Simple Analytical Equations Research Articles

Improved Determination of the Mobility and Built-In Voltage in Asymmetric Single-Carrier Devices

The steady-state charge-carrier mobility in semiconductors can be extracted from the space-charge-limited current in single-carrier devices. However, in many cases, a built-in voltage is present, which should be known accurately to obtain the correct mobility. Here, it is demonstrated that band bending at the injecting electrode has important consequences for the built-in voltage and the analytical description of the current-voltage characteristics. It is shown that the built-in voltage can be accurately determined from the slope of the current-voltage characteristics on a semilogarithmic scale. Knowing the effect of band bending on the injected current, a simple analytical equation for the drift-diffusion space-charge-limited current above the built-in voltage is derived, which allows for improved determination of the charge-carrier mobility from experimental data.

Analysis of three-dimensional mapping problems in incoherent digital holography.

Self-interference digital holography (SIDH) and Fresnel incoherent correlation holography (FINCH) are recently introduced holographic imaging schemes to record and reconstruct three-dimensional (3-D) information of objects by using incoherent light. Unlike conventional holography, a reference wave in incoherent holography is not predetermined by an experimental setup, but changes with target objects in incoherent holography. This makes the relation between the 3-D position information of an object and those stored in a measured hologram quite complicated. In this paper, we provide simple analytic equations for an effective 3D mapping between object space and the image space in incoherent holography. We have validated our proposed method with numerical simulations and off-axis SIDH experiments.

Analytical framework for the assessment and modelling of multi-junction solar cells in the outdoors

The assessment of multi-junction solar cells often relies on numerically intensive computations. Specifically, the power conversion efficiency strongly depends on the interplay between optical and electrical properties of different materials. Here, a compact and highly accurate analytical framework is proposed, facilitating the analysis of multi-junction solar cells; explicit yet simple analytical equations allow to assess the power conversion efficiency as a direct function of the cell’s parameters, without restrictive assumptions. They are first used to compare the performance of the industrial state-of-the-art to multi-junction approaches. Therefore, minute data products are obtained from free satellite-services for different climatic zones over 14 years. Any variations in the operating temperature, sunshine duration, Sun’s position, meteorological condition or atmospheric chemistry are thereby accounted for. Similarly, a strong site dependency is found for perovskite-on-silicon tandem cells under real-world conditions. For this, a scattering-matrix treatment is formulated based on incoherent sunlight as the relevant case. While this study gives new theoretical insights about the impact of the cell’s parameters on the conversion efficiency, it also presents a powerful analytical tool for the design and assessment of more efficient solar cells in the outdoors.

On the effect of the ply stacking sequence on the failure of composite pipes under external pressure

The use of high performance structural composites has become very important over the last decades, especially where weight is an essential factor. Particularly in the oil and gas industry, several designs of composite pipes for deep water applications have been recently proposed as competitive solutions against traditional steel pipes. Thus, it is important to assess the performance of composite pipes under high external pressure in order to avoid pipe failure or overconservative designs. In this paper, experimental tests of different composite pipe configurations are performed and then compared to analytical and numerical predictions. Unlike the case of internal pressure loads, the collapse pressure of composite pipes depends on the initial ovality and on the ply stacking sequence. The collapse resistance of different composite pipes is firstly studied through simplified analytical equations combined with different failure criteria. Then, a finite element model is developed using a progressive failure criterion [1]. Both analytical and numerical failure predictions were compared to experimental tests carried out on four composite pipes produced with different ply stacking sequence by the filament winding method [2]. An experimental-numerical-analytical comparison shows that numerical and analytical models provide results in good agreement with those obtained experimentally. Finally, a parametric analysis is carried out to show the effect of ovality and ply stacking sequence on the failure pressure of composite pipes.

Moiré patterns: a simple analytical model

By considering a straightforward geometrical construction in reciprocal space we derive simple analytical equations that describe the geometry (period and angle) of the moiré patterns (MPs) formed by superposition of any 2D material on any substrate (including other 2D materials). These equations should be valuable tools for interpreting experimentally observed MPs and for designing van der Waals heterostructures in which MPs are used to modulate the topological and/or electronic states of a device.

