Analysis of Coil Parameters and Comparison of Circular, Rectangular, and Hexagonal Coils Used in WPT System for Electric Vehicle Charging

: In this paper, the major factors that affect the performance of wireless power transfer systems, such as coil inner radius and coil number of turns, are discussed. A comparison of three coil shapes covering the coreless case, the case with ferrite, and the case with ferrite and aluminum is also carried out. Another comparison is proposed by addressing the combination of different coil shapes in the wireless power transfer (WPT)system. The analysis covers the coupling coefﬁcient, the mutual inductance, and the self-inductance. Due to the complexity of calculating these parameters, the ﬁnite element analysis (FEA) method is adopted by using the Ansys Maxwell software. An introduction to the typical WPT system for electric vehicle charging is also presented.


Introduction
As the large demand for fossil fuels for road transportation continues to grow and the percentage of pollution coming from conventional vehicles increases, the need for an alternative solution has become a major necessity. The electric vehicle presents a suitable solution for road transportation by offering multiple advantages: zero gas emissions, reduced noise pollution, low maintenance, and lower running costs [1].
The biggest challenge that electric vehicles face is the battery pack capacities, which give a short driving range because of their limited life and high cost [2]; currently, the battery charging operation for the electric vehicle is carried out by the charging cables. On the other hand, the wireless power transfer system charging method is already used in a variety of systems and applications, such as biomedical devices, industrial manufacturing, including robotic platforms, and a few consumer electronics (watches, smartphones, etc.) [3].
The magnetic resonant wireless power transfer (WPT) technology is a method proposed by Kurs [4] where the energy is wirelessly transferred between the transmitting coil and the secondary coil under resonant conditions. It is suitable for electric vehicle applications. The electric vehicle wireless charging operation is carried out in stationary or dynamic mode. In the first mode, the vehicle is parked whereby the transmitting coil is parallel to the receiving coil. Then, the charging operation can be started. In the second mode, the vehicle is charged while moving on the road occupied by transmitting coils. Thereby, the wireless charging method is easier, with no need for human intervention, safer, and needs less maintenance compared to the conductive method. However, the wireless charging method presents some drawbacks-for example, the large distance between the transmitting and receiving coil, resulting in low coupling and human safety due to the exposure to the magnetic field. The coil design is one of the crucial factors in designing covers the self-inductance, mutual inductance, and the coupling coefficient. Section 5 provides a conclusion to the present paper.

WPT System Basic Principles
The electric vehicle presents a perfect solution for green transportation. However, the major downside of this type of vehicle is related to the energy storage technology. The wireless charging technology offers multiple advantages towards the battery weight, cost, and size. Figure 1 illustrates the concept of WPT charging for the electric vehicle. The WPT system is enabled when the electric vehicle enters the charging zone. The energy is transferred then wirelessly from the power source to the battery of the vehicle using the magnetic field created between the transmitting coil and receiving coil. The charging system is composed of two parts: the on-board and the off-board parts. The first one is located on the vehicle, starting from the receiving coil located under the vehicle to the battery of the vehicle. The second one is located outside the vehicle, starting from the utility AC power source until the transmitting coil.
The SAE J2954 standard [19] proposes three possibilities for the transmitting coil position: above the ground surface, flush with the ground surface, or buried. Moreover, the suggested maximum permissible protrusion above the surface of the ground is 70 mm. This distance can be more or less than the value stated above depending on the local installation rules. For now, the transmitting coil above the ground is the case adopted here. Additionally, to classify the WPT systems based on the estimated maximum ground clearance, three Z-classes are defined: Z1 = 100-150 mm, Z2 = 140-210 mm, and Z3 = 170-250 mm.
The most prevalent WPT technologies are inductive coupling and magnetic resonant coupling. Inductive coupling is a technique commonly used in wireless charging applications. Its range of operation is limited to short distances of less than a centimeter [20]. Moreover, its performance falls dramatically when there is a misalignment between the transmitting coil and the receiving coil, even when the misalignment is only a few centimeters [21].
