Current-Division-Aware Modeling and Detuning Design of a Receiver-Side LCC-Type Three-Coil Wireless Power Transfer System for Fixed-Frequency CC/CV Charging

Abstract

To satisfy the constant-current/constant-voltage (CC/CV) charging requirements of electric-vehicle batteries, this paper investigates the extension of an established fixed-frequency reconfigurable three-coil wireless power transfer system using a modified receiver-side LCC-type network. The receiver-side shunt capacitor introduces a current-dividing path, so the receiver-coil current differs from the equivalent rectifier-load current and the strict self-resonant value of the L₂—C₂ branch no longer provides the intended CC regulation. A current-division-aware fundamental-harmonic model is established, and a closed-form correction gives C₂ = 21.950 nF instead of the strict resonant value of 22.825 nF at 85 kHz. Calculated results show that the maximum CC target error over RL = 15–35 Ω is reduced from 8.948% to 0.031%, while the calculated CC/CV transition resistance is restored from 39.25 Ω to 34.97 Ω. Comparative analytical results are provided for the series-compensated baseline, the strictly resonant LCC-type configuration, and the corrected LCC-type configuration, together with receiver-current and coupling-sensitivity assessments. At RL = 35 Ω, the retained prototype measurements give an output voltage of 109.14 V, an output current of 3.121 A, an output power of 0.341 kW, and a DC-DC efficiency of 93.120%. The prototype measurements confirm the nominal operating target of the corrected design, while the analytical and numerical results clarify the modeling, correction, and current-stress tradeoffs introduced by the modified receiver network.

1. Introduction

Wireless power transfer (WPT) provides a contactless and automated means of delivering electrical energy and has been extensively investigated for electric vehicles (EVs), autonomous parking systems, public charging infrastructure, and other applications in which repeated mechanical connection is undesirable [1,2,3,4,5,6]. Compared with conductive charging, an inductive charging interface can reduce connector wear and improve operating convenience. For EV-oriented applications, however, the power-transfer stage must provide not only high efficiency but also predictable output characteristics as the battery operating point changes.

Lithium-ion batteries are commonly charged through a constant-current (CC) stage followed by a constant-voltage (CV) stage. During the CC stage, the charging current is maintained near its prescribed value while the battery terminal voltage increases. After the terminal voltage reaches its upper limit, the charger enters the CV stage, in which the voltage is maintained and the charging current gradually decreases [4,7,8,9,10]. Because the equivalent battery resistance varies substantially during this process, a WPT charger must either provide approximately load-independent CC/CV outputs or employ an additional closed-loop regulation stage.

Existing CC/CV WPT methods can be broadly classified into frequency-tuning control, active modulation or downstream DC-DC regulation, compensation-topology reconfiguration, and multi-coil or high-order compensation approaches [4,7,8,9,10,11]. Frequency tuning can access different current-gain and voltage-gain operating points, but a wide frequency range may move the converter away from its intended zero-phase-angle condition and may be constrained by the allowable operating band. Active modulation and additional DC-DC stages provide flexible regulation, but they increase the number of semiconductor devices, sensing requirements, control complexity, and conversion losses. Reconfigurable passive networks can retain a fixed operating frequency, although their component count, switch arrangement, and soft-switching conditions depend strongly on the selected topology.

Three-coil WPT systems provide an additional magnetic and circuit degree of freedom and have been investigated for impedance transformation, transfer-distance extension, misalignment tolerance, compact receiver design, load-independent output, metal-cover smartphone charging, and offshore USV charging [12,13,14,15,16,17]. Representative CC/CV approaches include an S/S/P-compensated three-coil system operating at two zero-phase-angle frequencies [7], switchable LCC-S-S and S-S-S networks with an explicit parameter-design method [8], a negative-polarity partial-power-conversion system for high-power EV charging [4], an S-S-LCLCC network aimed at light-load efficiency and misalignment performance [11], and a hybrid receiver-side series/LCC compensation method [10]. These studies demonstrate that neither the use of three coils nor the use of an LCC-type compensation network alone constitutes a sufficient distinction from prior work. The reported methods instead differ in operating-frequency scheduling, switch and passive-component count, receiver complexity, cross-coupling treatment, input-impedance characteristics, and robustness to load or coupling variation.

Among the reported approaches, the fixed-frequency reconfigurable three-coil system in Ref. [9] provides the closest baseline to the operating principle considered in this study. In that system, two transmitter-side windings and a single relay are used so that the power-transfer path operates as a two-coil structure for CC charging and as a three-coil structure for CV charging. The receiver is series compensated, and the receiver-coil current is directly associated with the equivalent rectifier-load current under the adopted fundamental-harmonic model. Accordingly, the fixed-frequency two-coil/three-coil reconfiguration mechanism is treated as an established baseline in the present work rather than as a new contribution.

Receiver-side LCC compensation has also been used in three-coil WPT systems to realize load-independent output characteristics [10]. However, when the series-compensated receiver of the reconfigurable baseline is replaced by the modified LCC-type network considered here, the parallel capacitor introduces an additional current path. The receiver-coil current is then no longer identical to the current delivered to the equivalent rectifier-load branch. This current division changes the CC output expression and the transition condition, and a strictly resonant value of the L₂—C₂ branch does not necessarily minimize the load-dependent CC-current error. Furthermore, once C₂ is detuned, the nonzero branch reactance must also be retained when evaluating the CV output and the input-impedance angle. The specific problem addressed in this study is therefore the current-division-aware modeling and parameter correction required when extending the established reconfigurable architecture to this receiver-side network.

