IEC 62428 Edition 1 0 2008 07 INTERNATIONAL STANDARD NORME INTERNATIONALE Electric power engineering – Modal components in three phase a c systems – Quantities and transformations Energie électrique –[.]
IEC 62428 Edition 1.0 2008-07 INTERNATIONAL STANDARD Electric power engineering – Modal components in three-phase a.c systems – Quantities and transformations IEC 62428:2008 Energie électrique – Composantes modales dans les systèmes a.c triphasés – Grandeurs et transformations LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU NORME INTERNATIONALE THIS PUBLICATION IS COPYRIGHT PROTECTED Copyright © 2008 IEC, Geneva, Switzerland All rights reserved Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either IEC or IEC's member National Committee in the country of the requester If you have any questions about IEC copyright or have an enquiry about obtaining additional rights to this publication, please contact the address below or your local IEC member National Committee for further information Droits de reproduction réservés Sauf indication contraire, aucune partie de cette publication ne peut être reproduite ni utilisée sous quelque forme que ce soit et par aucun procédé, électronique ou mécanique, y compris la photocopie et les microfilms, sans l'accord écrit de la CEI ou du Comité national de la CEI du pays du demandeur Si vous avez des questions sur le copyright de la CEI ou si vous désirez obtenir des droits supplémentaires sur cette publication, utilisez les coordonnées ci-après ou contactez le Comité national de la CEI de votre pays de résidence About the IEC The International Electrotechnical Commission (IEC) is the leading global organization that prepares and publishes International Standards for all electrical, electronic and related technologies About IEC publications The technical content of IEC publications is kept under constant review by the IEC Please make sure that you have the latest edition, a corrigenda or an amendment might have been published Catalogue of IEC publications: www.iec.ch/searchpub The IEC on-line Catalogue enables you to search by a variety of criteria (reference number, text, technical committee,…) It also gives information on projects, withdrawn and replaced publications IEC Just Published: www.iec.ch/online_news/justpub Stay up to date on all new IEC publications Just Published details twice a month all new publications released Available on-line and also by email Electropedia: www.electropedia.org The world's leading online dictionary of electronic and electrical terms containing more than 20 000 terms and definitions in English and French, with equivalent terms in additional languages Also known as the International Electrotechnical Vocabulary online Customer Service Centre: www.iec.ch/webstore/custserv If you wish to give us your feedback on this publication or need further assistance, please visit the Customer Service Centre FAQ or contact us: Email: csc@iec.ch Tel.: +41 22 919 02 11 Fax: +41 22 919 03 00 A propos de la CEI La Commission Electrotechnique Internationale (CEI) est la première organisation mondiale qui élabore et publie des normes internationales pour tout ce qui a trait l'électricité, l'électronique et aux technologies apparentées A propos des publications CEI Le contenu technique des publications de la CEI est constamment revu Veuillez vous assurer que vous possédez l’édition la plus récente, un corrigendum ou amendement peut avoir été publié Catalogue des publications de la CEI: www.iec.ch/searchpub/cur_fut-f.htm Le Catalogue en-ligne de la CEI vous permet d’effectuer des recherches en utilisant différents critères (numéro de référence, texte, comité d’études,…) Il donne aussi des informations sur les projets et les publications retirées ou remplacées Just Published CEI: www.iec.ch/online_news/justpub Restez informé sur les nouvelles publications de la CEI Just Published détaille deux fois par mois les nouvelles publications parues Disponible en-ligne et aussi par email Electropedia: www.electropedia.org Le premier dictionnaire en ligne au monde de termes électroniques et électriques Il contient plus de 20 000 termes et définitions en anglais et en franỗais, ainsi que les termes ộquivalents dans les langues additionnelles Egalement appelé Vocabulaire Electrotechnique International en ligne Service Clients: www.iec.ch/webstore/custserv/custserv_entry-f.htm Si vous désirez nous donner des commentaires sur cette publication ou si vous avez des questions, visitez le FAQ du Service clients ou contactez-nous: Email: csc@iec.ch Tél.: +41 22 919 02 11 Fax: +41 22 919 03 00 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU IEC Central Office 3, rue de Varembé CH-1211 Geneva 