2.1. Optimization of Coil Structure

Figure 3 shows the generalized structure of two-layer 3D coil. Coil layers can be distinguished by their color; the first layer is shown in black and the second layer in red. Second coil layer is placed at a depth D ₂ relative to the first coil layer, and it has narrower width W ₂ compared to width of the first coil layer W. 3D coil structure has a length L with both coil layers folded to a depth D at the coil ends. Grey surface represents the referent plane placed at the distance h from the coil.

FIGURE 3: Two-layer 3D coil structure.

Coil structure optimization is a two-step procedure. In step one, the goal is to ensure a uniform field distribution across a width of the coil, i.e., over cross-section of the referent plane named cs ₁ in Figure 3. In step two, the goal is to ensure a uniform field distribution across a length of the coil, i.e., over cross-section of the referent plane named cs ₂ in Figure 3.

Figure 4 shows the proposed optimization framework. The maximal allowed ripple value of magnetic field (r) and first coil layer dimensions (width W and length L) are considered as inputs. The optimization outputs are optimal transfer distance (h), second coil layer variables (D ₂, W ₂), and end fold depth D, for which the magnetic field strength and uniform surface size are maximized.

FIGURE 4: Optimization framework.

Extensive analysis is conducted, resulting in mathematical model for rectangular coil optimization. Conducted analysis and mathematical model are explained in the next two sections.

2.2. Optimization of Second Coil Layer Variables

To ensure a uniform magnetic field distribution over cross-section cs ₁ (width of the coil), a position of the second coil layer must be optimized. Perfectly uniform magnetic field distribution cannot be achieved, so one parameter that must be taken into account is the maximal ripple of the magnetic field. The second parameter is the distance h of the referent plane from the coil. For different distances h, different values of the second coil layer variables (W ₂ and D ₂) obtain “most” uniform field distribution. Similarly, for different allowed ripple values of a magnetic field, different values of the second coil layer variables (W ₂ and D ₂) result in different shortest distance of the referent plane h.

Optimization deals with the following problem: finding values of the second coil layer variables to generate the most uniform field possible at a shortest distance between TX-coil and a referent plane. However, such optimization problem misses some important aspects. Namely, the short distance generally results in high magnetic field strength. Certain values of the second coil layer variables that ensure uniform field at short distances do that at the cost of a lower magnetic field strength. This would decrease the overall performance of WPT system. The second important aspect is the percentage of the cross-section over which the uniform field is achieved. It is quite easy to get uniform field at short distance with high field strength, but only over a small fraction of total coil width.

Accordingly, the first part of the optimization process is finding values of the second coil layer variables to get uniform field on most of the cross-section, but also with a highest magnetic field strength value. Therefore, as a figure of merit (FoM₁), we propose the product of average magnetic flux density (calculated only for part of the cross-section where magnetic field strength is uniform with respect to defined ripple value) and percentage of cross-section on which the uniform field is achieved.

The first step in second coil variables’ optimization is the FoM₁ analysis. This is a 4-dimensional problem. FoM₁ is affected by the referent plane distance h (first dimension), maximal allowed ripple of the uniform field (second dimension), and variables (W ₂, D ₂) of second coil layer (third and fourth dimension, respectively). Since the referent plane distance h must be observed with respect to coil width W, their ratio (h/W) will be used in FoM₁ analysis. We assume the same current magnitude and direction in both coil layers.

The four parameters are varied in the following ranges: h/W ratio ranges from 1% to 20% (20 steps), ripple value ranges from 1% to 20% (6 steps), width of the second coil layer W ₂ ranges from 0 to the coil width W (50 steps), and depth of the second coil layer D ₂ ranges from 0 to half of the coil width W/2 (50 steps).

The following methodology was used: for fixed values of h/W ratio and ripple, magnetic field strength was calculated for each possible value of the second coil layer variables. Namely, magnetic field strength for each value of D ₂ and W ₂ was calculated over the cs ₁ cross-section of the coil, and magnetic field shape is analyzed. In this step of the coil optimization, the coil length L is set to be at least 10 times larger than the coil width W, and the ends of the coil are not folded down to the depth D. Figure 5 shows results of the analysis of a magnetic field shape for four different positions of second coil layer. Position of the second coil layer is completely determined by the values of the second coil layer variables, D ₂ and W ₂.

FIGURE 5: Examples of magnetic field shape analysis: (a) ripple = 1%, h/W = 0.08, D ₂/W = 0.18, W ₂/W = 0.7; (b) ripple = 1%, h/W = 0.08, D ₂/W = 0.12, W ₂/W = 0.6; (c) ripple = 1%, h/W = 0.08, D ₂/W = 0.24, W ₂/W = 0.7; (d) ripple = 1%, h/W = 0.08, D ₂/W = 0.1, W ₂/W = 0.7.

