Nameplate and parameters of the simulated separately excited DC motor.
\\n\\n
\\n"}]',published:!0,mainMedia:{caption:"Milestone",originalUrl:"/media/original/124"}},components:[{type:"htmlEditorComponent",content:'
Barely three months into the new year and we are happy to announce a monumental milestone reached - 150 million downloads.
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\n\n\n\n\n'}],latestNews:[{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"},{slug:"intechopen-identified-as-one-of-the-most-significant-contributor-to-oa-book-growth-in-doab-20210809",title:"IntechOpen Identified as One of the Most Significant Contributors to OA Book Growth in DOAB"}]},book:{item:{type:"book",id:"9993",leadTitle:null,fullTitle:"Earthquakes - From Tectonics to Buildings",title:"Earthquakes",subtitle:"From Tectonics to Buildings",reviewType:"peer-reviewed",abstract:"Earthquakes - From Tectonics to Buildings is a compilation of eight chapters that explain tectonics and seismicity. The book presents state-of-the-art techniques for calculating ground motion amplification due to sediments above the engineering bedrock employing strong-motion recordings and microtremor data. It also explains liquefaction phenomena through interpretive structural modeling techniques. Finally, the book presents pile foundations’ seismic behavior on liquefiable soils and remedial countermeasures against earthquake attacks.",isbn:"978-1-83962-429-2",printIsbn:"978-1-83962-424-7",pdfIsbn:"978-1-83962-430-8",doi:"10.5772/intechopen.87816",price:119,priceEur:129,priceUsd:155,slug:"earthquakes-from-tectonics-to-buildings",numberOfPages:224,isOpenForSubmission:!1,isInWos:1,isInBkci:!1,hash:"1f9859a0a16af53d80bf3952fba7a272",bookSignature:"Walter Salazar",publishedDate:"May 19th 2021",coverURL:"https://cdn.intechopen.com/books/images_new/9993.jpg",numberOfDownloads:2821,numberOfWosCitations:1,numberOfCrossrefCitations:7,numberOfCrossrefCitationsByBook:0,numberOfDimensionsCitations:9,numberOfDimensionsCitationsByBook:0,hasAltmetrics:0,numberOfTotalCitations:17,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"June 9th 2020",dateEndSecondStepPublish:"September 10th 2020",dateEndThirdStepPublish:"November 9th 2020",dateEndFourthStepPublish:"January 28th 2021",dateEndFifthStepPublish:"March 29th 2021",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6,7",editedByType:"Edited by",kuFlag:!1,featuredMarkup:null,editors:[{id:"236461",title:"Dr.",name:"Walter",middleName:null,surname:"Salazar",slug:"walter-salazar",fullName:"Walter Salazar",profilePictureURL:"https://mts.intechopen.com/storage/users/236461/images/system/236461.jpg",biography:"Dr. Walter Salazar is a structural civil engineer who obtained a doctoral degree in Engineering Seismology from the Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan, in 2004. Dr. Salazar has been active in site-effects and seismic hazard research, producing several peer-reviewed maps for El Salvador, Jamaica, and the Eastern Caribbean. He has published sixty articles in peer-reviewed journals, books, and international conferences. In 2011, he received a Distinguished Salvadoran National Award. He is a peer reviewer for several scientific journals. Currently, Dr. Salazar is a Professor of Structural Engineering at the Catholic University of El Salvador.",institutionString:null,position:null,outsideEditionCount:null,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"1",institution:null}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"654",title:"Seismology",slug:"seismology"}],chapters:[{id:"74492",title:"Tentative Intracontinental Seismic Activity in South Siberia and Russian Far East",doi:"10.5772/intechopen.95073",slug:"tentative-intracontinental-seismic-activity-in-south-siberia-and-russian-far-east",totalDownloads:287,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"Overwhelming majority of minor, strong and major earthquakes in south Siberia and Russian Far East coincide with relatively narrow intracontinental zones on the boundaries of blocks and lithosphere plates. The geodynamic activity of these zones connects, besides the plate interaction, with deep lithosphere structure and anomalies of the different geophysical fields as well as with blocks’ kinematics. Authors’ located areas of the seismic centers origin and the possible manifestations of the high seismicity based on the distribution of the maximal volumes of releasing seismic energy. We established these areas, with certain care, in the northeast Altai and adjacent part of the west Sayany, in the west of the east Sayany, around the Baikal Lake and in northwest Transbaikalia, in the east of Transbaikalia between the Vitim River and upper stream of the Aldan River, and in the north of the Sakhalin Island. The majority of minor and strong, rarely major, earthquakes took place in these areas. Deep and near surface structural peculiarities influence on these areas’ geodynamics and allow establishing possible levels of seismic energy releasing. We draw areas of intensive seismic energy releasing with its calculating for each from investigated regions. They gravitate towards interblock zones, which separate crust blocks and the North Eurasian Lithosphere Plate. The fulfilled investigation allows establishing specific areas of the increased seismicity in south Siberia, Russian Far East and adjacent territories.",signatures:"Yuriy Gatinsky and Tatiana Prokhorova",downloadPdfUrl:"/chapter/pdf-download/74492",previewPdfUrl:"/chapter/pdf-preview/74492",authors:[{id:"325581",title:"Dr.",name:"Yuriy",surname:"Gatinsky",slug:"yuriy-gatinsky",fullName:"Yuriy Gatinsky"},{id:"325590",title:"Dr.",name:"Tatiana",surname:"Prokhorova",slug:"tatiana-prokhorova",fullName:"Tatiana Prokhorova"}],corrections:null},{id:"74480",title:"Tectonics and Seismicity in the periAdriatic Zones: Implications for Seismic Hazard in Italy",doi:"10.5772/intechopen.94924",slug:"tectonics-and-seismicity-in-the-periadriatic-zones-implications-for-seismic-hazard-in-italy",totalDownloads:369,totalCrossrefCites:2,totalDimensionsCites:4,hasAltmetrics:0,abstract:"The recognition of the seismic zones most prone to next major earthquakes in Italy would considerably help the choice of the most efficient prevention plan. This work describes an attempt to gain reliable information about that problem by exploiting the knowledge about the short-term development of the ongoing tectonic processes in the study area and its influence on the spatio-temporal distribution of major shocks. In the periAdriatic zones, such distribution is connected with the progressive northward displacement of the Adria plate, that is controlled by the progressive activation of the decoupling fault systems in the surrounding belts (Dinarides, Apennines and Eastern Southern Alps). The reliability of this hypothesis is evaluated by analysing the seismic histories of the periAdriatic zones. The regularity patterns that are tentatively recognised in such histories are used to identify the most probable location of next major shocks. Further insights into the present seismic hazard in the Southern Apennines and Calabria are tentatively inferred from tectonic connections between these regions and other periAdriatic zones, suggested by the seismic histories in the last 2–4 centuries and the geodynamic/tectonic context in the central Mediterranean area.",signatures:"Enzo Mantovani, Caterina Tamburelli, Daniele Babbucci, Marcello Viti and Nicola Cenni",downloadPdfUrl:"/chapter/pdf-download/74480",previewPdfUrl:"/chapter/pdf-preview/74480",authors:[{id:"326290",title:"Prof.",name:"Enzo",surname:"Mantovani",slug:"enzo-mantovani",fullName:"Enzo Mantovani"},{id:"336991",title:"Dr.",name:"Caterina",surname:"Tamburelli",slug:"caterina-tamburelli",fullName:"Caterina Tamburelli"},{id:"336993",title:"Dr.",name:"Daniele",surname:"Babbucci",slug:"daniele-babbucci",fullName:"Daniele Babbucci"},{id:"336994",title:"Dr.",name:"Marcello",surname:"Viti",slug:"marcello-viti",fullName:"Marcello Viti"},{id:"336996",title:"Dr.",name:"Nicola",surname:"Cenni",slug:"nicola-cenni",fullName:"Nicola Cenni"}],corrections:null},{id:"74466",title:"Seismological Data Acquisition and Analysis within the Scope of Citizen Science",doi:"10.5772/intechopen.95273",slug:"seismological-data-acquisition-and-analysis-within-the-scope-of-citizen-science",totalDownloads:488,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,abstract:"From 2017 till 2020 a low cost seismic sensor network was built in the southern Vienna Basin, Lower Austria, as a part of ongoing educational and citizen science projects. The purpose of the project is to inform society about the seismic activity in this area and to include authorities and interested citizens into data acquisition and exploitation. Near real time (NRT) seismic data are made accessible online. Seismic events are detected and archived automatically. The visualization of these events online facilitates instantaneously estimates of the extent of the shaking area and potential damage. Peak ground velocities (PGV) are related to macroseismic intensities (EMS-98) derived from reports about ground motion felt in the vicinity of the network stations. Observed amplitudes and travel times are modeled by simple, but effective relations. Traditional and innovative localization methods based on travel times and amplitudes are applied and analyzed with respect to data quality and localization accuracy. All results are accessible online and the computer code is open and applicable, e.g. for educational purposes.",signatures:"Ewald Brückl, Peter Carniel, Stefan Mertl and Rita Meurers",downloadPdfUrl:"/chapter/pdf-download/74466",previewPdfUrl:"/chapter/pdf-preview/74466",authors:[{id:"324000",title:"Emeritus Prof.",name:"Ewald",surname:"Brückl",slug:"ewald-bruckl",fullName:"Ewald Brückl"},{id:"339983",title:"Mr.",name:"Peter",surname:"Carniel",slug:"peter-carniel",fullName:"Peter Carniel"},{id:"339987",title:"Dr.",name:"Stefan",surname:"Mertl",slug:"stefan-mertl",fullName:"Stefan Mertl"},{id:"339993",title:"MSc.",name:"Rita",surname:"Meurers",slug:"rita-meurers",fullName:"Rita Meurers"}],corrections:null},{id:"75465",title:"Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism Indicating Causation by Nearby Oil Production",doi:"10.5772/intechopen.94923",slug:"seismicity-at-newdigate-surrey-during-2018-2019-a-candidate-mechanism-indicating-causation-by-nearby",totalDownloads:276,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,abstract:"During 2018–2019, oil was intermittently produced from the Late Jurassic Upper Portland Sandstone in the Weald Basin, southeast England, via the Horse Hill-1 and Brockham-X2Y wells. Concurrently, a sequence of earthquakes of magnitude ≤3.25 occurred near Newdigate, ∼3 km and ∼8 km from these wells. The pattern, with earthquakes concentrated during production from this Portland reservoir, suggests a cause-and-effect connection. It is proposed that this seismicity occurred on a patch of fault transecting permeable Dinantian limestone, beneath the Jurassic succession of the Weald Basin, hydraulically connected to this reservoir via this permeable fault and the permeable calcite ‘beef’ fabric within the Portland sandstone; oil production depressurizes this reservoir and draws groundwater from the limestone, compacting it and ‘unclamping’ the fault, reaching the Mohr-Coulomb failure criterion and causing seismicity. In principle this model is fully testable, but required data, notably the history of pressure variations in the wells, are not currently in the public domain. Quantitative estimates are, nonetheless, made of the magnitudes of the variations, arising from production