Optimization of Coreless PCB Coils Based on a Modified Taguchi Tuning Method for WPT of Pedelec

Abstract

The printed circuit board (PCB) winding coil offers advantages such as small size, high precision, high repeatability, and low cost, making it conducive to the miniaturization of electronic equipment and a popular choice in wireless power transmission systems. This paper aims to clarify the correlation between induction parameters and inductive capabilities using the orthogonal array of the modified Taguchi method for Pedelec applications. The conventional Taguchi method typically achieves only local optimization; however, this paper considers practical application conditions and combines experimental data to establish the initial values of the orthogonal array, thereby achieving global optimization. Additionally, the tuning process of the Taguchi method replaces physical experiments with simulations, enhancing optimization speed and reducing hardware implementation costs. The performance index for the proposed modified Taguchi tuning method is selected as a combination of the quality factor (Q) and coupling coefficient (k) to minimize AC resistance and improve system efficiency. To validate the proposed method, the designed coils were implemented and tested in a WPT system based on S–S compensation with a half-bridge topology. The experimental results demonstrate that the optimized PCB coil parameters derived from the proposed tuning method accurately validate the method’s effectiveness and accuracy. From the measured results with the proposed modified tuning method, the system efficiency is increased by 43.87% and the system transmitting power is increased by 28.51%.

1. Introduction

Printed circuit board (PCB) coils have the advantages of high wiring density, small volume, light weight, beneficial to the miniaturization of electronic equipment, can replace traditional solenoid coils, and can be applied to wireless charging technology. Due to the low cost and high repeatability at low power and mass production, choosing a wireless power transfer system with a PCB as the coil is better than choosing a single-wire or multi-strand cable. Therefore, this paper will focus on the application of PCB coils to Wireless Power Transfer (WPT).

References [1,2] use mathematical formulas to derive the physical parameters of various types of coils, which can verify their accuracy, and the coil design parameters involve the AC conduction characteristics of wires, electric energy and magnetic field conversion, electromagnetic coupling characteristics, etc. To design and optimize a wireless power transmission system, a compact model of mutual inductance between two planar inductors is presented [3].

The optimal design methods and discussions in the literature [4,5,6,7,8] mainly focus on the skin effect and proximity effect of the coil according to its PC shape, line width spacing, etc. Coreless PCB transformers with different geometrical parameters are tested, and their characteristics are discussed [4]. The AC resistance of the PCB windings is optimized by the width of the PCB conductors [5]. Discuss the relationship between the two parameters of PCB layer spacing and line width with resonant frequency [6,7].

A WPT system at a high operating frequency up to hundreds of kHz will take up the skin effect and proximity effect, resulting in a drastic increase in AC losses [5,7]. As a consequence, the minimized coils must reduce the influence of AC resistance to reduce the losses and maintain the performance of the transmission. To overcome the above limitations, some reviews of AC resistance optimizing methods include interleaved winding and multi-strand winding, which can effectively reduce the AC resistance [9,10,11]. However, it is difficult to design the interleaved winding, and the multi-strand winding is not significantly optimized. Additionally, the WPT system throughout power is generally with an emphasis on high quality factor (Q), and an optimal matching condition relies on coupling coefficient (k). The reference [12] shows that the WPT system should be based upon maximizing the k value rather than the Q value. Therefore, this paper attempts to optimize the transmission effect by adjusting the parameters of the winding based on Q and k as the performance index to obtain the PCB coil wireless transmission optimization method.

Various literatures propose different optimization methods. As shown in literature [13,14], the design of multi-strands or Litz style structure is adopted, and the analysis results in the paper show that the multi-strand structure is relatively simple, but the effect is limited. As for the Litz wire structure, although it can effectively reduce the AC loss, it is difficult to design due to the complex design. This paper aims to study the optimized design of PCB coils to improve transmission performance and application in WPT systems. To clarify the correlation between induction parameters and induction ability of transfer power and efficiency, this paper first analyzed the influence of the winding structure of the coil, then made eleven different PCB coils with designed parameters. After testing all sets of the PCB coils with the WPT system based on S–S compensation with symmetrical half-bridge topology, the relationships between coil parameters and Q value and k value were obtained. To verify the reliability of the experimental results and analyze the optimization trend of the coil parameters, this paper conducts the Taguchi-tuning method [15,16,17].

