Armature reaction and armature leakage in salient-pole type synchronous machines

Armature Leakage in Salient-pole .^ss

Type Synchronous Machines

PROEFSCHRIFT

T E R V E R K R I J G I N G VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL T E DELFT, OP GE-ZAG VAN D E RECTOR MAGNIFICUS, D R O. BOT-TEMA, HOOGLERAAR IN DE AFDELING DER ALGEMENE WETENSCHAPPEN, VOOR EEN COM-MISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 7 OCTOBER 1953, DES NAMIDDAGS

T E T W E E UUR

HENRICUS CORNELIS TOHANNES DE JONG

GEBOREN TE ROTTERDAM

dit proefschrift, zijn verricht in de fabriek van de N.V. H E E M A F te Hengelo (O.). Voor deze waardevolle hulp betuig ik gaarne mijn dank.

pour la pratique.

(The true object of electrical machinery theory is to furnish a clear general understanding and grasp of the phenomena, and from this to provide cal-culating methods sufficiently accurate for practical use).

page

CHAPTER I. Introduction 1

CHAPTER I I . Steady-state Balanced Operation 17

CHAPTER I I I . The Transient Reactance 64 CHAPTER IV. Measuring the Armature Reaction and Armature

Pole-axis Leakage 73

Bibliography 103 List of Symbols 106

INTRODUCTION

page

1.1 Scope of the Investigation 2 1.2 Steady-state Balanced Operation 4 1.3 Three-phase Sudden Short-circuit 14

1.4 Conclusions 16

1.11 The type of synchronous machine dealt with in this paper

comprises an induced part or armature carrying a symmetrical three-phase winding, and an inducing part provided with a direct current excited winding. The armature winding is distributed over the armature slots. The D.C. winding is concentrated in a number of coils, each of which encloses one of the salient poles.

This is the common design for synchronous machines with four or more poles and with frequencies ranging from a few to some hundreds of cycles per second. Usually, the stationary part or stator is provided with the induced poly-phase windings, the rotor carrying the poles. In order to distinguish the machine under consideration from the synchronous machines with cylindrical rotor, it is classified as salient-pole type synchronous machine.

1.12 The saliency effect is the cause of many difficulties in the electro-magnetic calculations. In machines such as the asynchro-nous induction motor and the synchroasynchro-nous generator with cylindrical rotor having windings distributed over the stationary and rotary parts, and an air gap of constant width between rotor and stator, it is possible to accurately determine the distribution in space and the periodic variations of the magnetic fluxes and electric currents and other related electrical and mechanical quantities under the various operational conditions of these machines. For salient-pole type synchronous machines such calculations can only be made by starting from approximations which are bound to strongly influence the results and render these inaccurate in spite of com-plicated methods of calculation. This is due to the considerable variation in the air gap clearance along the periphery of stator and rotor, and to the fact that the rotor windings are concentrated in but a few pole coils.

This is the reason why with this type of machines, more than with the other A. C. machines mentioned, one has to rely on simplified calculations which can only yield useful results under certain limited conditions and for a restricted purpose. Though in the course of the years, these calculations have been improved,

the generally accepted methods still involve a number of approxi-mations and omissions to an extent not tolerated for other A.C. machines. Consequently, calculations have not only to he checked but

also to be completed in a large measure by experiment.

It is the object of this investigation to indicate some of these shortcomings and to bridge the existing gaps by providing data for a more accurate calculation of the factors which determine the operation of the synchronous machine.

1.13 In order to gain an insight into the properties of an elec-trical machine, its behaviour is analyzed under certain conditions corresponding to such operational conditions or sequence of con-ditions as will be encountered in actual performance. For syn-chronous machines the principal investigations pertain to:

(1) The steady-state balanced operation with which the alternator (in itself assumed to be symmetrical *)) is charged with a constant and symmetrical load, while in the case of a synchronous motor it will be charged with a constant load and connected to a sym-metrical mains voltage.

(2) The effect of asymmetry in the load impedances in the case of an alternator and in the mains-voltages in the case of a motor. An excellent aid in solving the relating problems consists in resolv-ing the asymmetrical systems into symmetrical components.

(3) The conditions of steady-state short-circuit, in which one, two or three phases are short-circuited. These conditions are special cases of those mentioned sub (1) and (2).

