Mutual inductance calculation of the one-primary multiple-secondary coil system

Abstract

It is of great significance to accurately calculate the mutual inductance of one-primary multiple-secondary coil system (OPMSC) in order to improve energy transmission efficiency, measurement accuracy of magnetic sensor, etc. In this paper, a method based on time harmonic field to calculate such mutual inductance is proposed. Both the high frequency alternating current model and the time-harmonic electromagnetic field calculation method are used to ensure engineering practicability of the method. The current of each coil and the relative position between coils are considered in the proposed method. Moreover, according to the full derivation of a new analytical method. the extremum phenomenon of mutual inductance is described. And this method is verified by both finite element simulations and experiments, which proves that the method is effective for optimizing the relative position of the multiple coils when mutual influence among all the coils is taken into account.

Keywords

Electromagnetic analysis mutual inductance multi-coil system

1. Introduction

Nowadays, more and more multi-coil systems are utilized for energy exchange or information transmission, such as the multi-coil wireless power transfer system [1], contactless sensor [2], and multi-coil magnetic field sensor [3]. In these practical engineering systems, the mutual inductance among the multiple coils is a key parameter. However, when designing these systems, the mutual inductance between coils is determined based upon experimental parameter instead of theoretical calculation. Generally, the finite element method can be used to evaluate the required magnetic field and inductance, but it is difficult to solve the so-called “inverse problem”, i.e. designing the geometric dimensions (such as diameters, size, positions, etc.) of coils according to the required electromagnetic parameters. In short, it is time-consuming and laborious to design a sensor or a wireless power transmission system without the guidance of theoretical calculation.

In order to calculate the mutual inductance between coils analytically, some methods have been proposed. Grover and Babic [4–7] utilized Neumann’s formula and elliptical integral to calculate the mutual inductance at any desired positions. However, these calculations are based on the static magnetic field, which cannot accurately reflect actual circumstances. Many other methods [8–10] have been proposed to consider the influence caused by coils in multi-coil system, the mutual inductance for the system is still an undetermined parameter which means these multi-coil systems cannot be directly used in practical engineering program. In the reference [11], the mutual inductance has been calculated by the usage of Neumann’s formula, but the mutual inductance between the two coils is an independent parameter relative to the multi-coil system. Moreover, in the aspect of calculating the mutual inductance between two coils in a multi-coil system, it is invalid because the mutual influence between any two coils cannot be fully considered.

In this paper, a new analytical method to calculate mutual inductance for the multi-coil system has been proposed. Different from other methods, the analysis in this paper is based on a time-harmonic electromagnetic field instead of a static magnetic field. In the most practical applications, such as WPT system and non-contact sensor, it is the high frequency alternating current that excites the working magnetic field environment. Therefore, the time-harmonic electromagnetic field is much closer to the practical applications. On the other hand, this analytical method is suitable for the multi-coil system. Moreover, this analytical calculation method can explain the phenomenon that other coils’ influence on the mutual inductance between two coils in the multi-coil system. And a phenomenon of the optimum position for coil3 has been discovered, which also proved the existence of mutual influence among multiple coils. In order to verify the correctness of this calculation method, the simulation is executed by the finite element simulation tools COMSOL Multiphysics, and the experiment has been carried out. Comparing the results of calculation, simulation and experiment, it can be obtained that this kind of calculation is effective and suitable for practical application. Furthermore, it can also be utilized to give a pre-experiment guidance for engineers to solve the problem of the multi-coil system in practical applications.

Fig. 1.

The diagrammatic sketch of OPMSC system.

2. Model of system

The multiple circular filament coils has been shown in Fig. 1, where multiple coils have been placed along the Z^′ axis. The time harmonic electromagnetic field calculation method has been adopted to analog the actual high frequency alternating current (AC) working condition of contactless sensor or wireless power transmission system. Therefore, the sinusoidal alternating current flowing through each coil has been described as i_n(t), and the RMS (Root mean square) of this current is expressed as I_n. The subscript n represents the order of coils from bottom to top, the direction of current is shown in the Fig. 1. Similarly, the radius of each coil has been described as 𝜌_n. Because the whole system is an axisymmetric model, so the position of each coil can be kwon as the parameter z_n. A cylindrical coordinate system (O-𝜌Φz) has been established for the above geometric model.