Magnetic resonant coupling is a technique that was developed by [4]. Its range of operation can reach longer distances, and the energy is transmitted from the transmitting coil to the receiving one with high power transfer efficiency and with minimal energy losses. Figure 2 presents a block diagram of the unidirectional magnetic resonance wireless power transfer for electric vehicle charging. Firstly, the utility AC power is converted to DC power by using a rectifier containing a power factor correction; then, a high-frequency inverter is used to power the compensation network and the transmitting coil. The high current flowing in the transmitting coil generates an alternating magnetic field; this magnetic field induces a voltage in the receiving coil.
Secondly, the voltage induced is rectified using an AC/DC converter, which is used to charge the battery through the battery management system (BMS). The resonance between the two compensation networks improves the transferred power and efficiency [22].

The Magnetic Resonant WPT System
In this section, the induction part of the charging system and the magnetic resonant system's equivalent circuit are discussed.
The working theory of wireless power transfer is regulated by the laws of Ampere and Faraday. Ampere's law states that a magnetic field is generated when an electric current flows through a conductor in free space. (1) indicates the relation between the magnetic field B T created and the electric current flowing in the transmitting coil I 1 , where ∆l is the unit length of the conductor and N 1 is the number of turns of the transmitting coil.
Faraday's law states that when a time-varying magnetic field relates to a conductor, it will cause a voltage to be induced in the conductor. (2) indicates the relation between the voltage e and the rate of change in magnetic flux φ B , where N 2 is the number of turns of the receiving coil.
The WPT principle is based on two coils isolated by an air gap. The transmitting coil located in the ground is powered by the AC with high frequency. As a result, a timevarying magnetic field is created according to Ampere's law. The generated magnetic field is captured by the receiving coil located below the electric vehicle and, due to Faraday's law, the magnetic field induces a voltage. The coupling coefficient k between the transmitting coil and the receiving coil plays a significant role in achieving high efficiency [2]. It is related to the mutual inductance by the following expression (3): (3) Figure 3 shows the equivalent circuit of the magnetic resonant WPT system model for the electric vehicle, where L 1 , R 1 , and C 1 are, respectively, the inductance, the resistance, and the capacitor of the primary side; L 2 , R 2 , and C 2 are, respectively, the inductance, the resistance, and the capacitor of the secondary side; RL is the equivalent load resistance and M is the mutual inductance between coils. The magnetic field generated must be sufficient to connect the secondary coil; for this reason, a large current is needed. Otherwise, the inductive component must be canceled. Moreover, due to the large separation between the coils, the leakage inductance is high, as well as the proximity effect and winding resistance. Therefore, compensation capacitors are added on both the primary and secondary sides.
Different compensation topologies have been suggested in the literature to regulate the coils to work at the same frequency. These compensation networks are located between the inverter and the transmitting coil on the off-board side and between the receiving coil and the rectifier on the on-board side. The most common topologies are the four basic topologies: series-series (SS), series-parallel (SP), parallel-series (PS), and parallelparallel (PP), as illustrated in Figure 4. Moreover, new other topologies are proposed in the literature, such as LCL compensation and LCC compensation. The SS topology is the only one whose primary capacitance does not depend on the coupling coefficient variation [8,23]. Moreover, at low mutual inductance, it achieves a high and stable transfer efficiency, making this topology the best choice for use with variable load conditions [24].
The calculated capacitors for the SS compensation technology are given by (4) and (5): The present analysis reveals the relationship between the efficiency η and the mutual inductance M. The voltage equations of the transmitter and receiver sections can be expressed as: where Z T1 and Z T2 are the transmitter and receiver coil impedances, respectively, which can be expressed as: When the system reaches the resonance frequency, and according to (4)-(6), the voltage V 1 and the current circulating in the second section I 2 are rewritten as: The input power delivered by the voltage source P in and the output power of the load P out are: The power transfer efficiency η is then: If we consider the following condition, R L R 2 , the efficiency is rewritten as: Consequently, the high frequency and the large mutual inductance imply high efficiency.

Impacts of Coil Parameters and Coil Position on the WPT System Performance
In WPT systems for EV applications, the design of the coils used for transferring and receiving the energy must be optimized to improve the system performance. Therefore, it is essential to study what has the most impact on the coupling and efficiency of coils.