(4) Deviations from the synchronous speed of rotation of the rotor, accompanied by pulsations in mains and other currents and produced by irregularities in the torque of the driving or driven tool. (5) The phenomena produced by a sudden change in load, particularly by a sudden short-circuit of the synchronous machine whether or not under load.

*) A synchronous machine with 2p poles will be considered symmetrical when:

(1) the magnetic circuit can be divided into 2p mutually equal sections, each symmetrical with respect to its axis.

(2) the armature windings of a phase consist of p equal groups uniformly distributed over the armature circumference.

(3) the windings of the various phases are made up of the same number of turns and have the same distribution.

(4) the spacing between windings of each pair of consecutive phases is uniform.

For a detailed description of the phenomena mentioned sub (2) to (5), American authors have introduced a great number of impe-dances, reactances and time constants in order to make allowance for the effect of the amortisseur and eddy-currents in the iron, apart from that of the salient poles. Beyond doubt, these per-fections in the calculation of synchronous machines constitute a valuable improvement both from the designer's and the user's point of view. Yet, this trend is not without risks, for many of these newly introduced quantities are defined "externally": a certain voltage system is applied and the resulting current measured, or conversely: currents are generated in the circuit and the resulting voltages measured. Though attention has also been given to the calculation of these impedances etc. from the electro-magnetic data, it must be stated t h a t the calculation technique has not kept pace with actual developments. Consequently, the new quantities remain nothing but a number of separate units without any mutual relationship.

Whereas with other types of machines the endeavour has always been to use as few parameters as possible to explain their behaviour, with salient-pole type synchronous machines there seems to be a tendency to introduce an ever increasing number of parameters.

The focussing of attention cJn other problems has, moreover, led to a neglect of the theory of the steady-state balanced operation mentioned sub (1). The general practice is to simplify the subject by employing some short-cuts t h a t are partly based on experience. In this paper, on the contrary, the steady-state balanced condition will receive major attention since it is felt t h a t its theory and practice need improvement and completion.

We wiU also consider the transient part of the short-circuit phenomena indicated sub (5). For the plain type of machine we are to examine — one without an amortisseur and with laminated poles through which no eddy currents can flow — we shall find a remarkable relationship between the two conditions of steady-state balanced operation and sudden short-circuiting. An armature-reactance component which hitherto has received too little attention will appear to play an important role in the two so divergent conditions. Thus our investigation results in an "integration" t h a t may more or less counterbalance the "differentiation" referred to above.

1.2 Steady-state Balanced Operation

diagram showing voltage and current vectors and their mutual

relation for a definite steady load condition plays an important part. The aspect of this diagram is directly related to the method adopted in reducing the complexity of phenomena for any machine or transformer to an intelligible system of fundamental elements, while neglecting secondary characteristics.

Thus the diagram of the salient-pole type synchronous machine, like any other diagram, is based on a number of conventions which, though having proved their practical usefulness during more than half a century, should yet be considered critically in an investigation such as we propose to make. We will therefore start with accurately recording the customary conceptions to get a basis for the investi-gation to be made in the next chapter.

1.22 When the currents in the various phases of the armature are considered in their field generating and voltage inducing effect as one single system (which is possible in the case of the symmetrical conditions assumed here) we can distinguish two current carrying systems in the synchronous machine under load; namely the stator system and the rotor system. For either system only the time-fundamentals of the voltages generated in the armature winding will be considered. Only then it is possible to draw a diagram, because vectors represent quantities changing sinusoidally and with equal frequencies.

The magnetic phenomena produced by the armature currents, which are assumed to constitute a symmetrical three-phase system, are divided into a "main"field and a "leakage"field *). The first mentioned component is quantitatively the more important one. This division is commonly applied to all electrical machines, trans-formers and a great deal of other electrical equipment. It has proved to be very useful but difficult of exact definition. The synchronous machine, far from being an exception in this respect, is in fact one of the machines giving rise to the greatest confusion.

In the synchronous machine the main field due to the armature currents is commonly termed "armature reaction" and is defined as a magnetic flux rotating with the poles and linked with the pole windings. The armature leakage field is said to consist of the mag-netic fluxes that are not linked with the pole windings and

there-*) In this paper the term "field" will only be used when referring to the integral phenomenon, i.e. when not specifically dealing with magnetic induction (B), magnetomotive force (M) or m.m.f. gradient (H).

fore have their paths either entirely inside or in the vicinity of the armature.

