1 Introduction

Various electrical transportation systems, such as electric vehicles, and electric ships, are being used today [1, 2]. Because they operate using a battery, they must be recharged, and their charging method is an important design consideration. Typically, these methods can be divided into wired charging and wireless charging approaches [3, 4]. Wired charging has been used in various applications and provides both short charging times and high efficiency. However, a leakage current can be caused by old cracked cables, and it can be difficult to connect due to heavy cables [5, 6]. To solve these problems, wireless power transfer (WPT) is being developed for wireless charging. It does not cause a leakage current, and charging can be easily accomplished without a charging cable connection [7, 8]. For these advantages, WPT is a suitable option for battery charging [9,10,11].

WPT systems have been widely employed for low power applications such as implantable devices, and high-power applications, such as maritime and mobile vehicles [12,13,14].

Due to structural limitations and environmental factors, several types of misalignments can occur in the WPT system between the Tx and Rx coils, including lateral, longitudinal, and angular misalignments.

Figure 1 illustrates the concept of angular misalignment between the Tx and Rx coils. The angular misalignments are common in various applications, including medical implants, electric vehicles, and home appliances [15,16,17]. Especially, when the WPT is applied to electric ships or submarines, it is difficult to maintain a parallel condition or a perfect alignment angle between the Tx and Rx coils [18,19,20]. In these cases, when using symmetrical coils, mutual inductance is decreased when the misaligned angle exceeds a certain value. Because power transfer efficiency is an important performance factor in WPT systems, obtaining the maximum mutual inductance, which is highly related to power transfer efficiency, is also important [21, 22].

Fig. 1
figure 1

The concept of angular misalignment between the Tx and Rx coils in the WPT system

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This paper introduces asymmetrical WPT coils, trapezoidal Tx coils and a rectangular Rx coil to achieve the maximum mutual inductance and power transfer efficiency. Although WPT systems that include ferromagnetic materials achieve higher power transfer efficiency, this research excludes magnetic materials for simpler calculations. The mutual inductance between the two coils is theoretically derived under the angular misalignment condition using a geometrical approach, based on the Biot-Savart law [23]. By using the perspective of using trapezoid coil, which is a component of asymmetrical coils, it is able to avoid a complicated conventional integral formula. The proposed formula is verified by comparing the simulated and measured results. As a result, it is confirmed that the trapezoidal coil is suitable for the inclined WPT coil condition.

2 Theoretical Analysis

In order to derive the mutual inductance between two coils with a geometrical approach, a simplified trapezoidal coil is introduced and analyzed in four segmented parts. The analysis of the magnetic field vector is followed by mutual inductance investigation.

2.1 Theoretical Analysis of Magnetic Field Vectors

In order to derive the mutual inductance, the magnetic field must be calculated, which can be accomplished using the Biot-Savart law. The magnetic field at an arbitrary position in free space generated by a straight current carrying wire is given by (1)

$$\begin{array}{*{20}c} {\vec{B} = \frac{{\mu_{0} }}{4\pi }\int {\frac{{id\vec{l} \times \hat{a}_{r} }}{{\left| {\vec{r}} \right|^{2} }}} } \\ \end{array} ,$$
(1)

where the \(\left| {\vec{r}} \right|\) is the magnitude of the position vector at the observation point, and the vector \(\hat{a}_{r}\) is the unit vector of \(\vec{r}\) derived from the differential element of wire \(\vec{l}\), which is the current direction of the wire. The scalar \(\mu_{0}\) is the permeability of free space, and \(i\) is the current in the wire.

In order to apply the Biot-Savart law to a coil, it can be divided into a number of individual m wires. In particular, the trapezoidal coil can be separated into four wires, and here, each wire is designated wire 1 to wire 4. In addition, the ratio between the upper side wire and the lower side wire is defined as the length ratio, which is expressed by \(r^{*}\).

