Machine-learning based characteristic estimation method in printed circuit board production lines

Based on the referential literature [15], the fundamental Archimedean spiral was described by (1) in the Cartesian coordinate, in which a, b, and t were constant, constant, and sector angle, respectively. To meet the EM operation requirements (frequency of 2–6 GHz that supports Wi-Fi and Bluetooth applications), a, b, and t were defined as 0.591, 1, and 4.28π–12.86π, respectively; and the resulting pattern was 15 cm-long with line width (w) and line space (s) both 0.93 mm. When two identical Archimedean spirals were interlaced however with a 180° rotational difference, a self-complementary Archimedean spiral antenna (SCASA) was achieved (figure 1(a)). Although the idea was proposed for PCB industry as an example, imitating reasonable pattern distortions resulted by rheology that appeared in other printed microelectronics (such as by inkjet printing) was also the aim of the present work. Thus, SCASA was the evaluation patterns and this work introduced its distorted counterparts through mathematical methods for comparison

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} {x\left( t \right) = at{\text{cos}}\left( {bt} \right)} \\ {y\left( t \right) = at{\text{sin}}\left( {bt} \right)} \end{array}.} \right.\end{align} \tag{ 1 }$$

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In printed microelectronics, inks or solid contents accumulate owing to surface tensions and other rheological properties [16]. Regular ink accumulations could be found not only because of the pattern design [17] but also the printing sequence of the system [18]. As a result, repeated bulges and neckings appear on the patterns and literature described this effect by duplicating them through sinusoidal equations [15] with (2). The constant 25 in (2) indicated that there were 25 bulges along one arm. It was clear that a point P varies its location on the Cartesian coordinate due to the changing parameters of not only the fundamental spiral but also additional factors of BA and displacement (d, in unit of mm) from the smooth edge. Large BA was resulted by large d from its original location defined only by (1). Straightforwardly d = 0 led to a BA = 0.00 or 0% distortion; d = 0.93 led to a BA = 1.00 or 100% distortion. In short BA = 0.00 was the reference and the two arms touch with each other when BA = 1.00. Even though bulges and neckings were repeatedly generated through (2), the pattern was impractical because bulges enlarge as the spiral develops with the sector angle (figures 1(b) and (c)). Consequently, additional correction on generating constant bulges in a SCASA was needed.

$$\begin{align}P & = \left\{ at\cos \left( {bt} \right) - d\frac{{1 + BA \times \sin \left( {25t} \right)}}{{{{\left( {{b^2}{t^2} + 1} \right)}^{\frac{1}{2}}}}}\right.\nonumber\\ & \qquad \times [\sin \left( {bt} \right) + bt\cos \left( {bt} \right)],at\sin \left( {bt} \right)\nonumber\\ & \qquad \left.+\, d\frac{{1 + BA \times \sin \left( {25t} \right)}}{{{{\left( {{b^2}{t^2} + 1} \right)}^{\frac{1}{2}}}}}[\cos \left( {bt} \right) - bt\sin \left( {bt} \right)] \right\}.\end{align} \tag{ 2 }$$

In the aforementioned regard, this work introduced some correction factors, which helped on suppressing the enlarging bulge along the enlarging sector angle. As described in (3), the constant 25 in the sine operation in (2) was revised as a variable ωt, in which ω represent the angular frequency. As a result, along the increment of t, BA was suppressed. With appropriate ω, which was set to 0.3 in this work, similar bulge dimensions could be found regardless of the sector angle, reflecting practical cases in solid content accumulation during inkjet printing. In this work, BAs of 0.00 (figure 2(a)), 0.15, 0.30 (figure 2(b)), 0.45, 0.60 (figure 2(c)), and 0.75 were evaluated

$$\begin{align}P & = \left\{ at\cos \left( {bt} \right) - d\frac{{1 + BA \times \sin \left( {\omega {t^2}} \right)}}{{{{\left( {{b^2}{t^2} + 1} \right)}^{\frac{1}{2}}}}}\right.\nonumber\\ & \qquad \times [\sin \left( {bt} \right) + bt\cos \left( {bt} \right)],at\sin \left( {bt} \right)\nonumber\\ & \qquad \left.+ d\frac{{1 + BA \times \sin \left( {\omega {t^2}} \right)}}{{{{\left( {{b^2}{t^2} + 1} \right)}^{\frac{1}{2}}}}}[\cos \left( {bt} \right) - bt\sin \left( {bt} \right)] \right\}.\end{align} \tag{ 3 }$$

