Selection of Ferrite Depending on Permeability and Weight to Enhance Power Transfer Efficiency in Low-Power Wireless Power Transfer Systems

Abstract

With advancements in the field of electrical engineering, various low-power portable electronic devices have been commercialized. To eliminate and unify different types of cables, inductive wireless power transfer (WPT) technology, which uses magnetic fields to transfer energy, is being applied in numerous applications. Low-power devices typically have small coils and loads, leading to low power transfer efficiency even over short distances. Magnetic materials such as ferrites are used to improve power transfer efficiency (PTE). It is well known that high-permeability ferrites with low magnetic reluctance are ideal for achieving strong magnetic coupling. However, continuous increases in permeability raise the cost and weight of the ferrite, making it necessary to select ferrites with appropriate permeability from a mass-production perspective. This paper models and analyzes the changes in mutual inductance and power transfer efficiency with varying ferrite permeabilities using magnetic circuits, providing guidelines for the selection of suitable ferrites considering efficiency improvements. The proposed method is validated through 3D electromagnetic simulations and experiments, showing a power transfer efficiency difference of up to 0.6% between the experimental and calculated results.

1. Introduction

Recently, various wearable devices have been developed to enhance user convenience, becoming increasingly miniaturized. Due to diverse applications, users must charge each device with different cables, leading to a growing interest in wireless power transfer (WPT) technology to maximize user convenience. Consequently, efforts to apply WPT technology to the increasing variety of devices have intensified, and the market for the wireless charging of small, low-power devices is expected to expand further [1,2,3].

WPT systems utilize magnetic coupling between antennas to transfer power from a transmitting coil (TX) to a receiving coil (RX). This technology has been widely applied in inductive power transfer (IPT) systems due to its high efficiency and suitability for miniaturization. Inductive WPT technology can be applied to a variety of applications, ranging from low-power devices, such as wearable devices and implantable medical devices, to medium-power devices, such as automated guided vehicles (AGVs), robots, and drones, and even high-power applications like electric vehicles [4,5,6].

Among these, WPT technology for low-power devices has been commercialized in many cases due to its short power transfer distance, making it easier to comply with magnetic field regulations and electromagnetic compatibility (EMC)/electromagnetic interference (EMI) requirements [7,8]. Generally, inductive WPT technology achieves higher power transfer efficiency (PTE) as the physical distance between the TX and RX coils decreases, increasing the magnetic coupling coefficient. However, in low-power applications, the PTE can be low despite the short transmission distance because the current consumption is low, leading to a very high load impedance [9]. To address this issue, various wearable devices use ferrites to maximize PTE, even with short charging distances [10].

Ferrites, as soft magnetic materials, exhibit a property where their polarity changes in response to external magnetic fields. With higher permeability compared to air, ferrites can guide magnetic fields, increasing the coupling coefficient between coils and preventing magnetic field leakage. In other words, using ferrites with higher permeability enhances the performance of WPT systems [11]. However, it is not feasible to increase the permeability of ferrites indefinitely. As the permeability increases, the density of the magnetic material powder also increases, leading to the higher costs and weight of the ferrite [12]. In product design, it is crucial to select ferrites that maintain a high PTE while also being cost-effective and lightweight, as various design specifications exist beyond just PTE. Although the cost and weight may not be significant issues in single-product manufacturing, they become critical in mass production, where large quantities of ferrites are used. Therefore, selecting appropriate ferrites is particularly important for cost reduction and weight minimization.

Various ferrite products with different properties are available on the market, and their usable frequency range and permeability vary based on their characteristics [13,14]. MnZn ferrites have a wide range of permeabilities, from hundreds to thousands, and exhibit low magnetic losses at low frequencies, making them suitable for low-frequency WPT technology [13]. NiZn ferrites, on the other hand, have lower permeabilities, generally below 1000, and low losses at high frequencies, making them ideal for high-frequency WPT systems [14]. Additionally, recently, nanocrystalline ferrites with permeabilities exceeding 10,000 have gained attention [15,16]. Among the various ferrites, MnZn ferrites are primarily used for low-power wearable devices due to their suitability for low-frequency applications.

In WPT technology, the placement of ferrites and differences in permeability can cause various characteristic changes, which have been explored in existing studies [13,14,15]. For instance, [17] analyzed the electrical parameters in wireless charging systems for electric vehicles, focusing on how the placement of ferrite tiles affects these parameters. The spacing between ferrite tiles alters the air gap, thereby changing the effective permeability, which, in turn, can affect the coil inductance and mutual inductance. Additionally, studies have compared and analyzed the PTE based on the laminated structure and permeability of ferrite sheets [18], and research has been conducted to increase PTE using ferrite rod structures [19].

In [20], the performance of WPT systems was evaluated based on different ferrite structures and permeabilities, aiming to maximize efficiency and achieve lightweight designs. The study designed systems considering power losses and included analyses using nanocrystalline ferrites in addition to MnZn ferrites. Furthermore, research has been conducted on selecting ferrites with permeability that satisfies over 90% PTE while minimizing EMF. It was found that achieving minimal EMF does not always occur under conditions of maximum permeability and minimum loss tangent [21].

