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Classical Mechanics
Classical Mechanics Hyoungsoon Choi Spring, 2014 Contents 1 Introduction4 1.1 Kinematics and Kinetics . .5 1.2 Kinematics: Watching Wallace and Gromit ............6 1.3 Inertia and Inertial Frame . .8 2 Newton's Laws of Motion 10 2.1 The First Law: The Law of Inertia . 10 2.2 The Second Law: The Equation of Motion . 11 2.3 The Third Law: The Law of Action and Reaction . 12 3 Laws of Conservation 14 3.1 Conservation of Momentum . 14 3.2 Conservation of Angular Momentum . 15 3.3 Conservation of Energy . 17 3.3.1 Kinetic energy . 17 3.3.2 Potential energy . 18 3.3.3 Mechanical energy conservation . 19 4 Solving Equation of Motions 20 4.1 Force-Free Motion . 21 4.2 Constant Force Motion . 22 4.2.1 Constant force motion in one dimension . 22 4.2.2 Constant force motion in two dimensions . 23 4.3 Varying Force Motion . 25 4.3.1 Drag force . 25 4.3.2 Harmonic oscillator . 29 5 Lagrangian Mechanics 30 5.1 Configuration Space . 30 5.2 Lagrangian Equations of Motion . 32 5.3 Generalized Coordinates . 34 5.4 Lagrangian Mechanics . 36 5.5 D'Alembert's Principle . 37 5.6 Conjugate Variables . 39 1 CONTENTS 2 6 Hamiltonian Mechanics 40 6.1 Legendre Transformation: From Lagrangian to Hamiltonian . 40 6.2 Hamilton's Equations . 41 6.3 Configuration Space and Phase Space . 43 6.4 Hamiltonian and Energy . 45 7 Central Force Motion 47 7.1 Conservation Laws in Central Force Field . 47 7.2 The Path Equation . -
Pioneers in Optics: Christiaan Huygens
Downloaded from Microscopy Pioneers https://www.cambridge.org/core Pioneers in Optics: Christiaan Huygens Eric Clark From the website Molecular Expressions created by the late Michael Davidson and now maintained by Eric Clark, National Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 . IP address: [email protected] 170.106.33.22 Christiaan Huygens reliability and accuracy. The first watch using this principle (1629–1695) was finished in 1675, whereupon it was promptly presented , on Christiaan Huygens was a to his sponsor, King Louis XIV. 29 Sep 2021 at 16:11:10 brilliant Dutch mathematician, In 1681, Huygens returned to Holland where he began physicist, and astronomer who lived to construct optical lenses with extremely large focal lengths, during the seventeenth century, a which were eventually presented to the Royal Society of period sometimes referred to as the London, where they remain today. Continuing along this line Scientific Revolution. Huygens, a of work, Huygens perfected his skills in lens grinding and highly gifted theoretical and experi- subsequently invented the achromatic eyepiece that bears his , subject to the Cambridge Core terms of use, available at mental scientist, is best known name and is still in widespread use today. for his work on the theories of Huygens left Holland in 1689, and ventured to London centrifugal force, the wave theory of where he became acquainted with Sir Isaac Newton and began light, and the pendulum clock. to study Newton’s theories on classical physics. Although it At an early age, Huygens began seems Huygens was duly impressed with Newton’s work, he work in advanced mathematics was still very skeptical about any theory that did not explain by attempting to disprove several theories established by gravitation by mechanical means. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
A Guide to Space Law Terms: Spi, Gwu, & Swf
A GUIDE TO SPACE LAW TERMS: SPI, GWU, & SWF A Guide to Space Law Terms Space Policy Institute (SPI), George Washington University and Secure World Foundation (SWF) Editor: Professor Henry R. Hertzfeld, Esq. Research: Liana X. Yung, Esq. Daniel V. Osborne, Esq. December 2012 Page i I. INTRODUCTION This document is a step to developing an accurate and usable guide to space law words, terms, and phrases. The project developed from misunderstandings and difficulties that graduate students in our classes encountered listening to lectures and reading technical articles on topics related to space law. The difficulties are compounded when students are not native English speakers. Because there is no standard definition for many of the terms and because some terms are used in many different ways, we have created seven categories of definitions. They are: I. A simple definition written in easy to understand English II. Definitions found in treaties, statutes, and formal regulations III. Definitions from legal dictionaries IV. Definitions from standard English dictionaries V. Definitions found in government publications (mostly technical glossaries and dictionaries) VI. Definitions found in journal articles, books, and other unofficial sources VII. Definitions that may have different interpretations in languages other than English The source of each definition that is used is provided so that the reader can understand the context in which it is used. The Simple Definitions are meant to capture the essence of how the term is used in space law. Where possible we have used a definition from one of our sources for this purpose. When we found no concise definition, we have drafted the definition based on the more complex definitions from other sources. -