It is usually assumed t h a t the magnetic induction of the arma-ture leakage field is proportional to the current, and the voltage inducing effect in the armature winding is expressed as a reactance: the armature leakage reactance. More difficulties are encountered in calculating the armature reaction flux, because its value is largely dependent upon the relative position of the armature reaction and the poles, i.e. upon the phase angle between the armature current and the armature e.m.f. due to the pole windings.

1.23 The resolving of the armature reaction into a pole-axis and a quadrature-axis component — ascribed to BLONDEL i) ^) —• meant a great improvement both for the understanding and the calculation of the phenomena. The maximum of the pole-axis component coincides with the pole axis, whereas the quadrature-axis component has its maximum just between two adjacent poles. This provides an easy means of accounting for the difference in the possibility of flux generation depending on the position of the armature reaction in relation to the poles.

The pole-axis m.m.f. of the armature reaction is compared, as far as the flux-producing and voltage-inducing effect of the armature winding is concerned, to the pole m.m.f., because both affect the magnetic main circuit. In view of the saturation which generally occurs in this circuit, the voltage-inducing effect of the armature pole-axis field is usually expressed as a pole m.m.f. which would induce the same voltage.

In a similar way the voltage-inducing effect of the quadrature-axis field may be expressed as an equivalent pole m.m.f. However, the quadrature-axis field takes quite another path, so that such a comparison lacks the physical basis of the conversion of the pole-axis field into an equivalent rotor m.m.f. For this reason, the voltage-inducing effect of the quadrature-axis field is often ex-pressed as a reactance: the quadrature-axis main reactance; because the quadrature-axis fluxes chiefly pass through the air, a proportionate relation is assumed to exist between current and flux. This assumption does not always correspond to the facts, for the saturation of pole shoe and pole core may result in a mutual influencing of the pole-axis and quadrature-axis fields^). However, this phenomenon is not within the scope of our investigation and for simplicity's sake such a mutual influencing of pole-axis and quadrature-axis fields will here be neglected.

1.24 The single-phase diagram relating to the steady-state symmetrical load condition of the salient-pole type synchronous machine as based on the above considerations, is shown in fig. 1 for an alternator generating a lagging current. We will now briefly explain this diagram.

Fig. 1. Voltage vector diagram of salient-pole type synchronous machine.

Starting from a given terminal voltage per phase [U) and a given value and phase angle of the armature current (/, cos (p), we con-secutively draw the vectors: U, the resistance voltage drop in a stator phase {IR) (which in the diagram has been drawn ex-cessively large but is generally of little importance and may here be neglected), and the stator leakage reactance voltage drop (/Xj).

The process of resolving the armature reaction in a pole-axis and a quadrature-axis component may be imagined to be effected by dividing up the current into two components displaced 90 degrees relative to one another, which excite the two field compo-nents. Supposing f to be the "internal" phase angle between pole e.m.f. and armature current, the pole-axis component will have the value / sin rp and the quadrature component / cos f.

The quadrature-axis main voltage vector is at right angles to the pole e.m.f. vector; the magnitude of this voltage E^ can be calculated from the quadrature-axis main reactance X^:

£ , = / X , c o s v . (1.1) The voltage-inducing effect of the pole-axis field is expressed

as an equivalent pole m.m.f. Designating the amplitude of the space-fundamental of the air gap m.m.f. due to the armature reaction as M^, the amplitude of its pole-axis component will be M„ sin ip and the equivalent pole m.m.f. ^j,M„ sin y), in which

K<

equi-valent pole m.m.f. is adopted, the latter may be written as

k^M„ cos ip in which k^ < 1 and in general k^ ^ k^.

The voltage diagram shown in fig. 1 may be utilized to determine the excitation current at a given condition of load. The excitation of the rotor should be such that, after subtraction of the pole-axis component of the armature reaction, the remaining pole-axis m.m.f. generates a flux which induces a voltage £,• in the armature winding, £^ being found by adding the vectors U, IR, IX^ and £ „ . 1.25 In order to determine the armature reaction, the factors

k^ and k^ have to be known, or instead of k^ the quadrature-axis

main reactance X^. The now almost universally accepted definition of the armature reaction, which forms the basis of methods aiming at the determination of these quantities, reads *^) '*) ^) ') ^):

The armature reaction comprises the space-fundamental component of that air gap flux which is produced by the space-fundamental component of the air gap m.m.f. due to the armature currents and rotating in synchrononism with the poles.

