Finite element method and coupled mode theory coupling for accurate analysis of frequency splitting in wireless power transmission

Abstract

Wireless power transmission technology avoids the problem of towed wires in the process of using electric energy, and increases the flexibility of using electricity. It is a hot research topic at present. In order to improve the system performance, a field-circuit coupling algorithm is proposed to analyze the system performance, with the help of the concept of supercomputing. Frequency splitting is a phenomenon in wireless power transfer (WPT) system when the coupling distance is less than the splitting point, the load power changes from a single-peak curve to a double-peak curve driven by two non-intrinsic resonant frequencies. Asymptotic coupled mode theory (CMT) method is used to analyse the frequency splitting phenomena in WPT system. It provides detailed information about interaction of field strength under different coupling states through coupled solution of FEM and CMT. Over coupling, critical coupling and under coupling are three typical states classified by frequency splitting. Experimental results are acquired by two helical resonators. The overall system reaches the critical coupling state when resonators space 1.5 m and the total power on the load is 110 W. Therefore, it is an efficient way to forecast transmission characteristics by using this method.

Keywords

magnetic resonant coupling frequency splitting coupled mode theory critical coupling coupled solution

1 Introduction

As a novel technology for energy conversion and transmission, wireless power transfer (WPT), which has wide prospects on electric vehicles, wireless sensors, consumer electronics, biomedical and communications technology, is paid increasing attention in recent years [1 –7]. It is based on strongly coupled magnetic resonance via non-radiative near field between resonators. To transfer power spaced several meters, system has to operate under high frequency (at the range of several MHz) with high quality factor (at the range of several hundred), usually known as magnetic resonant coupling wireless power transfer (MRC-WPT) [8 –10]. Then, there is a phenomenon that when the coupling between resonators increases and is greater than critical coupling, the load power changes from a single-peak curve to a double-peak curve. It can be called as frequency splitting which is a key issue related to system transmission capability and efficiency [11, 12].

In [13], it states that if the coupling rate is much larger than all loss rates, the normal modes of system are split by twice the coupling coefficient. However, the impact of frequency splitting is not deeply analysed. In [14] and [15], S-parameter and eigenmodes are employed to describe the system performance considering the impact of frequency splitting via the circuit model. Their common feature is that frequency splitting is derived as a multivariate function in which the contribution of coupling coefficient, quality factor and angular frequency is coupled with each other and is too obscure for practical application. In [16], a circuit theory method is used to analyse frequency splitting which is expressed as a tenth-order equation. Consequently, it is difficult to estimate the contribution of system parameters to the splitting point. Furthermore, there is a blind spot of coupling situation under different coupling states. Because the energy exchange in near field cannot be reflected directly.

In this paper, a higher-order mathematical model of resonant coupling system has been proposed to explain the essential reason of the frequency splitting. The effect of coupling coefficient and quality factor on system performance are discussed when frequency splitting occurs. Moreover, the distribution of flux density and electric strength in surrounding space are achieved by combined solution of FEM and ODE equations. The discipline of frequency splitting is validated by a wireless power transfer prototype composed of helical resonators. The load power is measured when frequency is tuned automatically and compared with the ones driven by fixed frequency, so that the optimum working condition is achieved.

2 The principle of frequency splitting

A general structure of MRC-WPT system is shown in Figure 1 where transmitting end is composed of source coil and resonator a₁ while receiving end composed of resonator a₂ and device coil. The energy on source coil or device coil is transferred or absorbed through induction from the resonators. V_1,2, I_1,2, C_1,2 and R_1,2 are equivalent voltage, current, capacitance and resistance of resonator a_1,2 respectively.

Fig. 1

Directly field-circuit model of a MRC-WPT system.