3. Calculation method

3.1. General case

In this section, the mutual inductance calculation method for three-coil system with the metal medium will be introduced in detail. Firstly, Maxwell Equations in time harmonic electromagnetic field is $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}{\nabla}\times \boldsymbol{H}=\boldsymbol{J}_{s}+({\gamma}+j{\omega}{\varepsilon})\boldsymbol{E}\\ {\nabla}\times \boldsymbol{E}=-j{\omega}\boldsymbol{B}\\ {\nabla}\cdot \boldsymbol{B}=0\\ {\nabla}\cdot \boldsymbol{ D }={\rho}.\end{array} & & \displaystyle\end{eqnarray}$ (1)

In this analysis model established in Section 2, all the objects has good axisymmetric properties, the cylindrical coordinate system is adopted. Therefore, the constraint equation of circumferential component of magnetic vector potential can be written $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}^{2}\boldsymbol{A}_{i{\phi}}+\left({k_{i}}^{2}-\frac{1}{{\rho}^{2}}\right)\boldsymbol{A}_{i{\phi}}=0\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {k_{i}}^{2}=-j{\omega}{\mu}_{i}({\gamma}_{i}+j{\omega}{\varepsilon}_{i})\quad i=1,2\ldots 6.\end{eqnarray}$ (3)

In the (4), the A _iΦ is the magnetic vector potential of the ith region. In the (6), the parameter k_i is represent the properties of dielectric materials. In the analytical calculation of electromagnetic field for multi-coil system, the boundary conditions can be written as below [12] $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle \lim _{z\rightarrow z_{i}-0}\boldsymbol{A}_{i{\phi}}=\lim _{z\rightarrow z_{i}+0}\boldsymbol{A}_{(i+1){\phi}}\\ \displaystyle \lim _{z\rightarrow z_{i}-0}{\displaystyle \frac{\partial \boldsymbol{A}_{i{\phi}}}{\partial z}}-\lim _{z\rightarrow z_{i}+0}{\displaystyle \frac{\partial \boldsymbol{A}_{i{\phi}}}{\partial z}}={\mu}_{0}I_{j}{\delta}({\rho}-{\rho}_{j}){\delta}(z-z_{j}).\end{array} & & \displaystyle\end{eqnarray}$ (4)

The boundary conditions of metal material medium $\begin{eqnarray}\displaystyle \lim _{z\rightarrow z_{i}-0}\frac{\partial \boldsymbol{A}_{i{\phi}}}{\partial z}-\lim _{z\rightarrow z_{i}+0}\frac{\partial \boldsymbol{A}_{i{\phi}}}{\partial z}=0. & & \displaystyle\end{eqnarray}$ (5)

According to the constraint equations and boundary conditions, the general solution of analytical model can be expressed as below $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\boldsymbol{A}_{i}({\rho},z)=\displaystyle \int _{0}^{\infty }[C_{1}J_{1}({\lambda}{\rho})+C_{2}Y_{1}({\lambda}{\rho})][C_{3}e^{uz}+C_{4}e^{-uz}]d{\lambda}\\[12.0pt] u=\sqrt{{\lambda}^{2}-k_{i}^{2}}.\end{array} & & \displaystyle\end{eqnarray}$ (6)

The parameter C₁, C₂, C₃, C₄ are the undetermined coefficient in (8). J₁(𝜆𝜌) and Y₁(𝜆𝜌) are first-order Bessel functions of the first and second kind, respectively. In order to calculate the mutual inductance between coils, the magnetic flux of one single-turn coil will be expressed as below $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{\it\Phi}=\iint _{S_{S}}\boldsymbol{B}_{i}d\boldsymbol{S}_{s}=\iint _{S_{S}}({\nabla}\times \boldsymbol{A}_{i})d\boldsymbol{S}_{s}=\oint _{l_{s}}\boldsymbol{A}_{i}\cdot d\boldsymbol{l}_{s}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle d\boldsymbol{l}_{s}={\rho}_{i}d{\varphi}\boldsymbol{e}_{{\varphi}}.\end{eqnarray}$ (8)

The mutual inductance between kth coil and jth coil can be written $\begin{eqnarray}\displaystyle M_{kj}=\frac{\boldsymbol{\it\Phi}_{k}}{I_{j}}=\frac{\displaystyle \oint _{l_{s}}\boldsymbol{A}_{i}\cdot {\rho}_{i}d{\varphi}\boldsymbol{e}_{{\varphi}}}{I_{j}}. & & \displaystyle\end{eqnarray}$ (9)