A Novel Approach for the Determination of Surface Tilt Angles in Two-Dimensional Digital Image Correlation Experiments

Two-dimensional digital image correlation (2D-DIC) is increasingly being used in a wide variety of applications. This technique is capable of measuring the in-plane deformations and strains of planner surfaces. A basic condition for using this technique is having the camera to observe the target surface perpendicularly. If the camera is not perpendicular to the surface, errors will be introduced in the measured displacements and strains. Small amounts of camera misalignment can easily go undetected during the initial setting of the experimental setup. In this paper, a novel approach for verifying the perpendicularity of the camera with respect to the target surface in 2D-DIC experiments is introduced. If the camera is not perpendicular to the surface, the tilt angles about both of the two in-plane axes can be calculated using this novel approach. A simple in-plane rigid-body-translation of the specimen in any of the two in-plane directions (horizontal or vertical direction relative to the camera image), is used for detecting camera non-perpendicularity and calculating the tilt angle(s). This approach uses the normal and shear strain errors, induced in DIC results due to camera non-perpendicularity, to determine the surface tilt angle(s), if any is present. The tilt angle(s) are calculated using simple analytical equations that are developed based on the pinhole camera model. The proposed approach is validated using experiments performed while the target surface is tilted at angles ranging from 0.5° to 4°. The experimental results show that this approach is able to confidently detect camera misalignment for tilt angles of 1° or larger and the tilt angles can be calculated with less than 0.3° error.

Prediction of Eddy Current Losses in Cooling Tubes of Direct Cooled Windings in Electric Machines

The direct coil cooling method is one of the existing cooling techniques for electric machines with concentrated windings, in which cooling tubes of conductive material are inserted between the windings. In such cases, eddy current losses are induced in those cooling tubes because of the time variant magnetic field. To compute the cooling tubes losses, either a transient finite element simulation (mostly based on commercial software), or a full analytical method, which is more complex to be constructed, is required. Instead, this paper proposes a simple and an accurate combined semi-analytical-finite element method to calculate the losses of electric machines having cooling tubes. The 2D magnetostatic solution of the magnetic field is obtained e.g., using the free package “FEMM”. Then, the eddy current losses in the tubes are computed using simple analytical equations. In addition, the iron core losses could be obtained. In order to validate the proposed method, two cases are investigated. In Case 1, a six-toothed stator of a switched reluctance machine (SRM), without rotor, is employed in which six cooling tubes are used while in Case 2 a complete rotating SRM is studied. The proposed method is validated by a 2D transient simulation in the commercial software “ANSYS Maxwell” and also by experimental measurements. Evidently, the proposed method is simple and fast to be constructed and it is almost free of cost.

Response of pile groups in sand due to lateral cyclic loading

This paper considers data from centrifuge model tests on pile groups in silica sand involving lateral one-way cyclic loading. The influence of cyclic load amplitude, pile spacing within the pile group and number of cycles on the observed response are highlighted. Investigating the contribution of each pile on the lateral capacity shows leading row piles exhibit highest bending moments and attract larger loads due to ‘shadowing’ effects. Cyclic loading results in a subtle change of the load distribution within the pile group, with the load gradually shifting from the leading to the trailing piles as the number of cycles increases. The effect of cyclic loading on the initial stiffness of the p–y curves is more significant; initial stiffness reduces with increasing number of loading cycles at a rate that is higher in the leading row than in the trailing row. The changes in load distribution are quantified using simple analytical equations that may be used in design.

Phonon-limited mobility of Dirac fermions in three-dimensional Dirac semimetal Cd3As2

A theoretical model is presented for the phonon-limited mobility of the Dirac fermion gas in three-dimensional (3D) Dirac semimetal Cd3As2, with consideration of the scattering from both the acoustic and optical phonons. Screening effects are taken into account and it is found that they lead to significant enhancement of the mobility. Simple analytical equations and power laws are obtained for both the Bloch–Grüneisen and equipartition regimes. The dependence of the mobility on the temperature T and electron density ne is investigated. The optical-phonon-limited mobility μop is found to dominate over the acoustic-phonon-limited mobility μap in the higher-temperature region. There is a crossover of μap and μop, and the crossover temperature Tc shifts to a higher value with increasing ne. Numerical calculations of the mobility are conducted for recently reported experimental samples and good agreement is obtained with the experimental results. A comparison is also made with the results for a conventional 3D electron gas in a degenerate semiconductor.

