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Showing papers in "IEEE Antennas and Propagation Magazine in 2023"


DOI
TL;DR: In this article , the authors describe the technical jargon and concepts from physics that are unfamiliar to a classically trained electromagnetics engineer, as a result, this work is often difficult for engineers to become engaged in.
Abstract: One of the most popular approaches being pursued to achieve a quantum advantage with practical hardware is the use of superconducting circuit devices. Although significant progress has been made over the previous two decades, substantial engineering efforts are required to scale these devices so they can be used to solve many problems of interest. Unfortunately, much of this exciting field is described using technical jargon and concepts from physics that are unfamiliar to a classically trained electromagnetics engineer. As a result, this work is often difficult for engineers to become engaged in.

7 citations


Journal ArticleDOI
TL;DR: In this article , the authors revisited the reasons behind the cross-polarized (XP) radiation in microstrip patches and focused on alleviating a long-standing deficiency in terms of a comprehensive knowledge about the same.
Abstract: This article revisits the reasons behind the cross-polarized (XP) radiation in microstrip patches and focuses on alleviating a long-standing deficiency in terms of a comprehensive knowledge about the same. This study has a twofold objective. First, it explores all possible components of the surface fields as the correlating factors over all of the radiation planes, especially across the diagonal or skewed axes. An extensive study involving varied patch geometries along with different feed networks has been executed. The resulting huge volume of data has been analyzed to identify the ideal surface field requirements for the best possible XP performance.

5 citations


DOI
TL;DR: In this paper , the authors revisited the reasons behind the cross-polarized (XP) radiation in microstrip patches and focused on alleviating a long-standing deficiency in terms of a comprehensive knowledge about the same.
Abstract: This article revisits the reasons behind the cross-polarized (XP) radiation in microstrip patches and focuses on alleviating a long-standing deficiency in terms of a comprehensive knowledge about the same. This study has a twofold objective. First, it explores all possible components of the surface fields as the correlating factors over all of the radiation planes, especially across the diagonal or skewed axes. An extensive study involving varied patch geometries along with different feed networks has been executed. The resulting huge volume of data has been analyzed to identify the ideal surface field requirements for the best possible XP performance.

4 citations


DOI
TL;DR: In this article , a recurrent convolutional neural network (RCNN) is designed for full-wave electromagnetic (EM) modeling, which is equivalent to the finite difference time domain (FDTD) method.
Abstract: In this study, a recurrent convolutional neural network (RCNN) is designed for full-wave electromagnetic (EM) modeling. This network is equivalent to the finite difference time domain (FDTD) method. The convolutional kernel can describe the finite difference operator, and the recurrent neural network (RNN) provides a framework for the time-marching scheme in FDTD. The network weights are derived from the FDTD formulation, and the training process is not needed. Therefore, this FDTD-RCNN can rigorously solve a given EM modeling problem as an FDTD solver does.

4 citations


Journal ArticleDOI
TL;DR: In this paper , the authors present an overview of the 5G rectifying antenna and its primary elements for applications in millimeter-wave (mmWave) energy harvesting (EH) and wireless power transmission (WPT).
Abstract: In this article, we present an overview of the 5G rectifying antenna and its primary elements for applications in millimeter-wave (mm-wave) energy harvesting (EH) and wireless power transmission (WPT). The wide spectrum available for 5G communication bands have attracted significant attention for extensive applications. The power received by the harvesting antenna relies on the size of the antenna. Hence, the realization of antenna and rectenna systems with good efficiency at 5G mm-wave is a challenge. This review article highlights the recent advances in 5G rectenna systems for different applications at the component and structure levels. The primary objectives of the article are 1) to explore the potential advances of mm-wave rectenna systems and the feasibility of their designs to attain desired characteristics and 2) to present a comparative assessment of performance parameters of existing rectenna systems.

