5.1. Calculation of PCB Coil Inductance
In order to assess how much the coil inductance varies due to the irregularities described in the previous section, we need to be able to calculate the inductance of the PCB coil. However, an accurate calculation of the PCB coil inductance is not a straightforward task when the coil patterns are irregular. For a coaxial pair of circular filaments having the radii of
a and
r, and axially displaced by
z, Maxwell derived an expression of mutual inductance as [
12]
where
and
are complete elliptic integrals of the first and second kind, respectively, and where
Extending (
4) to the case of a coil pair with rectangular cross-sections is rather complex, as it involves an evaluation of an indefinite integral of Bessel functions [
8,
13]. Numerous approximate methods have been proposed (for example, [
14,
15,
16,
17]) In this paper, we used the approximation that replaces the coil with a pair of filaments displaced by “geometric mean distance” (GMD) and using (
4) to compute the inductances between all filaments (Lyle’s method) [
18]. The essence of this method is illustrated in
Figure 10. For the left coil with
, the coil is replaced with two filaments (1 and 2) with radii
and
[
18,
19]:
where
R is the geometric mean of the inner and outer radii of the coil. If
, as shown in the right of
Figure 10, the coil is again replaced with two filaments (3 and 4) having the same radius of
r at a distance
on either side of the mid-plane of the coil.
where
A is the geometric mean radius. The mutual inductance between two coils is given by
where
is computed from (
4) and (
5) with
z replaced with the geometric mean distance (GMD).
If PCB coils are ideal as illustrated in
Figure 6, we can utilize the formula proposed by Mohan et al. [
7] to calculate the self-inductance of a single-layer spiral coil, which is given as
where
is the permeability of free space,
the average diameter,
N the number of turns, and
the fill ratio
The mutual inductances between layers can be obtained from the coupling factor obtained from experiments [
20]. The total inductance of a multi-layer PCB coils can be calculated by
where
is the self-inductance of
i-th layer and
is the mutual inductance between the layers
j and
m.
If the pattern of the PCB coils is irregular, it is not possible to use (
12). For the irregularities in
Figure 2,
Figure 3 and
Figure 4, Lyle’s method can be used to calculate the inductance by the procedure illustrated in
Figure 11.
The above procedure cannot be applied to the case of layer misalignment (
Figure 5), as it assumes that the layers are coaxial. For the pair of circular filaments arbitrarily positioned in space, the method proposed by Babic et al. [
9] can be used. This method computes the mutual inductance between two filaments as illustrated in
Figure 12.
However, this method is unable to handle the coils with rectangular cross-sections. In this paper, we extended the method of [
9] to the case of PCB coils by combining it with Lyle’s method. Therefore, the procedure to calculate the inductance of PCB coils with layer-to-layer misalignment can be summarized in
Figure 13.
As summarized in
Table 4, the validity of the inductance calculations used in this paper is confirmed by investigating two cases of spiral coils. The first is the four-layer coils presented in [
8]. The second is the sensor coil design described in
Table 1. For the example case of [
8], the results by Lyle’s method are exactly the same as the results by the extended Babic’s method. Reference [
8] provides the FEA result as 628 nH, which is very close to our results. However, the result by Mohan’s formula is 44% of the FEA. This means that Mohan’s formula is not suited to a case where the number of turns is small (three, as in the example of [
8]).
For the sensor coil design in
Table 1, Lyle’s method again produces exactly the same inductance as the extended Babic’s method. Mohan’s formula calculates the inductance close to other methods (about 4% difference). Overall, it is evident that both Lyle’s method and the extended Babic’s method can calculate the inductance of PCB coils accurately. Furthermore, these methods can handle the variations of coil geometries, while Mohan’s formula assumes uniform configuration. It is also noted that the skin effect on coil inductance is negligible, as the calculated inductance is not very difference from the average of measurements (56.0
H vs. 54.2
H). Since the skin depth is around 66
m at the self-resonant frequency of 1 MHz, the cross-sectional area of a single trace is small enough to be influenced by the skin effect.
5.2. Assessment of Inductance Variation Due to PCB Irregularities
Using the aforementioned inductance model, it is possible to assess how much the inductance changes due to the irregularities observed in actual PCB coils. First, a reference design is determined, based on the measurements. The width and the spacing of trace are 83
m and 140
m, respectively. The inter-layer distances are:
m,
m, and
m.