3.2. Special case

In order to explain the effect of other coils motion on mutual inductance analytical calculation of two coils, and verify the validity of this new formulas, we present the following set of examples. In the model from Fig. 1, the positions of coil1 and coil2 are fixed at z₁ and z₂, respectively. And z₂ > z₁. The position of coil3 can be moved up and down along the z^′ axis. In order to imitate the motion of coil3, the above model is discussed in three cases, and when z₃ < z₁ < z₂, the boundary condition of one-primary two-secondary coil system can be written as follow $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle \lim _{z\rightarrow z_{3}-0}A_{1{\phi}}=\lim _{z\rightarrow z_{3}+0}A_{2{\phi}}\\[12.0pt] \displaystyle \lim _{z\rightarrow z_{3}-0}{\displaystyle \frac{\partial A_{1{\phi}}}{\partial z}}-\lim _{z\rightarrow z_{3}+0}{\displaystyle \frac{\partial A_{2{\phi}}}{\partial z}}={\mu}_{0}I_{3}{\delta}({\rho}-{\rho}_{3}){\delta}(z-z_{3})\\[14.0pt] \displaystyle \lim _{z\rightarrow z_{1}-0}A_{2{\phi}}=\lim _{z\rightarrow z_{1}+0}A_{3{\phi}}\\[14.0pt] \displaystyle \lim _{z\rightarrow z_{1}-0}{\displaystyle \frac{\partial A_{2{\phi}}}{\partial z}}-\lim _{z\rightarrow z_{1}+0}{\displaystyle \frac{\partial A_{3{\phi}}}{\partial z}}={\mu}_{0}I_{1}{\delta}({\rho}-{\rho}_{1}){\delta}(z-z_{1})\\[14.0pt] \displaystyle \lim _{z\rightarrow z_{2}-0}A_{3{\phi}}=\lim _{z\rightarrow z_{2}+0}A_{4{\phi}}\\[14.0pt] \displaystyle \lim _{z\rightarrow z_{2}-0}{\displaystyle \frac{\partial A_{3{\phi}}}{\partial z}}-\lim _{z\rightarrow z_{2}+0}{\displaystyle \frac{\partial A_{3{\phi}}}{\partial z}}={\mu}_{0}I_{2}{\delta}({\rho}-{\rho}_{2}){\delta}(z-z_{2}).\end{array} & & \displaystyle\end{eqnarray}$ (10)

As a result, the mutual inductance in the first case can be obtained as below $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}M_{12}^{1}={\pi}{\rho}_{2}\displaystyle \int _{0}^{\infty }\left[{\displaystyle \frac{{\mu}_{0}{\rho}_{1}J_{1}\left({\lambda}{\rho}_{1}\right)e^{uz_{1}}}{e^{uz_{2}}}}+{\displaystyle \frac{{\mu}_{0}{\rho}_{2}I_{2}J_{1}({\lambda}{\rho}_{2})}{I_{1}}}+{\displaystyle \frac{{\mu}_{0}{\rho}_{3}I_{3}J_{1}({\lambda}{\rho}_{3})e^{uz_{3}}}{e^{uz_{2}}I_{1}}}\right]J_{1}({\lambda}{\rho}_{2})d{\lambda}\\[16.0pt] M_{23}^{1}={\displaystyle \frac{\boldsymbol{\it\Phi}_{3}^{1}}{I_{2}}}={\pi}{\rho}_{3}\displaystyle \int _{0}^{\infty }\left[\frac{{\mu}_{0}{\rho}_{1}I_{1}J_{1}({\lambda}{\rho}_{1})e^{uz_{3}}}{I_{2}e^{uz_{1}}}+{\displaystyle \frac{{\mu}_{0}{\rho}_{2}J_{1}({\lambda}{\rho}_{2})e^{uz_{3}}}{e^{uz_{2}}}}+{\displaystyle \frac{{\mu}_{0}{\rho}_{3}I_{3}J_{1}({\lambda}{\rho}_{3})}{I_{2}}}\right]J_{1}({\lambda}{\rho}_{3})d{\lambda}.\end{array} & & \displaystyle\end{eqnarray}$ (11)

