Theorizing a Simple Ferrite Cored Coil Using Image Coils in Wireless Power Transfer

The technology of wireless power transfer, which wirelessly transmits power to any device, has been widely studied. Wireless power transfer using magnetic field resonance involves the use of coils and a material called ferrite, which improves the transmission characteristics. Until now, electromagnetic field analysis has been used to design ferrite cored coils, but electromagnetic field analysis requires a long analysis time and a computer with a large amount of memory. In this study, the ferrite shield and coil are represented as a image coil under the condition that the positional relationship between the ferrite shield and the coil does not change. This enabled the derivation of mutual inductance only by numerical analysis, which had been difficult up to now. In addition, experiments have shown that when the coil size does not change, mutual inductance can be derived only by numerical analysis using arbitrary coil parameters, regardless of the number of turns and pitch. The average error in deriving the design value was 5.2%. This facilitated the design of ferrite cored coils at 85 kHz and contributed greatly to the design of the system.


I. INTRODUCTION
Wireless power transfer is a technology for transmitting power wirelessly to all kinds of devices. Since power can be transmitted without cables, it has been actively researched in recent years from the viewpoints of expected convenience, utilization of devices in new environments [1], and durability of cables against abrasion, etc. The challenges imposed on wireless power transfer include lower transmission power due to the wireless connection, lower transmission efficiency, adverse effects such as noise on neighboring devices, and harmful effects on the surrounding human body, and there are regulatory values for magnetic field strength [2], [3], [4], [5]. In wireless power transfer using magnetic field resonance, a widely recognized means of solving these problems is to attach a core called a ferrite in the coil. This allows the magnetic field generated by the coil to be organized and suppresses unnecessary spreading, which is expected to improve inductance and lower magnetic field leakage. However, the The associate editor coordinating the review of this manuscript and approving it for publication was Diego Masotti . design of coils with ferrite core has so far been based on electromagnetic field analysis because it is difficult to capture the phenomenon. Electromagnetic field analysis can represent the power transmission environment very accurately, but it has the problem of requiring a lot of PC memory and analysis time [6]. And it requires expensive analysis software. The actual design parameters for wireless power transfer include not only coil size, number of turns, pitch and transmission distance, but also the input voltage of the circuit, loading conditions, circuit system and many other factors, making it difficult to perform parametric analysis of many parameters using electromagnetic field analysis. Therefore, it is very significant to design various parameters simultaneously at high speed by numerical analysis. It is also very important to express and think theoretically in terms of design.
In references [7], [8], [9], [10], [11], and [12], the high permeability material such as ferrite attached in the coil is represented by a image coil, which exists in a plane symmetrical to the ferrite boundary plane. In reality, however, the ferrite shield is of finite size and thickness, which introduces errors in the design values. In [13], the concept of a virtual mirror image plane is introduced to correct the position of the image coil under realistic conditions. Furthermore, it is shown that the position of the virtual mirror image plane does not change when the coil size does not change. However, there is an issue that the representation of the image coil into the equivalent circuit is different in magnetic materials. In addition, although it states that the position of the virtual mirror image plane does not change when the coil size does not change, the shape change in the number of turns and pitch is not large, and there are questions about the design in the case of even larger changes. Reference [14] similarly replaces coil design with electromagnetic field analysis, but it requires an algorithm.
In this study, in order to design a ferrite cored coil in a more realistic situation, the method of determining the virtual mirror image plane is presented, and it is verified by producing several coils that the position of the virtual mirror image plane does not change. In Chapter II, the method of representing the ferrite shield in the image coil is explained, in Chapter III, the design method of the ferrite cored coil and comparison with experimental values are shown, and in Chapter IV, the conclusion is presented.

II. EXPRESSION OF FERRITE CORES IN IMAGE COILS
The self-inductance and mutual inductance of a coil greatly affect the characteristics of wireless power transfer, so appropriate design is necessary. While the installation of a ferrite core in a coil improves the characteristics of the coil, it also makes it difficult to design the coil by numerical analysis. Here, a ferrite-cored coil is represented by a coreless coil and image coils, and the self-inductance and mutual inductance are derived by numerical analysis, assuming that only an coreless coil exists in the space.

