Research on Tradeoff between Power and Efficiency of Wireless Power Transfer via Magnetic Resonance Coupling

Abstract

Focusing on the tradeoff problem between the output power and transmission efficiency of magnetic coupling resonance wireless power transmission (MCRWPT) system, this paper firstly analyzes the series–series topology system by using the mutual inductance theory of concentrated parameters, and proposes the concept of power-efficiency composite function by using the method of normalization of output power and then the WPT system is assigned a weight factor. Based on that, the calculation expression of the optimal resonant frequency is obtained. The correctness of theoretical analysis is verified by simulation and experimental platform, which provides a useful reference for further improvement of power-efficiency synchronization and energy saving.

1 Introduction

In 2007, the research team of Massachusetts Institute of Technology proposed magnetic coupling resonance wireless power transfer (MCRWPT), which has received widespread attention [1, 2]. The technology of MCRWPT adopts strong magnetic coupling resonance, which can achieve higher power and efficiency, break through the limit of tiny transmission distance, and has no harm to the human body. At present, numerous scholars have conducted in-depth analysis of the transmission power and efficiency of MCRWPT system. However, a common phenomenon found in MCRWPT is the fact that frequency splitting occurs in the over-coupled region [3] and to solve the problem of system transmission performance degradation caused by it, methods such as frequency reduction control, frequency and load compound regulation are generally used [4,5,6]. However, the MCR-WPT system is relatively complicated with additional frequency tuning circuits. Maximum efficiency tracking control is the most common solution in which the drive duty cycle is regulated to ensure the maximum efficiency of WPT system [7]. In this method, the duty cycle is looked for repeatedly to ensure that the system always works at the operating point. But it requires precise control and is difficult to realise. Based on coupling coefficient, a novel maximum efficiency tracking control scheme is proposed, which is regulating the DC link current of secondary side by PI feedback controller to maintain the highest efficiency of WPT system when coupling coefficient varies [8]. This method solved the problem of maximum efficiency when the transmission distance changes dynamically. In literature [9], a simple and efficient topology for a transmitting system called the DC bus system is introduced. Its simplicity maximizes the potential transmitting efficiency. In addition to the research of control strategies, a metamaterial array having dimensions of 20–30 cm is positioned near the load coil to concentrate energy of magnetic field [10] and a dynamic parameter adjustment method based on chaos optimization algorithm is proposed [11], which can make several key parameters of the system achieve optimal matching.

Besides, the optimal output power also becomes a hot research topic in the recent several years and many scholars have proposed a series of remarkable innovation. The paper presents a method for achieving maximum power on the load by matching capacitance in a WPT system with given two-coupled-coils [12]. Literature [13] proposed a novel Maximum Power Efficiency Tracking (MPET) method. It uses k-Nearest Neighbors to estimate the system's coupling coefficient and uses an adaptive converter control to realize the maximum power tracking. MPET method is a common solution to maximize output power. In theoretical research, internal conditions of keeping the maximum power are studied in literature [14]. It is pointed out that the maximum power can be transmitted only when the resistance product of the transmitting circuit and the receiving circuit is small. In the aspect of impedance matching, the maximum achievable power of WPT system with the given load impedance is estimated, and the condition of source impedance for the maximum power is proposed [15]. However, the disadvantage is that the method is computationally complex and requires additional matching circuits. Furthermore, the research group in [16] is dedicated to studying the maximum power of system with the maximum efficiency, i.e. the power-efficiency synchronization. The relationship between transmission efficiency and power versus load is analyzed, and the probability of system synchronization state is verified.

In the above research, it is often pursued to maximize the transmission efficiency, maximize the output power, or further require the system to achieve the power-efficiency synchronization. However, in practical engineering applications, for a fixed MCRWPT system, when the transmission distance is determined, impedance matching cannot be used to achieve power-efficiency synchronization and how to quantitatively design the resonant frequency of MCRWPT system with given output power and efficiency proportion is of great practical significance, which avoids the waste of resources caused by constant resonant frequency increasing to meet some demands. According to the designed resonant frequency, the corresponding capacitance is matched. In allusion to the above problem, this paper makes a theoretical analysis of the relationship between the resonant frequency and output power, the resonant frequency and transmission efficiency, and discusses about the tradeoff between the output power and transmission efficiency. The optimal resonant frequency of the system under appointed conditions is derived. Finally, the theoretical analysis was verified by experimental results.

