Fast and Accurate Source Reconstruction for TSV-Based Chips via Contribution-Driven Dipole Pruning

Abstract

Electromagnetic compatibility (EMC) diagnostics for high-density through-silicon via (TSV)-based chips face significant challenges due to complex three-dimensional electromagnetic coupling and inefficient source reconstruction workflows. This paper proposes a universal contribution-driven dipole preprocessing technique tailored for dipole array-based source reconstruction methods, addressing the critical efficiency-accuracy trade-off inherent in traditional approaches. The core innovation is an influence factor-based evaluation-elimination mechanism that extracts effective dipole components aligned with the structural characteristics of TSV-based chips and multilayer printed circuit boards, while eliminating redundant dipoles independently of the downstream source reconstruction algorithm. Validation on a multilayer PCB (1 GHz) and a TSV-based chip (4 GHz) demonstrates that the technique maintains high reconstruction accuracy, with error increase limited to ≤0.2% for the simulated PCB and ≤0.05% for the physically measured TSV-based chip. Computational time is reduced by 28–61% for the PCB and 20–28% for the TSV chip compared to traditional source reconstruction without preprocessing. For TSV-based chips exhibiting complex electromagnetic behavior, the technique delivers consistent performance across different dipole configurations, providing a fast, robust, and universal EMC diagnostic tool for high-density electronic devices.

1. Introduction

Radiation source reconstruction (SR) is indispensable for electromagnetic compatibility (EMC) diagnostics of printed circuit boards (PCBs) and integrated circuits (ICs) [1]. As electronic devices rapidly miniaturize and achieve high-density integration, localized electromagnetic interference (EMI) sources—such as signal crosstalk in PCBs [2] and parasitic radiation from IC pins are increasingly difficult to identify through traditional far-field measurements [3]. SR addresses this challenge by reconstructing equivalent sources (including dipole arrays) from near-field data, enabling precise localization of EMI hotspots and quantitative evaluation of radiated emissions [4,5].

The advent of 3D integration technologies, particularly through-silicon via (TSV), has introduced new layers of complexity to EMC diagnostics. TSV-based chips enable vertical stacking of multiple active dies, which drastically increases device functionality within a small footprint. However, this dense 3D integration also intensifies electromagnetic coupling between layers and through vertical interconnects, creates intricate near-field patterns that are challenging to characterize [6]. Traditional far-field methods fail to resolve these localized, near-field interactions, making advanced SR techniques not just beneficial but necessary for the EMC analysis of TSV chips [7,8].

For equivalent dipole array reconstruction, researchers have developed traditional methods like regularization [9], singular value decomposition (SVD) [10,11], particle swarm optimization (PSO) [12], and differential evolution (DE) [13,14,15]. While these lay the foundation for SR, they have inherent limitations: heavy reliance on case-specific parameter tuning, high computational costs in complex multi-source scenarios, and potential fine-scale field information loss during processing. These limitations are exacerbated in TSV chip scenarios, where the 3D source distribution and strong coupling effects demand a high number of dipoles for accurate modeling, leading to a significant increase in computational complexity.

In recent years, neural networks (NNs) technology has emerged as a powerful alternative with intelligent computing development. Leveraging strong self-learning and nonlinear mapping, NNs model complex radiated field-dipole array relationships in intricate environments, overcoming traditional methods’ linearity constraints. Combining near-field scanning with machine learning for efficient SR has become an EMC research hotspot [16,17,18,19]. For instance, Ref. [20] uses an NN as an optimizer, incorporates a square function for phaseless near-fields, and offers a new pathway for complex sources requiring many dipoles. The work in Ref. [21] proposes a wave equation-informed DNN (WE-DNN) for EMI measurement near-field reconstruction, improving interpolation/extrapolation by integrating wave equation residuals into training. Compared to traditional algorithms, NNs have faster speed, better adaptability to complex fields, and stronger nonlinear EMI modeling—suited for high-density electronics.

However, existing SR methods, whether traditional or NN-based, share a key limitation: suboptimal handling of the equivalent dipole array. Most adopt uniformly arranged dipole arrays, whose distribution ignores the spatial variation in the target device’s electromagnetic field. This leads to two critical issues: (1) redundant dipoles often exist in weak-field regions, increasing the dimensionality of the linear system without boosting accuracy; (2) increasing the dipole count to capture fine-grained features (e.g., in complex PCBs, multi-layer ICs, or TSV structures) triggers a sharp, often prohibitive surge in computational resources. These issues create a pronounced SR efficiency-accuracy trade-off, severely limiting practical application—especially for modern high-density devices like TSV chips [22,23].

To address this fundamental trade-off and expand the practicality of SR technology in next-generation high-density devices, research on dipole reduction or sparsity-driven methods is urgently needed to better achieve source reconstruction. In this field, multiple approaches currently exist. Ref. [24] relies on image processing to perform iterative morphological erosion on field magnitude maps for dipole localization (positions and orientations), serving as a front-end standalone tool for general near-field scanning scenarios (e.g., PCBs) and outputting identified dipole locations. Work in Ref. [25], represented by algorithms such as differential evolution, achieves full-parameter inversion through global stochastic search to solve for dipole number, positions, and moments, acting as a standalone solver suitable for simple structures (e.g., microstrip lines) with sparsity as a natural outcome of the inversion process. The method in Ref. [26] is specialized for electromagnetic modeling in shielded enclosures, refining the model by replacing electric dipoles with magnetic dipoles via matrix analysis and contribution evaluation, and functions as a dedicated internal step in cavity modeling rather than a standalone solver.

