A Monotonic and Continuous Frequency Control Method Covering Constant-Current and Constant-Voltage Charging Processes for Series-Series WPT Systems

Abstract

In this paper, a monotonic and continuous frequency control method is proposed for series-series (SS) compensated wireless power transfer (WPT) systems with wide misalignment. The requirements for the operating frequency path through the whole constant-current (CC) and constant-voltage (CV) charging process and the whole coupling range are given as follows: (1) maintain a monotonic and continuous regulation; (2) maintain zero-voltage switching (ZVS) of the inverter; (3) the operating frequency of the CC mode should be closed to the resonant frequency for a higher power transfer efficiency. Based on mathematical derivations, the conditions for enabling an operating frequency path meeting the above requirements are visualized with graphical representations. Then, a comprehensive design framework and implementation steps are provided with the identified conditions. Finally, a prototype is designed with the proposed method, and then, it is built for experimental measurements with power ratings of 3.3 kW and 2.4 kW, respectively, and a coupling coefficient range of 0.1–0.3.

1. Introduction

Wireless power transfer (WPT) has been widely used in different scenarios, such as implantable devices [1], industrial robots [2], and electric vehicles [3]. Usually, a WPT system is used for charging a battery, which typically includes two modes: constant current (CC) and constant voltage (CV). On the other hand, the most attractive feature of WPT is the freedom of the receiver. However, this freedom leads to coupling variation in coils and imposes additional control complexity. Many methods have been proposed to realize the CC/CV control for WPT systems. Generally, they can be classified into the following groups:

(1)

Additional DC/DC converters can be used in the WPT system to regulate the output voltage/current [4], which is a simple method in terms of control but will increase power conversion stages.

(2)

Phase-shift modulation or pulse density modulation can be applied to the inverter for output regulation. As a basic control method, phase-shift control [5] is widely used in WPT systems. However, it is challenging to maintain soft-switching in the whole charging process and under various coupling conditions for phase-shift control. Pulse density modulation is used to regulate the output power of a WPT system in [6]. However, pulse density modulation tends to introduce low-order harmonics and eventually may lead to large fluctuations in the output current.

(3)

Active rectifier can be used to regulate output power [7], and it can achieve a fast response compared to passive rectifier [8]. This can address the delay issue caused by the wireless communication between the transmitter side and the receiver side. However, synchronization between the two sides will increase the complexity of the control system, which may degrade the robustness of the system operation.

(4)

Variable parameters (usually inductance [9], capacitance [10], or the combination of inductance and capacitance [11]) and variable topology [12] are introduced to adjust the output characteristics of WPT systems for different charging conditions. However, this kind of method suffers from extra switches and also high voltage/current stresses of the additional switches.

(5)

Frequency control is another widely adopted method for realizing output regulation [13,14,15,16,17,18,19,20,21,22] for WPT systems, especially for series-series (SS) compensated WPT systems. Compared with other methods, frequency control requires no additional devices and may achieve zero-voltage switching (ZVS) in the whole charging process and in a wide coupling range.

However, for an SS WPT system, frequency splitting (FS) [13] occurs as the coupling strength increases and/or the equivalent load resistance decreases, which turns the output-power-versus-operating-frequency curve from a single-peak curve to a double-peak-single-bottom curve. This means there are two operating frequencies, which correspond to two peaks of the output power. These two frequencies are termed the even splitting frequency (higher than the resonant frequency) and the odd splitting frequency (lower than the resonant frequency) [14]. For simplicity, even and odd splitting frequencies are called higher and lower splitting frequencies in this study.

For the single-peak power curve, the peak frequency is close to the resonant frequency, and ZVS can be achieved when the operating frequency is higher than the resonant frequency. Therefore, by restricting the operating frequency in the region higher than the resonant frequency, a monotonic control with ZVS is realized. However, a monotonic control becomes complicated with a double-peak-single-bottom power curve, especially when considering ZVS and high transfer efficiency [15]. Generally, existing solutions can be categorized into the following four groups:

(1)

The WPT system is designed to avoid frequency splitting. Thereby, a non-monotonic control is also avoided. For example, frequency splitting is avoided in [16] by limiting the inductance and the receiving coil. While this method ensures that frequency splitting will not occur in the strong coupling region, it will also weaken the coupling, which will eventually lead to a higher voltampere rating of the inverter and lower power transfer efficiency.

(2)

Restricting the operating frequency below the lower splitting frequency or above the higher splitting frequency. Since operating below the lower splitting frequency usually results in hard switching of the inverter, restricting the frequency above the higher splitting frequency is preferred [17,18]. However, as the equivalent load resistance decreases, the splitting frequencies move farther away from the resonant frequency of the resonators, inevitably leading to a significant amount of reactive power transfer between the primary and secondary sides. Therefore, even though a monotonic control with ZVS is achieved through the whole charging process and in the coupling range, the power transfer efficiency of the CC mode (low equivalent load resistance) is degraded when using this approach.