An alternative definition extends the division to the coil-end fields *) ^'') with the object of also differentiating a synchronous component of the armature coil-end fields and including this in the armature reaction. This extension should be considered as a theoretical perfection. It is of no importance for our investigation and of little value for practical calculations. We shall therefore confine ourselves to the first-mentioned definition and the cal-culation of armature reaction and armature leakage based on it.

Formulae for the analytical determination of the armature reaction have been given by SCHOUTEN i^), who considers the air gap m.m.fs. of armature and pole to be equivalent when they excite a space-fundamental flux of the same value and thereby induce equal time-fundamental voltages in the armature windings. Expressing this in other terms: the pole air gap m.m.f. equivalent with the armature air gap m.m.f. has a value such that with an armature without current this pole air gap m.m.f. induces the same time-fundamental voltage as the pole-axis (or quadrature-axis) component of the space-fundamental air gap m.m.f. excited by the armature current alone.

Following Schouten's example, most authors suppose the synchronous machine to be unsaturated in the concrete working out of the general formulae of SCHOUTEN *") i^) i3«) w») i ^ all formulae the normal component of the air gap induction at the

armature surface due to the rotor current or armature currents, is calculated as the product of the m.m.f. and the air gap permeance. Formally, this is correct, if the relation between the m.m.f. distri-bution and the permeance is accounted for, and thus to each m.m.f. space-function is assigned its own permeance space-function ^5). In practice, however, an average permeance distribution is assumed, which is the same for all m.m.fs., and may be determined from a simplified flux map.

This approximation is unnecessary when using the graphical

method, in which the flux distribution in the air gap and between

the poles is determined, neglecting the magnetic reluctance of the iron. Plotting a field map when current passes through the rotor only, and another one when there is only the pole-axis component of the armature current, the factor kj, can be found b y comparing the two space-fundamentals of the normal components of the air gap induction at the armature surface ^^). k^ is determined in a similar manner.

1.26 Numerous formulae have been developed for the cal-culation of the armature leakage reactance, as has been done for other machines, especially for the asynchronous induction motor. Our remarks will be confined to some essential differences in the calculating methods.

Properly speaking, the slot field is the only field which is unani-mously considered to contribute in its entirety to the armature leakage reactance, and for the calculation of which practically the same formulae are used. This is not the case with the coil-end field. As noted, some authors uphold the differentiation of a synchronous component which should have to be included in the armature reaction. As a rule, however, the entire armature coil-end field is considered to belong to the leakage field. The complicated paths of the fluxes render an exact calculation impossible, and thus various attempts have been made to arrive at workable formulae through simplified assumptions.

The greatest differences are found, however, in the calculation of the air gap leakage, which term implies all leakage fields whose paths run partly through the air gap. The obstacles that prevent exact calculation are related to those encountered in the asyn-chronous machine, though in our case the situation is different because of the different construction of the rotor. The armature leakage in the salient-pole type synchronous machine is usually

considered to include either a tooth-tip leakage *'') s') i^*) or the space-harmonic (and time-fundamental voltages inducing) fluxes due to the armature current ®) ''), or these are introduced together into the calculation ^").

A useful aid in eliminating the uncertainties in the treatment of the air gap leakage is given in recent literature by HUMBURG ^'). We will refer to this again later on.

1.27 Although, as appears from the aforegoing, the definitions and calculating methods show some divergencies in the manner of distinguishing between armature reaction and armature leakage and in the calculation of these two components, yet there is essen-tially a large measure of agreement between the various views. The conversion of the armature reaction into a field m.m.f. as given by SCHOUTEN has been maintained practically unchanged.

A totally different viewpoint is expounded by TOLLENAAR i^) in

a paper in which he attacks the fundamental principles on which all the usual conceptions are based. Since this publication constitutes

the starting point of our paper as far as it is concerned with the steady load condition, we will now more closely consider Tolle-naar's theory.

TOLLENAAR criticizes the treatment of the space-fundamental pole-axis component of the air gap m.m.f. due to the armature currents. Schouten's and aU related methods have in common that the entire flux due to this pole-axis component, as far as it induces a time-fundamental voltage, is included in the armature reaction and expressed as an equivalent pole m.m.f.

