Wireless power transfer (WPT) based on the magnetic resonant coupling scheme has recently become widely used in many applications, such as electric vehicles, robotics, and medical devices [1, 2]. The WPT using magnetic resonant coupling can improve the power transfer efficiency (PTE) or extend the service range compared with inductively coupled systems. To improve the PTE, researchers have used various methods, such as impedance matching [3, 4], adjusting the coupling among multiple transmitters (TXs) or multiple receivers (RXs) [5], and applying the optimum condition of the values of phase difference and amplitude ratio between two adjacent TXs [6], among others.

Adding more TXs and RXs overcomplicates the circuit and renders it difficult to apply in practice. The most widely considered configuration of the WPT system is using two parallel coaxial circular coils: a single TX coil and a single RX coil. The PTE is known to decrease sharply when angular misalignment or lateral misalignment occurs [1, 7–9]. A WPT system with two parallel coaxial coils can achieve maximum PTE compared with other configurations [10]. However, the experimental result in [11] showed that the maximum transmission efficiency is reached when two coaxially located circular coils have an angle difference of 45°. The PTE can be improved by changing the tilt angle of coils for the given position and the radii of coils.

In this study, we analyze the effect of the tilt angle of the coil on the PTE of the WPT system. A mutual inductance between two coils with the tilt angle located at an arbitrary position in a 3D coordinate is extracted using numerical analysis through the Neumann equation. Using the extracted mutual inductance and the equivalent circuit model of the WPT system, the PTE depending on the tilt angle is extracted. We propose an optimal tilt angle to maximize the efficiency of the WPT system. The angle at which the efficiency is maximized depends on the radii of the two coils and their relative position.

This paper is organized as follows. Section II describes the equivalent circuit model of the WPT system based on magnetically resonant coupling, and the coupling coefficient analysis considering angular misalignment is introduced. Section III shows the simulated and experimental efficiencies of the WPT system for three different RX coil radii, and the efficiency of the WPT system is analyzed when angular misalignment or lateral misalignment occurs. We also discuss the relationship among the radii of RX, the tilt angle of the coil, and the PTE. Section IV presents the findings.

An equivalent circuit model of a magnetic resonant coupled WPT system with a single TX coil and a single RX coil is shown in Fig. 1.

In Fig. 1, k is the coupling coefficient between a TX coil and an RX coil; V is the phasor of the voltage source in the TX; L_TX and L_RX are the self-inductances of the two coils; and R_P1 and R_P2 are the parasitic resistances of the two coils. C₁ and C₂ are the capacitances of TX and RX, respectively, to satisfy ω0=1/LTXC1=1/LRXC2, where ω₀ is the resonant frequency of the WPT system. R_S is the source resistance, and R_L is the load resistance.

The circuit model can be analyzed by Kirchhoff’s voltage law equations for two coils as follows:

where I_TX and I_RX are the phasors of the currents in the TX and RX, respectively.

The PTE is defined as the ratio between the power delivered to a load and the power supplied by the source as follows:

Eq. (3) can be rewritten as η = Ak² / (B + Ck²), where A=ω²L_TXL_RXR_L, B=R_P1(R_P2 +R_L)², C=ω² L_TXL_RX (R_P2 +R_L), and k ∈[0,1]. The derivative of η for k is as follows:

By setting Eq. (4) to 0, k will be 0. Therefore, k=0 is an extreme point of function η. When k∈[0,1], η′ is greater than zero, η is monotonically increasing in this range, and η increases when k increases.

To calculate the coupling coefficient between two coils, we consider the two coils located in the 3D space as shown in Fig. 2.

The coupling coefficient k is acquired using

where M is the mutual inductance between the two coils.

