Modeling and decoupling analysis for an integrated coupler and its non-compensated wireless power transfer system

Combining advanced modeling and analysis methods, wireless power transfer (WPT) technology effectively addresses the issue of unsafe and inflexible access associated with conventional charging methods^[1−5]. To increase the efficiency and power transfer capability of the WPT system, the existing research generally adopts two basic methods: improving the compensation structure and its parameter design method^[6,7]; optimize the electromagnetic coupling structure^[8,9]. The first method can raise the system output power, realize constant voltage or current output, suppress high-order harmonics, and reduce the reactive power component of the input. The second method has advantages in increasing the efficiency and power density, improving the misalignment ability, maintaining the system security, and reducing the system cost.

Four typical compensation topologies (S/S, S/P, P/S, and P/P) were utilized to improve the performance of inductive power transfer (IPT) systems, but they are prone to load sensitivity issues^[10]. Regarding capacitive power transfer (CPT) systems, the coupler can be equivalent to two series coupling capacitors or six cross-coupling capacitors^[11,12]. In centimeter-range power transmission applications, the method of using a large inductance for series reactive power compensation comes at the expense of system security and environmental adaptability^[13]. In response to the challenges faced by IPT and CPT systems, a range of high-order compensation topologies, such as LCL (inductor capacitor inductor), LCC (inductor capacitor capacitor), CLC (capacitor inductor capacitor), and LCLC (inductor capacitor inductor capacitor), have been proposed. This paper presents a novel parameter tuning method based on an LCL compensation topology. The proposed approach provides enhanced flexibility in the design of IPT systems, along with improved suppression of higher-order harmonics and a reduction in coil current^[14]. A dual-side LCC compensation topology along with its resonant scheme has been designed to ensure that the resonant frequency of the IPT system remains immune to variations in coupling coefficient and load, thereby maintaining stable and predictable operation^[15]. A π-CLC compensation structure at the transmitter and T-CLC compensation structure at the receiver are adapted in the CPT system, which achieves constant current output when switching between multiple receivers^[16]. To minimize the voltage stress imposed on the coupler to enable high-efficiency power delivery across long distances, a CPT system with double-sided LCLC compensation structure is put forward^[17]. To continuously improve WPT system performance, scientists have carried out in-depth studies on electromagnetic coupling structures. A significant amount of focus has been placed on the development of a hybrid inductive and capacitive WPT (HWPT) system^[18]. A novel HWPT system is presented that achieves a favorable trade-off between enhanced efficiency and a compact coupler design^[19]. In addition, an inductive and capacitive integrated coupler, made up of bent aluminum plates, is introduced^[20], offering a novel concept and approach for wireless charging. A 10 kW HWPT system exhibits a significant performance advantage, outperforming both previously published IPT and CPT systems across a range of misalignment conditions^[21]. However, all of the above couplers have to be compensated by high-order network. In general, the magnetic-field coupler is typically constructed from Litz wire, whereas the electric-field coupler is composed of metal plates. To enhance performance, Litz wire, metal plates, and high-order compensation topologies are often incorporated. However, this approach comes at the expense of greater volume, weight, and cost, creating a critical barrier for WPT technology's progress and market adoption. In conclusion, the majority of existing research has focused on enhancing the output power, the efficiency, and the transfer distance of WPT systems, while neglecting the concomitant increases in volume, weight, and cost.

Aimed at resolving the problems outlined previously, this study puts forward a novel integrated copper-foil electromagnetic coupler and a comprehensive methodology for its modeling and decoupling. Fabricated from copper foil, the four planar square spiral coils that constitute the electromagnetic poles are designed with the objective of alleviating the skin effect, consequently minimizing ohmic losses. Cross-coupling is generated among the four copper-foil coils; consequently, mutual inductances and cross-coupling capacitors are established. Compared with the previous research work, the proposed hybrid electromagnetic coupler is very lightweight and low cost, and the WPT system is very simple and practical for electric vehicle. The remainder of this paper is organized into the following sections. This paper first introduces the coupler structure and its decoupled model. The equivalent circuit of the coupler is analyzed and simplified, and the couplers' inherent electrical parameters are designed to make it self-resonance. Then the working conditions of both load-independent constant current (CC) or constant voltage (CV) output and input ZPA are obtained. In the following, the structure of the coupling electromagnetic pole is optimally designed based on Maxwell to balance the electric field and the magnetic field, and an efficient and portable electromagnetic coupler is constructed. Simulation and experimental results validating the proposed coupler and WPT system are discussed in verification part, concluding with the findings in the end.