TOLLENAAR, on the other hand, asserts that from this part of the armature reaction another (small) portion should be differen-tiated and included in the armature leakage. For the armature pole-axis field this means more armature leakage and less armature reaction, or in other words: a decrease of kj, and an increase of X^. The author justifies this shifting with the thesis that only that part of the air gap m.m.f. due to the armature current can be expressed as an equivalent pole m.m.f. — and thus neutralized by this pole m.m.f. — which causes the same m.m.f. gradient distribution on the armature surface as is produced by the pole m.m.f. This thesis is worked out for the special case of a synchro-nous machine with constant air gap under the poles and negligible permeance between poles, and it appears that the result of his calculations can be interpreted as indicated above.

useful method of dealing with the problem of armature reaction and armature leakage of the salient-pole type synchronous machine shows two deficiencies which call for further investigation, viz.: (1) The thesis on which the author bases his divergent method of

differentiation is not free from arbitrariness. So we believe that the method of differentiation proposed by TOLLENAAR is correct in principle, but that it should be placed on another theoretical basis in order to assure it of a firmer position in the further unimpaired theory.

(2) The method should be worked out in order to provide data with which the calculation of armature reaction and armature leakage reactance may be made for any synchronous machine with whatever shape of poles.

The required additions and improvements we propose to provide in chapter I I , which deals with the steady-state balanced operation of the synchronous machine.

1.28 Remains to be mentioned one more difficulty which also confuses the problem of armature reaction and armature leakage in the salient-pole type synchronous machine, namely the question whether two values should be assigned to the armature leakage reactance: a "pole-axis" and a "quadrature-axis" leakage reactance.

The difference between these two leakage reactances is supposed to be due to the contribution of the air gap field to the armature leakage, i.e. the tooth-tip or space-harmonic fields (or both) which are presumed to have different values under the pole and between the poles. Some authors do in fact make this distinction.

DOHERTY and NICKLE *) include in the armature leakage all

space-harmonic air gap fluxes inducing time-fundamental volt-ages. They show that these fluxes have different values for the pole-axis and the quadrature-axis components of the armature currents.

For the tooth-tip leakage also, it makes a difference whether or not it can take its path through the pole shoe. I t has been observed, however, t h a t with solid pole shoes eddy currents may practically cancel this difference i*').

KARAPETOFF 1®) adds to the confusion by including the entire quadrature-axis field in t h e armature leakage.

We will have to consider this point too, in order to bring more clarity into the whole problem.

experi-mental determination of armature reaction and armature leakage, after we have first stated what are in our opinion the right prin-ciples. However, we shall make an exception for the Potier method, on account of the important role it plays in all discussions on this subject 2").

Following the Potier method it is assumed that the zero power-factor saturation curve, i.e. the relation between armature voltage and rotor excitation with a constant armature current and zero power factor lagging, can be derived from the no-load saturation curve by means of a triangle as shown in fig. 2. The vertical side

Fig. 2. Potier relation between no-load and zero p.f. saturation curve.

of this "Potier" triangle represents the armature leakage voltage drop IX^ corresponding to the constant armature current, while the horizontal side represents the equivalent pole m.m.f. which here has the value kj,M^ because f is 90 degrees *).

The Potier triangle can be determined when the no-load and the zero power-factor saturation curves have been plotted, provided the measurements of these curves are extended into the saturated part.

The fact that the Potier method plays such an important part in the literature on salient-pole type synchronous machines may be explained, first, by its close relationship with the orthodox method of deahng with this type of machines, and, secondly, by a property of most salient-pole type synchronous machines which endangers the practical usefulness of the Potier method. This property is the saturation of the rotor iron, due to which the

*) POTIER assigned to the sides of the triangle the very general notations

Oil and XI and showed how a and A could be determined from the no-load

and inductive-load characteristics. For the actual calculation of a and A POTIER did not have sufficient means at his disposal.

difference in pole leakage flux of the machine at no-load and with an inductive load disturbs the simple relation between both charac-teristics, and this particularly in the curved part where the Potier triangle has to be found.

The Potier method has several variants which are partly based on the same simplified assumptions as the original method ^i) and partly put forward in order to more or less counterbalance its drawbacks ^) ^^). It is typical for the situation and also for the American way of treating the synchronous machine with salient poles, t h a t American authors distinguish a separate reactance: the Potier reactance, whose value is deduced from the Potier triangle This Potier reactance is used to determine the excitation at any given condition of load and especially at rated full load. The A.I.E.E. Test CODE ^*) describes a method of determining the full load

excitation from the short-circuit characteristic, the no-load charac-teristic and a load measurement with zero power factor lagging. This A.I.E.E. method has a variant in which the salient-pole type synchronous machine is treated as a machine with cylindrical rotor whose armature leakage reactance equals the Potier reactance and whose armature reaction corresponds to that resulting from the horizontal side of the Potier triangle. Such methods have no theoretical foundation. They are purely empirical and therefore cannot enter into our discussion. They deserve mention here, however, because they provide another typical example of the way in which salient-pole type synchronous machines are usually dealt with.