(3)

Restricting the operating frequency between the lower splitting frequency and the resonant frequency of the resonator [19,20]. Within this range, the output power varies monotonically with the operating frequency while maintaining ZVS for the inverter. However, the minimum achievable output power in this frequency range increases as the coupling weakens. This means that, at weaker coupling positions, CV charging under light load conditions may not be realized through frequency control alone. Additional inverter phase shift modulation or pulse width modulation is required, and this results in the loss of ZVS.

(4)

The CC/CV charging process transitions from the region between two splitting frequencies to the frequency-splitting-free region [21,22]. At the beginning of the CC mode, both the equivalent load resistance and the load voltage are low, and the operating frequency is set between the two splitting frequencies to achieve higher transfer efficiency. As the equivalent load resistance and load voltage increase, the system working region gradually transitions from the frequency-splitting region to the frequency-splitting-free region [21], theoretically deriving the condition to avoid frequency splitting under the constant-voltage load condition, thereby preventing non-monotonic control. However, only the case with identical primary and secondary resonant frequencies is considered. In practice, the primary resonant frequency is usually slightly lower than the secondary resonant frequency to facilitate ZVS, which is adopted in [22]. In [22], a transition from the region between two splitting frequencies to the frequency-splitting-free region is achieved in the CV mode. However, both [21,22] do not provide enough information for designing a system to ensure CC charging can be realized in the operating frequency range between two splitting frequencies and, at the same time, to ensure ZVS of the inverter.

The main objective of this study is to achieve a seamless transition from the region between two splitting frequencies to the frequency-splitting-free region during the CC/CV charging process while maintaining ZVS. By restricting the operating frequency for the CC mode between two splitting frequencies, a higher power transfer efficiency can be achieved. By combining mathematical derivations and graphical representations, the constraints under which the CC/CV charging process can transition from the region between two splitting frequencies to the frequency-splitting-free region without losing control monotonicity and ZVS are given. A comprehensive design framework and implementation steps are provided, through which a WPT system adaptable to various charging specifications and various coupling conditions can be developed with optimized power transfer efficiency. To compare the differences between this article and previous research,

Table 1. Comparison of this work and previous articles.

References	Coupling Range	Rated Power	Efficiency	Charging Range	The fs-fp Relationship	The Key Condition for Monotonic Frequency	System Parameters Design Guideline
[14]	0.194	1 kW	-	CC and CV	fs = fp	yes	no
[15]	0.3–0.48	288 W	-	CC	fs = fp	no	yes
[19]	0.15–0.25	250 W	70–90.51%	CC and CV	fs > fp	no	yes
[20]	0.23–0.41	288 W	92.1–93.2%	CC	fs = fp	no	yes
This work	0.1–0.3	3.3 kW	81.8–97.2%	CC and CV	fs > fp	yes	yes

is provided below. In Table 1, f_p and f_s represent the primary-side and secondary-side resonant frequencies, respectively.

The rest of this paper is organized as follows. In Section 2, the theory of zero-phase angle (ZPA) boundary, power variation, and charging frequency path with frequency control is introduced. Section 3 gives the design procedure of system parameters based on the selected frequency path. Section 4 adopts the proposed theory to design and implement a ZVS and high efficiency wireless power transfer system, which supports both 3.3 kW and 2.4 kW charging. Experimental results are also given in this section. Finally, concluding remarks are given in Section 5.

2. Zero-Phase Angle and Power Characteristics

2.1. Zero-Phase Angle of Different τ_c

Figure 1a is the circuit diagram of the SS WPT system. In this model, V_DC is the input DC voltage source; L₁, L₂ are the self-inductances of the primary and secondary coils, respectively; M is the mutual inductance (i.e., $M=k\sqrt{L_1{L}_2}$, where k is the coupling coefficient); C₁, C₂ are the compensating capacitances; R₁ and R₂ are the parasitic resistances of the primary and secondary resonators, respectively; R_LDC is the equivalent DC load resistance; and v₁ and i₁ are the inverter output voltage and output current, respectively. According to [19], fundamental harmonic approximation can be applied to the SS WPT system, and the simplified equivalent circuit is given in Figure 1b. V_in and R_L are the sinusoidal input voltage and the equivalent AC load resistance, respectively. As mentioned in [18], $V_{\mathrm{in}}=2\sqrt{2}{V}_{DC}/\pi$ and $R_L={8R_{LDC}/\pi }^2$. As reported in [5], the input impedance Z_in is given by

$$Z_{in}={\displaystyle \frac{{\omega }^2{M}^2\left(R_2+R_L\right)}{{\left(R_2+R_L\right)}^2+X_2^2}}+R_1+j\left[X_1-{\displaystyle \frac{{\omega }^2{M}^2{X}_2}{{\left(R_2+R_L\right)}^2+X_2^2}}\right]$$

