I. Introduction
The advancement of industrialization, the Fourth Industrial Revolution, and the emerging artificial intelligence-driven industrial ecosystem have led to an ever-increasing dependence on information and electronic equipment across most national and private systems. This high dependency, however, poses a significant risk given that unexpected exposure to high-power electromagnetic (HPEM) waves can result in substantial damage.
HPEM waves can be generated intentionally for military, security, or terrorist purposes, or can occur unintentionally due to interactions within electronic systems or interference from electronic devices. Depending on their range and purpose, HPEM fields can have severe impacts on various systems, be it on a broad scale or within a localized area. To mitigate such potential damage, it is essential to analyze the electromagnetic vulnerability of critical electronic equipment and their operational environments so that appropriate protective measures can be developed [1–5].
Conventional electromagnetic analysis methods present numerous limitations when employed to examine critical electronic equipment and their operational environments. Typically, electronic equipment comprise an enclosure with various apertures that houses internal PCBs composed of complex circuits and components. Moreover, critical electronic systems are often located and operated in specific rooms inside buildings. As a result, analyzing the effects of externally generated electromagnetic waves on specific locations of internal PCBs requires substantial computational resources.
This high demand for resources also stems from the fact that most analysis methods rely on mesh discretization based on the maximum frequency of interest. Given this context, analyzing the effects of high frequencies, such as 10 GHz, using conventional computing power is infeasible. To overcome these computational limitations, research on resource-independent analysis techniques is essential [6, 7].
The core concept of resource-independent analysis techniques is partitioned analysis, which can be implemented in various ways. One approach involves physically subdividing the entire analysis model using conventional electromagnetic analysis techniques, while the other approach divides the model into its constituent elements, analyzes each element separately, and then recombines the results [8, 9].
A representative method for the recombination approach is the electromagnetic topology approach, which involves decomposing a complex system based on relevant electromagnetic phenomena, analyzing each subcomponent using different techniques, and then applying topological interrelationships to reconstruct the overall electromagnetic response. Notably, since this method allows for partitioning either physically or by components, it effectively addresses computational resource limitations. However, when the analysis target size approaches the building scale, the issue of increasing resource demand may still persist. In such a case, another resource-independent partitioned analysis method that can be applied is probabilistic electromagnetic analysis [14]. One example of a probabilistic approach is the power balance (PWB) method, which evaluates the power transfer and internal impact of external electromagnetic waves on large structures. This method statistically analyzes power losses due to apertures, internal walls, and objects within an enclosure using the Q-factor. By estimating electromagnetic losses as a function of frequency based on structural dimensions, PWB has emerged as an effective technique for assessing the characteristics of electromagnetic interference [15, 16]. Another probabilistic method for electromagnetic interference analysis is the random coupling method (RCM). RCM is based on random matrix theory (RMT), which is commonly used to analyze the statistical properties of large ensembles of matrices. When applied to electromagnetic coupling problems, this approach is referred to as electromagnetic chaos theory (ECM). RCM assumes a random wave environment to derive scattering matrices for modeling electromagnetic coupling within enclosures, and then computes ensemble-averaged eigenvalues to extract coupling characteristics [17]. However, while probabilistic methods are useful for estimating approximate coupled field strengths within enclosures, they have limitations. Since these methods rely on statistical averaging, they do not provide detailed spatial field distributions or phase information, which makes them less suitable for precise enclosure penetration analyses or frequency-dependent coupling evaluations.
This paper proposes a partitioned-recombination analysis method that incorporates the concept of Huygens’ box to simultaneously address the limitations of both electromagnetic simulation resource efficiency analysis and probabilistic spatial analysis in evaluating field strength and phase characteristics. The proposed method is applied to analyze electromagnetic wave coupling to an enclosure with apertures and an internal PCB. A simulation model representing an electronic device with an enclosure and an internal PCB, including apertures, is analyzed. Additionally, an office environment with windows is simulated to model the operational electromagnetic environment and utilized for further analysis.