The coupling coefficient, the mutual inductance, and the coil self-inductance are the key factors in ensuring high coupling between coils. They are impacted by the coil-to-coil relative position, the addition of materials, and the coil structural parameters.
The analysis of these factors by calculation has limitations; the finite element analysis (FEA) method is the best way to execute these calculations.
The finite element method (FEM) is a numerical method based on determining approximate solutions. The main aspect of this method is to subdivide a complex problem into small pieces to generate a system of simple equations. The solutions of these equations are combined to reach the final solution.
Ansys Maxwell is a powerful finite element analysis (FEA) software program that offers many solvers to evaluate the electromagnetic component design. The system is subdivided into elements according to the mesh (a three-dimensional grid that defines the elements). The mesh can define elements of uniform size and shape or elements of different shapes and sizes in different parts of the domain.
The boundary conditions in the Ansys Maxwell software determine the magnetic field behavior at the interfaces or the edges of the problem region. For the wireless power transfer system, the boundary conditions in the eddy current solver are determined by the air region and/or box whereby the four terminals of two coils touch the face of the boundary. Accordingly, the default parameters can be explained as follows (Table 1)  In this section, a reference coil model is modeled using Ansys Maxwell software to study the system.

The Two-Coil Vertical Position
The coil-to-coil relative position is a crucial factor that impacts the system coupling. The variation in the vertical distance between the coils affects positively or negatively the coupling. This section clarifies the effect of varying the vertical distance between the coils on the coil's main characteristics. The analysis covers the coupling coefficient and the mutual inductance between the transmitting and receiving coils. A reference rectangular model is chosen. It is modeled using Ansys Maxwell software; see Figure 5. The vertical distance between the transmitting and receiving coil depends on the vehicle type and its ground clearance. It differs from one vehicle to another. The gap is the vertical distance between the upper extremity of the transmitting coil and the lower extremity of the receiving coil.
In this study, to simplify the system analysis, the WPT consists of two identical rectangular coils. Figure 6 represents the three cases of rectangular coil model for gap = 100 mm. The simulation of the coreless model ( Figure 6a) was carried out using Ansys Maxwell software by varying the z-distance from 50 to 300 mm.
The overall measurement results of coupling coefficient and mutual inductance for the gap variation (50-300 mm) are summarized in Table 2 and Figure 7.
The simulations indicate that the value of coefficient coupling and the mutual inductance decrease as the vertical distance between the transmitting and receiving coil increases. This implies that low power is transferred to the output as the distance increases. This type of system is referred to as a loosely coupled WPT system because the coupling coefficient between the transmitting coil TX and receiving coil RX coil is less than 0.2 [25].
The coreless transformer represents many advantages in terms of price and weight. However, the low efficiency and the decrease in system performance due to the higher core loss [26] limit its application in electric vehicle charging. Hence, specific materials are usually added to the system to improve its efficiency.

The Introduction of Ferrite and Aluminum to the Coil System
Several studies have been carried out to enhance the weak coupling resulting from the two coils inductively coupled; one of the efficient methods is the integration of new materials into the system. These materials should not affect the electrical performance of the system. The main objective is to enhance the coupling between the coils and minimize the leakage flux by guiding the flux from the transmitter coil to the receiver coil.
The relative permeability of ferrite is high and can reduce the reluctance path [26]. Moreover, according to [27], the hysteresis loop of Mn-Zn ferrite is narrow, which helps to minimize hysteresis loss. As a result, ferrite bars or plates are often used to guide magnetic flux and provide magnetic shielding.
A simple analysis is performed using a ferrite plate to analyze the coupling concerning the gap variation. The same rectangular shape is chosen. Figure 6b illustrates the rectangular coil with ferrite for gap = 100 cm. Table 3 and Figure 8 depict the simulation results of the coupling coefficient and mutual inductance for the gap variation (50 to 300 mm).
