Calculation of mutual inductance between arbitrarily positioned planar spiral coils for wireless power applications

Abstract

Mutual inductance is one of the main parameters required to determine the power link’s performance (output voltage, efficiency) in wireless power transfer. The coils are often misaligned angularly in these applications, which affects the mutual inductance and thus the performance. Hence, an accurate calculation of mutual inductance is necessary to decide the working region of the coil. This paper presents an analytical calculation of mutual inductance between two planar spiral coils under angular misalignment conditions. By solving the Neumann integral formula, mutual inductance is derived for constant current-carrying coils, and the final mutual inductance value is calculated numerically. The influence of angular misalignment of the coil, which can be due to nutation and spin angles, on mutual inductance is studied in detail. The mutual inductance of the spiral coil is calculated for different misalignment cases. The accuracy of the calculation results is verified by comparing it with conventional formulas (mainly the Liu, the Babic formula, and the Poletkin formula) and by simulation using the finite element method. The proposed method is a more generalized and simpler one that can be used to calculate the mutual inductance of any size of coils, either spiral or circular, with any lateral and angular misalignments. Finally, a couple of spiral coils are fabricated to validate it experimentally. The comparison of the simulation and experiment results with the calculation result shows its accuracy. Thus, the proposed method can be applied to compute mutual inductance in any angularly misaligned coupling coils for the optimization of the wireless power transfer and their design.

Keywords

Neumann integral formula mutual inductance angular misalignment wireless power transfer spiral coil

1. Introduction

Recently, the spiral coil has triggered tremendous interest owing to its simple design, making it favorable to utilize in various wireless power transfer systems such as consumer electronics [1-4] and biomedical implant charging system [5–7]. Power is transmitted between these coils using wireless power technology (WPT). For this purpose, WPT uses different techniques like inductive coupling [8], and magnetic resonance coupling [9,10]. It consists of the primary transmitter coil and the secondary receiver coil. Depending on the application, it can have more than one primary or secondary coil [11]. For estimating the power link performance (power transfer efficiency, output power), mutual inductance is one of the parameters required to determine accurately.

In the literature, many methods have been proposed to calculate the mutual inductance of the coil. To estimate the mutual inductance between planar circular coils, [12] adopted the elliptic integral, [13] Biot–Savart law, [14] approximated formula, [15] Heuman’s lambda formula, and magnetic flux density formula [16]. However, these methods have calculated for only perfectly aligned coils without considering the misalignment.

Mutual inductance between two noncoaxial circular coils for the lateral and angular misalignment has been calculated using Grover’s formula [17], Bessel functions [18]. However, these methods consider a circular coil instead of a spiral coil, which is used in wireless power transfer.

In [19], only the nutation angle is considered in the angular misalignment case, and a general misalignment which includes both the lateral and angular misalignment is not studied. Mutual inductance between circular spiral coils is derived using the magnetic vector potential approach [20]. This approach requires a lot of mathematical calculations.

Based on the Neumann formula, the mutual inductance between two coupling spiral coils has been investigated by considering the spiral coil as a group of concentric circles [21,22]. As a result, this method is more complex mathematically and requires more tedious work to solve the mutual inductance. Moreover, only the influence of lateral misalignment on mutual inductance is considered. However, the mutual inductance of the coil is also affected by its angular misalignment, which occurs in many applications.

In [23], mutual inductance between two Archimedean spiral coils has been determined for both lateral and angular misalignment of the coil; however, only one rotation angle (nutation angle) is computed despite the fact the spin angle also affect the mutual inductance. This problem is solved in the paper [24], where mutual inductance for the planar spiral coil is calculated, including all the rotational angles. However, the influence of the precession angle on the mutual inductance is negligible in the case of a circular coil, and this angle makes the calculation unnecessarily complex. In addition, this approach solved the mutual inductance in rectangular coordinates, which requires more calculation steps. Therefore, an easy and more simplified method is needed.

The finite element method (FEM) using Maxwell 3-D electromagnetic software is one of the most suitable methods to compute mutual inductances for all shapes of the coils; however, it requires a long computational time and a high-speed computer.

In this paper, a more compact, general, and simplified method is presented for calculating the mutual inductance between two arbitrarily positioned planar spiral coils. The proposed method used a cylindrical coordinates system which not only reduces the derivation steps but also reduces the number of variables in the final equation. As a result, the equation becomes more simplified, and computational efficiency increases. The mutual inductance is calculated for different positions of the coil by changing the translational distance, spin angle, and the nutation angle of the secondary coil relative to the primary coil. The calculation results are compared with the conventional methods. In addition, the calculation results are also verified with “3D ANSYS Maxwell 15 software” and validated experimentally.