In contrast to the aforementioned works, this paper proposes a contribution-driven dipole preprocessing method. The method proposed seeks to balance computational accuracy and execution speed. It employs a physics-informed metric as a front-end preprocessor, performing proactive pruning and sparsification of the source array via a coarse solve and spectral projection based on an influence factor. This significantly reduces the dimensionality of the subsequent reconstruction problem. Its role is that of a universal front-end preprocessor, integrable with any downstream solver, and applicable across a broad range of scenarios from PCBs to high-density 3D ICs. The output is a sparsified dipole array, prepared for efficient and detailed final reconstruction.

2. Materials and Methods

The proposed method combines dipole preprocessing with neural network-based radiation source reconstruction techniques. It optimizes the equivalent dipole through preprocessing algorithms and applies the updated dipole array to the source reconstruction process as shown in Figure 1.

2.1. Equivalent Dipole-Based SR: Basic SR

According to the equivalent dipole theory, the source of radiation generated by the Device Under Test (DUT) can be replaced by an array of equivalent dipoles. Based on source reconstruction theory, assuming that there are a certain number of dipoles set in the dipole plane and scanning points on each scanning surface, the electromagnetic field at the scanning point location can be derived as follows

$$\left[\begin{array}{l}H\left(r_i\right)\\ E\left(r_i\right)\end{array}\right]={\displaystyle \sum _{j=1}^N\left[\begin{array}{l}{G}^h\left(r_i,r_j\right)\\ 0\end{array}\right.}\left.\begin{array}{c}0\\ G^e\left(r_i,r_j\right)\end{array}\right]\left[\begin{array}{l}{M}^h\left(r_j\right)\\ P^e\left(r_j\right)\end{array}\right]$$

(1)

where $H\left(r_i\right)$ denotes the magnetic field generated by all N dipoles located at $r_j$ (i = 1, 2, …, N) at the scanning point $r_i$ (i = 1, …, M) and $E\left(r_i\right)$ is the electromagnetic fields at the scanning point $r_i$. $G^e(r_i,r_j)$ and $G^h(r_i,r_j)$ are the corresponding electric and magnetic field Green’s function matrices, where $P^e\left(r_j\right)$ and $M^h\left(r_j\right)$ are the corresponding dipole moments.

For most EMI sources in ICs, the radiation source is modeled as an array of infinitesimal dipoles (electric $P_z$ and magnetic $M_x$, $M_y$ and z-axis perpendicular to the chip surface). In order to simplify the whole process, this paper primarily uses the magnetic field component $H_x$ as the analytical quantity; other components can be studied similarly.

The contribution of a combined dipole at $r_j$ to $H_x$ at $r_i$ can be written as:

$$H_x\left(r_i\right)=G_{xx}^{hM}\left(r_i,r_j\right)M_x\left(r_j\right)+G_{xy}^{hM}\left(r_i,r_j\right)M_y\left(r_j\right)+G_{xz}^{hP}\left(r_i,r_j\right)P_z\left(r_j\right)$$

(2)

where $G_{xx}^{hM}$ to $G_{xz}^{hP}$ are the numerical Green’s function matrices corresponding to the magnetic field (complex-valued, split into real parts $RE\left(G\right)$ and imaginary parts $IM\left(G\right)$ during training).

The Green’s function components are derived from free-space propagation [23].

$$\left\{\begin{array}{l}{G}_{xx}^{hM}\left(r_i,r_j\right)=K_h{k}_0\left(\begin{array}{l}-{\displaystyle \frac{Y^2+Z^2}{R^2}}{f}_1\left(R\right)+f_2\left(R\right)-\\ {\displaystyle \frac{Y^2+{Z^{\prime }}^2}{{R^{\prime }}^2}}{f}_1\left(R^{\prime }\right)+f_2\left(R^{\prime }\right)\end{array}\right)\\ G_{xy}^{hM}\left(r_i,r_j\right)=K_h{k}_0\left({\displaystyle \frac{XY}{R^2}}{f}_1\left(R\right)+{\displaystyle \frac{XY}{{R^{\prime }}^2}}{f}_1\left(R^{\prime }\right)\right)\\ G_{xz}^{hM}\left(r_i,r_j\right)=K_h\left(-{\displaystyle \frac{Y}{R}}{f}_3\left(R\right)-{\displaystyle \frac{Y}{R^{\prime }}}{f}_3\left(R^{\prime }\right)\right)\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{K}_e=-j{\displaystyle \frac{k_0\eta }{4\pi }},K_h={\displaystyle \frac{k_0}{4\pi }},\eta =\sqrt{{\displaystyle \frac{{\mu }_0}{{\epsilon }_0}}}\approx 120\pi \\ k_0=\omega \sqrt{{\epsilon }_0{\mu }_0}={\displaystyle \frac{2\pi }{\lambda }}={\displaystyle \frac{2\pi }{3}}\times {10}^{-8}f\\ {\mu }_0=4\pi \times {10}^{-7},{\epsilon }_0={\displaystyle \frac{1}{36\pi }}\times {10}^{-9}\\ f_1\left(r\right)=\left({\displaystyle \frac{3}{{\left(k_{0}r\right)}^2}}+j{\displaystyle \frac{3}{k_{0}r}}-1\right){\displaystyle \frac{e^{-j{k}_{0}r}}{r}}\\ f_2\left(r\right)=\left({\displaystyle \frac{2}{{\left(k_{0}r\right)}^2}}+j{\displaystyle \frac{2}{k_{0}r}}\right){\displaystyle \frac{e^{-j{k}_{0}r}}{r}}\\ f_3\left(r\right)=\left({\displaystyle \frac{1}{k_{0}r}}+j\right){\displaystyle \frac{e^{-j{k}_{0}r}}{r}}\end{array}\right.$$

