Omnidirectional Wireless Power Transfer System Based on Rotary Transmitting Coil for Household Appliances

Abstract

An omnidirectional magnetically coupled resonant wireless power transfer (WPT) system based on rotary transmitting coil is presented. The proposed scheme can ease the variations of the transfer efficiency and output power caused by the deviation of transfer direction, and improve the unbalanced power distribution phenomenon between the receivers, which are still not fully achieved in current WPT systems. The modified coupled-mode model is built first to describe the non-rotary multi-receiver WPT system. The analysis indicates that the transfer efficiency and output power of the system can be expressed as functions of the deviation angle between the transmitting coil and receiving coil, which has a non-negligible influence on the system performances. Then, the modified high order coupled-mode model containing time-varying parameters about the deviation angle is derived for the proposed omnidirectional WPT system. Theoretical analysis and simulated results indicate that this system can transfer power to multiple receivers around the transmitter synchronously and evenly, which is very suitable for wireless charging for household appliances indoors. The scheme feasibility and theoretical analysis are verified by experimental results.

1. Introduction

These years WPT field has witnessed a great progress in many fields, especially in low power applications (such as medical implants [1,2] and mobile devices [3,4]). Wireless charging for medium and high power applications (such as mobile robots [5] and electric vehicles [6]) has also been developing. Specially, Kurs et al. [7] proposed a new kind of WPT technology based on magnetic-resonance in 2007, which extended the transfer distance of WPT system to a large extent.

As WPT technology is getting mature, many products equipped with the function of wireless charging have matched the related commercial and technical standards, and widely entered into human life. For example, both the latest iPhone 8 and iPhone X have installed chips of wireless charging, which enhances their ability to hold a charge. It can be imagined that in the near future, most of household appliances and portable devices can be charged by wireless power, which will realize a true efficient, clean and green wireless home environment.

However, current WPT systems are limited by the directionality of the coils, that is, only when the transmitting coil and receiving coil are coaxial can the system transmission capacity get its peak [8]. Absolutely this feature quite blocks the valid area of the system. Besides, due to the limitation brought by the directionality, generally only two receivers can be charged by one transmitting coil simultaneously, which brings a great waste of resources. There is thereby an urgent need but it is still a significant challenge to build a system to enable the omnidirectional WPT that multiple receivers can be charged wirelessly at any direction.

Nevertheless, much work just focuses on point-to-point WPT systems to achieve high efficiency and long distance, only a few articles study the directional limitation of WPT. Hwang et al. [9] designed an autonomous coil alignment system for providing energy wirelessly to electric vehicles. Lin et al. [10,11,12] proposed 2-D and 3-D WPT systems to realize omnidirectional wireless charging with multiple transmitters. These schemes utilize load detecting technology, which needs complicated control method to achieve the aim, and only one receiver can pick up power with high efficiency. In [13], the authors designed a ball-joint system, which can also realize the function of omnidirectional WPT. However, this scheme only applies to the mechanical systems with single-receiver movable mechanical parts and is unsuitable for household environment. Chabalko et al. [14] used the natural electromagnetic modes of hollow-metallic structures to produce uniform magnetic fields which can simultaneously power multiple small receiver coils contained almost anywhere inside, where the charging area is seriously limited.

On the other hand, some work has been done to study the unbalanced power distribution phenomenon in multi-receiver WPT systems [15]. Li et al. [16] designed a WPT system based on dual transmitters and dual receivers, but their aim is to upgrade the system power capacity. Several wireless charging platforms for multi-receiver applications were proposed [17,18,19], while their planar structures limit the receiver working flexibility when being charged.