As evident from these studies, it is generally well-known in IPT systems that increasing the permeability of ferrites increases mutual inductance and the magnetic coupling coefficient, thereby enhancing PTE. However, PTE does not increase linearly or exponentially with increasing permeability; instead, it saturates beyond a certain permeability threshold. Existing studies have examined this phenomenon and prioritized certain parameters or suggested structures and permeabilities to achieve maximum performance. However, these approaches are inductive (not deductive) and have limitations, as engineers cannot predict PTE performance based on permeability in the design phase without simulating every possible case. Additionally, while higher permeability ferrites improve PTE, they also have higher magnetic material density, leading to cost and weight constraints.

This paper proposes a method to select ferrites based on PTE performance analysis relative to permeability during the design phase. This approach not only reduces design costs and time but also offers significant advantages in cost reduction and weight minimization during production.

In this paper, the coils of a low-power WPT system with a short distance between the transmitting and receiving coils and a small load are modeled using magnetic circuits. The changes in electrical parameters with varying permeability are determined using the modeling results. The calculated parameters are used to compute the PTE, and a design guide for selecting permeability is proposed based on this analysis.

The paper is structured as follows: Section 1 introduces the background and necessity of this research. Section 2 analyzes the low-power IPT system and its characteristics with and without ferrites. Section 3 provides the magnetic circuit modeling of the WPT system for the proposed ferrite selection method and a design guide considering efficiency improvement. Section 4 and Section 5 verify the proposed method through simulations and experiments, respectively. Section 6 discusses cost, weight, and other considerations. Finally, Section 7 summarizes the conclusions of this paper.

2. Analysis of Low-Power WPT System

In this section, the circuit of the WPT system and the PTE of a low-power WPT system are analyzed using mathematical equations. The effects of using ferrites to mitigate PTE degradation in low-power applications are compared, and the relationship between mutual inductance and efficiency is explained.

2.1. Power Transfer Efficiency of Low-Power WPT System

An IPT system uses a compensation network along with TX and RX coils to maximize PTE at the resonant frequency. When the inductance of the coils is paired with capacitors in the compensation network, resonance occurs. Operating the system at this resonant frequency, where the input impedance is minimized, allows voltage to be induced in the RX coil by Faraday’s law of electromagnetic induction, thereby transferring power. Figure 1 shows a series–series topology WPT system and the definitions of the electrical parameters used are listed in

Table 1. Electrical parameters of the circuit.

Symbol	Parameters	Symbol	Parameters
$I_1, I_2$	Current flowing in TX and RX	${\mathrm{L}}_1, {\mathrm{L}}_2$	Inductances of TX and RX coils
$R_1, R_2$	Resistance of TX and RX coils	${\mathrm{C}}_1, {\mathrm{C}}_2$	Matching capacitances of TX RX coils
$R_L$	Load resistance	M	Mutual inductance of TX and RX coils

.

Typically, wireless charging systems used in small, low-power devices have low PTE despite high coupling due to the miniaturization of the antennas and power output limitations. To specifically analyze this, the power transfer efficiency of the circuit in Figure 1 can be expressed as Equation (1) [1].

$$\eta ={\displaystyle \frac{{\left|I_2\right|}^2{R}_L}{{\left|I_1\right|}^2{R}_1+{\left|I_2\right|}^2{R}_2+{\left|I_2\right|}^2{R}_L}}\times 100={\displaystyle \frac{{\omega }^2{M}^2}{R_1{R}_L+{\omega }^2{M}^2}}\times 100, where R_2\ll R_L$$

(1)

Here, η represents the PTE, which is maximized when the compensation capacitors are set such that the loop impedance of the TX and RX coils is minimized at the operating frequency of the power source. This is PTE under resonant conditions, and ω is the angular frequency in Equation (1). In low-power applications, the current is very small compared to the charging voltage, resulting in a very high load impedance. Therefore, the coil resistance can be considered negligible compared to the secondary load resistance, allowing PTE to be simplified, as shown in Equation (1). From Equation (1), it can be observed that increasing the $R_L$ decreases efficiency, which explains why PTE is low in low-power applications.

To understand the influence of the coupling coefficient (k) and mutual inductance on PTE, Equation (1) can be expressed in terms of the quality factor (Q-factor) and k. The Q-factor is an indicator of the energy stored versus the energy lost in the coil and is defined as shown in Equation (2). A lower Q-factor physically implies reduced PTE. The coupling coefficient, which represents the degree of coupling between the two coils, ranges from 0 to 1 and is defined as shown in Equation (3). The coupling coefficient increases as the physical distance between the TX and RX coils decreases, leading to an increase in the flux linkage [1].

$$Q_1={\displaystyle \frac{\omega L_1}{R_1}},Q_2={\displaystyle \frac{\omega L_2}{R_L}},where R_2\ll R_L$$

