A Parameter Design Method for a Wireless Power Transmission System with a Uniform Magnetic Field

Abstract

In a wireless power transfer (WPT) system, misalignment on the transceiver side has a major impact on the transmission efficiency. However, a uniform magnetic field can mitigate the detrimental impact of offsets on the system performance. In this paper, an optimized double-D (DD) coil is proposed by dividing the wires into two groups. The parametric design approach proceeds from the outside to the inside of the coil. The uniformity of the magnetic field of the improved design is increased by 20.7% compared to that of a regular DD coil. The average transmission efficiency of the final test prototype can reach 85%, and the offset rates in the X and Y directions are only 3.21% and 3.43%, respectively, when misalignment occurs. This design can effectively improve the anti-offset ability of the system and can be generalized to the design of other types of coils.

1. Introduction

With the diversification of electricity consumption scenarios and the rapid development of intelligent instruments, the traditional wired power transmission method is no longer applicable under some working conditions.

Wireless power transfer (WPT) refers to the process of transmitting electrical energy from a transmitting side to a load side through a medium without electrical contact. In this process, the transmitting coil is connected to an ac inductor to generate an alternating magnetic field. The receiving coil generates an induced current, which then passes through a rectifier circuit to reach the load. WPT is safer, more flexible, and more stable than traditional wired transmission and has been widely adopted in electric vehicles, medical electronics, and other fields [1,2,3,4,5].

As a significant component of a WPT system, the coupling mechanism has a great influence on performance parameters such as the coupling coefficient, transmission efficiency, and power [6,7]. One of the major topics to be investigated in this field is the anti-offset ability. When there are relative offsets between the transmitting and receiving sides of the coupling mechanism, which may include both angular and lateral offsets, the magnetic flux coupled to the receiving side changes, which makes it difficult to maintain a stable working state of the WPT system and can even cause security issues [8,9].

To improve the anti-offset performance of the coupling mechanism, the double-D (DD) coil structure was proposed, which improves the offset tolerance in the X and Y directions and has an effective charging area that is more than four times larger than that of a traditional circular coil [10]. Based on the DD coil structure, bipolar and DDQ coils were subsequently proposed to improve the interoperability of a WPT system while also increasing the amount of winding [11]. The anti-offset characteristics of a WPT system can be further improved by adding a concave–convex structure in the vertical direction [12]. The mutual inductance between the transceiver sides is related only to the distribution of and change in the magnetic field [13,14]. Notably, the above studies focused on the shape of the coil but ignored the fine-tuning of the wire. Their main concern was the mutual inductance as an electrical parameter [15]. Theoretically, however, by regulating the distribution of the magnetic field in space, the mutual inductance can be effectively controlled. The more uniform the magnetic field in the receiving plane is, the smaller the change in the mutual inductance. This method relies only on the design of the transmitting coil, which generates the magnetic field; consequently, the design steps are considerably simplified [16,17].

The study of how to establish a uniform magnetic field can be divided into two types of design schemes: single coils and composite coils. Some scholars have introduced genetic algorithms into single-coil designs [18,19,20], but most studies have focused only on rectangular coils, and the amount of computation necessary is very large. By extending the structure of a single coil in the vertical direction, a uniform magnetic field can be achieved more easily at the expense of space occupancy [21]. However, the freedom in designing a single coil is limited, so it is difficult to achieve a breakthrough with this scheme. Hence, an increasing number of research teams are turning their attention to the design of composite coil structures.

By combining spiral and planar coils, the corresponding convex and concave magnetic fields can be mutually compensated, and a uniform magnetic field superior to that of either of the two original coils is formed [22,23]. When the number of wires is equal, increasing the number of unit coils and reducing the size of the coils can lead to a more uniform axial magnetic field than that of a single coil [24,25]; however, the effective transmission distance is reduced, and the design process is more complex. At present, studies on composite coils tend to focus on the number of coils, and there is little research on the arrangement of the distribution of wires. However, the magnetic field of a composite coil can be more finely optimized by suitably designing the wire arrangement.

