Power Injection and Free Resonance Decoupled Wireless Power Transfer System with Double-Switch

Abstract

This article presents a type of power injection and free resonance decoupled wireless power transfer (WPT) system, the double-switch independent power injection and free resonance wireless power transfer (IPIFR-WPT) system working in CCM. Based on the stroboscopic mapping model, the theoretical results show that the operation point of the proposed WPT system is determined by itself instead of the switching control strategy. Specifically, once the voltage on the primary capacitance does not decrease to the input voltage in the free-resonance process, the diode in series would not turn on and the system would not switch to the power injection process. Therefore, there is a wide soft-switching margin to ensure the system operating in soft-switching states. Another characteristic of the proposed WPT system is the monotonicity between output power and operation cycle, which presents a simple power control method. And since the soft-switching margin may have intersection under dynamic coupling coefficient, the proposed system maintains soft-switching states with a fixed switching strategy and presents advantage to resist the dynamic change of coupling coefficient. All the characteristics of the proposed WPT system mentioned above have been verified in both theory and experiment.

1. Introduction

The wireless power transfer (WPT) system is mainly based on the principle of electromagnetic induction and high-frequency inverter technology to transfer power wirelessly from a primary coil to one or more moveable secondary coils across air gaps [1]. There is no direct physical connection between the primary coil and secondary coil and they are coupled by electromagnetic fields. Hence, WPT technology is an ideal solution for connector-free, moveable or rotatable devices, especially electric vehicles, robots, portable devices and even underwater devices [2,3,4,5,6].

Usually, WPT systems including LC, LCL and LCC systems using a variety of compensation networks are fully resonant types, that is, all working modes of the systems are in resonant state [7,8,9,10,11]. Power injection and system resonance are completed simultaneously. The fully resonant type WPT system works well under static operation condition. However, since the primary and secondary coils are not rigidly connected, the coil dynamic offset and vibration could be caused by environmental impact. By changing the system coupling coefficient and load resistance during operation, the impedance of the fully resonant type WPT system will change and the output power and operation efficiency will be greatly reduced. In order to ensure fully resonant type WPT systems work at the maximum power operation point or the maximum efficiency operation point under dynamic parameters, some closed-loop methods have been proposed, such as the impedance compensation methods and the frequency tracking methods [12,13]. However, the required operation points of the fully resonant type WPT system may exist in non-unique solutions, which brings difficulty for these closed-loop methods [14,15,16].

The independent power injection and free resonance wireless power transfer (IPIFR-WPT) system has been proposed, whose processes of power injection and resonance are decoupled instead of happening at the same time like the fully resonant type [17,18,19]. Article [17] described the performance of an IPIFR-WPT system with six switches working in discontinuous current mode (DCM) and put forward the corresponding equivalent linear models. Similar to the DCM mode of the BUCK circuit, there are three modes in one steady-state period in the six-switch topology, that is, power injection mode, self-resonance mode and stop mode. In the stop mode, the current through the primary-side inductor is zero. Hence, the stop mode has no contribution to transfer power and the power density is not very high. Then, a new IPIFR-WPT system with six switches working in continuous current mode (CCM) was proposed [18]. Similar to the CCM mode of the BUCK circuit, there are two modes in one steady-state period, that is power injection mode and self-resonance mode. Working in CCM, the current through the primary-side inductor is continuous and does not remain zero. However, in both articles [17,18], there was no theoretical method to determine the periods of the linear subsystems.

To simplify the modes and decrease the number of switches, article [19] proposed another type of IPIFR-WPT system with two switches working in DCM and described the easy power control method and the wide range of soft-switching operation. But only an approximate model had been put forward and the calculated result was not totally precise. As the case when the system parameters changed has not been discussed, the wide range of soft-switching operation could not be proved fully.

This article is trying to study the characteristics of self-determining operation point and open-loop adaptability in the double-switch type of IPIFR-WPT system working in CCM. The main work of this article is to present a theoretical method based on the stroboscopic mapping to determine the precise operation point of each linear subsystem when the system achieves a steady state [20]. All the performance can be predicted with the determined precise operation point. Furthermore, the characteristics of the proposed WPT system including the self-determining soft-switching operation point and the monotonicity of power curve about the cycle can all be verified.

In

Table 1. The comparison between the proposed system and similar systems.