2 Equivalent Circuit Modeling and Theoretical Analysis

MCRWPT technology belongs to the near-field strong magnetic coupling transfer technology, and its energy transmission part is composed of energy transmitting circuit and receiving circuit. In order to ensure the efficient transmission of power, the resonant angular frequency between the transmitting circuit and the receiving circuit should be consistent. In MCRWPT system, the resonance state is achieved by connecting the resonant capacitor in the transmitting circuit and the receiving circuit, and the energy migration is realized by resonance coupling of the space magnetic field between the transmitting coil and the receiving coil [17]. According to the connection pattern of coils and resonance capacitors, the resonant topology is divided into series-series(SS), series–parallel(SP), parallel-series(PS) and parallel-parallel(PP) [18]. The coupling resonance system of series-series(SS) is analyzed below, and the equivalent circuit of WPT system is shown in Fig. 1.

Fig. 1

Equivalent circuit model

As shown in Fig. 1, $\dot{U}_{S}$ is the AC excitation source; R₀ is the internal resistance of the power source; L₁ and L₂ are the self-inductance of coils; R₁ and R₂ are the self-impedance of coils; C₁ and C₂ are the resonant capacitance; R_L is the load impedance. M is the mutual inductance between transmitting and receiving coil loops. Denote the coupling coefficient between the transmitting and the receiving coils as k, which is given by [2]

$$k = \frac{M}{{\sqrt {L_{1} L_{2} } }}$$

(1)

For transceiver coils with the same structure and parameters, the calculation formula of coil equivalent inductance L [19] and the relationship between transmission distance and mutual inductance are [20]

$$L = N^{2} \cdot r \cdot \mu_{0} \left[ {\ln \left( \frac{8r}{a} \right) - 2} \right]$$

(2)

$$M = \frac{{\pi \mu_{0} Nr^{4} }}{{2D^{3} }}$$

(3)

Among them, D is the distance between the receiving and transmitting coils, r is the radius of the coils, N is the number of turns of the coils, a is wire diameter, and $\mu_{{_{0} }}$ is the permeability of vacuum.

R₁ and R₂ will increase with the increase of the operating frequency of the system. There are coil ohmic loss resistance R_i and radiation loss resistance R_r, and their approximate calculation formulas are respectively for [17, 21]

$$R_{i} = \sqrt {\frac{{w\mu_{0} }}{2\sigma }} \frac{{n_{i} r_{i} }}{{a_{i} }} = m_{i} \sqrt w \begin{array}{*{20}c} {} & {i = 1,2} \\ \end{array}$$

(4)

$$R_{r} = \sqrt {\frac{{\mu_{0} }}{{\varepsilon_{0} }}} \left[ {\frac{\pi }{12}n^{2} \left( \frac{wr}{c} \right)^{4} + \frac{2}{{3\pi^{2} }}\left( \frac{wh}{c} \right)^{2} } \right]$$

(5)

Suppose $\dot{I}_{1}$ and $\dot{I}_{2}$ are the current of transmitting and receiving terminal loops in the equivalent circuit. Based on Kirchhoff’s voltage law, its loop voltage equation can be expressed in

$$\left[ {\begin{array}{*{20}c} {\dot{U}_{s} } \\ 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{0} + Z_{1} } & {jwM} \\ {jwM} & {Z_{2} + R_{L} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{I}_{1} } \\ {\dot{I}_{2} } \\ \end{array} } \right]$$

(6)

In which

$$Z_{1} = R_{1} + jwL_{1} + \frac{1}{{jwC_{1} }},\;\;\;Z_{2} = R_{2} + jwL_{2} + \frac{1}{{jwC_{2} }}$$

(7)