The rest of this paper is structured as follows. Section II provides an overview of the fundamental concept of the Huygens’ box and its applications. Section III describes the partitioned analysis approach for the enclosure and the PCB, explains the recombination method based on topological analysis, and compares the simulation results. Section IV presents the partitioned analysis method and results for the enclosure model in a building. Finally, Section V concludes this study.
II. Concept and Application of Huygens’ Box
Huygens’ principle is a fundamental concept in electromagnetism that explains how waves, such as light or radio waves, propagate through space. As illustrated in Fig. 1, Huygens’ principle states that a wavefront reaching a particular point in space acts as a new source of secondary waves, which propagate while retaining the same characteristics as the original wavefront. In 1936, Schelkunoff [14] generalized this phenomenon as the electromagnetic equivalence principle, which has since been widely applied in various electromagnetic analyses. The Huygens’ box is constructed by dividing a homogeneous wavefront into six parts, forming a complete closed volume. It presents a method for analyzing the scattering and propagation of electromagnetic waves, allowing for the determination of electromagnetic phenomena either inside or outside a given closed space (box) based on the distribution of electromagnetic fields within that space. In particular, the current and charge distributions on the surface of the closed space are derived to reconstruct the surrounding field as an equivalent current source [18].
Since the analysis in this study targets electronic equipment located inside a building, the Huygens’ box is set as the enclosure around the equipment. The proposed method involves the installation of a virtual (Huygens’) box, with waveguide ports provided by the CST-Microwave Studio tool applied to each face of the box. Through this configuration, the scattering parameters (S-parameters) for all six faces of the box are computed. Additionally, to analyze the coupling of external electromagnetic waves to the internal PCB of an electronic device, a model simulating an enclosure consisting of a PCB is used, where the arbitrary cuboid space between the enclosure and the PCB is defined as the Huygens’ box. This space is modeled as a homogeneous electromagnetic wavefront.
The proposed analysis method is not a classical full-wave approach based on meshing. Instead, it calculates the S-parameters by considering each virtual surface as a waveguide port. Moreover, since the focus of this analysis is solely on the frequency-domain transmission characteristics, the proposed method requires less memory and computational time than classical approaches that use non-uniform meshing. However, since waveguide ports were employed, the accuracy decreased at frequencies below the cut-off frequency of the defined ports. Nevertheless, in practice, the physical dimensions of the target ports were significantly smaller than those of the defined waveguide ports. As a result, the frequency bands of interest from the target’s perspective were typically higher than the cut-off frequency of the assigned ports. Therefore, this limitation had a minimal impact on the overall frequency response. Subsequently, the interrelationships among the six wavefronts constituting each closed space are analyzed and modeled using the electromagnetic topology approach.
III. Partitioned Analysis Approach for Enclosure Model and Results
1. Partitioning of the Enclosure Model
Fig. 2 illustrates a model of the enclosure containing a PCB, simulating an electronic device. On one side of the enclosure is an aperture, while inside it is a simplified PCB layout. The enclosure is made of a perfect electric conductor (PEC) with dimensions of approximately 240 mm × 220 mm × 65 mm (width × length × height) and a thickness of 10 mm. Furthermore, the aperture size is approximately 27 mm × 18 mm. The PCB inside the enclosure is a microstrip line with a dielectric constant of 4.3, and a line width ensuring a central transmission frequency of 5 GHz. The dimensions of the dielectric are 10 mm × 10 mm × 0.05 mm, while those of the microstrip line are 1 mm × 10 mm × 0.03 mm.
The partitioning of this enclosure model was conducted in two stages. The first stage involved dividing the enclosure into six surfaces, each forming a Huygens’ box, as shown in Fig. 2. Meanwhile, the internal Huygens’ box, with a size of approximately 56 mm × 40 mm × 21.5 mm, was assumed to completely enclose the PCB. It was installed by maintaining a small gap from the physical edge of the internal PCB. The planewave incident from outside the enclosure was calculated as the electromagnetic field reaching all the intersection points of a fine uniform grid on the surface of the internal Huygens’ box by passing through the aperture. The electromagnetic field was analyzed using time-domain analysis based on the finite-difference time-domain (FDTD) method. Furthermore, as shown in Fig. 2, the second partitioning model consists of the same Huygens’ box as before, with the PCB inside the Huygens’ box. The coupling voltage at the 50 - Ω port installed at the terminal of the internal PCB was calculated again using the FDTD method by accounting for the electromagnetic field on the surfaces of the internal Huygens’ box.
