introduced many problems, such as frequency splitting, the power curve peak limit, and the strict switch strategy. To solve these problems, this paper proposed a new six-switch topology based on the full–-bridge converter. With the unique structures containing two capacitor-isolated switches and a source-isolated diode, the system decouples the converter and the resonant network, and its modes have been decoupled, called the independent power injection and free resonance WPT (IPIFR–WPT) system. The capacitor-isolated switches and the source-isolated diode make the converter operate only when the voltage on the primary capacitor is equal to the source voltage, and the source will be isolated by the diode when the capacitor voltage is great than the source, which provides a wide time margin for the switches of the converter to turn on in advance. In this margin, the operation point is self-determined the same whenever the switches turn on so that the system’s performance is consistent. Based on this characteristic, the system can self-adapt a dynamic change in system parameters, with at least 15% tolerance for the coupling coefficient and 14% for the load resistance.

1. Introduction

Wireless power transfer (WPT) technology uses electromagnetic fields for contactless power transfer [1]. Because there is no direct physical connection, it has higher security and better flexibility than traditional wired power transfer [2]. Therefore, WPT technology has developed very fast in recent years, especially on movable and rotatable devices, such as robots, electric vehicles, underwater devices, portable applications and implantable medical equipment [3,4,5,6,7].

The fully resonant wireless power transfer (FR–WPT) system has attracted the widest research in recent decades [8]. Due to the high coupling of the converter and the compensation network, FR–WPT systems really require the converter’s switching frequency to be equal to the fixed frequency of the resonant network to achieve stable and efficient operation. Since the converter is fully coupled with the resonant network, the FR–WPT system has faced some issues, such as the power peak limit, the parameter change sensitivity, frequency splitting, and so on [9,10,11]. To solve these issues, some closed-loop strategies including impedance matching and frequency tracking are proposed to improve the dynamic performance and ensure the system’s stable operation [12,13,14,15]. However, due to the limitation of the system’s structure, the closed-loop strategy is limited by the power peak limit. Furthermore, because of the frequency splitting phenomenon, it is easy to fall into local solutions and even make the system unstable, causing irreversible damage to the device [16].

The independent power injection and free resonance wireless power transfer (IPIFR–WPT) system uses a novel and unique topology structure that separates the switching converters from the resonant tanks [17,18]. The characteristic of this circuit structure is that the power injection process and the free resonance process are independent so that the converter and the resonance network are completely decoupled. In the IPIFR–WPT system, the power injection process is completely controlled independently, and the free resonance is also completely independent. This control strategy’s advantage is isolating the converter from the system and making the resonant network conduct passive-free resonance. Therefore, the frequencies of the converter and the resonant network are different, decoupling the resonant network and the converter, and simplifying the power control strategy [19]. At the same time, the operating frequency of the resonant network is also free to adapt to the influence of the change in the working parameters because it does not have to be affected by the operation of the converter, which solves the issues of frequency splitting and frequency tracking in the system.

Article [20] proposes a pulsed power injection IPIFR–WPT system. The principle is mainly to control the power injected into the coil inductance by controlling the turn-on time of the switch S₁. However, the system requires precise control for switch S₂, and its switching times must be strictly aligned with the zero crossing point of the resonant current. Otherwise, the soft switching condition will not be satisfied. This shortcoming causes strong adhesion between the converter and the resonance network, fails to achieve complete decoupling, has high requirements for the accuracy of the control system, is sensitive to parameter changes, and needs better dynamic performance. Article [21] proposes a full-bridge IPIFR–WPT system with six switches. The switches are used to periodically connect or isolate the resonant capacitor and the DC power supply into or out of the system to achieve power injection and free resonant mode switching. The single-switch IPIFR–WPT system uses only one switch to control mode switching, greatly reducing system switching losses and improving system efficiency [22,23]. The energy of the injection system can be adjusted by controlling the turn-on time of the switch to achieve the regulation of power, but the injection time cannot be too large. Furthermore, it is necessary to ensure that the current flowing through the primary inductor can be less than zero during the resonance process. Therefore, there is a theoretical power peak for the single-switch IPIFR–WPT system, which means the system cannot achieve wide-range power regulation.

Furthermore, a comparison of the technologies to improve the dynamic performance of the WPT system is summarized in

Table 1. Comparison of the technologies to improve dynamic performance.

Ref.	Dynamic Performance Improving Strategy	Inductors/Capacitors/Switches on the Converter	Have Any Voltage/Current Sensors?	Control Strategy	Achieve Decoupling?	Sensibility to the Parameters
[14]	Impedance matching with a capacitor matrix	1/(M × N)/(M × N + 1)	Yes	Close loop	No	Low
[15]	Frequency tuning loops	1/1/4	Yes	Close loop	No	High
[16]	Maximum power point tracking	1/1/4	Yes	Close loop	No	High
[20]	Switch-mode adjusting	2/3/2	No	Open loop	Partially	High
Proposed	Power injecting structure	1/1/6	No	Open loop	Yes	Low

. Normally, for the impedance matching method, a lot of components are used to adjust the equivalent input impedance [14]. Hence, the impedance matching method leads to the higher cost and larger size of the system. Moreover, the actual dynamic performance is discrete and approximate. Articles [15,16] use different frequency tracking methods to improve dynamic performance and eliminate frequency splitting. However, the high sensibility to the parameters needs precise design. Article [20] changes the structure of the converters to achieve the decoupling between the converter and the compensation network. However, the decoupling is incomplete, and the system is sensitive to the parameters.