(4)

where $X$, $Y$, and $Z$ are the three-dimensional coordinates of the i-th scanning point and $R$ is the distance between them. $k_0$ denotes the free-space wave number and ${\mu }_0$ is the vacuum permeability with the unit $\mathrm{N}/{\mathrm{A}}^2$. ${\epsilon }_0$ is the vacuum permittivity with the unit F/m. $\eta$ is the free space wave impedance with the unit Ω. $\omega$ is the angular frequency with the unit rad/s. f is the frequency with the unit $\mathrm{H}\mathrm{z}$. $K_e$, $K_h$, $f_1\left(r\right)$, $f_2\left(r\right)$, and $f_3\left(r\right)$ are intermediate parameters.

Equation (2) can be formulated for all scanning points as an approximate linear system

$$H_x\left(r_i\right)=f\left(\begin{array}{l}{G}_{xx}^{hM}\left(r_i,r_1\right),G_{xx}^{hM}\left(r_i,r_2\right),\dots ,G_{xx}^{hM}\left(r_i,r_N\right)\\ G_{xy}^{hM}\left(r_i,r_1\right),G_{xy}^{hM}\left(r_i,r_2\right),\dots ,G_{xy}^{hM}\left(r_i,r_N\right)\\ G_{xz}^{hM}\left(r_i,r_1\right),G_{xz}^{hM}\left(r_i,r_2\right),\dots ,G_{xz}^{hM}\left(r_i,r_N\right)\end{array}\right)=f(G)$$

(5)

where function $f$ (nonlinear dipole moment function, encompassing $M_x$, $M_y$, $P_z$ for dipoles) requires traditional methods or neural networks for efficient fitting. $G$ is the composite Green’s function matrix of size M × 3N.

Traditional methods solve the above approximate linear inverse problem via regularization or optimization but suffer from high complexity when N is large and there are redundant dipoles in weak-field regions. To address these issues, this letter uses backpropagation (BP) neural network as the core SR engine as shown in Figure 2. It excels at modeling nonlinear Green’s function-dipole moment relationships, boosting accuracy vs. traditional methods. Both data processing and model computation were performed on the MATLAB (version R2024a) platform.

Input data comprises the real and imaginary components of the Green’s function matrix $G$ (from $G_{xx}^{hM}$ to $G_{xz}^{hP}$). Output data corresponds to the $H_x$ field at scanning points. Phase effects are negligible given the chip’s small size relative to the wavelength. Training is designed to fit the dipole moment function $f$ for subsequent field prediction.

However, the initial dense dipole array inflates NN input dimension and adds irrelevant features, thus, our dipole preprocessing is integrated to eliminate redundancy and optimize NN performance.

2.2. Source Reconstruction with Dipole Preprocessing

Based on the difference in contribution of dipoles to the magnetic field response, the array size is reduced through an “evaluation-elimination” mechanism. Given that regularization-based approaches exhibit markedly faster computation speeds than NN-based modeling (yet deliver considerably inferior reconstruction performance), we first derive an initial dipole moment function matrix $M\left(r_j\right)$ via regularization-based fast source reconstruction. Next, we compute the eigenvectors E for the complex conjugate transpose product matrix ${\left(G^{*}\right)}^{T}G$, with $G$ denoting the Green’s function matrix that maps dipole moments to scanning points. Using these quantities, we define the influence factor $\omega \left(r_j\right)$ for the j-th dipole. The calculation is based on two inputs:

The initial dipole moment vector $M\left(r_j\right)={[M_x\left(r_j\right), M_x\left(r_j\right), P_z\left(r_j\right)]}^T$ obtained from the coarse reconstruction.

The eigenvectors $E=[e_1, e_2,\dots , e_{3N}]$ of the matrix ${\left(G^{*}\right)}^{T}G$, where each eigenvector ${\mathit{e}}_{\mathit{k}}$ (a column vector of length 3N) represents a principal transfer mode from the dipole space to the measurement space.

