1 INTRODUCTION. It is a well-known fact, supported by many experimental measurements, that the ends of the stator core of a turbogenerator increase in ...
Int. J. Elect. Enging Educ.; Vol. 20, pp. 5-14. Manchester V.P., 1983. Printed in Great Britain
THE INFLUENCE OF POWER FACTOR ON TURBOGENERATOR CORE END LOSSES - A SIMPLE PHYSICAL EXPLANATION D. C. NEWTON and P. J. TAVNER Central Electricity Generating Board, South Eastern Region, Scientific Services Department, Gravesend, England
LIST OF SYMBOLS E, e.mJ. behind synchronous reactance Eg e.mJ. behind leakage reactance H magnetic field strength H za axial magnetic field strength due to the stator end winding alone H zg axial magnetic field strength due to the magnetisation of the air gap Hzr axial magnetic field strength due to the rotor end winding alone H zres resultant magnetic field strength adjacent to the stator core end after summation of the components H za' H zg' H zr I current l, stator current I, rotor current Ma air gap m.mJ. component due to stator (armature) windings alone Mr air gap m.mJ. component due to rotor windings alone Mg resultant air gap m.mJ. V, stator terminal volts X at stator winding leakage reactance X, synchronous reactance (direct axis) X md reactance due to mutual inductance between rotor and stator winding (direct axis) () load angle L\ increment in load angle s angle between Eg and V, arising from leakage reactance 4> power factor 1 INTRODUCTION It is a well-known fact, supported by many experimental measurements, that the ends of the stator core of a turbogenerator increase in temperature under leading power factor operation". This phenomenon can be verified by magnetic leakage field calculations but from time to time the need arises for a simple physical explanation which this paper is intended to provide. The heating of the core end region of a turbogenerator has been shown from various investigations to be due mainly to the axial component of the magnetic leakage field at the core end. The way in which the leakage field of a turbo5
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generator changes with load is obscured by the complex geometry of the stator end region, but it can be shown that the causes of leakage flux may be represented in a simplified way. It is important to have a simple physical understanding of these causes because the effects of the leakage field on the temperature rise in an alternator end region are not necessarily obvious. In particular, recent measurements have shown that temperature rises in the clamping plate or screen, towards the core back, may decrease at leading power factor". Minors? has investigated the variation of axial leakage flux in terms of the rotor winding position and load angle, J. He has shown conclusively that the square of axial flux is linearly related to the cosine of the load angle. The work does not, however, show the influence of the power factor, cos 4J, on axial flux, nor does it show the spatial variation of axial flux over the core-end for a given electrical load. Singleton et al. 4.5 have used m.mJ. phasor diagrams to show the relative contributions to the core end axial leakage field due to core magnetisation, stator currents, and rotor winding currents. Their work was directed towards showing how axial leakage flux changes during transients in the alternator load angle, which subsequently result in pole-slipping. This paper shows the contribution to the axial leakage field of physically identifiable sources of magnetic field, including the stator winding. This allows the spatial distribution of axial field on the core-end to be demonstrated and the change in this distribution, with both load angle and power factor, to be seen. 2 THE SOURCES OF CORE END LEAKAGE FIELD The geometry of a typical turbogenerator end region is shown in Fig. 1. The magnetic sources which contribute to the axial component of the leakage
STAlOR END WINDING
/
AELD POINTS
ROTOR
FIG. I
Typical turbogenerator core end region showing sources ofaxial magnetic field.
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magnetic field can be subdivided into the 'applied field' and the 'reaction field'. The 'applied field' comprises: (i) Circumferential components of current in the rotor end winding (ii) Circumferential components of current in the stator end winding (iii) Magnetic surface polarities on the surfaces of the rotor and the stator bore as a result of magnetisation of those bodies. The 'reaction field' comprises: (iv) Magnetic surface polarities on the annular stator core end surface as a result of magnetisation of the stator. (v) Eddycurrents in the structural members at the core end, such as the core clamping plate and screen. The 'applied field' is the axial leakage field present in the absence of the core end polarity, (iv), and screen current sources, (v) and is the ph asor sum of contributions (i) to (iii).The 'total axial field' actually present on the core end is then the sum of this 'applied field' and the 'reaction field' of sources (iv) and (v). Magnetic polarities on the surfaces of the rotor and stator bore, item (iii) above, arise from the axial currents in rotor and stator conductor bars. These sources are all shown in Fig. I and have been discussed extensively elsewhere" 7.8. The sources of axial leakage field within the core, such as eddy currents in the core plate and volume polarities due to saturation and the laminar core construction are not considered, as their contribution in air has been shown to be small", The relative magnitudes and phases of the sources listed in (i) to (v) above depend on the air gap flux density in the alternator and on its stator and rotor currents. These in turn depend on the alternator load and power factor, which can be determined from the air gap phasor diagram, Fig. 2(a). The currents of (i) and (ii), are explicit.and can be found directly from the phasor diagram for a given load. The sources of field due to (i) and (ii) are deduced from these circumferential currents and the number of turns in the rotor and stator end windings respectively. Sources (iii),(iv) and (v) are all implicit and, in principle, are not known until the complete magnetic field of the alternator has been calculated. But source (iii) is fixed by the air gap flux density, which will be known if the winding currents are known, because of the cylindrical geometry of an alternator. The strengths and phases of sources (i) to (iii), therefore, can be deduced from the phasor diagram for the alternator and from the design of its windings. It has been shown 7.8 that core end surface polarities (iv) and screen eddy currents (v) can be deduced to a first degree of approximation, from the 'applied field'. Since the 'applied field' and 'reaction field' are linearly related, the 'total axial field' is therefore linearly related to the 'applied field'. It follows that by plotting the phasors of sources (i)-(iii) it can be shown how the 'applied field', and therefore the 'total axial field', varies with both load angle and phase angle.