As shown in Fig. 1, the electromagnetic poles of the proposed coupler take the form of square spiral windings fabricated from copper foil. By leveraging the previously ignored parasitic capacitance of the coil, a significant improvement in power transmission performance can be achieved. Under the influence of both electric and magnetic fields, this coupler inherently integrates a combination of self-inductance, mutual inductance, and cross-coupling capacitor. The three-dimensional model of the copper-foil electromagnetic coupler is presented in Fig. 1a. Its key components are four planar square spiral electromagnetic poles and an insulating dielectric layer. Figure 1b depicts the front view of the copper-foil electromagnetic coupler, where the four electromagnetic poles and insulating dielectrics are stacked. Electromagnetic pole P₁, dielectric D₁, electromagnetic pole P₂, electromagnetic pole P₄, dielectric D₂, and electromagnetic pole P₃ are lined successively from bottom to top. l₁ is the side length of P₁ and P₃, while l₂ is the side length of P₂ and P₄. d_t and d_s are the transfer distance and the distance between the same-side electromagnetic poles (also the thickness of the dielectric), respectively.

Figure 1. 

Basic structure of the copper-foil electromagnetic coupler. (a) 3D graph of the coupler. (b) Front view^[22].

System circuit topology and its equivalent simplification

With a view to structural simplification, this WPT system deliberately omits any additional compensation components. The copper-foil electromagnetic coupler utilizes its inherent capacitance and inductance characteristics to achieve self-resonant. A schematic illustration of the non-compensated circuit configuration is achieved in Fig. 2, where leakage field reduction is achieved by connecting the inner electromagnetic pole P₂ to the inverter's high-potential output terminal and the outer electromagnetic pole P₁ to the low-potential output terminal^[23]. For simplicity, the circuit characteristics are analyzed employing the Fundamental Harmonic Approximation (FHA), with ohmic losses in the coupler omitted. Due to the equivalence between the six-capacitor cross-coupling and three-capacitor π-type models^[24], the circuit from Fig. 2 can be simplified as depicted in Fig. 3, where $ {U}_{\text{in}}\left({U}_{\text{in}}={U}_{\text{dc}}\cdot \text{4/}\pi \right) $ and $ {R}_{\text{eq}}\left({R}_{\text{eq}}={R}_{L}\cdot 8/{\pi }^{2}\right) $ are defined as the inverter's fundamental output voltage component and the equivalent AC load resistance, respectively. The parameters in Figs 2 and 3 satisfy the following equivalent transformation relationship:

Figure 2. 

Circuit topology of WPT system without compensation.

Figure 3. 

System π-type equivalent model.

The electric field coupling coefficient is expressed as:

According to the reciprocity theorem, the π-type capacitor network composed of $ {C}_{\text{1}} $, $ {C}_{\text{2}} $, and $ {C}_{M} $ can be transformed into a T-type capacitor network composed of C_A, C_B, and C_C. Furthermore, the multi-coupled inductor circuit can be equivalently represented as a controlled-source circuit, wherein the mutual inductance voltages are replaced by current-controlled voltage sources (CCVS). Figure 4 shows the equivalent circuit. The capacitors satisfy the following equations:

Figure 4. 

Controlled source equivalent circuit.

Based on Kirchhoff's law, the input and output voltage can be expressed as:

By substituting I₁ − I₃ for I₂, the input voltage can be simplified as:

By substituting I₂ + I₃ for I₁, the output voltage can be simplified as:

where,

In this paper, the magnetic-field coupling coefficient is conceptualized as:

which is used to describe the magnetic coupling degree between coils.