On the advantages of operation in second-order diffraction of blazed gratings in soft X-ray monochromators.

The fact that a diffraction grating can provide twofold-smaller bandwidth when operated in second-order diffraction is long known and applied routinely in the laboratory for spectroscopy in the visible and ultraviolet spectral range. A similar routine operation in monochromators for the soft X-ray range is not reported yet. This study will thus address the feasibility of efficient diffraction of soft X-rays in the second order at reflection gratings when operated at grazing angles of incidence. The related systematic study could make profitable use of a recently introduced simple analytical equation for the prediction of the diffraction efficiency of blazed gratings with an ideal sawtooth profile. The predictions are then verified by use of rigorous calculations. The principle finding is that, by operation of gratings with lower groove densities, and thus with higher efficiencies, in higher order diffraction, one can extend the tuning in existing instruments with mechanical/optical limitations to larger photon energies. The performance in terms of transmission and spectral resolving power can be very similar to the performance of a grating with a larger groove density, which would otherwise have to be used for accessing the same energy range. This would allow operation of a single highly efficient grating over a larger photon energy interval at a modern synchrotron radiation source, e.g. from 0.3 to 2.2 keV. Without any requirement for a sophisticated grating exchange scheme, a related instrument promises to be sufficiently stable for the needs imposed by the improvements in source point stability at diffraction-limited storage rings.

Axial segregation in crystal growth from a thin melt layer

During a crystal growth from a thin melt layer, the transport of dopant occurs mostly by diffusion. One can describe the dopant distribution by simplified analytical equations, as, for example, those previously obtained by Tiller.We performed numerical 2D axi-symmetric computations of heat and mass transfer in a thin melt layer (0.05<h/d<0.2), in order to find the limits of the validity of such approaches. Sb doped Ge, in a cylindrical volume with a diameter of 200 mm, was used as a model material. Compared with the heat transfer, the mass transfer was found to be much more sensitive to residual convection. Some restrictions in the application of the analytical equations for the description of the experimental data are revealed and necessary recommendations to improve the results are made. In order to provide the uniform axial dopant distribution in the very last portion of the ingot, too, it is necessary to vary the rate of solidification by changing the cooling rate with time.

On smart optimization of blazed soft X-ray gratings.

The first attempts to calculate the diffraction efficiencies of gratings in the soft X-ray range were made on a scalar model. The results were simple analytical equations, that always severely overestimated the performance of real objects. In this respect, computer programs were found to be more successful, which rigorously consider all diffracted and refracted waves. Consequently soft X-ray gratings are presently optimized using these tools, which requires rather extensive calculations for any instrument optimization as general trends are not immediately obvious. Here it will be shown that the results of the rigorous calculations for gratings with blaze or sawtooth profile can be approximated rather well with a simple analytical equation. This equation contains three multiplicative factors, which deal independently with the effect of the reflectivity, the blaze angle and the groove density. This opens the possibility to initially ignore the effects of the blaze angle and thus to start an optimization in a very general way. Such optimization can be based on isoreflectivity curves and it can then provide `blaze maximum efficiency maps', i.e. simple images. In these latter images, one can identify directly the optimum parameters for a grating, i.e. the groove density providing best efficiency for a requested spectral resolving power. Only successively will the blaze angle have to be fixed. Its choice is then not the result of an extensive optimization process but of a simple calculation applied for the photon energy at which maximum efficiency performance is requested. The maps presented here are used for the optimization of a medium-resolving-power soft X-ray monochromator, which can scan the photon energy range 300-2000 eV.

Design Considerations for Radiofrequency Whole-Body and Head Coils

It is shown that the transverse-plane ${\bf{B}}_1^ + $ homogeneity in TEM whole-body and head RF coils, which is responsible for acquiring uniformly illuminated MR images, is frequency independent and solely depends on achieving high excitation purity of the fundamental $m = 1$ mode. This necessitates that the capacitive coupling introduced by the interconnecting lumped capacitors should maximize the separation between the resonance frequency of the fundamental mode and that of all other higher order modes. Alternatively, full-orthogonal excitation of the coil, if possible, would mitigate this design challenge, especially if high- Q operation cannot be guaranteed. In this contribution, a field theoretical technique for the optimization of the capacitive coupling is introduced and applied to the birdcage configuration as a representative of TEM coils. A fairly general analysis of these coils is also introduced and/or revisited, which underlies the proposed technique and offers simple analytic design equations. Simulation results are presented to validate the considered approach. The presented analysis deals with the homogeneously filled coils and therefore does not consider the ${\bf{B}}_1^ + $ inhomogeneities caused by the loading. The latter will be the subject of future publications.