The influence factor is computed as:

$$\omega \left(r_j\right)={\displaystyle \sum _{k=1}^{3N}{\displaystyle \sum _{d=1}^3{M}^{(d)}\left(r_j\right)\cdot E_{3(j-1)+d,k}}}$$

(6)

with d = 1, 2, 3 corresponding to $M_x, M_x, P_z$, respectively. $E_{3(q-1)+d,k}$ is the element of the eigenvector matrix $\mathit{E}$ at row 3(j − 1) + d and column k. The row index maps the d-th component of the j-th dipole to its corresponding position in the global eigenvector. k is the number of dominant eigenvectors considered. This factor projects the strength of each dipole’s initial estimate onto the dominant transfer modes of the system. Dipoles are then ranked by their influence factor $\omega \left(r_j\right)$. An elimination coefficient determines the fraction of dipoles to be removed. The number of dipoles retained after preprocessing is

$$N_p=N\cdot (1-\eta )$$

(7)

where η denotes the elimination coefficient (e.g., η = 0.2 corresponds to retaining the top 80% of dipoles).

The selection of the initial number of dipoles N follows two primary guidelines. First, it should correspond to the physical dimensions of the DUT and the spatial resolution of the near-field scan, essentially adhering to a spatial Nyquist criterion. Second, it should be based on a prior understanding of the DUT‘s electromagnetic behavior, such as the approximate number and distribution of radiating sources. Setting N excessively higher than the number of actual physical sources results in an over-parameterized model. This does not improve accuracy but can introduce spurious solutions and exacerbate overfitting.

The elimination coefficient η provides a practical trade-off between speed and accuracy. Based on our extensive simulations and measurements, values within the range of 0.1 to 0.3 typically preserve effective reconstruction accuracy while significantly reducing computation time. For most diagnostic purposes, starting with η = 0.2 is recommended, with minor adjustments possible based on the desired balance. The performance under different η has been verified by experiments, which can ensure the reproducibility of the method; the systematic selection criteria for η will be established as an adaptive model combined with device parameters in future research.

The full source reconstruction process integrates preprocessing in three steps:

Coarse source reconstruction: Initialize a dense dipole array with N dipoles and solve for initial $M\left(r_j\right)$ via fast regularization.

Dipole preprocessing: Compute $\omega \left(r_j\right)$ for all dipoles, eliminate redundant ones to get $N_p$ effective dipoles.

Fine source reconstruction: Reconstruct the final dipole moments using only $N_p$ dipoles via neural networks, reducing the inverse problem dimension from N to $N_p$ for cutting complexity.

2.3. Theoretical Analysis of Contribution-Driven Screening

The efficacy of the proposed preprocessing stems from its physical and mathematical grounding. The electromagnetic forward model is based on the Green’s function matrix $G$, which has dimensions M × 3N, where M is the number of scanning points and N is the initial number of dipoles (each dipole has three components: $M_x$, $M_y$, and $P_z$). This matrix quantifies the electromagnetic coupling between every dipole-scanner pair, making its size and the subsequent computations’ scale directly linked with the product of M and N.

The proposed contribution-driven screening reduces the effective number of dipoles from N to $N_p$ (where $N_p$ < N). This directly reduces the column dimension of the Green’s function matrix from 3N to 3$N_p$. Thus, the dimension of the inverse problem is reduced from M × 3N to M × 3$N_p$, significantly decreasing the computational load in subsequent reconstruction steps.

The eigenvectors of ${\left(G^{*}\right)}^{T}G$ (where ${\left(G^{*}\right)}^T$ denotes the conjugate transpose) represent orthogonal modes of energy transfer from the dipole array to the measurement domain. The influence factor ${\omega (r}_j)$, derived from these eigenvectors and the initial moment estimate ${M(r}_j)$, effectively projects the strength of each dipole onto these dominant transfer modes. A dipole with a high ${\omega (r}_j)$ significantly excites one or more key modes that are essential for reconstructing the observed near-field pattern. Conversely, a dipole with a negligible ${\omega (r}_j)$ contributes little to these dominant modes; its field contribution is either minimal or can be expressed as a linear combination of contributions from other, more influential dipoles. Eliminating these low-contribution dipoles prune the solution space without materially affecting the representation fidelity of the observable field, thereby achieving dimensionality reduction while preserving accuracy.

2.4. Performance Metric

To quantify reconstruction accuracy, a relative error metric is defined as

$${\sigma }_H={\displaystyle \frac{1}{M}}\cdot {\displaystyle \sum _{i=1}^M\frac{\left|H_{pre}(i)\right|-\left|H_{ref}(i)\right|}{\left|H_{ref}(i)\right|}}$$

(8)

where $H_{pre}(j)$ and $H_{ref}(j)$ are the predicted and reference (simulated or measured) magnetic fields, respectively. A lower ${\sigma }_H$ indicates better prediction accuracy. M represents the number of scanning points.

Computational efficiency is evaluated by comparing the computation time required before and after applying the preprocessing step.

2.5. Explicit Comparison with Sparsity-Driven Dipole Reduction Methods

To precisely position the novelty of our contribution-driven preprocessing, it is imperative to differentiate it from other state-of-the-art methods that also actively reduce the number of equivalent dipoles (i.e., promote sparsity) under the common constraint of near-field measurements.

