Data-Selective Learning Algorithm Using Resonance Parameters Based on Stacked Data Augmentation for Wideband Impedance Prediction of Printed Spiral Coils

Article information

J. Electromagn. Eng. Sci. 2025;25(2):190-201

Publication date (electronic) : 2024 December 17

doi : https://doi.org/10.26866/jees.2025.3.r.261

Joojoong Kim, Eakhwan Song

Department of Electronics and Communications Engineering, Kwangwoon University, Seoul, South Korea

^*Corresponding Author: Eakhwan Song (e-mail: esong@kw.ac.kr)

Received 2024 April 21; Revised 2024 April 30; Accepted 2024 August 4.

Abstract

I. Introduction

The density of wireless systems has increased continuously over the years owing to the requirement for miniaturization and multi-functionalization of wireless devices. Consequently, the near-field noise and radio-frequency interference (RFI) generated by parts of products have led to electromagnetic compatibility (EMC) problems [1–5]. The electromagnetic (EM) noise radiated from digital modules, IC chips, and I/Os in mobile devices causes RFI interference in antenna communication bands, such as global positioning system (GPS) and Wi-Fi, thereby increasing the bit error rate of wireless communication devices and reducing receiver sensitivity, as shown in Fig. 1 [6]. To address this, noise measurement technology comprising a near-field probe was deployed to improve EMC performance by analyzing and specifying the RFI noise source [7–13]. Notably, the printed spiral coil (PSC) structure is widely used as the current collector design structure for near-field probes to measure the magnetic field noise primarily generated by parts and board wiring. However, the measurement band is limited by the parasitic components in the PSC structure [14–16]. Moreover, as the operating frequency of a digital circuit increases rapidly, a frequency component of noise is generated over a wide band. To measure and identify this wideband noise component, it is crucial to improve the bandwidth of the near-field measurement probe through wideband modeling of the PSC impedance. The PSC structure can be modeled using a lumped element-based equivalent circuit, which comprises inductance and resistance resulting from coil-type wiring [17]. However, although the lumped-element-based equivalent circuit has a simple structure and can be easily modeled, it would be difficult to achieve wideband modeling owing to the simplicity of the model. To address this issue, a distributed element-based PSC equivalent circuit comprising multiple lumped elements and an analytical model based on equations was proposed [18]. This model achieved excellent high-frequency modeling accuracy compared to existing lumped element-based equivalent circuits. However, for a PSC structure with a large number of turns, modeling complexity increases exponentially due to the presence of parasitic components, which have a significant impact on PSC impedance in the high-frequency range. Therefore, a three-dimensional (3D) field simulation method was proposed to accurately analyze PSC characteristics, particularly at high frequencies. This method enabled the realistic modeling of the 3D structures, thus enhancing the accuracy of the results. However, achieving this accuracy requires considerable time and computing resources. Therefore, to address the shortcomings of equivalent circuit-based 3D EM models for PSCs, a wideband modeling approach using neural network learning for multiturn PSC structures was introduced [19]. This method effectively demonstrates the use of neural networks in predicting PSC impedance. However, the learning algorithm relied solely on numerical errors in the impedance amplitude data, leading to inefficiencies in modeling time and cost.

Fig. 1

Radio-frequency interference in a highly integrated mobile device.

In this work, we propose a neural network-based wideband impedance modeling method to boost efficiency by utilizing a data-selective learning algorithm that incorporates the resonance parameters of the PSC impedance and a stacked data augmentation technique. In the proposed method, a multilayer perceptron (MLP) neural network is employed and trained using combinations of multiple PSC design variables, as well as the frequency-dependent impedance values corresponding to each combination, as the training data for an impedance prediction model. To train the prediction models efficiently, we introduce a data selection algorithm that utilizes the difference in the resonance parameters of the PSC. Furthermore, we employ a stacked data augmentation method that continuously utilizes the selected data to enhance model learning stability and generalization performance. Validation results showed that the proposed method achieved an accuracy of up to 6 GHz compared to the EM field simulation results. Furthermore, it was confirmed that the amount of data required for neural network learning using the proposed method is significantly less than that required by conventional MLP models, resulting in a learning efficiency improvement of 54.4% over the conventional approach.