20 Switzerland Email: inmail@iec.ch Web: www.iec.ch IEC 62428 Edition 1.0 2008-07 INTERNATIONAL STANDARD Electric power engineering – Modal components in three-phase a.c systems – Quantities and transformations Energie électrique – Composantes modales dans les systèmes a.c triphasés – Grandeurs et transformations INTERNATIONAL ELECTROTECHNICAL COMMISSION COMMISSION ELECTROTECHNIQUE INTERNATIONALE PRICE CODE CODE PRIX ICS 01.060; 29.020 ® Registered trademark of the International Electrotechnical Commission Marque déposée de la Commission Electrotechnique Internationale S ISBN 2-8318-9921-4 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU NORME INTERNATIONALE –2– 62428 © IEC:2008 CONTENTS FOREWORD Scope .5 Normative references .5 Terms, definitions, quantities and concepts 3.1 General 3.2 Terms and definitions Modal transformation 4.1 General 4.2 Power in modal components 4.3 Established transformations 10 Decoupling in three-phase a.c systems 16 5.1 Decoupling in case of steady-state operation with sinusoidal quantities 16 5.2 Decoupling under transient conditions 19 Bibliography 23 Figure – Circuit, fed by a three-phase voltage source with U L1Q , U L2Q , U L3Q at the connection point Q and earthed at the neutral point N via the impedance Z N = RN + j X N 16 Figure – Three decoupled systems which replace the coupled three-phase a.c system of Figure under the described conditions (see text) 19 Table – Power-variant form of modal components and transformation matrices 11 Table – Power-invariant form of modal components and transformation matrices 12 Table – Clark, Park and space phasor components – modal transformations in the power-variant form 13 Table – Clark, Park and space phasor components – Modal transformations in the power-invariant form 14 Table – Transformation matrices in the power-variant form for phasor quantities 15 Table – Transformation matrices in the power-invariant form for phasor quantities 15 Table – Modal voltages and impedances in case of phasor quantities 18 Table – Modal voltages and inductances under transient conditions 22 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU 62428 © IEC:2008 –3– INTERNATIONAL ELECTROTECHNICAL COMMISSION ELECTRIC POWER ENGINEERING – MODAL COMPONENTS IN THREE-PHASE AC SYSTEMS – QUANTITIES AND TRANSFORMATIONS FOREWORD 2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international consensus of opinion on the relevant subjects since each technical committee has representation from all interested IEC National Committees 3) IEC Publications have the form of recommendations for international use and are accepted by IEC National Committees in that sense While all reasonable efforts are made to ensure that the technical content of IEC Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any misinterpretation by any end user 4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications transparently to the maximum extent possible in their national and regional publications Any divergence between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in the latter 5) IEC provides no marking procedure to indicate its approval and cannot be rendered responsible for any equipment declared to be in conformity with an IEC Publication 6) All users should ensure that they have the latest edition of this publication 7) No liability shall attach to IEC or its directors, employees, servants or agents including individual experts and members of its technical committees and IEC National Committees for any personal injury, property damage or other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC Publications 8) Attention is drawn to the Normative references cited in this publication Use of the referenced publications is indispensable for the correct application of this publication 9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of patent rights IEC shall not be held responsible for identifying any or all such patent rights International Standard IEC 62428 has been prepared by IEC technical committee 25: Quantities and units The text of this standard is based on the following documents: FDIS Report on voting 25/382/FDIS 25/390/RVD Full information on the voting for the approval of this standard can be found in the report on voting indicated in the above table This publication has been drafted in accordance with the ISO/IEC Directives, Part LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU 1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising all national electrotechnical committees (IEC National Committees) The object of IEC is to promote international co-operation on all questions concerning standardization in the electrical and electronic fields To this end and