The magnetic field vectors at the arbitrary point using the equation of a straight line is represented in Fig. 2, and a to d illustrate the magnetic field vectors generated by wire 1 to wire 4, respectively. Here, the equation of a straight line of \(m{\text{th}}\) wire can be expressed in the form of \(a_{m} x + b_{m} y + c_{m} = 0 \left( {m = 1, 2, 3, 4} \right).\) It is assumed that the distance between the arbitrary point \(P^{\prime } \left( {x_{0} ,y_{0} ,0} \right)\) and the wire is \(r_{m}\), and the distance between the arbitrary point \(P\left( {x_{0} ,y_{0} ,z_{0} } \right)\) and the wire is \(R_{m}\). Also, each wire has a length of \(2L_{m}\), and the distance between the center point of each wire and the foot for the perpendicular on the wire at point \(P^{\prime }\) is \(l_{m}\). Wire 1 and wire 3, which correspond to the side of the coil, are tilted by \(\theta\) from the y-axis, and as a result, the magnitude of the magnetic field vector \(\left| {\vec{B}_{m} } \right|\) generated at arbitrary point P by each wire can be calculated as (2)

$$\begin{array}{*{20}c} { \left| {\vec{B}_{m} } \right| = \frac{{\mu_{0} i}}{{4\pi R_{m} }}\left( {\frac{{L_{m} + l_{m} }}{{\sqrt {R_{m}^{2} + \left( {L_{m} + l_{m} } \right)^{2} } }} + \frac{{L_{m} - l_{m} }}{{\sqrt {R_{m}^{2} + \left( {L_{m} - l_{m} } \right)^{2} } }}} \right).} \\ \end{array}$$
(2)
Fig. 2
figure 2

The magnetic field vectors at an arbitrary point using the equation for a straight line. Magnetic field vectors generated by a wire 1, b wire 2, c wire 3, and d wire 4

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Please note that the magnetic field produced by each wire can be divided into magnetic field vectors in the xy-plane \(\vec{B}_{{xy_{m} }}\) and in the z-axis \(\vec{B}_{{z_{m} }}\). Then, \(\vec{B}_{{xy_{m} }}\) can be divided into the magnetic field vectors in the x-axis \(\vec{B}_{{x_{m} }}\) and in the y-axis \(\vec{B}_{{y_{m} }}\). The angle between \(\left| {\vec{B}_{m} } \right|\) and \(\vec{B}_{{xy_{m} }}\) is \(\phi_{m}\), and the angle between \(\vec{B}_{{x_{m} }}\) and \(\vec{B}_{{xy_{m} }}\) is \(\theta\). The magnetic field vectors in the x, y and z-axis, which are noted as \(\vec{B}_{x} ,\vec{B}_{y} ,\vec{B}_{z}\), respectively, can be calculated as (3-a), (3-b) and (3-c), where the unit vector of the x, y, and z-axis are noted as \(\hat{a}_{x} ,{ }\hat{a}_{y} , \hat{a}_{z}\), respectively. Since wire 2 and wire 4 are parallel to the x-axis, \(\vec{B}_{x}\) is neglected. When an angular misalignment exists between two coils, the mutual inductance can be easily derived by conducting two calculations, as illustrated in Fig. 3. The process of converting the coordinates is illustrated in Fig. 3a. It is assumed that the two points with different signs of the y-axis coordinate under the alignment condition are called \(P_{{k_{1} }}\) and \(P_{{k_{2} }}\), converted into two points \(P_{{k_{1} }}{\prime}\) and \(P_{{k_{2} }}{\prime}\) under the angular misalignment condition, and can be obtained as (4) and (5)

