In this report, we investigate the effects of conductor losses in a multipole-based cable braid magnetic penetration model. Our multipole model uses a mesh of the actual cable geometry, which enables us to model more complicated structures. After summarizing the first principles model formulation, we consider a one-dimensional array of wires, for which an analytical solution is known in the lossless case. We extend this solution to the lossy case by using a complex-valued radius. We also model this structure analytically using a conformal-mapping solution. We then compare both the self-impedance and the transfer impedance results from our first principles cable braid electromagnetic penetration model to those obtained using the analytical solutions. An analysis for various frequencies (and skin depths) usually encountered in cable modeling is reported. These results are found in good agreement up to a radius to half spacing ratio of about 0.7, demonstrating a robustness needed for many commercial and non-commercial cables.
In this paper, we employ our first principles multipole-based cable braid electromagnetic penetration model to evaluate the transfer inductance of cables exhibiting meandering wires. We concentrate on a cable structure with two wires, and consider the dependence on the transfer inductance as a function of braid angle and amplitude of the meandering. We compare the results from the first principles model to analytical estimations, confirming the accuracy and correctness of our model. In turn, this makes the multipole-based model readily available for the modeling of realistic cable geometries by accounting for the full dependence on the actual cable geometry.
This report examines the localization of high frequency electromagnetic fields in general three-dimensional cavities along periodic paths between opposing sides of the cavity. The focus is on the case where the mirrors at the ends of the orbit are concave and have two different radii of curvature. The cases where these orbits lead to unstable localized modes are known as scars. The ellipsoidal coordinate system is utilized in the construction of the scarred modes. The field at the interior foci is examined as well as trigonometric projections along the periodic scarred ray path.
Capacitance/inductance corrections for grid induced errors for a thin slot models are given for both one and four point testing on a rectangular grid for surface currents surrounding the slot. In addition a formula for translating from one equivalent radius to another is given for the thin-slot transmission line model. Additional formulas useful for this slot modeling are also given.
A method used to solve the problem of water waves on a sloping beach is applied to a thin conducting half plane described by a thin layer impedance boundary condition. The solution for the electric field behavior near the edge is obtained and a simple fit for this behavior is given. This field is used to determine the correction to the impedance per unit length of a conductor due to a sharp edge. The results are applied to the strip conductor. The final appendix also discusses the solution to the dual-sided (impedance surface & perfect conductor surface) half plane problem.
This report examines the problem of magnetic penetration of a conductive layer, including nonlinear ferromagnetic layers, excited by an electric current filament. The electric current filament is, for example, a nearby wire excited by a lightning strike. The internal electric field and external magnetic field are determined. Numerical results are compared to various analytical approximations to help understand the physics involved in the penetration.
An iteration method is introduced to obtain the asymptotic form of the impedance per unit length of a rectangular conductor when the half side lengths are large compared to the skin depth. The first terms of the asymptotic expansion are extracted in closed form. The manner in which the corner corrections fit into the expansion are illustrated. The asymptotic results are compared to a numerical solution in the square limit. The odd corner correction for a right angle edge is also discussed.
We provide corrections to the slot capacitance and inverse inductance per unit length for slot gasket groove geometries using an approximate conformal mapping approach. We also provide corrections for abrupt step changes in slot width along with boundary discontinuity conditions for implementation in the various slot models.
This report examines the localization of high frequency electromagnetic fields in general three-dimensional convex walled cavities along periodic paths between opposing sides of the cavity. The report examines the three-dimensional case where the mirrors at the end of the orbit have two different radii of curvature. The cases where these orbits lead to unstable localized modes are known as scars.
Using the power balance method we estimate the maximum electric field on a conducting wall of a cavity containing an interior structure supporting eccentric coaxial modes in the frequency regime where the resonant modes are isolated from each other.
We examine coupling into azimuthal slots on an infinite cylinder with a infinite length interior cavity operating both at the fundamental cavity modal frequencies, with small slots and a resonant slot, as well as higher frequencies. The coupling model considers both radiation on an infinite cylindrical exterior as well as a half space approximation. Bounding calculations based on maximum slot power reception and interior power balance are also discussed in detail and compared with the prior calculations. For higher frequencies limitations on matching are imposed by restricting the loads ability to shift the slot operation to the nearest slot resonance; this is done in combination with maximizing the power reception as a function of angle of incidence. Finally, slot power mismatch based on limited cavity load quality factor is considered below the first slot resonance.