The rest of the paper is organized as below: Section 2 describe the derivation of the mutual inductance of planar spiral coil under angular misalignment and both the lateral and angular misalignments. Comparison results of the proposed methods to the conventional methods under different misalignment conditions are explained in Section 3. The verification of the calculation results with the simulation result using ANSYS 3D Maxwell 15 and the experimental method using Gwinstek LCR-6020 are discussed in Section 4. Section 4 is divided into multiple subsections. The behavior of mutual inductance with the variations of the nutation angle of the secondary coil is discussed in Section 4.1. Section 4.2 illustrates the behavior of mutual inductance by the variations of both the nutation and the spin angle. The behavior of mutual inductance under different conditions of both the lateral and angular misalignments is explained in Section 4.3. Finally, Section 5 concludes this research work.

2. Calculation of mutual inductance between arbitrary positioned planar spirals coils

The general equation of the planar spiral coil is described graphically in Fig. 1 and is represented mathematically by Eq. (1). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R=R_{i}+a{\theta},\quad {\theta}_{i}\leq {\theta}\leq {\theta}_{0}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}=2{\pi}N\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle a=\frac{s}{2{\pi}}\end{eqnarray}$ (3) where R is the outer radius, R _i is the inner radius, s is the gap between turns, a is the pitch factor which affects the distance between the turns, 𝜃 is the angle of rotation. 𝜃_i and 𝜃₀ are the initial and final angles of rotation and N is the number of turns.

Fig. 1.

Planar spiral coil.

In order to derive the mutual inductance equation, a couple of arbitrarily positioned spiral coil is considered as shown in Fig. 2.

Fig. 2.

A couple of arbitrarily positioned circular spiral coils.

It is assumed that the primary coil C ₁ lies on a xy plane with the center of the coil at origin O, while the secondary coil C ₂ is located on the X∕Y∕Z∕ coordinate plane at a distance h, having a center O ₁. A perpendicular line is drawn from the O ₁ on the xy plane to obtain the displacement distances x _o and y _o. Two arbitrary points, P and Q, are considered at the outer radiuses of C ₁ and C ₂. The differential elements at point P are dl ₁ and dl ₂ at point Q. The distance between P and Q is R. 𝜃₁ and 𝜃₂ are the angle of rotation of the coil C ₁ and C ₂ respectively.

The initial position of coil C ₂ lies at the XYZ plane, which is parallel to the xyz plane of C ₁. The arbitrary position of any circular spiral coil can be denoted by the nutation angle and the spin angle. Therefore, the nutation angle is achieved by rotating coil C ₂ around an axis passing through the diameter of the coil, with an angle alpha in a counterclockwise direction. Then, rotating the C ₂ around Z-axis in a counterclockwise direction with an angle 𝛽, the spin angle is obtained. To find the differential elements dl ₁ and dl ₂ of the C ₁ and C ₂, a cylindrical coordinates system is used to reduce the mathematical calculation. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle dl_{1}=R_{A}(-\text{sin}∼{\theta}_{1}\hat{x}+\cos {\theta}_{1}{\hat{y}})∼d{\theta}_{1}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle dl_{2}=R_{B}(-\text{sin}∼{\theta}_{2}\cos {\alpha}\hat{x}+\cos {\theta}_{2}{\hat{y}}+\sin {\theta}_{2}(\sin {\beta}+\cos {\beta}\sin {\alpha})\hat{z})∼d{\theta}_{2}.\end{eqnarray}$ (5) Using (1), the equation of the spiral coil for the primary C ₁ and the secondary coil C ₂, can be expressed in Eqs (6) and (7) respectively. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{A}=R_{i1}+a_{1}{\theta}_{1}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{B}=R_{i2}+a_{2}{\theta}_{2}\end{eqnarray}$ (7) where R _A and R _B are the outer radius of C ₁ and C ₂ respectively. R _i1 is the inner radius of C ₁, R _i2 is the inner radius of C ₂, a ₁ is the pitch factor of C ₁ and a ₂ is the pitch factor of C ₂. The dot product of dl ₁ and dl ₂ is calculated in Eq. (8). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle dl_{1}\cdot dl_{2}=R_{A}R_{B}(\sin {\theta}_{1}\sin {\theta}_{2}\cos {\alpha}\cos {\beta}+\cos {\theta}_{1}\cos {\theta}_{2})∼d{\theta}_{1}∼d{\theta}_{2}.\end{eqnarray}$ (8) The distance (R) between points P and Q is determined as shown in Eq. (9). $\begin{eqnarray}\displaystyle R\text{} & =\text{} & \displaystyle (R_{B}\cos {\theta}_{2}\cos {\alpha}\cos {\beta}-R_{A}\cos {\theta}_{1})^{2}+(R_{B}\sin {\theta}_{2}-R_{A}\sin {\theta}_{1})^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(h-R_{B}\cos {\theta}_{2}(\sin {\beta}+\cos {\beta}\sin {\alpha}))^{2}.\end{eqnarray}$ (9) Now, the Mutual inductance between the arbitrarily positioned spiral coils can be obtained by putting the parameters of Eqs (8) and (9) into Neumann’s integral formula, shown in Eq. (10). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M=\frac{{\mu}_{0}}{4{\pi}}\oint _{C_{1}}\oint _{C_{2}}\frac{dl_{1}dl_{2}}{R}\end{eqnarray}$ (10) where 𝜇₀ is the vacuum permeability.