(2)

$$k={\displaystyle \frac{M}{\sqrt{L_1{L}_2 }}}={\displaystyle \frac{M}{L_1}}, where L_1=L_2$$

(3)

Using Equations (2) and (3), PTE can be re-expressed as shown in Equation (4). It is evident that higher coupling coefficients or higher Q-factors result in increased efficiency.

$${\eta }^{\prime }={\displaystyle \frac{1}{1+\frac{1}{k^2{Q}_1{Q}_2 }}}\times 100=\eta$$

(4)

As the load resistance $R_L$ increases, $Q_2$ decreases. In low-power WPT systems, the small coil size results in low inductance, which causes the Q-factor to decrease. Consequently, the efficiency of low-power WPT systems is reduced. To address this issue, the coupling coefficient needs to be increased. This can be achieved by increasing the mutual inductance, typically through the use of ferrites [1,10].

2.2. Low-Power WPT System with or without Ferrite

As analyzed in Section 2.1, to increase the efficiency of low-power WPT systems, the mutual inductance must be increased to enhance the coupling coefficient. The mutual inductance grows with the flux linkage between the transmitting and receiving coils, thereby increasing the induced voltage on the secondary side.

Ferrite, a material with lower magnetic reluctance than air, guides the magnetic field through the magnetic material, reducing the magnetic flux leakage into the air [6,7]. This method increases mutual inductance and the coupling coefficient by minimizing the leakage of the magnetic field.

Table 2. Comparison of efficiency and current in WPT systems with and without ferrites.

@110 kHz	W/o Ferrite	W/Ferrite
k	0.598	0.733
$\eta \left[\%\right]$	39.54	78.87
$I_{tx} \left[\mathrm{A}\right]$	9.866	4.140
$I_{rx} \left[\mathrm{mA}\right]$	63

shows a comparison of the efficiency and current of the WPT system designed according to the simulation setup in Chapter 4. The operating frequency is 110 kHz, and the received power is 160 mW for both cases. When comparing a system with a ferrite that has a real part permeability of 100 to a system without ferrite, the efficiency shows a difference of approximately 39.33%, and the TX coil current decreases by about 58%. This change in the coupling coefficient is also observed in Table 2. Therefore, despite potential issues with volume and weight increase due to the use of ferrite, achieving an over 30% efficiency improvement makes ferrite usage essential in low-power WPT systems.

3. Selection of Ferrite Depending on Permeability

In this chapter, we propose a method for selecting ferrites in low-power WPT systems by considering PTE and permeability based on the analysis conducted in Section 2. To model the changes in mutual inductance with varying permeability, we applied magnetic-circuit theory to model the WPT system. In addition to improving efficiency, we provide guidelines for selecting magnetic materials by considering the saturation phenomenon of ferrites.

3.1. Magnetic Circuit Modeling

To model the mutual inductance changes with varying permeability, we first need to understand the relationship between these two parameters. Mutual inductance refers to the change in magnetic flux in one coil induced by the change in current in another coil. This relationship can be expressed mathematically, as shown in Equation (5) [22].

$$M={\displaystyle \frac{N{\Phi }_{12}}{I_1}}={\displaystyle \frac{N\oint \mathit{B}d\mathit{s}}{I_1}}={\displaystyle \frac{\mu N\oint Hd\mathit{s}}{I_1}}$$

(5)

Here, $M$ represents mutual inductance, $N$ is the number of turns in the coil, $I_1$ is the current flowing through coil 1, ${\Phi }_{12}$ is the magnetic flux linking coil 1 to coil 2, and $\mu$ denotes permeability. It can be observed that mutual inductance is proportional to permeability. Therefore, by using magnetic circuit theory to derive the changes in equivalent permeability, the corresponding changes in mutual inductance can be determined. To calculate the equivalent permeability, we first modeled the TX and RX coils, including ferrite, using magnetic circuit theory [10,22].

Figure 2 illustrates the magnetic circuit representation using the side dimensions of the IPT coil. The magnetic circuit can be modeled using magnetic reluctance and magnetic flux, where magnetic reluctance is inversely proportional to permeability and the cross-sectional area of the path and directly proportional to the length of the path.

Table 3. Parameters of the magnetic circuit.

Symbol	Parameters	Symbol	Parameters
${\Re }_a$	Reluctance of the air gap	${\Re }_t$	Reluctance of the ferrite
$l_a$	Air gap	$l_t$	Radius of a coil
${\varphi }_1$	Magnetic flux of loop 1	${\varphi }_2$	Magnetic flux of loop 2
$V_m$	Magnetomotive force

lists the parameters used in the magnetic circuit.