In this paper, based on the traditional bridge circuit, a six-switch IPIFR–WPT system is proposed. Its unique capacitor-isolated switches and power-source-isolated diode mean the power supply and parallel resonant capacitor can be periodically connected or isolated from the system. When the power supply is connected to the system and the resonant capacitor is isolated, it is in the power injection mode. The longer the power injection mode duration, the more power is injected into the system. When the resonant capacitor is connected to the system and the power supply is isolated, the resonant network will use the energy stored in the inductance and capacitor to conduct free resonance. The resonant frequency is the fixed frequency of the resonant network and does not need to be controlled by the switches. Therefore, the six-switch IPIFR–WPT system realizes the mode decoupling of power injection and free resonance and the decoupling of the converter and resonant network. In addition, due to the source-isolated diode, the power supply can only inject energy into the system in one direction. Thus, the switches have been turned on in advance to ensure that the power supply will not be connected to the system in advance, only when the voltage on the primary resonant capacitor drops to equal the power supply voltage, and the leading time range is relatively wide, which is called the soft switching margin. At the same time, the parasitic diode on the switch can ensure the zero voltage switching (ZVS) condition.

2. Circuit Structure and Mode Analysis

2.1. Circuit Structure and Mode Analysis

The prototype of the six-switch IPIFR–WPT system is shown in Figure 1. In Figure 1, a unidirectional power supply comprises an isolation diode D₀ and a DC power supply E_dc. The purpose of a one-way power supply is to ensure that energy entering the system does not return to the power supply. Switches S₁–S₄ form a full-bridge converter to connect power E_dc to the system and inject energy into the system. Switches S₅ and S₆ let the primary compensation capacitor C_p be isolated from or connected to the system. Diodes D₁–D₆ are the bypass diodes of switches S1–S6, wherein bypass diodes D₁–D₄ do not function during the entire working process due to their different directions from isolation diode D₀; the bypass diodes D₅ and D₆ are used for continuous current conduction during the resonance process and can isolate C_p from the system. The self-inductances of the primary and secondary coils are L_p and L_s, while M is the mutual inductance of the coils. R_p and R_s are the parasitic resistance of the primary and secondary coils, respectively. The load R₀ consumes the energy received by the secondary circuit after being rectified by the back-end rectifying circuit of the secondary circuit. Supposing the inductive characteristics and the voltage drop of the back-end rectifier circuit are ignored, in that case, the equivalent load R_L can be used to convert the load resistance R₀ to the input terminal of the rectifier circuit, and the conversion relationship is as follows:

$$R_L=\frac{8R_0}{{\pi }^2}.$$

(1)

Figure 2 is the schematic diagrams of the operation modes of the six-switch IPIFR–WPT system. When the system operates, there are eight operating modes. Mode 1 connects the power supply E_dc to the system in a forward direction, a forward power injection mode. Modes 2–4 are resonant modes, also called free resonance processes, due to the isolation of the power supply E_dc from the system. Note that Modes 2 and 4 are transitional modes, both of which are unidirectional resonance processes. According to the symmetry of the full-bridge converter operation, Mode 5 reversely connects the power supply E_dc to the system as a reverse power injection mode. Furthermore, Modes 6–8 are resonant processes opposite to the working processes of Modes 2–4.

Figure 3 shows the waveform diagram of the source current i₁, the primary capacitor voltage u_p, and the primary inductor current i_p of the six-switch IPIFR–WPT system under steady-state operation. According to Figure 2 and Figure 3, the operation modes of the six-switch IPIFR–WPT system can be analyzed as follows:

Mode 1 [t₀, t₁]: This is a forward power injection mode with a duration of ξ₁ = t₁–t₀. Before the time t₀, the switches S₁ and S₄ have been turned on in advance, and the switches S₂ and S₃ remain turned off. At this time, the primary capacitor voltage u_p is greater than E_dc, forcing D₀ to be turned off, isolating the power supply E_dc from the system, thereby preventing the energy from being injected into the system. Until the time t₀, the voltage u_p drops to equal to E_dc, causing diode D₀ to turn on, and the current i_p flows from the branch D₆–C_p–S₅ to the switches S₁ and S₄ that have been turned on in advance. During the period [t₀, t₁], while the primary capacitor C_p is isolated from the system, the DC power supply E_dc directly injects energy into the primary inductor L_p.

Mode 2 [t₁, t₂]: This is a transitional mode of the first free resonance process. At the moment t₁, the switches S₁ and S₄ turn off, and at the same time, the current i_p will continuously conduct through the bypass diode D₆. At this time, the turning off of switches S₁ and S₄ meets the ZVS condition. In Mode 2, the primary inductor L_p and the primary compensation capacitor C_p form a resonant cavity and begin passive free resonance. The duration of Mode 2 is very short and can generally be taken as 1–2 μs, even a few hundred nanoseconds.