(1)

where ω = 2πf is the angular frequency, f is the operating frequency, and $X_i=\omega L_i-1/\omega C_i$ (i = 1 or 2). For WPT systems, zero-phase angle is the boundary that determines whether the primary-side inverter can achieve ZVS, which can be obtained when the imaginary part of Z_in is 0. By neglecting the effect of R₁ and R₂ on the input impedance angle, when $R_L^2+X_2^2\ne 0$, the zero-phase angle boundary can be simplified as

$$R_L^2={\displaystyle \frac{4{\pi }^2{L}_2^2\left(1-\frac{f_s^2}{f^2}\right)\left[k^2-\left(1-\frac{f_p^2}{f^2}\right)\left(1-\frac{f_s^2}{f^2}\right)\right]}{\frac{1}{f^2}\left(1-\frac{f_p^2}{f^2}\right)}}$$

(2)

where f_p and f_s are primary resonating frequency and secondary resonating frequency. It can be seen from (2) that the zero-phase angle boundary is related to both f_p and f_s. A parameter τ_c introduced in [22] (i.e., $\sqrt{x_c}$ in [23]) is adopted in this paper, whose definition is given by

$$\left\{\begin{array}{l}{\tau }_c={\displaystyle \frac{f_s}{f_p}}\\ f_s={\displaystyle \frac{1}{2\pi \sqrt{L_2{C}_2}}}\\ f_p={\displaystyle \frac{1}{2\pi \sqrt{L_1{C}_1}}}\end{array}\right.$$

(3)

With different values of τ_c, the zero-phase angle boundary can be summarized into three conditions as shown in Figure 2.

In Figure 2, positive input impedance angles as well as ZVS can be achieved in the shaded areas. As described in [24], f_τZL and f_τZH are the zero-phase angle frequencies when R_L equals 0, i.e.,

$$\left\{\begin{array}{l}{f}_{\tau \mathit{ZL}}=\sqrt{{\displaystyle \frac{2f_p^2{f}_s^2}{f_p^2+f_s^2+\sqrt{{\left(f_p^2-f_s^2\right)}^2+4k^2{f}_p^2{f}_s^2}}}}\\ f_{\tau \mathit{ZH}}=\sqrt{{\displaystyle \frac{2f_p^2{f}_s^2}{f_p^2+f_s^2-\sqrt{{\left(f_p^2-f_s^2\right)}^2+4k^2{f}_p^2{f}_s^2}}}}\end{array}\right.$$

(4)

It should be noted that, at the three points (0, f_s) in Figure 2a, (0, f₀) in Figure 2b, and (0, f_s) in Figure 2c, the condition $R_L^2+X_2^2\ne 0$ is no longer satisfied, and the imaginary part of Z_in at these points is not 0. Therefore, the zero-phase angle curve is marked with hollow points at these three points.

As described in [4], the efficiency of the system is given by

$$\eta ={\displaystyle \frac{{\omega }^2{M}^2{R}_L}{R_1\left[{\left(R_2+R_L\right)}^2+{\left(\omega L_2-\frac{1}{\omega C_2}\right)}^2\right]+{\omega }^2{M}^2\left(R_2+R_L\right)}}$$

(5)

With (5), the optimal frequencies at which maximum efficiencies can be achieved can derived as

$$f=\sqrt{{\displaystyle \frac{1}{\frac{1}{f_s^2}-2{\pi }^2{C}_2^2{\left(R_2+R_L\right)}^2}}}$$

(6)

In Figure 2, optimal frequency path (OFP) is defined by (6). In CC mode, the optimal frequency is close to f_s. To achieve ZVS within the whole CC/CV charging process, the operating frequency of the system should be adjusted continuously in the shaded areas. Thus, τ_c > 1 is the only case in which ZVS and high CC efficiency (the operating frequency close to f_s) can be simultaneously achieved.

2.2. Power Variation with τ_c > 1

By neglecting R₁ and R₂, the output power of the system can be derived as [4]

$$P={\displaystyle \frac{{\omega }^2{M}^2{V}_{in}^2{R}_L}{{\left({\omega }^2{M}^2-X_1{X}_2\right)}^2+R_L^2{X}_1^2}}$$

(7)

Taking f and R_L as the variables, the 3D and 2D plots of the output power are given in Figure 3. On this power surface, there are two “mountains” with two peaks (denoted as Peak1 and Peak2 in the figure, corresponding to higher and lower splitting frequency, respectively) and one “valley” with one bottom line (denoted as Bottom in the figure). This aligns with the theory of frequency splitting, which is frequency splitting appears when the equivalent load resistance becomes lower, for a given coupling position (i.e., given k) [13]. Therefore, when R_L is low, the output power has two peaks while scanning the operating frequency as shown in Figure 3. With larger R_L, the output power has only one peak. The exact curves that describe the peaks and the bottom can be obtained by setting ∂P/∂f equal to 0, which is given by (8), and the detailed derivation process of (8) is provided in Appendix A.

$$R_L^2={\displaystyle \frac{L_2^2\left[k^2-\left(1-\frac{f_p^2}{f^2}\right)\left(1-\frac{f_s^2}{f^2}\right)\right]\left(k^2-1-\frac{f_p^2+f_s^2}{f^2}+\frac{3f_p^2{f}_s^2}{f^4}\right)}{\frac{f_p^2}{2{\pi }^2{f}^4}\left(\frac{f_p^2}{f^2}-1\right)}}$$

(8)

In Figure 3, point V is the cross point of the Peak1 curve and the Bottom curve. The zone where R_L < R_V is the frequency splitting region, characterized by a double-peak-single-bottom feature, while the remaining area constitutes the frequency splitting-free zone exhibiting a single peak.

