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EE Springer 2017 Sorrentino Maduro Modeling of DC Contactor

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           1 3 Electrical Engineering Archiv für Elektrotechnik  ISSN 0948-7921Volume 99Number 1 Electr Eng (2017) 99:335-344DOI 10.1007/s00202-016-0421-2 Electromechanical modeling of a contactor with dc coil  Elmer Sorrentino & Armando Maduro           1 3 Your article is protected by copyright andall rights are held exclusively by Springer-Verlag Berlin Heidelberg. This e-offprint isfor personal use only and shall not be self-archived in electronic repositories. If you wishto self-archive your article, please use theaccepted manuscript version for posting onyour own website. You may further depositthe accepted manuscript version in anyrepository, provided it is only made publiclyavailable 12 months after official publicationor later and provided acknowledgement isgiven to the srcinal source of publicationand a link is inserted to the published articleon Springer's website. The link must beaccompanied by the following text: The finalpublication is available at link.springer.com”.  Electr Eng (2017) 99:335–344DOI 10.1007/s00202-016-0421-2 ORIGINAL PAPER Electromechanical modeling of a contactor with dc coil Elmer Sorrentino 1 ·  Armando Maduro 2 Received: 8 May 2014 / Accepted: 6 September 2016 / Published online: 16 September 2016© Springer-Verlag Berlin Heidelberg 2016 Abstract  The purpose of this article is to model a contac-tor with dc coil using simple electromechanical equations.These models do not need a detailed description of inter-nal characteristics of the contactor, and the parameters areestimated from external measurements (current in main coil,and operation time of contacts). Four models are formulatedto obtain the operation time of contacts and to describe thecurrent in contactor coil. Two ways for modeling the con-tactor have a constant equivalent area for airgap (with andwithout considering friction). The other two ways for mod-elingthecontactorhaveavariableequivalentareaforairgap;these two ways only have different parameters for models.The results of these models are compared with experimen-tal measurements. Although the proposed models includeseveral simplifications, these models are very accurate forobtaining the operation time of contacts for eight differentvalues of the dc source voltage. The three transient stages of the current in contactor coil are obtained with these models,with good accuracy for the time of occurrence of the localminimum of the current. These facts are important, becausethey imply a good representation of the behavior of the con-tactor.Theobtainingofthethreetransientstagesofcurrentincontactor coil, using the electromechanical models, had not B Elmer Sorrentinoelmersor@usb.ve 1 Departamento de Conversión y Transporte de Energía,Universidad Simón Bolívar, Apdo. Postal 89.000,Caracas 1080, Venezuela 2 Electrical Engineering Department, WorleyParsons,Edmonton, Canada beenshownintheliterature.Thisarticleshowsthefeasibilityof obtaining such computed results, which are in agreementwith experimental measurements. Keyword  Contactor with dc coil 1 Introduction The basic electromechanical equations for the descriptionof the transient behavior of contactors with dc coil can befound in some textbooks (e.g., [1]). However, some simula-tion details for contactors are only described in specializedliterature, for example: (a) the non-linear effect due to thelimitsforarmaturedisplacement[2,3],(b)thediscontinuities in the armature path, due to the changes in the springs andmasses in motion [4,5], and (c) the effects of small airgaps between armature and core when the contactor is closed [5].Ontheotherhand,theapplicationofcorrectionfactorsforthe equivalent lengths and the equivalent areas of the mag-neticfieldpathsisusualintheanalysisofmagneticcircuitsof transformers [6,7] and inductors [8,9]. Although the appli- cation of similar factors is not usual for contactors, there areexamples of their use for contactors with ac coil [10,11]. A correction factor for the equivalent area of the airgap is usedin this article and its effect is analyzed.The models based on the estimation of magnetic fields ineachpointofthecontactorgeometry(usingthefinite-elementmethod or other methods) require a very detailed descriptionoftheinternalpartsofthecontactor(geometryandcharacter-istics of the materials). In contrast, the models of this articlerequire an estimation of their parameters, and such estima-tion is based only on the external measurements: current inthe main coil and operation time of contacts. The electro-mechanical models and the models based on the calculation  1 3  336 Electr Eng (2017) 99:335–344 of magnetic fields usually have different applications for