Ripple of the magnetic field is used as input parameter in the optimization process. It is used to determine the size of the uniform section of magnetic field. Uniform section is defined as a section of the coil where the magnetic field deviation does not exceed the defined ripple value. The analysis of the magnetic field begins at the center of the cross-section and moves to the sides. The magnetic field strength at the center is used as an initial average field strength value. The two adjacent magnetic field strengths are compared to the initial value, and if they do not deviate from initial value by more than the defined ripple, they are considered to be in the uniform section of the magnetic field. New average field strength of uniform section is then calculated. The next two adjacent field values are then evaluated using the same methodology. At one point, the field values that deviate by more than the defined ripple value will be reached. These field values and all the remaining ones are not a part of the uniform magnetic field section. This can be seen in Figures 5(a) and 5(b). The section of the magnetic field shape that is considered uniform (with defined ripple) corresponds to the top section of the square waveform (Figures 5(a) and 5(b)).

In Figures 5(c) and 5(d), different shapes of the magnetic field that correspond to different positions of the second coil layer are shown. Such magnetic field shapes are not considered uniform because of the two peaks at the sides. While the center part of the magnetic field has uniform field distribution (Figure 5(c)), the overall magnetic field shape is considered nonuniform. The parts of the coil cross-section where the magnetic field strength exceeds the maximal allowed field strength (average strength + ripple value) can potentially be harmful for receiver circuits in WPT system, which are designed for uniform magnetic field. For that reason, Figures 5(c) and 5(d) have no square waveform representing the uniform section of the magnetic field.

To summarize, there are two possible outcomes of magnetic field shape analysis. The uniform section of the magnetic field is identified, or the magnetic field is considered nonuniform. For magnetic field shapes that have a uniform section, the FoM₁ is calculated:

(6) $\begin{array}{}{\text{FoM}}_1=B_{\text{avg}}\cdot \frac{\text{uniform section width}}{\text{coil width}},\end{array}$

where B _avg is an average value of magnetic flux density calculated only for the uniform section of magnetic field.

The same methodology is used to evaluate the magnetic field shapes for each possible position of the second coil layer. As a result, we get FoM₁ values for each position of the second coil layer. Figure 6 shows the FoM₁ values for five different h/W ratios with fixed ripple value.

FIGURE 6: FoM₁ values for different combinations of h/W ratio and ripple value of 2%: (a) ripple = 2%, h/W = 0.05; (b) ripple = 2%, h/W = 0.07; (c) ripple = 2%, h/W = 0.10; (d) ripple = 2%, h/W = 0.14; (e) ripple = 2%, h/W = 0.20.

For low h/W ratio value, majority of positions of second coil layer result in nonuniform field (dark blue areas). With higher h/W ratio values, more positions of second coil layer generate uniform magnetic field. For each fixed combination of h/W ratio and ripple value, there is an optimal position of the second coil layer which results in maximal FoM₁ value. This maximal FoM₁ value is represented as one dot in Figure 7. Figure 7 shows maximal FoM₁ values for all evaluated combinations of h/W ratio and ripple.

FIGURE 7: FoM₁ analysis results.

It can be seen that, for a higher ripple value, a higher FoM₁ can be achieved. For each ripple value, the maximal FoM₁ value is achieved at different h/W ratio (different distance between the referent plane and the coil). For higher values of h/W ratio, large uniform section can be easily achieved, but the average magnetic field strength is lower. For lower h/W ratio values, we have higher average magnetic field values, but narrower uniform sections. The maximal FoM₁ is achieved at optimal distance from the coil (optimal h/W ratio) with large uniform section and high average magnetic field strength. These optimal h/W ratios are shown with blue markers in Figure 8.

FIGURE 8: Optimal h/W ratios for different ripple values: simulated results are denoted by blue markers, and fitted mathematical model is represented by solid line.

For a given ripple value and known coil width W, optimal distance h of the referent plane can be calculated as

(7) $\begin{array}{}h=W\left(0.0191-0.013\cdot \ln \left(r\right)\right).\end{array}$

Once the distance of the referent plane is selected, the position of second coil layer can be determined. Figure 9 shows the positions of second coil layer that achieve maximal FoM₁, for different ripple values with referent plane at optimal distance.

FIGURE 9: Positions of second coil layer that achieve maximal FoM₁, for different ripple values with referent plane placed at optimal distance.

For the lowest evaluated ripple values (0.5%–2%), the optimal width of the second coil layer, W ₂, is 70% of the first coil layer width, W. For larger ripple values, the second coil layer width increases. The depth of second coil layer, D ₂, shows direct correlation with the ripple value (Figure 10).

FIGURE 10: Optimal depth of second coil layer (D)₂ for different ripple values: simulated results are denoted by blue markers, and fitted mathematical model is represented by solid line.