2.3. The Possible Charging Frequency Paths

Because the charging power is increasing in the CC mode as the battery voltage increases gradually, the operating frequency variation path of the CC mode should be like climbing one of the “mountains” in Figure 3. In other words, the operating frequency should be approaching one of the peak curves when the battery voltage increases (and also R_L increases). In contrast, as the charging power is decreasing in the CV mode, the operating frequency path of the CV mode should walk down from the mountain, which means the operating frequency should shift away from the “peak” curve.

In Figure 4, three possible scenarios are given. In each scenario, the operating frequency paths lie within the shaded area, which guarantees the realization of ZVS. Point A and point B are the beginning of the CC and CV modes, respectively. Point C is the end of the CV mode. Point E is the cross point of the Bottom curve and the zero-phase angle curve. However, point E may not be reached in some cases, which will be discussed in Appendix B. f_τPH is the frequency value of the cross point of the Bottom curve and vertical axis, which can also be obtained with (8).

In Figure 4a, the CC curve is above the Peak1 curve, and a monotonic control can be achieved. However, the frequency path is away from the optimal frequency path curve, which implies low system efficiencies.

In Figure 4b, the CC curve is between the Bottom curve and the Peak1 curve, while the CV curve shifts away from the Peak1 curve as R_L increases. Point B_CC and point B_CV are the end of the CC mode and the beginning of the CV mode, respectively, which have the same R_L value. At a certain coupling position with a certain coupling coefficient, point B_CC and point B_CV overlap on the Peak1 curve, and a monotonic and continuous control can be realized. However, at other different coupling positions, point B_CC and point B_CV are separated by the Peak1 curve, and thus, a continuous control becomes impossible.

In Figure 4c, both the CC curve and part of the CV curve are between the Bottom curve and the Peak2 curve, which makes it possible to achieve a monotonic and continuous control while staying close to the OFP curve. Therefore, the operating frequency path in the last scenario is chosen as the optimal one for the CC/CV control of SS WPT systems.

2.4. CC f-R_L Path of Charging Frequency Path

The CC f-R_L curve of the CC mode can be derived as

$${\displaystyle \frac{{\omega }^2{M}^2{V}_{in}^2{R}_L}{{\left({\omega }^2{M}^2-X_1{X}_2\right)}^2+R_L^2{X}_1^2}}=I_{cc}^2{R}_L$$

(9)

where I_cc is the constant current value. Then, (10) can be derived from (9), and the derivation process appears in Appendix C.

$$R_L^2={\displaystyle \frac{k^2{L}_2{V}_{\mathrm{in}}^2-4{\pi }^2{f}^2{L}_1{L}_2^2{I}_{\mathrm{cc}}^2{\left[k^2-\left(1-\frac{f_p^2}{f^2}\right)\left(1-\frac{f_s^2}{f^2}\right)\right]}^2}{L_1{I}_{\mathrm{cc}}^2{\left(1-\frac{f_p^2}{f^2}\right)}^2}}$$

(10)

A typical CC f-R_L curve (AB) is plotted in Figure 5 based on (10). Point A₁ is on the Bottom curve and has the same R_L as point A. Similarly, point B₁ has the same R_L as point B. In another case, when R_A < R_E, point A₁ should be on the zero-phase angle curve.

In the CC mode, as R_L increases, the CC f-R_L curve approaches the Peak2 curve, resulting in a rising output power. In Figure 5, when R_A ≤ R_V, point A is located within the region between the Bottom curve and the Peak2 curve, where f_A is close to f_s, which helps to increase the system efficiency.

Therefore, to achieve a monotonic frequency control and a high-efficiency operation in the CC mode, there are three constraints given as follows:

(1)

P_A1 ≤ P_start, where P_A1 means the output power at point A₁, which is the minimum output power that can be achieved by the system at R_A; P_start means the required output power at the start point of the CC mode.

(2)

P_B1 ≥ P_rated, where P_B1 means the output power at point B₁, which is the maximum output power that can be achieved by the system at R_B; P_rated means the required output power (i.e., the rated power) at the end of the CC mode.

(3)

R_A ≥ R_V.

2.5. CV f-R_L Path of Charging Frequency Path

The relationship between R_L and f in the CV mode is given by

$${\displaystyle \frac{{\omega }^2{M}^2{V}_{in}^2{R}_L}{{\left({\omega }^2{M}^2-X_1{X}_2\right)}^2+R_L^2{X}_1^2}}={\displaystyle \frac{U_{cv}^2}{R_L}}$$

(11)

where U_cv is the constant voltage value. Similarly, (12) can be derived from (11), and Appendix C gives the full derivation.

$$R_L^2={\displaystyle \frac{4{\pi }^2{f}^2{L}_1{L}_2^2{U}_{\mathit{cv}}^2{\left[k^2-\left(1-\frac{f_p^2}{f^2}\right)\left(1-\frac{f_s^2}{f^2}\right)\right]}^2}{k^2{L}_2{V}_{in}^2-L_1{U}_{cv}^2{\left(1-\frac{f_p^2}{f^2}\right)}^2}}$$

(12)

Four typical CV f-R_L curves are plotted in Figure 6 based on (12). Point I₁ and point I₂ are the cross points of the CV f-R_L curves with the Bottom curve and the Peak1 curve, respectively. Point C₁ is on the Bottom curve, and it has the same R_L as point C, while point D is on the CV curve, and it has the same R_L as point V.