In this paper, an optimized DD coil is proposed by analyzing the distribution of the magnetic field. In Section 1, the status of research on WPT systems with uniform magnetic fields has been analyzed. In Section 2, the definitions of the parameters and theoretical derivations relevant to the establishment of a uniform magnetic field are presented. The topological circuits are also introduced. In Section 3, the process of magnetic simulation and the coil design scheme are described. In Section 4, improved coil structures are presented and compared to regular coils. Section 5 discusses the development and testing of a prototype. In Section 6, conclusions and outlooks are given.

2. Magnetic Field Calculation Method

2.1. Calculation of Magnetic Induction Intensity

A DD coil can be simplified into two parts due to its axial symmetry, as shown in Figure 1a, where the physical structures of coil 1 and coil 2 can be seen to be the same. All of the coil parameters are provided in

Table 1. Parameters of the coil structure.

Symbol	Description	Range of Value
A	Outer diameter in X	[200, 600] mm
B	Outer diameter in Y	[200, 600] mm
a	Inner diameter in X	[96, 160] mm
b	Inner diameter in Y	[296, 360] mm
N	Number of turns	[6, 14]
p	Spacing of turns	[2, 10] mm
pi	Spacing of the inner group	[2, 6] mm
po	Spacing of the outer group	[2, 6] mm
pg	Spacing between the two groups	[25, 45] mm
β	Ratio of the inner and outer spacings	[1, 3]
H	Transmission distance	[100, 200] mm
S	Side length of the charging area	[100, 600] mm
I	Current amplitude	10 A
(x, y, z)	Coordinates of the source point	arbitrary
(m, n, o)	Coordinates of a field point	arbitrary
R	Distance between test points	R ≥ H
α	Index of uniformity	[0, 1]

. The amplitudes of the currents flowing in the two coils are the same, but their phases differ by 180 degrees. In Figure 1a, A and B refer to the outer diameter, and a and b refer to the inner diameter. The distribution of the wires is shown in Figure 1c; the radius of each wire is c, and the distance between every two wires is p. The active charging area is located on a plane H perpendicular to the coil, which is a square with a side length of S.

As shown in Figure 1b, the coordinates of an arbitrary field point are (m, n, o), the coordinates of the source point are (x, y, z), and the distance between the two points is R.

According to the Biot–Savart law, the magnetic induction intensity dB generated by the source point at a field point can be described as shown in (1). This value is inversely proportional to the square of the distance and directly proportional to the current amplitude and can describe point-to-point magnetic induction. µ₀ denotes the vacuum permeability, dl denotes the unit vector of the current, and e_R denotes the unit vector from the source point to the field point. R and e_R can be expressed as shown in (2) and (3), respectively.

$$d\stackrel{\rightharpoonup }{B}={\mu }_{0}Id\stackrel{\rightharpoonup }{l}\times {\stackrel{\rightharpoonup }{e}}_R/4\pi R^2$$

(1)

$$R=\sqrt{{(m-x)}^2+{(n-y)}^2+H^2}$$

(2)

$${\stackrel{\rightharpoonup }{e}}_R=\stackrel{\rightharpoonup }{R}/\left|\stackrel{\rightharpoonup }{R}\right|$$

(3)

The axial components of the magnetic induction intensity at a field point generated by the current elements along the X and Y directions can be expressed as shown in (4) and (5). Under ideal conditions, the current flows in the center of the cable, so the coil can be reduced to a side-current model, as shown in Figure 2. The coils can be considered equivalent to the superposition of multiple concentric loops, where each closed loop consists of four wire segments placed end to end. The axial component of the magnetic induction intensity generated at a field point by such a loop can be expressed as shown in (6). Because of symmetry, the axial magnetic induction intensity of the whole coil can be expressed as shown in (7) based on the superposition theorem. The indicator for comparing the degree of uniformity is defined as shown in (8), where B_max and B_min denote the maximum and minimum values of B, respectively, in the charging area.

$$d{B}_x=\frac{\pm {\mu }_{0}I(x-m)}{(4\pi {({(m-x)}^2+{(n-y)}^2+H^2)}^{1/2}{)}^3}$$

(4)

$$d{B}_y=\frac{\pm {\mu }_{0}I(n-y)}{(4\pi ({(m-x)}^2+{(n-y)}^2+H^2{)}^{1/2}{)}^3}$$