Circuit decoupled and parameter self-compensation

According to Eqs (5) and (6) in system circuit topology and its equivalent simplification, the controlled source circuit shown in Fig. 4 can be decoupled as a T-type model shown in Fig. 5. All inductances and capacitors are independent of each other. The simplified circuit can be divided into three impedance stages.

Figure 5. 

System T-type decoupled model.

Based on the AC impedance method, the impedances at each stage depicted in Fig. 5 can be expressed as:

For practical implementations, the electromagnetic coupler is often deliberately designed to be symmetric, specifically featuring identical dimensions between pole pairs P₁ and P₃, or alternatively, P₂ and P₄. Furthermore, the distance between P₁ and P₂ is equal to the distance between P₃ and P₄. Therefore, the circuit parameters in Fig. 1b have the following relationship:

According to Eqs (1), (3), and (6), the parameter relationships can be simplified as:

According to Eqs (3) and (11), the electric-field coupling coefficient k_CPT, and magnetic-field coupling coefficient k_IPT shown in Eqs (2) and (8) can be re-expressed as:

According to Eqs (9), (11), and (12), if the system works in the input ZPA condition, the imaginary part of the system input impedance should be equal to zero, so the parameters must satisfy one of the equations as follows:

or

Considering the above two conditions, some system parameters can be solved, as shown in Table 1，where A, B, and C are expressed as follows:

Since the current I_out is independent of R_eq under the first ZPA condition, the system operates in a CC output mode. For the second ZPA condition, the numerator and denominator of the output voltage U_out are conjugate complex numbers. The modulus of $ {\dot{U}}_{{\mathrm{out}}} $ can be expressed as:

It means the system works in constant voltage (CV) output mode, while the phase of the output voltage is opposite to the input voltage.

Employing finite element analysis (FEA) in ANSYS Maxwell, the characteristics of the coupler were analyzed with various dimensions and its spatial structure optimized to improve system performance, with the planar rectangular spiral electromagnetic pole's top view provided in Fig. 6. The self-inductance of the planar square spiral coil is calculated based on the equation as follows:

Figure 6. 

Top view of the single coil.

where, μ₀, N, D_avg, and ρ are the vacuum permeability, turns of the coil, average value of inner and outer diameter of the coil $ ({D}_{\text{avg}}=({D}_{\text{out}}+ {D}_{\text{in}})\text{/2}) $, and compression ratio of the coil $ \left(\rho \text{=}\left({D}_{\text{out}}-{D}_{\text{in}}\right)\text{/}\left({D}_{\text{out}}+{D}_{\text{in}}\right)\right) $, respectively. The values of c₁−c₄ are determined by coil layout and shape. When the coil is square, c₁ = 1.27, c₂ = 2.07, c₃ = 0.18, and c₄ = 0.13.

In addition, the theoretical formula of a capacitor can be expressed as:

where, $ {\varepsilon }_{\text{0}} $ and $ {\varepsilon }_{\text{r}} $ are the permittivity of vacuum and the relative permittivity of dielectric. The coupling capacitor between any two electromagnetic poles is positively correlated with their copper area, while negatively correlated with the distance between them. In summary, the self-inductance, mutual inductance, and cross-coupling capacitor are mainly influenced by D_out, D_in, N, and w.

In practice, the outer diameter D_out of the electromagnetic pole is generally determined by the specific application occasion. In this section, the outer electromagnetic pole (P₁ and P₃) with the size of 500 * 500 mm (D_out = 500 mm) is used for research. According to Eq. (16), when D_out is given, the self-inductance of the outer pole is mainly related to its width of copper foil w_out, number of turns N_out, and turns spacing s_out. In addition to the aforementioned parameters, the mutual inductances and cross-coupling capacitors are also governed by the width of the copper foil winding of the inner electromagnetic pole, the number of turns N_in, the turn spacing s_in, the transmission distance d_t, and the inter-pole distance on the same side d_s.

Influence of the poles' turns

To achieve larger cross-coupling capacitors, Polymethyl methacrylate (PMMA, relative permittivity 3.4) is employed as the dielectric medium separating the electromagnetic poles on each side. The system parameters, as defined in Table 2, are adopted for this investigation.