In coupled mode theory (CMT) domain, the energy of resonators exchanges between each other through resonant coupling and can be described by an asymptotic CMT model as (1) [17 –20]. $\begin{matrix} \frac{d}{dt} [\begin{matrix} a_{1} (t) \\ a_{2} (t) \end{matrix}] = [\begin{matrix} j ω_{10} - Γ_{1} \\ j κ \end{matrix} \begin{matrix} j κ \\ j ω_{20} - (Γ_{2} + Γ_{L}) \end{matrix}] \\ [\begin{matrix} a_{1} (t) \\ a_{2} (t) \end{matrix}] + [\begin{matrix} A_{s} e^{j ω t} \\ 0 \end{matrix}] \end{matrix}$ (1)

Where a₁(t) and a₂(t) denote the forward transfer mode of the resonators, respectively; ω_10,20 denote the natural resonant frequency of each resonator; κ denotes the mode coupling factor; Γ₁ and Γ₂ denote the intrinsic loss of resonators; Γ_L is the intrinsic loss due to the load resistance; A_se^jωt on resonator a₁ is a sinusoidal signal source with the amplitude A _s and the angular frequency ω.

When there is certain loss in system, its natural resonant angular frequency will shift in different degrees [21 –23]. If the quality factors of two resonators a₁ and a₂ are high enough, the first order Taylor expansion of the modified angular frequencies can be expressed as (2). $\begin{matrix} ω_{10, 20}^{'} = \\ ω_{10, 20} \sqrt{1 - \frac{1}{1 + Q_{1, 2}^{2}}} \to ω_{10, 20} (1 + \frac{j}{2 Q_{1, 2}}) \end{matrix}$ (2)

Where ω′_10,20 denote the modified angular frequencies; Q_1,2 are their quality factors and there is a relationship between Q_1,2 and the loss rate that Γ_1,2 = j/(2Q_1,2). Meanwhile, from the resonator side, there is Γ_L=j/(2Q_L) where Q_L denotes the quality factor of the load.

When MRC-WPT system is operating near the natural resonant angular frequency point ω₀, (1) can be simplified as (3) by using Q’₂^-1 to represent Q₂^-1 ⁺ Q_L^-1. $[\begin{matrix} a_{1} \\ a_{2} \end{matrix}] = [\begin{matrix} j (ω - ω_{10}^{'}) \\ - κ \end{matrix}] \frac{A_{s} e^{j ω t}}{(ω - ω_{10}^{'}) (ω - ω_{20}^{'}) + κ^{2}}$ (3)

A _s and κ in (3) can be expressed in circuit form as: A _s=jω₀L0.51I₁/2, κ=jω₀k₁₂/2, Where L₁ and I₁ denote the inductance and driven current of resonators a₁; k₁₂ denotes the coupling coefficient between a₁ and a₂. Therefore, active power on the load P_L is shown as: $\begin{matrix} P_{L} = ω_{0} {| a_{2} |}^{2} / Q_{L} = \\ \frac{ω_{0} L_{1} I_{1}^{2} k_{12}^{2} / (16 Q_{L})}{χ^{4} + χ^{2} [{(\frac{1}{2 Q_{1}})}^{2} + {(\frac{1}{2 Q_{2}^{'}})}^{2} - \frac{k_{12}^{2}}{2}] + {(\frac{k_{12}^{2}}{4} + \frac{1}{4 Q_{1} Q_{2}^{'}})}^{2}} \end{matrix}$ (4)

Where χ = 1 - ω/ω₀ denotes a detuning factor to describe the shift degree of ω. Consequently, P_L reaches the maximum value by making its partial derivative equal to zero which is shown in (5). Δ indicates the denominator of (4). $\begin{matrix} \frac{\partial P_{L}}{\partial χ} = \frac{- ω_{0} L_{1} I_{1}^{2} k_{12}^{2}}{8 Q_{L} Δ^{2}} \\ χ [2 χ^{2} + (\frac{1}{2 Q_{1}})^{2} + (\frac{1}{2 Q_{2}^{'}})^{2} - \frac{k_{12}^{2}}{2}] = 0 \end{matrix}$ (5)

If it is intended to achieve a sustained oscillation, there must be Δ≥0, and we can obtain: ${\begin{matrix} χ_{1, 2} = \mp \frac{1}{2} \sqrt{k_{12}^{2} - \frac{1}{2 Q_{1}^{2}} - \frac{1}{2} Q_{2}^{' 2}} \\ χ_{3} = 0 \end{matrix}$ (6)