3 citations



DOI
TL;DR: In this paper , the role of conduction current caused by a dielectric resonator antenna (DRA) on its metallic ground plane (GP) in influencing its radiation and also in predicting accurate directivity patterns was addressed.
Abstract: This article addresses the role of conduction current, caused by a dielectric resonator antenna (DRA) on its metallic ground plane (GP), in influencing its radiation and also in predicting accurate directivity patterns. Extraction of the surface current and its translation to the far field have been systematically studied using different methods. In contrast to what was reported earlier in [6], the conduction current alone is eventually found insufficient to predict the radiation successfully. An additional requirement of the passive DRA block to compensate fields over the front hemisphere has been identified, and an effective method of computation has been demonstrated. Measured and simulated data have been used as references to validate the method. A few examples have been furnished using different DRA shapes, feeding structures, and GP shapes with a special focus on different modes and polarizations. The provided method of computation will be helpful in understanding the physics of radiation from a DRA and interpreting its behavior under special conditions. This method shows that, by controlling the GP current and without disturbing the DRA resonance, the radiation patterns can be easily reconfigured.

2 citations


Journal ArticleDOI
TL;DR: In this article , the authors present a survey of state-of-the-art low-frequency ESAs, including passive resonant, tunable, multiband, and actively matched designs.
Abstract: Robust, secure, low-power wireless communications and networking are needed to support emerging applications in emergency and disaster response, robotics, autonomous vehicles, and the Internet of Things (IoT) in physically complex environments, such as indoor/outdoor, dense urban, and other challenging scenarios. These applications require a variety of wireless communications modalities, including high and very-high frequency (HF and VHF) bands. To effectively deploy compact wireless communications systems at these low frequencies, high-performance electrically small antennas (ESAs) are needed. In this article, we present a survey of state-of-the-art low-frequency ESAs, including passive resonant, tunable, multiband, and actively matched designs. While passive designs are subject to classic tradeoffs in size and efficiency, active and platform-integrated techniques offer potential performance-enhancing alternatives. And, though single ESAs tend to omnidirectional beam patterns, multiantenna arrays can achieve directionality, and designs include parasitic, biomimetic, and distributed approaches.

2 citations


Journal ArticleDOI
TL;DR: In this article , the achievable communication ranges of such links are theoretically and experimentally evaluated by considering the possible positions of the helmet wearer (standing or lying) on a flat field, representing a simple search and rescue (SaR) scenario.
Abstract: The use of the LoRa communication protocol in a new generation of transceivers is attractive for search and rescue(SaR) procedures because they can operate in harsh environmentscovering vast areas while maintaining a low power consumption.The possibility of wearing helmets equipped with LoRa-radiosand installing LoRa transceivers in unmanned aerial vehicles (UAVs) will accelerate the localization of the targets, probably unconscious. In this paper, the achievable communication ranges of such links are theoretically and experimentally evaluated by considering the possible positions of the helmet wearer (standing or lying) on a flat field, representing a simple SaR scenario.Simulations and experimental tests demonstrated that, for the standing position, the ground-bounce multi-path produces strong fluctuations of the received power versus the Tx-Rx distances. Such fluctuations can be kept confined within 100 m from the target by lowering the UAV altitude. Instead, for a more critical lying position, the received power profile is monotonic and nearly insensitive to the posture. For all the considered cases, the signal emitted by the body-worn transceiver can be exploited to localize the helmet wearer based on its strength, and it is theoretically detectable by the UAV radio up to 5 km on flat terrain.

2 citations


Peer ReviewDOI
TL;DR: A review article of the latest advances in 3D printing for enabling new materials and new geometries for radio-frequency (RF) devices, antennas, and metamaterials is presented in this paper .
Abstract: This is a review article of the latest advances in 3D printing for enabling new materials and new geometries for radio-frequency (RF) devices, antennas, and metamaterials. The article discusses the achievable material properties and various optimized applications that are achievable by creating new shapes in either dielectric or metal. This article demonstrates what is currently possible with additive manufacturing and the current limitations. Various additively manufactured RF devices are reviewed.

2 citations


Journal ArticleDOI
TL;DR: In this paper , a recurrent convolutional neural network (RCNN) is designed for full-wave electromagnetic (EM) modeling, which is equivalent to the finite difference time domain (FDTD) method.
Abstract: In this study, a recurrent convolutional neural network (RCNN) is designed for full-wave electromagnetic (EM) modeling. This network is equivalent to the finite difference time domain (FDTD) method. The convolutional kernel can describe the finite difference operator, and the recurrent neural network (RNN) provides a framework for the time-marching scheme in FDTD. The network weights are derived from the FDTD formulation, and the training process is not needed. Therefore, this FDTD-RCNN can rigorously solve a given EM modeling problem as an FDTD solver does.