The result shows that the coupling coefficient and the mutual inductance values decrease as the z-distance increases. Moreover, the values have been greatly increased compared to the coreless results. The ferrite is dedicated to providing an alternative path for the lines of the magnetic flux and changing its usual path [28].  Otherwise, among the advantages of using ferrite, it can reduce the serious risks for the user's health. Thereby, it is considered magnetic shielding. However, this type of shielding may not be sufficient to protect users and respect the guidance of the ICNIRP standard [29].
An aluminum plate is added to the system to analyze its influence on the coupling. Figure 6c shows the rectangular coil with ferrite and aluminum for gap = 100 mm.
The simulation results of the coupling coefficient and the mutual inductance are summarized in Table 4 and Figure 9.  It can be concluded that the coil with ferrite represents the highest coupling coefficient value among all three cases (case 1: coreless coil, case 2: a coil with ferrite, case 3: a coil with ferrite and aluminum). This result is due to the ferrite, which collects the flux and guides it along another path.
The introduction of aluminum to the rectangular coil with ferrite reduces marginally the coupling coefficient value while remaining relatively high [2]. According to [3,30], it is possible to reduce the variation in circuit parameters while maintaining the shielding effect if the ferrite is positioned between the coil and aluminum plate and the thickness of the aluminum is greater than the skin depth.
There are several types of shielding in the literature [3,31,32]. The working principle of conductive shielding can be explained as follows: the magnetic field generated by the coils induces eddy currents in the aluminum shield. These eddy currents in turn create an opposing magnetic field to attenuate or even cancel out the incident magnetic field [16,31].
Hence, by adding the ferrite and aluminum to the system, the characteristics are also merged. This means that the coupling is increased compared to case 1 and the magnetic field is better shielded compared to case 2.
The limitation and guidance of magnetic flux lines not only lead to a decrease in the leakage magnetic field but also an increase in coupling coefficient and mutual inductance.
In summary, both ferrite and aluminum are necessary to ensure good coupling and better shielding to meet the ICNIRP guidelines (ICNIRP).

Coil Main Parameters
The coupling coefficient, self-inductance, and the mutual inductance of two coils are considered as crucial factors determined by coil parameters that affect the system performance. Therefore, improvement of the coil parameters is a suitable means of enhancing the system coupling. The variation in coil parameters affects the system coupling by varying either the number of turns or the inner radius or other parameters. In this section, the impact of varying these parameters on the system coupling is analyzed. The simulations were performed by modeling a circular coil using Ansys Maxwell software ( Figure 10). Accordingly, the model characteristics can be explained as follows (Table 5): Figure 10. Reference circular coil model. In this analysis, the inner radius of the transmitting coil is kept constant, R TX = 20 mm, and that of the receiving coil R RX is changed from 10 to 40 mm. Figure 11 illustrates an example of two cases: (a) R RX = 10 mm and (b) R RX = 40 mm. The number of turns for both coils is equal to N RX = N TX = 20 mm. The simulation results of the coupling coefficient regarding the receiving coil inner radius variation for gap = 100 mm are illustrated in Table 6 and Figures 12 and 13.
The inner radius of the receiving coil in the first image is equal to 10 mm and the inner radius of the receiving coil in the second image is equal to 40 mm.
The same procedure is applied to the transmitting coil. The inner radius of the receiving coil is constant and we change that of the transmitting coil. The other parameters are kept the same. Figure 14 illustrates an example of two cases: (a) R TX = 10 mm and (b) R TX = 40 mm.
The simulation result of the coupling coefficient regarding the transmitting coil inner radius variation for gap = 100 mm is shown in Table 7 and Figures 15 and 16.       The results indicate that the coil inner radius variation has a significant effect on the system coupling.