Mode 3 [t₂, t₃]: This is a bidirectional free resonant mode. Since the switch S₆ turns on at t₂, the primary current i_p can achieve bidirectional free resonance in the resonant network L_p–C_p. During Mode 3, after the value of the current i_p resonates to less than zero, the flow direction of i_p in the branch is S₅–C_p–S₆. After that, at time t₃, the switch S₅ turns off, and the current i_p can still continuously conduct through the bypass diode D₅. The switch S₅ turns off to meet the ZVS condition. At the same time, at t₃, the switches S₂ and S₃ turn on. Due to the capacitance voltage u_p being less than −E_dc at this time, the isolation diode cannot be turned on, and the current cannot be exchanged from the branch D₅–C_p–S₆ to the switches S₂ and S₃. That is, before and after the turn-on of S₂ and S₃, the current flowing through them is zero, satisfying the zero current switching (ZCS) condition.

Mode 4 [t₃, t₄]: In this mode, the system remains in a state of free resonance. In addition, the current i_p continuously conducts through the branch D₅–C_p–S₆ during the time span until the time of t₄, when the value of u_p rises to equal to −E_dc through resonance. At time t₄, because the switches S₂ and S₃ have turned on in advance at t₃, the current i_p can be naturally switched from the branch D₅–C_p–S₆ to the switches S₂ and S₃, thereby switching to Mode 5.

Mode 5 [t₄, t₅]: This is a reverse power injection mode, which is reversed with Mode 1. Due to the natural commutation of the current i_p from the branch D₅–C_p–S₆ to the switches S₂ and S₃ that have turned on before t₄, the power supply E_dc is reversely connected to the system and injects energy into the primary inductor L_p.

Mode 6 [t₅, t₆]: This is a transitional mode of the second free resonance process, which is symmetric to Mode 2 and has a short duration. At time t₅, the switches S₂ and S₃ turn off. Since the current i_p will be continuously conducted through the bypass diode D₅, turning off switches S₂ and S₃ meets the ZVS condition. In Mode 6, the primary inductor L_p and the primary compensation capacitor C_p form a resonant network and begin free resonance.

Mode 7 [t₆, t₇]: This is a bidirectional free resonant mode symmetric to Mode 3. At t₆, the switch S₅ turns on. Due to the capacitance voltage u_p greater than E_dc, the current i_p continuously conduct through the bypass diode D₅ without commutation to S₁ and S₄. Therefore, the turn-on of switch S₅ satisfies the ZVS condition. Then the primary current i_p can achieve bidirectional free resonance in the resonant network L_p–C_p during [t₆, t₇].

Mode 8 [t₇, t₈]: This is a transitional mode symmetric to Mode 4. At t₇, switch S₆ turns off while turning on switches S₁ and S₄. Because of the voltage u_p being greater than E_dc, D₀ is turned off, and the current i_p continuously conducts through D₆ without commutation to S₁ and S₄. Therefore, S₆ meets the ZVS condition, while the opening of S₁ and S₄ satisfies the ZCS condition. The resonant network maintains resonance during [t₇, t₈]. Until t₈, the voltage up is equal to E_dc, and the current i_p naturally commutates from the resonant network to S₁ and S₄. Meanwhile, the system operates into Mode 1.

2.2. Calculation of System Soft Switching Operating Point

Based on the mode analysis of the six-switch IPIFR–WPT system in the previous section, the switching conditions for its eight operation modes are shown in Figure 4. The eight operation modes of a six-switch IPIFR–WPT circuit can be classified as four cyclic operating processes, namely, the forward power injection process, the first free resonance process, the reverse power injection process, and the second free resonance process. Among them, Mode 1 and Mode 5 are forward and reverse power injection processes, respectively. Modes 2, 3, and 4 jointly constitute the first free resonance process, and Modes 6, 7, and 8 jointly constitute the second free resonance process.

Figure 4 shows no switch operates when the free resonance process switches to the power injection process. The switching condition is to rely on the voltage u_p to form a voltage clamp with the power supply voltage E_dc through the diode D₀. In Mode 8, the switch groups S₁ and S₄ have turned on, but at this time, the voltage u_p is greater than E_dc, and the system remains in the free resonance process. Meanwhile, i_p > 0, that is, u_p will continue to decrease until u_p drops to equal to E_dc, and the diode D₀ reaches the critical conduction condition. At the same time, the current is commutated from branch S₆–C_p–S₅ to branch S₄–E_dc–D₀–S₁. The system switches from a free resonance process to a forward power injection process, and the voltage u_p on the primary capacitor C_p will maintain the voltage clamp at E_dc. Similarly, when switching from Mode 4 to Mode 5, it is necessary to clamp the voltage u_p on the primary capacitor C_p to −E_dc, and the free resonance process completes the switching to the reverse power injection process.