Accordingly, this paper addresses the following research question: how can an established fixed-frequency reconfigurable three-coil CC/CV architecture be extended with a modified receiver-side LCC-type network while correctly accounting for the parallel-capacitor current division and preserving acceptable CC/CV regulation? To answer this question, the receiver-coil current, parallel-capacitor current, and equivalent rectifier-load current are distinguished in the fundamental-harmonic model; the effect of the receiver network on the CC/CV transition is derived; and the L₂—C₂ branch is selected using an explicit load-sweep regulation criterion together with an input-impedance assessment.

The main contributions of this paper are summarized as follows:

(1)

A current-division-aware fundamental-harmonic model is established for the fixed-frequency reconfigurable three-coil system with a modified receiver-side LCC-type network. In contrast to the series-compensated baseline, the model explicitly distinguishes the receiver-coil current from the equivalent rectifier-load current and retains the current path introduced by the parallel capacitor C_f.

(2)

A systematic detuning procedure is formulated for the L₂—C₂ branch. The value of C₂ is selected through an explicit load-sweep current-regulation objective, while the resulting effects on the CC/CV transition, the CV-mode solution with nonzero branch reactance, and the input-impedance angle are assessed rather than treating the capacitance adjustment as an empirical choice.

(3)

The series-compensated baseline, the strictly resonant LCC-type configuration, and the detuned LCC-type configuration are comparatively evaluated using analytical and numerical results. Existing scaled-prototype measurements at the nominal transition load are used only as nominal-point experimental support, whereas all newly added load, coupling, and current-division results are explicitly identified as analytical or numerical evidence.

The remainder of this paper is organized as follows. Section 2 describes the baseline reconfiguration principle and establishes the current-division-aware equivalent model for the modified receiver-side LCC-type network. Section 3 presents the parameter-design procedure, including the C_f sensitivity assessment, the load-sweep selection of C₂, the CC/CV transition condition, and the input-impedance constraint. Section 4 reports the analytical and numerical comparisons together with the retained prototype measurements at the nominal operating point and discusses the limitations of the available experimental evidence. Section 5 concludes the paper.

2. Current-Division-Aware Modeling of the Reconfigurable Three-Coil WPT System

2.1. Baseline Reconfiguration Principle and Scope of the Present Extension

Figure 1 shows the fixed-frequency reconfigurable three-coil wireless power transfer (WPT) system considered in this study. The magnetic coupler contains two transmitter-side windings, L₁ and L₂, and one receiver coil, L₃. Their mutual inductances are denoted by M₁₂, M₂₃, and M₁₃. The inverter and relay states select the active power-transfer path for the constant-current (CC) and constant-voltage (CV) stages.

The basic two-coil/three-coil reconfiguration principle follows Ref. [9] and is treated here as the baseline operating mechanism. In the CC mode, the L₁-C₁ branch is excluded from the main excitation path and the system is primarily represented by the coupled L₂-L₃ subsystem. In the CV mode, L₁, L₂, and L₃ participate in the coupled network, producing the three-coil voltage-transfer relationship. The fixed operating frequency and the relay-based transition are therefore not claimed as new contributions of the present work.

The extension investigated here is located on the receiver side. Ref. [9] uses a series-compensated receiver, whereas the present circuit employs a series capacitor C_s, a compensation inductor L_f, and a shunt capacitor C_f connected in the actual order shown in Figure 1. Because C_f forms a parallel current path at the rectifier input, the receiver-coil current is no longer identical to the equivalent rectifier-load current. The resulting current division must be retained in the CC/CV model and in the transition-load calculation.

2.2. Configuration of the Modified Receiver-Side LCC-Type Network

The receiver network used in this work is arranged as

L₃-C_s-L_f-C_f‖R_eq

where R_eq is the fundamental-harmonic equivalent resistance of the rectifier and dc load. This arrangement differs from the conventional receiver-side LCC network reported in Ref. [13], in which the shunt capacitor is connected before the additional series inductor. The two arrangements are not generally interchangeable because the compensation inductor carries different branch currents.

For the conventional arrangement represented by Ref. [13], the receiver-side input impedance is

$$Z_{\mathrm{conv}}=R_3+j\omega L_3+{\displaystyle \frac{1}{j\omega C_{SS}}}+\left[{\displaystyle \frac{1}{j\omega C_K}}\parallel \left(R_K+j\omega L_K+R_{eq}\right)\right]$$

(1)

For the actual arrangement considered in this study, the series-path impedance and parallel-port impedance are respectively defined as

$$Z_s=R_{3\Sigma }+j\omega \left(L_3+L_f\right)+{\displaystyle \frac{1}{j\omega C_s}}$$

(2)

$$Z_p=R_{eq}\parallel {\displaystyle \frac{1}{j\omega C_f}}={\displaystyle \frac{R_{eq}}{1+j\omega C_f{R}_{eq}}}$$

(3)

where R_3Σ is the lumped series resistance of the receiver coil, compensation inductor, capacitor equivalent series resistance, and associated wiring. The total receiver-side impedance is therefore

$$Z_3=Z_s+Z_p$$

(4)

Equations (1)–(4) show that the two LCC arrangements are not generally equivalent. Accordingly, the term modified receiver-side LCC-type network is used throughout this paper, and all subsequent derivations are based on the actual circuit of Figure 1 rather than on an assumed conventional LCC equivalent.