from each well, in the state of stress on the seismogenic Newdigate fault. The general principles of this model, including the incorporation of poroelastic effects and effects of fault asperities into Mohr-Coulomb failure calculations, may inform understanding of anthropogenic seismicity in other settings.",signatures:"Rob Westaway",downloadPdfUrl:"/chapter/pdf-download/75465",previewPdfUrl:"/chapter/pdf-preview/75465",authors:[{id:"330592",title:"Dr.",name:"Rob",surname:"Westaway",slug:"rob-westaway",fullName:"Rob Westaway"}],corrections:null},{id:"74862",title:"S-Wave Site Amplification Factors from Observed Ground Motions in Japan: Validation of Delineated Velocity Structures and Proposal for Empirical Correction",doi:"10.5772/intechopen.95478",slug:"s-wave-site-amplification-factors-from-observed-ground-motions-in-japan-validation-of-delineated-vel",totalDownloads:309,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,abstract:"We first derived site amplification factors (SAFs) from the observed strong motions by the Japanese nationwide networks, namely, K-NET and KiK-net of National Institute of Earthquake Research and Disaster Resilience and Shindokei (Instrumental Seismic Intensity) Network of Japan Meteorological Agency by using the so-called generalized spectral inversion technique. We can use these SAFs for strong motion prediction at these observation sites, however, we need at least observed weak motion or microtremor data to quantify SAF at an arbitrary site. So we tested the capability of the current velocity models in Japan whether they can reproduce or not the observed SAFs at the nearest grid of every 250 m as the one-dimensional theoretical transfer functions (TTF). We found that at about one-half of the sites the calculated 1D TTFs show more or less acceptable fit to the observed SAFs, however, the TTFs tend to underestimate the observed SAFs in general. Therefore, we propose a simple, empirical method to fill the gap between the observed SAFs and the calculated TTFs. Validation examples show that our proposed method effectively predict better SAFs than the direct substitute of TTFs at sites without observed data.",signatures:"Eri Ito, Kenichi Nakano, Shigeki Senna and Hiroshi Kawase",downloadPdfUrl:"/chapter/pdf-download/74862",previewPdfUrl:"/chapter/pdf-preview/74862",authors:[{id:"330550",title:"Prof.",name:"Hiroshi",surname:"Kawase",slug:"hiroshi-kawase",fullName:"Hiroshi Kawase"},{id:"330553",title:"MSc.",name:"Eri",surname:"Ito",slug:"eri-ito",fullName:"Eri Ito"},{id:"330554",title:"Dr.",name:"Kenichi",surname:"Nakano",slug:"kenichi-nakano",fullName:"Kenichi Nakano"},{id:"342138",title:"Dr.",name:"Shigeki",surname:"Senna",slug:"shigeki-senna",fullName:"Shigeki Senna"}],corrections:null},{id:"74560",title:"Assessing Seismic Site Response at Areas Characterized by a Thick Buried Low-Velocity Layer",doi:"10.5772/intechopen.95277",slug:"assessing-seismic-site-response-at-areas-characterized-by-a-thick-buried-low-velocity-layer",totalDownloads:252,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"Earthquake ground motion is dependent on various factors, including local ground conditions. Whilst many studies have characterized the effect of having outcropping “soft” geological layers which have the ability to amplify ground motion, there is minimal literature on the effect of having such layers embedded between two harder layers. This situation creates a seismic wave velocity inversion. The Maltese islands (Central Mediterranean) present a good opportunity for the study of velocity inversion as almost half of the islands are characterized by a thick buried layer of clay. The results presented in this chapter are a combination of studies which have been conducted on the Maltese islands, using non-invasive geophysical prospecting techniques in areas characterized by a thick buried low-velocity layer, to characterize the response of earthquake ground shaking in such geological situations.",signatures:"Daniela Farrugia, Pauline Galea and Sebastiano D’Amico",downloadPdfUrl:"/chapter/pdf-download/74560",previewPdfUrl:"/chapter/pdf-preview/74560",authors:[{id:"58678",title:"Dr.",name:"Pauline",surname:"Galea",slug:"pauline-galea",fullName:"Pauline Galea"},{id:"330447",title:"Dr.",name:"Daniela",surname:"Farrugia",slug:"daniela-farrugia",fullName:"Daniela Farrugia"},{id:"339851",title:"Dr.",name:"Sebastiano",surname:"D'Amico",slug:"sebastiano-d'amico",fullName:"Sebastiano D'Amico"}],corrections:null},{id:"76179",title:"Elucidation of Seismic Soil Liquefaction Significant Factors",doi:"10.5772/intechopen.97278",slug:"elucidation-of-seismic-soil-liquefaction-significant-factors",totalDownloads:235,totalCrossrefCites:2,totalDimensionsCites:2,hasAltmetrics:0,abstract:"The paper develops a framework to analyze the interactions among seismic soil liquefaction significant factors using the interpretive structural model (ISM) approach based on cone penetration test. To identify the contextual relationships among the significant factors, systematic literature review approach was used bearing in mind the selection principle. Since multiple factors influence seismic soil liquefaction, determining all factors in soil liquefaction would be extremely difficult, as even a few seismic soil liquefaction factors are not easy to deal with. This study highlighted two main characteristics of seismic soil liquefaction factors. First, the seismic soil liquefaction factors–peak ground acceleration F2 (amax), equivalent clean sand penetration resistance F5 (qc1Ncs), and thickness of soil layer F11 (Ts) influenced soil liquefaction directly and were located at level 2 (top level) in the ISM model, meaning they require additional seismic soil liquefaction factors except thickness of soil layer F11 (Ts) to collaboratively impact on soil liquefaction potential. The multilevel hierarchy reveals that depth of soil deposit F10 (Ds) is formed the base of ISM hierarchy. Secondly, Matrice d’impacts croisés multiplication appliqués à un classement (MICMAC) analysis has been employed for evaluating these identified factors in accordance with driving power and dependence power. Factors with a higher driving power should be given special consideration. Autonomous soil liquefaction factors have no reliance on other soil liquefaction factors and interfere less. In order to identify the significant factors that affect seismic soil liquefaction susceptibility, the model built in this study clearly illustrates the complex relationships between factors and demonstrates the direct and indirect relationships.",signatures:"Mahmood Ahmad, Xiaowei Tang, Feezan Ahmad, Marijana Hadzima-Nyarko, Ahsan Nawaz and Asim Farooq",downloadPdfUrl:"/chapter/pdf-download/76179",previewPdfUrl:"/chapter/pdf-preview/76179",authors:[{id:"298331",title:"Dr.",name:"Mahmood",surname:"Ahmad",slug:"mahmood-ahmad",fullName:"Mahmood Ahmad"},{id:"329439",title:"Prof.",name:"Xiaowei",surname:"Tang",slug:"xiaowei-tang",fullName:"Xiaowei Tang"}],corrections:null},{id:"74449",title:"The Dynamic Behaviour of Pile Foundations in Seismically Liquefiable Soils: Failure Mechanisms, Analysis, Re-Qualification",doi:"10.5772/intechopen.94936",slug:"the-dynamic-behaviour-of-pile-foundations-in-seismically-liquefiable-soils-failure-mechanisms-analys",totalDownloads:605,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"This chapter presents a concise overview of the mechanics of failure, analysis and requalification procedures of pile foundations in liquefiable soils during earthquakes. The aim is to build a strong conceptual and technical interpretation in order to gain insight into the mechanisms governing the failure of structures in liquefaction and specify effective requalification techniques. In this regard, several most common failure mechanisms of piles during seismic liquefaction such as bending (flexural), buckling instability and dynamic failure of the pile are introduced. Furthermore, the dynamic response commentary is provided by critically reviewing experimental investigations carried out using a shaking table and centrifuge modelling procedures. The emphasis is placed on delineating the concept of seismic design loads and important aspects of the dynamic behaviour of piles in liquefiable soils. In this context, using Winkler foundation approach with the proposed p–y curves and finite-element analyses in conjunction with numerical analysis methods, are outlined. Moreover, the feasibility of successful remediation techniques for earthquake resistance is briefly reviewed in light of the pile behaviour and failure. Finally, practical recommendations for achieving enhanced resistance of the seismic response of pile foundation in liquefiable soil are provided.",signatures:"Rohollah Rostami, Slobodan B. Mickovski, Nicholas Hytiris and Subhamoy Bhattacharya",downloadPdfUrl:"/chapter/pdf-download/74449",previewPdfUrl:"/chapter/pdf-preview/74449",authors:[{id:"225058",title:"Dr.",name:"Slobodan B.",surname:"Mickovski",slug:"slobodan-b.-mickovski",fullName:"Slobodan B. Mickovski"},{id:"324691",title:"Ph.D. Student",name:"Rohollah",surname:"Rostami",slug:"rohollah-rostami",fullName:"Rohollah Rostami"},{id:"324955",title:"Prof.",name:"Subhamoy",surname:"Bhattacharya",slug:"subhamoy-bhattacharya",fullName:"Subhamoy Bhattacharya"},{id:"324956",title:"Prof.",name:"Nicholas",surname:"Hytiris",slug:"nicholas-hytiris",fullName:"Nicholas Hytiris"}],corrections:null}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},subseries:null,tags:null},relatedBooks:[{type:"book",id:"2227",title:"Tectonics",subtitle:"Recent Advances",isOpenForSubmission:!1,hash:"3b4c2f80af61284334fb3655852de9f7",slug:"tectonics-recent-advances",bookSignature:"Evgenii Sharkov",coverURL:"https://cdn.intechopen.com/books/images_new/2227.jpg",editedByType:"Edited by",editors:[{id:"32743",title:"Prof.",name:"Evgenii",surname:"Sharkov",slug:"evgenii-sharkov",fullName:"Evgenii 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by",editors:[{id:"64160",title:"Prof.",name:"Simona",middleName:null,surname:"Clichici",slug:"simona-clichici",fullName:"Simona Clichici"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},ofsBook:{item:{type:"book",id:"11439",leadTitle:null,title:"Crisis Management - Principles, Roles and Application",subtitle:null,reviewType:"peer-reviewed",abstract:"
\r\n\tThe word "crisis" can show up anywhere, anytime - it can be an acute urgency or a long-term response; it can happen at an individual level to a global scale. It is an interdisciplinary word that takes into account geographical, demographical, religious, social, and climate differences, and, by how it is managed, consequently, can have a difference in its results. We have seen how seemingly far-away issues became "my problem" in the past.
\r\n\r\n\tThe Subprime Mortgage Crisis affected a credit crash in the international financial market. The Fukushima nuclear power plant accident threatened health problems of the Asian countries, while the radioactivity problem remains an international crisis. The Syrian Civil War, from the country's internal economic recession, job disruptions, poor harvests affected by climate change, has raised a large refugee migration crisis. The COVID-19 pandemic has forced our community, from our small family group to a global scale, to adapt to unprecedented events to the continuing instability, anxiety, and uncertainty in our lives. There is a potential crisis with personal information or identity due to the misuse or insecurity of the data management. At the core of all, these seemingly unrelated types of crises are the resultant risk and burden of consequences that can affect all individuals.