• When R_C < R_V, the possible f-R_L curve is given in Figure 6a. In this case, the CV f-R_L curve lies between the Bottom curve and the Peak2 curve, and the power decreases as the frequency increases. Then, the constraints can be summarized as follows:

  (1)

  f_C ≤ f_C1.

  (2)

  P_C1 ≤ P_end, where P_C1 means the output power of the system at point C₁, which is the minimum achievable power of the system at R_C; P_end means the required output power at the end of the CV mode (i.e., at point C).

• When R_C ≥ R_V, there are three possible CV f-R_L curves:

  (1)

  For case2 (Figure 6b), the f-R_L curve penetrates the Peak1, which implies a continuous control is not possible.

  (2)

  For case3 (Figure 6c) and case 4 (Figure 6d), R_L and f increase simultaneously, and the CV f-R_L curves move away from the Peak2 curve. Case3 can be regarded as a critical case between case2 and case4.

• In summary, when R_C ≥ R_V, there is only one requirement to meet to realize monotonic and continuous frequency control, which is f_D ≤ f_V.

3. Parameter Design

According to the theory described in Section 2, the charging frequency path lies entirely within the shaded region; the problem of ZVS loss will consequently be solved. Hence, to achieve monotonic and continuous frequency control, there are a total of four constraints, given as

(1)

P_A1 ≤ P_start.

(2)

P_B1 ≥ P_rated.

(3)

R_A ≤ R_V.

(4)

P_C1 ≤ P_end when R_C < R_V or f_D ≤ f_V when R_C ≥ R_V.

In this paper, f_p is set to 80 kHz. The coupling coefficient k varies in the range of [0.1, 0.3]. With certain step sizes, every combination of [L₁, L₂, k, τ_c] will be tested through the flowchart in Figure 7 to judge whether the combination meets the above constraints. According to the finite element analysis simulation, the maximum inductance value that can be achieved after the coil is tightly wound is approximately 813.6 μH. To simplify the winding of coils, both L₁ and L₂ are assumed to be in the range of [300 μH, 810 μH]. The specifications of the prototype system are given in

Table 2. Specifications of the 3.3 kW experiment group.

Parameter	Symbol	Value
Rated power	Prated	3.3 kW
Initial power	Pstart	1.7 kW
Primary resonant frequency	fp	80 kHz
Input voltage	VDC	400 V
Output voltage (i.e., charging voltage)	Vo	200–400 V
Output current (i.e., charging current)	Io	1.7–8.3 A
Load range	RLDC	24–242 Ω
Coupling coefficient	kmin–kmax	0.1–0.3

.

From (4), it can be known that f_τzL increases as τ_c decreases, which means that the region between the Bottom curve and the Peak2 curve becomes narrower with a smaller τ_c, making the CC f-R_L curve more likely to approach the curve f = f_s and improve system efficiency. Therefore, the minimum τ_c is set to 1.01.

In this paper, the minimum value of L₂ required to achieve R_C < R_V is denoted as L_2C-V, while the minimum value of L₂ required to achieve R_A ≤ R_V is denoted as L_2A-V, which are given in

Table 3. The required L₂ for R_C < R_V and R_A ≤ R_V.

fp	τc	k	L2C-V	L2A-V
80 kHz	1.01	0.1	3310 μH	340 μH
		0.3	1070 μH	110 μH
	1.02	0.1	3530 μH	360 μH
		0.3	1090 μH	110 μH
	1.03	0.1	3630 μH	370 μH
		0.3	1100 μH	110 μH

. Table 3 indicates that, within the inductance range of [300 μH, 810 μH], only R_C ≥ R_V needs to be taken into consideration. The results of Figure 7 indicate that no parameter combinations can be found to meet the above four requirements when τ_c is 1.01 or 1.02. The constraint curves with τ_c values of 1.02 and 1.03 are shown in Figure 8 and Figure 9, respectively. The regions aligned with the arrow directions and enclosed by the constraint curves in Figure 8 and Figure 9 indicate the areas where the corresponding constraint conditions are met. Figure 2 indicates that τ_c = 1 represents the critical state of the zero-phase angle boundary, where the two ZVS regions just become connected. As τ_c increases, the connecting channel between two regions widens, resulting in an expanded design space. This explains why no feasible region can be identified until τ_c reaches 1.03. Eventually, a usable design is determined as L₁ = 785 μH and L₂ = 635 μH, which is within the usable region.