II. PSC Structure and Training Data Extraction

To train a neural network for predicting PSC impedance, we first established the relevant design parameters and compiled the training data, which included impedance values correlating with the parameters. This section outlines the PSC structure and the extraction process for the training data for wideband PSC impedance prediction using the proposed methodology. Fig. 2 depicts the conventional PSC structure and its design variables, including the number of turns (N), line spacing (S), line width (W), side length (D_out), substrate thickness (t), and substrate length (L). The PSC was printed on a 27-mm-long and 1.6-mm-thick square FR-4 epoxy substrate comprising copper traces of 35-μm thickness. The copper traces serve as the electrical circuitry for the PSC. Table 1 lists the values of the simulated PSC structure. Notably, in this study, we focused on varying the number of turns and the side length, both of which significantly affect PSC impedance, while keeping the other parameters constant. Table 2 presents the training data for the proposed model. The input data for the proposed model were structured as a 3D array, combining the PSC design variables with 600 frequency points ranging from 10 MHz to 6 GHz, incremented in steps of 10 MHz. Line spacing (S) and width (W), with a value of 0.1 mm, were the fixed parameters. Meanwhile, the variable parameters included the number of turns (N), which ranged from 2 to 5, and the side length (D_out), which varied from 5–20 mm in increments of 0.1 mm. The output data comprised frequency-dependent PSC impedance values corresponding to the input design parameters. Notably, the impedance values were extracted through HFSS simulations. The dataset included 362,400 data points representing various combinations of PSC design cases across the specified frequency range.

Fig. 2

A printed spiral coil structure.

Table 1

Design variables of the PSC

Table 2

Training data for the proposed model

III. Proposed Data Selective Learning Algorithm Using Resonance Parameters Based on Stacked Data Augmentation for Wideband Impedance Prediction of PSC

Conventional neural network learning methods cannot effectively improve prediction accuracy due to the increased amount and complexity of data [20]. In this context, a learning strategy that assigns a meaningful order to the training data can help improve both model performance and learning convergence speed [21, 22]. Therefore, this section proposes an efficient learning algorithm based on a neural network that combines data selection and stacked data augmentation. This approach leverages the impedance characteristics of PSCs to counteract the problem of model performance degradation due to the amount and complexity of the data. Specifically, the proposed algorithm utilizes differences in the resonance parameters of a PSC at high frequencies as indices for data selection.

1. Proposed Wideband PSC Impedance Prediction Model

Fig. 3 presents a block diagram of the proposed neural network-based data-selective learning algorithm for wideband prediction of PSC impedance. First, the training data for the proposed data selection algorithm were divided into the input and output data required for neural network learning. Notably, the input data comprised a combination of PSC design variables and frequency points, while the output data were composed of self-impedances corresponding to the PSC structure and frequency points. The input and output data together formed the dataset for the proposed model through labeling and shuffling. Subsequently, the proposed model employed an MLP neural network, which used multiple inputs and outputs to predict the impedances of 604 PSC structures containing eight hidden layers and 100 hidden nodes. In particular, the MLP neural network contributed to predicting the PSC impedance, calculating the loss between the predicted and target impedances within the selected frequency range in the proposed data selection algorithm, and learning iteratively to achieve the loss convergence condition. Notably, for instances where the loss value did not satisfy the convergence conditions, the proposed data selection algorithm was applied to a conventional MLP neural network for fast loss convergence and accuracy impedance prediction.

Fig. 3

Block diagram of the proposed neural network-based data-selective learning algorithm using the resonance parameters of PSC.

For data selection, the differences in the resonance parameters of the PSC within the selected frequency range were first defined and then utilized as indices. In other words, the data selection method involved considering the variations in the predicted resonance parameters as indices for selecting the data and groups, meaning that the data were selected based on these indices. Furthermore, the training data used for subsequent learning comprised stacked data, including the data obtained by applying the proposed data selection algorithm and the previously selected data. Therefore, the proposed algorithm, which combines the selected data with stacked augmentation, enables efficient neural network learning for a set selection period. The proposed model repeats the learning and selection processes until the convergence condition is satisfied.