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From (3), not only the fundamental (reference)SCASA but also SCASA with five BAs were generated as evaluation targets. The six SCASA patterns were fabricated by single-sided PCB process, and 200 samples were fabricated through identical process for each SCASA design. To appropriately control the process tolerance, the SCASAs were manufactured by a foundry service (Ping-Yi Electronics Co., Ltd), who assured a less than ±100 μm variation. The PCB fabrication was started from mask design, substrate (35 μm thick copper on 0.4 mm thick FR4) exposure, development, etching, before deionized water cleaning and nitrogen blow was conducted for the finished sample (figure 2(c)). Due to the limited fabrication tolerance, process variation did not lead to ambiguous analysis result, which will be explained later. In addition, because BA is positively related to line edge roughness (LER), this work also analyzed the LERs of SCASAs by an existing AOI method [14].

Because both arms of the SCASA were conductors and the material between the two arms was dielectric, the SCASA already exhibited passive capacitive response after fabrication. Consequently, six capacitances appeared in simulated results (figure 3(a)) as the theoretical values. In addition, this work also focused on active EM responses of SCASAs in terms of intensity and frequency. Thus, the six SCASA patterns were also simulated for their corresponding results (figures 4(a) and (b) for intensity and frequency, respectively). Besides simulated passive and active results, each sample was also measured for the labels, which will be used during AI establishment. Among the fabricated 1200 samples, 80% (960 pieces) and 20% (240 pieces) were defined as the training and testing group, respectively.

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From the LER on each sector angle on each edge of each arm, features for ML (such as line width, line space, and pattern area) can be collected. In addition to the features, labels (active intensity and frequency, and passive capacitance of the SCASA) defined for ML were also collected. The SCASA intensity and frequency were obtained through vector network analyzer (Agilent, E5071C) with Yagi–Uda transmitter/receiver [19], which will be explained later; and the SCASA capacitance was detected through the capacitance digital converter (Analogue Devices, AD7746) with the official evaluation kit.

ML was performed with a training group of samples through the lazy learning method in commercial software MATLAB (version 2021b) with toolbox of Regression Learner App, which returned various potential AI models. Besides the training group, the rest samples were defined as the testing group and their features were fed into those AI models to generate the predicted results (labels) with different R²s and RMSEs. In this work, the samples in both the training and testing groups had their corresponding measured features and labels, but only the samples in the testing group had their corresponding predicted labels through AI models. Consequently, the correctness of the AI model could be determined by comparing the difference between the measured and predicted labels of the samples in the testing group; and this work chose the results with the largest R² and the smallest RMSE as the AI model.

Additionally, this work also simulated the active and passive responses of the SCASAs with six BAs, which was defined as the simulated values. Because one SCASA design only had a single set of simulated values, the eligibility of the proposed method could be proved by comparing the simulated and predicted label sets.

Although the three SCASA labels (i.e. intensity, frequency, and capacitance) still need to be collected through separate systems; the aforementioned LER analyses, feature collection, ML, AI modeling, and characteristic prediction were all automatically performed through an integrated graphical user interface (GUI). By using the GUI, not only the operation simplicity was enhanced, efficiency and accuracy were simultaneously improved when compared with other works that introduced manual operations.

After analyzing the LERs of the fabricated SCASA patterns, results indicated an expected tendency: LER increased as BA enlarged (figure 5). This not only proved that the pattern distortion could be scientifically quantified and qualified but also the proposed workflow and algorithm were correct. Even though large variation existed in the reference (BA = 0.00) and BA = 0.15 groups, the rest four groups exhibited limited tolerance, indicating that the PCB fabrication was reliable. Furthermore, the differences between BA groups were noticeable: the largest LER in the former BA group never surpassed the least LER in the latter BA group. Consequently, these LERs were eligible to be included in the ML as the feature. Although the LER did not necessarily show a linear relationship to the BA, linear fit showed a R² = 0.966 when averaged LER in one BA group was used.