4. Experimental Verification

4.1. Description of the Prototype

According to Figure 1a, a prototype is built as shown in Figure 10. The SiC MOSFET C3M0021120D from CREE is used for the inverter, and the diode RURG80100 from ON Semiconductor is used for the rectifier. The system parameters are given in

Table 4. Parameters of the system.

Parameter	Symbol	Value
Primary coil inductance	L1	785 μH
Secondary coil inductance	L2	635 μH
Primary coil turns	N1	32
Secondary coil turns	N2	29
Primary resonant capacitance	C1	5.04 nF
Secondary resonant capacitance	C2	5.875 nF
Asymmetrical factor	τc	1.03

. The values of L₁ and L₂ were obtained in Section 3. With (3), the values of C₁ and C₂ can, therefore, be calculated. By performing finite element simulations with different numbers of turns, as shown in Figure 11, the primary and secondary coil turns, N₁ and N₂, that most closely match the required L₁ and L₂ values, respectively, can be determined.

Figure 11 shows the transmitting side and the receiving side of the prototype system. The parameters of the coils are given in

Table 5. Parameters of two coils.

Parameter	Transmitting Side	Receiving Side
Litz wire	ϕ1 mm × 800	ϕ1 mm × 800
Outer dimension (mm)	450 × 450	450 × 450
Inner dimension (mm)	180 × 180	180 × 180
Ferrite dimension (mm)	500 × 500 × 5	500 × 500 × 5
Aluminum dimension (mm)	500 × 500 × 4	500 × 500 × 4
dcf (mm)	2	2
daf (mm)	30.16	30.16

, where d_cf is the distance between the coil and the ferrite core and d_af is the distance between the aluminum layer and the ferrite core.

4.2. Experimental Results

To verify the proposed control and design method, 30 operation points (10 for k_max = 0.3, 10 for k_mid = 0.2, and 10 for k_min = 0.1) are selected for the 3.3 kW experiment group. R_LDC at 3.3 kW is 48.5 Ω, which is denoted as R_Lrated. R_LDC for different operating points are 0.5R_Lrated, 0.6R_Lrated, 0.7R_Lrated, 0.8R_Lrated, 0.9R_Lrated, R_Lrated, 2R_Lrated, 3R_Lrated, 4R_Lrated, and 5R_Lrated, respectively. These load resistance values can be accurately obtained using the constant resistance mode of the electronic load.

To enhance the practicality of the proposed method, the designed system should be able to work with different specifications. As introduced in Section 3, the charging process can be accomplished as long as the system meets the four constraints of the target charging condition. To verify this, another 30 operation points are conducted for the 2.4 kW experiment group. The CV mode of electronics load is used, which can better simulate the real charging process, and the parameters are listed in

Table 6. Specification of the 2.4 kW Experimental group.

Parameter	Symbol	Value
Rated power	Prated	2.4 kW
Initial power	Pstart	1.2 kW
Primary resonant frequency	fp	80 kHz
Input voltage	VDC	300 V
Output voltage (i.e., charging voltage)	Vo	150–300 V
Output current (i.e., charging current)	Io	1.6–8 A
Load range	RLDC	19–188 Ω
Coupling coefficient	kmin–kmax	0.1–0.3

.

Figure 12 and Figure 13 give the theoretical operating frequency curves and the measured operating frequency points for both 3.3 kW and 2.4 kW experiment groups, respectively. As shown in Figure 12, the OFP curve is presented. Compared to the cases of k = 0.3 and k = 0.2, the constant current curve for k = 0.1 is closer to the OFP curve. This characteristic, to some extent, mitigates the efficiency drop caused by weak coupling and demonstrates the necessity of the constraint R_A ≤ R_V. The theoretical P-f variation curves corresponding to the loads at both the CC charging initiation point and the experimental point near point V are also provided in Figure 12 and Figure 13, which verifies that the frequency path indeed crosses from the frequency splitting region between two splitting frequencies to the frequency splitting-free region. The measured CC/CV f-R_L curves are all located at the designated area, which guarantees the monotonicity and continuity of the control. The actual operating frequencies are slightly lower than the theoretical values because losses are not included in the calculation in (7). Compared to the cases with k = 0.2 and 0.3, the f-R_L curve of k = 0.1 in CC mode is closer to the Bottom curve (see the P-f curve with R_L/R_Lrated = 0.5 in Figure 12c), where the output power changes more slowly with the operating frequency. As a result, the power losses neglected in the calculation will lead to larger differences between the measured frequency values and the theoretical frequency values.