The mutual inductance M can be calculated using the Neumann formula [12]:

where N_TX and N_RX are the number of TX and RX coil turns, respectively, and μ₀ is the magnetic permeability of free space. In Fig. 2, r₁ represents the radius of TX, and r₂ is the radius of RX. The centers of the TX coil and RX coil are O₁(0, 0, 0) and O₂(0, y, d), respectively. The horizontal distance is represented by y, and d is the vertical distance from the center of the TX coil to the center of the RX coil. The point P₁ on the TX coil is (x_TX, y_TX, z_TX), and the point P₂ on the RX coil is (x_RX, y_RX, z_RX). The dl_TX and dl_RX are counter-clockwise directional differential length change vectors for the TX and RX coils, respectively. R is the distance between P₁ and P₂ as follows:

The parametric equations of the TX coil and the RX coil are described using φ and ϕ (0 ≤φ,ϕ ≤ 2π), respectively. When two coils have no angular misalignment, the parametric equations of the two coils can be represented as follows:

In Fig. 2, two coils are non-coaxial if y ≠0. The normal direction of the RX coil plane is z′, and the normal direction of the TX coil plane is z. The angle θ represents the tilt angle between the two coils, and it is defined as the angle between the normal vector of plane z and the normal vector of plane z′. On the basis of the TX coil plane, we build a Cartesian coordinate system xyz. According to the xyz coordinate system, the x′y′z′ coordinate system can be established by drawing x′ parallel to x in the RX coil plane and drawing y′ perpendicular to x′z′. The rotation matrix, which performs the rotation relationship between xyz and x′y′z′, can be represented as follows:

Therefore, the P₂ of the RX coil considering the tilt angle can be represented using Eqs. (9) and (10) as follows:

The dl_TX and dl_RX for P₁ in Eq. (8) and P₂ in Eq. (11), respectively, are as follows:

And

where x, y, z are the base vectors in the xyz coordinate. Substituting Eqs. (7), (8), (11), (12), and (13) into Eq. (6), the coupling coefficient between the TX coil and the RX coil can be calculated. If the centers and the radii of the two coils are given, then the coupling coefficient is a function of the tilt angle as follows:

where α1=NTXNRXμ04πLTXLRX, α₂ = r₁r₂ sinφ sinϕ, α₃ = r₁r₂ cosφcosϕ, α4=r12+r22+y2+d2-2r1y sin φ-2α3, α₅ =2r₂ sinϕy–2α₂, and α₆ = 2r₂ dsinϕ.

In Eqs. (3) and (4), we show that η increases when k increases. To achieve an optimal tilt angle θ_opt, which gives the maximum PTE for the given center positions and radii of the two coils, we set θ_opt as a value that satisfies ∂k(θ ) / ∂θ = 0 and ∂²k(θ ) / ∂θ ² < 0.

To verify the above analysis and to determine the maximum transmission efficiency, we implement a WPT system. The parameters of the implemented WPT system are listed in Table 1. We execute simulations and experiments for various cases. Three different receivers are considered; cases I, II, and III have different radii of the RX coils. The results show that the inductances and resistances of these three cases differ. Different capacitors are used to ensure that the LC resonance for each case occurs at the same resonant frequency as described in Table 1.

For the given vertical distance d, we simulate the PTE using Eqs. (3) and (14) by changing the horizontal distance (y) and the tilt angle of the RX coil (θ ). Fig. 3 shows the numerical extracted coupling coefficients for cases I, II, and III when O₂(0,0.05,0.15), respectively. Fig. 3 clearly shows that the coupling coefficient is related to the tilt angle of the RX coil.

To examine the effect of the tilt angle on the coupling coefficient for the coaxially parallel circular TX and RX coils, we extract the coupling coefficients using Eq. (14) versus the radii of the RX coil, r₂. In this simulation, we change the tilt angle of the RX coil in the range of −90° to 90° and change r₂ from 0 to 0.15 m. The simulated results are shown in Fig. 4.

The results show that the optimal tilt angle is not always zero and depends on the radius of the RX coil. The simulation results indicate that the angle between the TX and the RX for the maximum efficiency is 0° when the radius of the RX coil is less than 0.1 m for the case with d = 0.15 m. When the radius is larger than 0.1 m for this case, the angle for the maximum efficiency is non-zero. We plot the change of the optimal tilt angle for the various radii of the RX coil for the coaxially located TX and RX coils in Fig. 5.