Table 5 compares the inductance by Mohan’s formula, Lyle’ method, extended Babic’s model, and the average of measurements. Lyle’s method and the extended Babic’s model both agree quite well with Mohan’s formula and the average of measurements.
The procedure to assess the inductance variation is as follows. The parameter of interest is randomly varied within one standard deviation from the average value in
Table 3. For example, the widths of all 104 traces (26 turns times 4 layers) are randomly varied. Then the inductance is calculated. This is repeated 10,000 times. The distribution of inductances of the 10,000 trials is analyzed. Extended Babic’s model is used for the misalignment case, while Lyle’s method is used for all other cases. It is assumed that the distributions are normal, and also assumed that individual random variation is representative of actual coils. As shown later, one of the distributions is rather skewed, but the effect is not significant. Due to the second assumption, the variability may be underestimated. However, the purpose of this research is to identify the dominant factors contributing to the variability of inductance. Thus, it is important to maintain the same variability for all factors.
Figure 14 is the distribution of inductances when the layer distances are varied randomly. The mean is 56.23
H, which is slightly larger than the reference inductance calculated by both Lyle’s method and the extended Babic’s method. The standard deviation is 46 nH.
Figure 15 shows the distribution when the spacings between traces are varied randomly. The mean is 56.24
H, while the standard deviation is 59 nH. The inductance variation due to uncertainty in trace width is infinitesimal, as shown in
Figure 16. The standard deviation is only 2 nH. The distribution due to misalignment in
Figure 17 is somewhat unsymmetrical, since the inductance only increases irrespective of the direction of misalignments. The standard deviation is 18 nH.
From these results, several observations can be made. First, any irregularities increase the inductance. For the four types of irregularities, the average inductance increases to as much as 56.24
H from the base value of 56.0
H. It is also apparent that each irregularity type affects the uncertainty in different proportions.
Table 6 compares the sensitivity of irregularity types on the inductance variation. The normalized deviation is obtained by the ratio of the standard deviation to the average value using the data in
Table 3. The normalized uncertainty is defined as the ratio of standard deviation to the average inductance as shown in
Figure 14,
Figure 15,
Figure 16 and
Figure 17. The last column is the ratio of the normalized uncertainty to the normalized deviation. Clearly, the variation of trace width affects the uncertainty very little in inductance. On an absolute scale, the trace spacing affects the inductance variation the most. The most sensitive irregularity type is the layer distance. Image analysis shows that the layer distance is fairly consistent (3.6% variation at most). However, the sensitivity is the largest.
The results of uncertainty simulations can be compared with the measured variation. As shown in
Table 6, the layer-layer misalignment is not a dominant factor. Assuming that the nonlinearity in the distribution of misalignment does not affect too much, the total uncertainty is calculated from
Using the standard deviations of four types of irregularities, the total uncertainty is 0.074 H, which is less than the standard deviation of the measurements, which is 0.142 H. This is rather expected. Cutaway images show that irregularity is systematic while random variation is employed in simulations. Another factor contributing to large variations in inductance measurements is the uneven thickness of sensor coil. Due to the manufacturing process of PCB, it is very difficult to maintain the same thickness. As much as 8 m of difference in thickness across one coil is observed. Since the inductance is sensitive to the distance between the layers, uneven thickness would increase the variability of inductance.
What are the causes of the PCB irregularities?
Figure 18 lists two common issues of the PCB manufacturing process [
21]. On the left is the undercutting of PCB traces during the etching process. Undercutting is the main cause of the trapezoidal trace cross sections observed in the cutaway images. The etch factor defined as
can be improved by using heavier copper foil in spite of increased cost.
Figure 18b describes the foil outer stack-up for manufacturing multi-layer PCB. A fully-cured C stage is sandwiched between half-cured B stage epoxy. The outermost layers made of copper foil are located outside of this B stage. Then, pressure is applied while heating the platens. This process explains why the thickness of the middle layer (
is more consistent than
and
in
Figure 2. While pressurized, the B stages are squeezed much more than the fully-cured C stage. Consistency of layer thickness can be improved if the clad outer stack-up method is used, where only C stages are used.