2.3. Fundamental-Harmonic and Rectifier Equivalent Relationships

Unless otherwise specified, all ac voltages and currents in the following analysis are rms phasors and are denoted by a tilde. The dc output voltage and current are denoted by V_REC and I_REC without a tilde. The angular operating frequency is

$$\omega =2\pi f$$

(5)

For the half-bridge excitation involved in each active power-transfer path, the rms value of the fundamental voltage is

$${\tilde{U}}_s={\displaystyle \frac{\sqrt{2}}{\pi }}{V}_{INV}$$

(6)

where Ũ_s= Ũ₂ in the CC mode and Ũ_s= Ũ₁ in the CV mode. This definition refers to the bridge-leg fundamental involved in the equivalent circuit and avoids confusing it with the full bridge terminal waveform.

Under the fundamental-harmonic approximation, the diode rectifier and dc load R_L are represented by

$$R_{eq}={\displaystyle \frac{8}{{\pi }^2}}{R}_L$$

(7)

The fundamental voltage and current at the rectifier input satisfy

$$\left|{\tilde{U}}_{eq}\right|={\displaystyle \frac{2\sqrt{2}}{\pi }}{V}_{REC}$$

(8)

$$\left|{\tilde{I}}_{eq}\right|={\displaystyle \frac{\pi }{2\sqrt{2}}}{I}_{REC}$$

(9)

In the present receiver network, the current through L₃, C_s, and L_f reaches the output node and then divides between C_f and R_eq. Hence,

$${\tilde{I}}_3={\tilde{I}}_{eq}+{\tilde{I}}_{Cf}={\tilde{I}}_{eq}+j\omega C_f{\tilde{U}}_{eq}$$

(10)

Since Ũ_eq = R_eqĨ_eq,

$${\tilde{I}}_3=\left(1+j\omega C_f{R}_{eq}\right){\tilde{I}}_{eq}$$

(11)

and therefore

$${\tilde{I}}_{eq}={\displaystyle \frac{{\tilde{I}}_3}{1+j\omega C_f{R}_{eq}}}$$

(12)

Equation (12) is the central difference from the series-compensated receiver in Ref. [9], for which C_f = 0 and Ĩ₃ = Ĩ_eq.

2.4. Complete CC-Mode Model

In CC mode, the L₁-C₁ branch does not participate in the main power transmission path, and the circuit is modeled by the coupled L₂-L₃ subsystem, the equivalent diagram of which is shown in Figure 2.

The lumped impedance of the L₂—C₂ branch is

$$Z_2=R_{2\Sigma }+j\omega L_2+{\displaystyle \frac{1}{j\omega C_2}}$$

(13)

where R_2Σ includes the coil resistance, capacitor equivalent series resistance, and series wiring resistance. The phasor-domain equations are

$$\left[\begin{array}{c}{\tilde{U}}_2\\ 0\end{array}\right]=\left[\begin{array}{cc}{Z}_2& j\omega M_{23}\\ j\omega M_{23}& Z_3\end{array}\right]\left[\begin{array}{c}{\tilde{I}}_2\\ {\tilde{I}}_3\end{array}\right]$$

(14)

Solving (14) gives the receiver-coil current

$${\tilde{I}}_3=-{\displaystyle \frac{j\omega M_{23}{\tilde{U}}_2}{Z_2{Z}_3+{\omega }^2{M}_{23}^2}}$$

(15)

Combining (12) and (15), the equivalent rectifier-load current is

$${\tilde{I}}_{eq}=-{\displaystyle \frac{j\omega M_{23}{\tilde{U}}_2}{\left(Z_2{Z}_3+{\omega }^2{M}_{23}^2\right)\left(1+j\omega C_f{R}_{eq}\right)}}$$

(16)

The DC output current is consequently

$$I_{REC,CC}={\displaystyle \frac{2\sqrt{2}}{\pi }}\left|{\displaystyle \frac{j\omega M_{23}{\tilde{U}}_2}{\left(Z_2{Z}_3+{\omega }^2{M}_{23}^2\right)\left(1+j\omega C_f{R}_{eq}\right)}}\right|$$

(17)

The corresponding dc output voltage is

$$V_{REC,CC}=I_{REC,CC}{R}_L$$

(18)

The input impedance observed from the active CC-mode bridge leg is

$$Z_{in,CC}={\displaystyle \frac{{\tilde{U}}_2}{{\tilde{I}}_2}}=Z_2+{\displaystyle \frac{{\omega }^2{M}_{23}^2}{Z_3}}$$

(19)

Equation (19) will be used in Section 3 to quantify the effect of C₂ detuning on the input-impedance angle. A positive imaginary part indicates an inductive input condition, whereas a negative imaginary part indicates a capacitive input condition. A claim of zero-voltage-switching feasibility should only be made after evaluating (19) with the actual component values and load range.

For physical interpretation, if the receiver is reduced to the series-compensated baseline by setting C_f → 0, L_f → 0, and C_s = C₃, and if both coupled branches are ideally resonant with negligible losses, Formula (17) reduces to

$$I_{REC,CC}\approx {\displaystyle \frac{4V_{INV}}{{\pi }^2\omega M_{23}}}$$

(20)

Equation (20) is the established ideal CC expression of the baseline architecture. It is used only for initial design and interpretation; all parameter sweeps in the revised manuscript will use the complete expression in (17).

2.5. Complete CV-Mode Model with Nonzero Z₂

In CV mode, the three coils participate in the coupling network, and the equivalent diagram is shown in Figure 3.