Additionally, some applications with PCB for WPT systems are proposed [18,19,20]. The paper designed and demonstrated a smartwatch strap wireless charging system with a flexible PCB coil for the first time [18]. A recent study has suggested that magnetic energy around the transmission line can be harvested and delivered wirelessly to the monitoring equipment through PCB resonators embedded inside an insulation rod. This article presents a rigorous analysis, modeling, and optimization of such PCB resonators. A new closed-form quasi-static model of the PCB resonators is derived [19]. The complete system comprising the energy harvesting, wireless power transfer (WPT), and output power stages is practically evaluated using PCB resonators embedded in a 35 kV composite insulator [20].

The star-shaped coil design offers excellent rotation tolerance and uniform power distribution, making it ideal for free-moving applications, with the added advantage of being extendable for various uses without the need for complex feedback systems. However, it has a complex design that requires advanced manufacturing techniques, and efficiency drops significantly at extreme angles. Additionally, there may be biocompatibility concerns due to prolonged electromagnetic exposure [21]. In contrast, the third document’s maze-based WPT system is highly scalable and adaptable for small animal research, ensuring uniform power delivery and energy efficiency through a natural power localization mechanism. Despite these strengths, the system is complex to design and optimize, primarily suited for low-power applications, and may require overlapping Tx resonators, which adds to the implementation complexity [22]. The inductance, AC resistance, quality factor, and coupling coefficient of each coil were simulated by Maxwell electromagnetic field simulation software R2. The Q value and k value are then substituted into the practical designed PCB to calculate the performance index (PI). An analysis of variance was then performed to re-plan the experiment until the trend converged to obtain the best combination of coil designs. The Taguchi Method is chosen primarily for its ability to optimize multi-variable systems efficiently with fewer experiments, reducing costs and time. It enhances system robustness against noise and external variations, making it highly valuable in real-world applications. Additionally, the method is simple to apply, does not require complex statistical knowledge, and is flexible for multi-objective optimization. Additionally, the Taguchi Method does not have a formal mathematical formulation for optimization because it is a statistical method focused on experimental design and empirical optimization. Instead of relying on an explicit mathematical model, it leverages data-driven insights to find the optimal settings for system parameters, emphasizing robustness over precision in complex, real-world applications where traditional mathematical models may not be feasible or practical. Finally, this paper uses the Taguchi tuning method with different performance index weights to verify the optimization method of the PCB coil winding structure. Overall, the design in this paper offers effective solutions for simplifying the design and manufacturing process, reducing costs, and enhancing system flexibility.

2. PCB Planar Spiral Problem Formulation

In this paper, the WPT system is used for power transfer and communication of the torque sensor for pedal electric cycle (pedelec), and the system structure is shown in Figure 1. Because of the space limitations for the proposed WPT system, the coreless PCB coil is adopted in this paper. The transmission power is very low, and the transmission efficiency becomes important. Specifications of the WPT system are listed in

Table 1. Specifications of the WPT system.

Specifications	Value
Input and output voltage	5 V
Switching frequency	100 kHz
Minimum output power	350 mW
Distance between transmit and receive	5 mm

. A circuit diagram is shown in Figure 2. In order to provide the torque sensor and the circuit system on the receiving side with sufficient power supply, reducing the conduction loss is an important issue. The PCB coil has several parameters, such as coil inductance, Q value, and AC resistance, which will have various effects on the transmission power and efficiency.

In this paper, multiple sets of coils are designed according to parameters such as the number of layers, the number of turns, the width of the traces, and the spacing of the traces, and the transmission results are analyzed. The key parameters affecting transmission power and efficiency are selected through experimental results and compared in the following sections.

3. Experimental Architecture for Initial Values

In order to solve the local optimization problem of the conventional Taguchi method, the actual application of the circuit board conditions is considered, and the dispersion of its parameter values is used to achieve the global optimal effect. The four coil parameters of the experimental design include the number of layers, the number of turns, the trace width, and the spacing between traces and are divided into six groups from A~F according to these four parameters. A group (A) is the comparison benchmark; Group B (B1, B2) is to change the number of layers; Group C (C1, C2) is to change the number of turns; Group D (D1, D2) is to change the trace width; Group E (E1, E2) is to change the trace spacing; and Group F (F1, F2) is to change the trace width and trace spacing at the same time. The detailed parameter values of the six groups of experiments are shown in

Table 2. The detailed parameter values of the six groups.