For convenience, the switching conditions β₁, β₂, β₃, and β₄ between the four operation processes are shown in Figure 4 and written in parallel:

• β₁: S₁ = 0 and S₄ = 0;
• β₂: u_p = −E_dc and i_p < 0;
• β₃: S₂ = 0 and S₃ = 0;
• β₄: u_p = E_dc and i_p > 0.

Figure 5 shows the equivalent circuit diagram of the main operation process of the six-switch IPIFR–WPT system. Since the state space model of the system during the forward power injection process and the reverse power injection process are entirely consistent, the equivalent circuit model is shown in Figure 5a. However, the bus voltages during the forward power injection and reverse power injection processes are different due to the conduction switch sets of the full-bridge converters, and their input voltages are E_dc and −E_dc, respectively. At the same time, the system models for the first and second free resonance processes are also entirely consistent, with their equivalent circuit models shown in Figure 5b, and there is no power input in the free resonance process.

During the power injection process in Figure 5a, the primary compensation capacitor C_p is isolated from the system by the switches S₅ and S₆ being turned off. The DC power supply directly charges the primary inductance L_p through full-bridge forward conduction (switches S₁ and S₄) or reverse conduction (switches S₂ and S₃), with input voltages of E_dc or −E_dc, respectively. At the same time, due to the isolation diode D₀, the current i₁ only flows in one direction, i₁ > 0, and energy can only be injected into the system from the power source and cannot be returned to the power source. In the free resonance process shown in Figure 5b, the DC power source is isolated from the system, while the inductance L_p and capacitance C_p in the primary circuit are directly connected to form a resonance circuit. Hence, through mutual inductance coupling, electrical power is transferred from the primary circuit to the secondary circuit.

According to Figure 5, set the state variable as the voltage u_p on the primary capacitor C_p, the current i_p on the primary coil inductance L_p, the current i_s on the secondary coil inductance L_s, and the voltage u_o on the secondary capacitor C_s, that is, x = [u_p, i_p, i_s, u_o]^T. Because the secondary capacitance C_s and the equivalent load resistance R_L are in parallel, the voltage u_o can also be considered as the equivalent output voltage. Assuming the system input is u = [E_dc], differential equations can be listed as follows for each state variable according to Kirchhoff’s voltage law and current law:

Forward power injection process:

$$\dot{x}=A_{1}x+B_{1}u,$$

(2)

First free resonance process:

$$\dot{x}=A_{2}x,$$

(3)

Reverse power injection process:

$$\dot{x}=A_{3}x+B_{3}u,$$

(4)

Second free resonance process:

$$\dot{x}=A_{4}x,$$

(5)

where

$$A_1=A_3=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& \frac{L_s{R}_p}{\Delta }& \frac{M{R}_s}{\Delta }& \frac{M}{\Delta }\\ 0& \frac{M{R}_p}{\Delta }& \frac{L_p{R}_s}{\Delta }& \frac{L_p}{\Delta }\\ 0& 0& \frac{1}{C_s}& -\frac{1}{C_s{R}_L}\end{array}\right],$$

(6)

$$B_1=-B_3={\left[\begin{array}{cccc}0& \frac{-L_s}{\Delta }& \frac{-M}{\Delta }& 0\end{array}\right]}^T,$$

(7)

$$A_2=A_4=\left[\begin{array}{cccc}0& -\frac{1}{C_s}& 0& 0\\ \frac{-L_s}{\Delta }& \frac{L_s{R}_p}{\Delta }& \frac{M{R}_s}{\Delta }& \frac{M}{\Delta }\\ \frac{-M}{\Delta }& \frac{M{R}_p}{\Delta }& \frac{L_p{R}_s}{\Delta }& \frac{L_p}{\Delta }\\ 0& 0& \frac{1}{C_s}& -\frac{1}{C_s{R}_L}\end{array}\right],$$

(8)

$$\Delta =M^2-L_p{L}_s,$$

(9)

$$M=k\sqrt{L_p{L}_s}.$$

(10)

For a linear time-invariant system Σ: (A, B, C, D), if the system matrix A is invertible, and when the initial time is t₀, the corresponding time domain solution can be expressed as follows [24]:

$$x(t)=x_{zi}(t)+x_{zt}(t)=\Phi (t-t_0)x_0+A^{-1}\left[\Phi (t-t_0)-I\right]Bu,$$

(11)

where Φ(t) = exp{At}, I is the identity matrix, x_zi(t) is the zero input response, x_zt(t) is the zero state response, and x₀ is the initial state of the system at the initial time t₀.

