Modeling and analysis of magnetic resonance wireless transmission coil

Inductive power transmission technology is a novel, flexible, and practical power supply technology.¹ It is based on the principle of electromagnetic induction, the electromagnetic field as a medium, and modern control theory as an implementation tool. However, its transmission distance is still short and can only be realized in a relatively small range. Wireless power transmission technology based on the magnetic resonance mode has become a research focus for many scholars in China and abroad. It began in 2007 when MIT scholars proposed the principle of wireless power transmission based on the magnetic resonance mode, which successfully emitted light within a range of 2 m. The 60 W bulb tests have since sparked a research boom in the wireless transmission field.² 

In practical application systems, a well-designed transmission coil has a great impact on the performance of the system. Different coil systems have a different impact on the transmission power and efficiency of the transmission system due to the differences in self-inductance, mutual inductance, and their own internal resistance.^3–5 

In this paper, some methods to quantitatively analyze the coil characteristics that have a significant impact on the system are presented. By analyzing the influence of the distance between different coils on the mutual inductance between the coils, the relationship between the coil spacing and the efficiency and power of energy transmission is obtained; by studying the influence of the coil’s own wire diameter, length, and material on the coil’s self-inductance, internal resistance, and energy transmission efficiency and power; and by studying the different resonance frequencies the coil itself is designed for. Based on applying an external circuit, an experiment on the output characteristics of the transmission technology was carried out, and the relationship between the output power efficiency of the wireless transmission and the system resonance frequency was clarified.

The power transmission system in magnetic resonance mode is shown in Fig. 1. Compared with the induced electric energy mode, the magnetic resonance mode has the following two advantages:^6,7

1. The inverter frequency of the energy conversion link at the transmitter has been greatly increased, and its frequency has reached MHz.

2. Two resonance links are added. The coils used in the resonance link are all in an open state; their own distributed capacitance is used as a resonance capacitance and has a fixed topology, while the transmitting and receiving ends adopt different topologies.

The compensation of the winding inductance of the transmitter and receiver can reduce the reactive power capacity of the system so as to realize the maximum energy transmission of the system. Because the topological structure of the compensation link between the transmitter and receiver is different, it can be divided into four topological structures of PSSS, PSSP, SSSS, and SSSP, as shown in Fig. 1. In Fig. 1, P and S represent the parallel type and series type compensation, respectively. In the magnetic resonance mode, the wireless transmission of electric energy can be realized at a medium distance, which can reach several times the diameter of the coil. At this time, the resonance coil 1 and the secondary receiving coil, the transmitting coil and the resonance coil 2, and the mutual inductance between the transmitting coil and the secondary receiving coil can be ignored. The rough high-frequency inverter circuit is represented by U_i and I_i, respectively, U_i represents the voltage-type equivalent power supply, and I_i represents the current-type equivalent power supply. The inductance is represented by L_s, the resonance coil 2 is represented by L_r, the compensation capacitance of the inductance L_p at the transmitting end is represented by C_p, and the receiving end is represented by C_o; the equivalent series resistance of the transmitting end is represented by R_p, and the receiving end is represented by R_L. The effective series resistance is represented by R_s, the resonance coil 2 is represented by R_r, R_o is the load, and the reflection impedance from the receiving end to the resonance coil 2, the resonance coil 2 to the resonance coil 1, and the resonance coil 1 to the transmitting end is represented by Z_rl, Z_sr, and Z_ps. The mutual inductance between the transmitting end and the resonant coil 1, between the two resonant coils, and between the resonant coil 2 and the receiving end are represented by M_ps, M_sr, and M_rl, respectively.

The ohmic loss resistance R_o and the radiation loss resistance R_ra constitute the main loss resistance of the coil at high frequency. Because the ohmic loss resistance is much larger than the radiation loss resistance in the magnetic resonance mode power transmission system, the radiation loss is generally ignored.

It can also be clearly seen from Eq. (10) that the main factors that affect the output power and efficiency are the resonance frequency, the mutual inductance and self-inductance of the coil, and the internal resistance of the coil.

The biggest difference between the magnetic coupling wireless power transmission system and the traditional electromagnetic induction wireless power transmission technology is that the former can transmit electric energy efficiently at a medium distance (several times the length of the transmitting and receiving coil). In this section, through the output power and efficiency of the magnetic coupling wireless power transmission system deduced earlier, the principle of the magnetic coupling wireless power transmission system capable of high-efficiency transmission at medium distances is analyzed, and the main factors affecting the efficiency of the system are extracted. This provides a theoretical basis for simulation.