Series No.	Layer	Turn	Width (mm)	Spacing (mm)
A0	2	10	0.8	0.2
B1	1	10	0.8	0.2
B2	3	10	0.8	0.2
C1	2	12	0.8	0.2
C2	2	8	0.8	0.2
D1	2	10	0.9	0.2
D2	2	10	0.7	0.2
E1	2	10	0.8	0.1
E2	2	10	0.8	0.3
F1	2	10	0.9	0.1
F2	2	10	0.7	0.3

. The inductance value, AC resistance, and quality factor of the PCB coils were measured by LCR meter, as shown in

Table 3. Measurement Results at 100 kHz.

Series No.	Inductance (μH)	AC Resistance (Ω)	Quality Factor
A0	21.065	1.53	8.68
B1	5.81	0.734	4.95
B2	47.07	2.891	9.89
C1	25.87	1.677	9.63
C2	16.055	1.231	8.12
D1	19.63	1.287	9.56
D2	22.895	1.697	8.73
E1	22.684	1.514	9.48
E2	19.686	1.509	8.18
F1	21.473	1.354	9.93
F2	21.349	1.664	8.21

. The definition of the winding parameters of the PCB coil is shown in Figure 3. The parameters and size of the PCB coils for transmission and receiver are the same.

The transmission results are measured with a WPT system in a half-bridge circuit topology. Based on the requirements of low power consumption and high efficiency for the pedelec application, the impedance matching is designed with series–series (S–S) compensation, the operating frequency is set to 100 kHz according to the common wireless charging frequency band, and the transmission distance is 5 mm. The measurement results of the coil were obtained using an LCR meter, and the WPT system is depicted in Figure 4. The experimental results are shown in

Table 4. Experimental Result at 100 kHz.

Series	5 mm Coupling Coefficient	5 mm Power (W)	5 mm Efficiency (%)
A0	0.549	0.429	24.509
B1	0.535	0.212	22.517
B2	0.546	0.273	30.288
C1	0.556	0.454	31.277
C2	0.539	0.326	14.799
D1	0.554	0.526	28.441
D2	0.544	0.386	23.417
E1	0.543	0.443	26.859
E2	0.535	0.408	22.063
F1	0.556	0.387	23.464
F2	0.56	0.381	22.433

, and the best performance of each group of coils is selected, respectively, in the table with color. The best performances of each group in the experiment are B2, C1, D1, E1, and F1.

The wireless power supply system in this paper has limited vertical distance of transmission due to the low transmission power. Therefore, the WPT system needs to improve output power and efficiency to ensure sufficient transmit power. The relationship between transmission power and efficiency at 5 mm vertical distance is shown in Figure 5. and the experimental results are screened; the poor transmission performance of each group of coils is removed. It can be found that when the output voltage is regulated at 5 V and the load is at least 0.07 A. The D1 coil has a maximum output power of about 0.526 W, and the C1 coil has the highest transmission efficiency of about 31.45%.

To find out the optimization parameters of transmission power and efficiency, this paper analyzed the parameters, trace width, and turns of the PCB coils above. Regarding the number of layers of PCB, it can be seen from A0, B1, and B2 that its transmission power A0 (2 layers) is the best, and the transmission efficiency is B2 (3 layers) as the best. Considering that the number of layers of the PCB does not change much and is all integers, and the price of the 3-layer board is high, this paper will fix the number of PCB layers to 2 layers and do not participate in the optimization of the parameter operation of the orthogonal array.

The difference in the spacing between traces is too slight to find, and it is an important parameter of AC resistance. Lastly, the optimization parameters were selected as the factors of the proposed modified Taguchi-tuning method. In order to find the optimal parameters for transmission power and efficiency, the parameters of the above measured coils, trace width (Group D), and number of turns (Group C) are analyzed and discussed. However, the difference in trace spacing is too small to be found, an important parameter of AC resistance. Finally, the optimization parameters are selected as factors for the modified Taguchi tuning method.

From the above measured results of 11 experiments, we can see that the best number of turns is 12 (Group C), the best width is 0.9 mm (Group D), and the best spacing is 0.1 mm (Group E). Considering the basic requirements of the trace width, the space between traces, and the diameter of the PCB, the maximum number of turns is 12, which is the best value of the measured result. Therefore, level 3 of the turn parameter is 12 turns, level 2 is selected as 4 turns, and level 2 is 8 turns. The trace width is limited by current; 0.154 mm is the basic requirement, so this is regarded as the minimum value of the width variable (level 1), and the value 1 mm close to the optimal value of 0.9 mm is selected as its maximum value (level 3), and level 2 is selected as 0.5 mm. The optimal value of the parameter of the space between traces is 0.1 mm, but considering the requirements of PCB production, it is 0.1 mm, so the minimum value level 1 is selected as 0.1 mm. Level 2 is 0.2 mm, and level 3 is 0.3 mm. Therefore, the initial values of 3 parameters can be obtained taking into account the size of the actual PCB coil and the current requirements of the circuit, and it can avoid getting stuck in the local optimal range.

