VECTOR ANALYSIS OF A THREE-PHASE STATOR Ronald De Four Department of Electrical & Computer Engineering, The University of the West Indies St Augustine, Trinidad rdefour@eng.uwi.tt Emily Ramoutar University of Trinidad & Tobago, Point Lisas Campus, Trinidad eramoutar@tstt.net.tt Juliet Romeo Department of Electrical & Computer Engineering, The University of the West Indies St Augustine, Trinidad jnromeo@hotmail.com Brian Copeland Department of Electrical & Computer Engineering, The University of the West Indies St Augustine, Trinidad bcopeland@eng.uwi.tt Abstract Vector analysis is widely used for the analysis, modeling and control of electrical machines excited with sinusoidal supply voltages However, since the presentation of the theory to field, it is not evident that any attempt had been made to justify the existence and location of vector currents and voltages and the equality of scalar and vector current magnitudes This paper presents the development of an equivalent electric and magnetic circuit to show the production of a magnetic vector current from a scalar electric current in a winding, provide justification for the equality of scalar and vector current magnitudes and support the existence and location of voltage vectors In addition, the development of the equivalent electric and magnetic circuits of a three-phase stator would provide a platform for vector © Copyright 2007 by Ronald De Four All rights reserved No part of this publication may be reproduced in any material form without the written permission of the Copyright Owner, except in accordance with the provisions of the Copyright Act 1997 (Act No of 1997) or under the terms of a License duly authorized and issued by the Copyright Owner analysis of electrical machines excited with non-sinusoidal and dc supplies as is the case with brushless dc machines Keywords: Vector Analysis; three-phase; stator Introduction Vector analysis of three-phase electrical machines was developed by Kovacs and Racsz in [1] and is widely used in the modeling, transient analysis and control of these machines The technique developed in [1] has been documented in many books of reputed authors [2-5], and has many advantages over other methods, some of which are: a reduction in system equations, easier to control the machine, a clear conceptualization of machine dynamics and easier analytical solution of dynamic transients of machine variables [6] However, the detailed material presented by Kovacs in [7] for the development of the vector analysis theory of a three-phase stator raises some issues of great concern This paper is intended to present the issues of concern in the vector analysis theory presented in [7] and provide a basis for the existence of current and voltage vectors associated with the threephase stator of an electrical machine, through the development of an equivalent electric and magnetic circuit Space Vector Theory Issues The material presented by Kovacs in [7], for the development of the vector analysis theory of a three-phase stator, raises the following issues of concern: (1) It was reported by Kovacs in [7] that a current flowing through a stator phase winding produces a related current vector along the winding’s magnetic axis In addition, if the instantaneous value of the winding current is given by i, the magnetomotive force (mmf) produced was given by F = Ni (1) where, N represented the number of turns in the winding The mmf was reported to be a vector quantity, lying on the magnetic axis of the winding and given by r r F = Ni (2) r r which presents the existence of a current vector i which is collinear with the mmf F However, the current flowing through the winding was also regarded as a vector Hence, the mechanism and process by which a scalar current flowing through the winding produces a vector current along the winding’s magnetic axis were not clearly presented (2) It was stated that the magnitude of the scalar current i in the winding was equal to that of the r vector current i lying on the winding’s magnetic axis, but no proof was given for this equality (3) The current vectors produced by each phase winding of a three-phase, two-pole stator, whose phase windings were separated from each other by 120 electrical degrees, were added vectorially to produce the resultant current vector, which was represented by r r r2 i r = i a + a ib + a i c (3) r where, i r represents the resultant current vector, i a , i b and ic are the instantaneous values of r r currents in phase windings a, b and c respectively Vectors a and a are unit position vectors representing the position of magnetic axes for windings b and c respectively It was then inferred that the resultant voltage vector could be produced in a similar manner by the vector addition of the voltage vectors produced by each phase winding and given by r r r v r = v a + a vb + a v c (4) r where, v r is the resultant voltage vector and v a , vb and vc are the instantaneous values of phase voltages for windings a, b and c respectively Although the above equation exists, it was not proven how scalar supply phase voltages were transformed into vector supply phase voltages and how these vector supply