\r\n\r\n\t
\r\n\tThe purpose of this book is to provide the readers with an understanding of the characteristics of the crisis itself, recognize the wide range and multi-layer of the crisis from a real situation, give ideas on how to minimize the damage, and find ways to increase resilience in the future. To adapt to the rapidly and diversely changing world, the necessary experience and appropriate management for all kinds of crisis issues will be discussed as well. At the same time, it is intended to suggest elements such as verified scientific and empirical knowledge and applicable technologies; more effective risk management operation; modeling of the risks, manuals, management plans, and strategies.
\r\n\t
In a coincidence with the title of this book, we can start with explaining what a term of analysis means: in general, it is a system of methods by means of which properties of investigated matters are gained. Here the properties of electrical machines are analyzed; therefore, it is welcomed to introduce the methods on how to do it.
To proceed in the investigation of transients and steady-state condition, it is necessary to know equivalent circuit parameters (resistances and inductances). The first method that is given below in Chapter 1 is the method of parameters calculated based on the design process in which geometrical dimensions and material properties must be known (see [1, 2, 3]). The other method on how to get the equivalent circuit parameters is to make measurements and testing, but this can be done only on the fabricated pieces. It is a very welcomed method on how to verify the calculated parameters gained during the design process. The method of measurement is not given here and can be found, e.g., in [4, 5, 6].
The other approach is if there is a real fabricated machine but without any documents and data. Then it is very useful to make the so-called inverse design calculation. It means to take all geometrical dimensions which can be seen on the real machine and get data from the real machine nameplate and catalogues, e.g., voltage, current, power, speed, pole numbers, slot shape, number of slots, number of conductors in the slot, etc. and to continue in the calculation of air gap magnetic flux density, etc. to the required parameters.
The general theory of electrical machines is presented in Chapter 2. It is possible to investigate transients and steady-state conditions of electrical machines by means of this theory, see [7, 8, 9]. The transients are solved on the basis of differential equations in which the parameters of equivalent circuits, i.e., resistances and inductances, are needed. The accuracy and reliability of the simulation results depend on the accuracy of the parameter values. Therefore, determination of the parameters must be done with the highest care.
Chapter 3 is formulated in this sense. This chapter is devoted to the modern computer method called finite element method (FEM), see also [10]. This method enables to investigate not only some parameters, mainly magnetizing inductances, but also the other properties such as losses, air gap developed torque, ripple torque, efficiency, etc., see [11, 12, 13].
It is important to add that verification of the calculated and simulated values and waveforms is made by measurement on a real machine if it exists. It is recommended to create a reliable simulation model. It means the gained simulated outputs verify on a real machine, and if the coincidence of measured and simulated values is satisfactory, such model can be employed to optimize geometrical dimensions or a concrete configuration, like slot shape, rotor barriers, etc., and the simulation outputs are considered as reliable. In such a way, it is possible in pre-manufacturing period to optimize the construction of the machine to the required properties, e.g., maximal torque, minimal ripple torque, maximum efficiency, and so on.
The authors would like to point out that all simulation models of electrical machines analyzed in this book are shown at the end as Appendixes A, B, C, and D, in the MATLAB-Simulink program. These models were employed in the appropriate chapters during the investigation of the simulation waveforms of all electrical machines.
In the rotating electrical machines, the total magnetic flux can be divided into two components: main flux (air gap flux) and leakage flux. The main flux enables electromagnetic energy conversion, but a proportion of the total flux does not participate in energy conversion, and this part is called leakage flux. The main flux must cross the air gap of rotating machine and its function is electromagnetically connected to both stator and rotor windings. The leakage flux is linked only with this winding in which it was created.
The main magnetic flux in air gap
Magnetizing inductance is the most important inductance of electrical machine and is defined by the winding, geometrical dimensions of the magnetic circuit, and the employed materials. It is derived from m-phase machine on the basis of magnetic flux density distribution on the surface of the rotor and its maximal value
Distribution of the magnetic flux density fundamental harmonic in the air gap over a pole pitch and on the length of the
This relationship is valid for machines without ventilating ducts and takes into account the magnetic flux distribution on the end of the machines [1].
Maximal value of the air gap magnetic flux can be calculated as a surface integral of magnetic flux density
where
The sinusoidal distributed waveform of the magnetic flux density is
An expression “maximal value of the magnetic flux” means maximal flux which penetrates the surface created by the coil and therefore creates one phase maximal linkage flux of the winding on the stator (the subscript “ph” is used), with number of turns
Magnitude of air gap magnetic flux density can be expressed on the basis of current linkage and the relationship between magnetic flux density and intensity of magnetic field. In the most simple case,
Then the magnitude of the magnetic flux density is:
and after the substitution to the expression of flux linkage, it yields:
By dividing the result by the peak value of the current, which in this case is magnetizing current, we obtain the magnetizing inductance of a single-phase winding (the main inductance):
If the winding is multiphase, the magnetic flux is created by all the phases of the winding, with the corresponding instantaneous values of the currents. As it is known, the three-phase winding creates the magnitude of the air gap magnetic flux density equal to 1.5 multiple of that which is created by the single phase. The magnetizing inductance of an
and after the substitution of pole pitch, the expression is obtained, in which its dependence on the parameters and geometrical dimensions of the machine is seen:
Here it is seen that the magnetizing inductance depends on magnetic circuit saturation, i.e.,
Effective air gap
Magnetizing inductance is not constant but depends on the voltage and the torque. The higher voltage activates the higher magnetic flux density; this activates higher saturation of magnetic circuit, and this requires higher magnetizing current.
Leakage inductances are described by the leakage magnetic fields, which are linked only with the turns of that winding by which they were created. It means they do not cross the air gap.
In greater detail, it can be said that leakage magnetic fluxes include the following:
All components of magnetic field that do not cross air gap
Those components of magnetic field that cross the air gap, but they do not take part in the electromechanical energy conversion
The leakage fluxes that do not cross the air gap can be divided into the next components:
Slot leakage flux
Tooth tip leakage flux
End winding leakage flux
Pole leakage flux
Leakage fluxes that do not cross the air gap are included into the air gap magnetic flux
According to the electrical motor design tradition, leakage inductance
Skewing slot factor defines skewing leakage inductance:
where the factor of the skewing leakage
At the skewing by one slot, this factor is given by the expression:
where
Solution: As it is known,
The number of stator slots per phase per pole
1 | −17 | 19 | −35 | 37 | |
0.995 | 0.06 | −0.05 | −0.03 | 0.03 |
It can be seen that the lowest order stator harmonics (−17, 19, −35, 37) have very small skewing factors, and thereby their effects are eliminated to a great degree (3–6%). The fundamental harmonic is reduced only by 0.5%.
Electromotive force (induced voltage) is given by the magnetizing inductance as a result of a propagating fundamental component of air gap flux density. Because of a spatial slotting and winding distribution, the permeance harmonics induce voltage of fundamental frequency of the winding. The air gap leakage inductance, i.e., the harmonic leakage inductance components, takes this into account. In integer slot machines per phase per pole
The sum of all induced voltages from all harmonic components, fundamental included, gives the basis for the calculation of the total inductance, which is the sum of magnetizing and air gap leakage inductance:
The expression for the fundamental harmonic, i.e.,
The air gap leakage inductance can be expressed also by means of the air gap leakage factor:
Then the leakage inductance is given by:
Of course, in Eq. (16), only the harmonics that are created by the given winding are used.
This inductance is created by a real leakage flux, which is closed through the space of the slot. Magnetic permeance of the magnetic circuit is taken as infinite; therefore the length of the magnetic circuit force line is taken only by the width of the slot (in the slot there is nonmagnetic material, i.e., there the vacuum permeability is used). For the rectangular slot, the magnetic permeance is derived in [1]. Gradually integrating the magnetic force lines and magnetic flux density along the slot height
Geometrical dimensions of various slot shapes to define their permeance factors, the calculations of which are given in the text.
The permeance factor
For the rectangular slot (Figure 2a) with the slot width
In the slot area with the height
The sum
where
Equation (22) is derived from [1] and other references dealing with the machine design.
The expressions gained on the basis of magnetic permeance integration along the slot height of single-layer winding according to Figure 2 are given.
For the slot shapes b, c, d, e, and f, the permeance factor will be calculated as follows:
For the slot in Figure 2g, the next expression is known:
and for the round slot from Figure 2h, the next expression is valid:
The slot leakage inductance of the double-layer winding, according to [1], on the basis of Figure 3, the appropriate expression is derived. It is necessary to consider that in some slots there are coil sides of different phases.
Geometry of the slot with double-layer winding.
where
The factor
but the factors
Similar to Eq. (26), also the equation for double-layer winding can be written for the slots in Figure 2 from (b) till (f):
and for slots from (g) till (i):
These expressions are valid also for the slots with single-layer winding, if
If the winding is short pitching , factors
For three-phase winding:
where
For two-phase winding:
Compare the results with the slot leakage inductance of a corresponding double-layer full-pitch winding.
and slot leakage inductance is according to Eq. (22):
for a double-layer full-pitch winding
The slot leakage inductance is now:
It is seen that the phase shift of the different phase coil sides in the double-layer winding causes a smaller slot leakage inductance for the short-pitched winding than the full-pitch winding. The slot leakage inductance in this case is about 10% smaller for the short-pitched winding.
The tooth tip leakage inductance is determined by the magnitude of leakage flux flowing in the air gap outside the slot opening. This flux is illustrated in Figure 4. The current linkage in the slot causes a potential difference between the teeth on opposite sides of the slot opening, and as a result a part of the current linkage will be used to produce the leakage flux of the tooth tip.
Tooth tip flux leakage around a slot opening, creating a tooth tip leakage inductance.
Tooth tip leakage inductance can be determined by applying a permeance factor:
where
In the machines with salient poles, the air gap is taken at the middle of the pole, where the air gap is smallest. If the air gap is selected to be infinite, a limit value of 1.25 is obtained, which is the highest value for permeance factor
The factor
Then the factor of tip tooth permeance is:
and the tooth tip leakage inductance is:
In Example 2, the slot leakage inductance was 0.34 mH. As the air gap in a rotor surface magnet machine is long, the tooth tip leakage has a significant value, about 70% of the slot leakage inductance.
End winding leakage flux results from all the currents flowing in the end windings. The geometry of the end windings is usually difficult to analyze, and, further, all the phases of polyphase machines influence the occurrence of a leakage flux. Therefore, the accurate determination of an end winding leakage inductance would require three-dimensional numerical solution. On the other side, the end windings are relatively far from the iron parts; the end winding inductances are not very high. Therefore, it suffices to employ empirically determined permeance factor.
According to Figure 5, the end winding leakage flux is a result of influence of all coil turns belonging to the group coils
Leakage flux and dimensions of the end winding.
If according to Eq. (22) this
The average length of the end winding
where
and the length of the end winding is:
The product of the length and permeance factor is:
and end winding leakage inductance is:
The slot leakage inductance of the 5/6 short-pitched winding in Example 2 is 0.304 mH, so it is seen that the slot leakage inductance is much higher than end winding leakage inductance.
Not only inductances but also resistances are very important parameters of electrical machines. In many cases winding losses are dominant components of the total loss in electrical machines. The conductors in electrical machines are surrounded by ferromagnetic material, which at saturation can encourage flux components to travel through the windings. This can cause large skin effect problems, if the windings are not correctly designed. Therefore, this phenomenon must be considered.