Table 1. A comparison of sparsification or reduction approaches for dipole source.

Methods	Approach	Applicability	Output
Morphology-Based Search [24]	Iterative erosion of field maps to locate dipoles.	General near-field scanning (e.g., PCBs).	Dipole locations.
Global Optimization [25]	Full-parameter stochastic inversion for sparse solutions.	Simple structures (e.g., microstrip lines).	Full dipole parameter set.
Two-Step SRM [26]	Matrix analysis and dipole replacement for cavity models.	Shielded enclosures only.	Cavity-accurate source model.
Proposed Preprocessing	Physics-informed pruning using Influence Factor.	General-purpose (PCBs, 3D ICs, etc.).	Sparsified dipole array for reconstruction.

provides a detailed comparison focusing on three such prominent approaches and our method.

3. Results

3.1. Validation with a Multilayer PCB Simulation

To quantitatively evaluate the proposed technique, a multilayer printed circuit board (PCB) operating at 1 GHz was first employed as a validation platform. A full-wave simulation model was constructed in Ansys HFSS (version 22.2.0), with co-simulation performed in SIwave (version 22.2.0) to obtain near-field radiation data during circuit operation, as shown in Figure 3. The magnetic field component $H_x$ on planes at heights of 4 mm and 5 mm above the PCB surface was extracted for training. The field data on a plane at 6 mm served as the ground truth for validation.

It should be noted that 1 GHz is chosen as the validation frequency for the PCB because this PCB is designed for 1 GHz differential signal transmission, which corresponds to its actual working scenario. Single-frequency validation is a common practice in the source reconstruction field (e.g., Refs. [14,16] also use the operating frequencies of target devices for experiments), allowing direct focus on EMC interference issues in the device’s working state.

Figure 4 presents a visual comparison of the predicted magnetic field distribution at the 6 mm plane, showing the ground truth, the prediction from the basic NN-based SR method without preprocessing, and predictions from the proposed method with different elimination coefficients (η). Qualitatively, all methods successfully reconstruct the major field patterns. However, the proposed method maintains visual fidelity comparable to the basic method even as the number of active dipoles is reduced.

The quantitative performance metrics are summarized in

Table 2. Error metrics (${\sigma }_H$) and computation time predicted by two methods for the PCB under different dipole counts (η = 0.1).

Number of Dipoles	Methods	${\mathit{\sigma }}_{\mathit{H}}$	Computation Time (s)
16	Basic Method	4.56%	6.76
	Proposed Method	4.69%	4.85
25	Basic Method	4.35%	17.33
	Proposed Method	4.42%	15.52
36	Basic Method	3.64%	66.83
	Proposed Method	3.83%	26.35

. The results demonstrate the core advantage of the proposed preprocessing technique: a significant reduction in computational time with only a negligible sacrifice in accuracy. For instance, with an array of 36 dipoles, the basic method requires 66.83 s to achieve a relative error of 3.64%. By applying the proposed preprocessing with an elimination coefficient of η = 0.1, the computation time is reduced by approximately 61% to 26.35 s, while the relative error increased merely by 0.19% to 3.83%. This trend is consistent across different dipole array sizes (16 and 25 dipoles), confirming the method’s effectiveness in eliminating redundant dipoles.

A further analysis of the trade-off between the elimination coefficient, computation time, and reconstruction error is depicted in Figure 5. As the elimination coefficient increases (meaning more dipoles are removed), the computation time decreases substantially. Crucially, within a practical coefficient range (e.g., up to 0.4), the associated increase in reconstruction error remains marginal and acceptable for EMC diagnostic purposes. This provides engineers with a flexible parameter to balance the need for speed against the required precision in different application scenarios.

3.2. Experimental Validation with a TSV-Based Chip

3.2.1. TSV-Based Chip with Peripheral Circuit

To verify the robustness and practical applicability of the proposed method for high-density integrated circuits, experimental validation was conducted using a through-silicon via (TSV)-based dual-sideband hybrid filter chip. A validation frequency of 4 GHz is chosen for the TSV chip because this chip is designed to operate at 4 GHz for FBAR-related functions, which aligns with the needs of practical EMC diagnostics. The chip, with dimensions of 2.5 mm × 2.5 mm, is fabricated based on a film bulk acoustic resonator (FBAR) model and incorporates redistribution layers (RDL) and TSV structures to minimize parasitic effects [23]. The internal 3D structure of the chip is illustrated in Figure 6. The chip was mounted on a dedicated test board alongside its peripheral driving circuit, as shown in Figure 7.

The scanning volume above the chip was 12 mm × 8 mm × 2 mm, with a step resolution of 0.1 mm in the x-y plane and 0.5 mm in the z-axis, resulting in five measurement planes with 9801 points each. The magnetic field data from the two planes (at 0.7 mm and 1.2 mm above the chip) were used as inputs for the source reconstruction process. The field data from the remaining planes were reserved for validation and this test follows the EMC test norms of IEC 61967-3 [27].