4. Proposed Modified Taguchi Tuning Method

In this paper, the proposed modified Taguchi tuning method is used as the optimal parameter search method, which is mainly used to deal with multi-factor and multi-level experiments and select representative experimental conditions in all experiments to simplify the number of experiments. From Figure 5a,b, we can see that the transmitted power and transmission efficiency do not necessarily have a linear relationship. Therefore, the proposed tuning method can obtain the optimal combination conditions of different parameters in the WPT system, which can be used to analyze the correlation between the transmit power and efficiency with the coil parameters and find the optimal parameters with the proposed tuning method. The performance index of the proposed modified Taguchi tuning method is selected as the combination of quality factor Q and coupling coefficient k to reduce the AC resistance and improve the system efficiency. In order to find the correlation between coil parameters and transmit power and efficiency, the proposed method uses Texas Instruments’ Coil Designer software 4.0 to design coils. The Q and k parameters are calculated with the TI software to increase the speed of optimization and the cost of hardware implementation.

The proposed tuning procedure is outlined as follows:

(1)	Using the experimental results in Section 4, the factors (number of turns, trace width, and spacing between traces) and levels of optimization are selected, and the initial values of the control factors are set as shown in Table 5. Levels of control factors. Levels Factors X Y Z Turns Width (mm) Spacing (mm) Level 1 $X_1=$ 4 $Y_1=$ 0.154 $Z_1=$ 0.1 Level 2 $X_2=$ 8 $Y_2=$ 0.5 $Z_2=$ 0.2 Level 3 $X_3=$ 12 $Y_3=$ 1 $Z_3=$ 0.3 . For example, Xn in the table, X represents the tuned coil parameters, and the subscript n represents its level n.
(2)	List the orthogonal array to reduce the set of experiments as Table 6. Orthogonal array of the proposed Taguchi-Tuning Method. Experiment No. Factors Performance Index X Y Z Turns Width Spacing 1 $X_1$ $Y_1$ $Z_1$ ${PI}_1$ 2 $X_1$ $Y_2$ $Z_2$ ${PI}_2$ 3 $X_1$ $Y_3$ $Z_3$ ${PI}_3$ 4 $X_2$ $Y_1$ $Z_2$ ${PI}_4$ 5 $X_2$ $Y_2$ $Z_3$ ${PI}_5$ 6 $X_2$ $Y_3$ $Z_1$ ${PI}_6$ 7 $X_3$ $Y_1$ $Z_3$ ${PI}_7$ 8 $X_3$ $Y_2$ $Z_1$ ${PI}_8$ 9 $X_3$ $Y_3$ $Z_2$ ${PI}_9$ .
(3)	Analysis of parameter effects using Equations (1)–(9) to explore the most influential parameters.
(4)	A new orthogonal array of new levels for each factor is obtained; the new levels are derived from Table 7. Deriving new factors from best level. Factor New Level Best Level 1st Level 2nd Level 3rd Level X $X_1^{n+1}$ $X_1^n-0.5·(X_2^n-X_1^n)$ $0.5·(X_1^n+X_2^n)$ $X_3^n-0.5·(X_3^n-X_2^n)$ $X_2^{n+1}$ $X_1^n$ $X_2^n$ $X_3^n$ $X_3^{n+1}$ $X_1^n+0.5·(X_2^n-X_1^n)$ $0.5·(X_2^n+X_3^n)$ $X_3^n+0.5·(X_3^n-X_2^n)$ Y $Y_1^{n+1}$ $Y_1^n-0.5·(Y_2^n-Y_1^n)$ $0.5·(Y_1^n+Y_2^n)$ $Y_3^n-0.5·(Y_3^n-Y_2^n)$ $Y_2^{n+1}$ $Y_1^n$ $Y_2^n$ $Y_3^n$ $Y_3^{n+1}$ $Y_1^n+0.5·(Y_2^n-Y_1^n)$ $0.5·(Y_2^n+Y_3^n)$ $Y_3^n+0.5·(Y_3^n-Y_2^n)$ Z $Z_1^{n+1}$ $Z_1^n-0.5·(Z_2^n-Z_1^n)$ $0.5·(Z_1^n+Z_2^n)$ $Z_3^n-0.5·(Z_3^n-Z_2^n)$ $Z_2^{n+1}$ $Z_1^n$ $Z_2^n$ $Z_3^n$ $Z_3^{n+1}$ $Z_1^n+0.5·(Z_2^n-Z_1^n)$ $0.5·(Z_2^n+Z_3^n)$ $Z_3^n+0.5·(Z_3^n-Z_2^n)$ . The superscript of the new level value symbol is n + 1, and the superscript n represents the old level value.
(5)	Repeat the experiment with a new orthogonal array until the value converges.