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3 THE CONTRIBUTIONS OF THE SOURCES TO THE LEAKAGE FIELD
3.1 General At any particular point in the end region, a phasor diagram can be drawn which combines the contributions to the axial leakage field due to all the sources listed in Section 2. It has been shown 7 that for an alternator on open circuit the end region field contribution due to the rotor end winding is small compared to that due to the air gap and core end polarities. On an alternator with both rotor and stator windings energised, however, their end winding contributions become significant because they are not in phase with one another or with the contribution due to the air gap. The total contribution of the windings therefore changes markedly with load. The precise contribution a source makes to the magnetic field, at a field point some distance from the source, depends on the magnitude of the source, its geometrical extent and the distance of the field point from it. Clearly for a field point on the stator core end, close to the air gap, the contributions due to that gap and the rotor winding will be important, but towards the core back those contributions become less significant. Analytical expressions can be derived" 7 for the axial component of field due to each of the sources listed in Section 2. This paper, however, is not concerned with precise fields but with the relative magnitudes and phases of the various contributions. The following Sections explain how an axial leakage field phasor diagram can be built up for a field point in the end region on the core end. They also show, by inspection of that phasor diagram, how the axial leakage field alters with changes in load. 3.2 Relationship between the axial leakage field and the air gap phasor diagram Consider the alternator shown in Fig. 1 operating at lagging power factor, cos , by which the rotor body is skewed to the main air gap flux on load. This can be seen in Fig. 3. Fig. 7 shows the axial field contribution due to the stator winding. Now the stator end winding is conical and the circumferential currents are therefore distributed over a conical surface. The sign of the stator winding phasor therefore depends on the radial position of the field point relative to the electrical centre of gravity of the circumferential current. This can be seen in Fig. 4. In Fig. 7 the position of field point X has a radius less than the electrical centre of gravity, therefore the stator winding contribution, H za, is directed into the core, as shown in the figure, and is in phase with M a. The phasor H za on Fig. 8 is therefore drawn parallel to M a, in Fig. 2(a), and in the same direction. It can be seen from Fig. 8, therefore, that for field point X the stator winding axial contribution H za' tends to aid that of the rotor winding, H zr • The resultant of the field due to the three sources, in Fig. 8, is therefore Hz res and this is the axial 'applied field', described in Section 2. It can be seen from Fig. 8 that the magnitude of this 'applied field' will be a function of the magnitudes of the contributions and of the size of the load angle, f>, and power factor, cos cjJ. The magnitudes of the individual phasors can be obtained from analytical expressions, as mentioned in Section 3.1, but a study of the phasor diagram will show how the 'applied field' changes with load.
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FIG. 7
Axial leakage field contribution due to the stator (armature) winding.
4 THE EFFECT OF POWER FACTOR AND LOAD Fig. 8 was drawn for a specific field point, X, when the turbogenerator was delivering a given power. If this power is held constant and there are small perturbations of power factor, due for example to tap-changes on the generator transformer, then the phasor H za' due to the stator winding, changes its phase angle by a small amount too. The locus of the tip of H za' for these small
lA(jG1N;
P. F.
FIG.8
(a)
p
(b)
Axial magnetic leakage field phasor diagramsfor field points X and Y shown in Figs. 1 and 3. (a) for field point X. (b) for field point Y
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p
(0)
p
FIG. 9 Axial magnetic field phasor diagrams for field points X and Y. with load angle increased by increment L\. (a) field point X, cf Fig. 8( a), (b) field point Y, cf Fig.8(b).