The final mutual inductance equation is shown in Eq. (11) while the equation of mutual inductance under both lateral and angular misalignment is given in Eq. (12). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M=\frac{{\mu}_{0}}{4{\pi}}\iint _{2{\pi}N}\frac{(R_{i1}+a_{1}{\theta}_{1})(R_{i2}+a_{2}{\theta}_{2})(\sin {\theta}_{1}\sin {\theta}_{2}\cos {\alpha}\cos {\beta}+\cos {\theta}_{1}\cos {\theta}_{2})∼d{\theta}_{1}∼d{\theta}_{2}}{\sqrt{(R_{B}\cos {\theta}_{2}\cos {\alpha}\cos {\beta}-R_{A}\cos {\theta}_{1})^{2}+(R_{B}\sin {\theta}_{2}-R_{A}\sin {\theta}_{1})^{2}+(h-R_{B}\cos {\theta}_{2}(\sin {\beta}+\cos {\beta}\sin {\alpha}))^{2}}}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M=\frac{{\mu}_{0}}{4{\pi}}\iint _{2{\pi}N}\frac{(R_{i1}+a_{1}{\theta}_{1})(R_{i2}+a_{2}{\theta}_{2})(\sin {\theta}_{1}\sin {\theta}_{2}\cos {\alpha}\cos {\beta}+\cos {\theta}_{1}\cos {\theta}_{2})∼d{\theta}_{1}∼d{\theta}_{2}}{\sqrt{(x_{o}+R_{B}\cos {\theta}_{2}\cos {\alpha}\cos {\beta}-R_{A}\cos {\theta}_{1})^{2}+(y_{o}+R_{B}\sin {\theta}_{2}-R_{A}\sin {\theta}_{1})^{2}+(h-R_{B}\cos {\theta}_{2}(\sin {\beta}+\cos {\beta}\sin {\alpha}))^{2}}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (12) It can be seen from Eq. (12), eleven parameters are enough to fully describe the mutual inductance of a couple of spiral coils under both lateral and angular misalignment. The parameters are s ₁, s ₂, N ₁, N ₂, R _i1, R _i2, x _o, y _o, h, 𝛼, 𝛽. The first six parameters are related to the size of the coils while the remaining five parameters indicate the relative position of the coil. This equation can also be used for perfectly aligned and lateral misaligned spiral coils by making the parameters, x _o, y _o, h, 𝛼, and 𝛽 to zero. Furthermore, if the s ₁ = s ₂ = 0, and N ₁ = N ₂ = 1, the mutual inductance between the spiral coils’ Eq. (12) reduces to the mutual inductance between the circular coils. Therefore, it is suitable to calculate the mutual inductance of any size of coils, either spiral or circular, with any lateral distance and angular positions.

3. Comparison of the calculation result with the conventional equations of mutual inductance

To verify the correctness of the calculation results in this work, it has been compared with the results of conventional mutual inductance equations.

It is necessary to compare the proposed method with different coil sizes and their misalignments for validation. Therefore, a few random examples solved in the conventional articles are calculated numerically, and their results are compared to the proposed method. Each example has various coil sizes and different lateral and angular (nutation and spin) displacements. The comparison detail and their results are shown below in tabular form.

3.1. Comparison of mutual inductance between spiral coils

The calculation result of Eq. (12) is compared with the coil parameters and their respective relative position parameters given in Table VII [24]. The coils size parameters are shown in Table 1 while the arbitrary position parameters and their results are shown below in Table 2.