Thus, increasing the inner radius of the transmitting or the receiving coil while keeping the other parameters constant enhances the value of the coupling coefficient. The same results are found for the mutual inductance. Therefore, although the two coils present different functions (transmitting and receiving the energy), their attitude towards the self-inductance is the same. Note also that the coil whose inner radius changes its selfinductance also changes and remains constant if no other parameters are modified, i.e., the inductance follows the variation in the inner radius. Figure 17 presents a cross-sectional view of a flat circular coil. Here, D out is the outer diameter, D in is the inner diameter, w is the wire diameter, and p is the spacing between the adjacent wire. From the figure, we can deduce the relation between coil radius R and coil inner radius R in as: If a is a constant, this means that the spacing between adjacent wire p, the wire diameter w, and the coil number of turns N is constant, and any increase in the inner radius R in leads to an increase in the coil radius. Therefore, any augmentation in the coil inner radius or the radius of the coil increases the inductance of the coil. These results are consistent with those found in [33], where the inductance is a monotonic function of the coil radius.
However, because the space taken up by the receiving coil in the vehicle chassis is limited, the coil cannot be completely increased, even if the large inner radius gives good coupling results. This space differs from one vehicle to another; therefore, if the pad design is dedicated to a specific vehicle, the inner radius must be large, depending on the spacing offered. On the other hand, if the pad is designed to be suitable for different types of vehicles, its parameters, including the inner radius, must follow dedicated standards.

The Coil Number of Turns
In this analysis, the number of turns of the transmitting coil is preserved constantly N TX = 10 and we change that of the receiving coil from 5 to 16. The inner radius for both coils is equal to R RX = R TX = 20 mm. The simulation results of the coupling coefficient regarding the receiving coil number of turns for gap = 100 mm are illustrated in Table 8 and Figures 18 and 19.   The same procedure is applied to the transmitting coil. The inner radius of the receiving coil is constant and that of the transmitting coil is changed. The other parameters are kept the same.
The simulation result of the coupling coefficient regarding the transmitting coil number of turns variation for gap = 100 mm is shown in Table 9 and Figures 20 and 21.   The results indicate that the coil number of turns affects the system coupling. Increasing the number of turns of the transmitting and receiving coils while keeping the other parameters constant enhances the coupling coefficient and the mutual inductance values. Furthermore, for the coil whose number of turns increases, its inductance also increases and remains constant if no parameters are modified, i.e., the inductance follows the variation in the number of turns.
Moreover, if the number of turns increases, the outer diameter increases effectively by default. Here, the concept of misalignment is introduced. The misalignment is a major factor that we should take into consideration while designing the coils. According to SAE J2954 (SAE), the lateral misalignment tolerance among the x-axis (vehicle movement) is ±100 mm, so if we consider that the outer diameter of the transmitting coil is named D TXout , the diameter of the receiving coil must then be D RXout : (17) This expression ensures that the misalignment is reduced according to the misalignment tolerance.
Therefore, increasing the number of turns of the coil remains an effective method to improve the inductance of the coil and to improve the coupling coefficient in the wireless power transfer system, taking into account the notion of misalignment.
It can be concluded that the coupling coefficient, mutual inductance, and coil inductance depend on the coil parameters and the relative position of the coils. Effectively, the study results reveal that the optimized design of the coil inner radius and number of turns is beneficial in improving the coupling and mutual inductance abilities.

The Three Coil Shapes' Comparison
In this section, three coil shapes are presented: circular, rectangular, and hexagonal ( Figure 22). The main objective is to compare the coupling coefficient, the mutual inductance, and the coil inductance for the three coil shapes. The comparison is set for air gaps of z = 50 mm up to z = 300 mm with steps of 50 mm and for zero misalignments (d x = d y = 0). For a reliable and reasonable comparison, the three coils' main parameters are designed to be equal, namely the number of turns, the inner diameter, the outer diameter, pitch, and the spacing between the adjacent wire. The dimensions of the coils are given in Table 10. The simulations of the three coils topologies are performed using Ansys Maxwell software and compared regarding the coupling coefficient, mutual inductance, and coil inductance. The simulations are conducted using an eddy current solution with a frequency of 85 kHz and a constant current of the transmitting coil. In addition, the direction of the current in the internal winding is opposite to that of the external winding and the coil size remains the same for all simulations.
The simulation results of the three geometries for the three cases-case 1: circular coil, case 2: rectangular coil, case 3: hexagonal-are presented in Tables 11 and 12.