Further analysis reveals the changing pattern of input impedance Z_in with respect to N_in and R_eq, as illustrated in Fig. 7^[22]. It can be observed that Z_in is primarily determined by R_eq, but the variation with N_in is relatively small. When Req assumes a small value (10−100 Ω), Z_in becomes quite high, leading to a minimal amount of power picked up at the load end. When R_eq takes on a large value (10³−10⁴ Ω), Z_in ranges from 10 to 100 Ω; however, under CV operating mode, the current picked up by the load remains quite small.

Figure 7. 

Variation of the input impedance with N_in and R_eq^[22].

As derived from the preceding analysis, the first ZPA operating mode is exclusively suited for applications characterized by an exceptionally high equivalent load resistance and a minimal load current. Since most practical implementations feature load resistances below 100 Ω, this paper consequently examines the electromagnetic coupler's performance under the second ZPA condition for subsequent analysis and design.

With varying N_in, the self-resonance frequency of the coupler differs significantly, leading to distinct ground voltages of the four electromagnetic poles. The dependence of the ground-referenced voltage at P₃ and P₄ on N_in is examined under two distinct operating frequencies, f₁ and f₂, as depicted in Fig. 8^[22]. A marked disparity is observed, with U_P3 and U_P4 demonstrating a substantial increase when the system operates at frequency f₂ compared to frequency f₁. Therefore, the selection of f₁ as the system operating frequency serves to reduce the coupler's fringing electric field and improve overall safety. Subsequent sections are dedicated to the analysis and design of a WPT system targeting operation at the resonant frequency f = f₁.

Figure 8. 

Ground-referenced voltage curve of P₃ and P₄^[22]. (a) f = f₁. (b) f = f₂.

Based on the parameters listed in Table 2, the dependence of the self-inductances, mutual inductances, and cross-coupling capacitors of the four electromagnetic poles on N_in can be systematically investigated through Maxwell simulations. Then, the regularity of the magnetic-field coupling coefficient k_IPT and electric field coupling coefficient k_CPT changing with N_in is obtained as shown in Fig. 9. It can be seen that k_IPT is negative, which is related to the setting of the homonymous end of the coupling coil. The determination of the homonymous end is affected by the current direction in the coil, which is related to the winding mode of the coil.

Figure 9. 

Variation of magnetic-field and electric-field coupling coefficient.

Influence of the poles' width

With the parameters of the coupler listed in Table 3 held constant, the influence of the copper foil width on the system parameters is investigated. In order not to change the occupied area of the outer electromagnetic pole (15 turns, 500 * 500 mm), and the inner electromagnetic pole (nine turns, 330 * 330 mm), the sum of copper foil width and turns spacing in the table is fixed as 15 mm. If the copper foil width changes, the copper-foil turns spacing changes accordingly. Through parametric scanning analysis on Maxwell, The functional dependence of both the magnetic and electric field coupling coefficients on the geometric parameters (copper foil widths w_in/w_out or turn spacings s_in/s_out) is illustrated in the accompanying contour diagrams, as shown in Fig. 10. A clear correlation is observed whereby the absolute value of k_IPT correlates positively with both w_in and w_out, whereas k_CPT increases with larger w_in but decreases with larger w_out. Moreover, Fig. 11 shows the regularity of the first resonant frequency f₁ varying with w_in and w_out in the case of R_eq = 40 Ω. It is known that the self-inductances, mutual inductances, and cross-coupling capacitors of the electromagnetic poles increase with the increase of w_in and w_out, so the system resonance frequency decreases, as shown in Fig. 11.

Figure 10. 

The variation of coupling coefficient with w_in and w_out (or s_in and s_out).

Figure 11. 

The variation of the system resonance frequency f₁ with w_in and w_out (or s_in and s_out).