Table 1
Coil parameters

Coil Turns N Turn gap s (mm) Wire diameter (mm) Inner radii R _i (mm)

C ₁ 7 10 2 20

C ₂ 5 10 3 40

Coil	Turns N	Turn gap s (mm)	Wire diameter (mm)	Inner radii R _i (mm)
C ₁	7	10	2	20
C ₂	5	10	3	40

Table 2

Relative position of spiral coil and their mutual inductance comparison to the proposed method

Parameters						Results
Case	x _o (mm)	y _o (mm)	z _o (mm)	𝛼 (rad)	𝛽 (rad)	Eq. (12) (μH)	[24 ] equation (μH)
A	0	0	100	𝜋∕4	0	0.489	0.483
B	0	0	100	0	𝜋	0.429	0.426
D	0	50	100	𝜋∕4	0	0.239	0.233
F	50	0	100	𝜋∕3	0	0.338	0.333

In this comparison, the cases including the yaw angle are not considered due to their little influence on the spiral coil of circular shape. Table 2 shows that the mutual inductance values are consistent for all cases discussed above. However, compared to the conventional mutual inductance equation, the proposed mutual inductance equation is easy to derive and more simplified, i.e., it reduces the calculation complexity.

3.2. Comparaison of mutual inductance between circular coils

Mutual inductance is calculated for circular coils instead of spiral ones using Eq. (12). It is computed for different configurations of the coils placed at various relative distances and orientations. The parameters of the coils are taken from the examples mentioned in [25]. Grover and Babic’s article [17] also solved these examples. A few of them are given in Table 3.

Table 3
Relative position of spiral coil and their mutual inductance comparison to the proposed method

Parameters Results

Example C ₁ R _i1 (mm) C ₂ R _i2 (mm) z _o (mm) y _o (mm) 𝛼 (rad) 𝛽 (rad) M (nH) M _c (nH)

1 250 200 100 0 0 0 248.54 248.78

3 100 100 40 0 0 0 134.94 135.07

5 250 200 80 0 0 0 288.75 289.04

6 150 150 160 120 0 0 45.28 45.33

9 100 80 120 160 0 0 4.45 4.46

12 200 40 13.2 15 0 0 15.98 15.99

13 100 25 0 0 𝜋∕3 0 6.040 6.043

19 160 100 17.5 43.30 𝜋∕3 𝜋∕4 16.98 16.81

19 160 100 175 43.30 𝜋∕3 11𝜋∕6 14.79 14.46

Parameters	Results
1	250	200	100	0	0	0	248.54	248.78
3	100	100	40	0	0	0	134.94	135.07
5	250	200	80	0	0	0	288.75	289.04
6	150	150	160	120	0	0	45.28	45.33
9	100	80	120	160	0	0	4.45	4.46
12	200	40	13.2	15	0	0	15.98	15.99
13	100	25	0	0	𝜋∕3	0	6.040	6.043
19	160	100	17.5	43.30	𝜋∕3	𝜋∕4	16.98	16.81
19	160	100	175	43.30	𝜋∕3	11𝜋∕6	14.79	14.46

M represents the calculation result of Eq. (12), and M _c is the result of the mutual inductance equation derived in [25]. Both calculations have almost equivalent results. Thus, the accuracy of Eq. (12) is verified in the case of the coil of a single turn. This conventional formula calculates the mutual inductance of a single-turn circular coil only, i.e., it is limited to the coil of a single turn. However, most wireless power applications use multi-turn (spiral) coils for higher power transmission. Moreover, the influence of the lateral distance x _o on the mutual inductance is not considered. However, Eq. (12) contains all the parameters that affect the mutual inductance, and it can be used for both circle (single-turn) coil and the spiral (multi-turn) coil.

3.3. Comparison of mutual inductance between circular coils under misalignments

In this section, we have compared the calculation result of Eq. (12) with another conventional equation for mutual inductance under lateral and angular misalignment, which is solved in [26]. This section has been divided into three subsections: lateral misalignment, angular misalignment, and both lateral and angular misalignment. These comparisons are shown in the following tables.

3.3.1. Lateral misalignment

For the coil parameters given in [26], h = z _o = 10 mm, R _i1 = 10 mm, R _i2 = 3 mm, N ₁ = N ₂ = 1, Eq. (12) is compared at various misalignment distances. The pitch angle 𝛼 and the spin angle 𝛽 are taken to zero. M represents the calculation result of Eq. (12), and M _c1 is the conventional result of [26]. Their comparison results are shown in Table 4.

Table 4
Comparison result of mutual inductance

x _o (mm) M (pH) M _c1 (pH)

0 610 610

2 600 600

4 550 550

6 480 480

8 380 380

10 270 270

x _o (mm)	M (pH)	M _c1 (pH)
0	610	610
2	600	600
4	550	550
6	480	480
8	380	380
10	270	270

Table 5

Comparison result under angular misalignment

𝛼 (rad)	M (pH)	M _c1 (pH)
0	610	610
𝜋∕12	590	590
𝜋∕6	540	540
𝜋∕4	450	450
𝜋∕3	320	320
5𝜋∕12	170	170

Table 6

Comparison result of mutual inductance under both lateral and angular misalignment

𝛼 (rad)	M (pH)	M _c1 (pH)
0	150	148
𝜋∕12	145	145
𝜋∕6	140	140
𝜋∕4	130	130
𝜋∕3	100	100
5𝜋∕12	70	70
𝜋∕2	30	30

3.3.2. Angular misalignment

The mutual inductances are compared at various angular misalignments for the parameters of the same coil as above. In this case, the spin angle 𝛽 is zero. M _c1 is the calculation result of the conventional equation. The comparison results under angular misalignment are shown in Table 5.