in addition to other activities, IEC publishes International Standards, Technical Specifications, Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “IEC Publication(s)”) Their preparation is entrusted to technical committees; any IEC National Committee interested in the subject dealt with may participate in this preparatory work International, governmental and nongovernmental organizations liaising with the IEC also participate in this preparation IEC collaborates closely with the International Organization for Standardization (ISO) in accordance with conditions determined by agreement between the two organizations –4– 62428 © IEC:2008 The committee has decided that the contents of this publication will remain unchanged until the maintenance result date indicated on the IEC web site under "http://webstore.iec.ch" in the data related to the specific publication At this date, the publication will be • • • • reconfirmed; withdrawn; replaced by a revised edition; or amended LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU 62428 © IEC:2008 –5– ELECTRIC POWER ENGINEERING – MODAL COMPONENTS IN THREE-PHASE AC SYSTEMS – QUANTITIES AND TRANSFORMATIONS Scope This International Standard deals with transformations from original quantities into modal quantities for the widely used three-phase a.c systems in the field of electric power engineering Normative references The following referenced documents are indispensable for the application of this document For dated references, only the edition cited applies For undated references, the latest edition of the referenced document (including any amendments) applies IEC 60050-141, International Electrotechnical Vocabulary (IEV) – Part 141: Polyphase systems and circuits 3.1 Terms, definitions, quantities and concepts General Quantities in this standard are usually time-dependent These quantities are for instance electric currents, voltages, linked fluxes, current linkages, electric and magnetic fluxes For quantities the general letter symbol g in case of real instantaneous values, g in case of complex instantaneous values and G in case of phasors (complex r.m.s values) are used NOTE Complex quantities in this standard are underlined Conjugated complex quantities are indicated by an additional asterisk (*) Matrices and column vectors are printed in bold face type, italic 3.2 Terms and definitions For the purposes of this document, the terms and definitions given in IEC 60050-141 and the following apply 3.2.1 original quantities quantities g or G of a three-phase a.c system NOTE Subscripts 1, 2, are generally used in this standard; additional letters may be put, for instance L1, L2, L3 as established in IEC 60909, IEC 60865 and IEC 61660 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU The examination of operating conditions and transient phenomena in three-phase a.c systems becomes more difficult by the resistive, inductive or capacitive coupling between the phase elements and line conductors Calculation and description of these phenomena in three-phase a.c systems are easier if the quantities of the coupled phase elements and line conductors are transformed into modal quantities The calculation becomes very easy if the transformation leads to decoupled modal systems The original impedance and admittance matrices are transformed to modal impedance and admittance matrices In the case of decoupling of the modal quantities, the modal impedance and admittance matrices become diagonal matrices –6– 62428 © IEC:2008 3.2.2 modal components quantities g M , g or G M found by a transformation from the original quantities according to M Clause NOTE Additional subscripts 1, 2, are used 3.2.3 column vector of quantities column matrix containing the three original quantities or modal components of a three-phase a.c system NOTE Column vectors are described by g or g M and G or G M , respectively NOTE The transformation can be power-variant or power-invariant, see Tables and 3.2.5 inverse modal transformation solution g M = T − g of the modal transformation that expresses a column vector g M containing the three modal quantities as a matrix product of the inverse transformation matrix T − by a column vector g containing the three original quantities 3.2.6 transformation into symmetrical components Fortescue transformation linear modal transformation with constant complex coefficients, the solution of which converts the three original phasors of a three-phase a.c system into the reference phasors of three symmetric three-phase a.c systems — the so-called symmetrical components — , the first system being a positive-sequence system, the second system being a negative-sequence system and the third system being