3.2.2. Near-Field Scanning Principle and Setup

The electromagnetic near-field radiation from the chip was acquired using an automated near-field scanning system. The system operates on the principle of Faraday’s law of induction. A high-resolution magnetic field probe, functioning as a miniature inductive loop, was positioned in the near-field region of the operating chip. The time-varying magnetic field (H) generated by the chip’s internal currents induces a proportional voltage in the probe. The system utilizes probes sensitive to the $H_x$, $H_y$ and $H_z$ field components to capture the full vector field characteristics. A four-axis (X, Y, Z, and rotation) precision positioning system with a 20 µm positioning accuracy moved the probe in an automated, pre-programmed raster scan pattern with a minimum step size of 0.1 mm, enabling sub-millimeter resolution volume scans above the chip surface.

The test platform comprised a vector network analyzer (VNA) to provide the 4 GHz input signal, a spectrum analyzer to record the probe’s output, and a motorized scanner as shown in Figure 8. The connection diagram of the test platform used is shown in Figure 9. The DUT was fixed to the scanner’s ground plane. The scan volume was defined as 12 mm × 8 mm × 2 mm, with step resolutions of 0.1 mm in the x-y plane and 0.5 mm along the z-axis. This resulted in five measurement planes parallel to the chip surface, each containing 9801 measurement points. The probe’s initial height was set to 0.7 mm above the chip. The VNA used is the ROHDE & SCHWARZ ZVA8, the spectrum analyzer is the ROHDE & SCHWARZ FSV7, the scanner is the FLS 106, and the DC power supply is the RIGOL DP832A.

3.2.3. Data Acquisition and Processing

The effectiveness of the proposed method is based on reliable near-field data. Random noise in the measurements was preprocessed via Gaussian filtering [28,29]. Although systematic errors such as probe positioning uncertainty are inherent challenges in near-field scanning, their impact was minimized through system calibration and repeated measurements. The more general influence of these factors on reconstruction models has been discussed in [23]. This paper focuses on the algorithmic innovation, with the aforementioned data quality control providing the necessary foundation for validating its performance.

For the source reconstruction process, the measured magnetic field component $H_x$ data from the two planes (at heights of 0.7 mm and 1.2 mm) were used as the input datasets. The data from the remaining planes served as the validation set. The acquired raw data underwent post-processing to suppress measurement noise before being used for the reconstruction algorithm. This near-field scanning methodology provides a direct, high-fidelity mapping of the complex electromagnetic near-field distribution emanating from the TSV chip’s 3D structure, forming the essential experimental basis for validating the source reconstruction method.

3.2.4. Reconstruction Results

The reconstruction performance for the TSV chip is presented in

Table 3. Error metrics (${\sigma }_H$) and computation time for the TSV-based chip under different dipole counts (Proposed Method: η = 0.25).

Configuration/Field Pattern	Basic Method	Proposed Method (η = 0.25)
Measured magnetic field $H_x$ above the DUT (Reference)
Predicted magnetic field $H_x$ with 4 Dipoles	Error metrics ${\sigma }_H$ = 5.33% Computation Time (s): 14.87	Error metrics ${\sigma }_H$ = 5.37% Computation Time (s): 11.93
Predicted magnetic field $H_x$ with 9 Dipoles	Error metrics ${\sigma }_H$ = 4.84% Computation Time (s): 21.02	Error metrics ${\sigma }_H$ = 4.89% Computation Time (s): 15.18

. The table compares the predicted $H_x$ field patterns and the corresponding error metrics (${\sigma }_H$) between the basic method and the proposed method (η = 0.25) for different numbers of equivalent dipoles (4 and 9). The reference field pattern, obtained from direct measurement, shows the complex field distribution arising from the chip’s 3D structure.

The results confirm that the proposed dipole preprocessing technique maintains high reconstruction accuracy even for the challenging case of a TSV chip with complex internal coupling. For both 4-dipole and 9-dipole configurations, the relative error of the proposed method is nearly identical to that of the basic method (e.g., 4.89% vs. 4.84% for 9 dipoles), with differences ≤0.05%. Simultaneously, the proposed method achieves a consistent 20% to 28% reduction in computation time. This demonstrates that the contribution-driven preprocessing successfully identifies and retains the dipoles essential for modeling the chip’s radiation characteristics, while discarding ineffective ones, thereby enhancing computational efficiency without compromising the fidelity of the source reconstruction for advanced packaging technologies.

3.2.5. Robustness Analysis and Discussion

Detailed test data are presented in

Table 4. Detailed performance analysis (average ${\sigma }_H$ and computation time) for the TSV-based chip under different dipole counts and elimination coefficients (NN-based).