3.3.3. Lateral and angular misalignment

In this case, the mutual inductance is calculated at the coils separating distance z _o = h = 20 mm under both the lateral and angular misalignment. The lateral distance is taken x _o = 5 mm, and pitch angle 𝛼 is changes from 0 rad to 𝜋∕2 rad. The angle 𝛽 is zero. The comparison results are shown in Table 6.

These comparisons tables show that the calculation results of both the conventional and the proposed methods are similar; however, the former has not included all the parameters, which influence the inductance, such as lateral distance y _o, and the spin angle. Furthermore, it is limited to only a circular coil, while the spiral coil is adopted in many practical wireless power transfers. In contrast, the proposed method is more general, and it can be applied to any size of a spiral coil or circular coil, with any lateral distance and orientation.

3.4. Comparison analysis

The proposed method was compared with three conventional approaches for calculating the mutual inductance between the coils. In comparison with the first approach [24], it was demonstrated that the proposed method was simplified using cylindrical coordinate approach instead of rectangular coordinate approach which make the derivation unnecessarily long.

In the second and third approaches [25,26], the mutual inductance was calculated using the Kalantarov–Zeitlin method and verified by simulation and other renowned methods such as Grover and Babic formula. The mutual inductance between two angularly misaligned coils was calculated under different nutation and spin angles. However, these methods calculate the mutual inductance between circular coils. It is unable to use them for spiral coils which are adopted in wireless power transfer. While in the third approach, besides calculating the mutual inductance between circular coils, only one lateral and angular misalignment, x _o, and nutation angle, was considered. The mutual inductance can also be affected by spin angle and y _o direction displacement. In contrast to these conventional methods, the proposed method was derived using the Neumann formula for spiral (multi-turn) coils; however, it can also be applied to calculate the mutual inductance between the circular (single turn) coils. The above comparison Tables are the evidence of this claim. The proposed method can be called an extension of the mutual inductance of the circular coil for the spiral coil. Moreover, the proposed method has included all the geometric, lateral, and angular parameters that influence mutual inductance.

4. Simulation and experimental verification

To verify Eq. (12), the mutual inductance between the primary coil C ₁ and secondary coil C ₂ is calculated numerically for various lateral and angular misalignment cases. The calculation results are compared with the simulation and the measurement result. The variation of calculation results between the simulation and measurement results is also determined.

The simulations are performed using the finite element method (FEM) ANSYS 3D Maxwell 15 version. The solution type is chosen as magnetostatics. The coils are excited with a uniform current at their cross-section to distribute the magnetic flux equally. The mesh type is length-based, and its maximum length is the default. The maximum number of passes for the solution setup is set 10.

The measurements are conducted using the Gwinstek LCR-6020 model. The measurement method is as follows. The inductance is measured in forwarding mode by connecting the inner terminals of the primary coil C ₁ and the secondary coil C ₂ and measuring the inductance between their outer terminals. Then, inductance is measured in backward mode by connecting the inner terminal of the primary coil C ₁ with the outer terminal of secondary coil C ₂ and measuring the inductance between the outer terminal of the C ₁ and the inner terminal of C ₂. Finally, the mutual inductance can be obtained by subtracting the backward mode inductance from the forward mode inductance and dividing the resultant inductance value by 4. Mathematically, the forward and backward inductance is written as Eqs (13) and (14). The mutual inductance equation is shown in Eq. (15). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L_{f}=L_{p}+L_{s}+2M\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L_{b}=L_{p}+L_{s}-2M\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M=\frac{(L_{f}-L_{b})}{4}.\end{eqnarray}$ (15)

Where L _p, L _s, L _f, L _b, and M are the inductance of the primary coil, secondary coil, forward inductance, backward inductance, and mutual inductance respectively.

Two coils of equal sizes are fabricated on a bobbin designed in a 3D printer. The filament type for the 3D printer is PLA (polylactic acid). The inner radius R _i of the coil is 10 mm, the number of turns N is 16, the gap between the turn s is 4.68 mm, and the outer radius R _o is 85 mm. The measuring device and the fabricated coil are shown in Fig. 3.

Fig. 3.