a zero-sequence system NOTE The transformation into symmetrical components is used for example for the description of asymmetric steady-state conditions in three-phase a.c systems NOTE See Tables and 3.2.7 transformation into space phasor components linear modal transformation with constant or angle-dependent coefficients, the solution of which replaces the instantaneous original quantities of a three-phase a.c system by the complex space phasor in a rotating or a non-rotating frame of reference, its conjugate complex value and the real zero-sequence component NOTE The term “space vector” is also used for “space phasor” NOTE The space phasor transformation is used for example for the description of transients in three-phase a.c systems and machines NOTE See Tables and 3.2.8 transformation into αβ0 components Clarke transformation linear modal transformation with constant real coefficients, the solution of which replaces the instantaneous original quantities of a three-phase a.c system by the real part and the LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU 3.2.4 modal transformation matrix equation T g M = g for a column vector g M containing the three unknown modal quantities, where g is a column vector containing the three given original quantities and T is a × transformation matrix 62428 © IEC:2008 –7– imaginary part of a complex space phasor in a non-rotating frame of reference and a real zero-sequence component or replaces the three original phasors of the three-phase a.c system by two phasors ( α and β phasor) and a zero-sequence phasor NOTE The power-variant form of the space phasor is given by given by g s = g s = g α + j g β and the power-invariant form is ( g α + j gβ ) NOTE The αβ0 transformation is used for example for the description of asymmetric transients in three-phase a.c systems NOTE See tables and NOTE The power-variant form of the space phasor is given by given by g = r g = g d + j g q and the power-invariant form is r (gd + j gq ) NOTE The dq0 transformation is normally used for the description of transients in synchronous machines NOTE See Tables and 4.1 Modal transformation General The original quantities g 1, g , g and the modal components g , g , g are related to each M1 M2 M3 other by the following transformation equations: t 13 ⎞ ⎛⎜ g ⎞⎟ ⎟ M1 t 23 ⎟ ⎜ g ⎟ ⎜ M2 ⎟ t 33 ⎟⎠ ⎜ g ⎟ ⎝ M3 ⎠ ⎛ g ⎞ ⎛ t 11 t 12 ⎜ ⎟ ⎜ ⎜ g ⎟ = ⎜ t 21 t 22 ⎜ g ⎟ ⎜t ⎝ ⎠ ⎝ 31 t 32 (1) or in a shortened form: g =T g (2) M The coefficients t ik of the transformation matrix T can all be real or some of them can be complex It is necessary that the transformation matrix T is non-singular, so that the inverse relationship of Equation (2) is valid g M =T −1 g (3) If the original quantities are sinusoidal quantities of the same frequency, it is possible to express them as phasors and to write the transformation Equations (2) and (3) in an analogue form with constant coefficients: LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU 3.2.9 transformation into dq0 components Park transformation linear modal transformation with coefficients sinusoidally depending on the angle of rotation, the solution of which replaces the instantaneous original quantities of a three-phase a.c system by the real part and the imaginary part of a complex space phasor in a rotating frame of reference and a real zero-sequence component 62428 © IEC:2008 –8– ⎛ G1 ⎞ ⎜ ⎟ ⎜G ⎟ ⎜ ⎟ ⎝G3 ⎠ 4.2 ⎛ t 11 t 12 ⎜ = ⎜ t 21 t 22 ⎜ ⎝ t 31 t 32 t 13 ⎞ ⎛ G M1 ⎞ ⎟⎜ ⎟ t 23 ⎟ ⎜ G M2 ⎟ ⎟⎜ ⎟ t 33 ⎠ ⎝ G M3 ⎠ (4) G = T GM (5) G M = T −1 G (6) Power in modal components Transformation relations are used either in the power-variant form as given in Table or in the power-invariant form as given in Table The instantaneous power p expressed in terms of the original quantities is defined by: p = u1i1∗ + u i 2∗ + u i3∗ = (u1 u NOTE ⎛i∗ ⎞ ⎜1⎟ u ) ⎜ i 2∗ ⎟ = u T i ∗ ⎜ ∗⎟ ⎜ i3 ⎟ ⎝ ⎠ The asterisks denote formally the complex conjugate of the currents i1 , (7) i , i3 If these are real, i1∗ , i 2∗ , i3∗ are identical to i1 , i , i3 If the relationship between the original quantities and the modal components given in Equation (2) is introduced for the voltages as well as for the currents: u = T u M and i = T i M (8) T T u T = (T u M )T = u M T , (9) taking into account the power p expressed in terms of modal components is found as: T T ∗ ∗ p = uM T T iM (10) For the power-variant case where T T T ∗ is not equal to the unity matrix an example is given at the end of this section In case of T ∗ T T =E (11) LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU For the power-invariant form of transformation, the power calculated with the three modal components is equal to the power calculated from