$$m_{X1}={\displaystyle \frac{1}{3}}({PI}_1+{PI}_2+{PI}_3)$$

(1)

$$m_{X2}={\displaystyle \frac{1}{3}}({PI}_4+{PI}_5+{PI}_6)$$

(2)

$$m_{X3}={\displaystyle \frac{1}{3}}({PI}_7+{PI}_8+{PI}_9)$$

(3)

$$m_{Y1}={\displaystyle \frac{1}{3}}({PI}_1+{PI}_4+{PI}_7)$$

(4)

$$m_{Y2}={\displaystyle \frac{1}{3}}({PI}_2+{PI}_5+{PI}_8)$$

(5)

$$m_{Y3}={\displaystyle \frac{1}{3}}({PI}_3+{PI}_6+{PI}_7)$$

(6)

$$m_{Z1}={\displaystyle \frac{1}{3}}({PI}_1+{PI}_6+{PI}_8)$$

(7)

$$m_{Z2}={\displaystyle \frac{1}{3}}({PI}_2+{PI}_4+{PI}_9)$$

(8)

$$m_{Z3}={\displaystyle \frac{1}{3}}({PI}_3+{PI}_5+{PI}_7)$$

(9)

The Q factor and coupling coefficient k have a significant impact on the transmitted power and transmission efficiency in wireless power transfer systems. The Q factor reflects the level of energy loss in the system. A higher Q factor indicates lower energy loss, making energy storage and transfer more efficient. A high Q factor typically improves transmission efficiency, especially in resonance conditions, where it helps to maximize energy transfer efficiency. The coupling coefficient k represents the degree of magnetic coupling between two coils. When the k value is higher, the coupling between the two coils is stronger, which usually leads to higher transmitted power and better transmission efficiency. However, if the k value is too low, transmission efficiency may significantly decrease, as the efficiency of energy transfer depends on the degree of coupling between the coils.

In summary, the Q factor and k value together determine the transmitted power and efficiency in a WPT system. Increasing the Q factor and k value generally enhances system performance, but the optimal match between them must be considered to achieve the best efficiency. A higher Q factor indicates lower losses in the inductive coil, meaning more energy is stored relative to the energy lost. Under resonance conditions, the transmission efficiency η can be expressed as:

$$\mathsf{\eta }={\displaystyle \frac{k^2{Q}_1{Q}_2}{{(1+\sqrt{1+k^2{Q}_1{Q}_2)}}^2}}$$

(10)

where k is the coupling coefficient, and Q₁ and Q₂ are the quality factors of the transmitting and receiving coils, respectively. This formula shows that a higher Q factor can improve the system’s transmission efficiency. A higher coupling coefficient k indicates stronger magnetic coupling between the two coils, which increases the transmitted power P and efficiency η. The transmitted power P can usually be expressed as:

$$P={\displaystyle \frac{k^2{Q}_1{Q}_2{V}^2}{R}}$$

(11)

where V is the voltage applied to the transmitting coil. Since transmission efficiency η is proportional to $k^2{Q}_1{Q}_2$, increasing both the Q factor and the coupling coefficient k helps improve transmission efficiency. However, an excessively high Q factor may lead to poor frequency stability in some cases, so a balance between these two parameters is necessary to achieve optimal performance. In summary, these formulas clearly show how the Q factor and coupling coefficient k influence the transmitted power and efficiency in a WPT system.

The matrix experiment consists of three parameters: number of turns (X), trace width (Y), and spacing between traces (Z), which are selected as important parameters of the winding structure. Among them, the coil is set to be double-layered, the trace width is 0.154 mm according to the minimum withstand current standard, the manufacturing technology limits the minimum spacing to 0.1 mm, the copper thickness is 1 ounce, and the board thickness is 1 mm. The performance index of the proposed tuning method is selected as shown in Equation (12).

$$PI=W_1·Q+W_2·k$$

(12)

where $W_1$ and $W_2$ are weighting functions, and $W_1+W_2=1$. For the first experiment, the weighting functions were set both 0.5 to observe the trend. The orthogonal array is shown in Table 7. The Q value and k value are simulation results of the parameters determined by orthogonal array, and the average error of the Maxwell software R2 simulation value compared with the measurement result is about 1.734%. Therefore, using the simulated data to perform the search for the best parameters can speed up its tuning speed, reduce the cost, and be similar to the actual experimental results. Additionally, to reduce the cost in mass production, the transmit and receive PCB coils are set to be exactly the same.