According to Eq. (6), the corresponding input impedance can be calculated as

$$Z_{in} = R_{1} + \left( {jwL_{1} + \frac{1}{{jwC_{1} }}} \right) + \frac{{\left( {wM} \right)^{2} }}{{R_{2} + R_{L} + \left( {jwL_{2} + \frac{1}{{jwC_{2} }}} \right)}}$$

(8)

The resonant angular frequency during system operation is $w_{R}$ shown in Eq. (9) [22]. According to the principle of magnetic coupling resonance, when the self-excited oscillation frequency of the transmitting end is consistent with that of the receiving end, the transmitting end and the receiving end will resonate and become a pair of resonant dipoles. The magnetic field energy is continuously exchanged between the resonant coils. As can be known from the principle of mutual inductance, the equivalent input impedance of resonant loop is

$$w_{R} = \frac{1}{{\sqrt {L_{1} C_{1} } }} = \frac{1}{{\sqrt {L_{2} C_{2} } }}$$

(9)

$$Z_{in} = R_{1} + \frac{{\left( {w_{R} M} \right)^{2} }}{{R_{2} + R_{L} }}$$

(10)

Suppose U₁, I₁ and I₂ are the valid value of input voltage, input current and output current, the corresponding input and output current can be calculated as

$$I_{1} = \frac{{U_{1} }}{{Z_{in} + R_{0} }} = \frac{{U_{1} }}{{R_{0} + R_{1} + \frac{{w_{R}^{2} M^{2} }}{{R_{L} + R_{2} }}}}$$

(11)

$$I_{2} = \frac{{U_{1} w_{R} M}}{{(R_{1} + R_{0} )(R_{2} + R_{L} ) + w_{R}^{2} M^{2} }}$$

(12)

The input power and output power of MCRWPT system can be given by

$$P_{in} = U_{1} I_{1} = \frac{{U_{1}^{2} \left( {R_{2} + R_{L} } \right)}}{{(R_{L} + R_{2} )R_{0} + R_{1} (R_{L} + R_{2} ) + \left( {w_{R} M} \right)^{2} }}$$

(13)

$$P_{out} = I_{2}^{2} R_{L} = \frac{{U_{1}^{2} \left( {w_{R} M} \right)^{2} R_{L} }}{{[(R_{2} + R_{L} )R_{0} + R_{1} (R_{2} + R_{L} ) + \left( {w_{R} M} \right)^{2} ]^{2} }}$$

(14)

Assuming that the structure and parameters of MCRWPT system are certain, the analysis of Eq. (14) shows that the system has a resonant angular frequency $w_{R}$ that enables the system to output maximum power. Take a derivative with the output power in Eq. (14).

$$\frac{{\partial P_{out} }}{{\partial w_{R} }} = \frac{{2M^{2} U_{1}^{2} R_{L} w_{R} \left( {R_{0} (R_{2} + R_{L} ) + R_{1} (R_{2} + R_{L} ) - w_{R}^{2} M^{2} } \right)}}{{[R_{0} (R_{2} + R_{L} ) + R_{1} (R_{2} + R_{L} ) + w_{R}^{2} M^{2} ]^{3} }}$$

(15)

When $\partial P_{out} /\partial w_{R} = 0$, the maximum output power can be calculated as:

$$P_{{out{ - }\max }} = \frac{{U_{1}^{2} R_{L} }}{{4(R_{2} + R_{L} )(R_{1} + R_{0} )}}$$

(16)

The corresponding resonant angular frequency can be calculated as

$$w_{R1} = \frac{{\sqrt {(R_{0} + R_{1} )\left( {R_{2} + R_{L} } \right)} }}{M}$$

(17)

According to Eq. (17), $w_{R1}$ is related to the internal resistance of the power source R₀, the internal resistance of coils R₁ 、R₂, the mutual inductance M and the load impedance R_L. Denote the normalized output power as $\lambda$ [23].

$$\lambda = \frac{{P_{out} }}{{P_{{out{ - }\max }} }} = \frac{{4R_{2} + R_{L} (R_{0} + R_{1} )w_{R}^{2} M^{2} }}{{[R_{2} + R_{L} (R_{0} + R_{1} ) + w_{R}^{2} M^{2} ]^{2} }}$$