the original quantities of a three-phase a.c system with three line conductors and a neutral conductor, where u1 , u and u3 are the lineto-neutral voltages and i1 , i2 and i3 are the currents of the line conductors at a given location of the network In a three-phase a.c system with only three line conductors, u1 , u and u3 are the voltages between the line conductors and a virtual star point at a given location of the network 62428 © CEI:2008 – 34 – Tableau – Composantes modales conservant la puissance et leurs matrices de transformation Composantes modales composantes symetriques (composantes de Fortescue) (composantes de Clarke) composantes dq0, référentiel tournant (composantes de Park) Indice: Première M1 T Deuxième M2 a Troisième M3 séquence directe (1) séquence inverse (2) séquence homopolaire (0) ⎛1 ⎜ ⎜a 3⎜ ⎝a b α α β β séquence homopolaire axe longitudinal d axe transversal q séquence homopolaire T −1 3 ⎛ ⎜ ⎜ ⎜− ⎜ ⎜− ⎜ ⎝ ⎛c ⎜ ⎜ ⎜ c2 ⎜ ⎜ c3 ⎝ a a ⎞ ⎟ ⎟ ⎟ 2⎟ ⎟ ⎟ 2⎠ − ⎛1 a ⎜ ⎜1 a 3⎜ ⎜1 ⎝ 1⎞ ⎟ 1⎟ 1⎟⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ 2⎠ − s1 − s2 − s3 3 ⎛ ⎜ ⎜ ⎜ ⎜⎜ ⎝ ⎛ c c2 ⎜ ⎜− s − s ⎜ 1 ⎜ ⎝ phaseur spatial s conjugué complexe du phaseur spatial s* référentiel fixe séquence homopolaire phaseur spatial r transformation en phaseur spatial, conjugué complexe du phaseur spatial r* référentiel tournant séquence homopolaire a ⎛1 ⎜ ⎜a 3⎜ ⎝a a a ⎛ e jϑ ⎜ jϑ ⎜a e 3⎜ ⎜ a e jϑ ⎝ ⎛1 a ⎜ ⎜1 a 3⎜ ⎜1 ⎝ 1⎞ ⎟ 1⎟ 1⎟⎠ e − jϑ a e − jϑ a e − jϑ 1⎞ ⎟ 1⎟ ⎟ 1⎟⎠ ⎛ e − jϑ ⎜ jϑ ⎜e 3⎜ ⎜ ⎝ c Toutes les matrices de transformation 2 − − ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠ c3 ⎞⎟ − s3 ⎟ ⎟ ⎟ ⎠ c c transformation en phaseur spatial, 2 − a ⎞ ⎟ a⎟ ⎟⎟ ⎠ a e − jϑ a e jϑ 2 a ⎞ ⎟ a⎟ ⎟⎟ ⎠ a e − jϑ ⎞ ⎟ a e jϑ ⎟ ⎟⎟ ⎠ c T données ici satisfont les conditions: t 11 + t 21 + t 31 = , t 12 + t 22 + t 32 = , t 13 = t 23 = t 33 b Les normes CEI 60909, 60865 et 61660 ont introduit les indices (1), (2), (0) pour les composantes symétriques ne conservant pas la puissance, pour éviter toute confusion si les indices 1, 2, sont utilisés la place de L1, L2, L3 c c1 = cos ϑ , c = cos (ϑ − s = sin (ϑ + 2π ), 2π ), ∗ 2π ), s1 = sin ϑ , s = sin (ϑ − 2π ), a = ej π / , a = a , + a + a = Dans le cas des machines synchrones, rotor c = cos (ϑ + ϑ est donné par ∫ ϑ = Ω (t ) dt , où Ω est la vitesse angulaire instantanée du Les Tableaux et donnent les relations pour les composantes αβ0 et dq0 avec les composantes en vecteurs spatiaux sous forme ne conservant pas la puissance et la conservant LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU composantes αβ0, référentiel fixe Composante: c = cos (ϑ − ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠ c ⎞⎟ − s3 ⎟ ⎟⎟ ⎠ 2π ), ϑ ( − 3 gβ est donné par ∫ ⎞⎟ 0⎟ ⎟ 1⎟ ⎠ s1 = sin ϑ , j e -jϑ - j e jϑ j e jϑ - j e - jϑ s = sin (ϑ − 2π ), ⎞⎟ 0⎟ ⎟ 1⎟ ⎠ s = sin (ϑ + ⎛ j 0⎞ ⎜ ⎟ ⎜ − j 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ e jϑ ⎜ ⎜ e - jϑ ⎜ ⎜ ⎝ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ 2π ) ( s T ) ⎞⎟ 0⎟ ⎟ 2⎟ ⎠ ( • g r ⎛ e -jϑ ⎜ ⎜ ⎜ ⎜ ⎝ e jϑ 0 ⎞⎟ 0⎟ ⎟ 1⎟ ⎠ e jϑ j e jϑ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ a a e-jϑ g e -jϑ j e - jϑ a e-jϑ a e-jϑ r g∗ e - jϑ 0 ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ e jϑ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞⎟ 0⎟ ⎟ 1⎟ ⎠ ⎞⎟ 0⎟ ⎟ 2⎟ ⎠ 2⎞ ⎟ 2⎟ ⎟ ⎟⎠ T ) ⎛ 1 0⎞ ⎟ 1⎜ ⎜ − j j 0⎟ 2⎜ ⎟ ⎝ 0 2⎠ ⎛ e jϑ ⎜⎜ − j e jϑ 2⎜ ⎜ ⎝ ⎛ 1 0⎞ ⎟ 1⎜ ⎜ − j j 0⎟ 2⎜ ⎟ ⎝ 0 2⎠ g ⎛ e jϑ ⎜ jϑ ⎜a e 2⎜ ⎜ a e jϑ ⎝ s g∗ 2⎞ ⎟ 2⎟ ⎟⎠ ⎛1 1⎜ ⎜a 2⎜ ⎝a • g ⎛ e -jϑ ⎜⎜ − j e - jϑ 2⎜ ⎜ ⎝ ϑ = Ω (t ) dt , où Ω est la vitesse angulaire instantanée du rotor 2π ), ⎛ e -jϑ ⎜ ⎜ e jϑ ⎜ ⎜ ⎝ ⎛ j 0⎞ ⎜ ⎟ ⎜ − j 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ c1 s1 ⎞ ⎜ ⎟ ⎜ − s1 c1 ⎟ ⎜ 0 ⎟⎠ ⎝ ) g0 T − s1 1⎞ ⎟ − s 1⎟ − s 1⎟⎠ gq ⎛ c1 − s1 ⎞ ⎜ ⎟ ⎜ s1 c1 ⎟ ⎜0 ⎟⎠ ⎝ ⎛ c1 ⎜ ⎜ c2 ⎜c ⎝ ⎞ 1⎟ 1⎟⎟ ⎟ 1⎟ ⎠ ( • gd ) g0 T ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ ⎜ ⎜− ⎜ ⎜⎜ − ⎝ • gα c = cos (ϑ + a e -jϑ ⎞ ⎟ a e jϑ ⎟ ⎟⎟ ⎠ a ⎞ ⎟ a⎟ ⎟ ⎟ ⎠ − − Dans le cas des machines synchrones, jϑ c1 = cos ϑ , ae a e -jϑ ⎛1 a 2⎜ ⎜1 a 3⎜1 ⎜2 ⎝ ⎛g ⎞ ⎜ s⎟ ⎜ g∗ ⎟ = ⎜ s⎟ ⎜g ⎟ ⎝ 0⎠ ⎛ e -jϑ ⎜ jϑ ⎜e 3⎜ ⎜ ⎝ ⎛ c1 c2 2⎜ ⎜ − s1 − s 3⎜ 1 ⎜ ⎝ ⎛ gd ⎞ ⎜ ⎟ ⎜ gq ⎟ = ⎜g ⎟ ⎝ 0⎠ ⎛g ⎞ ⎜ r⎟ ⎜ g∗ ⎟ = ⎜ r⎟ ⎜g ⎟ ⎝ 0⎠ ⎛1 − ⎜ 2⎜ 3⎜1 ⎜⎜ ⎝2 2 g )T ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ gα ⎞ ⎜ ⎟ ⎜ gβ ⎟ = ⎜g ⎟ ⎝ 0⎠ ⎛ g1 ⎞ ⎜ ⎟ ⎜ g2 ⎟ = ⎜g ⎟ ⎝ 3⎠ g2 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU • (g Tableau – Composantes de Clarke, de Park et en phaseurs spatiaux – Matrices de transformation pour les composantes modales sous forme ne conservant pas la puissance 62428 © CEI:2008 – 35 – c = cos (ϑ − ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠ 2π ), ϑ ( − gβ 3 g0 est donné par ∫ ⎞⎟ ⎟ ⎟ 2⎟ ⎠ s1 = sin