Reference	Number of Switches	Operation Mode	Maximum Efficiency (%)	Number of Control Modes
[17]	6	DCM	93.6	10
[18]	6	CCM	89.6	10
[19]	2	DCM	88.4	6
the proposed system	2	CCM	91.6	4

, the comparison between the proposed system and the similar systems has been listed. In the CCM operation mode, the proposed system has only four control modes, which means its control strategy is simpler than the other topologies.

Note that the power injection process and the free-resonance process of the IPIFR-WPT system are independent. The self-determining soft-switching operation point means that the IPIFR-WPT system can automatically adjust the real intervals of each process under both the static and dynamic system parameters, which allows the switching strategy that has a wide soft-switching margin to let the control process be flexible. And the monotonicity of power curve about cycle means that the input and output power of the proposed WPT system can be controlled continuously instead of being limited by the power peak, which could simplify the power control method.

2. Mode Analysis and Operation Point Calculating Method

2.1. Mode Analysis

The main circuit diagram of the IPIFR-WPT system is shown in Figure 1. The diode D₀ is used to prevent current backflow. The switch S₁ is used to connect the dc voltage source E_dc to the circuit of the primary side. Besides, there is a switch S₂ control to access the resonant capacitor C_p on the primary side. The primary self-inductance L_p with internal resistance R_p and the secondary self-inductance L_s with internal resistance R_s constitute the mutual inductance coil, the main power transfer element. Additionally, between the primary coil and the secondary coil, M is the mutual inductance and k is the coupling coefficient. C_s is the filter capacitor on the secondary side. In addition, a rectifier bridge in parallel with a relatively large capacitor C₀ in the secondary side to filter a dc output, and the real load R and the equivalent load R_L have the following relationship [21]:

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

(1)

The operation modes of the proposed system in CCM are shown in Figure 2. On Figure 2, u_p, i_p, i_s and u_o are the state variables on C_p, L_p, L_s and C_s, respectively. And i₁ is the current across the diode D₀. The power injection process only includes Mode 1, shown in Figure 2a. The self-resonance process consists of Mode 2, Mode 3 and Mode 4, which are shown in Figure 2b–d, respectively. Since the performance on the primary side matters the stability of converter directly, only the operation curves of i_p and u_p are shown in Figure 3. The operation time of power injection process and self-resonance process are ξ₁ and ξ₂, respectively.

Mode 1 [t₀, t₁]: Mode 1 is the power injection process, which is shown as Figure 2a. Before t₀, the switch S₁ has been already turned on and the switch S₂ has been turned off. But before t₀, u_p > E_dc and the diode D₀ was not conducting, meanwhile the current i_p flowed through the bypass diode D₂. At the time t₀, u_p decreases to E_dc and the diode D₀ is conducting so that the current i_p is commutated from the bypass diode D₂ to S₁. Meanwhile, C_p will be isolated from the system and the dc supply E_dc will charge the inductor L_p directly. The primary current i_p across L_p is unidirectional to ensure the turning-on of the diode D₀, namely, i_p > 0. Note that in mode 1, i₁ = i_p.

Mode 2 [t₁, t₂]: Mode 2 in Figure 2b is an incomplete self-resonance process. At t₁, the switch S₁ turns off and the switch S₂ remains off, but the current i_p > 0 so that the current i_p is commutated from the switch S₁ to the bypass diode D₂ naturally, meanwhile i₁ = 0. Since the operation time of Mode 2 is very short, the current i_p will satisfy that i_p > 0 during Mode 2.

Mode 3 [t₂, t₄]: Mode 3 in Figure 2c is a complete self-resonance process. At the time t₂, the switch S₂ turns on to let the capacitor C_p and the inductor L_p connect to each other to form a resonant network. Since i_p > 0, the diode D₀ is conducting. If the tube voltage drop of D₂ is ignored, the turning-on process of S₂ meets the opening zero voltage switching (ZVS) condition.

Mode 4 [t₄, t₅]: Mode 4 in Figure 2d is an incomplete self-resonance process. At the time t₄, the switch S₁ turns on meanwhile S₂ turns off. Since i_p > 0, the current i_p remains conducting in the diode D₂ to let the turning-off of S₂ meet the ZVS condition. Meanwhile as u_p > E_dc, the diode D₂ is not conducting so that no current flows through the switch S₁. Hence, the turning-on of S₁ meets the zero current switching (ZCS) condition. And the current i_p will not be commutated from the diode D₂ to the switch S₁ until the time t₅ where u_p = E_dc.

2.2. Calculating Method of the Operation Point

In the IPIFR-WPT system working in the CCM mode, the processes can be divided into two independent subsystems including the power injection Σ₁ and the self-resonance Σ₂, whose equivalent circuits are shown in Figure 4.