Peer ReviewDOI
TL;DR: In this article , the authors present a survey of state-of-the-art low-frequency ESAs, including passive resonant, tunable, multiband, and actively matched designs.
Abstract: Robust, secure, low-power wireless communications and networking are needed to support emerging applications in emergency and disaster response, robotics, autonomous vehicles, and the Internet of Things (IoT) in physically complex environments, such as indoor/outdoor, dense urban, and other challenging scenarios. These applications require a variety of wireless communications modalities, including high and very-high frequency (HF and VHF) bands. To effectively deploy compact wireless communications systems at these low frequencies, high-performance electrically small antennas (ESAs) are needed. In this article, we present a survey of state-of-the-art low-frequency ESAs, including passive resonant, tunable, multiband, and actively matched designs. While passive designs are subject to classic tradeoffs in size and efficiency, active and platform-integrated techniques offer potential performance-enhancing alternatives. And, though single ESAs tend to omnidirectional beam patterns, multiantenna arrays can achieve directionality, and designs include parasitic, biomimetic, and distributed approaches.

Journal ArticleDOI
TL;DR: In this paper , a foldable series-fed patch array designed on a strictly rigid printed circuit board (PCB) is presented, which does not require any external mechanical folding appendages (i.e., brackets, tapes, or offsets).
Abstract: Combining antenna arrays with physical reconfigurability (i.e., origami) allows for additional degrees of freedom in operation and enables larger structures to be folded into smaller volumes. The packaging ability of origami antennas is of great interest for CubeSat applications in particular. However, traditional origami is based on the manipulation of thin sheets, making physical reconfigurability with thick substrates very difficult. In this article, we present a foldable series-fed patch array designed on a strictly rigid printed circuit board (PCB). Notably, the array does not require any external mechanical folding appendages (i.e., brackets, tapes, or offsets), and it can be manufactured using commercially available PCBs and equipment. The PCB itself is strategically cut to form lamina emergent torsion (LET) joints. When fully deployed, the array operates at 5.7 GHz. A significant shift is not seen in the operational frequency until the folding exceeds 60° (5.7–5.61 GHz). This shows that the array, even in a state of near extreme faulty deployment, will operate as intended. An 8 × 6 prototype array was fabricated using a Rogers DiClad 880. When fully deployed, the array is extended in a surface area of 280 × 198 × 1.524 mm and can be folded into a 35.5 × 198 × 12.5-mm compartment. The array held its integrity well after 100 cycles. (One cycle starts at a state of full deployment, moves to a state of being fully packaged, and goes back to a state of full deployment.) Measurements show that the active reflection coefficients are in good agreement with the finite array simulations. The measured realized gains at 5.7 GHz for 0° and 30° folds were 22.4 and 21.4 dBi, respectively. At the 60° fold, a realized gain of 20.6 dBi was achieved at the frequency of 5.61 GHz.

Peer ReviewDOI
TL;DR: In this paper , the authors present an overview of the 5G rectifying antenna and its primary elements for applications in millimeter-wave (mmWave) energy harvesting (EH) and wireless power transmission (WPT).
Abstract: In this article, we present an overview of the 5G rectifying antenna and its primary elements for applications in millimeter-wave (mm-wave) energy harvesting (EH) and wireless power transmission (WPT). The wide spectrum available for 5G communication bands have attracted significant attention for extensive applications. The power received by the harvesting antenna relies on the size of the antenna. Hence, the realization of antenna and rectenna systems with good efficiency at 5G mm-wave is a challenge. This review article highlights the recent advances in 5G rectenna systems for different applications at the component and structure levels. The primary objectives of the article are 1) to explore the potential advances of mm-wave rectenna systems and the feasibility of their designs to attain desired characteristics and 2) to present a comparative assessment of performance parameters of existing rectenna systems.

Journal ArticleDOI
TL;DR: An overview of dielectric resonator (DR)-based sensing elements and their applications in RF energy-harvesting (RFEH) and wireless power transmission (WPT) systems is presented in this paper .
Abstract: This article presents an overview of dielectric resonator (DR)-based sensing elements and their applications in RF energy-harvesting (RFEH) and wireless power transmission (WPT) systems. With increased wireless applications, the demand for electrical energy goes up, thereby enabling the development of various energy sources. RF energy is widely available and the most efficient energy source. Although DR antennas (DRAs) have been studied extensively in the last few decades, they have not been employed in RFEH and WPT applications. The intention of the proposed article is 1) to provide an overview of the DRA for RFEH and WPT applications; 2) to accommodate various performance enhancement approaches for the DRA; and 3) to highlight the research gap for developing a complete rectenna system that helps future researchers. We believe that this survey may help the DRA.

