### I. Introduction

### II. Efficiency Analysis of the WPT Tilted Coil

*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*

*and*

_{TX}*L*

*are the self-inductances of the two coils; and*

_{RX}*R*

_{P}_{1}and

*R*

_{P}_{2}are the parasitic resistances of the two coils.

*C*

_{1}and

*C*

_{2}are the capacitances of TX and RX, respectively, to satisfy

*ω*

_{0}is the resonant frequency of the WPT system.

*R*

*is the source resistance, and*

_{S}*R*

*is the load resistance.*

_{L}*I*

*and*

_{TX}*I*

*are the phasors of the currents in the TX and RX, respectively.*

_{RX}*η*=

*Ak*

^{2}/ (

*B*+

*Ck*

^{2}), where

*A*=

*ω*

^{2}

*L*

_{TX}*L*

_{RX}*R*

*,*

_{L}*B*=

*R*

_{P}_{1}(

*R*

_{P}_{2}+

*R*

*)*

_{L}^{2},

*C*=

*ω*

^{2}

*L*

_{TX}*L*

*(*

_{RX}*R*

_{P}_{2}+

*R*

*), and*

_{L}*k*∈[0,1]. The derivative of

*η*for

*k*is as follows:

*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.

*k*is acquired using

*M*is the mutual inductance between the two coils.

*M*can be calculated using the Neumann formula [12]:

*N*

*and*

_{TX}*N*

*are the number of TX and RX coil turns, respectively, and*

_{RX}*μ*

_{0}is the magnetic permeability of free space. In Fig. 2,

*r*

_{1}represents the radius of TX, and

*r*

_{2}is the radius of RX. The centers of the TX coil and RX coil are

*O*

_{1}(0, 0, 0) and

*O*

_{2}(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

*P*

_{1}on the TX coil is (

*x*

*,*

_{TX}*y*

*,*

_{TX}*z*

*), and the point*

_{TX}*P*

_{2}on the RX coil is (

*x*

*,*

_{RX}*y*

*,*

_{RX}*z*

*). The*

_{RX}*dl*

*and*

_{TX}*dl*

*are counter-clockwise directional differential length change vectors for the TX and RX coils, respectively.*

_{RX}*R*is the distance between

*P*

_{1}and

*P*

_{2}as follows:

*φ*and

*ϕ*(0 ≤

*φ*,

*ϕ*≤ 2

*π*), respectively. When two coils have no angular misalignment, the parametric equations of the two coils can be represented as follows:

*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:

*P*

_{2}of the RX coil considering the tilt angle can be represented using Eqs. (9) and (10) as follows:

*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:

*α*

_{2}=

*r*

_{1}

*r*

_{2}sin

*φ*sin

*ϕ*,

*α*

_{3}=

*r*

_{1}

*r*

_{2}cos

*φ*cos

*ϕ*,

*α*

_{5}=2

*r*

_{2}sin

*ϕ*

*y*–2

*α*

_{2}, and

*α*

_{6}= 2

*r*

_{2}

*d*sin

*ϕ*.

### III. Simulated and Experimental Results

**.**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.

*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

*O*

_{2}(0,0.05,0.15), respectively. Fig. 3 clearly shows that the coupling coefficient is related to the tilt angle of the RX coil.

*r*

_{2}. In this simulation, we change the tilt angle of the RX coil in the range of −90° to 90° and change

*r*

_{2}from 0 to 0.15 m. The simulated results are shown in Fig. 4.

*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.

*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.

*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 (

*r*

_{2}= 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 (

*r*

_{2}= 0.1 m,

*r*

_{2}= 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.

*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:

*η*

*is the ratio of the real power dissipated in the load,*

_{P}*P*

*, to the power provided by the source driving the TX coil,*

_{RX}*P*

*;*

_{RX}*V*

*,*

_{TX}*I*

*,*

_{TX}*V*

*, and*

_{L}*I*

*are the voltage and current amplitudes of the source and the load; and*

_{L}*γ*

*and*

_{TX}*γ*

*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*

_{L}*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.