Where χ_1,2 indicates that there are two extreme value with the increase of k_1,2. When system structure is fixed, χ_1,2 is a function of k_1,2 and parameters in circuits. Then, the two splitting angular frequencies ω_1,2 are acquired as (7). $ω_{1, 2} = (1 \pm \frac{1}{2} \sqrt{k_{12}^{2} - \frac{1}{2 Q_{1}^{2}} - \frac{1}{2 Q_{2}^{' 2}}}) ω_{0}$ (7)

And corresponding active power is expressed as (8). $\begin{matrix} P_{1, 2} = \\ {\frac{ω_{0} L_{1} I_{1}^{2} k_{12}^{2} / Q_{L}}{{(k_{12}^{2} + \frac{1}{Q_{1} Q_{2}^{'}})}^{2} - {(k_{12}^{2} - \frac{1}{2 Q_{1}^{2}} - \frac{1}{2 Q_{2}^{' 2}})}^{2}} |}_{ω = ω_{12}} \end{matrix}$ (8)

With regard to χ₃ = 0 (e.g. ω₃=ω₀), it means that always keeping the excitation frequency and self-resonant frequencies of a₁ and a₂ the same. It is a traditional control strategy that maintains system working in resonant state, and we can obtain: ${P_{3} = \frac{ω_{0} L_{1} I_{1}^{2} k_{12}^{2} / Q_{L}}{(k_{12}^{2} + 1 / Q_{1} Q_{2}^{'})^{2}} |}_{ω = ω_{3} = ω_{0}}$ (9)

According to (8) and (9), the load power becomes a minimum value P₃ if system still working at its self-resonant frequency. Meantime, P₁ and P₂ are two maximum values once k₁₂ met the condition as (10). $k^{12} > \sqrt{\frac{1}{2 Q_{1}^{2}} + \frac{1}{2 Q_{2}^{' 2}}} ⩾ \sqrt{\frac{1}{Q_{1} Q_{2}^{'}}}$ (10)

As soon as k₁₂ reduced to $\sqrt{1 / Q_{1} Q_{2}^{'}}$ , splitting phenomenon disappeared and there is only one resonant point. Generally, Q_L is far less than Q₂, so the maximum value of P_Lmax is $\begin{matrix} P_{1} = P_{2} = P_{3} = \\ {\frac{Q_{1} Q_{L} I_{1}^{2} R_{L}}{4} = P_{L max} |}_{k_{12} = \sqrt{1 / Q_{1} Q_{2}^{'}} = k_{critical}} \end{matrix}$ (11)

At this moment, it is the optimum operating point of a MRC-WPT system, because the active power on the load reaches the peak value with the distance long enough. When k₁₂ continues to reduce, the active power on the load decreases rapidly and tends to zero as shown in (12), because k₁₂ is in negative relationship with the cube order of spacing distance. ${P_{L} = \frac{ω_{0} L_{1} I_{1}^{2} k_{12}^{2} (Q_{1} Q_{2}^{'})^{2}}{Q_{L}} |}_{k_{12} \to 0}$ (12)

The normalized active power as a function of k₁₂ and the ratio of angular frequency ω/ω₀ is shown in Figure 2. When system is working at the critical coupling point with the maximum value of P_Lmax. And when k₁₂ _> k_critical, which means the receiving end is close to the transmitting end continuously, there are two peak values and one valley value. The intervals of splitting frequency among them is broaden increasingly. On the other hand, the three-value above changes to one value and tend to zero eventually when k₁₂ _> k_critical. Therefore, it is divided into three states which are over coupling, critical coupling and under coupling.

Fig. 2

Normalized active power as a function of coupling coefficient and angular frequency.