Therefore, this paper proposes an omnidirectional magnetically-coupled resonant WPT system based on rotary transmitting coil, where the rotary coil is driven by an electric motor. This scheme can realize omnidirectionality simply and transfer power to multiple receivers simultaneously, evenly and efficiently, which can indeed be applied in household multi-application environment. Firstly, a high order coupled-mode model is built to describe the single-receiver and multi-receiver WPT systems based on non-rotary transmitting coil. The transfer efficiency and output power can be expressed as functions of the deviation angle related to the transmitter and receiver. The analysis indicates that the deviation angle has a non-negligible impact on the system performances. Secondly, a high order coupled-mode model containing time-varying parameters about the deviation angle is established to describe the proposed omnidirectional single-receiver and multi-receiver WPT systems. Theoretical analysis and simulations imply that the proposed system not only can transfer power to multiple receivers synchronously, but also can improve the problem of the unbalanced power distribution between each receiver. Furthermore, it is verified that utilizing two rotary transmitting coils can eliminate the dead zones of power supply. The scheme feasibility and theoretical analysis have been identified by the experimental results.

2. Analysis of Wireless Power Transfer System with Non-Rotary Transmitting Coil

Here, the WPT system of non-rotary transmitting coil refers to the WPT system whose transmitting coil remains still in working state. To compare with the system of rotary coil, in this part, theory and simulation analyses of transfer characters of the non-rotary single-receiver system and multi-receiver system are presented below, respectively.

2.1. Analysis of Single-Receiver System

A simplified diagram of the single-receiver system of non-rotary coil is shown in Figure 1, in which the red long rectangle represents the coil of transmitter T_s, the black long rectangle represents the coil of receiver R_X1, d_s1 is the distance between the centers of the two coils, and θ is the angle between central axises of the two coils (hereinafter referred to as “deviation angle”).

Differently from the ideal cases in considerable articles where the deviation angle is assumed to be zero when the system is working, in most of actual situations, directional misregistration is very difficult to be avoided, which has an impact on system performances. Thus, to analyze the system more precisely and closely to the reality, influence of the deviation angle should be considered in the corresponding model. The modified equations based on coupled-mode theory to describe the system are deduced, as shown in (1).

$$\{\begin{array}{l}\frac{d{a}_s}{dt}=-\left(j{\omega }_s+{\gamma }_s\right)a_s-j\kappa \left(\theta \right)a_1+F{u}_s\left(t\right)\\ \frac{d{a}_1}{dt}=-\left(j{\omega }_1+{\gamma }_1\right)a_1-j\kappa \left(\theta \right)a_s\end{array}$$

(1)

Here, a_s,1 are field amplitudes of the resonant cavities of T_s and R_X1 such that $\left|a_{s,1}\right|$² represent the energies stored in the cavities. ω_s and ω₁ are intrinsic angular frequencies of the two cavities. Fu_s(t) is power supply item [5]. As shown in (2) and (3), γ_s and γ₁ are intrinsic loss rates of T_s and R_X1, where L_s and L₁ are self-inductances of the transmitting coil and receiving coil, R_s and R₁ are resistances of the transmitting coil and receiving coil, and R_L is resistance of the load. In (3), γ₁ consists of loss rates made by R₁ and R_L, which are defined as γ₁₀ and γ_L, respectively.

$${\gamma }_s=\frac{R_s}{2L_s}$$

(2)

$${\gamma }_1={\gamma }_L+{\gamma }_{10}=\frac{R_L}{2L_L}+\frac{R_1}{2L_1}$$

(3)

The expression of the modified coupling rate κ(θ) is shown in (4), and it is a function of deviation angle θ.

$$\kappa \left(\theta \right)=\frac{{\omega }_1}{2}\frac{M\left(\theta \right)}{\sqrt{L_s{L}_1}}=\frac{{\omega }_1}{2}\frac{M_{\max }\left|\cos \theta \right|}{\sqrt{L_s{L}_1}}$$

(4)

M_max is maximal mutual inductance between the two coils (when θ = 0) and can be calculated by (5) [20,21], in which μ is air permeability, N_s and r_s are the number of turns and radius of the transmitting coil, and N₁ and r₁ are the number of turns and radius of the receiving coil, respectively.

$$M_{\max }=\frac{\pi \mu N_s{N}_1{r}_s^2{r}_1^2}{2{\left(r_s^2+d_{s1}^2\right)}^{3/2}}$$

(5)