2. Proposed Data Selection Algorithm using the Resonance Parameters of PSC

In this study, we also propose an efficient learning algorithm based on neural networks to predict the wideband impedance of PSC structures. The proposed algorithm integrates data selection and stacked data augmentation. It especially focuses on the high-frequency range, where lumped equivalent circuit models cannot accurately predict impedance. Fig. 4 presents a conceptual diagram of the impedance prediction results obtained from a conventional MLP neural network model using the proposed data selection algorithm. The proposed method was designed to achieve highly accurate PSC impedance prediction within the frequency range of 3 GHz (f₁)–6 GHz (f₂), along with secure wideband impedance prediction accuracy. During training, the proposed model successfully minimized the loss function between the predicted and target impedance values across the selected frequency range. Notably, loss functions include the mean squared error (MSE) and the global difference measure (GDM), as defined in the feature selective validation (FSV) method [23]. In the proposed method, the MSE was used as both the learning and performance metric for the neural network model. Meanwhile, the GDM, including the amplitude difference and trend information of the PSC impedance, was employed as the convergence metric for the model. When the losses converged, the training was considered complete. However, if convergence was not achieved, the proposed method went on to select a training data group deemed more helpful in terms of training efficiency improvement instead of training the entire dataset.

Fig. 4

Conceptual diagram for frequency range selection of the PSC impedance prediction model using the proposed algorithm.

Considering that PSC impedance exhibits multiple resonance peaks across a wide range of frequencies, resonance-related parameters were chosen as the primary criteria for data selection in the proposed method. Notably, the first resonance within a selected frequency range is predominantly influenced by the coil’s design parameters, while the higher-frequency resonances are partly engaged with parasitic components. Therefore, for data selection, the proposed method considers two key indices: variations in the first resonance frequencies within the selected frequency range and discrepancies in the impedances associated with the resonances. In Fig. 5, f_sim and f_pred denote the first resonance frequencies obtained from the EM simulation results and the proposed method, respectively, while Z_sim and Z_pred represent the corresponding resonance impedances, respectively. In the proposed method, Index A and Index B refer to the difference between the impedances and frequencies at the first resonance, as shown in the following equations:

Fig. 5

Resonance parameters for the proposed data selection algorithm.

(1) Index A=|Zsim-Zpred|,

(2) Index B=|fsim-fpred|.

Therefore, the resonance parameters for selecting the training data are defined as Index A and Index B, as shown in Eqs. (1) and (2), respectively. Fig. 6 presents 3D images of the data distribution calculated after initial training of the proposed model using the entire training data in terms of Index A and Index B, where the x-axis and y-axis represent the number of turns and the side length of the PSC, respectively. Index A and Index B for the total training data were first calculated, after which they were divided by the total number of training data and then averaged. The training data with A and B values lower than the average were classified into Group A (G_A) and Group B (G_B), respectively, according to the number of turns and side length of the PSC, as shown in Eqs. (3) and (4). Group A comprised datasets for which the difference in impedance amplitude between the prediction and the target at the first resonance frequency was below the training data average. Meanwhile, Group B comprised datasets in which the difference between the first resonance frequencies was below the training data average.

Fig. 6

Distribution of training data depending on (a) Index A and (b) Index B.

(3) GA(N,Dout)={X∈U|E(X)<1n(U)∑N=25∑Dout=520Index A(N,Dout)},

(4) GB(N,Dout)={X∈U|E(X)<1n(U)∑N=25∑Dout=520Index B(N,Dout)},

(5) ST(N,Dout)=U-GA(N,Dout)∩GB(N,Dout).