4. Validation

4.1. The validation of two-coil system

Because the traditional calculation of mutual inductance is limited to two-coil system, such as the classical computing method proposed by the reference [13]. In order to validate the correctness of proposed calculation method, the conventional coaxial two-coil system has been established. The radius of both circular filament coils is 80 mm, and the distance between the two coils varies from 1 mm to 88 mm at a step of 1 mm. The current in the coil1 is I₁. By using the calculation method proposed above, the mutual inductance between two coils can be obtained as below $\begin{eqnarray}\displaystyle M_{12}=\frac{\boldsymbol{\it\Phi}_{1}^{1}}{I_{1}}={\pi}{\rho}_{2}\int _{0}^{\infty }{\mu}_{0}{\rho}_{1}J_{1}({\lambda}{\rho}_{1})e^{-u|z_{2}|}d{\lambda}. & & \displaystyle\end{eqnarray}$ (12)

According to the reference [13], the mutual inductance between two coils can be obtained by the usage of following expression $\begin{eqnarray}\displaystyle M=\frac{{\mu}_{0}R_{s}}{{\pi}}\int _{0}^{2{\pi}}\frac{[p_{1}\cos {\varphi}+p_{2}\sin {\varphi}+p_{3}]{\Psi}(k)}{k\sqrt{V_{0}^{3}}}d{\varphi}. & & \displaystyle\end{eqnarray}$ (13)

By using the MATLAB, the calculation results has been showed in the Fig. 2. From the results, we find that the two curves are in good agreement, which proves the validity of the method proposed by this paper. Obviously, the absolute error between two analytical methods results is too small enough to be neglected, which means the proposed method is correct in a certain degree.

Fig. 2.

Comparison of two mutual inductance calculating methods.

Fig. 3.

The simulation model of three-coil system.

In the reference [13], by the usage of elliptic integrals and the calculation of vector magnetic potential, the mutual inductance between two coils has been obtained. However, because the mutual influence caused by the third coil cannot be totally taken into consideration, it is not applicable to calculate the mutual inductance among the multiple coils. Due to the limitation of traditional calculation method from the [13], the calculation method of multi-coil system is verified by finite element simulation.

4.2. Finite element simulation

In order to verify the correctness of proposed calculation, the simulation for the three-coil system by finite element software COMSOL has been executed. First of all, the simulation model is given as below in the Fig. 3.

By using the parametric scanning method in the COMSOL, the motion of coil3 can be effectively imitated. When the positions of coil1 and coil2 are fixed at −44 mm and 44 mm, the position of coil3 varies from −80 mm to 80 mm at the step of 1 mm. In this situation, the results of analytical calculation and finite element simulation are shown in Fig. 4. According to the calculation and simulation results of this three-coil system, the mutual inductance between coil1 and coil2 varies with the position of coil3. And obviously, there is an extremum phenomenon of mutual inductance. Moreover, the maximum value of M₁₂ appears at the situation that coil3 and coil2 are in the same plane. The existence of extremum phenomenon shows the influence of coil3 on mutual inductance between coil1 and coil2. In the study of the reference [14], the conclusion that the optimum position of the third coil for maximum power transfer is close to coil2 has been reached. Similarly, this same conclusion can be obtained by calculating the value of M₁₂ as shown above, which is a more basic way. This extreme phenomenon or the optimal coil position of the third coil is essentially the existence of an optimal energy transfer in an OPMSC system.

Fig. 4.

Comparison of M₁₂ calculation and simulation results.

Fig. 5.

Comparison of M₂₃ calculation and simulation results .

Similarly, the calculation of mutual inductance between coil2 and coil3 (M₂₃) and the results of simulation have been displayed in the Fig. 5. Different from the curve of M₁₂, two maximum points of M₂₃ occur when coil3 is in the same planes with coil1 and coil2, respectively. What’s more, the maximum mutual inductance M₂₃ at Z₁ position is larger than at Z₂ position. That is because the I₁ in coil1 is larger than the I₂ in coil2, which means the current through coil cannot be ignored in the mutual inductance calculation of multi-coil system. In the reference [15], the model of multiple coupled receivers with a transmitter in wireless power transfer system has been established. After the circuit analysis, the conclusion that high coupling between receivers would lower the peak efficiency value has been obtained. The high coupling between two receivers means the value of M₂₃ is large, which means the smaller M₂₃ value, the higher peak efficiency value. According to the calculation method proposed in this paper, the smallest M₂₃ value can be acquired by adjusting the relative position between coils. In a word, the method presented in this paper can guide the design of OPMSC system.