When both ω_s and ω₁ are equal to the angular frequency of power supply ω, the whole system meets the condition of resonance [22]. Solving the equations in (1), the analytic expressions of a_s and a₁ can be obtained, as shown in (6).

$$\{\begin{array}{l}{a}_s\left(\theta ,t\right)=\frac{{\gamma }_{1}F{u}_s\left(t\right)}{{\kappa }^2\left(\theta \right)+{\gamma }_s{\gamma }_1}\\ a_1\left(\theta ,t\right)=\frac{-jF\kappa \left(\theta \right)u_s\left(t\right)}{{\kappa }^2\left(\theta \right)+{\gamma }_s{\gamma }_1}\end{array}$$

(6)

The output power P_out and transfer efficiency η are determined by the power dissipated in the system resistance and load. Substituting (2)–(4) and (6) into the formulas of output power and transfer efficiency according to [5], (7) and (8) can be obtained.

$$\eta \left(\theta \right)=\frac{{\gamma }_L{\left|a_1\right|}^2}{{\gamma }_s{\left|a_s\right|}^2+{\gamma }_1{\left|a_1\right|}^2}=\frac{R_L{\omega }^2{M}_{\max }^2{\cos }^2\theta }{R_s{\left(R_1+R_L\right)}^2+\left(R_1+R_L\right){\omega }^2{M}_{\max }^2{\cos }^2\theta }$$

(7)

$$P_{out}\left(\theta ,t\right)=2{\gamma }_L{\left|a_1\right|}^2=\frac{8{\omega }^2{M}_{\max }^2{L}_s{L}_1{F}^2{u}_s^2\left(t\right){\cos }^2\theta }{{\left[{\omega }^2{M}_{\max }^2{\cos }^2\theta +R_s\left(R_1+R_L\right)\right]}^2}$$

(8)

(7) and (8) indicate that both output power and transfer efficiency can be expressed as functions of θ.

Figure 2 and Figure 3 present the curves of output power and transfer efficiency against time with different deviation angle values for single-receiver WPT system of non-rotary coil.

The parameters used for analysis are: self-inductance values of the coils are L_s = L₁ = 13 μH, compensation capacitance values are C_s = C₁ = 1.9 nF, resistances of the coils are R_s = R₁ = 0.5 Ω, the load resistance is R_L = 10 Ω, the maximal mutual inductance is M_max = 0.65 μH, the power supply item u_s(t) is 20cosωt and F = j/$\sqrt{2\mathit{Ls}}$ according to [22], the operating frequency f = 1 MHz (ω = ${6.28\times 10}^6$ rad/s), and the step size is h = ${1\times 10}^{-8}$ s.

The results of steady-state values in Figure 2 and Figure 3 are listed in

Table 1. Steady state solutions for single-receiver system.

$\mathit{\theta }$	$\left|\mathbf{c}\mathbf{o}\mathbf{s}\mathit{\theta }\right|$	Output Power	Transfer Efficiency
0	1.00	68.91 W	72.41%
$\mathsf{\pi }/10$	0.95	72.45 W	70.62%
$\mathsf{\pi }/5$	0.81	82.84 W	64.28%
$3\mathsf{\pi }/10$	0.59	93.92 W	49.80%
$2\mathsf{\pi }/5$	0.31	66.79 W	22.15%

. It reveals that the output power is influenced by the deviation angle. In addition, when the deviation angle gets larger, the output power does not always increase or decrease monotonously. On the other hand, a concomitant decrease in the transfer efficiency can be observed with the value of $\left|\cos \theta \right|$ getting smaller. All the analysis results imply that the deviation angle has a big impact on output power and transfer efficiency, that is, the system characters are quite influenced by different receiver positions.

2.2. Analysis of Multi-Receiver System

When there is more than one receiver, two assumptions are established to simplify the analysis:

• All the receivers are evenly placed around the transmitter. In other words, the linkages between centers of the receiving coils compose a regular polygon.
• All the receivers can be placed around the transmitter with their planes facing the center of transmitting coil.