Measuring device and the primary and secondary spiral coil.

In most wireless power applications, the operating frequency of the coil is in the megahertz range. Therefore, Litz wire is used for constructing the coil due to its lower AC resistance at a higher frequency than the solid conductor. In this work, Litz wire of 500 strands, with the diameter of each strand of 0.12 mm, is selected. The overall diameter of the Liz wire is 3.6 mm, which is decided empirically from Eq. (16). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{e}=1.6\left(\sqrt{\left(\frac{D_{s}}{2}\right)^{2}N_{s}}\right)2.\end{eqnarray}$ (16)

Where D _e is the estimated diameter, N _s is the total number of strands, and D _s is the dimeter of a single strand. The constant “1.6” is a value obtained by multiplying the diameter of one reel winding by an experience factor of 13.3. When the diameter of one reel winding is from 0.1 mm to 0.2 mm, an experience factor of 13.3 is used. The constant 2 is determined by considering the twist pattern of the Litz wire. In case bundles of twisted wire are twisted together with optional outer insulation, an experience factor of 2 is used.

4.1. Variation of nutation angle 𝛼 on the behavior of mutual inductance

For the separating distance h = 100 mm, and spin angle 𝛽 = 0°, the behavior of mutual inductance on the variation of nutation angle is observed. The separating distance is chosen such that both coils do not touch each other. Therefore, the value of h is 100 mm for each case studied in this work. In Fig. 4, the angle along the x-axis is the equivalent of the radian angle. It shows that the variation of mutual inductance is very small up to 𝜋∕4 radians, and it drastically decreases until 𝜋∕2 radians, which is almost zero. Hence, coil maximum angular misalignment should be at 𝜋∕4 radians for maintaining higher efficiency.

The calculation result error relative to the FEM result is higher than the error between the measurement and the calculation results. The comparison error at 𝜋∕2 is high, which is less than 9%. However, the mutual inductance also reduces to zero, and such a high tilt angle is not used normally in wireless power applications.

Fig. 4.

Mutual inductance at different angular misalignment 𝛼.

4.2. Variations of both nutation angle 𝛼 and spin angle 𝛽 on the behavior of mutual inductance

In this section, the behavior of mutual inductance is observed for different nutation and spin angle. Their comparison results are shown in Table 7. For a specific nutation angle, the mutual inductance reduces as the spin angle increases. Because as the angle increases, the position of the coil changes, which decreases flux linkage between the coils; as a result, mutual inductance drops. The mutual inductance remains the same at angles 0 and 2𝜋 due to the position of the coil being unchanged. The FEM and the measurement results also verify the calculation result. Error_C represents the calculation error relative to FEM and Error_M is the measurement error relative to the calculation result.

Table 7
Comparison of mutual inductance under different nutation and spin angles

Angles (rad) Results

𝛼 𝛽 Cal. (μH) FEM (μH) Meas. (μH) Error_C (%) Error_M (%)