(5)

$$\begin{array}{l}{B}_{nz}={\displaystyle \underset{L_1}{\oint }dB}={\mu }_{0}I(B_{1\to 2}(-d{B}_y)+B_{2\to 3}(-d{B}_x))/4\pi \\ +{\mu }_{0}I\cdot (B_{3\to 4}\cdot d{B}_y+B_{4\to 1}\cdot d{B}_x)/4\pi \end{array}$$

(6)

$$B={\displaystyle \sum _{N=1,2,3\cdots }[B_{nz}(x,y)+}{B}_{nz}(-x,y)]$$

(7)

$$\alpha =(B_{\max }-B_{\min })/B_{\max }$$

(8)

2.2. Calculation of Electrical Parameters

The LCC topology is selected for the compensation network of the system for a certain DD coil due to its higher power density and stable transmission performance [26,27,28,29,30], and the whole system can be simplified as shown in Figure 3. U_in represents the input voltage of the system; L₁ and L₂ are the inductances of the transmitting and receiving coils, respectively; L₃, C₁, and C₂ are the compensation components on the transmitting side; L₄, C₃, and C₄ are the compensation components on the receiving side; R_L is the load on the receiving side; M represents the mutual inductance between the transmitting and receiving coils; the impedance of the components on the transmitting side other than L₃ is equivalent to X₁; and the impedance of the components on the receiving side other than L₄ and the load is equivalent to X₂. I₁ is defined as the current flowing through L₃ and X₁, and I₂ is defined as the current flowing through L₄ and X₂. U is defined as the equivalent output voltage of the inverter, and R_E is the equivalent input resistance of the rectifier.

According to the working characteristics of the rectifier, the relationship between the load and the equivalent resistance can be obtained as shown in (9). According to Kirchhoff’s current law, the mesh current equation can be expressed as shown in (11). The power P and maximum efficiency η_max of the system can be expressed as shown in (12) and (13), respectively. Δ is equal to kQ₁Q₂, where k is a coupling coefficient that can be expressed as M/√L₁L₂ and Q₁ and Q₂ are the quality factors of L₁ and L₂, which can be expressed as ωL₁/r₁ and ωL₂/r₂, respectively, with the r parameters denoting the resistances of the respective coils. When the position of the transmitting coil remains unchanged, the magnetic induction intensity on the receiving plane gradually decreases from the center position to the surrounding area. When the magnetic field is ideally uniform in space, M can be approximated as shown in (10), where S₁ represents the magnetic flux area of the coil and I represents the current in the coil. Since the mutual inductance M between the coils is proportional to B, under the condition that S₁ remains unchanged, the greater the magnetic induction intensity B is, the greater the mutual inductance M [14]; consequently, when misalignment occurs, the mutual inductance M between the coils will gradually decrease, resulting in a decrease in the coupling coefficient k. According to (13), η_max is related only to k and Q; thus, when Q remains unchanged, η_max decreases with increasing misalignment of the coils, and effectively increasing the magnetic induction strength on the receiving plane can effectively weaken this phenomenon.

$$R_E={\pi }^2{R}_L/8$$

(9)

$$M\approx \frac{B\cdot S_1}{I}$$

(10)

$$\left[\begin{array}{cc}j\omega L_3+X_1& -j\omega M\\ -j\omega M& X_2+j\omega L_4+R_E\end{array}\right]\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]=\left[\begin{array}{c}U\\ 0\end{array}\right]$$

(11)

$$P=\omega M{I}_1{I}_2$$

(12)

$${\eta }_{\max }=\Delta /(\sqrt{1+\Delta }+1{)}^2$$

(13)

3. Outer-Layer Parameter Design Based on a Regular DD Coil

This section presents a parametric analysis of a regular coil to determine a suitable approach for improving the coil. The indicator for verifying the uniformity of the coil’s magnetic field is α, as defined in the previous section. The flow chart of the two-layer design process is shown in Figure 4. The outer layer of the process is designed to explore the influence of the outer and inner coil diameters on α. The inner layer is designed to explore the influence of the wire distribution in the coil. To study the variability of the coil, the external parameters H and S of the coupling mechanism are varied to simulate different charging scenarios. First, we explore the independent influence of each parameter on α, and then we further study the combined effects of the parameters.