Three-harmonic optimal multisine input power spectrum for bioimpedance identification

Objective: The attention towards optimal signals for measuring electrical bioimpedance (EBI) has increased recently due to the many advantages they offer such as, for example, reduced measuring time. Here, the design of a three-harmonic optimized multisine input power spectrum for measuring bioimpedance is considered. Approach: The approach is based on designing the input power spectrum multisine excitation by optimizing a scalar function of the information matrix using the Fricke–Morse model. Main results: Simple analytical equations are provided that can be adopted by the reader to optimize the multisine input power spectrum to satisfy any specific application-related EBI measurement. Significance: The presented signal may be a good choice of excitation signal in EBI applications where optimal excitation power spectrum is desired.

Transits of Inclined Exomoons—Hide and Seek and an Application to Kepler-1625

Abstract A Neptune-sized exomoon candidate was recently announced by Teachey &amp; Kipping, orbiting a 287 day gas giant in the Kepler-1625 system. However, the system is poorly characterized and needs more observations to be confirmed, with the next potential transit in 2019 May. In this Letter, we aid observational follow up by analyzing the transit signature of exomoons. We derive a simple analytic equation for the transit probability and use it to demonstrate how exomoons may frequently avoid transit if their orbit is larger than the stellar radius and sufficiently misaligned. The nominal orbit for the moon in Kepler-1625 has both of these characteristics, and we calculate that it may only transit ≈40% of the time. This means that ≈six non-transits would be required to rule out the moon’s existence at 95% confidence. When an exomoon’s impact parameter is displaced off the star, the planet’s impact parameter is displaced the other way, so larger planet transit durations are typically positively correlated with missed exomoon transits. On the other hand, strong correlations do not exist between missed exomoon transits and transit timing variations of the planet. We also show that nodal precession does not change an exomoon’s transit probability and that it can break a prograde-retrograde degeneracy.

Semiconductor Power Devices: Physics, Characteristics, Reliability [Book Review

The book covers today’s major power devices in 15 chapters and five appendices comprising 706 pages. Power conversion technology is driven by power density, efficiency, and reliability, and the semiconductor devices and their operation in the circuit play a key role. To operate devices to their limits in the service-oriented architecture, safe operating area (SOA) diagram engineers must acquire a deeper understanding of the intrinsic semiconductor structures, parasitic components added on because of packaging and chip-interfacing technologies, and destruction mechanism during operation in particular dynamic intervals. This is even more important when we move from fast-switching to ultrafastswitching power devices. The outstanding value of this text for all relevant devices used today is that the authors explain, in detail, the dynamic operation up to the limits, i.e., the failure and destruction mechanism in the case of electrical or thermal overstress. All of the authors have a very strong background in industry, which helps them demonstrate and explain realistic applications of these new device types; however, they are also excellent teachers who describe solid-state physics and the thermal performance and failure mechanism with simple analytical equations proven by experimental data generated in industry and research labs. This book serves as a guideline for engineers designing power converters, power device component development experts, and researchers looking ahead to the next generation of power semiconductor technologies. This publication has high value for the power electronics community, from students to experts in industry and academia, including decision makers in industry and government.

Analysis and Optimization of a Piezoelectric Energy Harvester

The paper aims to assess and improve the performances of a multilayer piezoelectric MEMS device for vibrations harnessing. Two operating modes are possible: at resonance and outside resonance. In some applications it is not possible to operate at resonance, functioning being mostly at low frequencies in a quasi-static regime. An Euler-Bernoulli classic beam theory mathematical model was studied for estimating the behaviour of multilayer piezoelectric generators, in terms of deflection and voltage, at functioning under resonance frequency. The analytical results were compared with the finite element method simulation in COMSOL Multiphysics. The main goal of this study is to obtain an accurate model for engineering design purposes, with simple analytical equations and ease of use, but with predictable errors. The study proved the usefulness of the derived model but also its limitations. It also proves the need to improve the model using plate theory, for sensors with high width/height ratio.