Since the system matrices A₁ and A₃ of the power injection process are irreversible, it is impossible to bring them into Equation (11) directly. Furthermore, the voltage u_p on the primary capacitor C_p is clamped at E_dc during the power injection process. Hence, the differential equation of u_p can be rewritten as follows:

$$\frac{d{u}_p}{dt}=-u_p+E_{dc}\equiv 0.$$

(12)

Note that Equations (2) and (3) only hold during the power injection process. The system matrices A₁, A₃ and input matrices B₁, B₃ of the power injection process will be rewritten to be reversible as follows:

$$A_1=A_3=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& \frac{L_s{R}_p}{\Delta }& \frac{M{R}_s}{\Delta }& \frac{M}{\Delta }\\ 0& \frac{M{R}_p}{\Delta }& \frac{L_p{R}_s}{\Delta }& \frac{L_p}{\Delta }\\ 0& 0& \frac{1}{C_s}& -\frac{1}{C_s{R}_L}\end{array}\right],$$

(13)

$$B_1=-B_3={\left[\begin{array}{cccc}1& \frac{-L_s}{\Delta }& \frac{-M}{\Delta }& 0\end{array}\right]}^T.$$

(14)

According to Equation (11), if the initial state of the four linear time-invariant systems at their respective initial time t₀ is x₀, the time domain solutions of the subsystems can be expressed as follows:

Forward power injection process:

$$x(t)={\Phi }_1(t-t_0)x_0+A_1^{-1}\left[{\Phi }_1(t-t_0)-I\right]B_1{E}_{dc},$$

(15)

First free resonance process:

$$x(t)={\Phi }_2(t-t_0)x_0,$$

(16)

Reverse power injection process:

$$x(t)={\Phi }_3(t-t_0)x_0+A_3^{-1}\left[{\Phi }_3(t-t_0)-I\right]B_3{E}_{dc},$$

(17)

Second free resonance process:

$$x(t)={\Phi }_4(t-t_0)x_0.$$

(18)

Due to the symmetry of the two energy resonance processes and the two free resonance processes, the time interval experienced by their subsystems is the same, so the time interval between the two power injection processes is assumed to be ξ₁. Set the time interval between two power injection processes as ξ₂. Let the initial value of the system in the nth cycle be x_n, and the terminal values of the four subsystems be x_n1, x_n2, x_n3, and x_n+1, respectively, where x_n+1 is the terminal value of the system in the nth cycle and also the initial value of the system in the (n + 1) cycle. The expressions for the terminal values x_n1, x_n2, x_n3, and x_n+1 of each subsystem are as follows:

$$x_{n1}={\Phi }_1({\xi }_1)x_n+A_1^{-1}\left[{\Phi }_1({\xi }_1)-I\right]B_1{E}_{dc},$$

(19)

$$x_{n2}={\Phi }_2({\xi }_2)x_{n1},$$

(20)

$$x_{n3}={\Phi }_3({\xi }_1)x_{n2}+A_3^{-1}\left[{\Phi }_3({\xi }_1)-I\right]B_3{E}_{dc},$$

(21)

$$x_{n+1}={\Phi }_4({\xi }_2)x_{n3}.$$

(22)

According to the modal analysis and the symmetry of the full-bridge system, when the system’s operating state reaches a steady state, there is x_n = −x_n2 = x_n+1. Generally, x_n = x_n+1 is called a fixed point, and the mapping relationship between x_n and x_n+1 becomes a fixed point mapping. To simplify the iterative process, take x_n = −x_n2 for fixed point mapping calculations, and the following equation can be obtained:

$$x_n=-x_{n2}=-{\Phi }_2({\xi }_2){\Phi }_1({\xi }_1)x_n-{\Phi }_2({\xi }_2)A_1^{-1}\left[{\Phi }_1({\xi }_1)-I\right]B_1{E}_{dc}.$$

(23)

After transforming and bringing ξ₂ = 1/2T − ξ₁ for Equation (23), we can obtain the following equation:

$$x_n=-{[I+{\Phi }_2(\frac{1}{2}T-{\xi }_1){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2(\frac{1}{2}T-{\xi }_1)A_1^{-1}\left[{\Phi }_1({\xi }_1)-I\right]B_1{E}_{dc}.$$

(24)

According to the modal analysis and subsystem switching conditions, the initial value of the system needs to meet the boundary β_4:u_p = E_dc and i_p > 0. Namely, the element up in the fixed point x_n is constant, and the following equation can be expressed:

$$-C_1{[I+{\Phi }_2(\frac{1}{2}T-{\xi }_1){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2(\frac{1}{2}T-{\xi }_1)A_1^{-1}\left[{\Phi }_1({\xi }_1)-I\right]B_1{E}_{dc}=E_{dc}.$$

(25)

Therefore, the soft switching operation point of the six-switch IPIFR–WPT system is the solution of Equation (25), and we can construct the following function g(ξ₁) to solve Equation (25):

$$g({\xi }_1)=1+C_1{[I+{\Phi }_2(\frac{1}{2}T-{\xi }_1){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2(\frac{1}{2}T-{\xi }_1)A_1^{-1}\left[{\Phi }_1({\xi }_1)-I\right]B_1.$$

(26)

The parameters of the six-switch IPIFR–WPT system are shown in

Table 2. The main parameters of the six-switch IPIFR–WPT system.