3.1. Influence of the Outer Coil Diameter on the Magnetic Field

For a given charging area, a small single-turn coil size of the transmitting coil will generate a convex magnetic field in space, while a larger single-turn coil size will give rise to a concave magnetic field. The transmitting coil consists of multiple single-turn coils of different sizes, and by suitably designing the size of each turn, both types of magnetic fields can be superimposed on each other, resulting in a relatively smooth magnetic field in the charging area. The magnetic field is smoothest when each single-turn coil is a square with a side length of approximately 400 mm [18]; thus, outer coil diameters in the range of 200 mm to 600 mm are considered.

This section examines the selection of the outer coil diameter parameter. The number of turns of the coil is fixed at 10, and the turn spacing is 4 mm. Figure 5 shows the distributions of α under different values of H, S, and A. α and S are positively correlated, while α and A are negatively correlated. Therefore, the optimal range of values for α is located in the lower right corner, with large values of A and small values of S. As H increases from 100 mm to 200 mm, the corresponding optimal value of A shifts in the direction of larger values. Suppose that the threshold for α is 0.5; then, A should be at least 400 mm, and S should be no more than 300 mm.

3.2. Influence of the Inner Coil Diameter on the Magnetic Field

This section examines the selection of the inner coil diameter parameter. When A is fixed at 400 mm, a is affected by both N and p, following the equation given in Figure 4. Figure 6 shows the influence of N on α. When S is larger than 300 mm, α is minimally affected, but the offset rate for α increases when S is smaller than 300 mm. Additionally, when the value of H is 100 mm, the optimal range of N is [10,12], and the interval shifts in the direction of larger numeric values with increasing H.

Figure 7 shows the influence of p on α. The trend of the change in the optimal range for p is similar to that for N, and with increasing H, the limitations on N and p become stricter. However, if N and p change simultaneously, the optimal combination of N and p cannot be observed directly from these figures.

Figure 8 shows the variation in α with both N and p. The optimal combinations of N and p are distributed in three regions, and the smallest value for α can be less than 0.3. In this case, the value of a is approximately 80 mm. The following analyses are based on the condition that the values of A and a are 400 mm and 80 mm, respectively. The values of H and S are also fixed at 100 mm and 200 mm, respectively.

The magnetic field distribution of the optimal regular DD coil is shown in Figure 9. The value of B is larger in the middle of the charging area, while it decreases as the test point shifts outward in all directions. To make the distribution flatter, the magnetic field in the middle region needs to be weakened, while the magnetic field in the outer region needs to be strengthened. Based on this understanding, two improved DD coil structures are proposed, as shown in Figure 10. The wire arrangements in these improved structures, which are referred to as a double-spacing coil and a combined coil, are modified in different ways.

4. Inner-Layer Parameter Design Based on a Regular DD Coil

Under the condition that the determined outer-layer design remains unchanged, this chapter proposes two inner-layer optimization schemes, namely, an inner and outer double-spacing coil structure and a combined coil structure. The two are compared with a regular DD coil based on the uniformity metric.

4.1. Double-Spacing DD Coil

The structure of the double-spacing coil can be seen in Figure 10, where p_i and p_o denote the spacings of the inner and outer groups of wires, respectively, and p_o is smaller than p_i. This design makes the wires more compact on the outside and sparser on the inside. This is equivalent to weakening the magnetic field in the center while strengthening the magnetic field on all sides. The greater the difference between p_o and p_i is, the more pronounced the change in the magnetic field.

4.2. Combined DD Coil

In this design, evenly distributed wires are divided into two groups. The turn spacing is equal to p in both groups, but the distance p_g between the two groups is much larger than the turn spacing within the groups. This design is equivalent to shifting the magnetic field in the center and on all sides. The larger p_g is, the farther the offset distance, and the more pronounced the change in the magnetic field.