Notably, the proposed model selected the training data (ST) by excluding a subset corresponding to the intersection of Groups A and B from the total training data, as shown in Eq. (5). The intersection of Groups A and B represents datasets with a low difference in index, as defined by the proposed data selection algorithm, along with highly accurate PSC impedance prediction results obtained by training the neural network model. In contrast, the selected training data were characterized by a large difference in the defined index, along with low-accuracy impedance prediction results of the neural network model. When using the proposed data selection algorithm, the neural network model prioritized datasets with high index differences and low accuracy compared to datasets with low index differences and high accuracy, thus selecting the data that require additional learning to efficiently perform wideband impedance prediction of the PSC. By prioritizing data with larger errors (higher values of Index A and Index B) for training, the model can achieve faster loss convergence. Accordingly, the initial biased data distribution can be resolved, and generalization performance can be given to the wideband impedance prediction results for the total training data.

3. Proposed Stacked Data Augmentation Method

Neural network learning models affect learning performance based on the presence or absence of bias in training data [24]. In this context, applying a data selection method that selects only the data with high index differences can produce imbalanced learning results due to bias in the distribution of the selected data. Notably, the data bias in the proposed data selection algorithm is closely related to the structure of a specific PSC. The generalization performance and impedance prediction accuracy of the proposed neural network model may decline if the data distribution is concentrated on the number of turns or the side length of the PSC. To address this issue, we present a stacked data augmentation method that integrates the data selection and data augmentation processes. Notably, the data selection method enables the efficient learning of a model and improved generalization performance, while the stacked data augmentation method enhances the stability of a model by reducing bias in the training data used. Additionally, as learning progresses, this method can be applied to Index A and Index B to update the impedance prediction results, thereby helping select only the data with large errors. As a result, small data can be utilized to accurately predict the characteristics of many PSC structures over a wideband frequency.

The application of the proposed stacked data augmentation method varies based on the number of data selections (k). Fig. 7 depicts the data distribution of the proposed stacked data augmentation method based on the number of data selections, where the x-axis and y-axis represent the side length and the number of turns of the PSC, respectively. Fig. 7(a) depicts the data distribution of Groups A and B and the selected training data when the number of data selected is one. Groups A and B are classified in terms of Index A and Index B, with the training data corresponding to the intersection of Groups A and B, characterized by a low index difference, excluded from the total training data (U). The remaining data—the data with high index differences and low impedance prediction accuracy in the initial training—were selected (ST_k). These selected training data were included in the subsequent data selection as previously selected data (P_k), with P_k denoting a set of the selected data obtained by applying the proposed data selection algorithm, as shown in Eq. (6). Fig. 7(b) depicts the data distribution of Groups A and B, the previously selected data, and the training data selected from the remaining data (C_k) when

Fig. 7

Data distribution for the proposed stacked data augmentation method depending on the number of data selections: (a) k = 1 and (b) k = 2.

(6) Pk(N,Dout)=∑k=1n-1STk(N,Dout),

(7) Ck(N,Dout)=U(N,Dout)-Pk(N,Dout),

(8) STk(N,Dout)={U(N,Dout)-GA,k(N,Dout)∩GB,k(N,Dout)(k=1)Ck(N,Dout)-GA,k(N,Dout)∩GB,k(N,Dout)(k>1).

Here, two distinct cases of data selection based on k are presented. If the number of data selections is greater than one, the proposed data selection algorithm is applied to the remaining data, excluding the previously selected data with high index differences. This process helps in continuously training the data with high index differences by maintaining the existing selected data and preventing any learning dependent on it. To apply the proposed data selection algorithm to the remaining training data, the training data consisting of an equivalent number of turns and side length of the PSC among the input variables of the neural network prediction model were compared with the previously selected data and then excluded, as shown in Eq. (7). Groups A and B within C_k were classified according to the defined indices, A and B, with the selected data excluding the intersections with low index differences, as in Eq. (8). The final selected data were used as the training data for the next epochs with P_k and as the data with a high index difference and low impedance prediction accuracy, corresponding to the number of data selections.

Therefore, the proposed model utilizes both the selected training data and the previously selected data acquired through data selection and stacked data augmentation methods as the training dataset for the MLP neural network. As a result, the proposed data selection algorithm based on the stacked data augmentation method can prevent the selected data from becoming overly concentrated at a specific number of turns or side lengths when predicting the impedance of the PSC structure, thus reducing bias in the training data and improving the stability of the neural network model.