The L₁—C₁ branch impedance is

$$Z_1=R_{1\Sigma }+j\omega L_1+{\displaystyle \frac{1}{j\omega C_1}}$$

(21)

The complete phasor—domain model is

$$\left[\begin{array}{c}{\tilde{U}}_1\\ 0\\ 0\end{array}\right]=\left[\begin{array}{ccc}{Z}_1& j\omega M_{12}& j\omega M_{13}\\ j\omega M_{12}& Z_2& j\omega M_{23}\\ j\omega M_{13}& j\omega M_{23}& Z_3\end{array}\right]\left[\begin{array}{c}{\tilde{I}}_1\\ {\tilde{I}}_2\\ {\tilde{I}}_3\end{array}\right]$$

(22)

To retain the actual detuned value of C₂, Z₂ is not set to zero in the numerical solution. Define the determinant

$$\Delta =Z_1{Z}_2{Z}_3+{\omega }^2\left(M_{12}^2{Z}_3+M_{13}^2{Z}_2+M_{23}^2{Z}_1\right)-2j{\omega }^3{M}_{12}{M}_{13}{M}_{23}$$

(23)

The complete receiver—coil current obtained from (22) is

$${\tilde{I}}_3={\displaystyle \frac{\left(-{\omega }^2{M}_{12}{M}_{23}-j\omega M_{13}{Z}_2\right){\tilde{U}}_1}{\Delta }}$$

(24)

The rectifier—load current and the dc output voltage are therefore

$${\tilde{I}}_{eq}={\displaystyle \frac{\left(-{\omega }^2{M}_{12}{M}_{23}-j\omega M_{13}{Z}_2\right){\tilde{U}}_1}{\Delta \left(1+j\omega C_f{R}_{eq}\right)}}$$

(25)

$$V_{REC,CV}={\displaystyle \frac{\pi }{2\sqrt{2}}}\left|R_{eq}{\tilde{I}}_{eq}\right|$$

(26)

The complete input impedance in the CV mode is

$$Z_{in,CV}={\displaystyle \frac{{\tilde{U}}_1}{{\tilde{I}}_1}}={\displaystyle \frac{\Delta }{Z_2{Z}_3+{\omega }^2{M}_{23}^2}}$$

(27)

Equations (23)–(27) retain the nonzero Z₂ branch, the signed cross-coupling M₁₃, the receiver-side current division, and symbolic branch-resistance terms. In the numerical results reported in Section 3 and Section 4, these resistance terms are set to zero because traceable coil-resistance and capacitor-ESR records were unavailable. The equations are used to verify that the selected C₂ improves CC regulation without causing an unacceptable CV error or input-impedance angle.

For interpretation only, if Z₁ ≈ Z₂ ≈ 0, M₁₃ ≈ 0, the receiver series path is approximately tuned, and losses are neglected, the induced receiver-port voltage is approximately proportional to the mutual-inductance ratio. Equation (26) then reduces to

$$V_{REC,CV}\approx {\displaystyle \frac{M_{23}}{2M_{12}}}{V}_{INV}$$

(28)

Equation (28) explains the approximate load-independent CV behavior of the baseline architecture. It is not used as a substitute for the complete solution when Z₂ ≠ 0.

2.6. CC/CV Transition Condition

At the CC-to-CV transition, the output voltage produced by the CC mode reaches the CV-mode voltage setting. With the complete model, the transition resistance is determined by the positive root of

$$F\left(R_L\right)=I_{REC,CC}\left(R_L\right)R_L-V_{REC,CV}\left(R_L\right)=0$$

(29)

where I_REC,CC is calculated from (17) and V_REC,CV is calculated from (26). Because both quantities depend on R_L, the transition resistance is obtained numerically rather than by directly dividing two ideal constants.

For initial design only, substituting the ideal expressions (20) and (28) gives

$$R_{tr,0}\approx {\displaystyle \frac{V_{REC,CV}}{I_{REC,CC}}}={\displaystyle \frac{{\pi }^2\omega M_{23}^2}{8M_{12}}}$$

(30)

Equation (30) provides an initial estimate close to 35 Ω for the present parameters. The final value reported in this paper is obtained from (29) using the actual C_f, C₂, and mutual-inductance values under the stated nominal low-loss approximation.

2.7. Model Consistency and Limiting Cases

The revised model is checked using the following limiting cases.

Series-receiver baseline: Setting C_f → 0, L_f → 0, and C_s = C₃ makes Z_p → R_eq and Ĩ₃ = Ĩ_eq, reducing the receiver to the series-compensated topology of Ref. [9].

Ideal CC relation: If the CC branches are tuned and losses are neglected, (17) reduces to (20).

Ideal CV relation: If Z₁ ≈ Z₂ ≈ 0, M₁₃ ≈ 0, and losses are neglected, (26) reduces to (28).

Actual detuned calculation: For C₂ = 21.950 nF, the numerical analysis retains Z₂ ≠ 0 in both CC and CV modes. The ideal equations are not used to generate the final regulation curves.

These checks establish the relationship between the proposed current-division-aware model and the previously reported series-compensated baseline while avoiding the unsupported assumption that the present receiver network is equivalent to a conventional LCC arrangement.