For a given ripple value and known coil width W, with referent plane placed at optimal distance, the depth of second coil layer can be calculated as

(8) $\begin{array}{}{D}_2=W\left(0.041-0.03\cdot \ln \left(r\right)\right).\end{array}$

Figure 11 shows resulting magnetic field with optimized cs ₁ cross-section, for 0.5% ripple.

FIGURE 11: Simulated magnetic field distribution generated by two-layer coil with optimized cs ₁.

2.3. Optimization of Coil Depth D

To achieve uniform magnetic field across cs ₂ (Figure 3), the narrower sides of the coil structure are folded downwards to the depth D. The goal is to get magnetic field shape from cross-section cs ₁ across the length of the coil L. If the coil ends are not folded down, at the ends of the coil there is a significant increase in the magnetic field strength (Figure 11).

The second part of the optimization process is determining the depth, D, to which the sides have to be folded down. The methodology to achieve this is the same as in the first part of the optimization process, but with one significant difference. The magnetic field shape throughout the coil length should be consistent, meaning that the magnetic field strength should not deviate across cs ₂ cross-section. If the same ripple value as in cs ₁ optimization would be allowed during cs ₂ optimization, the result would have unwanted field increase at the coil ends, as shown in Figure 12(a). Thus, surface enclosed by W and L of the coil has a minor part of uniform magnetic field, as shown in Figure 13(a). In this stage of optimization, modified figure of merit is adopted:

(9) $\begin{array}{}{\text{FoM}}_2=B_{\text{avg}}\cdot \frac{\text{uniform section surface}}{\text{coil surface}}.\end{array}$

FIGURE 12: Magnetic field shape (top view) for different ripple values: (a) cs ₁ 5%, cs ₂ 5% (D/W = 0.19); (b) cs ₁ 5%, cs ₂ 0.5% (D/W = 0.43); (c) cs ₁ 0.5%, cs ₂ 5% (D/W = 0.19); (d) cs ₁ 2.5%, cs ₂ 2.5% (D/W = 0.26).

FIGURE 13: Uniform field section (top view) for different combinations of cs ₁ and cs ₂ ripple values: (a) uniform magnetic field section (49.81%) for magnetic field shape from Figure 12(a), FoM₂ = 0.0015 T; (b) uniform magnetic field section (75.78%) for magnetic field shape from Figure 12(b), FoM₂ = 0.0021 T; (c) uniform magnetic field section (75.92%) for magnetic field shape from Figure 12(c), FoM₂ = 0.0019 T; (d) uniform magnetic field section (79.85%) for magnetic field shape from Figure 12(d), FoM₂ = 0.0021 T.

It is not practical to try to optimize cs ₂ cross-section for 0% ripple, but using a ripple value 10 times lower than that used during cs ₁ optimization gives good enough result (Figure 12(b)).

Due to high D/W ratio (0.43), the uniform section of the magnetic field at the coil end is quite irregular (Figure 13(b)). Such shape of the uniform field was obtained by optimizing cs ₁ for 5% ripple and cs ₂ for 0.5% ripple.

Alternative approach is to first optimize cs ₁ for 0.5% ripple and then optimize cs ₂ for 5% ripple. The resulting magnetic field is given in Figure 12(c). Uniform surface is 0.14% larger, and FoM₂ value is 10% lower, but the shape of the surface with uniform magnetic field follows the rectangular shape significantly better, as shown in Figure 13(c).

There is a trade-off between FoM₂ value and the shape of the surface with uniform magnetic field. The best results are obtained when both the cs ₁ and the cs ₂ optimizations are done for ripple value equal to one-half of the desired ripple. Figure 12(d) shows the magnetic field when cs ₁ and cs ₂ are optimized for 2.5% ripple. The uniform surface (Figure 13(d)) is drawn for ripple value of 5%. It results in the highest FoM₂ value and largest uniform surface. This approach is chosen as optimal.

The depth of the end fold D is a function of the ripple value (r) and the length to width ratio (L/W) of the coil. Figure 14 shows the end fold depths for different ripple values and L/W ratios. The fold depth increases for coils with larger L/W ratio and with lower allowed ripple value.

FIGURE 14: Required end fold depth (D) for different ripple values and L/W ratios.

Results of cs ₂ optimization are given as markers, and solid lines represent mathematical model, (9), which can be used to calculate required folding depth.

(10) $\begin{array}{}D=W\cdot \frac{1}{20.4\cdot \sqrt{r}}\left(1-e^{-3.25\sqrt{r}\left(L/W\right)}\right).\end{array}$

The proposed optimization method is developed for rectangular coil where the percentage of uniform surface increases with higher L/W ratio of the coil (Figure 15).

FIGURE 15: Percentage of uniform field surface for different L/W ratios.