During the power injection process, which is shown in Figure 4a, the capacitor C_p is isolated from the system and the dc supply E_dc is connected to the system. The power is injected to the system, meanwhile up remains satisfying u_p = E_dc during the power injection process.

During the self-resonance process, which is shown in Figure 4b, E_dc is isolated from the system while L_p and C_p are connected to each other to form a resonant network. As a result, the power will transfer from the primary side to the secondary side during the self-resonance process.

Let x = [u_p, i_p, i_s, u_o]^T and u = [E_dc] be the state vector and the input vector of the system, respectively. And the reference directions of every state have marked been on Figure 4. According to the Kirchhoff’s voltage and current laws, the equivalent circuit shown in Figure 4a of the two subsystems Σ₁ and Σ₂ can be described by the differential equations as follows:

Σ₁: the equation for power injection is

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

(2)

Σ₂: the equation for self-resonance is

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

(3)

where:

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

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

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

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

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

The solution of linear system Σ₁ and Σ₂ can be expressed as follows:

Σ₁: the equation for power injection is

$$x(t)={\Phi }_1(t)x_0+{{\displaystyle \int }}_{t_0}^{t+t_0}\Phi (t_0+t-\tau )B_1{E}_{dc}d\tau$$

(4)

Σ₂: the equation for self-resonance is

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

(5)

where x₀ = x(t)|_{t= 0}, Φ₁(t) = exp{A₁t} and Φ₂(t) = exp{A₂t}.

Note that the integral term in (4) is a zero-input response, whose value is a single valued function about time. Let Y_zi(t) replace this integral term to simplify writing and Equation (4) can be rewritten as follows:

$$x(t)={\Phi }_1(t)x_0+Y_{zi}(t)$$

(6)

The iterative relationship of the IPIFR-WPT system in nth operation cycle is shown in Figure 5. The intervals of subsystems Σ₁ and Σ₂ are ξ₁ and ξ₂, respectively. x_n is the initial state variables. After the time spans ξ₁ and ξ₂ of the two subsystems, an intermediate state variable x_n1 and the terminal state variable x_n+1 can be described as follows:

$$x_{n1}={\Phi }_1({\xi }_1)x_n+Y_{zi}({\xi }_1)$$

(7)

$$x_{n+1}={\Phi }_2({\xi }_2){\Phi }_1({\xi }_1)x_n+{\Phi }_2({\xi }_2)Y_{zi}({\xi }_1)$$

(8)

Once the system operates in steady state, the state variables must be periodic, namely, x_n+1 = x_n. Equation (8) is a mapping from x_n to x_n+1, therefore x* = x_n = x_n+1 can be described as a fixed point in this mapping, whose equation is presented as follows:

$$x^{*}={[I-{\Phi }_2({\xi }_2){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2({\xi }_2)Y_{zi}({\xi }_1)$$

(9)

Let the operation cycle T be a constant and bring ξ₂ = T − ξ₁ in (9), then the equation can be rewritten as follows:

$$x^{*}={[I-{\Phi }_2(T-{\xi }_1){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2(T-{\xi }_1)Y_{zi}({\xi }_1)$$

(10)

Based on the mode analysis, the boundary β₁ and β₂ shown in Figure 5 can be described as follows:

When Σ₂ switches to Σ₁, the boundary is β₁: u_p = E_dc & i_p > 0,

When Σ₁ switches to Σ₂, the boundary is β₂: S₁ = 0.

The boundary β₁ is determined by the value of state variables instead of the actions of switches. Concretely, before the boundary β₁, the system works in the free-resonance process, since the voltage u_p is greater than E_dc, the resonant current i_p flow through the resonant tank consists of C_p and L_p even the switch S₁ has been already turned-on. At the moment of β₁, the voltage u_p decreases to E_dc and the current i_p completes commutation from D₂ to S₁. Therefore, the voltage u_p equals to E_dc at the boundary β₁ and the equation can be expressed as follows:

$$C_1{[I-{\Phi }_2(T-{\xi }_1){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2(T-{\xi }_1)Y_{zi}({\xi }_1)=E_{dc}$$

(11)

where, C₁ = [1, 0, 0, 0].

Make a new function as follows:

$$f({\xi }_1)=C_1{[I-{\Phi }_2(T-{\xi }_1){\Phi }_1({\xi }_1)]}^{-1}{\Phi }_2(T-{\xi }_1)\frac{Y_{zi}({\xi }_1)}{E_{dc}}-1$$

(12)

The result of f (ξ₁) = 0 will meet the steady-state condition. The parameters of the IPIFR-WPT system are shown in

Table 2. The -parameters of the double-switch IPIFR-WPT system.