3. Parameter Design and Current-Division Correction

3.1. Design Assumptions and Nominal Parameters

The parameter design is carried out using the current-division-aware model derived in Section 2. The operating frequency and input voltage are fixed at 85 kHz and 110 V, respectively. The nominal magnetic parameters are L₁ = 5.0 μH, L₂ = 153.6 μH, L₃ = 185.3 μH, M₁₂ = 13.48 μH, M₂₃ = 26.75 μH, and M₁₃ = 3.83 μH. The implemented receiver network uses L_f = 10 μH, C_s = 17.95 nF, and C_f = 30 nF.

The compensation design in this section is intentionally separated from the later robustness assessment. The analytical correction is derived under the tuned, low-loss, and small-cross-coupling design approximation, consistent with the initial output-design equations. The complete model of Section 2 is subsequently used to assess parameter perturbations. Thus, the calculated results reported in this section should not be interpreted as additional experimental measurements. Because traceable winding-resistance and capacitor-ESR data were unavailable in the retained records, the numerical results set R₁Σ = R₂Σ = R₃Σ = 0; these terms are retained symbolically in Section 2 only to show how measured losses can be incorporated in future work. Internal properties are shown in

Table 1. Coil and mutual-inductance parameters used in the analytical design.

Parameter	Value (μH)	Parameter	Value (μH)
L1	5.0	M12	13.48
L2	153.6	M23	26.75
L3	185.3	M13	3.83

.

3.2. Strict Resonant Value of C₂

The strict series-resonant capacitance of the L₂—C₂ branch is

$$C_{2r}={\displaystyle \frac{1}{{\omega }^2{L}_2}}$$

(31)

According to (31), for f = 85 kHz and L₂ = 153.6 μH, the strict resonant value is C₂r = 22.825 nF. This value cancels the self-reactance of L₂ but does not compensate for the load-dependent reactive term introduced by the receiver-side shunt capacitor C_f.

3.3. Analytical Current-Division Correction

When the L₃—Cs—L_f series path is approximately tuned and losses are neglected for the initial parameter design, the receiver impedance and the L₂—C₂ branch impedance can be written as

$$Z_3\approx {\displaystyle \frac{R_{\mathrm{eq}}}{1+j\omega C_f{R}_{\mathrm{eq}}}}, Z_2=j{X}_2$$

(32)

Substituting (32) into the complete CC expression (17) yields

$$\left(Z_2{Z}_3+{\omega }^2{M}_{23}^2\right)\left(1+j\omega C_f{R}_{\mathrm{eq}}\right)={\omega }^2{M}_{23}^2+j{R}_{\mathrm{eq}}\left(X_2+{\omega }^3{M}_{23}^2{C}_f\right)$$

(33)

The imaginary part in the denominator of (33) depends on the load through R_eq. Therefore, the load-dependent term is eliminated when

$$X_2^{*}=-{\omega }^3{M}_{23}^2{C}_f$$

(34)

Using X₂ = ωL₂ − 1/(ωC₂) in (34), the corrected capacitance is obtained as

$$C_2^{*}={\displaystyle \frac{1}{{\omega }^2\left(L_2+{\omega }^2{M}_{23}^2{C}_f\right)}}$$

(35)

For C_f = 30 nF and M₂₃ = 26.75 μH, (35) gives C₂* = 21.950 nF. Relative to C₂r = 22.825 nF, the correction is −3.834%. The implemented capacitance of 21.95 nF therefore follows directly from the analytical cancelation condition rather than from empirical trial and error.

3.4. Numerical Verification of the Detuning Value

A one-dimensional sweep is performed to verify the analytical result. The rms normalized current-error objective is defined as

$$J_I\left(C_2\right)=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle {\displaystyle \sum }_{k=1}^N}{\left[{\displaystyle \frac{I_{\mathrm{REC},\mathrm{CC}}\left(R_{L,k},C_2\right)-I_{\mathrm{ref}}}{I_{\mathrm{ref}}}}\right]}^2}$$

(36)

The numerical sweep based on (36) uses R_L,k from 15 Ω to 35 Ω in 1 Ω increments, N = 21, and I_ref = 3.12 A. C₂ is swept from 20.5 nF to 23.5 nF. The calculated results are shown in Figure 4.

The strict resonant value produces an rms current error of 5.509% and a maximum target error of 8.948%. With C₂ = 21.950 nF, both the rms and maximum target errors are approximately 0.031% over the specified 21-point load sweep.

The resulting CC output current before and after the current-division correction is compared in Figure 5. Without the proposed correction, the output current gradually decreases as the load resistance increases. By contrast, the corrected design maintains an approximately constant output current of 3.121 A over the investigated load range.

3.5. Sensitivity to the Receiver-Side Parallel Capacitor

The parallel capacitor also determines the receiver-current stress. From the current-division relationship in (11), the current-amplification ratio and phase displacement are

$${\displaystyle \frac{\left|{\tilde{I}}_3\right|}{\left|{\tilde{I}}_{\mathrm{eq}}\right|}}=\sqrt{1+{\left(\omega C_f{R}_{\mathrm{eq}}\right)}^2}, {\theta }_{\mathrm{div}}={\tan }^{-1}\left(\omega C_f{R}_{\mathrm{eq}}\right)$$

(37)

At R_L = 35 Ω and C_f = 30 nF, |I₃|/|I_eq| = 1.098 and the phasor displacement is 24.44°. Therefore, increasing C_f requires a smaller corrected C₂ and increases the reactive current carried by the receiver coil. The implemented value of 30 nF is retained as a practical design value and is not claimed to be a universal optimum. The relationship is shown in Figure 6.