Influence of same-side spacing and transfer distance

The transfer distance d_t and the same-side electromagnetic poles spacing are set as variables, and the other parameters shown in Table 4 are taken as invariants, then the three-dimensional curved surface of k_IPT and k_CPT varying with d_t and d_s can be obtained, as shown in Figs 12 and 13. With the increase of d_s, k_CPT, and the absolute value of k_IPT increase gradually. However, as d_t increases, k_CPT and the absolute value of k_IPT decrease gradually. Generally, the larger the coupling coefficient is, the more conducive to the power transmission. Therefore, it is crucial to carefully balance the relationship between the coupling coefficient and the volume of the coupler when designing the coupler for practical application, which involves selecting appropriate values for d_s and d_t.

Figure 12. 

The variation of k_IPT with d_s and d_t.

Figure 13. 

The variation of k_CPT with d_s and d_t.

Moreover, the changes in self-inductance, mutual inductance, and cross-coupling capacitance with respect to d_s and d_t can also be obtained, leading to the derivation of the relationship between f₁ and d_s, as well as d_t, as illustrated in Fig. 14. As d_s and d_t increase, the mutual inductances and cross-coupling capacitances decrease, so the system resonance frequency increases.

Figure 14. 

The variation of f₁ with d_s and d_t.

According to the previous literature and experience, the frequency is generally around 1 MHz, so the value of inductance and capacitance will be less affected by the surrounding environment. In addition, the magnetic-field and electric-field coupling coefficient should be balanced, since they determine the system power transmission capability. To systematically optimize the coupler's geometry, the relationship between its parameters and the circuit parameters is investigated. Therefore, the self-inductances, mutual inductances, and cross-coupling capacitors can be obtained by Maxwell. Based on the above analysis in analysis and design of coupler parameters and circuit parameters, if the geometric parameters of the coupler are designed as in Table 5, the operation frequency equals 1.056 MHz, and the magnetic-field and electric-field coupling coefficients are calculated as −0.318 and 0.024, respectively. All the circuit parameters are shown in Table 6. Based on the circuit topology in Fig. 2, and the parameters in Table 6, the system simulation is conducted on MATLAB/Simulink.

The phase synchronization observed between the inverter output voltage and current in Fig. 15 confirms that the system achieves a unity input power factor by satisfying the ZPA condition. A measured input current amplitude of 2.484 A for the system compares well to the calculated value of 2.5 A. With a variable AC load, the system output voltage is presented in Fig. 16, which shows an acceptable constant voltage output. When the AC load R_eq increases by 100% (from 40 to 80 Ω), the output voltage increases by 0.4%. When R_eq decreases by 100% (from 40 to 20 Ω), the output voltage decreases by 1.36%. Table 7 shows the simulation voltages to ground of P₁−P₄ and the currents through them. P₁ and P₂ are directly connected with the output port of the inverter, so their amplitude values of the voltage to ground are equal to the inverter output voltage. The currents through P₁ and P₂ are equal, as well as the currents through P₃ and P₄, which can be also deduced from the structure shown in Fig. 2. According to IEEE human safety standard^[25], the permissible levels for human exposure to electric and magnetic fields are defined by the operating frequency f_s in the following manner:

Figure 15. 

Simulation waveforms of inverter output voltage and current.

Figure 16. 

Simulation waveforms of system output voltage varying with R_eq.

where, E_RMS (V/m) and H_RMS (A/m) are the root mean square of the electric field strength and magnetic field strength, respectively. Taking the voltages and currents in Table 7 as the excitation sources, finite element analysis is conducted on ANSYS Maxwell, and the electric field and magnetic field around the coupler are obtained, as shown in Fig. 17a and b, where the safety distance is presented.

Figure 17. 

(a) Electric field distribution around the coupler. (b) Magnetic field distribution around the coupler.

Experimental results

Based on the WPT system in Fig. 2 and its equivalent circuit in Figs 3 and 5, the experimental prototype is built with the coupler's dimension parameters in Table 5, as shown in Fig. 18. The coupler is made up of four square planar spiral copper-foil coils (P₁−P₄) arranged vertically. The copper foil strip is spirally wounded on the surface of acrylic plates. To reduce losses caused by skin effect, we should choose the copper foil with a suitable thickness. So the skin depth of the copper foil should be calculated, which can be expressed as:

Figure 18. 