Parameter	Edc (V)	Lp (μH)	Cp (μF)	Rp (Ω)	Ls (μH)
Value	100	660	0.4	0.2	585
Parameter	Cs (μF)	Rs (Ω)	k	RL (Ω)
Value	0.4	0.2	0.5	10

and the operation cycle T = 100 μs.

The curves of the function f (ξ₁) with T = 100 μs are shown in Figure 6. There are two results of f (ξ₁) = 0, that is, ξ_1a = 13.09 μs and ξ_1b = 36.71 μs. Then, bring ξ_1a and ξ_1b into (10) to solve the corresponding fixed point, respectively, and the results are shown in

Table 3. The fixed point of the double-switch IPIFR-WPT system.

ξ1 (μs)	Fixed Point x*
	up (V)	ip (A)	is (A)	uo (V)
13.09	100.0	5.774	3.289	28.72

.

According to the calculated result of fixed point, for ξ_1a = 13.09 μs, u_p = E_dc and i_p = 5.774A > 0 are both satisfied, which means the boundary β₁ is meeting and this result ξ_1a = 13.09 μs can be accepted. As for ξ_1b = 36.71 μs, u_p = E_dc is satisfied but i_p = −2.927A < 0, which means the boundary β₁ is not meeting and this result will be rejected.

Now, the operation point has been solve out and fixed point has been determined. Furthermore, the continuous solution of state variables in one steady cycle [nT, (n + 1)T] can be presented as follows:

$$x(t)=\left\{\begin{array}{l}{\Phi }_1(t-nT)x^{*}+Y_{zi}(t-nT),t\in [nT,\hspace{0.33em}nT+{\xi }_1]\\ {\Phi }_2(t-nT-{\xi }_1)x_{n1}^{*},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}t\in (nT+{\xi }_1,\hspace{0.33em}(n+1)T)\end{array}\right.$$

(13)

Bring ξ₁ = 13.09 μs and the corresponding fixed point shown in Table 3 into (13), then the waveforms of i_p(t) and u_p(t) with can be calculated and plotted in Figure 7. Ξ₁ and ξ₂ is the intervals of subsystems Σ₁ and Σ₂, respectively.

3. System Characteristics

3.1. The Wide Soft-Switching Margin Characteristic

The switching condition for Σ₂ switching to Σ₁, namely, the boundary β₁, is not determined by the switching actions, which is a remarkable characteristic different from the other WPT system. In the mode analysis shown in Figure 2, S₁ has already been turned-on in Mode 4 and the system mode will not switch to Mode 1 until the voltage u_p decreases to E_dc. Hence, the switching strategies are not strictly corresponding to the real operation points. Holding the switching cycle T = 100 μs unchanged, the curves of three switching control methods with different pulse widths, which is the simulation results by Matlab Simulink, are shown in Figure 8.

Based on the analysis above, once u_p > E_dc and i_p > 0 are both satisfied at the moment of turning-on S₁ while turning-off S₂, the boundary β₁ will meet and the operation point will be self-determined by the system. In Figure 8a, the maximum duty of S₁ has been taken at the critical position of i_p = 0. In Figure 8c, the minimum duty of S₁ has been taken at the boundary position of u_p = E_dc. Hence, once the duty of S₁ takes any values in the margin [D₃, D₁], the operation points of the double-switch IPIFR-WPT system will be self-determined to be the same and the waveforms of u_p(t) and i_p(t) are also the same. And the margin [D₃, D₁] is called the soft-switching margin.

In the case that the system parameters take the value in Table 2 and T = 100 μs, the soft-switching margin is [13.09 μs, 29.07 μs], which takes 15.98% of the cycle T. No matter what value in [13.09 μs, 29.07 μs] has been taken as the duty of S₁, the intervals of power injection and self-resonance are 13.09 μs and 86.91 μs, respectively, namely, the same operation point has been determined by the system itself instead of the switching strategies. Hence, there is a great margin to ensure the system work is steady and the difficulty of control strategies can be reduced greatly. And note that the value of minimum duty equals to the determined interval ξ₁ of power injection process.