(18)

The functional relationship shown in Eq. (18) is presented in Fig. 2. Obviously, the value range of $\lambda$ is [0,1). The working frequency range of the WPT system in this article is 0–120 kHz, and the skin effect of the coil wire is negligible. Therefore, this study assumes that the coil impedance does not change with frequency. As shown in Fig. 2, $\lambda$ first increases and then reduces with the increase of $w_{R}$, and $\lambda$ has a maximum value.

Fig. 2

Frequency response curve of normalized output power

According to Eqs. (13) and (14), the transmission efficiency $\eta$ can be calculated as

$$\eta_{{}} = \frac{{P_{{{\text{out}}}} }}{{P_{{{\text{in}}}} }} = \frac{{\left( {w_{R} M} \right)^{2} R_{L} }}{{\left( {R_{2} + R_{L} } \right)\left[ {(R_{2} + R_{L} )R_{0} + R_{1} (R_{2} + R_{L} ) + \left( {w_{R} M} \right)^{2} } \right]}}$$

(19)

For the medium transmission distance condition, $w_{R}$ is in the range of 0 kHz ~ 20 MHz, at this time R_r <  < R_i, R_r can be ignored, and Eq. (19) is rewritten as

$$\eta_{{}} = \frac{{P_{{{\text{out}}}} }}{{P_{{{\text{in}}}} }} = \frac{{\left( {w_{R} M} \right)^{2} R_{L} }}{{\left( {m\sqrt w + R_{L} } \right)\left[ {(m\sqrt w + R_{L} )(m\sqrt w + R_{0} ) + \left( {w_{R} M} \right)^{2} } \right]}}$$

(20)

Under normal circumstances, the order of magnitude of m is 10⁻⁵ [17]. When $w_{R}$ is about 100 kHz studied in this paper, as shown in Fig. 3, $\eta$ increases monotonically with $w_{R}$ increasing, and the rate of increasing first fast, then slow, and finally flat.

Fig. 3

Transfer efficiency curve of low-frequency

However, when $w_{R}$ is in the range of about 10 MHz, due to the existence of the high-frequency skin effect, the coil ohmic loss resistance R_i will increase with the increase of $w_{R}$, so the curve between $\eta$ and $w_{R}$ is shown in Fig. 4. The system transmission efficiency first increases and then decreases with the increase of $w_{R}$, and there is a maximum value.

Fig. 4

Transfer efficiency curve of high-frequency

Combining Figs. 2 and 3, it can be observed that when $w_{R} = w_{R1}$, the system output power reaches the maximum value, but the system transmission efficiency is still rising at this time, and when the system transmission efficiency gradually flattens to reach the maximum value, the system output power is reduced to a lower value, the output power and the transmission efficiency cannot reach the synchronization. And if $w_{R}$ of MCRWPT system is MHz or even GHz, the skin effect of the coils wire cannot be ignored, and the coils resistance will increase greatly. As shown in Fig. 4, the system transmission efficiency has an extreme value. Take into account $P_{out - \max }$, the optimal resonant angular frequency for power-efficiency synchronization will be obtain.

3 Tradeoff Analysis between the Output Power and Efficiency

After considering the optimization function in mathematics, denote a whole new power-efficiency composite function as $F\left( {w_{R} ,\alpha } \right) = \alpha \lambda + \left( {1 - \alpha } \right)\eta$, $\alpha$ of which is the power-efficiency weight factor. The value of $\alpha$ is 0–1.

$$F(w_{R} ,\alpha ) = \frac{{4\alpha (R_{2} + R_{L} )(R_{0} + R_{1} )w_{R}^{2} M^{2} }}{{[(R_{2} + R_{L} )(R_{0} + R_{1} ) + w_{R}^{2} M^{2} ]^{2} }} + \frac{{\left( {1 - \alpha } \right)\left( {w_{R} M} \right)^{2} R_{L} }}{{\left( {R_{2} + R_{L} } \right)\left[ {(R_{2} + R_{L} )R_{0} + R_{1} (R_{2} + R_{L} ) + \left( {w_{R} M} \right)^{2} } \right]}}$$