Energy Harvesting for Low-Power Sensor Systems
White Paper – RX100 Microcontroller Family Energy Harvesting for Low-Power Sensor Systems By: David Squires, Squires Consulting; Forrest Huff, Renesas Electronics America Inc. Feb. 2015 Abstract This white paper details important design issues associated with designing the very-low-power remote sensors that play important roles in the feedback control loops of embedded electronic systems. Using a basic configuration for a standalone sensor system as a reference, the discussion covers sensor/transducer types, power budgets, power sources (especially devices that harvest ambient energy) and energy-storage solutions. It also summarizes microcontroller requirements and highlights the latest power-management chips and wireless ICs. Special emphasis is placed on the concept and reality of energy harvesting as a viable method for powering standalone embedded systems for extended periods of time. An example security alarm design is described and explained to provide engineering context and perspective. Index I. Introduction – III. Additional Insights on Very-Low-Power Sensor Products 2 Sensor Components 7 II. Basic System Design Issues 2 • Sensors 7 • Sensors 3 • Energy Storage Devices 7 • Power Budgets 3 • Energy Harvesting Solutions 12 • Energy Storage Devices 3 • MCUs 15 • Energy Harvesting Solutions 4 • Power Management Devices 18 • Microcontrollers (MCUs) 6 • Wireless Connectivity Modules 20 • Power Management Devices 6 IV. Example Design: Glass Break Sensor 22 • Wireless Connectivity Modules 6 V. Summary 28 VI. Appendix 28 White Paper – Energy Harvesting for Low-Power Sensor Systems Page 1 of 29 I. Introduction – Very-Low-Power Sensor Products Modern low-power sensors enable precise local, remote or autonomous control of a vast range of products. Their use is rapidly proliferating in vehicles, appliances, HVAC systems, hospital intensive care suites, oil refineries, and military and security systems. -
Planar Dual Polarized Metasurface Array for Microwave Energy Harvesting
electronics Article Planar Dual Polarized Metasurface Array for Microwave Energy Harvesting Maged A. Aldhaeebi 1 and Thamer S. Almoneef 2,* 1 Department of Electronics and Communication Engineering, Hadhramout University, Mukalla 50512-50511, Yemen; [email protected] 2 Electrical Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia * Correspondence: [email protected] Received: 26 October 2020; Accepted: 18 November 2020; Published: 24 November 2020 Abstract: The design of a dual polarized metasurface collector based on the metamaterial full absorption concept for electromagnetic energy harvesting is introduced. Unlike previous metamaterial absorber designs, here the power absorbed is mostly dissipated across a resistive load rather than within the dielectric substrate. This is achieved by channeling the absorbed power to an optimal resistive load through a via hole. The simulation results show that a power absorption efficiency of 98% is achieved at an operating frequency of 2 GHz for a single unit cell. A super unit cell consisting of four cells with alternating vias was also designed to produce a dual polarized metasurface. The simulation results yielded a radiation to AC efficiency of around 98% for each polarization. Keywords: planar meatsurface; dual polarized absorbers; energy harvesting; rectenna 1. Introduction The microwave energy harvesting system (MEHS) has recently received much attention in the field of developing rectenna arrays based on metamaterials [1]. The two main components of any microwave energy harvesting system are an antenna and a rectification circuit. The antenna is used for collecting incident electromagnetic radiated power and converting it to AC (Radiation to AC efficiency), whereas a rectification circuit is used for converting the collected AC power to DC (AC to DC efficiency) [2]. -
A Review on Thermoelectric Generators: Progress and Applications