4.3. Comparison of the Magnetic Field Distributions

Figure 11 shows the distributions of B within the charging area in the different coil designs, with [220, 300] μT set as the acceptable range. The effective areas of both new designs are larger than that of the regular design, and fault phenomena in B appear in the regular design at 220 μT. In addition, the degree of uniformity in the effective charging area appears to be different for each design, which can be analyzed by comparing α. A comparison of the α values for the different coils is shown in Figure 12. N is 15 for all three coil structures. β represents the ratio of the outer turn spacing to the inner turn spacing, which is 1 except for the double-spacing coil, whereas p_g represents the distance between the two sets of wires, which is zero except for the combined coil. α is 0.29 for the optimal regular coil, while the minimum values of α in the double-spacing design and the combined design are 0.28 and 0.23, respectively. Thus, the uniformity of the magnetic field of the combined design is increased by 20.7% compared to that of the regular coil, which means that the magnetic flux through the receiving coil shows less spatial variation compared to that of the regular coil.

Table 2. Comparison of calculated and simulated data.

Design Type	Maximum Point	Calculated Data (μT)	Simulated Data (μT)	Deviation
Regular	(−0.046,0)	3.65 × 10-4	3.81 × 10-4	−4.1%
Double	(3.846,−0.051)	3.28 × 10-4	3.36 × 10-4	−2.4%
Combined	(0.046,0.032)	3.03 × 10-4	3.22 × 10-4	−5.9%
Design type	Minimum point	Calculated data (μT)	Simulated data (μT)	Deviation
Regular	(−0.068,−0.1)	2.6 × 10-4	2.76 × 10-4	−5.8%
Double	(−0.068,−0.02)	2.3 × 10-4	2.39 × 10-4	−3.8%
Combined	(−0.046,0.006)	2.34 × 10-4	2.48 × 10-4	−5.6%

shows a quantitative comparison of relevant data from calculations and simulations. The deviation between the simulated and calculated results is within an acceptable range, which indicates that the DD coil structure is suitable for establishing a uniform magnetic field within the charging area. These simulation results demonstrate the feasibility of the design flow shown in Figure 4, which results in the optimal combination of the coil parameters under multiple conditions.

5. Experimental Verification

Based on the above theoretical analysis and simulations, a set of WPT prototypes was built. To test the anti-offset performance of the combined DD coil, it was compared against the other DD coil designs. The experimental setup is depicted in Figure 13. The structures of the transmitting and receiving coils are shown in Figure 14, and the parameters of the coils can be seen in

Table 3. Coil parameters.

Parameter	Value
Outer diameter of the transmitting coil	400 mm
Wire diameter of the transceiver coils	2 mm
Number of turns of the transmitting coil	12
Outer diameter of the receiving coil	250 mm
Number of turns of the receiving coil	13
Inductance of the regular transmitting coil	91.804 μH
Inductance of the combined transmitting coil	111.483 μH
Inductance of the receiving coil	44.945 μH

. The parameters of the compensation network are shown in

Table 4. Parameters of the compensating network.

Parameter	Value	Parameter	Value
L3	46 μH	C1	10 nF
L4	36.8 μH	C2	29.5 nF
C3	27 nF	C4	23.6 nF

.

The prototype system consists of an electronic source, an inverter, a rectifier, a transmitting coil, a receiving coil, a load, and several capacitances and inductances used for compensation. Ferrites can guide magnetic induction lines to become more concentrated in space, thereby enhancing the degree of coupling between transceiver coils [12]. In this experiment, a PC44 ferrite sheet with a side length of 50 mm and a height of 1 mm was used for magnetic leakage shielding. The ferrite sheet was placed below the transmitting coil and above the receiving coil such that the coil was completely covered.

The experimental system operates as follows. First, an ac current at a frequency of 85 kHz is generated by the electronic source and inverter. This creates an alternating magnetic field in space as it flows through the transmitting coil. The alternating magnetic field then acts on the receiving coil to produce an induced current in the receiving circuit. Due to the design of the LCC compensation network, the loops of the transceiver coils are in a resonant state, which improves the transmission effect. Finally, the induced current is passed to the load, completing the WPT process.