(21)

Assuming that the structure and parameters of MCRWPT system are certain, set up parameters as follows: U_S = 12√2sinwV, R₁ = R₂ = 0.05Ω, k = 0.2, L₁ = L₂ = 60μH. Draw $F\left( {w_{R} ,\alpha } \right)$ shown in Eq. (21). It can be seen from Fig. 5 that when $\alpha$ is close to 1, the curve is close to the output power function curve and on the contrary, the curve is close to the transmission efficiency function curve. Based on this, it can be calculated with designated $\alpha$ according to actual engineering needs.

Fig. 5

Power-Efficiency compound function curve

In order to obtain the optimal angular resonant frequency value, the derivative of the power-efficiency composite function $F\left( {w_{R} ,\alpha } \right)$ shown in Eq. (21) is as follows:

$$\frac{\partial F}{{\partial w_{R} }} = \frac{{2(R_{0} + R_{1} )w_{R} M^{2} [(1 - \alpha )(R_{2} + R_{L} )(R_{0} + R_{1} )R_{L} + (1 - \alpha )w_{R}^{2} M^{2} R_{L} + 4\alpha (R_{2} + R_{L} )^{2} (R_{0} + R_{1} ) - 4\alpha (R_{2} + R_{L} )w_{R}^{2} M^{2} ]}}{{[(R_{2} + R_{L} )(R_{0} + R_{1} ) + w_{R}^{2} M^{2} ]^{3} }}$$

(22)

when $\partial F/\partial w_{R} = 0$, $F\left( {w_{R} ,\alpha } \right)$ reaches the maximum value and the optimal resonant angular frequency $w_{R3}$ can be given by

$$w_{R3} = \sqrt {\frac{{(R_{2} + R_{L} )(R_{1} + R_{0} )[(4\alpha R{}_{2} + (3\alpha + 1)R_{L} ]}}{{[4\alpha R_{2} + (5\alpha - 1)R_{L} ]M^{2} }}}$$

(23)

when the parameters of MCRWPT system and $\alpha$ are given, $w_{R3}$ can be obtained. It will simplify the experiment process and it is not necessary to increase $w_{R}$ continuously.

4 Experimental Verification

4.1 Experimental Testing

In order to verify the correctness of $w_{R3}$ obtained by the above-mentioned power and efficiency analysis, a practical MCRWPT system is fabricated as shown in Fig. 6. The system can be divided into four modules: charging host system, resonance network, rectifier filter module and output load. Among them, the input voltage at the transmitting terminal is an adjustable DC Power Supply. The high frequency inverter network is mainly composed of 4 switch tubes of the full-bridge, the resonant network adopts the SS resonance mode, the rectifier filter module is composed of a full-wave rectifier bridge and a filter capacitor, and finally there is an output load module. In the experiment, the transmitting coil and the receiving coil are wound with 200*0.1 mm Litz wire, and the parameters such as wire diameter, radius, and the number of turns are consistent and kept unchanged. The specific parameters are shown in Table 1. The physical platform is shown in Fig. 7.

Fig. 6

Structure of the experiment platform

Table 1 Parameters of the transceiver coils

Fig. 7

The MCRWPT system experiment platform

In Table 1, the value of N is 20. The parameters of the transceiver coils in this experiment are substituted into the Eqs. (2), (3) and (4), and according to the Eq. (20), the relationship curve between transmission efficiency and coil turns is obtained by Matlab simulation. It can be seen from Fig. 8 that 20 is more suitable for N. If the number of turns continues to increase, the transmission efficiency of the system will not improve a lot.

Fig. 8

The efficiency curve with turns

The transmitting coil and the receiving coil are coaxially placed in parallel, the excitation power is set to 12 V. Adjust the distance between transceiver coils and the value of load impedance. Observe the change of $P_{out}$ and $\eta$ through these dynamical adjustments. The experimental results are shown in Fig. 9.