It can be seen from

Table 8. The Orthogonal array of the experiment.

Experiment No.	Performance Index		m		Factor
1	${\mathit{P}\mathit{I}}_1$	0	${\mathit{m}}_{\mathit{A}1}$	0.309	${\mathit{X}}_1$	4
2	${\mathit{P}\mathit{I}}_2$	0.312	${\mathit{m}}_{\mathit{A}2}$	0.631	${\mathit{X}}_2$	8
3	${\mathit{P}\mathit{I}}_3$	0.615	${\mathit{m}}_{\mathit{A}3}$	0.494	${\mathit{X}}_3$	12
4	${\mathit{P}\mathit{I}}_4$	0.239	${\mathit{m}}_{\mathit{B}1}$	0.224	${\mathit{Y}}_1$	0.154
5	${\mathit{P}\mathit{I}}_5$	0.654	${\mathit{m}}_{\mathit{B}2}$	0.602	${\mathit{Y}}_2$	0.5
6	${\mathit{P}\mathit{I}}_6$	1	${\mathit{m}}_{\mathit{B}3}$	0.683	${\mathit{Y}}_3$	1
7	${\mathit{P}\mathit{I}}_7$	0.433	${\mathit{m}}_{\mathit{C}1}$	0.614	${\mathit{Z}}_1$	0.1
8	${\mathit{P}\mathit{I}}_8$	0.841	${\mathit{m}}_{\mathit{C}2}$	0.253	${\mathit{Z}}_2$	0.2
9	${\mathit{P}\mathit{I}}_9$	0.207	${\mathit{m}}_{\mathit{C}3}$	0.567	${\mathit{Z}}_3$	0.3

that the influence value m of the Y parameter (trace width) is greater than the m value of the X parameter (number of turns) and the Z parameter (spacing between coils). That is, in this proposed tuning method, the parameter of the trace width is the most influential parameter.

This paper normalized the experiment Q value and k value first to make sure the value will not affect the model weighting results. According to the experimental requirements, the larger the Q value and k value, the better the induction and coupling effect. Therefore, the quality characteristic is set as larger the better (LTB) characteristic as the performance index. The optimal parameters of power and efficiency optimization effects are obtained theoretically and experimentally. The tuning results of the proposed tuning method are shown in

Table 9. The tuning steps of the orthogonal array for experiments.

Factor	Level	Initial Level	1st Tuning Level	2nd Tuning Level
X (Turn)	$X_1^n$	4	6	9
	$X_2^n$	8	8	10
	$X_3^n$	12	10	11
Y (Width)	$Y_1^n$	0.154	0.75	1.125
	$Y_2^n$	0.5	1	1.25
	$Y_3^n$	1	1.25	1.375
Z (Spacing)	$Z_1^n$	0.1	0.1	0.1125
	$Z_2^n$	0.2	0.125	0.125
	$Z_3^n$	0.3	0.15	0.1375

.

Once the orthogonal array reaches the limitation of the parameter of turns, the evolution of the optimal parameters can be determined. The parameters after 2 times of tuning, the optimal parameters of the PCB coil are 10 turns, the width of 1.25 mm, and spacing of 0.1375 mm.

5. Experimental Results

The optimal coils were tested by S–S compensation with symmetrical half-bridge topology for pedelec, which is shown in Figure 6. The transmitter and receiver of the proposed WPT system board are shown in Figure 7a,b. The experimental configuration of this system uses a power supply to input a DC voltage of 5 V; the vertical Y-axis distance between the primary and secondary circuit boards is fixed to 5 mm; and the system is connected to an electronic load on the secondary side output. The transmission power and efficiency of the optimized coil at different vertical distances and the output voltage were stable at least 5 V, shown in Figure 8, which is a great improvement over the former designed C1 and D1 coils. It can be seen from Table 9 that the best transmission efficiency experiment is D1, and the best transmission power is C1.