In addition, we analyze the PTE for various cases with the horizontal offset between the TX and RX coils. We extract the PTE by rotating the RX coil in the range of −90° to 90° and by changing the horizontal distance, y, from 0 to 0.15 m along the y-axis positive direction. The simulated results are shown in Figs. 6(a), 7(a), and 8(a) for cases I, II, and III, respectively. The results show that the PTE strongly depends on the tilt angle of the RX coil. The condition for the maximum PTE and its value for each case are summarized in Table 2.

The simulated maximum PTE is denoted by η_max, which us obtained by utilizing the optimal values of the angle between the two coils and the horizontal distance. When the radius of the RX coil is small (r₂ = 0.075 m), such as in case I, the RX coil, which is coaxially aligned with the TX coil with no tilt angle, gives the maximum PTE. However, as the radius of the RX increases (r₂ = 0.1 m, r₂ = 0.15 m) in cases II and III, the maximum PTE is achieved by using the non-coaxially located RX coil with the non-zero tilt angle.

To verify the simulated results, we measure the PTE for the WPT system listed in Table 1 by changing the tilt angle for the three different horizontal offset positions (case A, y = 0 m; case B, y = 0.05 m; and case C, y = 0.1 m). The measured PTE is acquired using the following equation:

where η_P is the ratio of the real power dissipated in the load, P_RX, to the power provided by the source driving the TX coil, P_RX; V_TX, I_TX, V_L, and I_L are the voltage and current amplitudes of the source and the load; and γ_TX and γ_L are the phase difference between the voltage and the current signals in the source and the load. The measured results are plotted in Figs. 6(b), 7(b), and 8(b) for cases I, II, and III, respectively. Fig. 6(b) shows the simulated transmission efficiency with various tilt angles and a number of special horizontal distances from Fig. 6(a). The experimental results are presented in Fig. 6(b). Fig. 7 shows the simulated and experimental results when the radius of RX is 0.1 m. In Fig. 7(b), when the position of the RX coil is 0.05 m in the positive y-axis and the angle between the RX coil and the TX coil is −34.7667°, the maximum transmission efficiency is obtained. The simulation and experimental results are given in Fig. 8, where the radius of RX is the same as that of TX. In Fig. 8(b), the bifurcation phenomenon occurs when the RX coil is 0.1 m or 0.15 m in the positive y-axis. In this study, the bifurcation phenomenon is not considered. Fig. 8(b) shows that when the radii of the two coils are equal, a special angle and a special horizontal distance are given to ensure the maximum transmission efficiency. In this experiment, the maximum transmission efficiency is reached when the angle between TX and RX is about −56.4247° and the RX coil is located at 0.05 m in the positive y-axis.

The measured results are compared with those of the simulated ones. As the horizontal offset increases, the tilt angle of the RX coil for the maximum PTE also increases. The simulated PTEs agree well with the measured results as shown in Figs. 6(b), 7(b), and 8(b). The results show that the predicted maximum tilt angle for each case is accurate. The simulated and measured values of PTE for each case are presented in Table 3. Table 4 shows the accuracy between the simulated results and the experimental results using root mean square errors for the WPT system with one TX and one RX. As shown in Table 4, the experimental results agree with the measured ones.

We analyze the effect of the tilt angle of the coil on the PTE of the WPT system. Based on our analysis results, the PTE of a WPT with two circular coils can be optimized by changing the tilt angle, relative position, and radii of coils. To archive the optimized PTE, we propose a method to calculate the mutual inductance between the two coils with the tilt angle located at an arbitrary position in a 3D coordinate through the Neumann equation. Using the extracted mutual inductance and the equivalent circuit model of the WPT system, the coupling coefficient and the PTE depending on the tilt angle are extracted. We propose an analytical scheme to extract the optimal tilt angle that maximizes efficiency. To demonstrate our proposed theoretical method, we design experiments using a WPT with two circular coils. By changing the position, radius, and the tilt angle of the RX coils, we compare the transmission efficiency of the different cases. The suggested analysis effectively predicts the optimum tilt angle of the RX coil and the PTE for the various cases.