𝜋∕6 𝜋∕12 1.9167 1.9421 1.8792 1.31 1.95

𝜋∕6 13𝜋∕36 1.5434 1.5653 1.8792 1.39 1.33

𝜋∕6 𝜋 1.6946 1.7202 1.5228 1.48 1.30

𝜋∕6 2𝜋 1.6969 1.7156 1.6725 1.08 0.68

𝜋∕12 𝜋∕6 1.8925 1.9215 1.6853 1.50 1.69

𝜋∕12 𝜋∕3 1.6496 1.6832 1.8605 1.99 1.84

𝜋∕12 𝜋 1.6861 1.7125 1.6192 1.54 1.23

𝜋∕12 2𝜋 1.6873 1.7056 1.6652 1.07 1.46

𝜋∕4 𝜋∕3 1.7386 1.7696 1.7021 1.75 2.09

𝜋∕4 𝜋 1.6601 1.6852 1.6325 1.48 1.66

𝜋∕4 2𝜋 1.6639 1.6886 1.6403 1.46 1.41

𝜋∕3 𝜋∕12 1.8765 1.8995 1.8425 1.21 1.81

𝜋∕3 2𝜋 1.4646 1.4883 1.4521 1.59 0.85

Angles (rad)	Results
𝜋∕6	𝜋∕12	1.9167	1.9421	1.8792	1.31	1.95
𝜋∕6	13𝜋∕36	1.5434	1.5653	1.8792	1.39	1.33
𝜋∕6	𝜋	1.6946	1.7202	1.5228	1.48	1.30
𝜋∕6	2𝜋	1.6969	1.7156	1.6725	1.08	0.68
𝜋∕12	𝜋∕6	1.8925	1.9215	1.6853	1.50	1.69
𝜋∕12	𝜋∕3	1.6496	1.6832	1.8605	1.99	1.84
𝜋∕12	𝜋	1.6861	1.7125	1.6192	1.54	1.23
𝜋∕12	2𝜋	1.6873	1.7056	1.6652	1.07	1.46
𝜋∕4	𝜋∕3	1.7386	1.7696	1.7021	1.75	2.09
𝜋∕4	𝜋	1.6601	1.6852	1.6325	1.48	1.66
𝜋∕4	2𝜋	1.6639	1.6886	1.6403	1.46	1.41
𝜋∕3	𝜋∕12	1.8765	1.8995	1.8425	1.21	1.81
𝜋∕3	2𝜋	1.4646	1.4883	1.4521	1.59	0.85

4.3. Variations of both lateral and angular misalignment on the behavior of mutual inductance

The lateral misalignment can be due to the translation of coil C ₂ in the x _o direction, y _o direction, or combined x _o and y _o direction. Similarly, the angular misalignment can be due to nutation angle rotation and spin angle rotation. Therefore, it is necessary to know the influence of each translational and rotational motion on mutual inductance. Each case is studied separately.

4.3.1. Coil is translated in x _o direction

In this case, coil C ₂ is tilted at the angle 𝜋∕4 radian and translated along the x _o direction, then the behavior of the mutual inductance is observed, as shown in Fig. 5.

Fig. 5.

Mutual inductance at both lateral and angular misalignment.

Figure 5 shows that the mutual inductance slightly increases at a small lateral misalignment in the x _o direction and then it decreases with the increase in the distance increases. At a slight distance with a tilted angle, the lower part of the C ₂ is closer to the C ₁, which increases the flux linkage and thus the slight increase in mutual inductance. However, without tilting the coil the mutual inductance decreases with an even slight increase in the translational distance. Because the mutual coupling of the flux decreases when the increase in lateral displacement. This phenomenon is also explained in [27]. The relative error between the calculation result, FEM, and the measurement is less than 2%.

4.3.2. The coil is translated in both x _o and y _o direction

In this case, coil C ₂ is translated in both x _o and y _o directions at the nutation angle 𝜋∕4 and spin angle 0. The behavior of mutual inductance is shown in Fig. 6. It can be seen that the mutual inductance declines faster when the coil moves in both x _o and y _o directions compared to the single translation, which is described above. If the lateral distance is more than the length of the outer radius, the mutual inductance approaches zero. Hence, for better efficiency, the maximum lateral distance is limited to the length of the outer radius of the coil. The relative error of the calculation result is below 2.5%.

Fig. 6.

Mutual inductance at angular misalignment and lateral distance x _o and y _o.

4.3.3. The coil is fixed at lateral distance y _o and the nutation angle is changed

Fig. 7.

Mutual inductance at a fixed lateral misalignment and changing angular misalignment.

In this case, coil C ₂ is fixed at a lateral distance y _o = 20 mm, and the nutation angle is changed. The range of the nutation angle is zero to 𝜋 radian. Therefore, the mutual inductance is calculated at the nutation angle −𝜋∕2 to 𝜋∕2 radians. The variation of the mutual inductance is shown in Fig. 7. The pattern of mutual inductance is similar to the one described in Fig. 3. The only difference is in the value of mutual inductance, which is almost 5% less when the coil is misaligned both laterally and angularly. The error is high at the nutation angle 𝜋∕2 radian; however, mutual inductance is also approaching zero at this angle, thus high accuracy is not required.

Throughout the measurement process, it is found that the measurement value of the mutual inductance is a little smaller compared to the calculated value. This difference might be caused due to human error during the measurement process or the error in the measuring device. Similarly, the simulation result is slightly higher than the calculation result, which can be due to the usage of long terminals of the coils in ANSYS Maxwell.

The proposed method Eq. (12) is more time-efficient. Its computational time is less than 5 s to calculate the mutual inductance. While the simulation, which was performed in Ansys Maxwell 15 version, took more than more than 10 h. The simulation was done on an Intel (R) Core (TM) i5-4590 with 3.30 GHz of processor speed and 16 GB of installed RAM. Thus, it proves that calculation time is reduced significantly by utilizing the proposed method.