Here, the parameters used in the following tables are defined as: • k c , k r , and k h are the coupling coefficient of circular, rectangular, and hexagonal coils, respectively; • M c , M r , and M h are the mutual inductance of circular, rectangular, and hexagonal coils, respectively; • L 1c , L 1r , and L 1h are the inductance of circular, rectangular, and hexagonal transmitting coils, respectively; • L 2c , L 2r , and L 2h are the inductance of circular, rectangular, and hexagonal receiving coils, respectively.  For the coreless case, the results show that the rectangular shape represents the greatest coupling coefficient k and mutual inductance M compared to the circular and hexagonal shapes.
The circular coil is the second shape that has a good coupling coefficient, while the coupling coefficient of the hexagonal coil is also close to this value.
The introduction of ferrite and aluminum into the coils enhances the coupling coefficient values for all shapes, as can be concluded from Section 3. Notably, the rectangular shape still presents the best values.
Moreover, the transmitting coil inductance remains unchanged when the vertical distance is varied. Effectively, there is no change in the coil parameters and the current flowing in the transmitting coil is considered constant. When the ferrite is added, the inductance improves and remains constant over all gap variation. Moreover, the introduction of aluminum into the system composed of coil and ferrite marginally reduces the inductance value, which remains notably high. In this case, the rectangular coil presents the best value of L2 compared to the circular and hexagonal, especially after adding the ferrite.
On the other hand, the rectangular and hexagonal receiving coils' inductance in the coreless case decrease with the large vertical distances (300 mm). This influence diminishes by adding ferrite and aluminum. However, the receiving circular coil inductance remains a high value basically in the large vertical distances.
These results give the advantage to the rectangular shape, with high coupling and inductance values, over other shapes for the variation in the gap under the given conditions. Other factors that can alter these results are the static or dynamic type of charging of electric vehicles and the misalignment of coils along the x and y-axis. Therefore, these coils are of different natures and are suitable for specific applications.
Several studies have been carried out to determine the coil inductance expressions, namely the modified Wheeler formula, the expression based on current sheet approximation, and the data-fitted monomial expression [34]. Thereby, the ratio between the current flowing through the coil and the magnetic flux generated represents its self-inductance [35].
Hence, the self-inductance of the coil is expressed as follows [34]: where µ 0 is the permeability of free space, N is the number of turns, d out and d in are the outer and inner diameters of the coil, d avg = d in +d out 2 is the average diameter, ρ = d out −d in d out +d in is the fill factor. The coefficients C i are layout-dependent, as illustrated in Table 13.

The Nine Possibilities of the Three Shapes' Combination
There are different possible shapes for the receiving coil attached to the bottom of vehicles, the same as for the transmitting coils located at charging stations. For this, we propose in this part nine possibilities that can be encountered in wireless charging operations using the three coil shapes: circular, rectangular, and hexagonal. Each shape of the transmitting coil corresponds to the three shapes of the receiving coil. The topologies consist of air-core coils without any material to ensure equitable comparison.
To compare the nine possibilities of shaped coils, their self-and mutual inductances and coupling coefficients were calculated using FEM software.
The modeling of the different coils can be considered the same as in the section above. The 3-D models created in ANSYS Maxwell software for the nine shapes' cases can be described as follows: Case 1: circular transmitter coil and circular receiving coil; Case 2: circular transmitter coil and rectangular receiving coil; Case 3: circular transmitter coil and hexagonal receiving coil; Case 4: rectangular transmitter coil and circular receiving coil; Case 5: rectangular transmitter coil and rectangular receiving coil; Case 6: rectangular transmitter coil and hexagonal receiving coil; Case 7: hexagonal transmitter coil and circular receiving coil; Case 8: hexagonal transmitter coil and rectangular receiving coil; Case 9: hexagonal transmitter coil and hexagonal receiving coil.