perturbations, will be a circle of radius H za centred on P, as shown in Fig. 8(a). As the power factor goes leading the angle ljJ goes negative, so the tip of H za moves around the locus and the resultant axial leakage field Hz res increases in magnitude. Thus in the stator tooth region, where X is located, the core-end axial flux increases with decreasing leading power factor. As the power factor goes lagging, Hz res reduces in magnitude and the core end axial flux decreases. Fig. 8(b) shows the phasor diagram for a field point Y on the core back region, as shown in Figs. 1 and 3, where the stator and rotor winding axial fields oppose one another. Now the action of moving further into leading power factor operation, from Figure 8(b), decreases the core-end axial flux. When the load angle, ii, increases by a small amount, L\, for a given phase angle, ljJ, Fig. 8 alters to Fig. 9. It can be seen that the resultant axial leakage field, Hz res' again increases for both field points X and Y. This shows that increasing the load angle, ii, naturally increases core-end axial flux and this agrees with the results of Minors", 5 CONCLUSION This paper shows how the axial leakage field variation at a particular site of the core end of a turbogenerator can be deduced for both variations of power factor and load angle. The axial field is derived by considering the sources of leakage field in the end region and using phasor diagrams. These axial field variations alter according to the site on the core end being considered. The phasor diagrams show how leakage axial flux increases with leading power factor on the core end, but decreases at the core back. They also agree with Minors? result that axial flux increases with load angle.
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ACKNOWLEDGEMENT The authors acknowledge the interest and help of Professor P. Hammond, Southampton University, in this work. This paper is published with the permission of the Director-General of the South Eastern Region of the Central Electricity Generating Board. REFERENCES [I]
[2] [3] [4] [5]
[6] [7] [8]
Estcourt, V. F., Holley, C. H., Johnson, W. R. and Light, P. H., 'Underexcited operation of large turbine generators on the Pacific Gas and Electric Cos System', AlEE Trans; 72, pt. III, pp. 16-22 (1953) Tavner, P. J. and Newton, D. C; CEGB Internal Report Minors, R. H., 'Core end heating in turboaltemators: the importance ofload angle', CEGB Internal Report Singleton, R. C. C; Renew, D. C. and Marshall, P., 'Transient axial flux in a synchronous machine', 14th Universities Power Engineering Conference, Loughborough (April, 1979) Singleton, R. C. C, Marshall, P. and Steel, J. G., 'Axial magnetic flux in synchronous machines: the effect of operating conditions', Trans IEEE PES, Winter Meeting, New York (1980) Tavner, P. J., Penman, J., Stoll, R. L. and Lorch, H. 0., 'Influence of winding design on the axial flux in laminated stator cores', Proc. lEE, 125, (10), pp. 948-950 (1978) Tavner, P. J., Hammond, P. and Penman, J., 'Contribution to the study of leakage fields at the ends of rotating electrical machines', Proc lEE, 125, (12), pp. 1339-1349 (1979) Tavner, P. J., 'The influence on the end region field in large turbogenerators of clamping plate screens and damper rings', 15th Universities Power Engineering Conference, Leicester (March 25-27th, 1980)
ABSTRACTS-ENGLISH, FRENCH, GERMAN, SPANISH The influence of power factor on turbogenerator core end losses - a simple physical explanation The object of this paper is to provide a physical explanation for the variations of stator end region magnetic leakage field, and consequent temperature rises, in a turbogenerator. In particular the paper shows how the losses and temperature rises vary with the power factor and load angle of the turbogenerator.
L'influence du facteur de puissance sur Ies pertes des frootaIes des turbogeoenteun: one explication physique simple L'objet de cet article est de fournir une explication physique des variations du champ rnagnetique de fuite des regions terminales du stator, et l'elevation consecutive de temperature, dans un turbogenerateur, En particulier, cet article montre comment les pertes et l'elevation de temperature varient avec Ie facteur de puissance et l'angle de charge du turbogenerateur, Der Einfluss des Leistungsfaktors auf Stirnraumverluste von Turbogeneratoren - eine einfache physikalische Erklirung Aufgabe dieser Arbeit ist eine physikalische Erkliirung fiir die Anderungen des Statorkopfstreufelds eines Turbogenerators, und die dadurch verursachten Erwiirmungen, zu geben. Die Arbeit zeigt im besonderen, wie Verluste und Erwiirmung mit Leistungsfaktor und Polradwinkel des Turbogenerators variieren. La influencia del factor de potencia en las perdidas en los extremos del nUcleode los turbogeneradores. Una sencilla interpretacion fJsica El objeto de este articulo es presentar una interpretacion fisica de las variaciones del flujo magnetico de dispersion en los extremos del estator, y del consiguiente aumento de temperatura, en un turbogenerador. En particular, en el articulo se muestra como las perdidas y los incrementos de temperatura varian con eI factor de potencia y eI angulo de carga del turbogenerador.