A. DERIVATION OF SELF-INDUCTANCE L AND MUTUAL INDUCTANCE L m OF CORELESS COIL
In deriving the self and mutual inductance of a ferrite cored coil, it is necessary to derive the self and mutual inductance at the coreless coils. The respective derivation methods are described below.

1) DERIVATION OF THE SELF-INDUCTANCE L OF THE COIL
Accurate derivation of self-inductance is very important to characterize the circuit and to set the appropriate resonant capacitor. (1) is the formula for the derivation of the selfinductance L of a square spiral coil [15], where d a is the diameter of the Litz wire conductor and α is the length of the gap between the Litz wires, expressed as α = p − d a . The parameters used to derive the self-inductance are shown in FIGURE 1. Pitch p, the distance between Litz wires, is shown in FIGURE 2.

2) DERIVATION OF THE MUTUAL-INDUCTANCE L m OF THE COIL
During wireless power transfer, coupling occurs between the primary and secondary coils, the proportion of which is expressed by the coupling coefficient k. Together with the coupling coefficient and the self-inductance, the mutual inductance can be obtained, which is a very important parameter in circuit calculations. The equation that can directly express the mutual inductance is called the Neumann equation and can be expressed as (4) [16]. The parameters used in (4) are shown in FIGURE 3.

B. DERIVATION OF SELF AND MUTUAL INDUCTANCE WHEN FERRITE IS ATTACHED
The mirror image method is used as a simple method to capture the spatial distribution of electric and magnetic fields in a 8066 VOLUME 11, 2023    The current flowing in the image coil can be expressed by (5) using the permeability distributed in the space. Although ferrite is a material with high permeability, it is not possible to set the image coil in a position symmetrical to the boundary plane due to size and thickness limitations compared to the ideal situation as shown in FIGURE 4. Therefore, a virtual boundary surface is set as shown in FIGURE 5, and it is referred to as a virtual mirror image surface. The distance from the virtual mirror image plane due to the ferrite itself is expressed as d bottom .
Inductance of the standard coil and position of the virtual mirror image plane obtained by electromagnetic field analysis.
The ferrite cored coil L 1f has a main coil L 1a and a image coil L 11 ′ with the same inductance as the main coil, as shown in FIGURE 5. When current i flows in the main coil L 1a , current βi also flows in the image coil L 11 ′ . The ferrite cored coil can be considered that the magnetic flux from the image coil L 11 ′ also flows into the main coil L 1a . Therefore, the self-inductance L 1f can be expressed by (6) using the mutual inductance M 11 ′ between the main coil L 1a and the image coil L 11 ′ . The procedure for deriving the distance d bottom to the virtual mirror image plane is shown below.
Step1: Derive the self-inductance of the ferrite cored coil L 1f by electromagnetic field analysis.
Step2: Derive the self-inductance of the coreless coil L 1a with the ferrite removed.
Step3: Calculate the mutual inductance between the coreless coils L 1a and find the distance d bottom to the virtual mirror image plane that satisfies (6).
By determining the distance to the virtual mirror image plane, the number of turns and pitch can be freely designed, as long as the coil size does not change [12]. The fact that design is possible using the same virtual mirror image surface is explained in Chapter 3 through actual measurements.