The simulation results of the coils for the nine cases are presented in Tables 14 and 15. Here, the parameters used in the following tables are defined as follows: • k cc , k cr , and k ch are the coupling coefficients of the combination of transmitting circular coil and receiving circular, rectangular, and hexagonal coils, respectively; • M cc , M cr , and M ch are the mutual inductances of the combination of transmitting circular coil and receiving circular, rectangular, and hexagonal coils, respectively; • k rc , k rr , and k rh are the coupling coefficients of the combination of transmitting rectangular coil and receiving circular, rectangular, and hexagonal coils, respectively; • M rc , M rr , and M rh are the mutual inductances of the combination of transmitting rectangular coil and receiving circular, rectangular, and hexagonal coils, respectively; • k hc , k hr , and k hh are the coupling coefficients of the combination of transmitting hexagonal coil and receiving circular, rectangular, and hexagonal coils, respectively; • M hc , M hr , and M hh are the mutual inductances of the combination of transmitting hexagonal coil and receiving circular, rectangular, and hexagonal coils, respectively; • L 1cc , L 1cr , and L 1ch are the self-inductance of transmitting coils in the combination of transmitting circular coil and receiving circular, rectangular, and hexagonal coils, respectively; • L 2cc , L 2cr , and L 2ch are the self-inductance of receiving coils in the combination of transmitting circular coil and receiving circular, rectangular, and hexagonal coils, respectively; • L 1rc , L 1rr , and L 1rh are the self-inductance of transmitting coils in the combination of transmitting rectangular coil and receiving circular, rectangular, and hexagonal coils, respectively; • L 2rc , L 2rr , and L 2rh are the self-inductance of receiving coils in the combination of transmitting rectangular coil and receiving circular, rectangular, and hexagonal coils, respectively; • L 1hc , L 1hr , and L 1hh are the self-inductance of transmitting coils in the combination of transmitting hexagonal coil and receiving circular, rectangular, and hexagonal coils, respectively; • L 2hc , L 2hr , and L 2hh are the self-inductance of receiving coils in the combination of transmitting hexagonal coil and receiving circular, rectangular, and hexagonal coils, respectively.
The coupling coefficient and the mutual inductance decrease with the large gap values. First, we can notice that all the results of the coupling coefficient and mutual inductance appear similar.
The combination that gives the best coupling coefficient, mutual inductance, and coil inductance values is the rectangular shape for the combination of the transmitting and receiving coils. Moreover, for different coil combinations, the circular and the rectangular coils present the best results among all the combinations.
The inductance of the transmitting coil of all cases stays almost constant, but the inductance of the receiving coil experiences a change when the gap value is large.  These results open up potential discussions about the performance of these three shapes according to other variations. Likewise, the rectangular coil shape presents greater horizontal tolerance compared to the circular one and the hexagonal coil shape presents the maximum power transfer efficiency at the central position [36]. Therefore, pad design remains an important field to research.

Conclusions
This paper discusses the wireless transfer coil system and the factors that affect ot. The following points were discussed: first, the influence of adding a ferrite core and aluminum sheet to the coil system is discussed. The result shows that the ferrite and aluminum are necessary to guide the magnetic flux, reduce the leakage flux, protect the users around the charging operation, and meet ICNIRP limits. Second, the large air gap reduces the coupling coefficient and mutual inductance values. The main challenge while designing the wireless charger for the electric vehicle is the distance limitation between the transmitting coupler and receiving coupler. This distance depends on the electric vehicle ground clearance and the specifications given by SAE J2954 standard. Third, the coil parameters such as inner radius and number of turns have a significant effect on the system coupling, selfinductance, and mutual inductance. The largest coil inner radius and coil number of turns give the best values of k, L 1 , L 2 , and M. However, the space taken up by the receiving coil in the vehicle chassis is limited, and the coil cannot be totally increased even if the large inner radius gives good results. Finally, the comparison of three shapes, including ferrite and aluminum, gives priority to the rectangular shape, with a small difference compared to the circular and hexagonal shapes. Thus, several coil possibilities can be encountered during the wireless charging operation. In our case, with the use of the three coil shapes, circular, rectangular, and hexagonal, the combination of circular and rectangular coil gives the best results.
Further research could concentrate on the three-coil structure and the effect of the variation of its main parameters and the shape and the size of the ferrite core and aluminum sheet on the system efficiency for bidirectional WPT system application. Moreover, the human safety issue must be taken into consideration during the design process, especially when the power transfer is increased.