ϑ , ⎞ ⎟ ⎟ ⎟⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ − s3 − s2 − s1 gq g0 s = sin (ϑ − 2π ), ⎛1 j ⎜ ⎜1 − j 2⎜ ⎝0 0 ⎞⎟ ⎟ ⎟ 2⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ s = sin (ϑ + ⎞ ⎟ ⎟ ⎟⎠ j e jϑ - j e - jϑ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ 2 )T ⎛ c1 − s1 ⎞ ⎜ ⎟ ⎜ s1 c1 ⎟ ⎜0 ⎟⎠ ⎝ ⎛ e jϑ ⎜⎜ - jϑ e ⎜⎜ ⎝ ⎛c ⎜ ⎜ ⎜ c2 ⎜ ⎜ c3 ⎝ ( • gd ( s 2π ) a g ⎛ e -jϑ ⎜ ⎜ ⎜ ⎜ ⎝ e jϑ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ T ) ⎞ ⎟ ⎟ ⎟⎠ 1⎞ ⎟ 1⎟ 1⎟⎠ 0 ⎞⎟ 0⎟ ⎟ 1⎟ ⎠ e jϑ j e jϑ a ⎛1 1 ⎜ ⎜− j j 2⎜ ⎝0 s g∗ ⎛1 ⎜ ⎜a 3⎜ ⎝a • g ⎛ e -jϑ ⎜⎜ − j e - jϑ ⎜ 2⎜ ⎝ ϑ = Ω (t ) dt , où Ω est la vitesse angulaire instantanée du rotor 2π ), j e -jϑ - j e jϑ ⎛1 j ⎜ ⎜1 − j 2⎜ ⎝0 2 )T ⎛ c1 s1 ⎞ ⎜ ⎟ ⎜ − s1 c1 ⎟ ⎜ 0 ⎟⎠ ⎝ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ e -jϑ ⎜⎜ jϑ e ⎜⎜ ⎝ ⎛ ⎜ ⎜ ⎜− ⎜ ⎜− ⎜ ⎝ • gα c = cos (ϑ + a e-jϑ ⎞ ⎟ a e jϑ ⎟ ⎟⎟ ⎠ a ⎞ ⎟ a⎟ ⎟⎟ ⎠ c3 ⎞⎟ − s3 ⎟ ⎟ ⎟ ⎠ − 2 Dans le cas des machines synchrones, jϑ c1 = cos ϑ , ae a e -jϑ ⎛1 a ⎜ ⎜1 a 3⎜ ⎜1 ⎝ ⎛ e-jϑ ⎜ jϑ ⎜e 3⎜ ⎜ ⎝ 3 2 − ⎛ c c2 ⎜ ⎜− s − s ⎜ 1 ⎜ ⎝ 2 ⎛ ⎜ ⎜ ⎜ ⎜⎜ ⎝ − g )T ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛g ⎞ ⎜ r⎟ ⎜ g∗ ⎟ = ⎜ r⎟ ⎜g ⎟ ⎝ 0⎠ ⎛g ⎞ ⎜ s⎟ ⎜ g∗ ⎟ = ⎜ s⎟ ⎜g ⎟ ⎝ 0⎠ ⎛ gd ⎞ ⎜ ⎟ ⎜ gq ⎟ = ⎜g ⎟ ⎝ 0⎠ ⎛ gα ⎞ ⎜ ⎟ ⎜ gβ ⎟ = ⎜g ⎟ ⎝ 0⎠ ⎛ g1 ⎞ ⎜ ⎟ ⎜ g2 ⎟ = ⎜g ⎟ ⎝ 3⎠ g2 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU • (g ⎞⎟ ⎟ ⎟ 2⎟ ⎠ Tableau – Composantes de Clarke, de Park et en phaseurs spatiaux – Matrices de transformation pour les composantes modales sous forme conservant la puissance r 0 ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ e ⎞⎟ 0⎟ ⎟ 1⎟ ⎠ - jϑ ⎛ e jϑ ⎜ ⎜ ⎜ ⎜ ⎝ T ) ⎞ ⎟ ⎟ ⎟⎠ e -jϑ j e - jϑ a e -jϑ a e-jϑ e -jϑ g ⎛1 1 ⎜ ⎜− j j 2⎜ ⎝0 ⎛ e jϑ ⎜⎜ − j e jϑ ⎜⎜ ⎝ r g∗ ⎛ e jϑ ⎜ jϑ ⎜a e 3⎜ ⎜ a e jϑ ⎝ ( • g ⎞⎟ ⎟ ⎟ 2⎟ ⎠ 1⎞ ⎟ 1⎟ ⎟ 1⎟⎠ – 36 – 62428 © CEI:2008 62428 © CEI:2008 – 37 – Les Tableaux et donnent les relations entre les composantes modales, valables si les grandeurs originales sont sinusoïdales, de même fréquence, et peuvent être représentés par des phaseurs r.m.s Tableau – Matrices de transformation pour les phaseurs sous forme ne conservant pas la puissance • (G G ⎛ G1 ⎞ ⎜ ⎟ ⎜G ⎟ = ⎜G ⎟ ⎝ 3⎠ G )T ( • G (1) ⎛1 a 1⎜ ⎜1 a 3⎜ ⎜1 ⎝ a ⎞ ⎟ a⎟ ⎟⎟ ⎠ ⎛1 − ⎜ 2⎜ 3⎜1 ⎜⎜ ⎝2 ⎛G α ⎞ ⎜ ⎟ ⎜Gβ ⎟ = ⎜G ⎟ ⎝ 0⎠ 2 − − ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠ G (0 ) ( • Gα ⎛ ⎜ ⎜− ⎜ ⎜⎜ − ⎝ 1⎞ ⎟ 1⎟ 1⎟⎠ a a )T Gβ G0 ⎞ 1⎟ 1⎟⎟ ⎟ 1⎟ ⎠ − )T 3 ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ j 0⎞ ⎟ 1⎜ ⎜ − j 0⎟ 2⎜ ⎟ ⎝ 0 2⎠ ⎛ 1 0⎞ ⎜ ⎟ ⎜ − j j 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ Tableau – Matrices de transformation pour les phaseurs sous forme conservant la puissance • (G G ⎛ G1 ⎞ ⎜ ⎟ ⎜G ⎟ = ⎜G ⎟ ⎝ 3⎠ ( G )T • G (1) ⎛1 ⎜ ⎜a 3⎜ ⎝a ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛G ⎞ ⎜ (1) ⎟ ⎜ G ( 2) ⎟ = ⎜ ⎟ ⎜ G (0 ) ⎟ ⎠ ⎝ ⎛1 a ⎜ ⎜1 a 3⎜ ⎜1 ⎝ ⎛G α ⎞ ⎜ ⎟ ⎜Gβ ⎟ = ⎜G ⎟ ⎝ 0⎠ ⎛ ⎜ ⎜ ⎜ ⎜⎜ ⎝ 2 3 2 − G ( 2) a ⎞ ⎟ a⎟ ⎟⎟ ⎠ 2 − − G (0 ) a a 1⎞ ⎟ 1⎟ 1⎟⎠ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠ ⎛1 1 ⎜ ⎜− j j 2⎜ ⎝0 )T ( • Gα ⎛ ⎜ ⎜ ⎜− ⎜ ⎜− ⎜ ⎝ Gβ G0 − ⎛1 j ⎜ ⎜1 − j 2⎜ ⎝0 0 ⎞ ⎟ ⎟ ⎟⎠ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ )T 2 ⎞ ⎟ ⎟ ⎟⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU ⎛1 ⎜ ⎜a ⎜a ⎝ ⎛ 0⎞ ⎜ ⎟ ⎜0 0⎟ ⎜ 0 1⎟ ⎝ ⎠ ⎛G ⎞ ⎜ (1) ⎟ ⎜ G ( 2) ⎟ = ⎜ ⎟ ⎜ G (0 ) ⎟ ⎝ ⎠ G ( 2) 62428 © CEI:2008 – 38 – Découplage dans les systèmes a.c triphasés 5.1 Découplage en régime permanent avec des grandeurs sinusoïdales La Figure montre un exemple d’un système a.c triphasé avec un couplage inductif entre les trois conducteurs de ligne L1, L2, L3 ou les trois éléments de phase d’un moteur ou d’un générateur triphasé dont le point neutre est mis la terre par l’intermédiaire d’une impédance ZN NOTE Les indices L1, L2, L3 désignent les tensions étoilées aux points Q ou N et les courants de ligne (voir 3.2) IL1 R1 L2 IL2 R2 L3 IL3 R3 jX11 jX12 = jX21 jX23 = jX32 N jX22 jX13 = jX31 jX33 ZN UL1Q UL2Q UL3Q UN E IEC 1237/08 Figure – Circuit alimenté par une source triphasée de tension U L1Q , U L2Q , U L3Q au point Q et mis la terre au point neutre N par l’intermédiaire de l’impédance Z N = RN + j X N Il résulte de la Figure 1: ⎛ U L1 Q ⎞ ⎛ Z L1L1 Z L1L2 ⎜ ⎟ ⎜ ⎜ U L2 Q ⎟ = ⎜ Z L2 L1 Z L2 L2 ⎜ U L3 Q ⎟ ⎜ Z L3 L1 Z L3 L2 ⎝ ⎠ ⎝ Z L1L3 ⎞ ⎟ Z L2 L3 ⎟ Z L3 L3 ⎟⎠ ⎛ I L1 ⎞ ⎛U N ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ I L2 ⎟ + ⎜ U N ⎟ , ⎜I ⎟ ⎜U ⎟ ⎝ L3 ⎠ ⎝ N⎠ U LQ = Z L I L + U N (16) (17) En remplaỗant dans lEquation (17) les grandeurs originales par les composantes modales l’aide de l’Equation (2), on obtient: T U MQ = Z L T I M + T U MN (18) L’Equation (18) multipliée par T −1 donne l’Equation (19), car T T = E : −1 U MQ = T −1 Z L T I M + U MN = Z M I M + U MN (19) LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU Q L1 62428 © CEI:2008 – 39 – On peut par conséquent donner les composantes modales suivantes: Matrice modale d’impédance Z M = T −1 Z L T (20) U MQ = T −1 U LQ (21) I M = T −1 I L (22) U MN = T −1 U N (23) Vecteur modal de tension en Q Vecteur modal de courant Si Z N est finie, on déduit l’équation suivante de la Figure 1: ⎛U N ⎞ ⎛ Z N ⎜ ⎟ ⎜ ⎜U N ⎟ = ⎜ Z N ⎜U ⎟ ⎜ Z ⎝ N⎠ ⎝ N ZN ZN ZN ZN ⎞ ⎟ ZN ⎟ Z N ⎟⎠ ⎛ I L1 ⎞ ⎜ ⎟ ⎜ I L2 ⎟ ⎜I ⎟ ⎝ L3 ⎠ (24) U LN = Z N I L (25) U MN = T −1 Z NT I M = Z MN I M (26) L’Equation (23) devient: Le Tableau donne les tensions modales et les impédances modales en composantes symétriques et en composantes αβ0 sous forme ne conservant pas la puissance (invariance du conducteur de référence) et conservant la puissance dans le cas de phaseurs sous les conditions suivantes: • Système symétrique avec: U L2 Q = a U L1Q , U L3 Q = a U L1Q , d’où il résulte: U L1Q + U L2 Q + U L3 Q = • Matrice impédance cyclique et symétrique: Z L1L1 = Z L2 L2 = Z L3 L3 = Z A , Z L1L2 = Z L2 L1 = Z L1L3 = Z L3 L1 = Z L2 L3 = Z L3 L2 = Z B • Matrice impédance seulement cyclique: Z L1L1 = Z L2 L2 = Z L3 L3 = Z A , Z L1L2 = Z L2 L3 = Z L3 L1 = Z B , Z L1L3 = Z L2 L1 = Z L3 L2 = Z C LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU Vecteur modal de tension en N 62428 © CEI:2008 – 40 – • Le Tableau montre que le découplage est possible si la matrice impédance Z L est cyclique et symétrique en diagonale et, pour les composantes symétriques, également si la matrice impédance Z L est seulement cyclique La Figure montre ce résultat Le système homopolaire ne comporte pas de source de tension Tableau – Tensions et impédances modales pour des phaseurs Forme ne conservant pas la puissance Composantes modales αβ0 ⎛ U L1Q ⎞ ⎛ U M1Q ⎞ −1 ⎜ ⎟ ⎜ ⎟ T ⎜ U L2 Q ⎟ = ⎜ U M2 Q ⎟ = ⎜ U L3 Q ⎟ ⎜ U M3 Q ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ U L1 Q ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ U L1Q ⎞ ⎜ ⎟ ⎜ − j U L1Q ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ U N ⎞ ⎛ U M1N ⎞ ⎟ ⎜ ⎟ ⎜ T ⎜ U N ⎟ = ⎜ U M2 N ⎟ = ⎜U ⎟ ⎜U ⎟ ⎝ N ⎠ ⎝ M3 N ⎠ ⎛ ⎞ ⎛0 0 ⎞ ⎛ I M1 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜I ⎟ ⎜ ⎟ = ⎜0 ⎜U ⎟ ⎜ 0 Z ⎟ ⎜ M2 ⎟ N ⎠ ⎝ I M3 ⎠ ⎝ N⎠ ⎝ système symétrique de tensions de source (1) (2) (0) αβ0 ⎛ U L1 Q ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ U L1 Q ⎞ ⎜ ⎟ ⎜ − j U L1 Q ⎟ ⎜ ⎟ ⎝ ⎠ a −1 ⎛ ⎞ ⎛0 0 ⎞ ⎛ I M1 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟ = ⎜0 0 ⎟⎜I ⎟ ⎜U ⎟ ⎜ 0 Z ⎟ ⎜ M2 ⎟ N ⎠ ⎝ I M3 ⎠ ⎝ N⎠ ⎝ b T −1 Z L T = Z M avec ZL ⎛Z A ⎜ = ⎜ZB ⎜Z ⎝ B ZB ZA ZB Indépendant du type de composantes modales ZB ⎞ ⎟ ZB ⎟ Z A ⎟⎠ ZM c ⎛ Z M1 ⎜ =⎜ ⎜ ⎝ Z M2 0 ⎞ ⎛Z A − ZB ⎟ ⎜ 0 ⎟=⎜ ⎟ ⎜ Z M3 ⎠ ⎝ 0 ⎞ ⎟ ⎟ + Z B ⎟⎠ ZA − ZB ZA seulement pour les composantes symétriques (1) (2) (0) ⎛ Z M1 Z M = ⎜⎜ Z M2 ⎜ 0 ⎝ T −1 Z L T = Z M avec ZL ⎛Z A ⎜ = ⎜ZC ⎜Z ⎝ B ZB ZA ZC ZC⎞ ⎟ ZB ⎟ Z A ⎟⎠ ⎛ Z A + a2 Z B + a Z C ⎜ =⎜ ⎜⎜ ⎝ d a U L2 Q = a U L1Q , U L3 Q = a U L1Q , b U L1N = U L2 N = U L3 N = U N , voir Figure c A: L1L1 = L2L2 = L3L3, A: L1L1 = L2L2 = L3L3, d ⎞ ⎟⎟ Z M3 ⎟⎠ Z A + a ZB + a ZC U L1Q + U L2 Q + U L3 Q = B: L1L2 = L2L3 = L3L1 B: L1L2 = L2L3 = L3L1, ⎞ ⎟ ⎟ Z A + Z B + Z C ⎟⎟ ⎠ C: L1L3 = L2L1 = L3L2 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU (1) (2) (0) Forme conservant la puissance 62428 © CEI:2008 – 41 – ZM1 IM1 Q N UM1Q UM1N 0M1 ZM2 IM2 Q N UM2Q UM2N 0M2 N ZM3N = 3ZN UM3Q UM3N IEC 1238/08 Figure – Remplacement du système a.c triphasé couplé de la Figure par trois systèmes découplés sous des conditions déterminées (voir texte) Le découplage est également possible si une matrice admittance, par exemple pour les capacités ligne ligne et ligne terre, est donnée la place d’une matrice impédance 5.2 Découplage en régime transitoire On peut écrire les équations différentielles du système triphasé a.c couplé représenté la Figure 1: ⎛ uL1Q ⎞ ⎛ RL ⎜ ⎟ ⎜ ⎜ uL2 Q ⎟ = ⎜ ⎜ ⎟ ⎜ ⎝ uL3Q ⎠ ⎝ 0 RL 0⎞ ⎟ 0⎟ RL ⎟⎠ ⎡⎛ LL1L1 LL1L2 ⎛ iL1 ⎞ ⎢ ⎜ ⎟ d ⎜ ⎜ iL2 ⎟ + dt ⎢⎜ LL1L1 LL2 L2 ⎢⎜⎜ ⎜i ⎟ ⎝ L3 ⎠ ⎢⎣⎝ LL3 L1 LL3 L2 uLQ = RL iL + LL1L3 ⎞ ⎟ LL2L3 ⎟ ⎟ LL3 L3 ⎟⎠ ⎛ iL1 ⎞ ⎤ ⎛ uL1N ⎞ ⎟ ⎜ ⎟⎥ ⎜ ⎜ iL2 ⎟ ⎥ + ⎜ uL 2N ⎟ ⎜ i ⎟⎥ ⎜ ⎟ ⎝ L3 ⎠ ⎥⎦ ⎝ uL3N ⎠ d (LL iL ) + uLN dt (27) (28) En remplaỗant dans lEquation (28) les grandeurs originales par des composantes modales l’aide de l’Equation (2), on obtient: T u MQ = RL T i M + d (LL T i M ) + T uMN dt (29) En multipliant gauche l’Equation (29) par T −1 on obtient l’Equation (30), en tenant compte −1 de T T = E u MQ = T −1 RL T i M + d d (T −1 LL T i M ) + ( T −1 T ) T −1 LL T i M + u MN dt dt (30) LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU 0M3 ZM3 IM3 Q 62428 © CEI:2008 – 42 – On introduit la matrice des résistances modales: T −1 RL T = R M (31) et la matrice des inductances