Experimental prototype of the WPT system.

where, μ is the copper magnetic permeability and equals 4π × 10⁻⁷ H/m; σ is the copper conductivity and equals 5.8 × 10⁷ S/m; f is the operating frequency, and the unit is Hz. When f = 1.056 MHz, δ = 64.3 μm, so the copper-foil thickness is approximately selected as 2δ for the experiment. The power transmitter is formed by P₁ and P₂ attached to D₁, while the power receiver is formed by P₃ and P₄ attached to D₂. The self-inductances, mutual inductances, and cross-coupling capacitors are measured using an Agilent E4980A LCR meter, with the results being closely aligned with the simulations in Table 6. The inverter incorporated Cree C2M0080120D SiC MOSFETs, with a TMS320F28335 DSP serving as the digital controller.

The experimental waveforms of the inverter output voltage, current, and AC load voltage are shown in Fig. 19 when the equivalent AC load resistance R_eq equals 40 Ω. The phase of the inverter output current I_inv lags the inverter output voltage U_inv slightly, which means the input impedance is weakly inductive and zero voltage switching (ZVS) condition is achieved. The phase difference between the inverter output voltage and current is defined as^[16]:

Figure 19. 

Experimental waveforms of the inverter output and the AC load voltage.

where, C_oss and t_D represent the parasitic output capacitance and the switching dead time, respectively. It is critical that is fully discharged prior to the transistor turn-on. Furthermore, the inverter output voltage and the AC load voltage are 180° out of phase, while the fundamental component of U_inv is equal to U_{out_ac}, thereby validating the simulation results presented in Fig. 15. Figure 20 shows the experimental waveforms of the DC load voltage U_{out_dc} when R_L changes from 100 to 50 Ω, then to 25 Ω. With R_L increasing or decreasing 100%, U_{out_dc} changes within 2%, achieving an approximately constant voltage output under a wide load range. The experiment data are measured by a power analyzer when R_L equals 50 Ω, as shown in Fig. 21. The input power and output power are 123.83 and 111.84 W, respectively, so the system efficiency is 90.32%. The loss distributions of the experimental prototype are shown in Fig. 22. It can be seen that the system losses are dominated by the four copper-foil electromagnetic poles.

Figure 20. 

Experimental waveforms of the DC load voltage when R_L changes (100→50→25 Ω).

Figure 21. 

The experiment results shown in power analyzer.

Figure 22. 

Loss distributions of the experimental prototype.

The misalignment tolerance of the system was evaluated, and the corresponding results are presented in Fig. 23. It can be seen from the overall trend of change that the output power drops with the X or Y misalignment increasing. The system performance degrades significantly at a misalignment of 150 mm, with both output power and efficiency plummeting to around 30% of those achieved in the perfectly aligned state. When the misalignment increases to about 250 mm, the output power and efficiency are very low. But they rise slightly when the misalignment continues to increase. That's because, the coupling coefficients k_CPT and |k_IPT| are very small when the misalignment is in the range of (250, 300 mm), and only a little electric power can be transmitted. With the continuous increase of the misalignment, k_CPT and |k_IPT| return to increase slightly, but both of them are going to end up near zero.

Figure 23. 

(a) Test output power of the misalignment ability for this WPT system. (b) Experimental efficiency of the misalignment ability for this WPT system.

A copper-foil electromagnetic coupler characterized by the integration of inductance and capacitance, which yields self-resonance, is proposed in this work. The absence of an additional compensation circuit in the proposed hybrid WPT system yields significant benefits, including reduced cost, lighter weight, and structural simplicity. The absence of additional compensation components and the mitigation of the skin effect collectively enhance the overall system efficiency. The relationship between the coupler's geometric parameters and system electrical parameters is analyzed and designed. A wide range of constant voltage output, high power factor, and ZVS conditions are achieved. An experimental device is set up with an output power of 111.84 W and an efficiency of 90.32%.

In addition, the misalignment ability of the prototype should be strengthened in the future. Because the resonant frequency is closely related to the geometric parameters of the coupler, some communication technology and control methods will be applied to this system to improve the misalignment ability. The proposed coupler and system can be well extended to the field of electric vehicles' wireless charging.