It should still be observed that the use of the reactance derived from the Potier triangle as " t h e " leakage reactance implies the assumption that "pole-axis" and "quadrature-axis" reactance are equal. For the load characteristic is measured with zero power factor, a condition in which the armature air gap field is a pure pole-axis field. Thus only the "pole-axis" leakage reactance is actually measured and it is taken for granted that the "quadrature-axis" component has the same value, the ultimate result being termed " t h e " armature leakage reactance. Measurement, however,

does not provide evidence in support of this assumption.

*) Cf. the following English commentary ^^): "Potier reactance is not a fundamental synchronous machine quantity and there is no place for it in theoretical work".

The fact of greatest importance for our investigations concern-ing the application of the Potier method and all other methods based on the same differentiation of the armature flux producing effect, is this:

I The leakage reactance thus determined is considerably larger than

the sum of the slot-, coil-end- and air-gap leakage reactances which can be calculated or measured with sufficient accuracy.

This observation was one of the motives which led the author to undertake this study with the object of providing a closer agree-ment between test results and calculation.

1.3 Three-phase Sudden Short-circuit

1.31 It is generally known that sudden short-circuiting of an

alternator supposed to be running at no-load, sets up strong currents in stator and rotor, showing a typical course. As far as the stator currents are concerned, in each of the phases this current may be divided up into a low-frequency component (usually referred to as D.C. component), and a component whose frequency differs but little from mains frequency (A.C. component).

The latter component shows a high initial value which decreases rapidly at first, then slower, and eventually attains a constant value which is termed steady-state short-circuit current. Subtract-ing this constant value from the entire A.C. component 3delds an alternative current decreasing to zero, which apart from the initial cycles, disappears practically according to the exponential law.

If we suppose this exponential curve to be extended until the instant of short-circuit, and thus neglect the superimposed but rapidly disappearing current during the first cycles, then the ratio of the no-load terminal voltage before short-circuit to the initial value of the alternating current may be expressed as a reactance, commonly known as "transient reactance". The neglected "sub-transient" phenomenon during the initial periods is due to rapidly disappearing currents in the electrical rotor circuits outside t h e excitation winding, as well as to saturation of the iron parts through which the leakage fields can close their circuit.

We shall confine our discussions to an alternator with laminated rotor without an amortisseur and without conducting coil frames so t h a t rotor stray currents are excluded, and we will assume that the short-circuit currents are so small that no saturation of the leakage fields can occur. Thus we shall only have to deal with the "transient" phenomenon.

1.32 A large number of armature current oscillograms of synchronous machines at sudden short-circuit have been investi-gated by RüDENBERG 2^). He directed his attention in particular to the maximum instantaneous value of the current, which occurs at an instant half a cycle after the instant of short-circuit in that phase, which has zero voltage while short-circuiting. This maximum value / „ may be expressed in the r.m.s. value of the phase voltage

E before short-circuit, an effective short-circuit leakage reactance X^,^ and a damping coefficient x, see equation (1.2):

/ „ = 2 V 2 ~ ^ ^ . (1.2)

RüDENBERG observed that for machines with salient laminated poles without a damper winding, equation (1.2) yielded a value of X which for all machines examined showed comparatively little variation from an average of 0.9 if for X^^ was filled in the stator reactance with removed rotor, i.e. the stator reactance including the bore leakage reactance. (The bore leakage field is the field set up in the stator bore when the rotor has been removed 2')).

The value of the short-circuit current 7 „ in equation (1.2) is affected both by the saturation of the leakage fields and by the subtransient phenomena. This results in X,^ being smaller than the transient reactance. The results obtained by RÜDENBERG indicate that at a sudden short-circuit, apart from slot, tooth-tip and coil-end leakage of the armature, there must be still another leakage component considerably greater than the also effective rotor leakage.