theanalysisofelectricaldevices,andbothareusefulwithintheirown scopes.Thetransientcurrentinthecontactorcoilhasthreestages.Some research articles [12–19] show currents similar to the resultsobtainedforthisarticle,butthefinite-elementmethodis always necessary for their models. The purpose of thisarticle is to model a contactor with dc coil, using lumpedparameters(i.e.,withoutthedetailedcalculationofmagneticfields).Therefore,thecontactorismodeledherewithoutcon-sideringitsspecificgeometrybutonlyasimplifiedschematicrepresentation. The obtaining of the three transient stages of the current in the contactor coil, using the electromechan-ical models, had not been shown in the previous literature.Thisarticleshowsthefeasibilityofobtainingsuchcomputedresults, which are in agreement with the experimental mea-surements. In addition, computed results for the operationtime of contacts are accurate, and they are also obtained hereusing these electromechanical models.The electromechanical models with lumped parametersmight be useful as simple methods for analyzing the contac-tor when the applied voltage is varying in time, for example:(a) due to events in the electrical system, as short circuitsor starts of large motors; and (b) due to the search of differ-ent ways for energizing the contactor coil, using electroniccontrol [20,21]. On the other hand, these models might be necessary for some users of contactors, to avoid thesearch of the internal characteristics of the devices, becausethis could imply an undesired disassembling of its internalpieces.Four ways for modeling the contactor were developed inthis article: two ways have a constant equivalent area for theairgap, and the other two ways have a variable equivalentarea for the airgap. The main contribution of this article isto show that the simple electromechanical models can beapplied for obtaining the contactor operation time and thetransient behavior of the coil currents when the device istested with different dc source voltage values.An article about electromechanical modeling of a con-tactor with ac coil, using lumped parameters, was recentlypublished [22]. Both cases (ac and dc coils) are based on simple electromechanical models, and the computed resultsforthecontactoroperationtimeareaccurateinbothcases.Incaseofaccoil[22],themodelisconceptuallymorecomplex, due to the presence of shading rings. In case of dc coil (thisarticle), the obtaining of the correct behavior of the coil cur-rentwasmorelaborious(andthesearchofmodelparametersrequired many more attempts). 2 Basic simplifications The following simplifications were used for the model:(a) Only an equivalent spring was considered, whose forcevaries linearly with the armature movement.(b) The contactors usually have different springs, related tothe armature and the auxiliary contacts. The effect of these different springs was not considered.(c) The effect of the small collisions, produced in the coreduring the armature travel, was not considered.(d) Changesinthemassesinmotion,whichcouldoccurdur-ing the armature travel, were not considered.(e) Anequivalentmasswasconsideredthatrelatestheaccel-erating force with the armature acceleration.(f) Only a linear variation of the friction force with respectto the armature velocity was considered.(g) The non-linear characteristic of the ferromagnetic coreof the contactor was not considered.(h) The presence of the iron remanent flux at the energizinginstant was not considered.(i) Correction factors for the variables in the magnetic pathswere used only in the equivalent area of the airgap.Thissetofsimplificationsisforobtainingasimplemodel,without considering the specific details of the tested device.For example, the changes in the springs and the masses inmotion[4,5]mayimplythatphysicallythereisnotanyequiv- alentspring,orequivalent mass,butthesesimplificationsarenecessary to achieve a simple model. 3 Electromechanical models Figure 1 shows a simplified drawing of a contactor with dccoil.  x   is the armature position,  x   =  0 at the position limitwhen the coil is de-energized, and  x   =  d   at a theoreticalposition equal to the gap length. There is other position limitat the end of the motion and it occurs at a position (  x  )  whichis less than  d  ; however, for simplicity, the model considersthe that end of the motion is at a position equal to  d  .The magnetic force was assumed to be proportional to thesquare of the current and to the derivative of the inductancewith respect to the position. However, some authors recog-nize that the algebraic expression for the magnetic force is ArmatureCore φ  /2 iV  x x = 0 x = d  φ  /2 φ   /  2  φ  /2  φ  : magnetic flux   U  ArmatureCore φ  /2 iV  x x = 0 x = d  φ  /2 φ   /  2  φ  /2  φ  : magnetic flux   U  Fig. 1  Schematic representation of a contactor with dc coil  1 3
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