Journal ArticleDOI
TL;DR: In this article , the general theory of linear time-varying (LTV) systems and the means to properly account for frequency dispersion of nonstationary systems are discussed.
Abstract: During the last decade, possibilities to realize new phenomena and create new applications by varying system properties in time have gained increasing attention in many research fields. Although the interest in using time-modulation techniques for engineering electromagnetic (EM) response has become revitalized only in recent years, the field originates from the middle of the previous century, and a multitude of works have been published ever since. In this tutorial article, we provide a historical picture and review the basic concepts in this field. In particular, we introduce the general theory of linear time-varying (LTV) systems and discuss the means to properly account for frequency dispersion of nonstationary systems. Also, we elucidate models of time-varying electrical circuits and materials and discuss some useful effects that can be achieved by time modulation of circuit or material parameters.

DOI
TL;DR: In this paper , Artificial Neural Networks (ANNs) have been applied to build a surrogate model for the computation-intensive FDTD simulation and to bypass the numerous simulations required for UQ.
Abstract: Artificial neural networks (ANNs) have appeared as a potential alternative for uncertainty quantification (UQ) in the finite difference time-domain (FDTD) computation. They are applied to build a surrogate model for the computation-intensive FDTD simulation and to bypass the numerous simulations required for UQ. However, when the surrogate model utilizes an ANN, a considerable number of data are generally required for high accuracy, and generating such large quantities of data becomes computationally prohibitive. To address this drawback, a number of adaptations for ANNs are proposed, which additionally improves the accuracy of ANNs in UQ for the FDTD computation while maintaining a low computational cost. The proposed algorithm is tested for application in bioelectromagnetics, and considerable speed up, as well as the improved accuracy of UQ, is observed compared to traditional methods such as the nonintrusive polynomial chaos (NIPC) method.

DOI
TL;DR: In this paper , the effect that snow accumulation may have on the radome is analyzed in terms of de-pointing of the main beam, with a particular focus on the difference of depointing between the sum and delta modes, and a real-time, effective, solution is not yet available.
Abstract: Ground stations working at high frequencies, such as the K-band, installed at polar latitudes and protected by radomes, are increasingly used to support modern satellites. The effect that snow accumulation may have on the radome is consequently important, as at these frequencies it may jeopardize the satellite link. This article analyses an operative case, referred to as SNOWBEAR (Svalbard grouND StatiOn for Wide Band Earth observation dAta Reception), where a 6.4-m antenna has been installed at Svalbard, Norway, to track an Earth observation (EO) satellite, NOAA-20, for a period of two years. We demonstrate, using experimental data and numerical models, that a chief effect can be described in terms of de-pointing of the main beam, with a particular focus on the difference of de-pointing between the sum and delta modes, and that a real-time, effective, solution is not yet available.


DOI
TL;DR: In this article , the advantages and disadvantages of different manufacturing techniques for the millimeter band were evaluated using features such as manufacturing tolerances, accuracy, surface roughness, and cost, bearing in mind both the performance of the different manufacturing methods and the particular circuital topologies of the component under development.
Abstract: This article analyzes some of the key aspects of manufacturing techniques for passive components and antennas in the millimeter band, with emphasis on the W band. Even experienced microwave designers might face great technological challenges when upscaling their designs to higher-frequency bands. Thus, several manufacturing technologies are widely analyzed in this work by designing, manufacturing, and measuring different components in the W band. Subtractive manufacturing techniques, such as computer numerical control (CNC) milling and electrical discharge machining (EDM), as well as additive manufacturing techniques, including stereolithography (SLA), direct metal laser sintering (DMLS), and diffusion bonding (DB) are explored. The advantages and disadvantages of these technologies are evaluated using features such as manufacturing tolerances, accuracy, surface roughness, and cost. The purpose is to offer some guidelines for the selection of the most appropriate manufacturing technology in the W band, bearing in mind both the performance of the different manufacturing methods and the particular circuital topologies of the component under development.

