This section validates the proposed approach, step-by-step, using the FLCP for a particular set of turns and in a specific charging position. Then, a prototype is built and the mapping profile of self and mutual inductances is validated experimentally under different vertical and lateral displacements. Estimation errors and computational time gains of the proposed mapping methodology are evaluated and compared with existing literature.
4.1. Specifications and FEA Simulations
Table 1 presents the operation specifications and some physical constraints of a typical IPT application. The installation area hinges the MC geometry, and it imposes, inadvertently, the lateral tolerance limits and maximum size for the MC. On the other hand, the minimum admissible size for the MC depends on system specifications such as output power levels, air gap and lateral displacement values. One characteristic of non-polarized pads is the total decoupling between the transmitter and receiver pads when the lateral displacements exceed around 40% of the total diameter (
d) of the pads [
1]. This means the FLCP needs a minimum size of 400 mm to comply with the lateral tolerance of 150 mm, listed in
Table 1. An FLCP with a size of 650 mm was selected for evaluation, and it respects the maximum size limit of 800 mm imposed by
Table 1. The transmitter and receiver pads of FLCP have the same size, and its dimensions are shown in
Figure 2b.
The coils are wounded with Litz wire formed by 1050 strands, a cross-section of 4 mm and a rated current of 30 A. The value of is set at 2, whereas is set at 14 in order to avoid large induced voltage values at the coil terminals. The ferromagnetic core is modeled with the characteristics of the material N87 from Epcos.
A 3D model of the FLCP was created and simulated in an FEA tool called Flux from Altair. Each simulation has a second-order mesh with approximately 40,000 mesh nodes. The use of a second-order mesh increases the simulation time, but it provides accurate results, especially in ferrite-less geometries, such as the FLCP. The open-circuit test is performed in each coil, and the mutual and self-inductance values are determined using (
2) and (
3), respectively. This means that the same
in each
has to be simulated with three different electric circuits. The total number of FEA simulations needed for a full characterization of an MC is then determined by:
where
is the total number of coils in the MC, and it can take the values 2 or 3 for two- or three-coil systems, respectively. The parameters
and
correspond to the total number of required
and
, respectively.
Table 2 lists the minimum number of simulations required for different MCs based on (
14).
The runtime of each simulation ranges from 7 to 15 min, using a computer with an i7 4960X processor (max frequency of 4.00 GHz), 32 GB DDR3 at 2133 MHz and 2 TB HDD 7200 RPM Sata disk.
4.2. Self and Mutual Inductance Profiling
This section explains in detail how to obtain the value of
and
for an
mm and
mm of the FLCP with 650 mm. As identified in
Section 3.3, the FLCP and CP require the simulation results in six
(
) to extrapolate the mutual and self-inductance profiles. Since the number of turns is unknown, a total of seven
(
), identified in (
13), have to be simulated for each
. A total of 126 FEA simulations, according to (
14), are then needed to extract the mutual and self-inductance values. The charging positions are illustrated in the
side view of
Figure 5, and they have the following coordinates: (
,
):
1 = (100, 0),
2 = (250, 0),
3 = (100, 75),
4 = (250, 75),
5 = (100, 150) and
6 = (250, 150).
Table 3 lists the FEA simulation results from
A to
E, described in (
13), in each
for the FLCP with a size of 650 mm. The first step in the fitting approach method is the identification of
in all six
. The method described in
Section 3.1.1 is applied in detail to
1.
Figure 9 illustrates
in a 2D view for
1 with all significant values. The corners of the geometric figure correspond to the
values of
A to
D. The linear function between
A and
D corresponds to a fixed value of two turns (
) in the receiver coil, while the number of turns of the transmitter is varied, and it is given by:
The constants in (
15) are found using a curve-fitting tool, such as the
fit command in Matlab. The fitting process of two-dimensional functions, such as linear, exponential or Gaussian functions, requires the
x- and
y-point coordinates in two separate vectors. The curve-fitting tool then applies linear or nonlinear parametric regression to the inserted vectors, and it retrieves the respective constants. For example, the
x and
y vectors used in (
15) were
and [1.12 × 10
, 8.48 × 10
], respectively. The
x vector corresponds, in this particular case, to the values of
in
A and
D, whereas the
y vector is the correspondent
values in the same
. The same approach is also applied to discover the constant values in exponential and Gaussian functions.