3.2. Self-Determining Soft-Switching Operation Point under Dynamic Coupling Coefficient

Since the switching strategies of the proposed WPT system have a wide soft-switching margin under static system parameters, the soft-switching margin may have overlapping parts under different system parameters. Therefore, the operation point could be self-determined under dynamic system parameters but the same switching strategies. Of all parameters, the coupling coefficient k is the most prone to change dynamically due to the relative movement of the coils.

When k changes from 0.3 to 0.6, the curves of soft-switching margin [D_min, D_max] can be plotted as Figure 9 and Mar is defined as the percentage of this margin in cycle T as follows:

$$Mar=\frac{D_{max}-D_{min}}{T}\times 100\%$$

(14)

When the switching strategies remain in the shadow part between the curves of D_min and D_max on Figure 9, the system will keep working in soft-switching operation state under dynamic change of k. And the percentage Mar remains above 14.4%, which indicates that there is a wide soft-switching margin to ensure the system working under dynamic change of k.

According to Figure 9, if we let k equal to 0.45, 0.50 and 0.55, respectively, meanwhile the duty of S₁ remains as 20%, the system will work in a soft-switching margin under the three situations. Namely, when k changes from 0.45 to 0.55 dynamically, the switching strategies keep the same, T = 100 μs and D = 25%.

Then, as shown in Figure 10, the waveforms of i₁(t), i_p(t) and u_p(t) are plotted when k equals to (a) 0.45, (b) 0.50 and (c) 0.55, respectively. The determined intervals ξ₁ by the system itself are (a) 10.09 μs, (b) 13.09 μs and (c) 16.55 μs correspondingly and the boundary conditions β₁ are all meeting in the three figures. Therefore, when the switching strategies remain the same and the coupling coefficient k changes dynamically, the IPIFR-WPT system can keep working in soft-switching states by self-determining its operating point.

3.3. The Monotonicity of Power about Cycle

Since Equation (13) can solve the time series of the state variables x(t) of the IPIFR-WPT system in one steady-state cycle, it is simple to calculate the input and output power as follows:

$$P_{in}=\frac{E_{dc}}{T}{\displaystyle {\int }_0^{{\xi }_1}{i}_p(t)}dt$$

(15)

$$P_{out}=\frac{1}{T{R}_L}{\displaystyle {\int }_0^T{u}_o^2(t)}dt$$

(16)

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

(17)

where i_p(t) = [0, 1, 0, 0]x(t) and u_o(t) = [0, 0, 0, 1]x(t), η is the efficiency of the whole system.

According to the characteristic of a self-determining operation point, it is illustrated that the switching strategies of double-switch IPIFR-WPT system has a wide margin to ensure the soft-switching operation states and the duty of S₁ does not determine the operation point. The operation cycle T can be used to adjust the output power and the relationship of output power P_out with cycle T is shown in Figure 11.

In Figure 11, there is a monotonic relationship between the output power P_out and cycle T and the linear relationship between P_out and T becomes more obvious when the coupling coefficients k increase. Hence, it is simple to control the output power by adjusting the cycle T, namely, the frequency f of the IPIFR-WPT system.

In Figure 12, the corresponding efficiency η has been plotted and the efficiency η has few decreases with the increases of cycle T. Therefore, the efficiency η maintains few changes when the system control the output power by adjusting frequency.

4. Experimental Verification

4.1. Experimental Devices

Based on the main circuit of the double-switch IPIFR-WPT system, the experimental system has been built as shown in Figure 13a. The part of the DC voltage source consists of the AC voltage regulator and rectifier module (MDQ100A1600V). As shown in Figure 13b, the convertor is composed of the MOSFET driver (SI8233), the MOSFET (C2M0045120D), the diode D₀ (MURF2060) and the microcontroller (STM32F107VC). The values of L_p, C_p, L_s, C_s and R_L are selected as Table 2. And the oscilloscope and power analyzer are RTB2004 (Ronde & Schwarz, Munich, Germany) and WT500 (Yokogawa, Musashino, Japan), respectively.

The relationship between the coupling coefficient k and the distance d of the coils are shown in Figure 14. With the increases of the distance d from 5 cm to 20 cm, the coupling coefficient k decreases from 0.64 to 0.12. Notice, in the follow-up experiments, the coupling coefficient k is always changed via changing the distance of the coils.

4.2. Self-Determining Soft-Switching Operation Point

It can be observed from the experiment that there is always a wide soft-switching margin under dynamic coupling coefficient k. Once the duty of S₁ maintains in the soft-switching margin, the IPIFR-WPT system will work in the soft-switching states.