energies Review A Review on Thermoelectric Generators: Progress and Applications Mohamed Amine Zoui 1,2 , Saïd Bentouba 2 , John G. Stocholm 3 and Mahmoud Bourouis 4,* 1 Laboratory of Energy, Environment and Information Systems (LEESI), University of Adrar, Adrar 01000, Algeria; [email protected] 2 Laboratory of Sustainable Development and Computing (LDDI), University of Adrar, Adrar 01000, Algeria; [email protected] 3 Marvel Thermoelectrics, 11 rue Joachim du Bellay, 78540 Vernouillet, Île de France, France; [email protected] 4 Department of Mechanical Engineering, Universitat Rovira i Virgili, Av. Països Catalans No. 26, 43007 Tarragona, Spain * Correspondence: [email protected] Received: 7 June 2020; Accepted: 7 July 2020; Published: 13 July 2020 Abstract: A thermoelectric effect is a physical phenomenon consisting of the direct conversion of heat into electrical energy (Seebeck effect) or inversely from electrical current into heat (Peltier effect) without moving mechanical parts. The low efficiency of thermoelectric devices has limited their applications to certain areas, such as refrigeration, heat recovery, power generation and renewable energy. However, for specific applications like space probes, laboratory equipment and medical applications, where cost and efficiency are not as important as availability, reliability and predictability, thermoelectricity offers noteworthy potential. The challenge of making thermoelectricity a future leader in waste heat recovery and renewable energy is intensified by the integration of nanotechnology. In this review, state-of-the-art thermoelectric generators, applications and recent progress are reported. Fundamental knowledge of the thermoelectric effect, basic laws, and parameters affecting the efficiency of conventional and new thermoelectric materials are discussed. The applications of thermoelectricity are grouped into three main domains. -
Definitions Matter – Anything That Has Mass and Takes up Space. Property – a Characteristic of Matter That Can Be Observed Or Measured
Matter – Definitions matter – Anything that has mass and takes up space. property – A characteristic of matter that can be observed or measured. mass – The amount of matter making up an object. volume – A measure of how much space the matter of an object takes up. buoyancy – The upward force of a liquid or gas on an object. solid – A state of matter that has a definite shape and volume. liquid – A state of matter that has a definite volume but no definite shape. It takes the shape of its container. gas – A state of matter that has no definite shape or volume. metric system – A system of measurement based on units of ten. It is used in most countries and in all scientific work. length – The number of units that fit along one edge of an object. area – The number of unit squares that fit inside a surface. density – The amount of matter in a given space. In scientific terms, density is the amount of mass in a unit of volume. weight – The measure of the pull of gravity between an object and Earth. gravity – A force of attraction, or pull, between objects. physical change – A change that begins and ends with the same type of matter. change of state – A physical change of matter from one state – solid, liquid, or gas – to another state because of a change in the energy of the matter. evaporation – The process of a liquid changing to a gas. rust – A solid brown compound formed when iron combines chemically with oxygen. chemical change – A change that produces new matter with different properties from the original matter. -
On Space-Time, Reference Frames and the Structure of Relativity Groups Annales De L’I
ANNALES DE L’I. H. P., SECTION A VITTORIO BERZI VITTORIO GORINI On space-time, reference frames and the structure of relativity groups Annales de l’I. H. P., section A, tome 16, no 1 (1972), p. 1-22 <http://www.numdam.org/item?id=AIHPA_1972__16_1_1_0> © Gauthier-Villars, 1972, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincaré, Section A : Vol. XVI, no 1, 1972, 1 Physique théorique. On space-time, reference frames and the structure of relativity groups Vittorio BERZI Istituto di Scienze Fisiche dell’ Universita, Milano, Italy, Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy Vittorio GORINI (*) Institut fur Theoretische Physik (I) der Universitat Marburg, Germany ABSTRACT. - A general formulation of the notions of space-time, reference frame and relativistic invariance is given in essentially topo- logical terms. Reference frames are axiomatized as C° mutually equi- valent real four-dimensional C°-atlases of the set M denoting space- time, and M is given the C°-manifold structure which is defined by these atlases. We attempt ti give an axiomatic characterization of the concept of equivalent frames by introducing the new structure of equiframe. In this way we can give a precise definition of space-time invariance group 2 of a physical theory formulated in terms of experi- ments of the yes-no type. -
Thermoelectric Energy Harvesting for Sensor Powering