It can be seen from Eq. (11) that when the distance between the resonant coils increases, except for the M_sr, which decreases with the increase in distance, the other parameters do not change significantly due to their parameters and relative positions. In summary, with the increase in coil spacing, the output power of the power transmission system can also continue to increase, which enables the magnetic coupling wireless power transmission system to achieve and realize medium-distance and high-efficiency power transmission.¹¹ 

The parameter ω₀²M_sr²L_l² will be smaller when the distance continues to increase. When the coil mutual inductance M_sr and the coil internal resistance R_s and R_r have obvious effects on the output power, the expression of the output power is Eq. (8). At this time, the mutual inductance M_sr and the internal resistance R_s, R_r are both working, followed by the coil mutual inductance M_sr. There is an optimal value to maximize the output power.

At this time, the resonance coil exceeds the range of the middle distance, and as the distance increases, the mutual inductance M_sr decreases. Under the condition that other parameters are almost unchanged, it can be seen from Eq. (12) that the output power decreases as the distance increases. The system output efficiency in Eq. (9) is similar to the analysis in Eq. (12). Under the medium distance condition, the efficiency first increases and then decreases with the increase of the distance, that is, with the change of M_sr, the efficiency has an optimal value.

At the same time, it can be clearly seen from the simplified expression that the excitation frequency is the resonant frequency ω₀, and the mutual inductance of the resonant coil M_sr and the coil internal resistance R_r, R_s constitute the main factors affecting the output power and efficiency of the wireless power transmission system. Among them, the mutual inductance M_sr and resonant frequency ω₀ of the resonance coil can be changed through the optimal design of the circuit and the structure design of the coil. The higher the frequency, the greater the mutual inductance, and the longer the optimal and maximum distances for power transmission between coils can be; resistance R_r, R_s need to change the properties of the coil itself, such as changing the wire diameter materials to improve the properties. The smaller the internal resistance, the higher the efficiency. Through the above-mentioned analysis, it can be seen that the increase in mutual inductance between the resonant coils, the increase in resonant frequency, and the decrease in the internal resistance of the resonant coils can all achieve the extension of the power transmission distance.¹² 

Ansoft Maxwell is software based on Maxwell’s differential equations using finite element discrete form to transform electromagnetic field calculations in engineering into huge matrix solutions. It has the ability to establish two-dimensional and three-dimensional models of static magnetic fields, eddy current fields, transient magnetic fields, electrostatic fields, AC and DC conduction electric fields, and so on. This article mainly uses its 3D modeling and transient magnetic field solution functions. The four-coil resonance model established in this paper based on electromagnetic simulation software Ansoft Maxwell is shown in Fig. 2. The following will use this as a basis to verify the theory described in the previous article.

Both the resonant coil 1 and the resonant coil 2 are copper wires with a rectangular cross section and a circumscribed circle radius of 1.5 cm wound into a disk-shaped coil. The inner diameter of the resonant coil is 20 cm, 40 turns and 50 turns are wound, and the coil spacing between each turn is 0.1 cm.

First, simulate its own inductance parameters. Take a cross section on each coil and apply a current of 10 A to simulate the static magnetic field type. Set the distance D₁ between the two resonance coils as a parameter variable, and the self-inductance and mutual inductance parameters obtained by simulation are shown in Table I.

The items in Table I from left to right are the self-inductance of the transmitting coil, the mutual inductance between the transmitting end and the resonance coil 1, the self-inductance of the resonance coil, the mutual inductance between the resonance coil 1 and the resonance coil 2, and the self-inductance of the receiving coil.

It can be seen from Table I that the self-inductance of the receiving end of the transmitting end, the mutual inductance between the transmitting end and the resonance coil 1, the receiving end, and the resonance coil 2 do not change much with the distance, while the mutual inductance between the resonance coil 1 and the resonance coil 2 varies with the distance. This is also consistent with the theoretical analysis results in Sec. III, that is, when the distance between the resonant coils increases, except for M_sr, which decreases with the increase in distance, the other parameters have no obvious changes due to their parameters and relative positions. The output power of the system increases as the distance increases, which is an important reason why the magnetic coupling wireless power transmission system can transmit power at medium distances with high efficiency. It can be seen from Table I that its self-inductance is 53 µH. According to the simulation results, the following line chart can be generated for further analysis.

It can be seen from Fig. 3 that the relationship between the coil self-inductance and the distance between the coils is not great, which is consistent with the definition and nature of the self-inductance in the theory. Figure 4 shows that the mutual inductance between the transmitting end coil and the receiving end coil decreases linearly from the middle distance to the long distance, and the mutual inductance is basically zero when the distance between the two coils is 60 mm, indicating that the coil distance has a great influence on the system circuit parameters.