2) DERIVATION OF THE MUTUAL INDUCTANCE L m OF THE FERRITE CORED COIL
Consider the power transmission condition where a ferrite is attached in one of the coils. In this case, the ferrite can be considered to be attached at a distance from the coreless coil, so a virtual mirror image plane is set as shown in FIGURE 6, and the distance d top to the virtual mirror image plane by the ferrite relative to the other coil is set. The method for obtaining the distance d top is the same as in B.1). The positional relationship between the image coil L 1 ′ of the primary coil and the image coil L 2 ′′ of the secondary coil is summarized as in FIGURE 7. From FIGURE 5 and FIGURE 6 and the relationship in (6), the position of each image coil on the primary and secondary sides can be determined. VOLUME 11, 2023 Now that the positions of all the image coils have been determined, the derivation of the mutual inductance L m when a ferrite is attached on one side is explained. Equations (7) and (8) are circuit equations for the circuit shown in FIGURE 8. Equation (7) is represented by the selfinductance L 1a of the coreless coil of the primary coil, mutual inductance M 11 ′ of the primary coil with the image coil by its own ferrite, the mutual inductance M 12 considering the magnetic flux flowing into the primary coil L 1a from the secondary coil L 2a , the mutual inductance M 12 ′′ considering the magnetic flux flowing into the primary coil L 1a For M 12 and M 21 , the reciprocity theorem holds because they are coreless to coreless, but for (9) and (10), the reciprocity theorem for mutual inductance does not hold because image coils are used. However, since no difference in mutual inductance appears in actual wireless power transfer, the mutual inductance of the ferrite cored coil is expressed by (11). In this section, the power transmission environment in which ferrites are attached to both the primary and secondary side is considered, as in FIGURE 9. As explained in B.2), it is necessary to consider not only the image coils the image coils of the ferrites on the opposite sides. Furthermore, when ferrites are sandwiched between each other, image coils are continuously generated against the image coils. Equation (12) and (13) represent the position of the image coil [8]. Z ′ i,k represents the coordinates of successive image coils on its own ferrite side and Z ′′ i,k represents the coordinates of successive image coils on the side opposite its ferrite. i represents the primary or secondary side (i = 1 or 2) and k represents the level of the image coil. FIGURE 9 shows four types of image coils with up to 1st level, both primary and secondary. Although there are many successive level of image coils, it is sufficient to derive the mutual inductance up to 5th level [8]. Equation (14) shows the current i i,k flowing in each of the primary and secondary in the k th level of image coils.
Next, the representation on the equivalent circuit when multiple image coils are present is shown. Again, when considering the equivalent circuit of FIGURE 8, the circuit equations can be expressed as (15) and (16). For example, M 12 ′ k represents the mutual inductance between the primary coreless coil L 1a and the kth level of image coil on the bottom side of the secondary. M 21 ′′ k represents the mutual inductance between the secondary side coreless coil L 2a and the kth level of image coil on the top side of the primary.
From (15) and (16), the mutual inductance from the primary side and the secondary side are shown in (17) and (18), respectively. From this, the mutual inductance of the entire circuit can be expressed by (19). The positions of the image coils are obtained in (12) and (13), and can be easily derived by numerical analysis in (4), since they are replaced with each other by coreless coils. In other words, if the distance d 1_bottom , d 1_top , d 2_bottom , d 2_top to the virtual mirror plane can be obtained by electromagnetic field analysis once with the standard coil when the coil size, distance between the ferrite and the coil, and the ferrite size and transmission distance remain unchanged, it can be seen that the self-inductance and mutual inductance of a ferrite cored coil can be derived using only an coreless coil.

III. DESIGN PROCEDURES AND EXPERIMENTAL VALIDATION
In this chapter, the design method for coils with ferrites attached on both the primary and secondary sides is described according to the procedure, and the experimental and   theoretical values are compared. The goal is to optimize the coils, but since optimization can be done freely by showing how to derive self-inductance and mutual inductance, the goal of this paper is to achieve agreement between experimental and theoretical values.

A. SETUP OF POWER TRANSMISSION ENVIRONMENT
In order to represent a ferrite cored coil as a image coil and to derive it only by numerical analysis, it is necessary to set up a standard coil and determine the size and thickness of the ferrite, the distance between the ferrite and the coil, and the transmission distance. Assuming power transmission frequency at 85 kHz, a Litz wire with 500 strands and a strand diameter of 0.1 mm was used. The transmission distance was set at 103 mm. Table 1 summarizes the standard coil. FIGURE 10 and FIGURE 11 shows the standard coil size and power transmission environment.