5. Conclusions

This paper presents a compact and more general method for the computation of mutual inductance between the arbitrarily positioned planar spiral coils. All the parameters, which influence the mutual inductance, such as the coil’s geometry, and lateral and angular misalignments, are included in a single equation. Furthermore, it can be also used to calculate the mutual inductance of circular coils. Details of the mathematical derivation are presented. The mutual inductance of several coil configurations under different lateral and angular misalignments is calculated. The calculated results are verified against the results found in the literature (primarily Liu, Babic, and Poletkin formula). Compared to these conventional methods, the proposed mutual inductance equation is more compact and simpler, i.e., it reduces the calculation complexity. Furthermore, these conventional methods are limited to the calculation of the mutual inductance of a circular coil, while the spiral coil is adopted in many practical wireless power transfers. In contrast, the proposed method is more general, and it can be applied to any size of a spiral coil or circular coil, with any lateral distance and orientation. The proposed method can be called a simplified and extended version of conventional circular coil mutual inductance calculation methods for the spiral coil. In addition to the comparison with the conventional methods, the calculation results of the proposed method are also compared with the simulation and the experimental results, which verify their accuracy. Therefore, the proposed method can be considered for the optimization of wireless power transfer systems.

The limitation for angular misalignment is up to 𝜋∕3 radian, the mutual inductance decreases as the angle increases, such as it is zero at 𝜋∕2 radians, and such a large angular misalignment is not required because mutual inductance approaches zero.

In this article, mutual inductance was derived by supposing the uniform current flow in the coil. However, the inductance value could be different for un-uniform current flow. It requires further research in this regard. Moreover, the simplified mutual inductance equation is in double integral form. Its simplification into the single integral form will be considered in future work.

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Parameters							Results
Example	C ₁ R _i1 (mm)	C ₂ R _i2 (mm)	z _o (mm)	y _o (mm)	𝛼 (rad)	𝛽 (rad)	M (nH)	M _c (nH)
1	250	200	100	0	0	0	248.54	248.78
3	100	100	40	0	0	0	134.94	135.07
5	250	200	80	0	0	0	288.75	289.04
6	150	150	160	120	0	0	45.28	45.33
9	100	80	120	160	0	0	4.45	4.46
12	200	40	13.2	15	0	0	15.98	15.99
13	100	25	0	0	𝜋∕3	0	6.040	6.043
19	160	100	17.5	43.30	𝜋∕3	𝜋∕4	16.98	16.81
19	160	100	175	43.30	𝜋∕3	11𝜋∕6	14.79	14.46

Angles (rad)		Results
𝛼	𝛽	Cal. (μH)	FEM (μH)	Meas. (μH)	Error_C (%)	Error_M (%)
𝜋∕6	𝜋∕12	1.9167	1.9421	1.8792	1.31	1.95
𝜋∕6	13𝜋∕36	1.5434	1.5653	1.8792	1.39	1.33
𝜋∕6	𝜋	1.6946	1.7202	1.5228	1.48	1.30
𝜋∕6	2𝜋	1.6969	1.7156	1.6725	1.08	0.68
𝜋∕12	𝜋∕6	1.8925	1.9215	1.6853	1.50	1.69
𝜋∕12	𝜋∕3	1.6496	1.6832	1.8605	1.99	1.84
𝜋∕12	𝜋	1.6861	1.7125	1.6192	1.54	1.23
𝜋∕12	2𝜋	1.6873	1.7056	1.6652	1.07	1.46
𝜋∕4	𝜋∕3	1.7386	1.7696	1.7021	1.75	2.09
𝜋∕4	𝜋	1.6601	1.6852	1.6325	1.48	1.66
𝜋∕4	2𝜋	1.6639	1.6886	1.6403	1.46	1.41
𝜋∕3	𝜋∕12	1.8765	1.8995	1.8425	1.21	1.81
𝜋∕3	2𝜋	1.4646	1.4883	1.4521	1.59	0.85

Calculation of mutual inductance between arbitrarily positioned planar spiral coils for wireless power applications

Abstract

Keywords

1. Introduction

2. Calculation of mutual inductance between arbitrary positioned planar spirals coils

3.1. Comparison of mutual inductance between spiral coils

Table 1 Coil parameters Coil Turns N Turn gap s (mm) Wire diameter (mm) Inner radii R i (mm) C 1 7 10 2 20 C 2 5 10 3 40

3.3.1. Lateral misalignment

Table 4 Comparison result of mutual inductance x o (mm) M (pH) M c1 (pH) 0 610 610 2 600 600 4 550 550 6 480 480 8 380 380 10 270 270

3.3.3. Lateral and angular misalignment

3.4. Comparison analysis

4. Simulation and experimental verification

4.3.1. Coil is translated in x o direction

References

Table 1
Coil parameters

Coil Turns N Turn gap s (mm) Wire diameter (mm) Inner radii R _i (mm)

C ₁ 7 10 2 20

C ₂ 5 10 3 40

Table 4
Comparison result of mutual inductance

x _o (mm) M (pH) M _c1 (pH)

0 610 610

2 600 600

4 550 550

6 480 480

8 380 380

10 270 270

4.3.1. Coil is translated in x _o direction