Particle-particle heat transfer in thermal DEM: Three competing models and a new equation

Simulation of transient heat transfer from the wall of contact equipment (e.g. rotary drum) to a mechanically agitated bed of particles in gas atmosphere by the discrete element method (DEM) requires knowledge of the particle-particle heat transfer coefficient. A simple analytical equation is proposed in order to calculate this quantity from the effective thermal conductivity of the respective packed bed for the described class of problems. Comparison with other calculation methods that assign particle-particle heat transfer exclusively to either the solid or the gas phase shows that large differences can occur. It stresses the need for further work on the fundamentals of thermal DEM for such applications, and for a critical attitude until those have been clarified.

A New Simple Analytic Equation for Estimating Asymmetric Growth of Stimulated Reservoir Volume in Anisotropic Shales

Summary The objective of this paper is to present the development and application of a simple equation for calculating the asymmetric growth of the stimulated reservoir volume (SRV) in an anisotropic shale–petroleum reservoir using microseismic data, and the hydraulic diffusivities of the anisotropic shale. Calculation of the SRV is a problem tackled with solutions that involve different degrees of complexity. Because shale reservoirs are anisotropic, microseismic events generally develop 3D nonuniform asymmetric patterns around the injection points. This paper presents a new method with an easy–to–use analytic equation that allows for reproducing the asymmetric growth of microseismic events as a function of time by considering reservoir anisotropy. Asymmetric growth refers to the fact that propagation of the microseismic cloud in a given direction can be larger, equal, or smaller compared with the propagation in other directions. Accurate determination of the SRV asymmetric pattern is critical for use in specialized material–balance and reservoir–simulation models of shale–petroleum reservoirs. This determination allows for more–realistic projections of reservoir performance. The novelty of the method is the development of an easy–to–use approach for estimating SRV in a spatially nonuniform asymmetric anisotropic reservoir using octants in a coordinate system. The SRV is calculated from the volume of a symmetric ellipsoid divided by a constant value Vc. This is despite the fact that the point of injection of the fracturing fluids in the asymmetric reservoir can be at, close to, or far from the center of the ellipsoid. The development of Vc is presented in this paper. Use of the SRV calculation model is illustrated with real microseismic data of the Horn River Shale in Canada for a case where Vc is equal to 1.3722. Also presented are calculations of hydraulic diffusivities in this anisotropic shale.

Improvement of the Richardson-Zaki liquid-solid fluidisation model on the basis of hydraulics

One of the most popular and frequently used models for describing homogeneous liquid-solid fluidised suspensions is the model developed by Richardson & Zaki in 1954. The superficial fluid velocity and terminal settling velocity together with an index makes it possible to determine the fluid porosity in a straightforward way. The reference point for the Richardson-Zaki model is the terminal settling velocity at maximum porosity conditions. To be able to predict porosity in the proximity of minimum fluidisation conditions, either the minimum fluidisation velocity must be known or the Richardson-Zaki index must be very accurate. To maintain optimal process and control conditions in multiphase drinking water treatment processes, the porosity is kept relatively low. Unfortunately, the Richardson-Zaki index models tends to overestimate the minimum fluidisation velocity and therefore also results in less accurate predictions with respect to porosity values. We extended the Richardson-Zaki model with proven hydraulics-based models. The minimum fluidisation velocity is acquired using the model proposed by Kozeny (1927), Ergun (1952) and Carman (1937). The terminal settling velocity is obtained through the model developed by Brown & Lawler (2003), which is an improved version of the well-known model developed by Schiller & Naumann (1933). The proposed models are compared with data from expansion experiments with calcium carbonate grains, crushed calcite and garnet grains applied in drinking water softening using the fluidised bed process. With respect to porosity, prediction accuracy is improved, with the average relative error decreasing from 15% to 3% when the classic Richardson-Zaki model is extended with these hydraulics-based models. With respect to minimum fluidisation velocity, the average relative error decreases from 100% to 12%. In addition, simplified analytical equations are given for a straightforward estimation of the index n.