$$\begin{array}{*{20}c} {\vec{B}_{x} = \mathop \sum \limits_{m = 1}^{4} \vec{B}_{{x_{m} }} = \left( { - \frac{{\left| {\vec{B}_{1} } \right|z_{0} \cos \left( \theta \right)}}{{R_{1} }} + \frac{{\left| {\vec{B}_{3} } \right|z_{0} \cos \left( \theta \right)}}{{R_{3} }}} \right)\hat{a}_{x} } \\ \end{array}$$
(3-a)
$$\begin{array}{*{20}c} {\vec{B}_{y} = \mathop \sum \limits_{m = 1}^{4} \vec{B}_{{y_{m} }} = \left( {\begin{array}{*{20}c} {\frac{{\left| {\vec{B}_{1} } \right|z_{0} \sin \left( \theta \right)}}{{R_{1} }} - \frac{{\left| {\vec{B}_{2} } \right|z_{0} \sin \left( \theta \right)}}{{R_{2} }} + } \\ {\frac{{\left| {\vec{B}_{3} } \right|z_{0} \sin \left( \theta \right)}}{{R_{3} }} + \frac{{\left| {\vec{B}_{4} } \right|z_{0} \sin \left( \theta \right)}}{{R_{4} }}} \\ \end{array} } \right)\hat{a}_{y} } \\ \end{array}$$
(3-b)
$$\begin{array}{*{20}c} {\vec{B}_{z} = \mathop \sum \limits_{m = 1}^{4} \vec{B}_{{z_{m} }} = \left( {\frac{{\left| {\vec{B}_{1} } \right|r_{1} }}{{R_{1} }} + \frac{{\left| {\vec{B}_{2} } \right|r_{2} }}{{R_{2} }} + \frac{{\left| {\vec{B}_{3} } \right|r_{3} }}{{R_{3} }} + \frac{{\left| {\vec{B}_{4} } \right|r_{4} }}{{R_{4} }}} \right)\hat{a}_{z} } \\ \end{array}$$
(3-c)
$$P_{{k_{1} }}^{\prime } \left( {x_{{k_{1} }}^{\prime } , y_{{k_{1} }}^{\prime } ,z_{{k_{1} }}^{\prime } } \right) = P_{{k_{1} }}^{\prime } \left( {x_{{k_{1} }} , y_{{k_{1} }} \times \cos \left( \alpha \right), z - \left| {y_{{k_{1} }} \times \sin \left( \alpha \right)} \right|} \right)$$
(4)
$$P_{{k_{2} }}^{\prime } \left( {x_{{k_{2} }}^{\prime } , y_{{k_{2} }}^{\prime } ,z_{{k_{2} }}^{\prime } } \right) = P_{{k_{2} }}^{\prime } \left( {x_{{k_{2} }} , y_{{k_{2} }} \times \cos \left( \alpha \right), z + \left| {y_{{k_{2} }} \times \sin \left( \alpha \right)} \right|} \right),$$
(5)

where the misaligned angle is \(\alpha (\alpha > 0)\).

Fig. 3
figure 3

The magnetic field vector under angular misalignment. a Conversion of coordinates. b Calculation of vertical component magnetic field vector perpendicular to the Rx coil

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The magnitude of the magnetic field vector which is perpendicular to the Rx coil, noted as \(\left| {\vec{B}_{{V_{m} }} } \right|\), is illustrated in Fig. 3b. It is assumed that the angle between \(\vec{B}_{m}\) and y-axis is \(\beta_{m}\), and the angle between \(\vec{B}_{{V_{m} }}\) and \(\vec{B}_{m}\) is \(\gamma_{m}\). By using \(\alpha\) and \(\beta_{m}\), \(\gamma_{m}\) can be derived as \(\gamma_{m} = \pi /2 - \left( {\alpha + \beta_{m} } \right)\), where \(\beta_{m} = \tan^{ - 1} \left( {\left| {\vec{B}_{{z_{m} }} } \right|/\left| {\vec{B}_{{y_{m} }} } \right|} \right)\). As a result, \(\left| {\vec{B}_{{V_{m} }} } \right|\) can be derived as (6).

$$\begin{aligned} \left| {\vec{B}_{{V_{m} }} } \right| & = \sqrt {\left( {|\vec{B}_{{y_{m} }} |} \right)^{2} + \left( {\left| {\vec{B}_{{z_{m} }} } \right|} \right)^{2} } \times \cos \left( {\gamma_{m} } \right) \\ & = \sqrt {\left( {|\vec{B}_{{y_{m} }} |} \right)^{2} + \left( {\left| {\vec{B}_{{z_{m} }} } \right|} \right)^{2} } \times \cos \left( {\frac{\pi }{2} - \left( {\alpha + \beta_{m} } \right)} \right) \\ & = \sqrt {\left( {|\vec{B}_{{y_{m} }} |} \right)^{2} + \left( {\left| {\vec{B}_{{z_{m} }} } \right|} \right)^{2} } \sin \left( {\alpha + \tan^{ - 1} \left( {\frac{{\left| {\vec{B}_{{z_{m} }} } \right|}}{{\left| {\vec{B}_{{y_{m} }} } \right|}}} \right)} \right) \\ \end{aligned}$$
(6)