Then a simplified diagram of multi-receiver system can be shown in Figure 4 (a five-receiver system is shown here to explain), in which d_si is the distance between centers of the transmitting coil and the ith receiving coil, d_im is the distance between centers of the the ith and the mth receiving coils, θ_si is the deviation angle between the transmitting coil and the ith receiving coil, and θ_im is the deviation angle between the ith and the mth receiving coils (i, m = 1, 2, …, n, I ≠ m, and n is the number of receivers).

As the number of receivers n has increased, (1) should be derived to (n + 1) orders. In a similar way, defining a_i as field amplitude of resonant cavity of the ith receiver, γ_i as the ith receiver intrinsic loss rate, ω_i as the ith receiver intrinsic angular frequency, κ_n.si(θ_si) as modified coupling rate between the transmitter and the ith receiver, and κ_n.im(θ_im) as modified coupling rate between the ith and the mth receivers, the system can be described by (9).

$$\{\begin{array}{l}\frac{d{a}_s}{dt}=-\left(j{\omega }_s+{\gamma }_s\right)a_s-j{\kappa }_{n.s1}\left({\theta }_{s1}\right)a_1-\dots -j{\kappa }_{n.sn}\left({\theta }_{sn}\right)a_n+F{u}_s\left(t\right)\\ \frac{d{a}_1}{dt}=-j{\kappa }_{n.s1}\left({\theta }_{s1}\right)a_s-\left(j{\omega }_1+{\gamma }_1\right)a_1-\dots -j{\kappa }_{n.1n}\left({\theta }_{1n}\right)a_n\\ \dots \\ \frac{d{a}_{n-1}}{dt}=-j{\kappa }_{n.s\left(n-1\right)}\left({\theta }_{s\left(n-1\right)}\right)a_s-j{\kappa }_{n.1\left(n-1\right)}\left({\theta }_{1\left(n-1\right)}\right)a_1-\dots -\left(j{\omega }_{n-1}+{\gamma }_{n-1}\right)a_{n-1}-j{\kappa }_{n.\left(n-1\right)n}\left({\theta }_{\left(n-1\right)n}\right)a_n\\ \frac{d{a}_n}{dt}=-j{\kappa }_{n.sn}\left({\theta }_{sn}\right)a_s-j{\kappa }_{n.1n}\left({\theta }_{1n}\right)a_1-\dots -j{\kappa }_{n.\left(n-1\right)n}\left({\theta }_{\left(n-1\right)n}\right)a_{n-1}-\left(j{\omega }_n+{\gamma }_n\right)a_n\end{array}$$

(9)

The related expressions and parameters in (9) are explained below. Since the sum of the interior angles of an n-sided polygon is π(n − 2), the expressions of κ_n.si(θ_si) and κ_n.im(θ_im) can be derived and the results are shown in (10) and (11), where L_i and L_m are self-inductances of the ith and the mth receiving coils, M_nsi.max is maximal mutual inductance between the transmitting coil and the ith receiving coil, and M_nim.max is maximal mutual inductance between the ith and the mth receiving coils.

$${\kappa }_{n.si}\left({\theta }_{si}\right)=\frac{{\omega }_i}{2}\frac{M_{nsi.\max }\left|\cos {\theta }_{si}\right|}{\sqrt{L_s{L}_i}},{\theta }_{si}=\pi \left(n-2\right)\left(i-1\right)/n$$

(10)

$${\kappa }_{n.im}\left({\theta }_{im}\right)=\frac{{\omega }_i}{2}\frac{M_{nim.\max }\left|\cos {\theta }_{im}\right|}{\sqrt{L_i{L}_m}},{\theta }_{im}=\pi \left(i-m\right)\left(n-2\right)/n$$

(11)

The values of M_nsi.max and M_nim.max can be calculated by (12) and (13), where r_i and N_i are the radius and number of turns of the ith receiving coil, and r_m and N_m are the radius and number of turns of the mth receiving coil, respectively.

$$M_{nsi.\max }=\frac{\pi \mu N_s{N}_i{r}_s^2{r}_i^2}{2{\left(r_s^2+d_{si}{}^2\right)}^{3/2}}$$

(12)

$$M_{nim.\max }=\frac{\pi \mu N_i{N}_m{r}_i^2{r}_m^2}{2{\left(r_i^2+d_{im}{}^2\right)}^{3/2}}$$

(13)

According to cosine law, there is a relation between d_im and d_si, that is:

$$d_{im}=\sqrt{2d_{si}{}^2\left(1-\cos \frac{\left|i-m\right|2\pi }{n}\right)}$$

(14)

It indicates that the value of M_nim.max can be determined based on (12)–(14) once the values of M_nsi.max and d_si are set.