High-quality factor resonant cavities are challenging structures to model in electromagnetics owing to their large sensitivity to minute parameter changes. Therefore, uncertainty quantification (UQ) strategies are pivotal to understanding key parameters affecting the cavity response. We discuss here some of these strategies focusing on shielding effectiveness (SE) properties of a canonical slotted cylindrical cavity that will be used to develop credibility evidence in support of predictions made using computational simulations for this application.
A resonant cavity undergoes three distinct behaviors with increasing frequency: 1) fundamental modes, localized in frequency with well defined modal distribution; 2) undermoded region, where modes are still separated, but are sufficiently perturbed by small imperfections that their spectral positions (and distributions) are statistical in nature; and 3) overmoded region, where modes overlap, field distributions follow stochastic distributions, and the slot acts as if in free space. Understanding the penetration through slots in the overmoded region is of great interest, and is the focus of this article. Since full-wave solvers may not be able to provide a timely answer for very high frequencies due to a lack of memory and/or computation resources, we develop bounding methods to estimate worst-case average and maximum fields within the cavity. After discussing the bounding formulation, we compare its results to full-wave simulations at the first, second, and third resonance supported by the slot in the case of a cylindrical cavity. Note that the bounding formulation indicates that results are nearly independent of cavity shape: only the cavity volume, frequency, and cavity quality factor affect the overmoded region, making this formulation a powerful tool to assess electromagnetic interference and electromagnetic compatibility effects within cavities.
Metallic enclosures are commonly used to protect electronic circuits against unwanted electromagnetic (EM) interactions. However, these enclosures may be sealed with imperfect mechanical seams or joints. These joints form narrow slots that allow external EM energy to couple into the cavity and then to the internal circuits. This coupled EM energy can severely affect circuit operations, particularly at the cavity resonance frequencies when the cavity has a high Q factor. To model these slots and the corresponding EM coupling, a thin-slot sub-cell model [1] , developed for slots in infinite ground plane and extended to numerical modeling of cavity-backed apertures, was successfully implemented in Sandia's electromagnetic code EIGER [2] and its next-generation counterpart Gemma [3]. However, this thin-slot model only considers resonances along the length of the slot. At sufficiently high frequencies, the resonances due to the slot depth must also be considered. Currently, slots must be explicitly meshed to capture these depth resonances, which can lead to low-frequency instability (due to electrically small mesh elements). Therefore, a slot sub-cell model that considers resonances in both length and depth is needed to efficiently and accurately capture the slot coupling.
We analyze the coupling into a slotted cylindrical cavity operating at fundamental cavity modal frequencies overlapping with the slot’s first resonance frequency through an unmatched formulation that accounts for the slot’s absorption and radiation processes. The model is validated through full-wave simulations and experimental data. We then couple the unmatched formulation to a perturbation theory model to investigate an absorber within the cavity to reduce the interior field strength, also validated with full-wave simulations and experiments. These models are pivotal to understanding the physical processes involved in the electromagnetic penetration through slots, and may constitute design tools to mitigate electromagnetic interference effects within cavities.
Direct coupling of early-time high-altitude electromagnetic pulse (HEMP) to substation control cables is simulated for cable layouts based on surveys of seven electrical substations in the United States. An analytic transmission line modeling code is used to estimate worst-case coupled current at the terminations of cable segments in or near the control shack. Where applicable, an induced voltage due to cable shield grounding is also estimated. Various configurations are simulated, including cables with different elevations, lengths, radii, and terminations. Plots of the coupled HEMP effects are given, and general relationships between these effects and the substations geometric and material parameters are highlighted and discussed.
Sandia National Laboratories sponsored a three-year internally funded Laboratory Directed Research and Development (LDRD) effort to investigate the vulnerabilities and mitigations of a high-altitude electromagnetic pulse (HEMP) on the electric power grid. The research was focused on understanding the vulnerabilities and potential mitigations for components and systems at the high voltage transmission level. Results from the research included a broad array of subtopics, covered in twenty-three reports and papers, and which are highlighted in this executive summary report. These subtopics include high altitude electromagnetic pulse (HEMP) characterization, HEMP coupling analysis, system-wide effects, and mitigating technologies.
In a transmission line, the coupling between a line and a tower above ground is evaluated when the excitation is an E1 high-altitude electromagnetic pulse (HEMP). The model focuses on capturing correctly the effect of the coupling on the peak of the HEMP induced current that propagates along the line. This assessment is necessary to accurately estimate the effect of the excitation on the systems and components of the power grid. This analysis is a step towards a quantitative evaluation of HEMP excitation on the power grid. The results obtained indicate that the effect can be significant, especially for lines heights of 20 meters or more.