Number of Dipoles	Methods and Different η	${\mathit{\sigma }}_{\mathit{H}}$	Computation Time (s)	Standard Deviation (s)
4	0 (Basic method)	5.34%	14.8667	1.44
	0.25	4.69%	14.85	2.24
	0.35	5.37%	11.38	1.69
	0.45	5.51%	7.62	3.36
9	0 (Basic method)	4.35%	17.33	2.28
	0.25	4.42%	15.52	1.20
	0.35	5.08%	14.74	3.24
	0.45	5.30%	10.08	1.99
16	0 (Basic method)	5.66%	65.96	8.09
	0.25	5.64%	40.95	4.27
	0.35	5.86%	33.72	2.75
	0.45	5.94%	27.92	9.96

. To ensure statistical reliability and reproducibility, the results in the table represent the average of five independent experimental runs, with outliers carefully identified and excluded prior to averaging. The outlier exclusion was performed to mitigate the influence of non-ideal experimental variations and better represent the method’s consistent performance. The criteria for outlier removal were based on two primary considerations that reflect realistic computational and training conditions:

Firstly, the inherent stochasticity in training the BP neural network. The network’s convergence and final performance are sensitive to the random initialization of weights and thresholds. A suboptimal initialization can lead the model to converge to a poor local minimum, resulting in a reconstruction error that is not representative of the method’s typical capability. Secondly, the computational environment on a shared high-performance server. While efficient, such an environment is subjected to variable load from other concurrent processes. This can lead to fluctuations in the precise timing of computationally intensive operations (e.g., matrix inversions during the coarse reconstruction or gradient updates during NN training), potentially introducing minor non-deterministic variations in the numerical results and computation time for identical input parameters.

To objectively identify outliers stemming from these factors, we employed a statistical criterion: data points from the five runs where the reconstruction error fell outside the range of the mean ± 2 standard deviations were excluded. This threshold is a common statistical practice for identifying values that significantly deviate from the central tendency of a dataset. By applying this filter, we ensure that the averaged performance metrics presented in Table 4 are not skewed by rare, atypical runs, thereby providing a more robust and accurate reflection of the proposed method’s intrinsic characteristics and stable performance under normal operating conditions. The standard deviation of relative error in five repeated experiments is ≤0.2%, which proves the accuracy and stability of the results.

A noteworthy phenomenon, as shown in Table 4, is that when the initial number of dipoles exceeds a certain threshold (N ≥ 16 in this case), the reconstruction error ceases to improve and instead begins to increase. This phenomenon can be explained by fundamental principles in electromagnetic inverse problems and machine learning. First, it stems from the conflict between model complexity and limited data. The number of true physical radiation sources within a chip operating at a specific frequency is finite and sparse. When the number of equivalent dipoles is set far beyond the number of actual physical sources, the model becomes over-parameterized. Constrained by limited near-field data containing measurement noise, an overly complex model tends to fit the noise in the data rather than the true field distribution pattern—a phenomenon known as over-fitting—leading to performance degradation when generalizing the validation data [30,31]. Second, the ill-posedness of the problem is exacerbated. The condition number of the Green’s function matrix $G$ worsens as its dimension (particularly the number of columns, 3N) increases. This makes the solution to the inverse problem $H=G·M$ exponentially more sensitive to minor errors (e.g., measurement noise) in $H$, thereby reducing numerical stability and amplifying reconstruction errors [32,33]. Therefore, in practical application, selecting a number of dipoles that matches the actual physical complexity of the specific device—the “optimal modeling complexity”—is crucial, rather than assuming that “more is always better.”

3.3. Validation of Algorithm-Agnostic Property

To rigorously evaluate the claim that the proposed contribution-driven preprocessing is agnostic to the subsequent source reconstruction algorithm, its performance was tested with a fundamentally different type of solver: SVD, a direct linear algebraic method. This contrasts with the neural network approach used as the primary solver in this work. The multilayer PCB case served as the testbed.

Consistent with the NN solver mentioned previously, SVD also adheres to the following two configurations:

Baseline Configuration (SVD): The solver operates directly on the full, dense dipole array (size N).

Preprocessed Configuration (SVD-Pre): The solver operates on the pruned, sparse dipole array (size $N_p$) output by the proposed preprocessing module.

The source reconstruction results obtained using the SVD solver are shown in Figure 10, with detailed performance data provided in

Table 5. Detailed performance analysis (average ${\sigma }_H$ and computation time) for the TSV-based chip under different dipole counts and elimination coefficients (SVD-based).

Number of Dipoles	Methods and Different η	${\mathit{\sigma }}_{\mathit{H}}$	Computation Time (s)
16	0 (Basic method)	6.68%	5.3
	0.1	6.69%	4.6
	0.2	6.70%	4.4
	0.3	6.74%	3.7
25	0 (Basic method)	6.44%	8.1
	0.1	6.45%	7.4
	0.2	6.54%	6.5
	0.3	6.57%	5.9
36	0 (Basic method)	5.94%	11.3
	0.1	6.06%	10.1
	0.2	6.12%	9.2
	0.3	6.19%	8.0

.

Figure 10 visually demonstrates that the SVD solver can reconstruct magnetic field distributions highly consistent with the reference, regardless of whether the proposed preprocessing is applied. The key distinction is that the preprocessed SVD solver requires a significantly reduced number of active dipoles (from N to $N_p$), yet no visual degradation in the quality of the reconstructed fields is observed compared to using the full array.

The quantitative data in Table 5 further corroborate this observation. As demonstrated in the 36-dipole case with an elimination coefficient of η = 0.3, the total computation time for the SVD solver was reduced from 11.3 s to 8.0 s—a reduction of approximately 29.2%. Concurrently, the reconstruction error increased only marginally from 0.0594 to 0.0619, a relative increase of merely 4.2%. This controlled trade-off, where a significant acceleration is achieved with only a minor, acceptable variation in accuracy, is consistently observed across all tested configurations and initial array sizes. This result clearly indicates that the proposed preprocessing module, by pruning redundant dipoles and reducing problem dimensionality, provides a tangible and scalable benefit that can be leveraged by a solver like SVD, which is based on a completely different mathematical principle, without substantially compromising reconstruction accuracy.