2. Recombination of the Partitioned Analysis Results using Electromagnetic Topology Analysis
The transmission characteristics of the two partitioned models in Fig. 2 were recombined using electromagnetic topology analysis. The model for the topology analysis is shown in Fig. 3, where an arbitrary external point on the left side is defined as Junction 1 (J1), based on which the six surfaces of the Huygens’ box are defined as J2 to J7. Furthermore, the coupling analysis of the PCB’s terminal load is conducted based on the six surfaces of the Huygens’ box using six tubes and Junction 8 (J8).
The defined topology model was analyzed using the Baum-Liu-Tesche (BLT) equation [12, 19], which is formulated in terms of the propagation supermatrix (PSM), the scattering super matrix (SSM), and the source and result supervectors. The BLT equation can be expressed as follows:
where VS denotes the voltage source supervector, and U is the unit matrix. Furthermore, P and S stand for PSM and SSM, respectively. In the equation, the supermatrices and supervectors are denoted using brackets {} and [], respectively. The desired output voltage supervector V of the target location was obtained using the BLT equation.
PSM and SSM refer to the propagation and scattering matrices that reflect the overall characteristics of the entire network structure based on the given topology model, which is depicted in Fig. 3. Notably, the propagation matrix represents the transmission characteristics between pairs of junctions. In this study, only the transmission characteristics of the external signal source were considered, while the reverse characteristics were set to zero. Moreover, given that we assumed an arbitrary virtual Huygens’ surface in space, the reflection characteristics were also set to zero. Therefore, the characteristics of PSM and SSM can be formulated as follows:
(2)
(3)
The electric field distributed on the surface of a Huygens’ box represents the relative transmission characteristics of the external signal source across the sampled frequencies, assuming that the source is located at an infinite point. Notably, since the electric field is proportional to the applied voltage, the input and output of the BLT equations are defined in terms of voltage, and the transmission characteristics are expressed in terms of S-parameters. Furthermore, the subscripts of the voltage are defined in terms of the tube number and junction number. The supermatrices in Eqs. (2) and (3) are defined for the frequency range from 10 MHz to 10 GHz, with the input supervector bearing a value of 1 at the external infinite point.
The coupling analysis results are presented in Fig. 4, where the red solid line indicates the terminal voltage obtained through full-wave simulation, and the blue dashed line denotes the partitioned-recombined analysis results. It can be observed that despite the recombined analysis results showing some fluctuation in the frequency range below 2 GHz, the overall results are quite similar. Notably, the mean absolute percentage error (MAPE) between the two results over the entire frequency range was calculated to be 4.64%. In terms of computational resources, the full model required 627 seconds of computation time and 1,237,179 kB of memory. In contrast, the proposed method required only 448 seconds of computation time—solving the BLT equation in 4.47 seconds—and 880,347 kB of memory, demonstrating a reduction in computational time and memory usage by 28.6% and 28.8%, respectively. These results confirm that the proposed method allows for a significant reduction in resource requirements (Table 1).
IV. Partitioned Analysis Method and Results for the Enclosure Model in a Building
1. Partitioned Analysis Method for the Case of Electronic Equipment Placed Inside a Building
In general, most electronic equipment are operated within large enclosures, such as in aircraft or naval vessels, or placed inside buildings for operation. Therefore, to simulate a realistic scenario, the given electronic equipment was assumed to be located in a room within a building. The room in question contained a window. Similar to the previous analysis model, the electronic equipment featured a small enclosure with an internal PCB. In this case, when electromagnetic waves from outside the building entered the room through a window, the induced voltage on the internal PCB of the electronic equipment was calculated.