Figure 8 shows the experimental results with the proposed tuning method when the performance index W1 = W2 = 0.5. From the measured results with the proposed tuning method, when the spacing is 5 mm, the system efficiency is increased by 43.87% compared with the experiment D1, and the system transmitting power is increased by 28.51% compared with the experiment C1.

5.1. No Load

Figure 9a,b show the measured waveforms of the transmitter and receiver of the WPT system when the system operates under no-load conditions, respectively. Figure 9a shows the driving signals VGS1 and VGS2, resonant current itx, and coil voltage Vtx of the half-bridge converter of the transmitter circuit. Figure 9b shows the measured resonant current irx, coil voltage Vrx, and rectifier voltage Vo of the receiver circuit. The phase difference between itx and vtx is 90 degrees, and it can be seen that it does work at the resonance point. Similarly, it can be seen that irx and Vrx also work at the resonance point, so it can be seen that the compensation of s-s is correct. The final output voltage is 33.1 V after the full-bridge rectifier and filtering of the receiver circuit.

5.2. Rated Load of 0.07 A

When the system is operated at a rated load of 0.07 A and the vertical distance between two coils is 5 mm. The input is 5 V/0.34 A, the switching frequency of the driving circuit is 100.96 kHz, the voltage ${\mathrm{V}}_{\mathrm{t}\mathrm{x}\left(\mathrm{R}\mathrm{M}\mathrm{S}\right)}$ on the transmitter coil at resonance is 11.79 V, and the current ${\mathrm{i}}_{\mathrm{t}\mathrm{x}\left(\mathrm{R}\mathrm{M}\mathrm{S}\right)}$ is 0.92 A, shown in Figure 10a. On the receiver, the induced voltage ${\mathrm{V}}_{\mathrm{r}\mathrm{x}\left(\mathrm{R}\mathrm{M}\mathrm{S}\right)}$ of the coil is 6.44 V, and the current ${\mathrm{i}}_{\mathrm{r}\mathrm{x}\left(\mathrm{R}\mathrm{M}\mathrm{S}\right)}$ is 0.314 A; the voltage after rectification and filtering is 7.49 V; and the output power is 0.565 W under the condition of a stable output of 5 V, shown in Figure 10b.

Observation of the experimental results, it can be concluded that the WPT system with optimal coils can improve the transfer power and efficiency. The important feature is that the wider trace and the less spacing are proportional to the efficiency theoretically, but the trend of tuning is not. Comparing the former designed coils C1, D1, and the optimal coil with weighting functions W1 = 0.5 and W2 = 0.5, we can find that the optimal coil has a lower AC resistance of about 1.183$\mathsf{\Omega }$ and a higher Q value of about 11.15. Therefore, the AC losses and coil quality Q are the main cause of influence, and the coil structure could be well designed with the proposed tuning method.

Additionally, this paper continued the Taguchi tuning method in different weightings to verify the optimization method of the PCB coils with the technical limitations. The optimization results in other weighting are shown in

Table 10. Optimal parameters under different weighting functions.

${\mathit{W}}_1$	${\mathit{W}}_2$	Turns	Width	Spacing	Q	k
0.3	0.7	11	1.375	0.325	11.926	0.618
0.5	0.5	10	1.25	0.1375	10.745	0.577
0.7	0.3	10	1.25	0.1375	10.745	0.577

. When the weight functions W1 = 0.7 and W2 = 0.3 of the performance index are selected, the results are the same as the condition of W1 = 0.5 and W2 = 0.5. This is because it is limited by the current standard of PCB trace width and actual PCB size. However, when the weighting functions are selected as W1 = 0.3 and W2 = 0.7, the optimal parameters of the PCB coil are 11 turns, the width of 1.375 mm, and spacing of 0.325 mm. The parameter Q has a relative impact on the transmission power, and parameter k has a relative impact on the system efficiency, so the weight functions W1 and W2 can be selected appropriately to obtain the required requirements for the WPT system. From the experimental results, it can be concluded that the proposed method indeed reduces the cost of multiple hardware implementations and improves the overall efficiency of the Wireless Power Charging (WPC) system when applied to Pedelec.

5.3. Discussion and Comparison

Table 11. The comparison of the advantages and disadvantages of the Proposed Taguchi Method with other common optimization methods.