This is confirmed by a direct determination of the transient reactance from short-circuit oscillograms. To this end the instan-taneous values of the enveloping curve of the A. C. component of the short-circuit current less the steady-state value are plotted on a logarithmic scale as a function of time plotted on a linear scale. Except for the initial cycles, a straight line is obtained which is extended up to the instant of short-circuit, where it indicates the value of the transient current from which follows the transient reactance.

The value of the transient reactance appears to be considerably greater than can be calculated from the combined effect of the arma-ture slot-, tooth-tip- and coil-end leakage, and the pole leakage.

To find an explanation of this phenomenon, it will be necessary to analyse the short-circuit phenomenon, which analysis will have to result in the description of a leakage field for which an extra

t component will have to be added to the leakage reactance. We , have found but one author, who gives a formula for the calculation of such a leakage component, i.e. KILGORE i"). In chapter I I I we shall deal with the calculation of the transient reactance and also discuss Kilgore's formula.

1.4 Conclusions

From the foregoing survey it is apparent that the present theory

of the salient-pole type synchronous machine fails to explain:

(a) the additional leakage reactance component contained in the Potier reactance,

(b) the additional leakage reactance component which is effective at sudden short-circuit.

By "additional" reactance component is meant a leakage reactance which should be added to the "ordinary" armature leakage (slot, tooth-tip and coil-end) reactances, and sub (b) also the pole leakage reactance.

Since the theory offers no explanation, it obviously cannot provide the formulae required for the calculation. With the Potier reactance the calculation has been disregarded altogether. This reactance is defined empirically, t h a t is, from a few measurements and graphical treatment of the measured results. In the case of sudden short-circuit an extraneous reactance is resorted to, namely t h a t of the bore leakage field with removed rotor.

In the following pages we shall endeavour to find the missing explanations on which we then can base our formulae. In doing so, we shall make a remarkable discovery, namely, that for

salient-pole type synchronous machines with laminated salient-poles and without an amortisseur, one reactance can be described with which the existing gaps in theory and calculation can be filled for both the steady-state balanced operation and the sudden short-circuit behaviour.

STEADY-STATE BALANCED OPERATION

page

2.1 Introduction 18 2.2 On the Principles of Comparing the Flux-producing Effects

of Armature Currents with Those of Field Current . . 18

2.3 The Simplified Synchronous Machine 23 2.4 Differentiation of Armature Pole-axis Field and Rotor Field.

The Electrical Analogue 24

2.5 A Test Model 29 2.6 Formulae for the Differentiation of Armature Pole-axis

Field and Rotor Field. •— Armature Reaction and

Arma-ture Leakage Reactance 34 2.7 The Improved Voltage Diagram 46

2.8 Development of the Formulae for Armature Pole-axis

Field and Rotor Field 50 2.9 Calculations Referring to the Actual Synchronous Machine 52

Since in this chapter it is our object to improve one of the aspects of the classical theory of the salient-pole type synchronous machine, namely t h a t dealing with steady-state balanced operation, it is desirable to briefly state beforehand which conventions in this theory we shall adopt unchanged. These conventions are:

(1) The electromagnetic flux-generating and voltage-inducing effect of the armature currents is made up of three components:

armature leakage,

armature reaction in the quadrature-axis, armature reaction in the pole-axis.

(2) As far as the armature leakage and the armature quadrature-axis reaction are concerned, there is proportionality between the armature current and the voltage induced in the armature winding. The voltage-inducing effect of both components is therefore expressed in terms of a reactance.

(3) The flux of the armature reaction pole-axis component is affected by the saturation in the main circuit, and its flux-producing effect is expressed as an equivalent field m.m.f. / (4) Harmonics of currents, fluxes and voltages will not be

con-sidered.

(5) The pole-axis and quadrature-axis components of the armature reaction do not influence each other in their flux producing effect.

Our departure from the classical theory will consist in a modified method of determining leakage and reaction of the armature pole-axis component.

2.2 On the Principles of Comparing the Flux-producing Effects of

Armature Currents with Those of Field Current

2.21 When we want to divide up the flux-producing effect of the armature currents into armature leakage (whose voltage-inducing effect is expressed as a reactance) and armature reaction (whose voltage-inducing effect is converted into an equivalent rotor m.m.f.), the question arises with respect to the latter

com-ponent: how can the armature m.m.f. be expressed as a field m.m.f.? In more general terms, how can the flux-producing effect of a given system of windings be expressed in that of an other system which has an entirely different distribution in space and, moreover, a speed relative to the first system?