It is convention to define at first the DC resistance
Resistance is highly dependent on the running temperature of the machine; therefore a designer should be well aware of the warming-up characteristics of the machine before defining the resistances. Usually the resistances are determined at the design temperature or at the highest allowable temperature for the selected winding type.
Windings are usually made of copper. The specific conductivity of pure copper at room temperature, which is taken 20°C, is
The accurate definition of the winding length in an electrical machine is a fairly difficult task. Salient-pole machines are a relatively simple case: the conductor length can be defined more easily when the shape of the pole body and the number of coil turns are known. Instead winding length of slot windings is difficult, especially if coils of different length are employed in the machine. Therefore, empirical expressions are used for the length calculation.
The average length of a coil turn of a slot winding
where
When the voltage is between 6 and 11 kV, the next can be used:
After the average length is determined, the DC resistance may be calculated according to Eq. (39), by taking all the turns and parallel paths into account.
The alternating current in a conductor and currents in the neighboring conductors create an alternating flux in the conductor material, which causes skin and proximity effects. In the case of parallel conductors, also circulating currents between them appear. The circulating currents between parallel conductors can be avoided by correct geometrical arrangement of the windings. In this chapter, the skin and proximity effects will be dealt together and called the skin effect.
Skin effect causes displacement of the current density to the surface of the conductor. If the conductor is alone in the vacuum, the current density is displaced in all directions equally to the conductor surface. But in the conductors embedded in the slots of electrical machines, the current density is displaced only in the direction to the air gap. In this manner, the active cross-section area of the conductors is reduced, increasing the resistance. This resistance increase is evaluated by means of resistance factor. It is the ratio of the alternating current resistance and direct current resistance:
The direct consequence of the resistance increase is loss increase; therefore, the resistance factor can be expressed also by the ratio of the losses at alternating current and direct current:
Loss increase because of skin effect is the reason why it is necessary to deal with this phenomenon in the period of the machine design and parameter determination. Resistance and losses at alternating current can be calculated on the basis of Eq. (43) or (44), if the resistance factor is calculated by means of the equations given below.
In electrical machines the skin effect occurs mainly in the area of the slot but also in the area of end winding. The calculation in these two areas must be made separately, because the magnetic properties of the material in the slot and its vicinity and the end winding are totally different.
Analytical calculation of resistance factor which includes skin effect influence is given in many books dealing with this topic; therefore, here only expressions needed for resistance factor calculation are shown. An important role in the theory of skin effect is the so-called depth of penetration, meaning the depth to which electromagnetic wave penetrates into a material at a given frequency and material conductivity. The depth of penetration depends on the frequency of alternating current, specific electric conductivity of the conductor material
Determination of reduced conductor height (a) if in the slot with width
The conductor height
Note that the product of specific electric conductivity and the ratio of conductor width to the slot width express the reduced electric conductivity of the slot area
If in the slot there are rectangular conductors placed
The resistance factor of the
where the functions
Equation (48) shows that the resistance factor is smallest on the bottom layer and largest on the top layer. This means that in the case of series-connected conductors, the bottommost conductors contribute less to the resistive losses than the topmost conductors. Therefore, it is necessary to calculate the average resistance factor over the slot:
where
If
Equations above are valid for rectangular conductors. The eddy current losses (skin effect losses) of round wires are 0.59 times the losses of rectangular wire. If in the slot there are round conductors, resistance factor and also the eddy current losses are only 59% of that appeared in the rectangular conductors. Therefore, for the round conductors, Eq. (52) will have a form:
An effort of the designer is to reduce the resistance factor what would result in the reduction of the losses. Obviously, it is recommended to divide the height of the conductor: it means to make more layers
If the conductors are divided into parallel subconductors, which are connected together only at the beginning and at the end of winding, they must be also transposed to achieve effect of the reduction of the resistance factor and of the eddy current (skin effect) losses. Transposition must be made in such a way that all conductors are linked with the equal leakage magnetic field. It means that the changing of the conductor positions must ensure that all conductors engage all possible positions in the slot regarding the leakage magnetic flux. Without transposition of the subconductors, a divided conductor is fairly useless. Further details are given in [1, 2, 3].
If the height of the conductor is large, or if the winding is created only by one bar, as it is in the case of the squirrel cage of asynchronous machines, in the conductors with alternating current, skin effect appears. The skin effect is active also at the medium frequencies and has considerable influence on the resistance (see Section 3.1) and on the winding inductance too. That part of winding, which is positioned on the bottom of the slot, is surrounded by the higher magnetic flux than that on the top of the slot. Therefore, the winding inductance on the top of the slot is lower than that on the bottom of the slot, and therefore the time-varying current is distributed in such a way that the current density on the top of the slot is higher than that on the bottom of the slot. The result is that skin effect increases conductor resistance and reduces slot leakage inductance. To express the decrease of the inductance, the so-called skin effect factor
To calculate the skin effect factor, we need to define the reduced conductor height:
where
where
In the cage armature,
In the cage armature, it is usually
The permeance factor of the slot opening is:
The reduced height
Then the inductance skin effect factor is:
and permeance factor of the slot under the skin effect is:
The leakage inductance of a squirrel cage aluminum bar if skin effect is considered is:
The theory of individual types of electrical machines from the historical point of view was developed independently. Therefore, also terminology and signing of symbols and subscripts were determined independently. This theory was devoted to the investigation of steady-state conditions and quantities needed for design calculation of electrical machines.
However, the basic principles of electrical machines are based on common physical laws and principles, and therefore a general theory of electrical machines was searched. The first person, who dealt with this topic was Gabriel Kron, who asked the question: “Does a universal arrangement of electrical machine exist from which all known types of electrical machines could be derived by means of simple principles?” An answer to this question resulted in the fact that around the year 1935, G. Kron mathematically formulated general theory and defined universal electrical machine, which at various connections represented most of the known machines.
Kron’s theory employed tensor analysis and theory of multidimensional non-Euclidean spaces and therefore was misunderstood and non-employed by majority of the technical engineers. After the year 1950, the first works appeared, in which Kron’s theory was simplified and therefore better understood. But only after the personal computers (PC) were spread in a great measure and suitable software facilities were available, the general theory of electrical machines became an excellent working means for investigation of electrical machine properties. Nowadays it represents inevitable equipment of technically educated experts in electrical engineering.
A fundamental feature of the general theory of electrical machines is based on the fact that it generalizes principles and basic equations of all electrical machines on the common base, and in such a way it simplifies their explanation and study. Its big advantage is that it formulates equations of electrical machines in such a form that they are valid in transients as well as in steady-state conditions. In this theory the electrical machines are presented as a system of the stationary and moving mutual magnetically linked electrical circuits, which are defined by the basic parameters: self-winding and mutual winding inductances, winding resistances, and moment of inertia, see [7, 8, 9].
The general theory of electrical machines is general in such a sense that it is common for a majority of electrical machines and explains their basic properties and characteristics on the basis of common principles. Further it is applicable for various running conditions: steady state, transients, unsymmetrical, and if they are fed by frequency converter at a non-sinusoidal voltage waveform. On the other side, the electrical machines are idealized by simplifying assumptions.
These simplifying assumptions enable to simplify equations, mainly their solutions. Here are some of them:
It should be noted that slotting is not ignored totally. In cooperation with the finite element method (FEM), it is possible to receive waveform of air gap magnetic flux density, where the influence of slotting is clearly seen. It is a distorted waveform for which harmonic analysis must be made and to determine components of the harmonic content. For each harmonic component, the induced voltage can be calculated, and the total induced voltage is given by the sum of all components. In this case the slotting influence is included in the value of the induced voltage.
Next, an arrangement of the universal machine, on the basis of which the general theory was derived, will be given.
A two-pole commutator machine is taken. The theory spread to the multipole arrangement will be carried out if mechanical angles are converted to the electrical angles and mechanical angular speed to the electrical angular speed:
The typical phenomenon of the universal machine is that its windings are located in two perpendicular axes to each other: The direct axis is marked “d” and quadrature axis marked q (see Figure 7).
Design arrangement of the universal machine with marked windings.
(a) Replacement of the rotor’s winding by solenoid, (b) creation of the quasi-stationary q-winding, and (c) creation of the quasi-stationary d-winding.
Stator has salient poles with one or more windings on the main poles in the d-axis and q-axis. In Figure 7, windings f and D are in the d-axis and windings g and Q in the q-axis. These windings can represent field winding (external, shunt, series, according to the connection to the armature), damping, commutating, compensating, and so on (see chapter about the DC machines) or, as we will see later, three-phase winding transformed into the two-axis system d, q.
The rotor’s winding with commutator expressed oneself as the winding of the axis that goes through the brushes. If the rotor rotates, conductors of the coils change their position with regard to the stator and brushes, but the currents in the conductors which are located in one pole pitch have always the same direction. In other words, there exists always such conductor which is in a specific position, and the current flows in the given direction.
Therefore, the rotor’s winding with commutator seems to be, from the point of view of magnetic effect, stationary; it means it is quasi-stationary. The magnetic flux created by this winding has always the same direction, given by the link of the given brushes. It is valid for the winding in d-axis and q-axis.
On the commutator, there are two sets of the brushes: one, on terminals of which is voltage
The number of the equations is given by the number of windings. All windings are taken as consumer of the energy. Then the terminal voltage equals the sum of the voltage drops in the windings. The power in the windings is positive; therefore, the voltage and current have the same directions and are also positive. The voltage equations are written according to the 2nd Kirchhoff’s law and Faraday’s law for each winding. By these equations three basic variables of the given winding, voltage, current, and linkage magnetic flux, are linked. The voltage equation in general for j-th winding, where j = f, D, g, Q, is in the form:
where
The linkage magnetic fluxes of the windings are given by the magnetic fluxes created by the currents of the respected winding and those windings which are magnetically linked with them. In general, any winding, including rotors, can be written as:
For example, for f-winding, the following is valid:
Equations written in detail for all windings are as follows:
In these equations formally written in the order of the windings and their currents, it is shown also, which we already know, that their mutual inductances are zero, because their windings are perpendicular to each other, which results in zero mutual inductance.
Rotor winding is moving with an angular speed
Rotor winding is composed of two parts, one is located in the d-axis, leading up to the terminals in the d-axis, and the second in the q-axis, leading up to the terminals in the q-axis. In Figures 9 and 10, it is shown that not both windings have both voltage components from both linkage magnetic fluxes.
Illustration of the induced voltage in the q-axis.
Illustration of the induced voltage in the d-axis.
Transformation voltage created by the time variation of
Linkage magnetic flux
A movement of the q-winding in the marked direction (Figure 9) does not cause any rotating induced voltage from
Rotating voltage in q-winding is induced by crossing the magnetic force lines
whereby there the known expression from the theory of electrical machines was used:
This expression can be modified to the generally written mode; it means without regard which axis the winding is, in such a way that it will be written by means of electrical angular speed and linkage magnetic flux.
where half of the conductors
In this expression the effective number of turns of one parallel path
Now the relationship for the terminal voltage in q-axis can be written in the form:
As shown in magnetic flux directions, movement and direction of currents in Figure 9 are in coincidence with the rule of the left hand, meaning for motor operation (consumer). Therefore, all signs in front of the voltages in Eq. (74) are positive.