IV. Results and Validation

In this section, we demonstrate the improved model accuracy and efficient learning achieved by analyzing the GDM and impedance prediction results of the conventional MLP neural network, which yielded different outcomes based on the application of the proposed data selection algorithm. The proposed model demonstrated high impedance prediction accuracy compared to the 3D EM field simulation results. Furthermore, the learning efficiency of the proposed model was analyzed through a comparison of the number of training data used for the conventional MLP model corresponding to the selection index combining Index A and Index B.

Fig. 8 illustrates the simulation setup for estimating the self-impedance of the PSC using a High Frequency Structure Simulator (HFSS). In this simulation, the electrical flow starts at port 1, propagates along the copper trace, and then connects to the reference of port 1 through a perfect electric conductor (PEC) structure. Fig. 9 depicts the GDM convergence results of a conventional MLP model equipped with the proposed data selection algorithm in terms of the number of iterations, with the target convergence levels achieved at 0.8 and 0.4. Notably, in the FSV method, GDM values of 0.8 and 0.4 correspond to the boundary values for “Poor” and “Fair” and “Fair” and “Good,” respectively. At a GDM convergence level of 0.8, the proposed model with selection Index A, Index B, and a combination of both A and B required 58.2%, 51.5%, and 56.3% fewer iterations, respectively, than the conventional MLP model. However, at a GDM convergence level of 0.4, the results demonstrated a reduction of 31.2%, 24.8%, and 54.4% in the number of iterations for the models with selection Index A, Index B, and the combination of both A and B, respectively. Furthermore, the number of training data used by the proposed model, which applied the selection method combining Index A and Index B, was significantly reduced. These results demonstrate that the learning efficiency of the proposed model depends on both the combination of the selection indices and the target convergence level. Therefore, the proposed model, using a single selection index, can be trained for a specific PSC structure through data selection by considering only the difference in the first resonance frequency or impedance amplitude. However, this process may lead to data bias in training the neural network model and can cause slow convergence speed and low accuracy.

Fig. 8

The 3D electromagnetic field simulation setup for validating the proposed method.

Fig. 9

Difference in the measured validation results with target convergence levels of (a) GDM < 0.8 and (b) GDM < 0.4.

Fig. 10 depicts the PSC impedance results predicted by the conventional MLP neural network model using the proposed data selection algorithm. Fig. 10(a) and 10(b) depict the PSC impedance prediction results obtained based on the target convergence levels for the test data with N = 3 and D_out = 14.2 mm, Fig. 10(c) and 10(d) show the results for N = 3 and D_out = 8.7 mm, while Fig. 10(e) and 10(f) present the results for N = 5 and D_out = 14.2 mm. The proposed model not only exhibits higher impedance prediction accuracy than the MLP model but also shows high accuracy with regard to the HFSS simulation results from 10 MHz to 6 GHz. Moreover, as the number of turns (N) and the outer diameter (D_out) of the PSCs increase, the parasitic components within the coil also increase, leading to more complex impedance characteristics. Nonetheless, the proposed model offers improved prediction capabilities compared to conventional models by accurately capturing complex impedance characteristics.

Fig. 10

Impedance prediction results for PSC structures on using the proposed model at the target convergence level for (a) N = 3 and D_out = 14.2 mm with GDM < 0.8; (b) N = 3 and D_out = 14.2 mm with GDM < 0.4; (c) N = 3 and D_out = 8.7 mm with GDM < 0.8; (d) N = 3 and D_out = 8.7 mm with GDM < 0.4; (e) N = 5 and D_out = 14.2 mm with GDM < 0.8; and (f) N = 5 and D_out = 14.2 mm with GDM < 0.4.

Table 3 shows the GDM results of each model with regard to the PSC structure. When using the resonance parameters proposed in this paper, the proposed model demonstrated low sensitivity and high stability with regard to the number of turns (N) and the side length (D_out). Moreover, as the number of turns and the side length of the PSCs increased, the parasitic components within the coils also increased, leading to more complex impedance characteristics. The proposed model exhibited improved prediction capabilities compared to conventional models in terms of accurately capturing these complex impedance characteristics. In particular, the model combining Index A and Index B attained GDM interpretation values of “Good” and “Very good” at the target convergences of 0.8 and 0.4, showing stable and excellent results that are not sensitive to complex structures.