Substituting the data obtained in the above-mentioned simulation process into the output power and efficiency expression derived in the previous article, connecting the dots into a line to get the following broken line graph, which can clearly see that the excitation frequency and the resonance coil distance affect the system output power and the effect of efficiency, as shown in Figs. 5 and 6.

From Eq. (13), C₁ = C₂ = 1.2 pF, and the load resistance is 100 Ω. According to the above data, you can edit the circuit as shown in Fig. 7 in the Maxwell Circuit Editor software as required and connect it to the coil model.

According to the requirements, the same space placement method is introduced into the circuit according to different distances and different excitation (resonant) frequencies under the same distance of the coils. The simulation is carried out in the transient magnetic field with the power frequency as a variable, and the excitation frequency is changed in the external circuit. The model used in this article is three-dimensional, and it takes a certain time to run to get the result. If you need to change the operating efficiency and operating time, you can achieve the effect by changing the parameters such as the fault tolerance rate, the number of iterations, the grid size, and the calculation step length. For example, increasing the fault tolerance rate from the default 1%–10% and the number of iterations from the default 10 to 5 can shorten the simulation time, but at the same time, it also sacrifices the accuracy rate, which will bring a certain error.

In Fig. 2, the four coils are coaxially placed in parallel, the simplified distance is 25 cm, and the resonance frequency is set to 20 MHz. According to the transmission efficiency expression derived earlier, the following curve can be obtained according to the different simulation ideas. When the space is placed in the same way and the distance is different, make the two coaxial and parallel to be placed directly. When the distance is 1, 5, 10, 15, 20, 25, or 30 cm, by changing the sine excitation, the frequency that is obtained at different frequencies at the distance, and substituting into Eq. (9), the following graphs of transmission efficiency and excitation frequency at different distances can be obtained, as shown in Fig. 8.

As can be seen from the above-mentioned figure, when the distance between the transmitting end coil and the receiving end coil is small, which belongs to the short-distance transmission of electric energy, the transmission efficiency is high in the frequency range of 15–25 MHz, and it drops rapidly when the frequency is too high or too low. When the distance between the transmitting end coil and the receiving end coil is large, which is a medium-distance power transmission, the power transmission efficiency will reach a peak at a frequency of about 20 MHz, and the efficiency will decrease as the frequency gradually moves away from the peak frequency. This shows that under the condition of medium-distance power transmission, there is an optimal system resonance frequency, which maximizes the system transmission efficiency.

The following conclusions can be obtained by analyzing the self-inductance and mutual-inductance values of the transmission coil and comparing the above “output power/output efficiency-coil distance” and “excitation frequency-transmission efficiency” curves:

1. The wireless power transmission power increases first and then decreases with the increase in coil distance. When the coil distance reaches 80 mm, the transmission power reaches its peak value. This shows that in the simulation system in this article, the optimal distance of 70 mm to reach the power is consistent with the theoretical analysis result. The simulation shows that in the design process, it is necessary to select the closest coil parameters and the equivalent parameters of the system to obtain the best transmit power and transmission distance.

2. The wireless power transmission efficiency increases first and then decreases with the increase in coil distance. When the coil distance reaches 70 mm, the transmission efficiency reaches its peak, and the peak distance is close to the transmission power transmission distance. This shows that in the simulation system in this article, the resonance point is the biggest factor that affects the transmission power and efficiency of the system. For the same system, as long as the appropriate distance is selected to obtain the optimal resonance, both the transmission power and the transmission efficiency will correspondingly reach a good level, which is consistent with the theoretical results. The simulation shows that in the design process, according to the coil parameters and the equivalent parameters of the system, select the transmission distance close to the optimal transmission efficiency.

3. The magnetic field after coupling resonance is mainly concentrated near the coil and hardly diffuses into the surrounding space.

4. In this simulation, under the premise that there is a certain distance between the two coils (the distance between the resonant coils is 25 mm), the coupling of the magnetic field and the transmission of energy are still realized, which verifies that the magnetic coupling resonance is suitable for medium and short distance energy transmission. As the simulation time progresses, it can be seen that the coupling degree of the two coils gradually deepens and becomes stable, and the coupling reaches its strongest at the set resonant frequency.

5. The close distance between the two coils will produce the so-called “frequency splitting phenomenon,” which causes the two coils to fail to reach the maximum transmission efficiency at resonance, but it can make the two coils achieve greater transmission efficiency within a certain frequency bandwidth.

6. When the two coils are far apart and there is no “frequency splitting phenomenon,” the maximum transmission efficiency is reached at the resonance frequency, and as the distance increases, the maximum transmission efficiency gradually decreases.

The authors have no conflicts to disclose.

Liping He: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Writing – original draft (lead). Pengfei Lu: Investigation (equal); Methodology (equal); Validation (equal). Hongshun Liu: Resources (equal); Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.