As shown in Fig. 5, the dimensions of the room considered for this study are 3,400 mm × 7,800 mm × 2,600 mm, with the wall thickness being 100 mm. On one side of the room is a window measuring 2,700 mm × 1,700 mm. The dimensions of the enclosure and PCB are the same as those shown in Fig. 2. The aperture size of the enclosure containing the PCB was 25 mm × 10 mm. The electronic equipment was placed 6,000 mm from the window. Notably, based on the physical size of the target and the specifications of the analysis computer, the analysis frequency range was set from 0.1 to 7 GHz. The specifications of the computer used for the analysis are an Intel Xeon 1.90 GHz CPU with 256 GB RAM and an NVIDIA RTX A4000 graphics card.
Fig. 6 illustrates the strategy for conducting Huygens’ box-based partitioning to analyze the coupling and propagation environment of the electronic equipment located inside a room in a building. In this context, two main cases were considered. In the first case, a single Huygens’ box was used to surround the enclosure inside the room. With regard to the partitioning of the given model, it was advantageous to partition the electrically large and small sections. Notably, when performing electromagnetic analysis based on the FDTD method, mesh settings are crucial, because the size of the electrical model becomes an important factor and greatly influences the accuracy of the analysis results. In this study, since the modeled enclosure and PCB occupied a relatively small volume compared to the overall room size, setting such a partitioning strategy proved advantageous. Meanwhile, in the second case, another Huygens’ box was added to surround the PCB, as analyzed in Section II, with the existing enclosure already enclosed within a Huygens’ box, thus creating a two-stage Huygens’ box configuration. Since the smallest physical dimension in the entire analysis model was the width of the PCB traces, an additional Huygens’ box was set up to partition this section.
The electromagnetic topology network model for the first analysis model exhibited the structure depicted in Fig. 3, where J1 is the infinite point outside the building, J2–J7 surround the enclosure, and J8 is the termination load of the PCB. The electromagnetic topology model for the second analysis model—the 2-stage Huygens’ box—is shown in Fig. 7. In this case, a second Huygens’ box surrounds the PCB, as shown in Fig. 2, in addition to the 1-stage analysis model. Since two stages of Huygens’ surfaces are formed, all junctions could be sufficiently interconnected. The final termination, J14, denotes the load of the PCB. Notably, both electromagnetic topology models were designed to have the same PSM and SSM structures as defined in Eqs. (2) and (3), respectively.
2. 1-Stage Huygens’ Box Analysis Results
Simulations were performed for each partitioned model using the 1-stage Huygens’ box, and coupling analysis was conducted using the described topology model. First, an analysis of the characteristics of the electromagnetic wave propagated to the Huygens’ box from outside the building was conducted. The size of the installed Huygens’ Box 1 was 260 mm × 240 mm × 85 mm, and it was set 10 mm away from the outer boundary of the enclosure. The planewave used as the external signal source was set to 1 V/m, incident perpendicularly on the window, with horizontal polarization. The analysis time for this step was 17,481 seconds, the maximum number of cells used was 994,576,968, and the memory usage was 97,950,204 kB.
In the second step, the room structure was removed, with only the enclosure and internal PCB left for further analysis, and the voltage induced at the PCB termination in the Huygens’ box was observed. This process required 424 seconds of computation time. Furthermore, the maximum number of cells used for this step was 1,911,392, and the memory used was 4,940,264 kB. Therefore, the total analysis time for the 1-stage Huygens’ box was 17,905 seconds (approximately 5 hours), with the maximum number of cells being 994,576,968 and memory usage being 102,890,468 kB, as shown in Table 2.
For comparison, the response of the entire model was obtained by conducting a full-wave simulation through CST’s finite integration technique (FIT). In this process, the analysis time was 80 hours, 26 minutes, and 43 seconds, with the maximum number of cells being 1,255,703,040 and the memory used being approximately 123,657,220 kB. Notably, for verification of the Huygens’ box-based electromagnetic topology analysis method, which is a key issue in this study, it was judged that comparison with the full-wave simulation would be sufficient rather than actual experimental validation.