3.6. Input-Impedance and Transition Assessment

The input-impedance angles are evaluated using Z_in,CC and Z_in,CV from Section 2. The calculated input-impedance angles for the CC and CV operating modes are presented in Figure 7. At the 35 Ω operating point, the corrected design gives calculated phase angles of approximately +0.014° in the CC mode and −0.016° in the CV mode under the nominal design approximation. The corrected system is therefore close to the ZPA condition. Because the switching-node waveforms required to demonstrate device-level ZVS were not newly measured, the revised manuscript does not claim additional experimental ZVS verification.

The CC/CV transition resistance is calculated from (29), equivalently as

$$F\left(R_L\right)=I_{\mathrm{REC},\mathrm{CC}}\left(R_L\right)R_L-V_{\mathrm{REC},\mathrm{CV}}\left(R_L\right)=0$$

(38)

Solving (38) gives R_tr = 34.97 Ω for the corrected design, whereas the strict resonant design shifts the calculated transition to 39.25 Ω. The corresponding numerical comparison before and after the current-division correction is presented in Figure 8.

Thus, the current-division correction restores the intended transition near 35 Ω.

3.7. Final Compensation Parameters

The final compensation parameters obtained from the current-division correction are summarized in

Table 2. Final compensation and transition parameters.

Parameter	Value	Design Role
C1	701.1 nF	L1 branch compensation
Cs	17.95 nF	Tunes L3 + L_f series path
Cf	30 nF	Implemented shunt-capacitor value
C2r	22.825 nF	Strict series resonance
C2	21.950 nF	Current-division-corrected value
Rtr	34.97 Ω	Calculated CC/CV transition

.

4. Results and Discussion

4.1. Experimental Setup and Measured Nominal-Point Verification

A scaled laboratory prototype was used to verify the nominal operating point of the fixed-frequency reconfigurable three-coil wireless power transfer system. The platform consists of a dc source, a high-frequency inverter, a relay-based reconfiguration branch, the three-coil magnetic coupler, the modified receiver-side LCC-type network, a diode rectifier and filter stage, a resistive load, an oscilloscope, and a power analyzer. The dc input voltage and operating frequency were fixed at 110 V and 85 kHz, respectively. A load resistance of 35 Ω was selected because the analytical transition resistance obtained in Section 3 is 34.97 Ω. The prototype is shown in Figure 9.

Figure 10 and Figure 11 show the retained oscilloscope records for the CC and CV operating states at R_L = 35 Ω. In the CC state, U₂ is the high-frequency bridge-leg excitation, I₂ is the resonant current of the active transmitter branch, and V_REC and I_REC are the rectified dc output quantities. The measured output current is approximately 3.121 A, which is consistent with the target value of 3.12 A. In the CV state, U₁ is the high-frequency excitation and I₂ and I₃ exhibit the expected resonant-current behavior. The measured output voltage is approximately 109.14 V, in agreement with the designed value.

The retained power-analyzer record at the nominal operating point is shown in Figure 12 and the corresponding measured quantities are summarized in

Table 3. Measured nominal-point quantities retained from the original prototype.

Quantity	Symbol	Measured Value	Evidence
Input dc voltage	VIN	110.00 V	Power-analyzer display
Input dc current	IIN	3.325 A	Power-analyzer display
Input power	PIN	0.366 kW	Power-analyzer display
Output dc voltage	VREC	109.14 V	Power-analyzer display
Output dc current	IREC	3.121 A	Power-analyzer display
Output power	POUT	0.341 kW	Power-analyzer display
DC–DC efficiency	η	93.120%	Directly displayed value

. The displayed input voltage, input current, and input power are 110.00 V, 3.325 A, and 0.366 kW, respectively. The output voltage, output current, and output power are 109.14 V, 3.121 A, and 0.341 kW, respectively, and the directly displayed dc–dc efficiency is 93.120%. The ratio calculated from the rounded powers differs slightly from the displayed efficiency because the numerical values shown on the screen are rounded.

4.2. Controlled Comparison Among the Three Receiver Configurations

To isolate the influence of the receiver network and the C₂ correction, three configurations were evaluated using the same input voltage, operating frequency, magnetic parameters, and fundamental-harmonic model: (A) the series-compensated receiver used as the analytical baseline; (B) the modified LCC-type receiver with the strict L₂–C₂ resonant value C₂ = 22.825 nF; and (C) the same modified LCC-type receiver with the current-division-corrected value C₂ = 21.950 nF. For this controlled comparison, M₁₃ is initially set to zero so that the effects of C_f and C₂ can be separated from the signed cross-coupling effect. The influence of M₁₃ is discussed separately in Section 4.4. This controlled comparison isolates the compensation mechanism and is not intended as a direct numerical reproduction of the nominal prototype with its measured nonzero M₁₃.

Figure 13a shows that Configuration A maintains approximately 3.1205 A over the investigated CC load range. After the shunt capacitor is introduced without correcting C₂, Configuration B exhibits a systematic current decrease from 3.0630 A at 15 Ω to 2.8408 A at 35 Ω, giving a maximum target-current error of 8.948%. Configuration C restores the output current to approximately 3.1210 A, and the maximum target-current error is reduced to 0.031%. Thus, the relevant result is not that an LCC-type network alone guarantees constant current, but that its current-dividing effect must be included in the parameter design.