The curves of soft-switching margins with respect to the relative distance d of coils is shown in Figure 15. Notice, at the operation points P₁, P₂ and P₃, the distances of the coils are 6 cm, 7 cm and 9 cm, respectively, but the duty of S₁ is the same, 20%. And the waveforms working at P₁, P₂ and P₃ are shown in Figure 16.

When d equals to (a) 6 cm, (b) 7 cm and (c) 9 cm, respectively, the corresponding waveforms of v_gs1(t), i₁(t) and i_p(t) are shown in Figure 16. The determined intervals ξ₁ by the system itself are (a) 16.2 μs, (b) 12.8 μs and (c) 7.6 μs, respectively. Therefore, when the switching strategies remains the same and the relative distance d of coils changes dynamically, the IPIFR-WPT system can keep working in soft-switching states by self-determining its operating point, which operates the same as the theoretical results.

As shown in Figure 17, taking a fixed working cycle T = 100 μs, the curves of soft-switching margins with respect to the change in equivalent load resistance R_L are plotted.

As shown in Figure 17, when the equivalent load resistance R_L increases from 5 Ω to 20 Ω, although the soft switching range is somewhat narrowed, it can maintain a soft switching margin of nearly 13% or more. And in Figure 17, when the load resistance R_L undergoes dynamic changes within the range of 5 Ω to 20 Ω, the soft switching duty cycle range of the proposed WPT system still has a large overlap range, which makes the system have an adaptive ability to resist dynamic changes in load resistance R_L.

In Figure 18, the equivalent load resistance R_L is plotted when taking 10 Ω, 15 Ω, and 20 Ω and using the same switching strategy (D₁ = 25.0 μs and T = 100 μs). The waveform of the power supply current i₁ and the current i_p on the primary inductor. As shown in Figure 18, regardless of the values of R_L being 10 Ω, 15 Ω, and 20 Ω, the system can meet the boundary conditions under the same switching strategy β₂ (u_p = E_dc and i_p > 0), meaning that the system can still operate at its respective soft switching operating points, therefore the system can resist dynamic changes in load resistance R_L over a large range.

4.3. The Monotonicity of Output Power about Cycle

According to the analysis above, the output power shows monotonicity with respect to cycle T. The experimental curves are shown in Figure 19, when the distances d are (a) 6 cm, (b) 8 cm and (c) 10 cm, respectively, the output power increases from (a) 106 W, (b) 54 W, (c) 26 W to (a) 530 W, (b) 537 W, (c) 597 W monotonously with the rise of cycle T from 100 μs to 130 μs, which is pretty close to the calculated results in Figure 11.

The corresponding efficiencies η are recorded and plotted in Figure 20. The efficiency of the system maintains more than 80% when the coils distance less than 10 cm and achieves 90% under 6 cm coils distance.

5. Conclusions

This article presents a double-switch IPIFR-WPT system working in CCM, whose operation modes can be decoupled as two independent processes, the power injection and the free resonance. Different from the fully resonant type of WPT system, the power injection process and the resonance process do not proceed simultaneously. Instead, for the proposed WPT system, the intervals ξ₁ and ξ₂ of the two processes are independent and the power injects into the system only in the time of the interval ξ₁. It can be known from the theoretical analysis that the operation point of the proposed WPT system is not determined by the switching control strategy, hence there is a wide soft-switching margin to ensure the system operating in soft-switching states. Since the power injected into the system increases monotonously with respect to operation cycle T, thus the output power can be controlled simply by adjusting cycle T. The characteristic of a self-determining soft-switching operation point has been proved in both theory and experiment. Under a fixed switching control strategy, the system maintains soft-switching states without adjusting the switching strategy when the coupling coefficient k rises from 0.4 to 0.6 theoretically or when the relative coils distance d varies from 6 cm to 9 cm experimentally. Hence IPIFR system has a great advantage to resist the dynamic change of coupling coefficient and is more suitable for the reality situation where the environmental factors would cause the movement of coils.

As for the monotonicity of the power curve about cycle T, the output power rises from 106 W to 597 W with the increases of T from 100 μs to 130 μs at 10 cm coils distance meanwhile the efficiency almost maintains a constant (80%) and several curves with different coil distances present the monotonicity of output power about the cycle. Hence, the simple method for the IPIFR system to control the output power flexibly is adjusting the operation cycle under a constant coils distance d. Furthermore, if the coupling of the coils becomes stronger, the efficiency of the IPIFR system will rise. Concretely, when the coils distance changes from 10 cm to 6 cm, the average efficiency rises from 80% to 91%.