In d-winding there is induced transformation voltage done by time varying of
In Figure 10, it is seen that current, magnetic flux, and moving directions are in coincidence with the right-hand rule, which is used for generator as a source of electrical energy.
Therefore, the sign in front of the rotating voltage is negative.
Power and electromagnetic torque of the universal machine will be derived on the basis of energy equilibrium of all windings of the whole machine: We start with the voltage equations of stator and rotor windings, which are multiplied with the appropriate currents and time d
For the stator windings, Eq. (63) will be used for the terminal voltage of each winding. This equation will be multiplied by
For the rotor winding in the d-axis, Eq. (75) multiplied by
For the rotor winding in the q-axis, Eq. (74) multiplied by
Now the left sides and right sides of these equations are summed, and the result is an equation in which energy components can be identified:
The expression on the left side presents a rise of the delivered energy during the time d
The first expression on the right side presents rise of the energy of the Joule’s losses in the windings:
The second expression on the right side is an increase of the field energy:
The last expression means a rise of the energy conversion from electrical to mechanical form in the case of motor or from mechanical to electrical form in the case of generator:
The instantaneous value of the electromagnetic power of the converted energy can be gained if the expression for energy conversion will be divided by time d
where
If left and right sides of Eqs. (80) and (81) are put equal, an expression for the instantaneous value of the electromagnetic torque in general theory of electrical machines yields:
If the motoring operation is analyzed, it can be seen that at known values of the terminal voltages (six equations) and known parameters of the windings, there are seven unknown variables, because except six currents in six windings there is also angular rotating speed, which is an unknown variable. Therefore, further equation must be added to the system. It is the equation for mechanical variables:
in which it is expressed that developed electromagnetic torque given by Eq. (82) covers not only the energy of the rotating masses
Therefore, from the last two equations, the time varying of the mechanical angular speed can be calculated:
For the time varying of the electrical angular speed, which is directly linked with the voltage equations, we get:
These equations will be simulated if transients of electrical machines are investigated.
If the equivalent circuit of the universal machine and equivalent circuits of the DC machines are compared in great detail, it can be seen that the basic principle of the winding arrangement in two perpendicular axes is very well kept in DC machines. It is possible to find a coincidence between generally defined windings f, D, Q, g, d, and q and concrete windings of DC machines, e.g., in this way:
The winding “f” represents field winding of DC machine.
The winding “D” either can represent series field winding in the case of compound machines, whereby the winding “f” is its shunt field winding, or, if it is short circuited, can represent damping effects during transients of the massive iron material of the machines. However, it is true that to investigate the parameters of such winding is very difficult [1].
The windings “g” and “Q” can represent stator windings, which are connected in series with the armature winding, if they exist in the machine. They can be commutating pole winding and compensating winding.
Windings “d” and “q” are the winding of the armature, but in the case of the classical construction of DC machine, where there is only one pair of terminals, and eventually one pair of the brushes in a two-pole machine, only q-winding and terminals with terminal voltage
The modified equivalent circuit of the universal electrical machine applied to DC machine is in Figure 11.
Modified equivalent circuit of the universal machine applied on DC machine.
The field winding of the separately excited DC machine is fed by external source of DC voltage and is not connected to the armature (see Figure 12). Let us shortly explain how the directions of voltages, currents, speed, and torques are drawn: The arrowhead of the induced voltage is moving to harmonize with the direction of the magnetic flux in the field circuit. The direction of this movement means the direction of rotation and of developed electromagnetic (internal) torque. The load torque and the loss torque (the torque covering losses) are in opposite directions. The source of voltage is on the terminals and current flows in the opposite direction. On the armature there are arrowheads of voltage and current in coincidence, because the armature is a consumer.
Equivalent circuit of DC machine with separate excitation in motoring operation.
To solve transients’ phenomena, a system of the voltage equations of all windings is needed (for simplification g-winding is omitted):
where:
because the current in “d”-winding is zero, seeing that d-winding is omitted. In addition, the fact that mutual inductance of two perpendicular windings is zero was considered.
Then equation for electromagnetic torque is needed. This equation shows that electromagnetic torque in DC machine is developed in the form (again the member with the current
and that it covers the energy of the rotating mass given by moment of inertia, time varying of the mechanical angular speed, and load torque:
A checking of equation for electromagnetic torque of DC machines for steady-state conditions will be done if for
is introduced to Eq. (95) for the torque, whereby for the current the subscript “a” is employed and used for the armature winding and the number of the conductors
If the damping winding D, neither the windings in the quadrature axis Q, g, are not taken into account and respecting Eq. (90) for linkage magnetic flux, we get equations as they are presented below. The simplest system of the voltage equations is as follows:
The expression
We get from Eq. (96) the equation for calculation of the time varying of the mechanical angular speed:
and the electrical angular speed
In this way, a system of three equations (Eqs. (99), (100), and (102)), describing the smallest number of windings (three), was created. The solution of these equations brings time waveforms of the unknown variables (
If a DC machine is in motoring operation, the known variables are terminal voltages, moment of inertia, load torque, and parameters of the motor, i.e., resistances and inductances of the windings.
Unknown variables are currents, electromagnetic torque, and angular speed. Therefore, Eqs. (99) and (100) must be adjusted for the calculation of the currents:
The third equation is Eq. (102). It is necessary to solve these three equations, Eqs. (102)–(104), to get time waveforms of the unknown field current, armature current, and electrical angular speed, which can be recalculated to the mechanical angular speed or revolutions per minute:
As it was seen, a very important part of the transients’ simulations is determination of the machine parameters, mainly resistances and inductances but also moment of inertia. The parameters can be calculated in the process of the design of electrical machine, as it was shown in Chapter 1. However, the parameters can be also measured if the machine is fabricated. A guide how to do it is given in [8]. The gained parameters are introduced in equations, and by means of simulation programs, the time waveforms are received. After the decay of the transients, the variables are stabilized; it means a steady-state condition occurs. The simulated waveforms during the transients can be verified by an oscilloscope and steady-state conditions also by classical measurements in steady state.
Designers in the process of the machine design can calculate parameters on the basis of geometrical dimensions, details of construction, and material properties. If they use the above derived equations, they can predestine the properties of the designed machine in transients and steady-state conditions. This is a very good method on how to optimize machine construction in a prefabricated period. When the machine is manufactured, it is possible to verify the parameters and properties by measurements and confirm them or to make some corrections.
The derived equations were applied to a concrete motor, the data of which are in Table 1. The fact that the motor must be fully excited before or simultaneously with applying the voltage to the armature must be taken into account. Demonstration of the simulation outputs is in Figure 13. In Figure 13a–d, time waveforms of the simulated variables
Nameplate and parameters of the simulated separately excited DC motor.
Simulation waveforms of the separately excited DC motor. Time waveforms of (a) field current, (b) armature current, (c) speed, and (d) developed electromagnetic torque. Dependence of the speed on the torque in steady state, for (e) various terminal voltages, (f) various resistances connected in series with armature, (g) various field currents, and (h) dependence of speed on the armature current for the various field currents.
In Figure 13e–g, basic characteristics of
Figure 13h points to the fact that value of the armature current
Equations for universal machine are derived in general; therefore, they can be used also for generating operation. If the prime mover is taken as a source of stiff speed, then the time changing of the speed can be neglected, i.e.,
Equivalent circuit of the separately excited DC generator. The driving torque (
The constant speed of the prime mover equations will be changed in comparison with motoring operation. The armature current is in the opposite direction, because now the induced voltage in the armature is a source and current flows from the source. An electrical load is connected to the terminals; therefore the voltage and current on the load are in the same directions. The induced voltage is divided between voltage drops on the resistances and inductances of the winding and on the terminal voltage. Terminal voltage is given by the resistance of the load and its current. Equations are created in this sense:
and simultaneously the terminal voltage is given by equation of the load:
An electrical machine, the data of which are given in Table 1, is used also for simulation in generating operation. The dynamo is kept at constant speed and is fully excited before any loading occurs.
Simulation waveforms in Figure 15a–e show time dependence of the variables:
Simulation of the separately excited dynamo: time waveforms of (a) field current, (b) induced voltage, (c) armature current, (d) developed electromagnetic torque, (e) terminal voltage, and (f) dependence of the terminal voltage on the load current in steady-state conditions (external characteristic).
Dynamo is rotating by the rated speed, and at the time
This machine is so called because the field circuit branch is in shunt, i.e., parallel, with that of the armature. Figure 16 shows equivalent circuits of the shunt machines, in motoring and generating operation. As it is seen, the shunt motor differs from the separately excited motor because the shunt motor has a common source of electrical energy for armature as well as for field winding. Therefore, the field winding is connected parallel to the armature, which results in the changing of equations. In Eqs. (99) and (100), the terminal voltages in both windings are identical:
Equivalent circuits of the shunt DC machine in (a) motoring and (b) generating operation.
The power input is given by the product of terminal voltage and the total current
The power output is given by the load torque on the shaft and the angular speed. The developed electromagnetic torque is given by equation:
The time waveform of the electrical angular speed is given by Eq. (102).
To get simulations of DC shunt motor transients, it is necessary to solve equations from Eq. (107) to Eq. (110) and Eq. (102). Terminal voltage and parameters are known; currents and speed time waveforms are unknown (see Figure 17). Here the investigated motor has the same data as they are in Table 1.
Simulations of the DC shunt motor. Time waveforms of the (a) field current, (b) armature current, (c) speed, and (d) speed vs. torque for various terminal voltages and further in steady-state conditions waveforms of the (e) speed vs. torque for various rheostats connected in series with armature and (f) speed vs. torque for various field currents at constant terminal voltage.
Because this motor reaches its rated field current at the same field voltage
The control of the field current is carried out by variation of the field rheostat, which ensures decreasing of the field current
A shut generator (dynamo) differs from the separately excited dynamo by an essential way because a source for the field current is its own armature, where a voltage must be at first induced. To ensure this, some conditions must be filled. They are as follows: (1) some residual magnetism must exist in the magnetic system of the stator, which enables building up of the remanent voltage, if dynamo rotates, (2) resistance in the field circuit must be smaller than a critical resistance, (3) speed must be higher than a critical speed, and (4) there must be correct direction of rotation and connection between polarity of the excitation and polarity of induced voltage in the armature.
Because the field current depends on the terminal voltage, and this terminal voltage on the induced voltage, which again depends on the field current, this mutual dependence must be taken into account in simulations by magnetizing curve of the investigated machine, i.e., induced voltage vs. field current
Equation (107) is valid, but Eq. (108) is changed, because the terminal voltage is smaller than induced voltage because of the voltage drops, or opposite, induced voltage covers terminal voltage as well as voltage drops:
and armature current supplies field circuit as well as load circuit. Then the load current is:
whereby the terminal voltage is given by the load current and load resistance:
A machine, in which its data are in Table 1, was used for simulations of transients and steady-state conditions. In addition it is necessary to measure magnetizing curve
Magnetizing curve
During the simulation the machine is kept on the constant speed, and simulation starts with the connection of the armature to the field circuit. Because of remanent magnetic flux, in the armature there is induced small remanent voltage
Simulations of the shunt dynamo: time waveforms of (a) field current, (b) induced voltage, (c) armature current, (d) load current, (e) terminal voltage, and (f) terminal voltage vs. load current in steady-state conditions at the constant resistance in the field circuit when the load resistance is changed.