To better demonstrate the stability of the proposed method, the frequency errors between theoretical and measured frequency values, along with the corresponding uncertainty bands (grey shaded regions), have been added to Figure 12 and Figure 13. The grey shaded region surrounding the theoretical curve represents the 99% confidence interval of the predicted values, quantifying the combined uncertainties from measurement systems and parameter estimation. The upper and lower boundaries are calculated through the expressions P_upperbound = P + Z_value × std_error and P_lowerbound = P − Z_value × std_error, where P denotes the theoretical output power and std_error corresponds to a 20% relative uncertainty of the theoretical power value. This is because the minimum measured efficiency in the experiment is close to 80%, implying that a maximum of 20% of the power is dissipated as losses in the experimental points. Z_value = 2.58 defines the 99% confidence level under normal error distribution assumptions. With (7), the frequencies corresponding to P_upperbound and P_lowerbound, namely f_lowerbound and f_lowerbound, can be derived. The partial overlap of the grey shaded region’s boundary with the Peak2 curve indicates that even the power along Peak2 also cannot exceed 120% of the required power. Similarly, the overlap with the Bottom curve signifies that the power on the Bottom curve cannot fall below 80% of the required power. For the remaining boundaries that do not coincide with these curves, the corresponding frequencies have a 99% probability of delivering power precisely within the range of ±20% of the required value. All experimental points are in the grey shaded region. The experimental points that have large frequency errors correspond to load resistances where the output power changes very gradually with frequency. In such operating regimes, even minor variations in power loss can lead to proportionally larger shifts in the operating frequency. It can be found that the frequency errors for most experimental points are acceptable, which validates the effectiveness of the proposed method.

The DC-DC efficiencies for both 3.3 kW and 2.4 kW experiment groups are given in Figure 14 and Figure 15, respectively. The measured DC-DC efficiency ranges for rated power are 89.9–92.1% and 85.8–89.8%, respectively. The theoretical efficiencies for the 3.3 kW experiment group in Figure 14 are derived from the loss analysis presented in the following Part. Since the loss analysis for the 3.3 kW experimental group has been thoroughly conducted, the theoretical efficiency calculation is not repeated for the 2.4 kW experiment group.

Uncertainty bands and efficiency deviations have also been incorporated into Figure 14. The uncertainty bands are computed using the formula $y_{fit}\pm t_{\alpha /2,n-1}\cdot s\cdot \sqrt{1+1/n}$, where y_fit represents the values of the measured efficiency fitted curve; t_α/2,n−1 is the critical value from the Student’s t-distribution, where α = 0.01 (99% confidence level, t_α/2,n−1 = 3.25 in this paper) and n is the number of experimental data points (n = 10 in this study); s is the standard deviation of residuals between experimental data points and the fitted curve; and the term 1 + 1/n incorporates the additional uncertainty when predicting new observations beyond the existing dataset. There are only a few individual theoretical points, particularly in the low-resistance region. The reason is that the switching frequency in this work ranges from 72 to 91 kHz, whereas the equivalent series resistance (ESR) values of the coils and capacitors used for the efficiency calculation are measured at a fixed frequency of 80 kHz. The frequencies corresponding to the theoretical points outside the grey shaded region deviate from 80 kHz, leading to a discrepancy between the actual ESR and the value used in the theoretical efficiency calculation model. Consequently, a small number of theoretical efficiency points fall outside the grey uncertainty bands. It can be observed that the discrepancy between the theoretical and measured efficiency does not exceed 1.4% for all measurement points. This finding further attests to the effectiveness and reliability of the proposed method. To avoid redundancy, the efficiency errors and uncertainty bands are not calculated for Figure 15.

The measured and the theoretical input impedance angles, i.e., θ_in, for each operation point in the 3.3 kW experimental group are summarized in

Table 7. θ_in of the 3.3 kW experiment group.

Parameter	RL/RL,rated	kmax	kmid	kmin	RL/RL,rated	kmax	kmid	kmin
Theoretical	0.5	54°	53°	37°	0.6	47°	45°	35°
Measured		49°	50°	43°		40°	42°	42°
Theoretical	0.7	39°	38°	33°	0.8	30°	30°	32°
Measured		32°	36°	40°		23°	29	38°
Theoretical	0.9	20°	21°	30°	1	7°	14°	30°
Measured		15°	21°	37°		8°	15°	34°
Theoretical	2	7°	20°	61°	3	24°	52°	70°
Measured		8°	21°	60°		17°	47°	68°
Theoretical	4	45°	61°	75°	5	54°	67°	78°
Measured		38°	56°	70°		48°	63°	72°

. The measured range of θ_in is 8–72°, which is in good agreement with the theoretical values considering that the losses in the inverter, rectifier, shield, core, etc., are not included in the theoretical model. The minimum input impedance angle required for robust ZVS is determined by the energy balance during dead time. The critical angle θ_ZVSmin is calculated to ensure sufficient inductor current to discharge the MOSFET junction capacitances. The critical angle can be calculated as θ_ZVSmin = arcsin(2C_ossV_DC/t_deadA_m), where Coss is the MOSFET output capacitance, V_DC is the DC input voltage, t_dead is the dead time, and A_m is the amplitude of the primary current. In this paper, for the rated power, the four system parameters are given as C_oss = 200 pF, V_DC = 400 V, t_dead = 450 ns, and A_m = 13.4 A (13.4 A is the minimum measured current value). Then, θ_ZVSmin can be derived as 1.5°. From Table 7, it is clear that ZVS can be achieved over the entire coupling range and load range, as the minimum angle (8°) provides a sufficient margin to ensure its realization. The waveforms of the inverter output voltage and output current are given in Figure 16 at the rated power for the 3.3 kW experimental group, also verifying ZVS operation.