At last, the value of ${\gamma }_i$ can be obtained by (15).

$${\gamma }_i={\gamma }_{Li}+{\gamma }_{i0}=\frac{R_{Li}}{2L_i}+\frac{R_i}{2L_i}$$

(15)

Here, γ_i consists of γ_Li and γ_i0, which are loss rates made by R_Li and R_i. R_Li is resistance of the i_th load and R_i is resistance of the ith receiving coil.

Let the values of L_i, C_i, R_i, R_Li, M_nsi.max and ω_i equal to those of L₁, C₁, R₁, R_L, M_max and ω in Section 2.1, respectively. Besides, set d_si = 0.3 m and other values of parameters the same as those in Section 2.1. Then the results of steady state solutions for three-receiver, four-receiver and five- receiver systems are listed in

Table 2. Steady state solutions for multi-receiver system.

n	Abs(a)	Output Power	Transfer Efficiency
3	abs(a1) = 0.0069	P1 = 40.03 W	η1 = 50.81%
	abs(a2,3) = 0.0035	P2,3 = 10.05 W	η2,3 = 12.76%
4	abs(a1,3) = 0.0035	P1,3 = 10.27 W	η1,3 = 22.45%
	abs(a2,4) = 0	P2,4 = 0 W	η2,4 = 0%
5	abs(a1) = 0.0046	P1 = 17.57 W	η1 = 33.82%
	abs(a2,5) = 0.0014	P2,5 = 1.68 W	η2,5 = 3.23%
	abs(a3,4) = 0.0036	P3,4 = 10.72 W	η3,4 = 20.62%

.

The output power P_i and transfer efficiency η_i of the ith receiver is calculated by (16) and (17).

$$P_i=2{\gamma }_{Li}{\left|a_i\right|}^2$$

(16)

$${\eta }_i={\gamma }_{Li}{\left|a_i\right|}^2/\left({\gamma }_s{\left|a_s\right|}^2+{\displaystyle \sum _1^n{\gamma }_i{\left|a_i\right|}^2}\right)$$

(17)

The data in Table 2 demonstrates that when more than one receiver is around the non-rotary transmitter, even though there is no difference between the receivers, the energy, output power and transfer efficiency of the system are all distributed unevenly to each receiver as a consequence of directionality problem. In particular, when the receiving coil and transmitting coil are perpendicular, the receiver cannot pick up any power.

3. Analysis of Wireless Power Transfer System with Rotary Transmitting Coil

According to Section 2, there are many directional limitations in traditional WPT systems. Therefore, an omnidirectional WPT system based on rotary transmitting coil is proposed, which is especially suitable for charging wirelessly for household appliances indoors.

3.1. System Structure

A structure schematic of the system mentioned above is shown in Figure 5, and a working sketch is shown in Figure 6. In Figure 5, the transmitting coil is fixed on a rotor of the electric motor, connecting to the power supply by an electric slip ring. Thus the transmitting coil can rotate with the rotor in the same speed and all the appliances around it can be charged wirelessly. Besides, the rotating speed of the rotor is adjustable.

The mathematical model and analysis are presented in the following parts of this section.