The magnetic field distribution between WPT coils can be modeled using magnetic flux, as shown in Figure 2. This distribution can be expressed in terms of the ferrite radius, air gap length, and their respective permeabilities. The magnetic reluctances in Figure 2, representing the ferrite and air, are given by Equations (6) and (7), respectively. Table 3 lists the parameters used in the magnetic circuit representation.

$${\Re }_a={\displaystyle \frac{l_a}{{\mu }_{a}S}}$$

(6)

$${\Re }_t={\displaystyle \frac{l_t}{{\mu }_{t}S}}$$

(7)

Here, ${\mu }_a$ and ${\mu }_t$ represent the permeabilities of air and ferrite, respectively. In Equation (6), the area $S$ through which the flux passes is different for air and ferrite but is assumed to be the same for simplicity [22]. Using Equation (6), the equivalent magnetic reluctance of the magnetic circuit is derived, as shown in Equation (8).

$${\Re }_{eq}={\displaystyle \frac{V_m}{{\varphi }_1+{\varphi }_2}}={\displaystyle \frac{l_{eq}}{{\mu }_{eq}S}}={\displaystyle \frac{3}{2}}{\Re }_a+{\Re }_t$$

(8)

$l_{eq}$ and ${\mu }_{eq}$ represent the equivalent length and permeability derived using the definition of magnetic reluctance. Using the modeled circuit, we can now determine the mutual inductance variations with respect to changes in permeability. From Equation (5), it is evident that mutual inductance changes proportionally with equivalent permeability as the ferrite permeability changes. Thus, by calculating the equivalent permeability based on the changes in ferrite permeability and knowing the ratio between them, we can determine the corresponding changes in mutual inductance.

$$M_p=\alpha M_c, where \alpha ={\displaystyle \frac{{\mu }_{eqp}}{{\mu }_{eqc}}}$$

(9)

First, when there is a list of available ferrites, arrange them in order to increase permeability. Define the mutual inductance when using the ferrite with the smallest permeability as $M_c$, and the mutual inductance when using ferrites with other permeabilities as $M_p$. Let $\alpha$ be the ratio of the two mutual inductances, which can be expressed as the ratio of the equivalent permeabilities in the two cases.

${\mu }_{eqc}$ is the equivalent permeability when the permeability of the first ferrite in the ordered list ${\mu }_c$ is substituted into Equations (7) and (8), and ${\mu }_{eqp}$ is the equivalent permeability when the permeability of the other ferrites ${\mu }_p$, except for the first one, is substituted. Equations (10)–(12) show the process for calculating $\alpha$.

$${\Re }_{eqc}={\displaystyle \frac{l_{eq}}{{\mu }_{eqc}S}}={\displaystyle \frac{3}{2}}{\Re }_a+{\Re }_c, where {\Re }_c={\displaystyle \frac{l_t}{{\mu }_{c}S}}$$

(10)

$${\Re }_{eqp}={\displaystyle \frac{l_{eq}}{{\mu }_{eqp}S}}={\displaystyle \frac{3}{2}}{\Re }_a+{\Re }_p, Where {\Re }_p={\displaystyle \frac{l_t}{{\mu }_{p}S}}$$

(11)

$$\alpha ={\displaystyle \frac{{\mu }_{eqp}}{{\mu }_{eqc}}}={\displaystyle \frac{{\mu }_p\left(1.5l_a{\mu }_c+l_t{\mu }_a\right)}{{\mu }_c\left(1.5l_a{\mu }_p+l_t{\mu }_a\right)}}$$

(12)

Here, ${\mu }_c$ is the permeability of the initial ferrite model, and ${\mu }_p$ is the permeability of the ferrite being compared, where ${\mu }_p$ is greater than or equal to ${\mu }_c$. In Equations (10)–(12), all parameters can be obtained using the dimensions and material properties of the coil and ferrite, allowing for the calculation of $\alpha$. Using Equations (9) and (12), $\alpha$ can be determined, and the mutual inductance with varying permeability can be calculated, which can then be used to find the coil inductance. This is shown in Equation (13).

$$L_p=\left(L_c-M_c\right)+M_p$$

(13)

$L_c$ represents the coil inductance when using the ferrite model with permeability $M_c$, and $L_p$ represents the coil inductance when using the ferrite with permeability $M_p$.

In Section 3.1, the equivalent permeability variations due to changes in ferrite permeability were determined using the magnetic circuit modeling of WPT coils. This ratio was then used to calculate the coil inductance and mutual inductance. In Section 3.2, the obtained electrical parameters were used to estimate the efficiency of the WPT system and explain how to select ferrites based on efficiency changes.

3.2. PTE Depending on Ferrite Permeability

Using Equations (9)–(13), the electrical parameters that change with varying permeability can be obtained, allowing for the calculation of PTE. In Equation (1), which represents PTE, all parameters remain constant except for the mutual inductance when the permeability of the ferrite changes. Therefore, by using Equations (1) and (9), the efficiency variation due to changes in ferrite permeability can be calculated. This is expressed in Equation (14). Additionally, this equation does not consider core loss.