2.2 Derivation of the Mutual Inductance

The mutual inductance using division of the Rx coil is illustrated in Fig. 4. In order to avoid a complicated conventional integral formula, the Rx coil is divided into a number of k square-shaped differential areas from \(\Delta s_{1}\) to \(\Delta s_{k}\), which are placed inside the Rx coil without overlapping parts. Each area has a center point \(P_{k}\), and the calculation of the magnetic field vectors at the \(P_{k}\) generated by the \(m{\text{th}}\) wire is the same as in (2) and (3). Based on the magnetic field vectors and division of the Rx coil, the magnetic flux \(\Phi_{Tx}\) generated by a single-turn Tx coil under the angular misalignment condition is represented in (7).

$$\Phi_{Tx} = \mathop \sum \limits_{k = 1}^{25} \left( {\mathop \sum \limits_{m = 1}^{4} \left| {\vec{B}_{{V_{m} }} } \right|} \right)\Delta s_{k} = \mathop \sum \limits_{k = 1}^{25} \left[ {\mathop \sum \limits_{m = 1}^{4} \left\{ {\sqrt {\left( {|\vec{B}_{{y_{m} }} |} \right)^{2} + \left( {\left| {\vec{B}_{{z_{m} }} } \right|} \right)^{2} } \sin \left( {\alpha + \tan^{ - 1} \left( {\frac{{\left| {\vec{B}_{{z_{m} }} } \right|}}{{\left| {\vec{B}_{{y_{m} }} } \right|}}} \right)} \right)} \right\}} \right]\Delta s_{k}$$
(7)
$$\Phi_{{Tx_{ij} }} = \left[ {\mathop \sum \limits_{p = 1}^{25} \left[ {\mathop \sum \limits_{m = 1}^{4} \left\{ {\sqrt {\left( {|\vec{B}_{{y_{m} }} |} \right)^{2} + \left( {\left| {\vec{B}_{{z_{m} }} } \right|} \right)^{2} } \sin \left( {\alpha + \tan^{ - 1} \left( {\frac{{\left| {\vec{B}_{{z_{m} }} } \right|}}{{\left| {\vec{B}_{{y_{m} }} } \right|}}} \right)} \right)} \right\}} \right]\Delta s_{p} } \right]_{ij}$$
(8)
Fig. 4
figure 4

Division of the Rx coil into a number of k differential areas

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Accordingly, the mutual inductance between two single-turn coils can be expressed as \(M_{single} = N_{Rx} \Phi_{Tx} /I_{Tx}\), where \(N_{Rx}\) is the number of turns in the Rx coil, and \(I_{Tx}\) is the current in the Tx coil.

The results of the comparison of the calculated and simulated values of mutual inductance according to the number of differential areas are illustrated in Fig. 5. Calculations and simulations were performed assuming the length ratios of the Tx coils were 1:1 and 1:3, and a 3-D finite element method (FEM) analysis tool Ansys Maxwell was used for the simulation. As described, as the number of differential areas increased, the calculated values tended to converge to the simulated values, and the 25 differential areas were considered for the simple calculation.

Fig. 5
figure 5

Results of the comparison of the mutual inductance according to the number of differential areas

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Based on (7), the calculation of magnetic flux through the \(j{\text{th}}\) Rx coil generated by the \(i{\text{th}}\) Tx coil under angular misalignment condition, noted as \(\Phi_{{Tx_{ij} }}\), can be expressed as (8). As a result, the mutual inductance between two multi-turn coils can be obtained as (9)

$$M_{multi} = \mathop \sum \limits_{i = 1}^{{N_{Tx} }} \mathop \sum \limits_{j = 1}^{{N_{Rx} }} M_{ij} = \mathop \sum \limits_{i = 1}^{{N_{Tx} }} \mathop \sum \limits_{j = 1}^{{N_{Rx} }} \frac{{N_{Rx} \Phi_{{Tx_{ij} }} }}{{I_{Tx} }}.$$
(9)

The results of the comparison of the mutual inductance of the multi-turn Tx and Rx coils according to the angular misalignment are illustrated in Fig. 6. The length ratios of the Tx coil were 1:1, 1:2, 1:3, and 1:4. Also, the misaligned angle ranged from 0° to 60°. As a result, error rates between the simulation and calculation were lower than 2%, and these results validate the proposed process of calculating the mutual inductance. In addition, there are angle ranges in which a symmetrical coil is recommended, and an asymmetrical coil is recommended, from a specific misaligned angle (\(\upalpha\)).