3.2. Analysis of Single-Receiver System

As the transmitting coil is rotating, time-varying items should be added to the coupling rate, and it can be expressed by (18), in which ω_n is the rotating angular frequency of the motor rotor.

$$\kappa \left(t\right)=\frac{{\omega }_1}{2}\frac{M_{\max }\left|\cos {\omega }_{n}t\right|}{\sqrt{L_s{L}_1}}$$

(18)

Then the modified system model can be obtained, as presented in (19).

$$\{\begin{array}{l}\frac{d{a}_s}{dt}=-\left(j{\omega }_s+{\gamma }_s\right)a_s-j\kappa \left(t\right)a_1+F{u}_s\left(t\right)\\ \frac{d{a}_1}{dt}=-\left(j{\omega }_1+{\gamma }_1\right)a_1-j\kappa \left(t\right)a_s\end{array}$$

(19)

Setting ω_n equal to 314 rad/s (f_n = 50 Hz) and other parameters used for analysis the same as those in Section 2.1, the waveforms of $\left|a_{s,1}\right|$, output power and transfer efficiency are presented in Figure 7 with solid lines.

From the figures it seems that energy, output power and transfer efficiency of the rotary system all vary periodically with time. Here the distortion of output power waveform is because of the non-monotonous relation between the output power and coupling rate. Similarly, the distortion of $\left|a_1\right|$ waveform is because of the non-monotonous relation between $\left|a_1\right|$ and coupling rate. Letting θ = 0 and taking the derivative of M_max in (8), it is found that P_out will get its maximal value when M_max$\approx$0.365 μH. Setting M_max equal to this value, a new set of waveforms are obtained, as presented in Figure 7 with imaginary lines. On this occasion, the distortions discussed above are eliminated. It means that both output power and energy will get their maximal values when θ = 0.

In order to measure the validity of the proposed system, average output power ${\overline{P}}_{\mathit{out}}$ and average efficiency $\overline{\eta }$ are defined and can be calculated as follows.

$${\overline{P}}_{out}=\frac{{\omega }_n}{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{{\omega }_n}}{P}_{out}dt}=\frac{{\omega }_n}{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{{\omega }_n}}2{\gamma }_L{\left|a_1\right|}^{2}dt}$$

(20)

$$\overline{\eta }=\frac{{\omega }_n}{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{{\omega }_n}}\eta dt}=\frac{{\omega }_n}{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{{\omega }_n}}\frac{{\gamma }_L{\left|a_1\right|}^2}{{\gamma }_s{\left|a_s\right|}^2+{\gamma }_1{\left|a_1\right|}^2}dt}$$

(21)

Through numerical integration based on trapezoid formula, ${\overline{P}}_{\mathit{out}}$ and $\overline{\eta }$ can be calculated numerically. When the rotating frequency is 50 Hz, ${\overline{P}}_{\mathit{out}}$ = 70.19 W. Compared with the results in Table 1, it is lower than the value under the condition where the deviation angle is $3\mathsf{\pi }/10$ but higher than the values under some adverse conditions, e.g., where the deviation angle is $2\mathsf{\pi }/5$.

Figure 8 shows the curves of average output power and average efficiency under the conditions of different rotating frequencies from 20 Hz to 80 Hz. The average efficiency keeps constant at 48.62%, while the average output power keeps nearly constant with the value change no more than 0.07 W. The results reveal that the rotating frequency has little impact on transfer characteristics. Since rotating the transmitting coil is a kind of periodic behavior and the total energy pick up by the receiver per circle is fixed, the above results are reasonable.

3.3. Analysis of Multi-Receiver System

Similarly, add the time-varying items to the coupling rate κ_n.si, and its expression is shown in (22).

$${\kappa }_{n.si}\left(t\right)=\frac{{\omega }_i}{2}\frac{M_{nsi.\max }\left|\cos \left({\omega }_{n}t+\pi \left(n-2\right)\left(i-1\right)/n\right)\right|}{\sqrt{L_s{L}_i}}$$

(22)