Parameter	Value	Parameter	Value
Lp	660 μH	Edc	100 V
Ls	585 μH	k	0.5
Rp, Rs	0.2 Ω	RL	10 Ω
Cp, Cs	0.4 μF	T	110 μs

. The curve of the function g(ξ₁) with T = 110 μs is shown in Figure 6. There are two results of g(ξ₁) = 0, that is, ξ_1a = 11.11 μs and ξ_1b = 36.02 μs. Then, we bring ξ_1a and ξ_1b into (20) to solve the corresponding fixed point x_n, listed in

Table 3. The calculated result of the fixed point x_n and the intermediate variable x_n1.

Operation Point	xn				xn1
	up (V)	ip (A)	is (A)	uo (A)	up (V)	ip (A)	is (A)	uo (A)
11.11 μs	100.0	9.812	5.986	56.594	100.0	11.305	5.657	57.483
36.02 μs	100.0	−3.273	−1.260	−17.146	100.0	3.850	2.287	20.212

.

The fixed point x_n is both the initial and final values of the steady-state period. Moreover, the fixed point x_n must satisfy the boundary condition β₄ (u_p = E_dc and i_p > 0). In Table 3, when taking ξ_1a = 11.11 μs, the value of fixed point x_n satisfies the boundary condition β₄., and when taking ξ_1b = 36.02 μs, i_p = −3.272 A < 0 does not meet the boundary condition β₄. Therefore, the root ξ_1b = 36.02 μs could be abandoned (x_n = −x_n2 = x_n+1).

After calculating the fixed point x_n and the transition state quantity x_n1 of the six-switch IPIFR–WPT system, according to the proposed symmetry of the full-bridge converter, there are x_n = −x_n2 = x_n+1, x_n1 = −x_n3. Then, in order to more intuitively represent the trade-off between the two results mentioned above, by substituting x_n and x_n1 obtained in Table 3 into Equations (19)–(22), the curves of state variables in the complete steady-state cycle can be obtained, and the curves of current i_p and voltage u_p in them can be plotted in Figure 7.

As can be seen from Figure 7, taking ξ_1a = 11.11 μs, the simulation waveform is consistent with the mode analysis, and all boundary conditions can be satisfied within a steady-state cycle. However, taking ξ_1b = 36.02 μs, the calculated operating waveform is inconsistent with the expected, and the boundary conditions β₂ (u_p = −E_dc and i_p < 0) and β₄ (u_p = E_dc and i_p > 0) cannot be satisfied, so that ξ_1b can be abandoned.

3. System Characteristics and Experimental Results

3.1. Experimental Devices

In order to verify the characteristics of the six-switch IPIFR–WPT system, an experimental platform was built, as shown in Figure 8. As shown in Figure 8a, the DC power supply used in the system consists of three parts: an AC voltage regulator TDGC2–3kVA (Delixi, China), a single-phase rectifier module MDQ100A1600V, and a filter capacitor. The oscilloscope is RTB2004, produced by Rohde & Schwarz Company. The power analyzer is the WT500, produced by Yokogawa Company. The resonance capacitors C_s and C_p are MKPH–R 0.4 μF. In Figure 8b, the isolation diode D₀ used in the system is FR607 (1000 V, 6 A) and the switches S₁–S₆ are IXFN56N90P (900 V, 56 A).

The square coupling coils used in this system use litz wires while using a radial layout of ferrite to increase their self-inductances and mutual inductance. The size of the square coils is 32 cm times 40 cm. The self-inductances L_p and L_s of the primary and secondary coils are 660 μH and 585 μH, respectively. The internal resistances R_p and R_s are both 0.2 Ω. The relationship between the coupling coefficient k and the coil distance d is shown in Figure 9.

3.2. Soft Switching Margin Characteristic

The six-switch IPIFR–WPT system has a unique wide soft switching margin characteristic due to its particular structure. In Mode 4 and Mode 8, the last modes of the free resonance process, the full-bridge converter has been turned on in advance, but the voltage u_p is greater than E_dc so that the diode D₀ is turned off and the power supply is isolated from the system. Since the free resonance process will maintain until the voltage u_p is equal to E_dc, the switches of the full-bridge converter could turn on before in a very time margin when the voltage u_p is greater than E_dc. Hence, in this time margin, the system itself will achieve the soft switching condition and determine the operation point without any switching action. The time margin is called the soft switching margin.

Figure 10 is a schematic diagram of the wide soft switching margin of a six-switch WPT system. In Figure 10, the black curve is the voltage input of the full-bridge converter, and the orange and red curves represent the values of the primary capacitor voltage u_p and the primary inductor current i_p, respectively.