modales: T −1 LL T = L M (32) L’Equation (30) conduit à: d (LM i M ) + (T −1 d T ) LM i M + u MN dt dt (33) Le terme qui appart dans l’Equation (33) mais pas dans l’Equation (28) peut être déterminé comme suit partir du Tableau 2: • • composantes αβ0 et phaseur spatial dans un référentiel fixe: d T −1 T = dt composantes dq0: T • (34) −1 ⎛0 − 0⎞ ⎟ d dϑ ⎜ T = ⎜ 0⎟ dt dt ⎜ ⎟ ⎝0 0⎠ (35) phaseur spatial dans un référentiel tournant: T −1 ⎛ j 0⎞ ⎟ d dϑ ⎜ T = ⎜0 − j 0⎟ dt dt ⎜ ⎟ ⎝0 0⎠ (36) Si le point neutre de la Figure est mis la terre au travers d’une résistance RN et d’une inductance LN en série, le vecteur des tensions peut s’écrire: ⎛ uL1N ⎞ ⎛ RN ⎜ ⎟ ⎜ ⎜ uL2 N ⎟ = ⎜ RN ⎜⎜ ⎟⎟ ⎜ R ⎝ uL3 N ⎠ ⎝ N RN RN RN RN ⎞ ⎟ RN ⎟ RN ⎟⎠ ⎡⎛ LN ⎛ iL1 ⎞ d ⎢⎜ ⎜ ⎟ ⎜ iL2 ⎟ + dt ⎢⎜ LN ⎜i ⎟ ⎢⎣⎜⎝ LN ⎝ L3 ⎠ uLN = RN iL + d ( LL i L ) dt LN LN LN LN ⎞ ⎟ LN ⎟ LN ⎟⎠ ⎛ iL1 ⎞ ⎤ ⎜ ⎟⎥ ⎜ iL2 ⎟ ⎥ ⎜ i L3 (37) (38) En remplaỗant dans l’Equation (38) les grandeurs originales par les composantes modales l’aide de l’Equation (2), on obtient: T u MN = RN T i M + d (LN T i M ) dt (39) Du fait de T T = E , en multipliant gauche l’Equation (39) par T −1 on obtient l’Equation (40): −1 LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU u MQ = R M i M + 62428 © CEI:2008 – 43 – u MN = T −1 RN T i M + d d (T −1 LN T i M ) + (T −1 T ) T −1 LN T i M dt dt (40) En introduisant la matrice des résistances modales: T −1 RN T = R MN (41) T −1 LN T = L MN (42) et la matrice des inductances modales: u MN = R MN i M + d (L i ) + (T −1 d T ) LMN i M dt MN M dt (43) Le terme qui appart dans les grandeurs originales de l’Equation (43) mais pas dans celles de l’Equation (38) peut être déterminé comme suit partir du Tableau 2: u αN = , u dN = , u sN = , u rN = , u βN = , u qN = , u ∗sN = , u 0N = RN i0 + LN d i0 dt u ∗rN = , avec RMN = RN et LMN = LN Dans le cas du régime permanent, où u MQ → U MQ , i M → I M , d i dt M → j ω I M , u MN → U MN et si T est indépendant du temps, l’Equation (30) se ramène l’Equation (19) et l’Equation (43) l’Equation (26) Le Tableau donne les expressions des tensions et des inductances modales pour les composantes αβ0 , dq0 et phaseurs spatiaux en régime transitoire LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU l’Equation (40) implique: d c b c LB LA LB LB ⎞ ⎟ LB ⎟ L A ⎟⎠ LC ⎞ ⎟ LB ⎟ L A ⎟⎠ LM2 0 ⎞ ⎛ L A − LB ⎟ ⎜ ⎟=⎜ ⎜ ⎟ LM3 ⎠ ⎝ A: L1L1 = L2L2 = L3L3, A: L1L1 = L2L2 = L3L3, B: L1L2 = L2L3 = L3L1, B: L1L2 = L2L3 = L3L1 u L1N = u L2 N = u L3 N = u N ⎛ LM1 LM = ⎜⎜ LM2 ⎜ 0 ⎝ C: L1L3 = L2L1 = L3L2 L A − LB 2π ) ⎞ ⎛ LA + a LB + a LC ⎜ ⎟⎟ = ⎜ ⎜ ⎟ LM3 ⎠ ⎜ ⎝ LA + a LB + a LC 0 0 ⎞ ⎟ ⎟ LA + LB + LC ⎟⎟ ⎠ ⎞ ⎟ ⎟ L A + LB ⎟⎠ seulement pour les composantes ss*0 et rr*0: ⎛ LM1 ⎜ LM = ⎜ ⎜ ⎝ indépendant des composantes modales: ⎛ uˆ e j ((ω −Ω ) t +ϕQ −ϑ0 ) ⎞ ⎜ Q ⎟ − j ((ω − Ω ) t +ϕQ −ϑ0 ) ⎟ ⎜ ˆ R uQ e ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ uˆQ cos ((ω − Ω ) t + ϕ Q − ϑ0 )⎞ ⎜ ⎟ F ⎜ uˆQ sin ((ω − Ω ) t + ϕ Q − ϑ0 ) ⎟ ⎜ ⎟ ⎝ ⎠ R= rr*0 et dq0 ⎞ ⎛ i M1 ⎞ ⎛ 0 ⎞ ⎛ i M1 ⎞ ⎛ ⎞ ⎛0 ⎟ ⎜ ⎟ ⎟⎜ ⎟ d⎜ ⎜ ⎟ ⎜ ⎟ ⎜ i M2 ⎟ + ⎜ 0 ⎟ ⎜ i M2 ⎟ D ⎜ ⎟ = ⎜0 ⎜ u ⎟ ⎜ 0 R ⎟ ⎜ i ⎟ ⎜ 0 L ⎟ dt ⎜ i ⎟ N ⎠ ⎝ M3 ⎠ ⎝ N⎠ ⎝ N⎠ ⎝ ⎝ M3 ⎠ uL1 Q = uˆQ cos(ω t + ϕQ ) , uL2 Q = uˆQ cos(ω t + ϕ Q − 23π ) , uL3 Q = uˆQ cos(ω t + ϕQ + d LB LA LC L L T = L M avec ⎛LA ⎜ LL = ⎜ L C ⎜L ⎝ B T −1 ⎛ LA ⎜ LL = ⎜ L B ⎜L ⎝ B T −1 L L T = L M avec b ⎛ uˆ e j (ω t +ϕQ ) ⎞ ⎜ Q ⎟ − j (ω t +ϕQ ) ⎟ ⎜ ˆ R uQ e ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ss*0 F = et R = , Forme conservant la puissance: F = forme ne conservant pas la puissance: D = 1, forme conservant la puissance: D = ⎛ uˆ Q cos (ω t + ϕ Q ) ⎞ ⎜ ⎟ F ⎜ uˆ Q sin (ω t + ϕ Q ) ⎟ ⎜ ⎟ ⎝ ⎠ αβ0 Forme ne conservant pas la puissance: LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU a a ⎛ u N ⎞ ⎛ u M1N ⎞ ⎟ −1 ⎜ ⎟ ⎜ T ⎜ u N ⎟ = ⎜ u M2 N ⎟ = ⎜u ⎟ ⎜u ⎟ ⎝ N ⎠ ⎝ M3 N ⎠ −1 ⎛ u L1 Q ⎞ ⎛ uM1 Q ⎞ ⎜ ⎟ ⎜ ⎟ T ⎜ uL2 Q ⎟ = ⎜ u M2 Q ⎟ = ⎜u ⎟ ⎜u ⎟ ⎝ L3 Q ⎠ ⎝ M3 Q ⎠ Composantes modales Tableau – Tensions et inductances modales en régime transitoire – 44 – 62428 © CEI:2008 62428 © CEI:2008 – 45 – Bibliographie CEI 60865 :1986, Système de vidéodisque optique réfléchissant préenregistré « Laser vision » 50 Hz/625 lignes – PAL CEI 60909 :2001, Courants de court-circuit dans les réseaux triphasés courant alternatif – Partie : Calcul des courants CEI 61660 (toutes les parties), Courants de court-circuit dans les installations auxiliaires alimentées en courant continu dans les centrales et les postes LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU ELECTROTECHNICAL COMMISSION 3, rue de Varembé PO Box 131 CH-1211 Geneva 20 Switzerland Tel: + 41 22 919 02 11 Fax: + 41 22 919 03 00 info@iec.ch www.iec.ch LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU INTERNATIONAL