Table 3

GDM results of the proposed model according to PSC structures

Fig. 11 presents a comparison of the number of training data points used by a conventional MLP and the proposed models to achieve the target convergence-level GDM values of 0.8 and 0.4. In the case of a GDM convergence level of 0.8, the proposed models considering Index A, Index B, and a combination of Index A and Index B used 896,072, 1,037,768, and 935,608 training data points, demonstrating a 58.2%, 51.5%, and 56.3% reduction in the training data used, respectively, compared to the conventional MLP model. Meanwhile, at the GDM convergence level of 0.4, each proposed model (Index A, Index B, and a combination of Index A and Index B) used 2,875,000, 3,133,000, and 1,859,400 training data points, respectively. Additionally, the number of training data required was reduced by up to 54.4% compared to the conventional MLP model. In particular, the data selection method combining Index A and Index B significantly reduced the number of training data required for the neural network model, demonstrating that the difference between the first resonance frequency and impedance of the PSC is a useful index for selecting neural network training data for wideband impedance prediction. Notably, in the FSV method, the GDM values of learning convergence corresponding to “Good” were 0.262 for the conventional MLP model and 0.287, 0.274, and 0.310 for the proposed models (Index A, Index B, and a combination of Index A and Index B), respectively. This implies that the proposed models exhibit impedance prediction accuracies similar to that of the conventional MLP model.

Fig. 11

Number of training data of the proposed model.

Table 4 summarizes the results obtained by comparing learning efficiency and accuracy with the number of training data and GDM values at the convergence level of 0.4. The results indicate that the proposed model enables not only efficient neural network learning but also fast and accurate wideband impedance prediction of the PSC compared to the conventional MLP model. Table 5 lists the time elapsed by the EM field simulation and the proposed model to achieve the target convergence level of 0.4 with regard to the impedance prediction results for the PSC structures. While the EM field simulation required approximately 24 hours, the proposed model needed only 1 hour, using identical computing equipment. This demonstrates the potential of the proposed method for efficient analysis in predicting PSC impedance characteristics in terms of time consumption.

Table 4

Learning efficiency and accuracy comparison results

Table 5

Elapsed time for the field simulation and the proposed model

V. Conclusion

In view of the extensive research being conducted on neural network learning in various fields, this study proposes a neural network-based model for wideband impedance prediction of PSCs. To improve the learning efficiency of the neural network model for PSC impedance prediction, a data selection algorithm was proposed and applied to a learning model accounting for various PSC structures and frequency-dependent impedances. The proposed data selection algorithm selects a high-frequency range that is difficult to model and considers the resonance parameters of the PSC within the selected frequency range as indices for data selection. Additionally, a method integrating data selection and stacked data augmentation was proposed to reduce bias in the distribution of the selected data. The learning efficiency of the proposed model demonstrated a reduction of approximately 54.4% in the training data compared to the conventional MLP model. Furthermore, the model exhibited the same FSV interpretation values as the GDM results. In addition, the validation results of the proposed model showed high accuracy, up to a frequency of 6 GHz, between the impedance prediction results of the unlearned PSC structure and the EM field simulation results.

Overall, the performance of the proposed model depends on the selection of the frequency range, the definition of the resonance parameter for PSC impedance, and the method used to select the training data for the proposed algorithm. Notably, the results of this study can be applied to coil design for high-frequency noise measurement. Moreover, we believe that these results will contribute significantly to enhancing the utility of high-frequency EMC analysis.

This present research was funded by a research grant from Kwangwoon University in 2022. This work has also been supported by the Institute of Information and Communications Technology Planning and Evaluation (IITP) grant funded by the Korean government (MSIT) (No. 2020-0-00839, Development of Advanced Power and Signal EMC Technologies for Hyper-Connected E-Vehicle).

Article information Continued