$$\begin{gathered} {\eta }_p={\displaystyle \frac{{\omega }^2{M}_p^2}{R_1{R}_L+{\omega }^2{M}_p^2}}\times 100={\displaystyle \frac{{\omega }^2{\left(\alpha M_c\right)}^2}{R_1{R}_L+{\omega }^2{\left(\alpha M_c\right)}^2}}\times 100={\displaystyle \frac{A}{B}}\times 100 \\ where A={\omega }^2{\left({\displaystyle \frac{{\mu }_p\left(1.5l_a{\mu }_c+l_t{\mu }_a\right)}{{\mu }_c\left(1.5l_a{\mu }_p+l_t{\mu }_a\right)}}{M}_c\right)}^2, \\ B=R_1{R}_L+{\omega }^2{\left({\displaystyle \frac{{\mu }_p\left(1.5l_a{\mu }_c+l_t{\mu }_a\right)}{{\mu }_c\left(1.5l_a{\mu }_p+l_t{\mu }_a\right)}}{M}_c\right)}^2 \end{gathered}$$

(14)

Using Equation (14), $\alpha$ changes with permeability variations, and as permeability increases, ${\eta }_p$ also increases. Ideally, the efficiency increase saturates when the derivative of ${\eta }_p$ approaches zero. However, mathematically, it cannot be exactly zero. Therefore, a threshold value is selected based on the derivative result to determine the permeability at which the efficiency increase saturates. The derivative of ${\eta }_p$ is given by Equation (15).

$${\displaystyle \frac{d{\eta }_p}{d{\mu }_p}}=\left({\displaystyle \frac{100}{B^2}}\right)\left(B{\displaystyle \frac{dA}{d{\mu }_p}}-A{\displaystyle \frac{dB}{d{\mu }_p}}\right)$$

(15)

Using Equation (15), the efficiency variation with respect to ${\mu }_p$ can be determined, and the user can set a threshold value as a design parameter to select the appropriate permeability. Therefore, engineers can consider the weight, cost, and efficiency increase to select the most effective ferrite for their design.

However, selecting ferrites based solely on PTE may lead to unforeseen issues. It is crucial to consider ferrite saturation to ensure the system operates reliably. When the external magnetic field exceeds a certain threshold, most magnetic dipoles align, leading to saturation. In this state, the coil cannot store magnetic energy, resulting in system malfunction. As permeability increases, the slope of the B-H curve becomes steeper, reducing the saturation’s current range [22]. Therefore, while higher permeability can increase efficiency by reducing the current in the coil, it also raises the likelihood of saturation.

Saturation flux density and current vary with the magnetic material used, so it is essential to refer to the datasheet after selecting a ferrite to determine if saturation occurs. The saturation current can be calculated using Equation (16) [22].

$$I_{sat}={\displaystyle \frac{B_{sat}{S}_{f}t}{\mu N}}$$

(16)

In Equation (16), $I_{sat}$ is the saturation current, $B_{sat}$ is the saturation flux density, $S_f$ is the cross-sectional area of the ferrite, $t$ is the thickness of the ferrite, and $N$ is the number of turns in the coil. Since each magnetic material has a different saturation flux density and saturation current, after selecting a ferrite to improve efficiency, it is necessary to consult the datasheet to consider saturation in the post-processing stage. Therefore, to apply this method, it is necessary to verify whether the ferrite is saturated by considering the excitation current. The guidelines for selecting permeability, considering both the efficiency increase and ferrite saturation, are explained in Section 3.3.

3.3. Guidelines for Selection of Ferrite

In this section, we explain the process of selecting ferrites considering the increase in PTE and the magnetic saturation discussed in Section 3.2. This guide can be applied when a database of ferrite material properties with various permeabilities is established.

Figure 3 illustrates the flow chart for selecting ferrites. Once the target application is determined, the design constraints for the coil are established. The material properties of the available ferrite models can be represented using the example index chart shown in Figure 4, which arranges the ferrites in order of increasing permeability.

Based on the design constraints and the properties of the ferrite at the initial index, the specific dimensions of the TX and RX coils are determined. Using these dimensions, electrical parameters such as coil inductance, mutual inductance, and coil resistance can be obtained through a 3D EM simulation. With the acquired electrical parameters and using Equations (14) and (15), the PTE and its derivative can be calculated. By incrementally increasing the index, the PTE and its derivative can be determined for each increment in ferrite permeability.

After completing calculations up to the final index, the ferrite can be selected by identifying the index where the derivative of the efficiency is less than the product of the initial derivative result and the design parameter $\mathsf{\gamma }$. Here, $\mathsf{\gamma }$ is a user-defined design parameter with a value between 0 and 1. A smaller $\mathsf{\gamma }$ maximizes efficiency but increases the range of ferrite permeability, which can be adjusted by the engineer based on design priorities.

Although γ was set as a design parameter, it can also be selected by considering design factors such as weight and cost. Generally, as the permeability of the ferrite increases, the density of the magnetic material increases, resulting in an increase in weight [12]. For example, by considering the changes in weight with respect to permeability, the crossover point between the weight change and efficiency change can be chosen as γ, thereby determining the appropriate permeability. This can vary depending on the weight assigned to each design factor. In this paper, an example of selecting permeability considering weight is provided in Section 4.