Fig. 6
figure 6

Results of the comparison of the mutual inductance between the multi-turn Tx and Rx coils

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3 Experimental Verification

3.1 Mutual Inductance Measurement Setup

To verify the proposed process for calculating the mutual inductance, a small-scale measurement setup was configured.

The designed resonance frequency was 85 kHz, which is the standard for the wireless charging of an electric vehicle [24]. The air gap between the Tx and Rx coils was assumed to be 100 mm. Four Tx coils with length ratios of 1:1, 1:2, 1:3, 1:4, and one square-shaped Rx coil were fabricated to investigate the tendency of the length ratio of the Tx coils, as illustrated in Fig. 7. All the coils were fabricated using Litz wire of AWG-38 under the same internal area. For the experiment, the mutual inductance between two coils was measured according to the angular misalignment, which ranged from 0° to 60° at 5° intervals, using the LCR meter (HIOKI-IM3523) as illustrated in Fig. 8. The inductance of each coil, the capacitance of each matching capacitor, and the impedance of each matching circuit were measured. Detailed electrical parameters are listed in Table 1.

Fig. 7
figure 7

Dimensions of fabricated coils. a Tx coils, b Rx coil

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Fig. 8
figure 8

Experimental setup for measuring the mutual inductance

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Table 1 Electrical parameters of fabricated coils
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3.2 Mutual Inductance Measurement Results

The tendency of the mutual inductance for each misaligned angle is illustrated in Fig. 9. Measured values are compared with calculated and simulated values. Solid line, dashed line, and plotted triangle represent the calculated, simulated, and measured data, respectively. Each graph denotes the mutual inductance (M) according to the length ratio, which can be represented as a function of several parameters, as \(M = f\left( {r^{*} ,\alpha ,z_{0} \cdots } \right)\). In the graph above, the maximum mutual inductance can be obtained by \(\left| {\partial M/\partial r^{*} } \right| = 0\), and we defined the \(\left| {\partial M/\partial r^{*} } \right|\) as \(\zeta\). As a result, a critical length ratio, where the maximum mutual inductance can be evaluated, can be found from the minimum value of \(\zeta\). For example, when the misaligned angle is 40°, a Tx coil with a length ratio of 1:2.8 has the maximum mutual inductance. In addition, the same trend is observed when each graph is transformed according to the misaligned angle, as illustrated in Fig. 10.

Fig. 9
figure 9

Results of the comparison of mutual inductance according to the length ratio of Tx coils at the air gap of 100 mm

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Fig. 10
figure 10

Results of the comparison of mutual inductance according to the misaligned angle at the air gap of 100 mm

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Based on the simulated and measured mutual inductance, the coupling coefficient can be obtained using the formula k = M/\(\sqrt {{\text{L}}_{{{\text{Tx}}}} {\text{L}}_{{{\text{Rx}}}} }\), and is illustrated in Fig. 11. Similar to Fig. 10, as the angular misalignment varies, the length ratio achieving the maximum coupling coefficient also varies.

Fig. 11
figure 11

Results of the comparison of coupling coefficient to the various misaligned angle at the air gap of 100 mm

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3.3 Power Transfer Efficiency Measurement Setup

The fabricated coils were then used to measure power transfer efficiency. A matching circuit was attached to each Tx and Rx coil.

To generate the Gate-Source signal (Vgs) of MOSFET, a field programmable gate array (FPGA) was used, and monitored by an oscilloscope 1 (Tektronix TDS2014B). Transmit power, receive power, and power transfer efficiency between the Tx and Rx coils were monitored by oscilloscope 2 (Keysight DSOX4024A) with differential voltage probes (Keysight N2891A) and current probes (Keysight N2782B).

In order to measure the coil to coil power transfer efficiency, a rectifier was not considered in this experiment, and instead, a resistor (1 Ω) was used for a load. To compare the tendency according to the air gap, three cases of air gaps were considered, 50 mm, 75 mm, and 100 mm. The overall setup for measuring the power transfer efficiency is illustrated in Fig. 12.