phase voltages lie on the magnetic axes of the windings (4) And finally, the multiplication of a scalar voltage differential equation for a phase winding by a unit position vector representing the position of the magnetic axis of that winding, although mathematically correct, fails to show how these scalar voltages are physically transformed into vector quantities For example, Kovacs in [7] presented Eq (5) which represent the scalar voltage differential equation for winding b as vb = i b Rb + d λb dt (5) where, vb , i b and λb are the scalar phase supply voltage, scalar phase current and scalar phase flux linkage respectively and Rb the stator phase resistance for winding b Eq (5) was then r multiplied by a , the unit position vector indicating the direction of the positive magnetic axis of winding b, producing Eq (6) r r r d λb a vb = a i b Rb + a dt (6) Eq (6) is mathematically correct, however, no physical steps were taken by Kovacs in [7] to show how scalar voltages ib Rb and d λb in Eq (5) were transformed to vector quantities in Eq dt (6) It was also reported by Kovacs in [7] that the vector method is a simple but mathematically precise method that makes visible the physical background of the various machine phenomena This is absolutely correct, however, the issues presented in (1) to (4) above, has failed to use the physical background of the various phenomena in the development of the vector quantities and equations As a result, the full power and benefits of the vector method in the analysis of threephase machines were not realized Holtz in [8-10] has employed and presented the space vector theory developed by Kovacs and Racz [1] in the development of vector equations for electrical machines However, the issues raised in (1) to (4) were not addressed In fact, Holtz summarized the space vector notation as introduced by Kovacs and Racz by stating it represents the sinusoidal field by a complex vector He added, it is postulated that the causes and effects of such field, namely the currents and voltages, also have the property of space vectors owing to existing formal properties [10] Equivalent Circuits of a Three-Phase Stator A cross section of the stator windings of a two-pole, three-phase machine is shown in Fig The phase windings are shown to be displaced from each other by 120 degrees and the positive direction of current flowing through each winding is upwards through the non-primed side and downwards through the primed side of each winding Using this convention of current flow through the windings, positive magnetic axes were developed for each phase winding, along which all magnetic quantities exists Fig Stator Windings of a Two-Pole, Three-Phase Machine The analysis of electromagnetic systems has traditionally been performed with the production of two circuits, an electrical circuit for electrical analysis and a magnetic circuit for magnetic analysis [11] However, quantities in the electrical circuit affect quantities in the magnetic circuit and vice versa As a result of the dependence of both electrical and magnetic circuits on each other, the development of an equivalent circuit containing both electrical and magnetic quantities would prove to be very useful in the analysis of electromagnetic systems Since the three-phase stator is an electromagnetic system, then the development of an equivalent circuit containing both electrical and magnetic quantities would be a powerful tool in the vector analysis approach of this electromagnetic system For this analysis, one phase winding of the three-phase stator, winding aa' was selected for analysis This winding is represented by its center conductors and current flow through the winding is in the positive direction as shown in Fig 2(a) The phase winding possesses resistance which is an electrical quantity and inductance which is both an electrical and a magnetic quantity The winding resistance R a being an electrical quantity is removed from the winding together with La di a di , which is an electrical voltage These two quantities R a and La a are placed on dt dt the left (electric) side of the circuit with the supply voltage V a The winding with its magnetic quantities and electric current is on the right side of the circuit of Fig 2(a) This process was undertaken in order to separate the electrical quantities from the magnetic quantities The electric scalar current i a , which leaves the electric circuit flows through the winding and produces r vector magnetic field intensity H a along the positive magnetic axis of winding aa' as shown in r Fig 2(a) The magnitude of H a , is obtained by a Amperes Circuital Law and is given by ia N a r , where l a is the length of the path of H a and N a is the number of turns of winding aa' la r r r The magnetic field intensity H a produces flux density B a , which is collinear with H a , and r r whose magnitude is given by µ | H a | , where µ is the permeability of the medium in which B a r r r exists Flux density B a produces flux φa , which is also collinear with B a and whose r magnitude is given by B a Aa , where Aa is the cross-sectional area of concern The flux linking r r winding aa' is given by λ a , which is