Finally, it is essential to verify that the selected ferrite does not saturate in the system design. Using Equation (16) and the ferrite datasheet information, the saturation of ferrite can be assessed based on the current flowing through the coil. If the ferrite does not saturate, the selected ferrite can be finalized. However, if saturation occurs, the index can be decreased by one, and the saturation should be reassessed before making a final decision.

The guidelines proposed in this paper enable the prediction of PTE increases due to changes in permeability based on magnetic circuit modeling and mathematical formulation using only the 3D EM simulation results of the initial index. Typically, efficiency is calculated by changing the simulation conditions and performing 3D EM simulations case by case for each material property. Therefore, this method can reduce design time and mitigate costs and weight increases due to over-specified ferrites. Further details are discussed in Section 6.

4. Verification through Simulation

In this section, we validate the proposed process through simulations. The coil design was targeted for low-power devices, and the efficiency variation results due to changes in permeability were compared to verify the mathematical formulation. Using the permeability conditions of the initial index, the coil was modeled using ANSYS Maxwell3D 2023 (Canonsburg, PA, USA), which is a finite element method-based 3D EM simulation tool, to extract the electrical parameters. Subsequently, the simulated efficiency and the efficiency results derived from the proposed magnetic circuit modeling were compared.

4.1. Simulation Setup

The parameters and coil dimensions for the simulation were set, targeting low-power wearable devices such as smartwatches and wireless earbuds. The operating frequency of the WPT system was selected as 110 kHz, referring to the Qi standard range. The load resistance and received power were determined based on a scenario where a battery with a capacity of 200 mAh was charged at 4 V and 40 mA [21].

Figure 4a,b shows the 3D models of the coils from a perspective view and a side view, respectively.

Table 4. Design parameters for the simulation setup.

Parameters	$d_{wire} \left[\mathrm{mm}\right]$	$d_o \left[\mathrm{mm}\right]$	$d_i \left[\mathrm{mm}\right]$	$h_g \left[\mathrm{mm}\right]$	$h_f \left[\mathrm{mm}\right]$	# of turns
Value	1.3 (AWG 16)	40	27	2.5	5	5

lists the coil design and design parameters, while

Table 5. Electrical parameters obtained from simulation for ${\mu }_c$ = 100 (Index = 1).

Parameters	Value	Parameters	Value
$L_1\left(=L_c\right)$	2.425 $\mathsf{\mu }\mathrm{H}$	$R_1$	5 mΩ
$L_2\left(=L_c\right)$		$R_2$
$M\left(=M_c\right)$	1.777 $\mathsf{\mu }\mathrm{H}$	$k$	0.733

presents the electrical parameters obtained from the simulation under the initial index conditions. These results correspond to an initial permeability set to 100. The structures of the TX and RX coils, as well as the ferrites, are symmetrical. Thus, the inductance and resistance of the coils are the same for both the TX and RX coils.

Table 6. Circuit simulation setup for verifying PTE.

Parameters	$f\left[\mathrm{kHz}\right]$	$C_1\left[\mathrm{F}\right]$	$C_2\left[\mathrm{F}\right]$	$R_L\left[\mathsf{\Omega }\right]$
Value	110	${\displaystyle \frac{1}{{\left(2\pi f\right)}^2{L}_1 }}$	${\displaystyle \frac{1}{{\left(2\pi f\right)}^2{L}_2 }}$	${\displaystyle \frac{8R_{load}}{{\pi }^2}}$

shows the parameters of the WPT system depicted in Figure 1, where the mutual inductance and coil inductance vary according to the ferrite permeability. The operating frequency $f$ is 110 kHz, and the load resistance $R_{load}$ is 100 ohms, as determined by the charging conditions. The efficiency is calculated as the power delivered from the inverter output to the load, and the load resistance value is converted to the impedance magnitude as seen from the rectifier input, taking into account the effects of the square wave output of the rectifier as shown in Table 6 [5].

Table 7. Examples of weight variation with ferrite permeability (volume = 6.28 cm³).

Model	Vendor	Permeability	Weight [g]
Custom	Nopion	200	22
HM1400-300	Laird	650	30.61
33P2098-0M0		2300	37.16

lists the ferrites investigated to consider weight according to changes in permeability. Using these data, the interpolation of the weight curve based on permeability was performed. The results of selecting ferrites by simultaneously considering efficiency and weight are explained in Section 4.2.

4.2. Simulation Result

Figure 5a,b present the graphs showing the calculated coil inductance and mutual inductance as the ferrite permeability increases, alongside the results obtained from 3D EM simulations. Due to the symmetrical structure of the TX and RX coils, only one inductance value is displayed. The calculation results show that the differences in coil inductance and mutual inductance, compared to the simulation results, are 0.08% and 0.11%, respectively, when ${\mu }_p$ is 200.

Figure 6a compares the PTE calculated using Equation (14) with the mutual inductance derived from simulations and the PTE obtained from circuit simulations using the mutual inductance from 3D EM simulations. When the ${\mu }_p$ is 200, the difference between the simulation and calculated results is approximately 0.11%. The results indicate that the increase in PTE significantly diminishes for permeabilities above approximately 600, as confirmed by the derivative results in Figure 6b.