Fig. 12
figure 12

Experimental setup for measuring the power transfer efficiency

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3.4 Power Transfer Efficiency Measurement Results

The measured voltages and currents of both Tx coil and Rx coil are illustrated in Fig. 13. As this system used series-series WPT topology, the VTx and ITx has same phase part as LC resonance circuit is applied. The ringing is observed in during the measurement, which introduced by low input-impedance and parasitic capacitance in an inverter. This phenomenon affects power transfer efficiency degradations of compared to the simulation results.

Fig. 13
figure 13

Experimental results of VTx, ITx, VRx, and IRx according to misaligned angle \((\upalpha )\) and length ratio (r*) with 50 mm air gap. a r* = 1:1 with \(\upalpha = 0^{ \circ }\), b r* = 1:1 with \(\upalpha = 45^{ \circ }\), c r* = 1:1 with \(\upalpha = 60^{ \circ }\), d r* = 1:4 with \(\upalpha = 0^{ \circ }\), e r* = 1:4 with \(\upalpha = 45^{ \circ }\), and f r* = 1:4 with \(\upalpha = 60^{ \circ }\)

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For this paper consider the series-series topology with a resistive load (1 Ω), the power transfer efficiency can be derived as

$$\upeta = \frac{{\upomega ^{2} {\text{M}}^{2} {\text{R}}_{{\text{L}}} }}{{\left( {{\text{R}}_{{{\text{Rx}}}} + {\text{R}}_{{\text{L}}} } \right)\left( {\upomega ^{2} {\text{M}}^{2} + {\text{R}}_{{{\text{Tx}}}} \left( {{\text{R}}_{{{\text{Rx}}}} + {\text{R}}_{{\text{L}}} } \right)} \right)}},$$
(10)

where the ω, M, RTx, RRx, and RL represents angular frequency of VTx, mutual inductance, inner resistance of transmitting coil, inner resistance of receiving coil, and load resistance, respectively.

Please note that the absolute value of power transfer efficiency is not considered, as this experiment only validates the increase in power transfer efficiency of the asymmetrical coil in an inclined WPT system. Although the power transfer efficiency is observed to be around 50%, the results support the proposed idea that a trapezoidal coil including asymmetrical coils achieves higher power transfer efficiency compared to conventional symmetrical WPT coils.

Specifically, the results of the power transfer efficiency measurements according to the length ratio of the Tx coils are illustrated in Fig. 14. The calculated, simulated, and measured results have a similar tendency at the three air gaps of 50 mm, 75 mm, and 100 mm, as presented in (a) to (c), respectively. By calculating the minimum value of the \(\zeta\), a length ratio with the maximum transfer efficiency for various misaligned angles can be obtained. Unlike the other three cases, when the misaligned angle is 60°, it is expected to have a maximum power transfer efficiency for a length ratio greater than 1:4. These results showed that the asymmetrical coils exhibited the maximum mutual inductance at the above conditions, and therefore have the maximum power transfer efficiency. Additionally, according to the air gap, the range of misaligned angles for which symmetrical and asymmetrical coils are recommended, is depicted in Fig. 15. The results of three air gaps of 50 mm, 75 mm, and 100 mm are represented in (a) to (c), respectively.

Fig. 14
figure 14

Experimental results of power transfer efficiency according to length ratio. a Air gap = 50 mm, b Air gap = 75 mm, and c Air gap = 100 mm

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Fig. 15
figure 15

Experimental results of power transfer efficiency according to misaligned angle. a Air gap = 50 mm, b Air gap = 75 mm, and c Air gap = 100 mm

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It can be observed that as the air gap varies, the range of misaligned angle in which the asymmetrical coil is recommended varies. Also, as the air gap increases, the range of misaligned angle in which the asymmetrical coil is recommended becomes wider.

4 Conclusion

This paper proposed asymmetrical WPT coils for high efficiency under various angular misalignment conditions. Based on the Biot-Savart law, the mutual inductance was calculated by dividing the Rx coil into a number of k differential areas. Calculated results were compared with the simulated and measured results. Three asymmetrical Tx coils, one symmetrical Tx coil, and one Rx coil were fabricated for measurement. The experimental results show that the asymmetrical coils have the maximum power transfer efficiency by achieving the maximum mutual inductance as the angle between two coils is more inclined.