DOI
TL;DR: In this article , a foldable series-fed patch array designed on a strictly rigid printed circuit board (PCB) is presented, which does not require any external mechanical folding appendages (i.e., brackets, tapes, or offsets).
Abstract: Combining antenna arrays with physical reconfigurability (i.e., origami) allows for additional degrees of freedom in operation and enables larger structures to be folded into smaller volumes. The packaging ability of origami antennas is of great interest for CubeSat applications in particular. However, traditional origami is based on the manipulation of thin sheets, making physical reconfigurability with thick substrates very difficult. In this article, we present a foldable series-fed patch array designed on a strictly rigid printed circuit board (PCB). Notably, the array does not require any external mechanical folding appendages (i.e., brackets, tapes, or offsets), and it can be manufactured using commercially available PCBs and equipment. The PCB itself is strategically cut to form lamina emergent torsion (LET) joints. When fully deployed, the array operates at 5.7 GHz. A significant shift is not seen in the operational frequency until the folding exceeds 60° (5.7–5.61 GHz). This shows that the array, even in a state of near extreme faulty deployment, will operate as intended. An 8 × 6 prototype array was fabricated using a Rogers DiClad 880. When fully deployed, the array is extended in a surface area of 280 × 198 × 1.524 mm and can be folded into a 35.5 × 198 × 12.5-mm compartment. The array held its integrity well after 100 cycles. (One cycle starts at a state of full deployment, moves to a state of being fully packaged, and goes back to a state of full deployment.) Measurements show that the active reflection coefficients are in good agreement with the finite array simulations. The measured realized gains at 5.7 GHz for 0° and 30° folds were 22.4 and 21.4 dBi, respectively. At the 60° fold, a realized gain of 20.6 dBi was achieved at the frequency of 5.61 GHz.

Journal ArticleDOI
TL;DR: In this article , the expression x"r defines a constant function of rank r that yields the constant value x, and the specific constant functions mentioned can therefore be written alternatively as 0"_ and 9"_, etc.
Abstract: Other constant functions include _9: and _8: etc. to 9:. More generally, the expression x"r defines a constant function of rank r that yields the constant value x. The specific constant functions mentioned can therefore be written alternatively as _"_ and _9"_ and 0"_ and 9"_, etc.

Journal ArticleDOI
TL;DR: In this paper , the authors discuss the dualities of wave propagation across spatial and temporal interfaces, by recasting the telegrapher equations in a linear time-variant (LTV) transmission line (TL).
Abstract: Recent years have witnessed a surge of interest in exotic electromagnetic (EM) wave propagation in time-varying systems. An interesting concept is the one of a temporal interface, the time-analogue of a spatial interface, formed by an abrupt and uniform change of the EM properties of the host medium in time. A time interface scatters the incident wave in a dual fashion compared to its spatial counterpart, conserving momentum but transforming frequency and exchanging energy with the wave. This article provides an overview of the wave-scattering features induced by time interfaces from an antennas and propagation engineering perspective. We first discuss the dualities of wave propagation across spatial and temporal interfaces, by recasting the telegrapher equations in a linear time-variant (LTV) transmission line (TL). Then, we introduce the scattering matrix to describe temporal scattering processes, and derive the conditions for reciprocity and momentum conservation. Understanding temporal scattering through the lens of conventional microwave engineering theory may facilitate the analysis and implementation of next-generation metamaterials encompassing temporal degrees of freedom.

Journal ArticleDOI
TL;DR: In this article , the effect of snow accumulation on the radome of a ground station working at high frequencies is analyzed, where a 6.4m antenna has been installed at Svalbard, Norway, to track an Earth observation (EO) satellite, NOAA-20, for a period of two years.
Abstract: Ground stations working at high frequencies, such as the K-band, installed at polar latitudes and protected by radomes, are increasingly used to support modern satellites. The effect that snow accumulation may have on the radome is consequently important, as at these frequencies it may jeopardize the satellite link. This article analyses an operative case, referred to as SNOWBEAR (Svalbard grouND StatiOn for Wide Band Earth observation dAta Reception), where a 6.4-m antenna has been installed at Svalbard, Norway, to track an Earth observation (EO) satellite, NOAA-20, for a period of two years. We demonstrate, using experimental data and numerical models, that a chief effect can be described in terms of de-pointing of the main beam, with a particular focus on the difference of de-pointing between the sum and delta modes, and that a real-time, effective, solution is not yet available.