Then the coupled-mode equations for multi-receiver WPT system based on rotary transmitting coil can be deduced:

$$\{\begin{array}{l}\frac{d{a}_s}{dt}=-\left(j{\omega }_s+{\gamma }_s\right)a_s-j{\kappa }_{n.s1}\left(t\right)a_1-\dots -j{\kappa }_{n.sn}\left(t\right)a_n+F{u}_s\left(t\right)\\ \frac{d{a}_1}{dt}=-j{\kappa }_{n.s1}\left(t\right)a_s-\left(j{\omega }_1+{\gamma }_1\right)a_1-\dots -j{\kappa }_{n.1n}\left(t\right)a_n\\ \dots \\ \frac{d{a}_{n-1}}{dt}=-j{\kappa }_{n.s\left(n-1\right)}\left(t\right)a_s-j{\kappa }_{n.1\left(n-1\right)}\left(t\right)a_1-\dots -\left(j{\omega }_{n-1}+{\gamma }_{n-1}\right)a_{n-1}-j{\kappa }_{n.\left(n-1\right)n}\left(t\right)a_n\\ \frac{d{a}_n}{dt}=-j{\kappa }_{n.sn}\left(t\right)a_s-j{\kappa }_{n.1n}\left(t\right)a_1-\dots -j{\kappa }_{n.\left(n-1\right)n}\left(t\right)a_{n-1}-\left(j{\omega }_n+{\gamma }_n\right)a_n\end{array}$$

(23)

The analyses on output power and transfer efficiency of three-receiver, four-receiver and five- receiver systems are also implemented with f_n = 50 Hz and values of other parameters the same as those in Section 2.2. The results are shown in Figure 9, Figure 10 and Figure 11. The value of M_nsi.max has been selected properly, thus there is no distortion in the output power waveforms.

The average output power values pick up by every receiver in three systems are 20.07 W, 12.50 W and 8.51 W. It illustrates that the proposed system ensures all the receivers around the rotary transmitter can pick up power averagely, which quite improves the unbalanced power distribution phenomenon between receivers in the non-rotary system mentioned in Section 2.2.

3.4. Analysis of Dual-Supply System

According to the analysis in part 2 and part 3 of this section, both the output power and transfer efficiency have dead zones when there is only one rotary transmitting coil. This problem can be solved by increasing the number of transmitting coils appropriately. Here a dual-supply system is proposed. The only difference is that in this kind of system, there are two orthometric rotary transmitting coils working together so that when one of the transmitting coils is perpendicular to the receiving coil, the other one can supply power to the receiver. The system model is presented in (24).

$$\{\begin{array}{l}\frac{d{a}_{s1}}{dt}=-\left(j{\omega }_s+{\gamma }_s\right)a_{s1}-j{\kappa }_{s1}\left(t\right)a_1+F{u}_s\left(t\right)\\ \frac{d{a}_{s2}}{dt}=-\left(j{\omega }_s+{\gamma }_s\right)a_{s2}-j{\kappa }_{s2}\left(t\right)a_1+F{u}_s\left(t\right)\\ \frac{d{a}_1}{dt}=-\left(j{\omega }_1+{\gamma }_1\right)a_1-j{\kappa }_{s1}\left(t\right)a_{s1}-j{\kappa }_{s2}\left(t\right)a_{s2}\end{array}$$

(24)

Here, a_s1 and a_s2 are field amplitudes of the two transmitter resonant cavities. κ_s1(t) and κ_s2(t) are coupling rates between the receiver and two transmitting coils, respectively. Besides, let the expression of κ_s1(t) the same as (18) and then the expression of κ_s2(t) is:

$${\kappa }_{s2}\left(t\right)=\frac{{\omega }_1}{2}\frac{M_{\max }\left|\cos {\omega }_{n}t+\pi /2\right|}{\sqrt{L_s{L}_1}}$$

(25)

Note that because of the orthogonality of two transmitters, the coupling rate between a_s1 and a_s2 is zero, it does not appear in (24).

Figure 12 presents the output power waveform. The values of parameters for analysis are the same as those in Section 3.2. The results reveal this kind of systems can eliminate the aforementioned dead zones, and the average output power is 112.89 W. Furthermore, although increasing the number of transmitters will to some extent increase the setup cost, it can decrease the needed rotating speed of the motor.

4. Experimental Verifications

The omnidirectional WPT system based on rotary transmitting coil shown in Figure 5 has been fabricated and the setup photograph is shown in Figure 13.