Therefore, the combined evidence from Figure 10 and Table 5 demonstrates the algorithm-agnostic applicability of the proposed contribution-driven dipole preprocessing method. It can function as a universal front-end that provides downstream solvers with a sparsified, lower-dimensional equivalent source model, thereby delivering significant computational efficiency gains—as shown here for SVD—without sacrificing accuracy.

4. Discussion

The results from both the simulated multilayer PCB and the measured TSV-based chip consistently demonstrate that the proposed contribution-driven dipole preprocessing technique effectively addresses the efficiency-accuracy trade-off inherent in traditional source reconstruction methods. By introducing an influence factor derived from an initial coarse reconstruction and the eigen-analysis of the Green’s function matrix, the method provides a physics-informed mechanism to prune redundant dipoles before the computationally intensive fine reconstruction stage.

The key advantage of this approach is its algorithm-agnostic nature. While validated here with a neural network-based SR engine, the preprocessing step operates independently, filtering the dipole array based on its contribution to the observed near-field. This makes it readily integrable with other SR algorithms that rely on equivalent dipole models, such as optimization-based or regularization-based methods, offering a universal pathway to accelerate them.

For practical EMC diagnostics, especially in industrial settings where time is a critical factor, the achieved 20–61% reduction in computation time is significant. It enables faster iteration during design debugging or more efficient quality control screening. The minimal impact on accuracy (≤0.2% for PCB, ≤0.05% for TSV chip) ensures that this speed gain does not come at the cost of diagnostic reliability. The method’s proven effectiveness on the TSV chip is particularly noteworthy, as it extends the practical utility of efficient SR to the domain of advanced, high-density 3D integrated circuits, where electromagnetic coupling is more complex and challenging to model.

Looking towards industrial deployment, this preprocessing technique can be seamlessly integrated into existing EMC diagnostic workflows. It can function as a front-end module for commercial near-field scanning systems or EDA tools, providing an automated means to configure the equivalent source model optimally before detailed analysis. This reduces the need for expert manual intervention in setting up SR simulations, making advanced diagnostics more accessible.

The core logic of this method is frequency-agnostic; it can adapt to different frequencies by adjusting the frequency-dependent parameters of the Green’s function (a standard operation in broadband source reconstruction), and has theoretical broadband application potential, which will be further verified in future research.

A limitation of the current study is that the validation was performed at single frequencies (1 GHz and 4 GHz). Real-world EMI often spans broad frequency bands. The core logic of this method is frequency-agnostic; it can adapt to different frequencies by adjusting the frequency-dependent parameters of the Green’s function (a standard operation in broadband source reconstruction), and has theoretical broad-band application potential, which will be further verified in future research. Future work will focus on extending the dipole preprocessing framework to wideband source reconstruction scenarios. This may involve evaluating dipole contributions across a frequency range or developing frequency-adaptive elimination criteria. In this work, the initial dipole count N was set based on practical considerations of device size and scan resolution. The key contribution is the proposed preprocessing that efficiently screens this initial set to a dominant subset $N_p$. It is noted that deriving an optimal N directly from the DUT’s properties remains an open question. Formulating and automating this initial selection presents a valuable direction for future research, promising to further streamline the source reconstruction workflow. Therefore, another promising research direction is to establish guidelines or a methodology for determining the optimal dipole array size for different classes of DUTs (e.g., various PCB complexities, IC packages, or TSV configurations). This could involve systematic studies correlating the optimal dipole count with key device attributes, such as physical size, operating frequency, and estimated source sparsity, to move beyond empirical trial-and-error towards a more principled design of the SR model setup. Furthermore, exploring the integration of this preprocessing technique with other advanced SR architectures, such as deep convolutional neural networks, presents a promising direction for handling even more complex emission sources with higher efficiency.

5. Conclusions

This paper has presented a novel, contribution-driven dipole preprocessing technique designed to enhance the efficiency of neural network-based electromagnetic source reconstruction for EMC diagnostics. The core of the method is an evaluation-elimination mechanism that ranks equivalent dipoles based on their influence factor and removes redundant ones prior to the fine reconstruction stage. This significantly reduces the dimensionality of the inverse problem.

Comprehensive validation on a simulated multilayer PCB and a physically measured TSV-based chip confirms the method’s robustness and general applicability. The results demonstrate that the technique can reduce computational time from 20% to 61% compared to the standard approach, while limiting the increase in reconstruction error to a negligible margin (≤0.2% for PCB, ≤0.05% for the TSV chip). This successful balance between speed and accuracy makes the proposed method a highly practical tool for the rapid localization and quantification of EMI sources in both conventional PCBs and emerging high-density devices employing technologies like TSV. The proposed dipole preprocessing step is universal and can be integrated into various existing SR workflows to accelerate the diagnostic process for electronic systems.