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On passive responses, the capacitance values increased as the BA (or LER) increased (figure 3(a)), which followed the expectation. This result was straightforward because when the LER worsened, the arm length increased. In the SCASA, the arm length and Cu thickness (an independent but constant parameter in this work) formed the area of the capacitor drawn in figure 2(a). As a result, the prolonged arm length due to the worsened LER increased the area of the capacitor, which led to the enlarged capacitance value. Furthermore, the arm length was 441.45, 442.03, 443.57, 446.12, 449.66, and 454.16 mm for 0.00, 0.15, 0.30, 0.45, 0.60, and 0.75 BA designs, respectively; and a nonlinear relationship existed between LER and BA was found as shown in figure 5. However, as mentioned earlier, the arm length had a positive relationship with the area of the capacitor, which in turn had a positive relationship with the capacitance value. Consequently, high confidence levels (simulated and measured results showed R² of 0.997 and 0.990, respectively) were obtained from linear fits (figure 3(b)), indicating the correctness of the analyzed passive responses.

On active responses, the simulated results followed the expectation because the coupling efficiency increased as the capacitance values increased [15], which led to the decay of transmitted intensity S₂₁ (figure 6(a)) in the detection setup (figure 7). However, the measured results showed a reversed tendency (figure 6(b)). This could be attributed to three potential reasons. First possibility was the environmental tolerance, which failed to isolate surrounding noises around the detection setup. Secondly, the Yagi–Uda antenna emitted EM waves in a broadening manner instead of keeping them as plane waves throughout the detection system. Thirdly, as BA (or LER) worsened, imperfect Cu in the neckings appeared easier, leading to reduced coupling efficiency.

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In case of frequency, the operating point (in which the weakest intensity located) moved to lower frequency as the BA (or LER) worsened, which matched the tendency of simulated results with a negative relationship in an exponential manner (R² were 0.998 and 0.997 for simulated and measured frequency, respectively), as shown in figure 6(a). These environmental or operational tolerance could be avoided in the future when better setups are available.

To this point, the required feature (LER) and labels (capacitance, intensity, and frequency) were all collected from the samples of the training group for ML. Even though there were various ML methods and it was not the only solution to use ML to establish the AI model, this work introduced lazy learning when considering the complexity of the aforementioned relationships among all features and labels.

Lazy learning results indicated that the model developed with Gaussian progress regression with covariance function of exponential exhibited the largest R² and the smallest RMSE. Based on these quantified evidences, this result was considered the AI model of this work, which was an independent module from the AOI and GUI for simplicity of the system and practicality of hardware or software revisions when need.

To validate the accuracy of the established AI model, not only individual pattern variations in samples of the testing group was analyzed but also their active and passive responses were examined. When the LERs of individual samples were fed into the AI model, the proposed system output the predicted corresponding capacitances, intensities, and frequencies of the SCACAs. Results indicated that, on the passive response, the predicted and measured capacitance showed a positive relationship to the BA (figure 8). Although some results overlapped with each other in the BA groups of 0.00 and 0.15, the difference between the predicted and measured data was less than 0.481% in a single BA group (of 0.45). Taking the measured capacitance of the samples in the testing group as the reference, the difference between the predicted and measured results was merely 0.404% on average through (4). In addition, the result of linear fit exhibited a high confidence level with R² of 0.969, indicating that the established AI model was practical (table 1)

$$\begin{align}& {\text{Tolerance }}\left( \% \right)\nonumber\\ & \quad = {\text{ }}\left| {\frac{{{\text{Measured}}\;{} {\text{value}} - {\text{Predicted}}\;{} {\text{value}}}}{{{\text{Measured}}\;{} {\text{value }}}}} \right|{} \end{align} \tag{ 4 }$$

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Similarly, on the active responses, the predicted and measured intensity results also showed good linearities regardless of intensity (figure 9) or frequency (figure 10). As mentioned earlier, because the active responses were influenced by the passive responses, some data overlapped with each other when negligible distortions (BA of 0.00 and 0.15 groups) existed regardless of intensity or frequency. However, the accuracy of the AI model was compatible to that used for passive responses. The difference between the predicted and measured results for intensity and frequency was less than 7.11% (worst case appeared in the BA 0.30 group) and 0.192% (worst case appeared in the BA 0.45 group), respectively. Taking the measured intensity of the samples in the testing group as the reference, the difference between the predicted and measured results was only 4.925% on average with R² of 0.775. Similarly, taking the measured frequency of the samples in the testing group as the reference, the difference between the predicted and measured results was only 0.166% on average with R² of 0.982.

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To this point, the proposed AI-assisted characteristic prediction system for PCB production lines through AOI was demonstrated. The established AI model not only successfully predicted the three labels (capacitance, intensity, and frequency) of the SCASAs from the feature (LER) with different distortions in the testing group but also showed outstanding accuracy to the measured data, when compared with existing studies.