3 Coupled solution of FEM & CMTfor MRC-WPT system

For MRC-WPT system, the electromagnetic wavelength is several times of resonator size, so it can be seen as a quasi-static electromagnetic problem. Through there is a certain degree of radiation, it is a kind of low efficiency antenna whose radiation loss is represented by an equivalent resistance. For the FEM domain in Figure 1, it is described by the following equations as (13), when the displacement current is ignored. ${\begin{matrix} υ \nabla \times B = J_{s} + σ E \\ \nabla \times E = - \frac{\partial B}{\partial t} \end{matrix}$ (13)

Where B is the flux density (T); J _s is the source current density (A/m²); E is the electric field intensity in V/m; t is the time (s); ν and σ are the magnetic reluctivity and electrical conductivity, respectively. An electric scalar potential function ϕ is defined to determine the relationship with the electric field intensity E and magnetic vector potential A given as (14). $\nabla \times E = - \frac{\partial A}{\partial t} - \nabla ϕ$ (14)

Coulomb Gauge∇ A =0 is applied to (13) for guarantying the uniqueness of A . According to the continuity property of current, it is also necessary to add ∇ J =0 into the governing equations which can be expressed as (15). ${\begin{matrix} a_{1}, a_{2}, L_{D} : {\begin{matrix} \nabla \times (υ \nabla \times A) - \nabla \times (υ \nabla \cdot A) \\ + σ (\frac{\partial A}{\partial t} + \nabla ϕ) = 0 \\ \nabla \cdot (- σ \frac{\partial A}{\partial t} - σ \nabla ϕ) = 0 \end{matrix} \\ Air : {\begin{matrix} \nabla \times (υ \nabla \times A) - \nabla \times (υ \nabla \cdot A) = 0 \\ \nabla \cdot (- σ \frac{\partial A}{\partial t} - σ \nabla ϕ) = 0 \end{matrix} \\ L_{S} : \nabla \times (υ \nabla \times A) - \nabla \times (υ \nabla \cdot A) = J_{s} \end{matrix}$ (15)

For computer programs, different regions are automatically chosen by σ, so that (15) is re-combined into one equation. And it is intended to avoid spurious solutions which may arise from scalar basis function, that a weighted residual method with a vector testing function N_i is employed. As regard to CMT domain, (1) is solved by a three-order total differential equation solver. Then the current on each resonator is determined by (16). ${\begin{matrix} a_{1} = 1 / 2 \sqrt{L_{1}} (I_{1} + j ω_{10}^{'} C_{1} V_{1}) \\ a_{2} = 1 / 2 \sqrt{L_{2}} (I_{2} + j ω_{20}^{'} C_{2} V_{2}) \end{matrix}$ (16) ${PDE : \int \int \int N_{i} [\begin{matrix} \nabla \times (υ \nabla \times A) - \nabla \times (υ \nabla \cdot A) \\ + σ (j ω A + \nabla ϕ) - J_{s} \end{matrix}] d v = 0 ODE : {\begin{matrix} \frac{{da}_{1}}{dt} = j ω_{10}^{'} a_{1} + j κ a_{2} \\ \frac{{da}_{2}}{dt} = j κ a_{1} + j ω_{20}^{'} a_{2} \end{matrix}$ (17) Therefore, the coupled solution model of FEM & CMT for MRC-WPT system is shown as (17), where ODE and PDE functions are solved in proper sequence. Then the performance of MRC-WPT system under different coupling state is achieved.

4 Experimental validation of frequencysplitting

In order to verify the accuracy of this model, a MRC-WPT system is designed and its detailed parameters are shown in Table 1. The resonators are made into a helical structure and placed coaxially. Also, they have the same geometry parameters and natural resonant frequency. The source coil is made of the same material as the resonators, and it is a single turn. The silver-plated enamelled Teflon wire is chosen for the device coil which is placed concentrically with resonator a₁ and resonator a₂. For observing the frequency splitting under different spacing distance, two bulbs (220 V/60 W) are used as the load which is connected with device coil.