The electromagnetic topology analysis results are shown in Fig. 8(a), confirming that the results of the full model analysis and the proposed 1-stage coupling analysis model are very similar. The MAPE between the two results over the entire frequency range of Fig. 8(a) was calculated to be 3.83%. Despite a slight difference in the frequency range below 2 GHz, the frequency peaks and voltage levels are largely similar in shape. Effectively, the effectiveness of the partitioned analysis method using the Huygens’ box and the recombination analysis approach based on an electromagnetic topology model was sufficiently verified.
3. 2-Stage Huygens’ Box Analysis Results
When using the 2-stage Huygens’ box for analysis, the model was divided into three stages. The first stage involved creating Huygens’ Box 1, which is the same as the box used when only one Huygens’ box was applied. In the second stage, Huygens’ Box 2 was implemented. The size of Huygens’ Box 2, which surrounds the PCB, was set to 114 mm × 114 mm × 14.09 mm, maintaining a 7-mm gap at each end of the PCB terminals. The simulation for Huygens’ Box 2 took 25 seconds, with the maximum cell count being 319,332 and the memory usage being 1,178,064 kB. In the third stage, the voltage induced on the PCB’s terminal load by the newly created Huygens’ Box 2 was examined. This process was executed in 276 seconds, with a maximum cell count of 499,720 and memory usage of 1,830,196 kB. Therefore, the total analysis time when using the 2-stage Huygens’ Box was 17,782 seconds, with a maximum cell count of 944,576,968 and memory usage of 100,958,464 kB.
A comparison of the electromagnetic topology analysis results with those of the full-wave simulation is presented in Fig. 8(b), showing close agreement. The MAPE between the two results over the entire frequency range in Fig. 8(b) was calculated to be 4.21%. While there are slight voltage-level differences at certain frequency points, the similarity observed is sufficient to verify the validity of the proposed method.
Table 2 summarizes the resources required for the simulations. When using the 1-stage Huygens’ box, the analysis time was reduced by 93.82%, from 289,603 seconds to 17,905 seconds, compared to the full-wave simulation. The maximum cell count also decreased by 20.8%, and memory usage was reduced by 16.79%. Furthermore, when using the 2-stage Huygens’ box, the analysis time was shortened by 93.86% to 17,782 seconds, while the maximum cell count and memory usage also decreased by 20.73% and 18.36%, respectively.
V. Conclusion
This paper proposes a coupling analysis method for external electromagnetic waves using an electronic equipment simulation model, featuring an enclosure with an aperture and an internal PCB. To reduce analysis resources requirements, a partitioned analysis technique incorporating the concept of Huygens’ box was implemented, and the electromagnetic topology analysis method was employed to combine the results. First, the PCB was separated from the enclosure, and separate analyses were performed for each part of the simulation model. Subsequently, the analysis results were recombined, and then compared against those of the complete model. Overall, approximately 28% of resource savings were achieved. Furthermore, to simulate the propagation environment, a scenario in which the electronic equipment was located within a building was modeled and analyzed. For the analysis, 1-stage and 2-stage Huygens’ boxes and electromagnetic topology analysis methods were implemented. The results were compared with those obtained by conducting full-wave simulations, exhibiting strong agreement across most frequency bands. Additionally, the analysis time, maximum cell count, and memory usage were compared. In comparison to the full-wave simulation, a general reduction in the need for analysis resources was observed. It was found that most of the analysis resources were required to analyze the room, while relatively fewer resources were needed to analyze the housing and PCB, resulting in minimal resource differences. In conclusion, using just one Huygens’ box was sufficient to achieve effective analysis results.
Overall, this method can be widely applied in propagation environments where complex equipment are operated and in models that include external enclosures and internal PCBs, making it an effective and highly reliable approach for analyzing a broad range of frequency bands.
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