Analogously, the linear function between
C and
B corresponds to a fixed value of 14 turns (
) in the receiver coil while the number of turns of the transmitter is varied according to:
All admissible
values for every combination of
and
are in between Equations (
15) and (
16). To validate the methodology that finds
for a particular set of turns, the following conditions are assumed as an example:
and
. First, the
values for points
and
are determined by replacing
with the value 8 in (
16) and (
15), respectively. The impact of
is already taken into account in
and
for
and
, respectively. The linear function between
and
infers the impact of
in
, defined as:
The value
= 26.5
H for
1 is finally determined using (
17) and replacing
with 10. The same approach is applied to the remaining five
, and the results are listed in
Table 4Step 2 of the fitting approach method characterizes
using the values found in Step 1 for a particular set of turns. The estimated results of
for
and
, listed in
Table 4, are used as an example to validate Step 2 of the proposed approach in detail. First,
is characterized as a function of the
for
with the same
. The
results for
1 and
2, identified in
Table 4, are inserted in a curve-fitting tool to discover the constant values in (
7). In this particular case, the
x vector used in fitting tool is equal to
, whereas the
y vector is equal to
. The same approach is carried out for the
pair results (
3,
4) and (
5,
6), and they are defined as:
To determine
at a particular air gap value, the variable
is replaced in (
18) by the desired value. Therefore, these three exponential equations determine three new
values in specific charging positions, labeled from
to
. As an example, the
in (
18) is replaced by 185 mm, and the following
values are found:
= 14.1
H,
= 12.74
H and
= 9.64
H. These
values are valid for
,
and
mm, with lateral displacements of 0, 75 and 150 mm, respectively. The remaining values of
for charging positions with different lateral displacements are found using (
8). The constants in (
8) are obtained with the curve-fitting tool, using the values of
from
to
. The vectors used in the fitting tool were
[0, 75, 150] and
, respectively. The new equation, described in (
19), determines
value as a function of
for an FLCP with a size of 650 mm:
To find
in a particular lateral displacement, the variable
is replaced in (
19) with the desired value. As an example,
was replaced by 100 mm, and the value of
H was found.
The aforementioned process determined the specific value of
for
,
,
mm and
mm, but the proposed approach extends beyond the estimation of
in a particular set of conditions. For instance, the results illustrated in
Figure 9 show the profile of
in
1, and it allows the immediate extrapolation of
for all possible combinations of turns without additional FEA simulations. Furthermore, the exponential equations, listed in (
18), characterize
for a particular set of turns and as a function of
, and they extrapolate
for different air gap values, even those that are outside the specifications listed in
Table 1. In conclusion, the proposed approach profiles
individually as a function of different parameters, and it combines all individual profiles in an iterative way to form, ultimately, the volume of
shown in
Figure 4.
Concluding the validation process of the mutual inductance, the proposed approach is now applied to the self-inductance. The profiling of
is explained step-by-step as a guide reference, but the same fitting methodology extends to
and
. As described in
Section 3.2, the first step in profiling
is the identification of
in all
. The
results from
A,
B and
E, listed in
Table 3, are inserted in the curve-fitting tool to find the constants of a second-order polynomial function, given by (
11). A total of six equations are found for the
results of
1 to
6. The equations found for
2,
4 and
6 are described in (
20), and they were selected to show the impact of lateral displacement in
:
As can be observed, the quadratic constants across all equations in (
20) are similar, and they correspond to the permeance of the transmitter pad. These results indicate that the presence of the receiver pad has small impact in the magnetic flux distribution of the transmitter pad, within the evaluated air gap and lateral displacement values. The value of
for a particular
is found by replacing
in the corresponding second-order polynomial equations of each
. As an example, the value of
was assumed and the estimated
values are listed in
Table 5. The results show a maximum deviation of 3.2% between the minimum and maximum values, and they are in line with the existing literature. Nevertheless, the impact of the air gap and lateral displacements must be accounted for using the same approach as the identification of
. The set of
results in the same lateral displacements, i.e., the set of
values (
1,
2), (
3,
4) and (
5,
6) are used to find the constants in (
7). These exponential equations model
as a function of the air gap in three distinct lateral displacement values. As an example, using the same
value of 185 mm, the following
values are found in the specific charging positions:
= 81.4
H,
= 81.2
H and
= 80.8
H.
To account for the effect of lateral displacements, the obtained values of in
in
,
and
are used to find the constants in (
8) through a curve-fitting tool. The new function characterizes
as a function of
for an
mm and
. The variable
is then replaced with the desired value to find the final value for
. The process is repeated iteratively to build the profile
. The same approach is conducted for
and
.
Figure 10a illustrates the built prototype of a FLCP and test bench. A detailed view of both the transmitter and receiver pads is made in
Figure 10b with the following turns:
,
and
. The FEA simulation results from
Table 3 were used to extrapolate the fitting curves of
and
, illustrated in
Figure 11. For each solid line,
is constant and the
is varied. As such, the
x axis corresponds to the vertical displacements. For dashed lines, the analysis is reversed, i.e., the
is constant and
is varied along the
x axis. The experimental measurements correspond to square points, and the FEA simulations correspond to circle points. From the figure analysis, it is possible to confirm that both Gaussian and exponential decay functions can be used to mimic the behavior for
and
. Fitting-based methods have inherent estimation errors that depend on the numbers, quality and distance between the fitting points. Any difference can be mitigated by adjusting the fitting parameters of the curve-fitting tool to reduce the square errors in the worst charging positions, i.e., for the highest vertical and lateral displacements.