When h₁₃ (the distance between coil1 and coil3) and h₁₂ (the distance between coil1 and coil2) vary at the same time, the mutual inductance among the three coils can be obtained as below in Fig. 6. Obviously, there is an optimum value for each kind of mutual inductance, which means the system can be optimal designed by using the method proposed by this paper.

Fig. 6.

Results of mutual inductances.

4.3. The circuit experiment

The circuit experiment has been carried out for explanation of mutual influence among the multiple coils and validation for the calculation method above. As shown in the Fig. 7, the radius size of coil1 and coil2 is 43 mm and the radius size of coil3 is 30 mm. The diameter of used copper wire is 0.3 mm. The distance between coil1 and coil2 is 52 mm, therefore, the origin of z-axis coordinates is located at the center of the vertical direction of coil 1 and coil 2. Under such a set condition, the coordinates of coil3 change from −56 mm to 56 mm at the step of 8 mm.

Fig. 7.

Experimental setup for the three-coil system.

Figure 8 shows the measured mutual inductance between coil1 and coil2 with various position of coil3. Because of the inconsistency between the calculated and experimental models, the experimental values and the calculated results are inevitably different. However, the changing trend of the two curves is consistent. It is obvious that a maximum point of both curves occurs when the position of coil3 is in the same plane with that of coil2. According to the experimental results, the influence of the motion of coil3 on the mutual inductance between coil1 and coil2 cannot be neglected. Meanwhile, the calculation method of multi-coil mutual inductance proposed in the paper can well describe this kind of influence.

Fig. 8.

Results of circuit experiment.

Footnotes

Acknowledgements

The authors would like to acknowledge the financial supports from the NSFC (Natural Science Foundation of China, Nos. 51705241, 11802118), the NSFJP (National Science Foundation of Jiangsu Province, No. BK20170808).

Conflict of interest

All authors (the name of author) declare that they have no conflict of interest or financial conflicts to disclose.

References

Zhang

Zhu

, Compensation of cross coupling in multiple-receiver wireless power transfer systems, IEEE Transactions on Industrial Informatics12 (2016), 474–482.

Kallel

Trabelsi

and Kanoun

, Large air gap misalignment tolerable multi-coil inductive power transfer for wireless sensors, IET Power Electronics9 (2016), 1768–1774.

Tumanski

, A multi-coil sensor for tangential magnetic field investigations, Journal of Magnetism and Magnetic Materials242 (2002), 1153–1156.

Grover

F.W.

, Inductance Calculations, Van Nostrand Reinhold, 1964.

Babic

Sirois

Akyel

, Magnetic force between inclined circular filaments placed in any desired position, IEEE Transactions on Magnetics48 (2012), 69–80.

Grover

F.W.

, The calculation of the mutual inductance of circular filaments in any desired positions, Proceedings of the IRE32 (1944), 620–629.

Babic

S.I.

and Akyel

, New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air, IEEE Transactions on Magnetics42 (2006), 1661–1669.

Kong

Bae

Jung

D.H.

, An investigation of electromagnetic radiated emission and interference from multi-coil wireless power transfer systems using resonant magnetic field coupling, IEEE Transactions on Microwave Theory and Techniques63 (2015), 833–846.

Prasanth

Bandyopadhyay

Bauer

Analysis and comparison of multi-coil inductive power transfer systems, in: IEEE International Power Electronics and Motion Control Conference (PEMC), 2016, pp. 993–999.

10.

Ramrakhyani

A.K.

and Lazzi

, On the design of efficient multi-coil telemetry system for biomedical implants, IEEE Transactions on Biomedical Circuits and Systems7 (2013), 11–23.

11.

Chen

Liu

Analysis and optimization of 3-coil magnetically coupled resonant wireless power transfer system for stable power transmission, in: 2017 IEEE Energy Conversion Congress and Exposition (ECCE), 2017, pp. 2584–2589.

12.

G.Z.

, Principle of Engineering Electromagnetic Field (in Chinese), Higher Education Press, 2002.

13.

Babic

Sirois

Akyel

, Mutual inductance calculation between circular filaments arbitrarily positioned in space: alternative to Grover’s formula, IEEE Transactions on Magnetics46 (2010), 3591–3600.

14.

Ahn

and Hong

, A study on magnetic field repeater in wireless power transfer, IEEE Transactions on Industrial Electronics60 (2013), 360–371.

15.

Ahn

and Hong

, Effect of coupling between multiple transmitters or multiple receivers on wireless power transfer, IEEE Transactions on Industrial Electronics60 (2013), 2602–2613.