During the experiment, the transmitting coil was fixed, the distance between the transmitting and receiving coils was 100 mm, and the offset phenomenon was simulated by moving the receiving coil horizontally. The changes in the transmission efficiency of the system under different offset conditions were recorded. Figure 15 shows the variations in the simulated and experimentally observed transmission efficiencies of the regular and combined coil structures under different offsets. To compare the anti-offset performance of the coils, the offset rate of the transmission efficiency is defined as shown in (14).

$$({\eta }_{\max }-{\eta }_{\min })/{\eta }_{\max }$$

(14)

When the transmitting coil is a combined DD coil, the transmission efficiency of the system in either the X or Y direction is lower than that of a system with the optimal regular DD coil when no offset is present. However, when an offset occurs, the change in the performance of the regular coil is greater than that of the combined coil. When the offset is sufficiently large, the efficiency of the combined coil is higher than that of the regular coil. When the offset distance is within 75 mm, the overall offset rates of the efficiency of the combined and regular coil designs in the X direction are 3.21% and 11.1%, respectively. In the Y direction, the offset rates of the overall transmission efficiency of the two coils are 3.43% and 6.31%, respectively. Thus, the improved coil has a lower offset rate in both directions, although the optimization in the X direction is more substantial than that in the Y direction. Accordingly, these experiments show that the coil structure obtained through the design flow shown in Figure 4 can improve the anti-offset ability of the coil.

By comparing the simulation results with the experimental results, it can be seen that the overall trend of the change in the transmission efficiency is consistent with the calculations and simulations, but the absolute efficiency in the experimental results is considerably decreased compared to the expected value, while the offset rate of the efficiency is slightly larger. This may be attributable to the fact that in the equivalent circuit analysis, the losses caused by capacitance resistance and electromagnetic stress as well as the parameter deviations of the physical components were ignored. Moreover, real experiments will be influenced by environmental disturbances as well as disturbances in the inductance and mutual inductance of the coil over time.

The offset rates of the transmission efficiency of different DD coils presented in other studies are compared in

Table 5. Comparison of offsets in transmission efficiency.

	Design	[11]	[12]	This Paper
Axis
X		11.1%	7.7%	3.21%
Y		6.21%	6.25%	3.43%

. The offset rate of the combined DD coil proposed in this work is reduced by 71.1% in the X direction and 44.7% in the Y direction compared to that of the regular DD coil. The double-spacing coil, however, has little change in the arrangement in the Y direction; hence, the offset rate is reduced only in the X direction. The combined structure has similar offset rates in the X and Y directions and thus shows greater promise for practical applications.

In this experiment, the uniformity coefficient of the improved transmitting coil was 0.23. The experimental results show that the offset rate of the transmission efficiency is positively correlated with the uniformity coefficient. Subsequently, the relationship between the offset rate of the transmission efficiency and the uniformity coefficient can be more clearly defined by investigating samples with a wider range of values of the uniformity coefficient α.

6. Conclusions

An anti-offset WPT system that is designed to have a magnetic field that is as uniform as possible is proposed. As the transmitting coil, a combined coil design is adopted that can generate a uniform magnetic field in the receiving plane so that the mutual inductance between the receiving and transmitting coils will remain relatively constant when the receiving coil is offset in the X and Y directions in the charging plane.

First, our design was tested using simulation software to investigate its anti-offset performance. Then, experiments were performed to compare our design with designs presented in other studies. When the unilateral offset of the receiving coil in the X or Y direction is less than 75 mm, the proposed system can maintain more than 80% of its transmission efficiency, and the offset rates in the X and Y directions are less than 3.21% and 3.43%, respectively.

The anti-offset performance of the proposed coil has been effectively improved in both directions, and the simulated and experimental results are consistent with the theory. Since the proposed design involves optimizing only the wire arrangement, it is difficult to achieve an optimization effect comparable to that of multiple superimposed transmitting coils, but our approach ensures the simplicity of the structure. Furthermore, additional groups of wires could also be considered to achieve better performance.

In this study, a bridge is built between the anti-offset performance of a WPT system and its magnetic field uniformity coefficient through the design of the coil parameters, thereby simplifying the steps of the design process. The corresponding improvements in coil parameter design can be applied in WPT systems to improve the uniformity of their magnetic fields.