A waveform of
A DC series machine has its field winding connected in series with its armature circuit, as it is seen in Figure 20 for motoring and generating operation. This connection essentially influences properties and shapes of characteristics of the series machine and also equations needed for investigations of its properties.
Equivalent circuits of the series machine (a) in motoring and (b) in generating operation.
For a series DC motor, it is typical that the terminal voltage is a sum of the voltages in the field circuit and in the armature circuit:
but because of only one current flowing in the whole series circuit, the next is valid:
and Eq. (114) is simplified:
Equation (110) for electromagnetic torque is also changed because of only one current:
and angular speed is gained on the basis of the equation:
The time waveforms of the current, developed electromagnetic torque and angular speed, which can be recalculated to the revolutions per minute, are based on Eqs. (116)–(118). In Figure 20, there are simulated waveforms of the motor; the data of which are shown in Table 2.
Nameplate and parameters of simulated series motor.
Simulated waveforms in Figure 21a–f show time waveforms of the variables
Simulations of series motor. Time waveforms of (a) armature current and field current, (b) developed electromagnetic torque, and (c) speed and then speed vs. torque in the steady-state conditions for (d) various terminal voltages; (e) various resistances in series with armature circuit, at
For this reason, this motor in praxis cannot be in no-load condition and is not recommended to carry out its connection to the load by means of chain, or band, because in the case of a fault, it could be destroyed. In simulation the motor is after the steady condition at the instant
The approach to the simulations is the same as in previous chapters concerning the generating operations: the constant driving speed is supposed, induced voltage is a source for the whole circuit, and this voltage covers not only the voltage drops in the field and armature windings but also the terminal voltage. The current is only one
The magnetizing characteristic, i.e., no load curve
Data and parameters of a machine which was simulated in generating operations are in Table 2. Dynamo is kept at constant speed; at first in the no load condition, it means terminals are opened, and no current flows in its circuit. A small voltage is possible to measure at its terminals at this condition. This voltage is induced by means of remanent magnetic flux (Figure 23b,
Measured magnetizing curve for the investigated series machine
After the load is applied to the terminals at the instant
Simulations of the series dynamo and time waveforms of (a) armature current, (b) induced voltage, (c) terminal voltage, and (d) terminal voltage vs. load current in steady-state conditions at the changing of the load resistance.
As it is known, compound machines are fitted with both series and shunt field windings. Therefore, also simulations of transients and steady-state conditions are made on the basis of combinations of appropriate equations discussed in the previous chapters.
These motors, known as universal motor, can work on DC as well as AC network. Their connection is identical with series DC motors, even though there are some differences in their design. At the simulations, it is necessary to take into account that there are alternating variables of voltage and current; it means that winding’s parameters act as impedances, not only resistances.
Up to now we have dealt with DC machines, the windings of which are arranged in two perpendicular axes to each other. However, alternating rotating machines obviously have three-phase distributed windings on the stator, which must be transformed into two perpendicular axes, to be able to employ equations derived in the previous chapters.
In history, it can be found that principles of the variable projections into two perpendicular axes were developed for synchronous machine with salient poles.
A different air gap in the axis that acts as field winding and magnetic flux is created and, in the axis perpendicular to that magnetic flux, was linked with a different magnetic permeance of the circuit, which resulted in different reactances of armature reaction and therefore different synchronous reactances. It was shown that this projection into two perpendicular axes and variables can be employed much wider and can be applied for investigation of transients on the basis of the general theory of electrical machines.
On the other side, it is necessary to realize that phase values transformed into dq0 system have gotten into a fictitious system with fictitious parameters, where investigation is easier, but the solution does not show real values. Therefore, an inverse transformation into the abc system must be done to gain real values of voltages, currents, torques, powers, speed, etc. This principle is not unknown in the other investigation of electrical machines. For example, the rotor variables referred to the stator in the case of asynchronous machines mean investigation in a fictitious system, where 29 the calculation and analysis is more simple, but to get real values in the rotor winding a reverse transformation must be done.
Therefore, we will deal with a transformation of the phase variables abc into the fictitious reference k-system dq0 with two perpendicular axes which rotate by angular speed
A graphical interpretation of the transformation abc into the system dq0 is shown in Figure 24. This arrangement is formed according to the original letters given by the papers of R.H. Park and his co-authors (around 1928 and later), e.g., [14], although nowadays it is possible to find various other figures, corresponding to the different position of the axes d, q, and corresponding equations.
Graphical interpretation of the three-phase variable transformation abc into the reference k-system dq0, rotating by the speed
According to the original approach, if the three-phase system is symmetrical, the d-axis is shifted from the axis of the a-phase about the angle
According to Figure 24, the d-component of the
where
Also, projections into the q-axis are made in a similar way. It is seen that the projections to the q-axis are expressed by sinusoidal function of the phase variable with a negative sign, at the given +q-axis (see Eq. (125)).
The zero component is a sum of the instantaneous values of the phase variables. If the three-phase system is symmetrical, the sum of the instantaneous values is zero; therefore also the zero component is zero (see Eq. (126)). The zero component can be visualized in such a way that the three-phase variable projection is made in the 0-axis perpendicular to the plain created by the d-axis and q-axis, whereby the 0-axis is conducted through the point 0.
Then the equation system for the Park transformation from the abc to the dq0 system is created by Eqs. (124)–(126). To generalize the expressions, proportional constants
It is true that R.H. Park does not mention such constants in the original paper,because he solved synchronous machine, which will be explained later (Sections 8, 10, and 16). For the purposes of this textbook, it is suitable to start as general as possible and gradually adapt the equations to the individual kinds of electrical machines to get a solution as advantageous as possible. Therefore, the constants can be whichever except zero, though of such, that the equation determinant is not zero (see Eq. (127)). Then the inverse transformation will be possible to do and to find the real phase variables.
The determinant of the system is as follows:
Equations for the three-phase system transformation can be spread to the
Similarly, equations for the q- and 0-components are written. If a proportional constant 2/3 will be used for the three-phase system, then the corresponding constant for the
Equations for the inverse transformation are derived from the previous equations. Equation (124) is multiplied by expression
and from the third Eq. (126), the following is derived:
which is necessary to introduce to Eq. (130):
In this way, the equation for the inverse transformation of the a-phase variable is gained:
In a similar way, equations for the inverse transformation and also for b-phase and c-phase are derived:
Equations (133)–(135) create a system for the inverse transformation from dq0 to the abc system. These equations will be employed, e.g., for calculation of the real currents in the phase windings, if the currents in the dq0 system are known.
A space vector is a formally introduced symbol, which is illustrated in a complex plain in such a way that its position determines space position of the positive maximum of the total magnetic flux or magnetic flux density.
This definition is very important because as we know from the theory of electromagnetic field, neither current nor voltage is the vector. After the definition of the space vectors, it is possible to work with the currents and voltages, linked by Ohm’s law through impedance, but to image that it is a vector of the air gap magnetic flux density, which is by these currents and voltages created, which is very profitable. Therefore to distinguish a term “vector” as a variable which has a value and a direction, here the term “space vector” is used. The whole name “space vector” should be expressed and should not be shortened to “vector” because it can cause a misunderstanding, mainly between the people who do not work with investigation of transients.
To express that all three phases to which terminal voltages
where unit phasors
The subscript “s” means that it is a stator variable. Also, a proportional constant is marked with this subscript. In Figure 25, a complex plain with the stator axis is graphically illustrated, which is now identical with the axis of the a-phase winding. Then there is a rotor axis, which is shifted from the stator axis about the
Graphical illustration of the complex plain with the stator axis, rotor axis, and axis of the k-reference frame.
The transformation of the stator variables into the k-reference frame (
If this equation is split by means of the goniometrical functions into the real components, i.e., with cos-members, and into the imaginary components with sin-members,
then in a coincidence with Eqs. (139) and (141), the next two equations are gained:
As it is seen, these equations are identical with Eqs. (124) and (125), which were derived in general for three-phase circuits. In those equations separately marked constants in each axis were introduced, but here it is justified that it is enough to employ only one constant for both axes:
The third equation for the zero component, which is needed for investigation of the asymmetrical systems, can be added:
The same approach is used for the rotor variables with the subscript “r.” To distinguish them from the stator variables, the subscripts DQ0 will be used for the k-axis and ABC for the phase variables. The rotor variables must be multiplied by the expression
The variables of the k-system expressed in two perpendicular d-axis and q-axis are:
then Eq. (146) can be itemized into two equations:
and for the zero component:
Here is a system of equations for rotor variables of the three-phase system ABC transformed into the k-system DQ0. They differ from the stator variables by the angle
The same equations as for the terminal voltage of the universal machine (63) can be written also for the terminal voltage of the three-phase machines. For example, the stator windings, where the phases are marked with a, b, and c equations of the terminal voltage, are in the next form:
In the next only a-phase will be investigated. In Eq. (151), the variables
Now a derivation by time d
From Eq. (151), d
It was supposed that all three phases are identical and their resistances are equal:
The expressions from the inverse transformation are introduced also for
The left sides of Eqs. (155) and (158) are equal; therefore, also right sides will be equal. Now the members with the same goniometrical functions and members without goniometrical functions will be selected and put equal, e.g., members at
The equation for
Equations (160)–(162) are the voltage equations for the stator windings of the three-phase machines, such as asynchronous motors in k-reference frame, rotating by the angular speed
The instantaneous value of the input power in a three-phase system is a sum of instantaneous values of power in each phase (see also Section 2.2):
Instantaneous values of
Now it is necessary to multiply all members with each other, including the goniometrical functions, and after a modification the result is:
Variables
If an analysis in greater details is made, it is seen that an input power on the left side must be in equilibrium with the right side. It is supposed to be motoring operation. Therefore the input power applied to the terminals of the three-phase motor is distributed between the Joule’s resistance loss
Here it is seen that it is advantageous to choose the same proportional constants:
and, eventually,
This is the base expression for the power, which is converted from an electrical to a mechanical form in the motor or from a mechanical to an electrical form in the case of the generator. Next an expression for the electromagnetic torque is derived.
As it is known, an air gap power can be expressed by the product of the developed electromagnetic torque and a mechanical angular speed, now in the k-system:
or by means of electrical angular speed:
where
and after a reduction the torque is:
This is the base expression for an instantaneous value of developed electromagnetic torque of a three-phase machine. It is seen that its concrete form will be modified according to the chosen proportional constants. The most advantageous choice seems to be the next two possibilities:
Then:
Then:
It will be shown later that the first choice is more advantageous for asynchronous machines and the second one for synchronous machine.