4.3. Performance and Runtime
The estimation discrepancies of the proposed fitting approach method against FEA simulations and experimental data are quantified in this subsection as well as the time savings by the proposed mapping approach with the existing literature.
Figure 12 lists the errors between the fitting curves, the experimental data and FEA simulation results for
under vertical and lateral displacements. Within feasible displacements, the average error between the experimental data and the proposed fitting is below 3%. The error difference is higher in scenarios where the value of
is higher. For example, a charging position outside the displacement specifications listed in
Table 1 ((
,
) = (280, 200 mm)), corresponding to a coupling factor of 0.06, has an error of 6.8% between experimental and fitting curve (0.58
H). The estimation errors are higher (between 4.2 and 7%) for charging scenarios that exhibit low coupling values (between 0.04 and 0.078). Such charging positions are unfeasible for an efficient high-throughput energy transfer due to the high circulating currents required in the transmitter side. Similar error results are found for
, and for that reason, they are not displayed.
The estimation errors for
are between 2.4 and 3.2%. This error range is a consequence of an 1.4
H offset between the experimental data and the fitting approach and FEA simulation results, as depicted in the second graph of
Figure 11. Despite the offset value, the fitting curves follow the same pattern of the experimental data. Similar
values are obtained in different vertical and lateral displacements with difference errors in the range of 0.8 to 3.4%.
Figure 13 shows the error results between the fitting approach method and 3D FEA simulation results for
and
with an FLCP size of 650 mm. The comparison is made with the FLCP in six different charging positions and with different sets of turns, as identified in the top left corner of
Figure 13. The results show an average error in
around 4%, whereas the average error of
is inferior at 1%. The highest errors in
occur for higher lateral displacement values, such as points 1 and 2. In these cases, the values of
are inferior to 10
H and had a variation of just 0.5
H in the fitting approach, which lead to an error of 5%. The estimation of
, on the other hand, has average errors inferior to 1%, and in some conditions, such as point 2 and point 3, the error is negligible. This was due to the little effect of the air gap and lateral displacements in the self-inductance values, which reduces the estimation errors. Similar error results were obtained for
,
and
and for this reason are not depicted in
Figure 13.
One benefit of the proposed approach is the reduced number of FEA simulations required to create the self and mutual inductance profiles.
Table 2 lists the minimum number of simulations required for different MCs for known and unknown
. As explained in
Section 3.1, the profiling of the self and mutual inductance surfaces as a function of the number of turns requires the simulation of
in each
. The final number of required simulations in the unknown category is then affected by a factor that equals the number of
. Furthermore, non-circular-shaped MCs require twelve additional
, such as the BPP in
Table 2, to profile
and
as a function of lateral displacements along the
x and
y axes. These types of MCs require a total of 180 FEA simulations, if the set of turns is unknown, and 36 FEA simulations, if the set of turns is known. On the other hand, circular designs such as the CP only require six
, and the total number of simulations is reduced to one-third when compared with the BPP. In overall, the total number of FEA simulations that fully characterize an MC are comprised between 12 (for the CP) and 180 (for the BPP).
Table 6 shows the benefits of the proposed fitting approach, taking into account the presented case study, in comparison with the typical approach. In addition, a benchmark comparison is also made in
Table 6 with existing works in the literature. The table is subdivided into two groups: Literature and Proposed work. The first group shows the total number of evaluated MCs, whether the number of turns is known and the total number of simulations carried out in each work (
). The second group, identified in bold, shows the total number of simulations needed to characterize the MCs with the proposed fitting approach and the computational saving time in percentage. The first row in the table compares the profiling of the mutual and self-inductance values of the presented case study with the conventional approach and the proposed fitting methodology. To determine the number of simulations in the typical approach, the analysis of three different air gap values and five lateral displacements was established, making a total of 15 different charging positions. In terms of turns, six FLCPs were considered with different sets of turns, making a total of
× 5 × 6 ×
FEA simulations. These assumptions are in line with the existing literature to profile
and
. As can be observed, with the proposed fitting approach, for an unknown number of turns, the time savings are around
. These time savings only accounts for six different sets of turns, whereas the proposed approach takes into account all possible combination of turns between 1 and 14, which would increase the time savings by more than 90%. The remaining rows of
Table 6 compare the proposed approach with the existing literature. As expected, the total number of simulations considered in works [
9,
16] is drastically reduced using the proposed fitting approach with time savings around 80%. Optimization works of MCs, such as [
21,
22], can also take advantage of the proposed fitting approach. However, the lack of information regarding the total number of simulations, the air gap and lateral displacement intervals make the time savings estimation difficult. Still, if the simulation intervals for the air gap and lateral displacements are between 25 and 50 mm, the fitting approach could reduce the total number of simulations between 20% and 50% in the aforementioned works.