Thermoelectric Energy Harvesting for Sensor Powering Yongjia Wu Dissertation submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy In Mechanical Engineering Lei Zuo, Chair Thomas E. Diller Rui Qiao Michael D. Heibel 05/09/2019 Blacksburg, Virginia Keywords: Thermoelectric, energy harvesting, heat transfer, sensor powering Thermoelectric Energy Harvesting for Sensor Powering Yongjia Wu ABSTRACT The dissertation solved some critical issues in thermoelectric energy harvesting and tried to broaden the thermoelectric application for energy recovery and sensor powering. The scientific innovations of this dissertation were based on the new advance on thermoelectric material, device optimization, fabrication methods, and system integration to increase energy conversion efficiency and reliability of the thermoelectric energy harvester. The dissertation reviewed the most promising materials that owned a high figure of merit (ZT) value or had the potential to increase ZT through compositional manipulation or nano-structuring. Some of the state-of-art methods to enhance the ZT value as well as the principles underneath were also reviewed. The nanostructured bulk thermoelectric materials were identified as the most promising candidate for future thermoelectric applications as they provided enormous opportunities for material manipulation. The optimizations of the thermoelectric generator (TEG) depended on the accuracy of the mathematical model. In this dissertation, a general and comprehensive thermodynamic model for a commercial thermoelectric generator was established. Some of the unnecessary assumptions in the conventional models were removed to improve the accuracy of the model. This model can quantize the effects of the Thomson effect, contact thermal and electrical resistance, and heat leakage, on the performance of a thermoelectric generator. -
Feasibility of Harvesting Solar Energy for Self-Powered Environmental Wireless Sensor Nodes
electronics Article Feasibility of Harvesting Solar Energy for Self-Powered Environmental Wireless Sensor Nodes 1, 1, 1 2 2 Yuyang Li y, Ehab A. Hamed y, Xincheng Zhang , Daniel Luna , Jeen-Shang Lin , Xu Liang 2 and Inhee Lee 1,* 1 Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA; [email protected] (Y.L.); [email protected] (E.A.H.); [email protected] (X.Z.) 2 Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA; [email protected] (D.L.); [email protected] (J.-S.L.); [email protected] (X.L.) * Correspondence: [email protected] These authors contributed equally to this work. y Received: 5 November 2020; Accepted: 30 November 2020; Published: 3 December 2020 Abstract: Energy harvesting has a vital role in building reliable Environmental Wireless Sensor Networks (EWSNs), without needing to replace a discharged battery. Solar energy is one of the main renewable energy sources that can be used to efficiently charge a battery. This paper introduces two solar energy harvesters and their power measurements at different light conditions in order to charge rechargeable AA batteries powering EWSN nodes. The first harvester is a primitive energy harvesting circuit that is built using elementary off-shelf components, while the second harvester is based on a commercial boost converter chip. To prove the effectiveness of harvesting solar energy, five EWSN nodes were distributed at a nature reserve (the Audubon Society of Western Pennsylvania, USA) and the sunlight at their locations was recorded for more than five months. For each recorded illumination, the corresponding harvested energy has been estimated and compared with the average energy consumption of the EWSN with the most power consumption. -
Power Optimization Configurations in Piezoelectric Energy Harvesting Systems
Power Optimization Configurations in Piezoelectric Energy Harvesting Systems by Kristen Thompson Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering in the Electrical Engineering Program YOUNGSTOWN STATE UNIVERSITY December 2020 Power Optimization Configurations in Piezoelectric Energy Harvesting Systems Kristen Thompson I hereby release this thesis to the public. I understand that this thesis will be made available from the OhioLINK ETD Center and the Maag Library Circulation Desk for public access. I also authorize the University or other individuals to make copies of this thesis as needed for scholarly research. Signature: Kristen Thompson, Student Date Approvals: Frank X Li, Thesis Advisor Date Mike Ekoniak, Committee Member Date Eric MacDonald, Committee Member Date Dr. Salvatore A. Sanders, Dean of Graduate Studies Date ABSTRACT Energy harvesting research from vibration gained great interest for its potential to excel in lower power applications. Often piezoelectric devices are implemented and harness the vibrational frequency as a means to excite the component. The piezoelectric device converts mechanical strain into electrical charge and exists in various prototypes. The cantilevered beam and performance are dependent on the material configuration, size, shape, and layers. This thesis analyzes several piezoelectric components to determine the best way for power optimization and efficiency in this conversion. Store purchased piezoelectric components were soldered and assembled