A developed electromagnetic torque in the rotating electrical machines directly relates with equilibrium of the torques acting on the shaft. During the transients in motoring operation, i.e., when the speed is changing, developed electromagnetic torque
Unknown variables in motoring operation are obviously currents and speed, which can be eliminated from Eqs. (176) and (175). The mechanical angular speed is valid:
and electrical angular speed is:
The final expression for the time changing of the speed will be gotten, if for
Then
Then
If there is a steady-state condition,
The expression for the three-phase power in dq0 system is:
which was derived from the original expression for the three-phase power in abc system:
The expression can be modified by means of the constants
If
then
in which the principle of power invariance is not fulfilled, because the members in dq0 axes are figures, although it was derived from Eq. (163), where no figures were employed.
If
then
in which the principle of power invariance is fulfilled.
In Section 7, the three-phase system abc into the dq0 system was transformed, and expressions for
Consider the voltage symmetrical three-phase system:
where
Equations for transformation (124) till (126), derived in Section 4 for the variable
For the phase voltages, expressions from Eqs. (188)–(190) are introduced. At first, adjust expression for the voltage in the d-axis is as follows:
After the modification of the goniometrical functions and summarization of the appropriate members, in the final phase, it can be adjusted as follows:
In the transients, if the speed is changing, the angle
Here it is seen that the voltage in d-axis is alternating sinusoidal variable with the frequency which is the difference of the both systems: original three-phase abc system with the angular frequency
Now the same approach will be used for the q-axis:
The adjusting will result in equation:
which is finally accommodated to the form:
The voltage in the q-axis is shifted from the voltage in the d-axis about 90°, which is in coincidence with the definition of the d-axis and q-axis positions, which are perpendicular to each other. In transients when the speed is quickly changing, the angle
Finally, the equation for the zero component is adjusted as follows:
It is the sum of the voltage instantaneous values of the symmetrical three-phase system, which is, as it is known immediately, zero, or it is necessary to multiply all expressions for goniometrical functions, and after summarization of the appropriate members, the result is zero:
which is in coincidence with a note that the sum of the instantaneous values of variables, therefore also voltages, of the symmetrical three-phase system, is zero.
If the investigated three-phase system is not symmetrical, the zero component would have no zero value and would be necessary to add the equation for zero component to the dq0 system of equations. After the solution of dq0 variables, it would be necessary to make an inverse transformation on the basis of Eqs. (133)–(135), where component
Here the universality of the method of transformation is seen, because it is possible to investigate also unsymmetrical three-phase systems.
At the end of this chapter, the properties of the transformed sinusoidal variables are summarized, as shown in the above equations:
Variables d and q are alternating variables with a frequency which is given by the difference of the frequency of both systems: original three-phase system abc with the angular frequency
Transformed variables d and q are shifted about 90°, unlike the three-phase system, in which the axes are shifted about 120°.
Variables of the zero component, i.e., with the subscript 0, are in the case of the symmetrical system, zero. If the three-phase system is not symmetrical, it is necessary to take the zero component into account, to find its value and to employ it in the inverse transformation into the system abc.
Magnitudes of variables dq0 depend on the choice of the constant of the proportionality.
The voltages in d-axis and q-axis are adjusted to the form:
whereby
whereby
Here it is seen that if:
Equations for the voltages
Note that if there are supposed cosinusoidal functions of the three-phase system, i.e.,
after the same approach at derivation as for sinusoidal functions, equations for the variables in d-axis and q-axis are gotten:
As it will be shown in Section 19, this version of the voltage origin of the three-phase system definition is more suitable for a synchronous machine because of the investigation of the load angle.
The final form of the voltage equations in the system dq0 does not depend only on the choice of the proportionality constants but also on the position of the reference k-system and the angle
The k-system can be positioned totally arbitrary, but some of the choices bring some simplicity in the investigation, which can be employed with benefit. Here are some of the most used possibilities, which are marked with special subscripts.
This choice means that the k-system is identified with the axis of the stator a-phase winding, i.e., the k-system is static and does not rotate, much like stator a-phase winding.
This choice is distinguished from all others by subscripts. Instead of the subscripts d, q, 0, the subscripts
Equations for the voltages and currents are adjusted to be able to see how this choice brings benefits. From the equation in the previous chapter:
It is seen that if, simultaneously with the choice
which are very important findings, in that an instantaneous value of the voltage (and current) in the transformed system is identical with the instantaneous value of the voltage (or current) in phase a. This brings very simple situation, because it is not needed to make any inverse transformation.
Have a look at the voltage in the
which means that this voltage is fictitious and such voltage does not exist in the real abc system and is shifted about 90° from the voltage
It is the most important thing that in the same way the currents are transformed. It means that in the motoring operation, where the currents, together with the speed, are unknown,
Additionally here are equations for an electromagnetic torque and time varying of the angular speed. On the basis of Eqs. (179) and (180), derived for the
At the end it is necessary to say that this choice is not profitable only for the squirrel cage asynchronous motors (see Section 11) but also for asynchronous motors with wound rotor and for asynchronous generators.
This choice means that the k-system is identified with the rotor axis and the speed of the k-system with the rotor speed.
This transformation is employed with benefit for synchronous machines, because in equations for the voltage, there is a so-called load angle (see Eq. (218)), which is a very important variable in the operation of the synchronous machines. On the rotor of the synchronous machine, there is a concentrating field winding fed by DC current, which creates DC magnetic flux. Here the d-axis is positioned. Therefore the rotor system is not necessary to transform because the field winding is positioned directly in d-axis, and if the rotor has damping windings, they are decomposed into two axes, d-axis and q-axis, perpendicular to each other. Finally, as it was mentioned before, this transformation was developed for synchronous machine with salient poles; therefore, the subscripts d, q are left in the form, in which they were used during the whole derivation.
As in previous case, equations for the voltages and currents are again adjusted to be able to see advantage of this choice.
is seen that if
In Section 16 and 18 there will be derived, why in the case of synchronous machines the proportionality constants are chosen in this form:
Then the original voltage
and the voltage
Equations for the electromagnetic torque and time varying of the speed (181) and (182), derived for
Section 16 and others will deal with the synchronous machines in the general theory of electrical machines. These equations will be applied at the investigation of the properties of the synchronous machines in concrete examples, and also synchronous machines with permanent magnets will be investigated.
This choice means that the speed of k-system rotation
Adjust equations for the voltages, in which the advantage of this choice will be visible. From the equations in the previous chapter:
result that if
Similarly, for voltage
It is seen that both voltages in this system are constant DC variables, and it depends on the choice of the constants and initial value
If, for example, the initial position of the k-system is chosen to be zero,
If it looks uncomfortable that both voltages are negative values, it is enough, if derivation of transformation equations from abc to dq0 start with an assumption that:
In the steady-state condition, all variables on the stator and rotor are illustrated as DC variables. Therefore, the solution is very easy, but it is true that it is necessary to make an inverse transformation into the real abc system. This transformation system is very suitable for asynchronous motors.
The equation for torque is also very simplified, because the x-component of the current is also zero (
and equation for time varying of the speed is:
Although the whole derivation of transformed variables was made for the dq0 axis, because it was historically developed in such a way, and then the new subscripts were introduced by means of the special choice of the reference k-system position, it is seen that the subscripts dq0 are kept only for the synchronous machine, for which this transformation was developed. If it should be started now, perhaps two perpendicular axes to each other would be marked as u, v, 0. Nevertheless the original configuration of universal machine had windings in the axes d, q, and it is kept also for the future. However here introduced marking was not accepted by all experts dealing with this topic, and some authors used the system x, y, 0 instead of α, β, 0.
It is supposed that a reader is familiar with the basic design of asynchronous machine and its theory and properties. Now we will analyze the three-phase symmetrical system on the stator, marked abc and on the rotor, marked ABC, i.e., six windings together (Figure 26).
Illustration figure of an asynchronous machine with three windings abc on the stator and three phase ABC on the rotor. They are shifted from each other about the angle
Basic voltage equations for the terminal voltage can be written for each winding or by one equation, at which the subscripts will be gradually changed:
where j = abc, ABC.
If the system is symmetrical, then it is possible to suppose that:
Linkage magnetic flux can be also expressed by one equation as a sum of all winding contributions:
where j, k = abc, ABC, but because of transparency here is the whole sum of the members in details:
where:
The others are mutual inductances of stator and rotor windings. It is necessary to investigate if they depend on the rotor position or not.
1. Self-inductances of the stator windings
Self-inductance of stator single phase
This flux is divided into two parts: leakage magnetic flux
where
(a) Illustration of mutual inductance
2. Self-inductances of the rotor windings
These inductances are expressed similarly as the stator ones:
Have a note that in the whole general theory of electrical machines, rotor variables are referred to the stator side.
3. Mutual inductances of the stator windings—
Take an image that two stator windings have an identical axis, e.g., b-phase is identified with a-phase. Then their mutual inductance is
and this value is constant; it means it is always negative because the position of b-phase on the stator with regard to a-phase is stable.
4. Mutual inductances of the rotor windings—
The same analysis as in item 3 results in the finding that the mutual inductance of the rotor windings without the influence of the stator windings is always negative and equals (Figure 28).
Equivalent circuit of the asynchronous machine (a) in d-axis and (b) in q-axis. All rotor variables are referred to the stator.
All mutual inductances of the stator and rotor windings are expressed as follows:
where
All expressions are introduced into (239); therefore, the inductances, linkage fluxes, and currents can be written in the matrix form:
After these expressions are introduced into (235), six terminal voltage equations are obtained, with nonlinear, periodically repeated coefficients
Equivalent circuit of asynchronous machine for reference k-system: (a) in d-axis and (b) in q-axis for a chosen reference frame. All rotor variables are referred to the stator side to distinguish this case of transformation from the others, the axes are marked αβ0 (originally it was marked as αβγ), and also all subscripts of the currents, voltages, and linkage magnetic fluxes are with these subscripts. Then on the basis of the subscripts, it is possible to know what kind of transformation was used.
To solve such equations is very complicated; therefore, it is necessary to eliminate the periodically repeated coefficients. This is possible to do by various real or complex linear transformations. The most employed is Park linear transformation, mentioned in Section 4. In the next it is applied for this case.
On the basis of the equations for transformation into dq0 system, the equation for
The expressions for
If we consider that:
and these expressions are introduced into the equation above for all three phases, after modifications, some expressions that are zero are found, e.g.:
and others in which transformed variables are seen, e.g.:
or:
If these two expressions are summed, it results in:
where
This is evident also in Figure 27a, if a constant air gap of asynchronous machines is taken into account. Therefore, inductances in d-axis and q-axis are equal, and there is no need to mark separately magnetizing inductance in d-axis and q-axis. The members of Eq. (252), in which act rotor currents
In this expression there is used a knowledge, that: (1) mutual inductance of the stator and rotor winding with contribution of all three stator phases is 3/2 M what is marked
Then it is possible to write that the transformed current of the rotor system is
and the whole Eq. (252) can be written much more briefly:
where
The equation for
or briefly:
where:
and
Considering that in the asynchronous machine the air gap is constant around the whole periphery of the stator boring, there is no difference in the inductances in d-axis and q-axis; therefore, the following can be written:
and also
Zero component is as follow:
where:
The fact that the zero-component inductance
Linear transformation is employed also at rotor linkage magnetic flux derivations in the system DQ0:
eventually considering that the air gap is constant and the parameters in the d-axis and q-axis are equal:
The meaning of the rotor parameters is as follows:
Similarly, for the zero rotor component can be written as:
where:
Take into account that all rotor variables are referred to the stator side; eventually they are measured from the stator side.