As shown in Figure 13b, all three configurations provide an approximately load-independent CV output under the controlled M₁₃ = 0 design approximation. The calculated voltages are approximately 109.143 V, 109.126 V, and 109.128 V for Configurations A, B, and C, respectively. Therefore, the proposed C₂ correction recovers the CC target without introducing a significant CV deviation in the adopted model. Figure 13c further shows that the uncorrected shunt-capacitor receiver shifts the transition resistance from approximately 34.98 Ω to 39.25 Ω, whereas the corrected design restores it to 34.97 Ω.

The input-angle results in

Table 4. Calculated comparison among the three receiver configurations.

Metric	A: Series Baseline	B: Strict LCC-Type	C: Corrected LCC-Type
Receiver network	Series C3	Cs–Lf–(Cf‖R_eq)	Same as B
C2	22.825 nF	22.825 nF	21.950 nF
Maximum CC target error	0.017%	8.948%	0.031%
CV (35–60 Ω)	109.143 V	109.126 V	109.128 V
Calculated transition load	34.98 Ω	39.25 Ω	34.97 Ω
CC input angle at 35 Ω	−0.000°	+24.461°	+0.014°
CV input angle at 35 Ω	−0.003°	−24.464°	−0.016°
|I3|/|I_eq| at 35 Ω	1.000	1.098	1.098

show that strict resonance of the L₂–C₂ branch does not produce a near-zero input phase after C_f is introduced. The corrected value moves the input phase close to 0° in both operating states under the design approximation. This result supports a near-ZPA input condition, but it does not constitute experimental verification of zero-voltage switching. Device-level ZVS would require switching-node voltage and gate-signal records that were not newly measured.

4.3. Receiver-Side Current Division and AC Current Stress

The modified receiver-side network creates a shunt-capacitor current in addition to the current delivered to the equivalent rectifier-load branch. At R_L = 35 Ω, Configuration C gives calculated rms fundamental values of |I_eq| = 3.467 A, |I_Cf| = 1.576 A, and |I₃| = 3.808 A. The capacitor current leads the load-branch current by 90°, and the resulting receiver-coil current has a phase displacement of approximately 24.44° relative to I_eq. Figure 14 presents the analytically reconstructed fundamental waveforms and their phasor relationship.

The receiver-coil current in Configuration C is approximately 9.86% higher than the equivalent load-branch current. Compared with the series-compensated baseline, this additional current represents an important design tradeoff because it can increase coil copper loss, compensation-component current stress, and rectifier-side reactive current. Therefore, the modified LCC-type receiver is not claimed to be universally superior to the series-compensated receiver. The specific contribution is the current-division-aware correction required when this receiver network is adopted.

4.4. Receiver-Coupling and Signed Cross-Coupling Sensitivity

Because no new physical displacement experiment was available during revision, the effect of receiver coupling is evaluated using the normalized factor α = M₂₃/M₂₃,0 while M₁₂ is held constant. This representation avoids assigning an unsupported displacement distance to the calculated results. Figure 15 shows the calculated CC, CV, and transition resistance of the corrected design as α varies. The blue dotted line represents the target constant-current (or constant-voltage) reference used for comparison.

Both the CC and CV characteristics are sensitive to M₂₃. The calculated range that keeps both output deviations within ±5% is approximately 0.952 ≤ α ≤ 1.050. For example, α = 0.90 increases the CC to 3.448 A, decreases the CV to 98.216 V, and shifts the transition resistance to 28.43 Ω. Conversely, α = 1.10 decreases the CC to 2.828 A, increases the CV to 120.040 V, and shifts the transition resistance to 42.50 Ω. These results quantify coupling robustness but do not replace a physical misalignment test. A mapping from α to displacement would require measured or finite-element mutual-inductance data.

The complete CV model also contains the transmitter-to-receiver cross-coupling M₁₃. Its influence depends on the coil winding directions and dot convention; therefore, M₁₃ must be treated as a signed quantity. Defining ρ = M₁₃/M₂₃, Figure 16 shows that the sign of ρ can noticeably change both the CV output and the input-impedance angle. At |ρ| = 0.143, the calculated CV is 111.284 V for negative ρ and 99.151 V for positive ρ. Because the original dot convention and mutual-inductance phase record were not recoverable, the primary controlled comparison uses M₁₃ ≈ 0 and the signed-M₁₃ results are reported as a sensitivity range rather than a calibrated prediction of the prototype.

4.5. Efficiency, Loss Sources, and Power-Level Discussion

The measured dc–dc efficiency at the nominal transition point is 93.120%. Based on the rounded displayed powers, the aggregate input–output power difference is approximately 25 W. The main physical loss mechanisms are expected to include inverter conduction and switching losses, AC and dc copper losses in the three coils, dielectric and equivalent-series-resistance losses in the compensation capacitors, rectifier conduction loss, and wiring loss. A component-level loss allocation is not reported because the original ac resistances, capacitor ESR values, semiconductor switching data, and thermal measurements were not available for a traceable retrospective calculation.

The measured output power of 0.341 kW is lower than the 3–11 kW range commonly associated with practical EV charging. The present prototype should therefore be interpreted as a scaled platform for validating the compensation mechanism, the nominal CC/CV transition, and the current-division correction, rather than as a full-power automotive charger. Scaling the method to kilowatt-level operation requires re-selection of semiconductor devices, litz-wire conductors, compensation components, magnetic materials, insulation clearances, thermal management, electromagnetic shielding, and protection circuits. These implementation issues are outside the experimental scope of the present prototype.

4.6. Comparison with Existing Three-Coil CC/CV Methods

Table 5. Comparison with representative three-coil and high-order compensated charging methods.