Method	Advantages	Disadvantages	Applicable Scenarios
Proposed Modified Taguchi Method	- Simple and effective experimental design - Reduces the number of experiments, saving time and cost - Easily identifies major influencing factors	- Not suitable for highly nonlinear systems - Primarily applicable for optimizing a small number of parameters	- Suitable for cases with few parameters, where preliminary screening and optimization of influencing factors are needed
Response Surface Methodology (RSM)	- Capable of handling complex relationships with multiple variables - Can generate a global model to predict system behavior - Applicable for continuous design variables	- Computationally intensive, especially with many variables - Requires a large amount of experimental data to build an accurate model	- Suitable for moderate numbers of parameters, where system behavior can be approximated as continuous
Particle Swarm Optimization (PSO)	- Can find global optimum solutions - Suitable for highly nonlinear and complex optimization problems - Does not rely on mathematical models of parameters	- Computationally intensive, especially with large parameter spaces - Requires careful parameter tuning to achieve optimal results	- Suitable for high-dimensional, complex, and nonlinear optimization problems
Genetic Algorithm (GA)	- Efficiently handles nonlinear, multi-modal complex problems - Can avoid local optima and find global optimum solutions - Strong random search capability	- Requires substantial computational resources - Slower convergence rate	- Suitable for very large search spaces and optimization problems that are difficult to solve using traditional methods

shows that it compares the advantages and disadvantages of the proposed Taguchi Method with other common optimization methods in WPT applications. In this paper, the proposed modified Taguchi Method has been refined to address the global optimum challenge. This enhancement allows it to identify not only local optima but also the global optimum, ensuring that the optimal solution is reached even in intricate systems. In the application field of Pedelec with limited dimensions, the selection of better initial values through simulation experiments has successfully addressed the issue of the traditional Taguchi Method’s inability to achieve the global optimum [23,24,25,26,27,28,29,30]. It also allows for the identification of the parameters that have the most significant impact.

6. Conclusions

This paper uniquely focuses on the application of printed circuit board (PCB) winding coils in wireless power transmission (WPT) systems. The use of PCB coils, known for their small size, high precision, and repeatability, is particularly advantageous for miniaturizing electronic devices. While PCB coils are commonly used in electronics, their optimization for WPT systems, especially in Pedelec applications, is less explored. This paper bridges that gap by investigating and optimizing PCB coils specifically for this purpose. The unique contributions of this paper are outlined as follows:

I.

Modified Taguchi Method for Global Optimization:

A significant contribution of this paper is the modification of the traditional Taguchi method to achieve global rather than local optimization. The conventional Taguchi method often falls short in obtaining the best overall solution, as it is typically limited to local optimality. This research enhances the method by incorporating practical application conditions and experimental data to determine the initial values of the orthogonal array, thereby pushing the method towards global optimization. This is a novel approach that improves the applicability of the Taguchi method in complex engineering problems.

II.

Simulation-Based Tuning Process:

The paper introduces a unique approach to the tuning process of the Taguchi method by replacing physical experiments with simulations. This not only speeds up the optimization process but also significantly reduces the cost associated with hardware implementation. The shift from experimental to simulation-based tuning is an innovative step that enhances efficiency and practicality in optimizing WPT systems.

III.

Dual Performance Index for Optimization:

The performance index selected for the optimization process is another unique aspect of this paper. Instead of relying on a single metric, the paper uses a combination of the quality factor (Q) and coupling coefficient (k) as the performance index. This dual-metric approach is designed to simultaneously reduce AC resistance and improve system efficiency, providing a more comprehensive optimization criterion than methods focusing on a single parameter.

IV.

Experimental Validation with S–S Compensation and Half-Bridge Topology:

The paper does not just propose a theoretical method but goes further to experimentally validate the optimized PCB coil parameters within a WPT system using S–S compensation with half-bridge topology. This practical demonstration of the proposed method’s effectiveness adds significant value to the research, offering tangible evidence of its accuracy and applicability in real-world scenarios. From the measured results with the proposed modified tuning method, the system efficiency is increased by 43.87% and the system transmitting power is increased by 28.51%. It also points out the significant correlation of Q and k of the proposed method between transfer power and system efficiency. Additionally, the comparison table and explanation for Taguchi, RSM (Response Surface Methodology), PSO (Particle Swarm Optimization), and GA (Genetic Algorithm) based on performance metrics are shown in

Table 12. The comparison table for Modified Taguchi, RSM, PSO, and GA based on performance metrics.

Method	Convergence Speed	Accuracy/Precision	Scalability	Stability	Optimality Gap
Proposed Modified Taguchi Method	Medium	High	Medium	stable	Moderate
RSM	Fast	High	High	Very stable	Moderate
PSO	High-speed	High	High	Very stable	Low
GA	Slow	Moderate	Medium	Unstable	High

.