collinear with φa and whose magnitude is given by r r λa r r φa N a And the flux linkage λ a produces current vector i a whose magnitude is given by La r which is collinear with λ a , where L a is the inductance of winding aa' Hence magnetic r r r r r quantities H a , B a , φa , λ a and current vector i a all lie along the positive (+ ve) magnetic axis of winding aa' and are spatial vector quantities possessing both magnitude and direction Since r r current vector i a leaves the magnetic circuit, a similar current vector i a must also enter the series connected magnetic circuit of Fig 2(b) In addition, since the electric and magnetic circuits are connected in series, this implies that the scalar electric current i a is of same r magnitude as the vector magnetic current i a The separation of electric and magnetic circuits is shown by the dotted vertical line I in Fig 2(b) r The vector magnetic current i a , on entering the electric circuit, produces a scalar current i a in the electric circuit, and the scalar electric current i a on entering the magnetic circuit, produces a r vector magnetic current i a on the magnetic axis of winding aa' as shown in Fig 2(b) Hence r the magnetic axis of winding aa' completes the electric circuit making i a and i a of same magnitude Fig Equivalent Circuits of Winding aa' (a) Electric and Magnetic- Electric Equivalent Circuits (b) Electric and Magnetic Equivalent Circuits If i a is changing, then the effect of the magnetic circuit on the electric circuit is seen in the voltage La di a r which opposes the current i a Since the magnitude of i a and i a are equal, and dt di a r can be referred i a lies along the winding’s magnetic axis, then the voltages i a R a and La dt to the magnetic axis of winding aa' without changing their magnitudes The vector summation r dia r of i a R a and La along the magnetic axis of winding aa', produces the supply voltage dt r vector V a along the magnetic axis of winding aa' Applying Kirchhoff’s voltage law to the electric and magnetic sides of Fig 2(b) yields, V a = i a Ra + La di a dt for the electric side, (7) for the magnetic side (8) and r dia r r V a = i a R a + La dt The production of an equivalent circuit containing electric and magnetic quantities for the electromagnetic system represented by one phase winding of a three-phase stator, clarifies the issues raised in (1) to (4) above A summary of the benefits gained from the above analysis utilizing the equivalent circuit containing electric and magnetic quantities as it relates to the issues raised in (1), (2) and (4) are as follows [12-13] (5) It shows, when a scalar current i a of an electromagnetic system, leaves the electric circuit r and enters the magnetic circuit, it is converted into a vector quantity i a of the same magnitude as the scalar current This is as a result of the series nature of the electric and magnetic circuits resulting in the same magnitude of both scalar and vector currents The location of the current vector is along the magnetic axis of the winding, because all magnetic quantities are located on its magnetic axis (6) Since the vector current is of the same magnitude as the scalar current and this current vector lies on the magnetic axis of the winding, then, scalar voltages i a R a and La di a can be dt r dia r referred to the magnetic axis of the winding becoming i a R a and La dt respectively, with these vector voltages being of same magnitude as their scalar counterparts In addition, since the di a in the electric circuit results in the scalar supply dt r dia r voltage, then, the sum of the vector voltages i a R a and La results in the vector supply dt sum of the scalar voltages i a R a and La voltage, which is of the same magnitude as the scalar supply voltage (7) In addition to showing the process by which scalar voltages are referred to the magnetic circuit of the electromagnetic system, the analysis provides a scalar and a vector voltage differential equation as shown in Eqs (7) and (8) If scalar analysis is being performed, then the scalar voltage differential equation is utilized, while, if vector analysis is being performed on the electromagnetic system formed by the stator, then, the vector voltage differential equation would be utilized Magnetic and Electric Vectors of a Three-Phase Stator The application of the above technique to the three-phase, two-pole stator shown in Fig 1, whose phase windings are displaced from each other by 120 electrical degrees, produces the magnetic and electric quantities of each phase along the phase magnetic axes as shown in Fig Each magnetic or electric phase variable can now be added vectorially to produce the resultant of r r r that variable Hence the resultant magnetic field intensity H res , flux density B res , flux φres , r r r flux linkage λ res , current vector i res , stator resistance voltage drop i res R s , inductance voltage Ls r d i res dt r and supply voltage V res are given by the vector addition of their phase variables shown on the magnetic axes of Fig 3, which yield: r r r2 H res = H a + a H b + a H c (9) r r r2 B res = B a + a Bb + a B c (10) r r r φ res = φ a + a φ b + a φ c (11) r r r2 λ res = λ a + a