According to the modal analysis above, boundary conditions must be met when switching from the second free resonance process to the forward power injection process β₄ (u_p = E_dc and i_p > 0), and switches S₁ and S₄ have been turned on in advance. It can be known from the waveforms of voltage u_p and current i_p in Figure 10 that the timing of turn-on of S₁ and S₄ must meet the following conditions: u_p > E_dc and i_p > 0. Therefore, the margin that satisfies the soft switching condition is [t₀, t₁]. Namely, the maximum pulse width for S₁ and S₄ is D_max = t₂ − t₀, and the minimum opening pulse width is D_min = t₂ − t₁. Based on the symmetry of the full-bridge converter, there are D_max = t₅ − t₃ = t₂ − t₀ and D_min = t₅ − t₄ = t₂ − t₁. The pulse widths of switch groups (S₁, S₄) and (S₂, S₃) can meet the conditions by taking any value of [D_min, D_max]. Therefore, the time range [D_min, D_max] can be called the wide soft switching margin of a six-switch IPIRF–WPT system, and M_ZVS is defined as the wide soft switching margin of the system, and its calculation formula is as follows:

$$M_{ZVS}=\frac{D_{max}-D_{min}}{T}\times 100\%.$$

(27)

In Figure 11a–c, let the fixed coil spacing be d = 6 cm (k ≈ 0.5), the equivalent load resistance be R_L = 10 Ω, and the period be T = 110 μs. Then, taking the positive pulse widths of the switching strategy as (a) D = 15 μs, (b) D = 20 μs, and (c) D = 26 μs, the system can self-determine the same operating point, ξ₁ = 10.52 μs, very close to the theoretical value, ξ₁ = 11.11 μs. Meanwhile, the waveform shape and amplitude of the capacitor voltage u_p and the inductor current i_p are consistent. According to the measurement results in Figure 11, the soft switching margin under the above static parameters is D_min = 10.52 μs to D_max = 29.95 μs. In Figure 11, the three positive pulse widths of (a), (b), and (c) are all within the range of [D_min, D_max] and can enable the system to operate at the same operating point and meet soft switching conditions. Therefore, it can be proven that the unique topology of the six-switch IPIFR–WPT system enables it to have a wide soft switching margin, which provides a wide tolerance for the control system, improves system stability, and reduces control difficulty.

3.3. Dynamic Parameter Adaptation Characteristic

The resistance to dynamic parameter changes is a significant dynamic characteristic of WPT systems. Due to no fixed physical connection for wireless coils, environmental disturbances will affect the relative position of the wireless coils to a certain extent, resulting in dynamic changes in the coil coupling coefficient k. On the other hand, a WPT system’s equivalent resistance R_L will change as the charging process progresses. Therefore, the coil coupling coefficient k and the equivalent resistance R_L are the main dynamic parameters for research. When the power injection process switches to the free resonance process, the boundary condition depends on the values of the system state variables instead of the switches. The wide soft switching margin characteristic analyzed above, that is, any switching duty D₁ ∈ [D_min, D_max] allows the system to operate at ξ₁ = D_min, the soft switching operating point. When parameters change dynamically, its soft switching margin has a certain degree of overlap, which can ensure that the system has the characteristics of resisting changes in dynamic parameters, called dynamic parameter adaptive characteristics.

In Figure 12, under the same switching strategy, T = 110 μs and D = 110 μs, there are the curves of M_ZVS, D_min, and D_max concerning the coupling coefficient k. The system parameters are taken according to Table 2, and D_min and D_max are measured under different k. As seen from Figure 12, in the process of gradually increasing the coupling coefficient k from 0.30 to 0.60, although the soft switching margin has decreased, it has maintained more than 15%.

Due to the unique hardware structure, the six-switch IPIFR–WPT system can have the ability to dynamically change the coupling coefficient and equivalent load resistance without changing the switching control strategy in the whole process, that is, in the open-loop state.

In the experimental waveforms shown in Figure 13, the switching control strategy of the system is fixed: T = 110 μs and D = 25 μs. Increasing the coil spacing d (a) 6 cm, (b) 7 cm, (c) 8 cm, and (d) 9 cm, the corresponding coupling coefficients k are (a) 0.56, (b) 0.49, (c) 0.43, and (d) 0.38. From the waveforms of the current i₁, during the process of changing the coupling coefficient k, the system can determine its own soft switching operating point, and its power injection times ξ₁ are self-determined as (a) 12.26 μs, (b) 10.52 μs, (c) 9.26 μs, and (d) 7.86 μs. The waveform trends of the capacitor voltage u_p and the inductor current i_p are always consistent, and the soft switching boundary conditions are satisfied at the boundaries of the free resonance process and the power injection process. In particular, the capacitor voltage u_p can clamp at E_dc and −E_dc during the power injection process at different coupling coefficients k, which indicates the soft switching conditions have been met. Therefore, this experiment proves that the six-switch IPIFR–WPT system can adapt to the dynamic change of the coupling coefficient k in a great range.

Similarly, when the equivalent load resistance R_L changes dynamically, the six-switch IPIFR–WPT system can maintain its adaptive characteristics. As shown in Figure 14, there are curves of the soft switching margin D_min and D_max concerning the equivalent resistance R_L. In Figure 14, when the equivalent load resistance R_L increases from 5 Ω to 20 Ω, although the soft switching margin can maintain more than 14%, and when the equivalent load resistance R_L changes in the range of 5 Ω to 20 Ω, there is a significant overlap range, which allows the system to resist changes in load resistance over a large range.