Table 1
Parameters of MRC-WPT system composed of helical resonators

Parameters Resonator a₁, a₂ Source coil Device coil

Materials Red copper Red copper Oxygen-free copper

Conductivity (S/m) 5.56×10⁷ 5.56×10⁷ 6.06×10⁷

Number of turns 4.25 1 4

Radius of wire (mm) 3 3 2.5

Radius of the coil(cm) 30 30 10

Conductor Spacing (cm) 6±2.5 —— ——

DC resistance (Ω) 0.046 0.011 0.002

Resonant frequency(MHz) 13.5–14.5 —— ——

Quality factor 1000 1500 —— ——

Parameters	Resonator a₁, a₂	Source coil	Device coil
Materials	Red copper	Red copper	Oxygen-free copper
Conductivity (S/m)	5.56×10⁷	5.56×10⁷	6.06×10⁷
Number of turns	4.25	1	4
Radius of wire (mm)	3	3	2.5
Radius of the coil(cm)	30	30	10
Conductor Spacing (cm)	6±2.5	——	——
DC resistance (Ω)	0.046	0.011	0.002
Resonant frequency(MHz)	13.5–14.5	——	——
Quality factor	1000 1500	——	——

The whole system is solved by the coupled method at three typical distances: 1 m, 1.5 m and 2 m. There is an observation error about±2.5% to measure the quality factor, and the final results form a region represented by dark shaded area, as shown in Figure 3. In additions, the measured results of designed system are agreement with the theoretical results. The power that is induced from resonators reaches the maximum value when resonator a₁ and resonator a₂ operate in resonant state as stated in (9). Here, the critical coupling point is about 1.5 m by numerical analysis, while there is about 2.08% relative error in measurement value. When gradually reducing distance, so that k₁₂ _> k_critical, frequency splitting occurs as stated in (7). The two bulbs cannot be lightened at the same time, for the power is only acquired under new splitting frequencies which are far from their resonant frequency. In this case, the received power of the load decreases rapidly if system is still driven by the natural frequency. When the distance is enlarged to make k₁₂ _< k_critical, frequency splitting disappears and a small amount of power is transmitted to the load through a long distance.

Fig. 3

Power of the load PL under three typical distances.

In order to get the performance characteristics of MRC-WPT system under different coupling states, we simulated the entire system using the finite element simulation software COMSOL. Distribution of magnetic field strength |H| and electric field strength |E| under critical coupling state is shown in Figure 4. According to section cloud image, magnetic field strength |H| reaches the maximum value at the centre of the coil, while electric field strength |E| obtains the maximum value in adjacent area of coils. Distribution of magnetic field strength |H| under three typical distances is shown in Figure 5. It indicates that when the receiving end is close enough to the transmitting end, the magnetic field generated by the former interacts with that of the latter. The magnetic fields weaken mutually, so that receiving power is reduced eventually. For another case, when d = 2 m, power exchange still exists but very little because spacing distance is larger than the best working distance of MRC-WPT system. Figure 6 shows that the system operates in the best state with a frequency of 13.56 MHz and a distance of 1.5 m. Two 220 V/60 W bulbs are lightened simultaneously and the power absorbed by the bulbs is estimated to be 110 W by measuring the voltage with a high accuracy oscilloscope.

Fig. 4

Distribution of |H| and |E| under critical coupling state.

Fig. 5

Distribution of |H| under three typical distances.

Fig. 6

Experimental validation of the optimum operating point.

Therefore, it can be summarized into three kinds of states which are over coupling, critical coupling and under coupling according to frequency splitting.

5 Conclusion

The above theoretical analysis and experimental results show that frequency splitting is a specific phenomenon in MRC-WPT system which is mainly caused by the variation of coupling coefficient combined with high quality factor. It simplifies the analysis progress by CMT which catches the key characteristics of the frequency splitting. The coupled solution of FEM and CMT shows detailed information about interaction of field strength under different coupling states. All the theoretical derivation is validated by a MRC-WPT system composed of helical resonators with an error less than 5%. This demonstrates the accuracy and effectiveness of the proposed model. Operation in critical or under coupling state for high power and longer distance is suggested.

Footnotes

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 51677132, in part by the University Innovation Team Training Plan of Tianjin under Grant TD13-5040.

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