Ref.	CC/CV Realization	Receiver-Side Network	Frequency/Switching	Power	Efficiency	Coupling/Offset Evidence	Main Distinction	Scope
[3]	S/S/P at two ZPA points	Single P capacitor	Two operating frequencies	0.258 kW	91.8–96.0%	Parameter sensitivity	Compact receiver and dual-frequency CC/CV	Experimental
[4]	LCC-S-S/S-S-S switching	Series-compensated receiver	Topology switching	0.576 kW	92.4–94.9%	Efficiency > 90% under reported misalignment	Gain design independent of magnetic coupler	Experimental
[8]	NP-PPC with active mode control	Multiple rectifier/dc–dc paths	Active partial-power conversion	1.200 kW	94.156%	EV-oriented control validation	High-power partial-power regulation	Experimental
[9]	Two-coil CC/three-coil CV	Series compensation	Fixed frequency; one relay	1.0 kW	94.4%	Nominal alignment	Closest reconfigurable baseline	Experimental
[10]	CV-stage enhancement	S-S-LCLCC	Fixed-frequency compensated operation	0.420 kW	89.6%	Misalignment and light-load comparison	Improved CV light-load efficiency stiffness	Experimental
[13]	Hybrid CV/CC receiver	Series/LCC receiver	Two additional receiver switches	0.384 kW	90.8%	20 cm air gap	Receiver-side topology switching	Experimental
This work	Two-coil CC/three-coil CV	Modified LCC-type receiver	Fixed frequency; relay reconfiguration	0.341 kW	93.120% at 35 Ω	Calculated α-sensitivity; no new offset test	Cf current-division model and C2 correction	Scaled nominal-point experiment

compares the present work with representative three-coil or high-order compensated charging methods. The comparison is intended to clarify the technical position of the proposed analysis rather than to claim universal superiority. In particular, Ref. [9] is the closest operating baseline because it already establishes fixed-frequency two-coil CC and three-coil CV reconfiguration using a series-compensated receiver. The present work adds a modified LCC-type receiver and addresses the resulting C_f current division and C₂ correction.

Compared with Ref. [9], the modified receiver network introduces additional passive components and a higher receiver-coil current, so the present design should not be described as simpler or intrinsically more efficient. Its contribution is instead the explicit current-division model and the analytical correction of C₂. Compared with the high-order or actively regulated methods in Refs. [4,8,10,13], the present scaled study does not demonstrate higher power, broader regulation, or stronger misalignment tolerance. The comparison therefore supports a narrowly defined modeling and parameter-design contribution.

4.7. Limitations and Practical Implications

The experimental evidence of this study is limited to the nominal 35 Ω operating point retained from the original prototype. The load-sweep, receiver-coupling, current-division, and cross-coupling results added during revision are analytical or calculated results and are explicitly labeled as such. No synthetic data are presented as measurements.

Several limitations remain. First, physical misalignment and air-gap variation were not newly measured, and the normalized coupling factor has not been mapped to a displacement distance. Second, the exact voltage-probe, current-probe, and calibration records were not available for a traceable instrument-level uncertainty analysis. Third, the prototype used a resistive load to emulate the equivalent battery resistance; real-battery charging trajectory, charging time, voltage evolution, and protection behavior were not evaluated. Fourth, quantitative thermal and electromagnetic-field compliance tests were not performed. Finally, the sign of M₁₃ was not calibrated from the original coil dot convention. These limitations define the next experimental stage: rebuilding a traceable platform for multi-load, physical-offset, device-level soft-switching, temperature-rise, stray-field, and battery-charging verification.

Despite these limitations, the analytical results support the following engineering conclusion: when the modified receiver-side LCC-type network is used in the fixed-frequency reconfigurable three-coil system, the shunt-capacitor current cannot be neglected. Strict L₂–C₂ resonance produces a systematic calculated CC error and shifts the calculated transition load, whereas the current-division-corrected value C₂ = 21.950 nF restores the intended output characteristics under the nominal design approximation. The available prototype measurements confirm only the nominal operating target of the corrected design.

5. Conclusions

This paper investigated a fixed-frequency reconfigurable three-coil wireless power transfer system with a modified receiver-side LCC-type network for constant-current/constant-voltage battery charging. Based on the established two-coil CC and three-coil CV operating mechanism, a current-division-aware fundamental-harmonic model was developed to distinguish the receiver-coil current from the equivalent rectifier-load current. The analysis shows that the shunt capacitor C_f introduces an additional reactive-current path and therefore changes the compensation condition of the L₂–C₂ branch.

For the investigated 85 kHz system, the strict resonant value of C₂ is 22.825 nF, whereas the corrected value obtained from the analytical condition is 21.950 nF. Over the calculated CC load range of 15–35 Ω, the correction reduces the maximum output-current error from 8.948% to 0.031% and shifts the calculated CC/CV transition resistance from 39.25 Ω to 34.97 Ω. The calculated input-impedance angles remain close to zero in both operating modes. At R_L = 35 Ω, the prototype measurements provide an output voltage of 109.14 V, an output current of 3.121 A, an output power of 0.341 kW, and a dc–dc efficiency of 93.120%. These measurements confirm the nominal operating target of the corrected design but do not directly verify the full calculated improvement over the load range.

The modified receiver network provides an additional degree of freedom for receiver-side impedance shaping, but it also increases the receiver-coil current stress. Future work will include repeated multi-load and physical-misalignment experiments, switching-level soft-switching verification, thermal and leakage-field evaluation, real-battery charging tests, and higher-power implementation.