In these results, γ is set to 0.05, and the choice of ferrite can vary based on γ. From the simulation results, it is determined that using a ferrite with a permeability of around 600 is appropriate, leading to an approximately 1.4% increase in PTE compared to the initial permeability.

Figure 7 shows the weight curve fitted to the permeability using the data from Table 7, over-plotted with the PTE curve. The importance may vary depending on the weight and PTE, and the graph shows that a crossover point of PTE and weight occurs at a permeability of approximately 950. In this case, the permeability is selected as 950, but the results may vary depending on the importance of design factors such as weight and cost.

5. Verification through Experiments

In this section, the proposed method is validated by measuring the inductance and mutual inductance of coils with different ferrite permeabilities. Due to the limitations in sourcing commercial ferrites with a wide range of permeabilities, two sets of coils were manufactured using ferrites with two different permeabilities. The measured inductance and mutual inductance were compared with the simulation results, and the PTE under resonant conditions was compared using the measured data.

5.1. Experiment Setup

The dimensions of the coils and ferrites used for the measurements were the same as those shown in Figure 4. The ferrites used in the experiments had permeabilities of 200 and 3200, respectively. The permeability 200 product is a custom model made by Nopion (Suwon, Republic of Korea), and the permeability 3200 product is PM12 of Todaisu (Wonju, Republic of Korea).

Figure 8a,b show the setup of the coils used for the measurements and the weight of the ferrites used, respectively.

Table 8. Specifications of ferrites used in the experiment.

	Model	Vendor	Permeability	Weight [g]
Case 1	Custom	Nopion	200	22
Case 2	PM 12 TILE 50D	Todaisu	3200	30

lists the specifications of the two types of ferrites used. The coils were maintained at a 2.5 mm air gap using acrylic plates, and the inductance and mutual inductance of the TX and RX coils were measured using a vector network analyzer (VNA). The VNA used for the measurements was E5071C from Keysight (Santa Rosa, CA, USA), with a frequency bandwidth ranging from 9 kHz to 8.5 GHz. In addition, an impedance analyzer was used to measure coil resistance to eliminate the influence of the probes used during the VNA measurement. The impedance analyzer is the E4990A from Keysight (Santa Rosa, CA, USA), with a bandwidth ranging from 20 Hz to 10 MHz.

5.2. Experiment Result

This research presents a method for selecting ferrites based on permeability by analyzing the efficiency changes caused by the choice of magnetic materials for coils. To independently verify the PTE variations due to changes in ferrite permeability, the measured 3D EM simulation and analytically calculated coil inductance and mutual inductance were compared first.

Table 9. Comparison of measured, simulated, and calculated results.

	Case 1	Case 2
$L_p \left[\mathsf{\mu }\mathrm{H}\right]$ (Cal)	2.476	2.522
$L_p \left[\mathsf{\mu }\mathrm{H}\right]$ (Sim)	2.474	2.525
$L_{p,tx},L_{p,rx} \left[\mathsf{\mu }\mathrm{H}\right]$ (Meas)	2.657, 2.583	2.734, 2.745
$M_p \left[\mathsf{\mu }\mathrm{H}\right]$ (Cal)	1.826	1.872
$M_p \left[\mathsf{\mu }\mathrm{H}\right]$ (Sim)	1.824	1.871
$M_p \left[\mathsf{\mu }\mathrm{H}\right]$ (Meas)	1.79	1.91
$R_1,R_2 \left[\mathsf{\Omega }\right]$ (Sim)	5 mΩ
$R_1,R_2 \left[\mathsf{\Omega }\right]$ (Meas)	26.32 mΩ, 28.88 mΩ

shows the comparison results of inductances and resistances. Using the measured data from the table, capacitors that resonate at 110 kHz were selected for each case, and the PTE obtained from circuit simulations was compared.

The resistance of each coil and the load resistance were the same as the conditions mentioned in Section 4.1. The capacitors of each compensation network, selected based on the resonant frequency, were calculated using the equations in Table 6 and are shown in

Table 10. Compensation capacitances for verifying PTE.

	$\mathit{C} \left[\mathbf{n}\mathbf{F}\right]$ (Cal)	$\mathit{C} \left[\mathbf{n}\mathbf{F}\right]$ (Sim)	${\mathit{C}}_{\mathit{t}\mathit{x}}, {\mathit{C}}_{\mathit{r}\mathit{x}} \left[\mathbf{n}\mathbf{F}\right]$ (Meas)
Case 1	845.48	846.17	787.89, 810.46
Case2	830.06	829.07	765.7, 762.63

. Since the inductances derived from the calculations and simulations were symmetrical, the compensation capacitances for the TX and RX coils were identical. However, to verify the PTE using the measured inductances, different capacitances for TX and RX were selected based on the measured inductance results to achieve resonance at 110 kHz. The PTE was calculated using the capacitances in Table 10, and the inductance results in Table 9 were compared.

Figure 9 and

Table 11. Comparison of PTE.