Fig. 9

Comparison of the efficiency and the output power under different distances and loads. a D = 5 cm,R_L = 25Ω; b D = 10 cm, R_L = 25Ω; c D = 15 cm, R_L = 25Ω; d D = 5 cm, R_L = 50Ω

4.2 Data analysis

Assuming that this experiment requires the $P_{out}$ and $\eta$ of the system to have the same weight, take $\alpha$ = 0.5 and k = 0.2 into Eq. (18) and $w_{R3}$ is calculated as 51 kHz. According Eq. (3), D is about 5 cm. So observe Fig. 9a, it can be seen that when $w_{R}$ = 40 kHz, the output power reaches the maximum value and then decreases obviously with the increase of $w_{R}$. When $w_{R}$ reaches 120 kHz, the efficiency remains essentially constant, but the output power drops to the minimum. In a word, when $w_{R}$ is around 51 kHz, the system maintains a large output power and efficiency. When the load impedance is changed to 50Ω, $w_{R3}$ is calculated as 73.25 kHz according to Eq. (18). As shown in Fig. 9d, when $w_{R}$ reaches 60 kHz, the output power is the maximum, but the transmission efficiency is still rising. When $w_{R} = w_{R3}$, the transmission efficiency remains unchanged and the output power decreases only by 6%. In both cases, $w_{R3}$ meets the system requirement on the whole.

When the weight between $P_{out}$ and $\eta$ is 8:2, take $\alpha$ = 0.8, k = 0.1, and the corresponding resonant angular frequency $w_{R3}$ is calculated as 90.3 kHz. D is 15 cm approximately. As shown Fig. 9c, it can be seen that when $w_{R}$ = 80 kHz, the output power reaches the maximum value but the transmission efficiency is still rising slowly with the increase of $w_{R}$. When $w_{R}$ is 90.3 kHz, the output power of the system is still high, but the transmission efficiency is relatively low, which meets the requirement of $P_{out}$ and $\eta$ ratio is 8:2.

The above results show that the optimal resonant frequency design under the consideration of weight factor of the power and transmission efficiency in this paper is reasonable. It still holds true when the distance between the transmitting and the receiving coil changes, or say the coupling coefficient changes.

Based on frequency control, the maximum efficiency tracking control is generally used to improve the transmission efficiency of the system. The transmission efficiency after optimization is generally 80% [24], but the maximum transmission efficiency is obtained by ignoring the maximum output power of the system. Also in this paper, as shown in Fig. 5, when $\alpha$ is taken as 0, the $F\left( {w_{R} ,\alpha } \right)$ curve becomes an efficiency curve, that is, the requirement of output power is ignored. In this experiment, when D = 5 cm, the transmission efficiency is the highest, reaching 66% at 120 kHz, which is less than 80% in reference [24]. However, when D = 15 cm, the transmission efficiency is 24%, while the transmission distance in reference [24] is 10 cm, the efficiency has dropped to 25%. When the transmission efficiency and the output power are both considered, in [25], the improvement of efficiency is only 19% at first, and then it is optimized from the perspective of mutual inductance parameters. In practical application, it is more complex to obtain the accurate mutual inductance than to calculate the resonant angular frequency value [26]. But in this paper, for any ratio, the optimal resonant angular frequency can be obtained easily.

5 Conclusion

This paper uses the equivalent circuit theory to model the magnetic coupling resonance wireless power transmission system, analyzes the effect of the resonant angular frequency on the output power and efficiency of the pure resistive load system under the low-frequency condition, and proves that when the distance between the transmitting and receiving coils is determined, the optimal resonant angular frequency for power-efficiency synchronization does not exist theoretically. And on this basis, the tradeoff between the output power and the transmission efficiency is analyzed and then the best resonant angular frequency can be obtained by a given power-efficiency percentage. In addition, three sets of experiments verified the correctness of the theoretical analysis. This study provides relevant theoretical guidance for the energy saving of the MCRWPT system, so as to achieve the optimal transmission of WPT system.