λ b + a λ c r r r2 i res = i a + a ib + a i c (13) r r i res R s = i a R a + a i b R b + a i c R c Ls (12) (14) r d i res (15) dt r dia r r r d i b r2 d i c = La + a Lb + a Lc dt dt dt r r r v res = v a + a v b + a v c (16) where, R s = R a = Rb = Rc and L s = L a = Lb = Lc r r In Eqs (9) to (16), a and a are unit vectors representing the position of the positive magnetic axes of windings bb' and cc' respectively and the magnetic and electric variables on the right hand side of these equations are the instantaneous values of these variables for the particular winding In addition, the stator resistance and inductance of each phase winding are represented by R s and L s respectively The application of Kirchhoff’s law to the vector voltages on each magnetic axis yields, va = ia Ra + d λa dt r r r d λb r v b = a vb = a ib Rb + a dt r r r r d λc v c = a vc = a ic Rc + a dt (17) (18) (19) Eqs (16) to (19) show that both the phase vector supply voltage and the resultant voltage vector of the three phase windings were obtained by vector addition of vector voltages that exist on the axes of the phase windings of Fig which addresses the issue raised in (4) 10 Fig Magnetic and Electric Quantities of Two-Pole, Three-Phase Stator Shown on Magnetic Axes of the Windings Conclusion The development of equivalent magnetic and electric circuits for a phase winding of a threephase stator produced electrical quantities on the electric side of the circuit and magnetic quantities on the magnetic side The boundary separating the electric and magnetic circuits is interfaced by the scalar electric current on the electric side and the vector magnetic current on the magnetic side of the equivalent circuit This indicates that a scalar electric current flowing through a stator phase winding produces a vector magnetic current on the magnetic axis of the winding, with the reverse taking place when the vector magnetic current enters the electric circuit The equality of scalar electric current and the magnitude of vector magnetic current are also realized by the manner in which the electric and magnetic circuits are connected This 11 equality in scalar electric and vector magnetic current magnitudes permit the referral of scalar electric voltages, including the supply voltage, to the magnetic circuit and lying on the magnetic axis of the winding The absence of this equivalent electric and magnetic circuit and the above results derived from it has not inhibited growth in the area of transient analysis and modeling of electrical machines under sinusoidal excitation However, its introduction to the analysis of electrical machines under sinusoidal excitation reveals the background of the various phenomena occurring in the machine In addition, the introduction of the equivalent electric and magnetic circuits of a three-phase stator would provide a platform for vector analysis of electrical machines excited with non-sinusoidal and dc supplies as is the case with brushless dc machines References Kovács P K and Rácz I., Transient Phenomena in Electrical Machines, Verlag der Ungarischen Akademie der Wissenschaften, Budapest, 1959 Vas P., Sernsorless Vector and Direct Torque Control, Oxford University Press, 1998 Boldea I and S A Nasar S A., Vector Control of AC Drives, CRC Press, Boca Raton, 1992 Bose B K., Power Electronics and Variable Speed Drives, IEEE Press, New Jersey, 1996 Trzynadlowski A M., The Field Orientation Principle in Control of Induction Motors, Kluwer Academic Publishers, USA, 1994 Krishnan K., Electric Motor Drives: Analysis, Modeling and Control, Prentice Hall Inc., New Jersey, 2001 Kovács P K., Transient Phenomena in Electrical Machines, Elsevier Science Publishers, Amsterdam, 1984 Holtz J., The Representation of AC Machine Dynamics by Complex Signal Flow Graphs, IEEE Trans on Industrial Electronics, Vol 42, No 3, June 1995, pp 263- 271 Holtz J., Pulse Width Modulation for Electronic Power Conversion, Proceedings of IEEE, Vol.82, No.8, pp.1194-1214, Aug 1994 10 Holtz J., On thje Spatial Propagation of Transient Magnetic Fields in AC Machines, IEEE Transactions on Industry Applications, Vol 32, No 4, July/Aug 1996, pp 927-937 11 Slemon G R and Straughen A., Electric Motors, Assison-Wesley Publishing Company, Inc., USA, 1982 12 12 De Four R., “A Self-Starting Method and an Apparatus for Sensorless Commutation of Brushless DC Motors” WIPO Publication No WO 2006/073378 A1 July 13, 2006 13 De Four R., “A Self-Starting Method and an Apparatus for Sensorless Commutation of Brushless DC Motors” TT Patent TT/P/2006/00070, 27 October, 2006 13 ... Fig 2 (a) The magnitude of H a , is obtained by a Amperes Circuital Law and is given by ia N a r , where l a is the length of the path of H a and N a is the number of turns of winding aa'' la r r... winding aa'' without changing their magnitudes The vector summation r dia r of i a R a and La along the magnetic axis of winding aa'', produces the supply voltage dt r vector V a along the magnetic axis... scalar supply dt r dia r voltage, then, the sum of the vector voltages i a R a and La results in the vector supply dt sum of the scalar voltages i a R a and La voltage, which is of the same magnitude