	PTE [%] (Cal)	PTE [%] (Sim)	PTE [%] (Meas)
Case 1 ($R_1,R_2 :$Sim)	79.714	79.603	79.242
Case 2 ($R_1,R_2 :$Sim)	80.499	80.493	81.101
Case 1 ($R_1,R_2 :$Meas)	43.021	42.968	42.321
Case 2 ($R_1,R_2 :$Meas)	44.245	44.218	45.212

show the efficiency graph and table depending on permeability, respectively. The difference between the measurement and simulation results is about 0.6% in the maximum case (case 2), and the difference between the measurement and calculation results is up to 0.5% in case 1.

In addition to the simulated coil resistance, the PTE confirmed using the measured resistance is also provided in Table 11. The system discussed in this paper is a low-power WPT system, so the inductance of the coil is small. If the resistance increases, it can cause a significant decrease in efficiency due to a reduction in the Q-factor. The comparison of the calculated, simulated, and measured PTE using the measured resistance shows a maximum PTE difference of 0.97% in case 2. Additionally, the results indicate that when using case 2 with higher permeability, the maximum PTE increases by up to 2.9% compared to case 1.

The discrepancies between the measured and calculated or simulated results can be attributed to simple measurement errors as well as tolerance in permeability. Magnetic materials typically have a process tolerance of about 20–25% in permeability, which can cause such differences [23].

The experimental validation confirms that the proposed magnetic circuit modeling and efficiency calculation process for varying permeabilities enables the selection of ferrite permeability considering PTE.

6. Discussion

In this section, we discuss the expected benefits of the proposed method and its validation results. The method introduced in this paper focuses on selecting ferrite permeability by considering the point at which PTE saturates. Understanding why increasing ferrite permeability indefinitely boosts PTE is not ideal but is crucial. This question can be answered through three main factors: cost, time, and weight.

As mentioned in the introduction, increasing the permeability of ferrite materials requires more raw materials within the same volume, which leads to higher costs [12]. When more air is mixed into the magnetic material, the permeability decreases, and maintaining a high raw-material ratio within the same volume demands advanced processing techniques [12].

Table 12. Comparison of specifications for commercial ferrite tiles with different permeabilities (volume = 7.102 cm³).

Model	Vendor	Permeability	Cost	Weight [g]
HM1400-300	Laird	650	$2.48	35.27
33P2098-0M0		2300	$5.33	42

compares the cost and weight of products with permeabilities of 650 and 2300 for the same volume.

Time refers to the duration required for product design. Using the flow chart presented in this paper, efficiency based on permeability can be analyzed using mathematical formulations much faster than 3D EM simulations. This approach reduces the labor required to perform sweeps and analyses for each product iteration.

Weight is closely related to cost. Avoiding the use of excessively high-permeability ferrite can reduce the weight of the magnetic material. Lower-density magnetic materials result in reduced weight, as evidenced by the weight difference between the two models measured in Table 8 and shown in Table 12. This leads to overall product lightening. Additionally, an increase in PTE results in a reduced current flowing through the coil. This means that the diameter of the wire used to make the coil can be reduced. Consequently, not only the weight of the ferrite but also the weight of the coil can be decreased, contributing to the overall product’s weight reduction and reducing heat generation due to current flow.

Finally, the proposed method utilizes the physical properties of magnetic materials, so it is necessary to examine potential issues from the perspectives of core loss and temperature. The loss per unit of volume based on magnetic flux density can be obtained from the vendor, which allowed us to estimate the ferrite core loss indirectly when the current flowed through the coil. The power loss can be determined using the data sheet provided by the ferrite manufacturer, Todaisu, for the ferrite used in case 2 of the experiment [ref]. Although core loss has nonlinear characteristics, approximate losses can be obtained by referring to the datasheet. Based on the maximum magnetic flux density of the ferrite extracted from the 3D EM simulation results, the core loss for the proposed structure can be calculated to be approximately 0.2 mW. This would result in about a 0.1% decrease in PTE, considering an output power of 160 mW. Therefore, core losses can be neglected in low-power WPT systems like those presented in this paper.

In the same context, this study did not consider temperature issues. Generally, in low-power WPT systems at around 100 mW, the temperature change is minimal, typically less than 2 °C [8]. Most temperature-related studies have been considered in systems like wireless charging for electric vehicles [24]. Therefore, when applying the proposed method in this paper to high-power devices, temperature must be considered.

7. Conclusions

This paper proposes a method for selecting the permeability of ferrites in low-power IPT systems, which are currently being commercialized for multiple products. The impact of ferrite permeability on magnetic coupling and PTE in WPT coil systems was analyzed, confirming that PTE saturation occurs as permeability increases. Through magnetic circuit modeling, the process of calculating mutual inductance based on permeability can be derived, providing guidelines for selecting an efficient range of ferrite permeability considering PTE without the need for EM simulation.

The proposed method was validated through both simulation and experimental results, demonstrating changes in inductance, mutual inductance, and PTE. By using this method, engineers can easily select ferrites that balance cost and weight according to their specific needs.