4.3. Loss Analysis

Figure 17 gives the system loss breakdown of rated power at k_max, k_mid, and k_min for the 3.3 kW experimental group, which indicates that the main losses are from the resonators. The core loss and shield loss are obtained through the volume integral of ohmic loss over the ferrite core and aluminum shield with finite element analysis simulation. According to the RURG80100 datasheet, the voltage–current relationship of the diode at 100 °C can be expressed as (13), where I_F represents the diode forward current and V_F represents the diode forward voltage. Meanwhile, I_F also represents the sinusoidal current flowing through the secondary coil, which can be expressed as (14), where A_m denotes the current amplitude, and it can be calculated with the method introduced in [25]. For the 3.3 kW prototype, A_m is 12.96 A at rated power. Within a single operating cycle of one diode, the power loss can be calculated as the integral of I_F multiplied by V_F over the interval [0, π/ω]. During each full operating cycle, this heat generation process is repeated across four diodes. Therefore, (15) can be employed to compute the total rectifier losses. The ZVS turn-on is achieved; thus, the MOSFET turn-on losses are neglected. The amplitudes of i₁ in Figure 16a–c for 3.3 kW experimental group are 13.4 A, 14.6 A, and 17.4 A, respectively. The dead time accounts for a very small portion of the entire current variation cycle. For simplicity, the current during the dead time is treated as a constant value in this work. With the i₁ amplitudes and the θ_in values in Table 7, the MOSFET current values at turn-off can be calculated to be less than 10 A. According to the C3M0021120D datasheet, the turn-off losses of the MOSFET at these current values can also be neglected. The calculation methods for the conduction losses of the MOSFET P_con are given by (16). According to [26], the conduction losses of the parallel diode P_Dcon and the reverse recovery losses P_res are provided by (17) and (18). The loss of the capacitors and the coils can be directly calculated with the data given in

Table 8. Parameters for loss calculation.

Parameter	Symbol	Value
The equivalent series resistance of primary coil	RL1	0.6 Ω
The equivalent series resistance of secondary coil	RL2	0.5 Ω
The equivalent series resistance of primary capacitor	RC1	0.76 Ω
The equivalent series resistance of secondary capacitor	RC2	0.69 Ω
MOSFET conduction resistance	RDS(ON)	27.6 mΩ
Reverse recovery charge	Qrr	897 nC
Anti-parallel diode voltage at kmax	VDM	3.02 V
Anti-parallel diode voltage at kmid	VDM	3.19 V
Anti-parallel diode voltage at kmin	VDM	3.60 V
The dead time	tdead	450 ns

, using $P=I_{rms}^{2}R$, where R can represent the equivalent series resistance of coil or capacitor. The calculated loss values corresponding to k_max, k_mid, and k_min are 259 W, 289 W, and 367 W, respectively, while the measured loss values are 283 W, 314 W, and 371 W. The difference between the two is acceptable.

$$V_F=0.000152384I_F^3-0.00481I_F^2+0.06724I_F+0.57539$$

(13)

$$I_F=A_m\mathit{sin}\left(\omega t\right)$$

(14)

$$P_{rectifier}=2f{\int }_0^{\frac{\omega }{\pi }}{V}_F{I}_F\mathit{dt}$$

(15)

$$P_{con}=A_m^2{R}_{DS\left(ON\right)}-4A_m^2\mathit{sin}{\left(\theta \right)}^2{R}_{DS\left(ON\right)}{t}_{dead}f$$

(16)

$$P_{\mathit{Dcon}}=4A_m\mathit{sin}\left(\theta \right)V_{DM}{t}_{dead}f$$

(17)

$$P_{res}=2Q_{rr}{V}_{DM}$$

(18)

5. Conclusions

In this article, the secondary-side resonant frequency is designed to be higher than the primary-side resonant frequency to facilitate the realization of ZVS in the SS WPT system. Following the derivation of the power transfer characteristics, a key conclusion for achieving high efficiency is that the operating frequency in CC mode should be as close as possible to the secondary-side resonant frequency. Based on these principles, a systematic parameter design flow is established, and an experimental prototype is constructed. A monotonic and continuous frequency control can be achieved to output the required power in both CC and CV modes, even with coil misalignment. Using the same experimental setup, both 3.3 kW and 2.4 kW experiment groups are conducted to demonstrate the effectiveness and performance of the proposed method. The experimental results show that the WPT system can maintain ZVS across a coupling coefficient range of [0.1, 0.3] and a load resistance variation of up to 10:1. The overall efficiency of the 3.3 kW experiment group ranges between 81.8% and 97.2%, and the efficiency drop is only 2.2% under a 3:1 coupling variation at rated power. Several aspects of the present work can be further improved. For example, to ensure accuracy in theoretical efficiency calculations across a broad frequency range, the frequency-dependent ESR of components must be accounted for at each specific operating point. Using a fixed set of ESR values measured at one frequency can otherwise lead to substantial deviations from the measured results. For practical WPT applications, parameter drift and temperature variations are critical factors that must be considered. In future work, these aspects will be incorporated prior to the system parameter design phase.