B. DETARMINE THE VIRTUAL MIRROR IMAGE PLANE OF THE STANDARD COIL
In this section, the virtual mirror image plane of the standard coil is obtained from (9) and (10), and the coordinates of the image coil up to the 5th level are obtained from (12) and (13).
The initial permeability of PC95 used is 3300, which is much larger than 1. Therefore, since the current decrease rate β of the image coil is almost 1, the effect of β is neglected here. Table 2 shows the inductance of the standard coil obtained by electromagnetic field analysis and the position of the virtual mirror plane of the image coil. Since the transmission distance this time is 103 mm, the coordinates of VOLUME 11, 2023

C. DERIVATION OF SELF AND MUTUALINDUCTANCE OF VARIOUS COILS
In this section, several coils with the same size as the standard coil and with different numbers of turns and pitches are produced, and the inductance of the theoretically obtained by using the image coil obtained in section III-B is compared with the experimental value. Table 3 and Table 4 summarizes the coils produced this time. The coils produced are shown in FIGURE 13 and FIGURE 14.

1) DERIVATION OF THE SELF-INDUCTANCE L OF THE FERRITE CORED COIL
Since the produced coil has the same distance relationship to the virtual mirror image plane as the standard coil for both the primary and secondary sides, the self-inductance can be obtained both with and without the ferrite in (1), (4) and (6) Calculated and experimented values were compared with FIGURE 15and FIGURE 16 along the coil numbers listed in Table 3 and Table 4. The calculated and experimental values agreed very well.

2) DERIVATION OF THE MUTUAL INDUCTANCE L m OF THE FERRITE CORED COIL
Since all the produced coils have the same distance relationship to the virtual mirror image plane as the standard coil, the   mutual inductance of the ferrite cored coils can be obtained from (4) and (19) as in C.1). The measurement scene is shown in FIGURE 17. FIGURE 18 shows the mutual inductance of the ferrite cored coils of primary coils (number 4 to 11) for the coil with secondary coil number 1. Due to the small self-inductance, the average error rate was 7.0%. FIGURE 19 shows the mutual inductance of the ferrite cored coils of primary coils (number 4 to 11) for the coil with secondary coil number 2. The average error rate was kept at 5.2%. FIGURE 20 shows the mutual inductance of the ferrite cored coils of primary coils (number 4 to 11) for the coil with secondary coil number 3. The large self-inductance allowed us to reduce the average error rate to 3.5%. It can be seen that the calculated and experimental values are in good agreement for any combination of coils.  and 5 are used and the mutual inductance results shown in (18). As mentioned in section III-B, it can be seen that the mutual inductance converges to the mutual inductance obtained by the experiment as the number of steps increases, indicating that the calculation up to the 5th level is sufficient. Consequently, by obtaining the virtual mirror image plane of the standard coil, the self and mutual inductances could be accurately obtained only by numerical analysis when the coil size, ferrite size, coil and ferrite arrangement, and transmission distance remain the same.

IV. CONCLUSION
In this study, the ferrite-cored coil is expressed as only coreless coil by the theoretical equation derived from the circuit equation by the image coil and the equivalent circuit.
This method enables us to derive the self-inductance and mutual inductance with high accuracy only by numerical analysis. The proposed theory was confirmed by experiments to be in good agreement. The accuracy of the mutual inductance derivation was 5.2% on average for the data presented in this study. However, it is difficult to describe magnetic materials in a simple way, and the challenge is that conditions such as the ferrite being placed far away from the air in which the coil exists, or the ferrite being small, or the ferrite being thin, lead to significant deviations from the ideal situation as in FIGURE 4, and may not be well described by numerical analysis alone. Also, if the position is misaligned, the secondary ferrite will no longer exist directly above the primary coil, and the position of the virtual mirror image plane may be misaligned or distorted. As described above, the proposed method for deriving the virtual mirror image plane and the self and mutual inductances is not yet highly versatile, but it can be used to obtain a higher degree of freedom in coil design through numerical analysis alone, since the number of turns and pitch can be changed within a fixed coil size.
Since the analysis time by electromagnetic field analysis is very long in the case of ferrite coils, it is very important to derive a low cost by numerical analysis in order to consider various design parameters of the circuit. This study will generate new research themes, and it is expected that research on wireless power transfer will become more and more active.