Voltage equations of the asynchronous machines in the dq0 system are obtained by a procedure described in Section 7. There are equations for the stator terminal voltage in the form:
The rotor voltage equations are derived in a similar way as the stator ones but with a note that the rotor axis is shifted from the k-system axis about the angle
These six equations create a full system of the asynchronous machine voltage equations. Rotor variables are referred to the stator side; expressions for the linkage magnetic flux are shown in Section 12.
According to Sections 7 and 10, the reference k-system can be positioned arbitrarily, but some specific positions can simplify solutions; therefore, they are used with a benefit. One of such cases happens if the d-axis of the k-system is identified with the axis of the stator a-phase; it means
This system is obtained by phase variable projection into stationary reference system, linked firmly with a-phase. It is a two-axis system, and zero components are identical with the non-rotating components known from the theory of symmetrical components.
The original voltage equations of asynchronous machine derived in Sections 7 and 13 are as follows:
where:
Now new subscripts the following are introduced:
For currents and voltages:
For inductances:
The original equations, rewritten with the new subscripts, with the fact that
If transients are solved for motoring operation, then stator terminal voltages on the left side of the equations are known variables and are necessary to introduce derived expressions for sinusoidal variables transformed into dq0 system, now α, β-axes ((196) for
In the motoring operation, the terminal voltages are known variables, and unknown variables are currents and speed. Therefore it is suitable to accommodate the previous equations in the form where the unknown variables are solved. From Eq. (303), the following is obtained:
and from Eq. (305):
This equation is introduced into Eq. (307). Then it is possible to eliminate a time variation of the stator current in the α-axis:
The same way is applied for the other current components:
The last equation is for time variation of the speed. On the basis of Section 8, if in the equation for the electromagnetic torque the constants
After considering Eq. (176), the electrical angular speed is obtained in the form:
Mechanical angular speed is linked through the number of the pole pairs
For Eqs. (309)–(312), the next expressions are introduced for the voltages (see Sections 9 and 10):
which is displaced about 90° with regard to the
In the next chapter, solving of the transients in a concrete asynchronous motor with squirrel cage rotor and wound rotor is shown.
Equations derived in the previous chapter are applied on a concrete asynchronous motor with squirrel cage rotor. The rotor bars are short circuited by end rings; thus the rotor voltages
In Figure 30, simulation waveforms of the starting up of an asynchronous motor when it is switched directly across the line are shown. Parameters of the investigated motor are in Table 3.
Simulation waveforms of the asynchronous motor at its switching directly across the line, time waveforms of the (a) speed, (b) a-phase current
Nameplate and parameters of the investigated asynchronous motor.
Simulation waveforms in Figure 30a–c show time variations of the variables
Relatively large starting current can cause an appreciable drop in motor terminal voltage, which reduces the starting current but also the starting torque. If the supply voltage drop would be excessive, some kind of across-the-line starter that reduces the terminal voltage and hence the starting current is required. For this purpose, a three-phase step-down autotransformer may be employed. The autotransformer is switched out of the circuit as the motor approaches full speed. The other method of starting is by a star-delta switch or by inserting resistances into the stator winding circuit. In the industry, a special apparatus is used, the so-called softstarter, which enables the starting of the defined requirement. Softstarter contains solid-state elements (thyristors), which enable to vary the terminal voltage of the motor. The start up is carried out by limitation of the maximal value of current, which will not be gotten over during the starting. This control is ensured by the possibility to change the terminal voltage of the motor. The more sophisticated way is a frequency starting during which not only voltage but also frequency is gradually increased, whereby the ratio
Simulations of softstarter and frequency converter applications are shown in Figure 31. In both cases not only value of the starting torque is reduced, which is undesirable, but also the value of the starting current. The current does not cross the rated value and in this simulated case neither no-load current
Simulations of the asynchronous motor starting up by means of softstarter, time waveforms of (a) speed
Equations in Section 14 are the basis for the simulations. In this case, it is possible to feed the terminals of the wound armature on the rotor. This possibility is employed in applications with asynchronous generators, where feeding to the rotor serves as stabilization of the output frequency of the generator. Previously, the rotor terminals of the asynchronous motor were used for variation of the rotor circuit resistance by external rheostats. Such starting up is shown in this part. The nameplate and parameters of the investigated motor are in Table 4.
Nameplate and parameters of the investigated wound rotor asynchronous motor.
Simulations are shown in Figure 32a–c. There are time waveforms of the variables
Simulations of the rotor wound asynchronous motor during the switching directly across the line. Time waveforms of the (a) speed, (b) phase current
Time waveforms of the simulations during the starting up of the wound rotor asynchronous motor by means of rheostats added to the rotor circuits: (a) speed
At the instant
Simulation waveforms are very similar with those of the squirrel cage rotor (high starting current and torques). But in the case of wound rotor, there is a possibility to add external resistors and to control the current and the torque (Figure 33).
It is supposed that a reader is familiar with the basic knowledge of a synchronous machine theory, properties, and design configuration. The synchronous machine with salient poles on the rotor; symmetrical three-phase system a, b, c on the stator; field winding f in the d-axis on the rotor; and damping winding, split into two parts perpendicular to each other (D and Q on the rotor), positioned in the d-axis and q-axis, as it is seen in Figure 34, is analyzed. The d-axis on the rotor is shifted about the angle
Synchronous machine with salient poles on the rotor and three-phase winding a, b, c on the stator, field winding f, and damping winding split into two parts (D and Q) perpendicular to each other, positioned in the d-axis and q-axis on the rotor. The d-axis on the rotor is shifted about the angle
Basic equations for terminal voltage can be written for each of the winding separately, or briefly by one equation, in which the subscripts are gradually changed for each winding:
where
If symmetrical three-phase winding on the stator is supposed, then it can be supposed that their resistances are identical and can be marked by the subscript “s”:
Linkage magnetic flux can be also expressed briefly by the sum of all winding contributions:
where
Although it is known that mutual inductances of the windings that are perpendicular to each other are zero:
for computer manipulation is more suitable if the original structure is kept and all inductances appear during the analysis:
The rest of the inductances are mutual inductances of the stator and rotor windings:
It is important to investigate if inductances depend on the rotor position or not.
Self- and mutual inductances of the rotor windings
Investigate, for example, a-phase winding on the stator and field winding f on the rotor, as it is shown in Figure 34.
When sinusoidally distributed windings are assumed, i.e., coefficients of higher harmonic components are zero, then the waveform of mutual inductance is cosinusoidal, if for the origin of the system such rotor position is chosen in which the a-phase axis and the axis of the field winding are identical (see Figure 35).
(a) Illustration to express mutual inductance of the a-phase on the stator and field winding f on the rotor, (b) waveform of the mutual inductance
Then the mutual inductances can be expressed as follows:
similarly:
Expressions for Q-winding positioned in the q-axis are written according to Figure 36a, where it is seen that the positive q-axis is ahead about 90° of the d-axis. Hence if the d-axis is identified with the axis of the a-phase, the q-axis is perpendicular to it, and mutual inductance
(a) Illustration to express mutual inductance of the Q-winding on the rotor and a-phase on the stator and (b) waveform of the mutual inductances
Self-inductances of the stator depend on the rotor position if there are salient poles. Self-inductance of the a-phase is maximal (
(a) Illustration to express self-inductance
The magnitude of the self-inductance
The minimal value of the self-inductance is obtained if the axis of the salient pole is perpendicular to the axis of the stator a-phase, i.e.,
If the rotor rotates about
Mutual inductances of the stator windings depend on the rotor position only in the case of the salient poles on the rotor. These inductances are negative because they are shifted about 120° (see explanation in Figure 27b). The rotor is in a position where mutual inductance
(a) Illustration to express mutual inductance of the stator windings a, b, c and (b) waveform of the mutual inductance
It is possible to assume that for the sinusoidally distributed windings, the magnitudes of harmonic waveform
or
which better corresponds to the waveform in Figure 38.
Now all the expressions of these inductances are introduced into Eq. (65) and Eq. (318). They are equations with nonlinear periodically changed coefficients. To eliminate these coefficients, it is necessary to transform the currents, voltages, and linkage magnetic fluxes. The most suitable is Park linear transformation, which was explained in Section 4 and is applied again in the next chapter.
Terminal voltage equations of the synchronous machine stator windings in a system d, q, 0 are obtained by means of the procedure described in Section 7. The next equations were derived:
Equations (160)–(162) are voltage equations of the three-phase stator windings, in this case synchronous machine but also asynchronous machine, as it was mentioned in Section 13. They are equations transformed into reference k-system rotating by angular speed
Terminal voltage equations of the synchronous machine rotor windings are not needed to transform in the d-axis and q-axis, because the rotor windings are embedded in these axes, as it is seen in Figure 34, and are written directly in the two-axis system d, q, 0:
The next the expressions for linkage magnetic flux are investigated.
In Eq. (65) of linkage magnetic fluxes, expressions for inductances as they were derived in Section 16 are introduced. For example, for field winding with a subscript “f,” the equation for linkage magnetic flux is written as follows:
If this equation is compared with Eq. (124), written for a general variable
After this modification in Eq. (352), the following is obtained:
where it was taken into account that mutual inductance of two perpendicular windings f and Q is zero.
On the same basis, the linkage magnetic flux for damping rotor windings D and Q is received:
A derivation for the stator windings is made in the like manner. It is started with a formal transformation equation from system a, b, c into the d-axis and then into the q-axis. The equation in the d-axis is as follows:
If into this equation expressions from Eq. (321), for linkage magnetic fluxes of a, b, c phases, are introduced, and for inductances appropriate expressions from Section 16 are introduced, then after widespread modifications of the goniometrical functions and for a rotor position in d-axis, i.e., if
the following is received:
Here:
is a direct synchronous inductance. The other symbols are for mutual inductances between the stator windings transformed into the d-axis and rotor windings, which are also in the d-axis:
The linkage magnetic flux in the q-axis is derived in a similar way, which results in:
where:
is a quadrature synchronous inductance and
is the mutual inductance between the stator winding transformed into the q-axis and rotor winding which is in the q-axis . From Eqs. (359) and (363), it is seen that if there is a cylindrical rotor, then
which is a known fact.
Finally, a linkage magnetic flux for zero axis is derived in a similar way:
where:
is called zero inductance. It is seen that this linkage magnetic flux and inductance are linked only with variables with the subscript 0 and do not have any relation to the variables in the other axes. Additionally, also here a knowledge from the theory of the asynchronous machine can be applied that zero impedance is equal to the leakage stator inductance that can be used during the measurement of the leakage stator inductance. Here can be reminded equation Section 12:
Namely, if all three phases of the stator windings (in series or parallel connection) are fed by a single-phase voltage, it results in the pulse, non-rotating magnetic flux (see [8]).
If there is a request to make equations more simple, then it is necessary to ask for equality of mutual inductances of two windings, for example, inductance
Therefore from Eq. (354) for
Then:
and
The same value is obtained if expressions for