In this article, we examine the coupling into an electrically short azimuthal slot on a cylindrical cavity operating at fundamental cavity modal frequencies. We first develop a matched bound formulation through which we can gather information for maximum achievable levels of interior cavity fields. Actual field levels are below this matched bound; therefore, we also develop an unmatched formulation for frequencies below the slot resonance to achieve a better insight on the physics of this coupling. Good agreement is observed between the unmatched formulation, full-wave simulations, and experimental data, providing a validation of our analytical models. We then extend the unmatched formulation to treat an array of slots, found again in good agreement with full-wave simulations. These analytical models can be used to investigate ways to mitigate electromagnetic interference and electromagnetic compatibility effects within cavities.
We describe here diffusion models apt to construct a multipole-based, cable braid time domain model for conducting wires. Implementation details of both a ladder network valid for time-domain signals with all frequency content and an approximate single-stage circuit valid for low-frequency dominated time signals (such as electromagnetic pulses) are reported. This time domain model can be leveraged to treat system-generated electromagnetic pulse events, as well as used to further confirm the correctness of the multipole-based, cable braid frequency domain model.
In a transmission line, we evaluate the coupling between a line and a tower above ground when the excitation is an El high-altitude electromagnetic pulse (HEMP). Our model focuses on capturing correctly the effect of the coupling on the peak of the HEMP induced current that propagates along the line. This assessment is necessary to accurately estimate the effect of the excitation on the systems and components of the power grid. This analysis is a step towards a quantitative evaluation of HEMP excitation on the power grid.
We summarize the narrow slot algorithms, including the thick electrically small depth case, conductive gaskets, the deep general depth case, multiple fasteners along the length, and finally varying slot width.
We summarize here the double exponential and inverse double exponential approximations for two common EMP waveforms, the Bell Laboratories (Bell Labs) and the International- military standard (IEC-MIL-STD). Both models have been used frequently due to their relatively easy analytical expressions for both the time domain waveforms and their associated frequency domain spectra.
Placing microwave absorbing materials into a high-quality factor resonant cavity may in general reduce the large interior electromagnetic fields excited under external illumination. In this paper, we aim to combine two analytical models we previously developed: 1) an unmatched formulation for frequencies below the slot resonance to model shielding effectiveness versus frequency; and 2) a perturbation model approach to estimate the quality factor of cavities in the presence of absorbers. The resulting model realizes a toolkit with which design guidelines of the absorber’s properties and location can be optimized over a frequency band. Analytic predictions of shielding effectiveness for three transverse magnetic modes for various locations of the absorber placed on the inside cavity wall show good agreement with both full-wave simulations and experiments, and validate the proposed model. This analysis opens new avenues for specialized ways to mitigate harmful fields within cavities.
The goal of this paper is to present, for the first time, calculations of the magnetic penetration case of a first principles multipole-based cable braid electromagnetic penetration model. As a first test case, a one-dimensional array of perfect electrically conducting wires, for which an analytical solution is known, is investigated: We compare both the self-inductance and the transfer inductance results from our first principles cable braid electromagnetic penetration model to those obtained using the analytical solution. These results are found in good agreement up to a radius to half spacing ratio of about 0.78, demonstrating a robustness needed for many commercial and non-commercial cables. We then analyze a second set of test cases of a square array of wires whose solution is the same as the one-dimensional array result and of a rhomboidal array whose solution can be estimated from Kley’s model. As a final test case, we consider two layers of one-dimensional arrays of wires to investigate porpoising effects analytically. We find good agreement with analytical and Kley’s results for these geometries, verifying our proposed multipole model. Note that only our multipole model accounts for the full dependence on the actual cable geometry which enables us to model more complicated cable geometries.
Use of insensitive high explosives (IHEs) has significantly improved ammunition safety because of their remarkable insensitivity to violent cook-off, shock and impact. Triamino-trinitrobenzene (TATB) is the IHE used in many modern munitions. Previously, lightning simulations in different test configurations have shown that the required detonation threshold for standard density TATB at ambient and elevated temperatures (250 C) has a sufficient margin over the shock caused by an arc from the most severe lightning. In this paper, the Braginskii model with Lee-More channel conductivity prescription is used to demonstrate how electrical arcs from lightning could cause detonation in TATB. The steep rise and slow decay in typical lightning pulse are used in demonstrating that the shock pressure from an electrical arc, after reaching the peak, falls off faster than the inverse of the arc radius. For detonation to occur, two necessary detonation conditions must be met: the Pop-Plot criterion and minimum spot size requirement. The relevant Pop-Plot for TATB at 250 C was converted into an empirical detonation criterion, which is applicable to explosives subject to shocks of variable pressure. The arc cross-section was required to meet the minimum detonation spot size reported in the literature. One caveat is that when the shock pressure exceeds the detonation pressure the Pop-Plot may not be applicable, and the minimum spot size requirement may be smaller.