Journal ArticleDOI
TL;DR: The IEEE Antennas and Propagation Society (AP-S) Young Professional (YP) Committee started a new initiative, the AP-S YP Ambassador Program, which is now in its second year as discussed by the authors .
Abstract: In 2021, the IEEE Antennas and Propagation Society (AP-S) Young Professional (YP) Committee started a new initiative, the AP-S YP Ambassador Program, which is now in its second year. The objective of this program is to inspire and inform AP-S YPs on a variety of topics both technical and nontechnical to enhance their interest and engagement in the field of antennas and propagation by delivering talks at various Chapters/Sections “on-demand” virtually in addition to getting involved in various AP-S YP activities.

Journal ArticleDOI
TL;DR: In this paper , a design-focused project greatly influenced my decision to pursue a career in engineering, and I only recently started appreciating the link between my experience at science fairs and my present passion for research, writing, and public speaking.
Abstract: Participating in local science fairs was a highlight of my school years, teaching me about scientific investigation and discovery from a young age. A design-focused project greatly influenced my decision to pursue a career in engineering, and I only recently started appreciating the link between my experience at science fairs and my present passion for research, writing, and public speaking.

Journal ArticleDOI
TL;DR: In this paper , an elaborate figure of merit (FOM) is introduced based on the radiation characteristics of antennas and characteristic modes (CMs) are employed to provide another representation of this FOM.
Abstract: The performance of mobile phone antennas is strongly dependent on their environment. However, in the literature, there is no standard criterion to judge this environmental sensitivity. In this work, first, an elaborate figure of merit (FOM) is introduced based on the radiation characteristics of antennas. Then, characteristic modes (CMs) are employed to provide another representation of this FOM. This technique is able to deliver an in-depth understanding of the effect of the environment on the radiating modes of an antenna, which are strongly related to the sensitivity. This perception opens the possibility for designers to achieve less-sensitive antennas. To demonstrate the applicability of the proposed FOM, it is applied to a range of mobile phone antennas available in the literature.

Journal ArticleDOI
TL;DR: In this paper , an overview of state-of-the-art error vector magnitude (EVM) simulation and measurement methods applied to antennas and phased arrays in this new era of millimeter-wave communications is presented.
Abstract: The error vector magnitude (EVM) is a comprehensive measure of the amplitude and phase distortion of digitally modulated signals and a potential tool for characterizing the performance of active beamforming arrays, especially at millimeter-wave frequencies, because of the high level of integration and limited spacing between the radiating elements. A review of the EVM measurement process and fundamental principles for time-domain demodulation with a vector signal analyzer (VSA) and spectral correlation with a vector network analyzer (VNA) was presented in [1]. This article demonstrates how researchers have modeled and measured impairments from antenna and phased array signal transmission to characterize their influence on the EVM of single and multicarrier digital signals. It presents examples of simulation models and measurements of the distortion from ultrawideband (UWB) passive antenna transmission, power amplifier (PA) nonlinearities, and phased array beam squint and beam scan intersymbol interference (ISI). A brief survey of over-the-air (OTA) test ranges for EVM analysis is also provided. This overview of state-of-the-art EVM simulation and measurement methods is intended to help interested readers understand and enhance the EVM performance analysis methods applied to antennas and phased arrays in this new era of millimeter-wave communications.


Journal ArticleDOI
TL;DR: In this article , the authors used the case of a circular 2D wire cage to illustrate how the residual field inside such a cage can be visualized in terms of a spatial spectral consideration of the induced charge.
Abstract: The static shielding properties of Faraday wire cages are intuitive as for a small mesh size, the effect on the field can be expected to approach that of an ideal Faraday cage, i.e., a closed conductive surface. However, as it has been recently pointed out, the shielding efficiency is somewhat worse than one might expect and does not particularly conform to the simple approximation of an exponentially decaying field, as it is, e.g., described in The Feynman Lectures on Physics. In the present contribution, we use the case of a circular 2D wire cage to illustrate how the residual field inside such a cage can be visualized in terms of a spatial spectral consideration of the induced charge. It is shown how the residual field in the cage’s center is related to a single Fourier coefficient of this spectral expansion, and that the approximation of the induced charge as a sampled version of the induced charge of a corresponding ideal Faraday cage yields useful approximations for the residual fields close to the cage boundary. The latter also turn out to justify the exponential decay approximation, at least in this region.