As shown in Figure 15, the switching control strategy of the system is fixed to T = 110 μs and D = 20 μs. When the equivalent load resistances R_L changes, (a) 10 Ω, (b) 15 Ω, and (c) 20 Ω, the waveforms of the current i₁ indicate that the system can determine its own soft switching operating point and power injection times ξ₁ are self-determined as (a) 10.52 μs, (b) 11.29 μs, and (c) 11.94 μs. The waveform trends of the capacitive voltage u_p and the inductive current i_p are consistent. The larger the equivalent load resistance R_L, the smaller the amplitude of u_p and i_p, indicating that the larger the load resistance, the less energy the system inputs. At the same time, the boundary conditions for switching between the free resonance process and the power injection process can be satisfied. Significantly, the capacitor voltage u_p can be clamped at E_dc and −E_dc during the power injection process under different equivalent load resistances R_L, which means the soft switching conditions have been satisfied. Therefore, this experiment proves that the six-switch IPIFR–WPT system can adapt to the dynamic changes in the equivalent load resistance R_L over a large range.

In summary, even in open-loop states, the six-switch IPIFR–WPT system has good adaptability to dynamic changes in parameters such as coupling coefficient k and equivalent load resistance R_L.

3.4. Decoupling Characteristic of Power Injection and Free Resonance

With the changes in T, the curves of the self-determined operating points (ξ₁, ξ₂) are shown in Figure 16. In Figure 16, as the period T increases, the injection time ξ₁ rises significantly, and ξ₂ remains unchanged, increasing the power injected into the system. Furthermore, due to ξ₂ remaining unchanged, the power injection and free resonance processes exhibit significant decoupling characteristics, which makes the power regulation strategy of the six-switch IPIRF–WPT system extremely simple, that is, directly adjusting the length of the period T.

Under static parameters, adjusting the input and output power of the system by changing the length of the period T is actually achieved by changing the power injection time ξ₁. As shown in Figure 17, when the periods T of the system are (a) 100 μs, (b) 110 μs, (c) 120 μs, and (d) 130 μs, respectively, the determined power injection times ξ₁ are (a) 5.94 μs, (b) 10.52 μs, (c) 13.10 μs, and (d) 20.07 μs. Therefore, as the period T grows, the duration of the power injection time ξ₁ also increases. Meanwhile, the amplitudes of u_p and i_p both increase significantly with the rise of T, meaning the input power increases synchronously.

The calculation methods of the input power P_in, the output power P_out, and the efficiency η are expressed as follows:

$$P_{in}=E_{dc}{I}_1,$$

(28)

$$P_{out}=\frac{U_{o,rms}^2}{R_L},$$

(29)

$$\eta =\frac{P_{out}}{P_{in}}\times 100\%,$$

(30)

where I₁ is the average value of i₁, and U_o,rms is the root mean square (RMS) value of u_o.

The curves of the output power P_out and the operation efficiency of the six-switch IPIFR–WPT system varying with the period T are shown in Figure 18. It can be seen from Figure 18 that the output power monotonically increases as the period T increases.

In Figure 18b, under the same coil spacing d, the operation efficiency of the system remains stable, while with the decrease in coil spacing d, the operation efficiency of the system increases relatively.

In summary, the converter and the resonant network have been decoupled by the unique hardware structures of the six-switch IPIFR–WPT system so that the power injection and the free resonance processes are also decoupled in mode. Under static parameters, the power injected into the system is only determined by the power injection time ξ₁, and increasing T can make ξ₁ grow, which results in a monotonic relationship between the output power and the period T. Therefore, compared to the fully resonant WPT system, the six-switch IPIFR–WPT system eliminates the peak power point, significantly simplifying power regulation strategies, eliminating system detuning caused by frequency splitting, and improving system stability.

4. Conclusions

This paper completes the theoretical analysis and experimental verification of the dynamic parameter adaptive characteristics and the decoupling of power injection and free resonance in the six-switch IPIFR–WPT system. Firstly, the state space modal is established based on the modal analysis of the six-switch IPIFR–WPT system. Secondly, using the stroboscopic mapping method and boundary conditions, a method is proposed to calculate all possible operating points of the system. Finally, the system characteristics are verified and explained based on simulations and experiments. Unlike traditional FR–WPT systems, the unique topology enables the six-switch IPIFR–WPT system to have adaptive characteristics of dynamic parameters and accomplish decoupling of the power injection and free resonance in modes. The experimental results were consistent with theoretical analysis.

In summary, under open-loop operation states, the unique topology makes the six-switch IPIFR–WPT system have at least 15% and 14% tolerance to the dynamic changes in the coil coupling coefficient k and the equivalent resistance R_L, respectively. Furthermore, the power injection and the free resonance processes are decoupled. Namely, the converter and the resonant network are decoupled. Hence, the power injected into the system is completely controlled, and the curve of output power is monotonic with the injection time ξ₁ or the period T. Moreover, according to the experiments, the power loss is greatest on the coils instead of the diode D₀. However, in this paper, without precise parameter design, the system efficiency could still achieve 88%.

Furthermore, a suitable parameter design can decrease the internal resistance of coils to improve the system efficiency and reduce the voltage stress to promote the output power.