Electromagnetic shields are usually employed to protect cables and other devices; however, these are generally not perfect, and may permit external magnetic and electric fields to penetrate into the interior regions of the cable, inducing unwanted current and voltages. The aim of this paper is to verify a circuit model tool with our previously proposed analytical model [1] for evaluating currents and voltages induced in the inner conductor of braided-shield cables. This circuit model will enable coupling between electromagnetic and circuit simulations.
In this paper, we investigate the coupling from external electromagnetic (EM) fields to the interior EM fields of a high-quality factor cylindrical cavity through a small perturbing slot. We illustrate the shielding effectiveness versus frequency, highlighting bounds on the penetrant power through the slot. Because internal fields may become larger than external ones, we then introduce a small amount of microwave absorbing materials decorating the slot to improve shielding effectiveness considerably, as shown by both simulations and experiments. Although the cylindrical cavity is used for demonstration purposes in this paper, the conclusions presented here can be leveraged for use with more complex cavity structures.
Recently there has been a large interest in achieving metasurface resonances with large quality factors. In this article, we examine metasurfaces that comprised a finite number of magnetic dipoles oriented parallel or orthogonal to the plane of the metasurface and determine analytic formulas for their resonances’ quality factors. These conditions are experimentally achievable in finite-size metasurfaces made of dielectric cubic resonators at the magnetic dipole resonance. Our results show that finite metasurfaces made of parallel (to the plane) magnetic dipoles exhibit low quality factor resonances with a quality factor that is independent of the number of resonators. More importantly, finite metasurfaces made of orthogonal (to the plane) magnetic dipoles lead to resonances with large quality factors, which ultimately depend on the number of resonators comprising the metasurface. In particular, by properly modulating the array of dipole moments by having a distribution of resonator polarizabilities, one can potentially increase the quality factor of metasurface resonances even further. These results provide design guidelines to achieve a sought quality factor applicable to any resonator geometry for the development of new devices such as photodetectors, modulators, and sensors.
This report explores the coupling between EIGER simulations and a transmission line analytical model to enable end-to-end simulations to translate an exterior electromagnetic environment to assess effects on electronic system performance.
This report explores the potential for reducing the fields and the quality factor within a system cavity by introducing microwave absorbing materials. Although the concept of introducing absorbing (lossy) materials within a cavity to drive the interior field levels down is well known, increasing the loading into a complex weapon cavity specifically for improved electromagnetic performance has not, in general, been considered, and this will be the subject of this work. We compare full-wave simulations to experimental results, demonstrating the applicability of the proposed method.
Improving the sensitivity of infrared detectors is an essential step for future applications, including satellite- and terrestrial-based systems. We investigate nanoantenna-enabled detectors (NEDs) in the infrared, where the nanoantenna arrays play a fundamental role in enhancing the level of absorption within the active material of a photodetector. The design and optimization of nanoantenna-enabled detectors via full-wave simulations is a challenging task given the large parameter space to be explored. Here, we present a fast and accurate fully analytic circuit model of patch-based NEDs. This model allows for the inclusion of real metals, realistic patch thicknesses, non-absorbing spacer layers, the active detector layer, and absorption due to higher-order evanescent modes of the metallic array. We apply the circuit model to the design of NED devices based on Type II superlattice absorbers, and show that we can achieve absorption of ∼70% of the incoming energy in subwavelength (∼λ∕5) absorber layers. The accuracy of the circuit model is verified against full-wave simulations, establishing this model as an efficient design tool to quickly and accurately optimize NED structures.
This report compares ATLOG modeling results for the response of a finite-length dissipative buried conductor interacting with a conducting ground to a measurement taken November 2016 at the High-Energy Radiation Megavolt Electron Source (HERMES) facility. We use the ATLOG frequency-domain method based on transmission line theory. Estimates of the impedance per unit length and admittance per unit length for a cable laying in a PVC pipe embedded in a concrete block are reported. Current wave shapes from both a single conductor and composite differential mode and antenna mode arrangements are close to those observed in the experiments.
This report considers plane wave coupling to a transmission line consisting of a wire above a conducting ground. Comparisons are made for the two types of available source models, along with a discussion about the decomposition of the line currents. Simple circuit models are constructed for the terminating impedances at the ends of the line including radiation effects. Results from the transmission line with these loads show good agreement with full wave simulations.