Here, A is a signal generator used to generate signal of high frequency. B is a broadband power amplifier. C is an impedance matcher used to adjust the resonant frequency of the coils. D is a DC power supply for the electric rotating motor E. F is the transmitting coil connected to a 12-access 2A electric slip ring. G are the receiving coils with loads. H is a motor speed meter. I is a motor speed regulator.

The operating frequency is 1 MHz, the transmitting coil and receiving coil both have six turns, the diameters of the two coils are 32 cm, and the diameter of the copper wire is 2 mm. Self inductance and capacitance of the receiving coil are measured by the impedance analyzer and the values are 25.76 μH and 893.28 pF, which determines the coil intrinsic frequency of 1.04 MHz. The transmitting coil intrinsic frequency is also adjusted to 1 MHz by the impedance matcher.

4.1. Single-Receiver Rotating Experiment

In this part, there is only one receiver. The receiver consists of one receiving coil and one set of loads, which include three series of 3 × 3 W LEDs. The distance between the centers of two coils is 30 cm.

4.1.1. System with One Rotary Transmitting Coil

By adjusting the motor speed, it can be observed that the flicker frequency of LEDs varies accordingly. The more quickly the motor rotor rotates, the shorter LEDs’ off time is. When the speed reaches a certain value, human eye can no longer tell whether the LEDs are flickering or not, and at this time it can be thought that the brightness of the LEDs remains stable. The experimental results indicate that when the speed reaches 589 rpm, LEDs can supply stable light.

4.1.2. System with Two Rotary Transmitting Coils

In order to verify the function of two orthometric rotary transmitting coils analyzed in Section 3.4, a dual-supply experiment setup has been built, which is shown in Figure 14. Experimentally, all the LEDs keep luminous during the system operating time, certifying that the aforementioned dead zones have been eliminated successfully. Besides, the experimental results indicate that when the speed reaches 286 rpm, LEDs can supply stable light.

4.2. Multi-Receiver Rotating Experiment

Another set of LEDs and a receiving coil has been added. In this experiment, the distance between the centers of transmitting coil and each receiving coil is 20 cm. Regard the deviation angle of the first receiver as reference angle and record it as θ. Two receiving coils are placed perpendicularly such that the deviation angle of the second receiver is always π/2 larger than θ. According to the theoretical analysis, two sets of LEDs would glow alternately with the value of θ varying in cycles. The experimental phenomena with different values of θ are presented in Figure 15, which show a dynamic process of the working system.

At first, when θ = 0, the first set of LEDs (hereinafter referred to as “LEDs1”) get their greatest brightness while the second set of LEDs (hereinafter referred to as “LEDs2”) are totally extinct. It implies all the power from the transmitter has been transferred to LEDs1. When θ = π/6, all the bulbs are on with LEDs1 a litter brighter than LEDs2. At this moment, LEDs1 pick up more power than LEDs2. Then we can see LEDs1 continue to become darker and LEDs2 become brighter. When θ gets π/4, the transmitter distributes equal power to each receiver due to symmetry of the positions so that LEDs1 and LEDs2 have the same brightness. Then with the increasing of θ, the power transferred to LEDs1 becomes less and more power is pick up by LEDs2. Until θ = π/2, LEDs2 get their greatest brightness while LEDs2 are totally extinct. At this time, all the power is picked up by LEDs2 and no power is transferred to LEDs1. And so on, LEDs1 and LEDs2 glow alternately. Furthermore, two sets of LEDs supply stable light of the same brightness when the motor speed gets fast enough, which reveals the balanced power distribution between the two receivers.

The above behavior is in great agreement with our analysis results presented in Section 3.3.

5. Discussion and Conclusions

This paper presents a theoretical study of a wireless power transfer system based on rotary transmitting coil. This system can be utilized to achieve omnidirectional wireless power transfer. In particular, it applies to multi-receiver system and ensures each receiver can pick up power synchronously and evenly. Besides, utilizing two rotary transmitting coils can eliminate the dead zones of power supply. Simulations and experimental results support the theoretical analysis. Through this scheme, household appliances equipped with receiving coils can be charged wirelessly regardless of their positions, which can realize a clean, efficient and convenient indoor environment.