This problem is related to that of the magnetic leakage between two winding systems, which was so widely discussed in Europe around 1930 ^^) ^^). From these discussions we shall adopt the useful conception according to which the comparison of two winding systems has its roots in the dividing up of the flux set up by each of the winding systems in iron-core transformers into a main flux taking its path through the iron, and a leakage flux chiefly taking its path through the air ^'^). When considering the first flux compo-nent, both windings may be compared to one another, since for both systems it takes the same path.

The relative speed of both winding systems in the synchronous machine involves a complication which can be eliminated as follows. At least in so far as the main flux component in the common path;' is concerned, a fictitious substitute system is introduced for one of the two winding systems, generating the same common flux but being stationary with respect to the other winding system.

2.22 In order that the flux-producing effects of two different winding systems may be compared to one another, it is therefore necessary to subdivide the fluxes of each winding into two or more components. This subdivision is based on the principle of super-position according to which in a space with a permeability indepen-dent of the m.m.f. gradient, the magnetic field resulting from the simultaneous presence in one and the same space of two different magnetic fields is found by vectorial composition of the m.m.f. gradients or the magnetic inductions in any point of t h a t space. Conversely, any magnetic field may be divided up into two or more components, on condition that the combination of all com-ponents yields the original field.

This principle of superposition has proved extremely useful in electrical engineering. It is applied when two or more winding systems are carrying current at the same time and together generate the magnetic field, but also in the dividing up of the magnetic field produced by one winding, into a "main" field and a "leakage" field. However, particularly in the latter application, and especially in electrical machines, failure to provide an accurate definition of each component has led to confusion. Leakage components like

tooth-tip, zig-zag, doubly-linked and space-harmonic leakage, all of which occur in the air gap of electrical machines, have been introduced, though it is far from being clear whether, and to what extent, these components have anything in common.

An excellent method of preventing this confusion has been j given by H U M B U R G ^ ' ) . He introduced fictitious winding systems which make it possible to differentiate step by step different field components. The slot leakage field, for instance, is found as follows:

I n addition to the actual winding are supposed two winding systems having the same distribution in space, namely, a uniform distribution in front of and near the slot opening. Each winding system carries a current such t h a t the total m.m.f. per slot takes the same value as that of the actual winding; but whereas the current in one of the systems flows in the same direction as in the actual winding, in the other system it flows in the opposite direction.

The total flux-producing effect of the three winding systems, i.e. one actual and two fictitious systems, obviously equals that of the actual winding alone. Now the slot leakage field is supposed to be due to the combination of the actual winding with the fictitious winding carrying the opposite current. These two winding systems are shown in fig. 3a in which the actual winding i s assumed to be

! 1 1

1

i

1

Fig. 3. Fictitious winding systems for differentiating:

a. slot leakage field; h. tooth-tip leakage field.

uniformly distributed over the slot area (wire and slot insulation being neglected). The remaining fictitious system, in which the current has the same direction as in the actual winding, is again combined with another fictitious set of windings, and so on, u n t u in the end a fictitious winding is left which produces exclusively the main field.

Owing to the intricate shape of the magnetic circuits in electrical machines, the calculation of the contribution to the reactance by any field component is not always an easy matter, but may lead to matherriatical complications. The great advantage of this method is, however, t h a t by introducing the fictitious winding systems every field component is clearly defined, thus excluding all confusion.

2.23 We now apply the Humburg method to our salient-pole type synchronous machine and consecutively differentiate the field components as described below.

First of all we differentiate the slot leakage field in the way described above, see fig. 3a. Next, we apply two equal fictitious winding systems through which opposite currents pass, the wires of a slot being uniformly distributed along the armature surface covering one tooth pitch. The combination of the winding in the slot opening carrying a positive current and the winding distributed over the tooth pitch carrying a negative current (see fig. 3b) generates a kind of tooth-tip leakage.

Of the now remaining winding, the wires of one phase are uni-formly distributed over one phase zone. Subtracting the m.m.f. harmonics generated by this winding, leaves the space-funda-mental m.m.f. due to a sinusoidally distributed phase winding. The resultant of the three space-fundamental m.m.fs. of the three phases shows also a sinusoidal distribution and moves at constant speed along the armature periphery. This enables us to introduce a stator winding system rotating with the rotor. We therefore assume that this m.m.f. is generated by a sinusoidally distributed winding rotating at synchronous speed and then dif-ferentiate this into two, likewise sinusoidally distributed compo-nents whose m.m.f. maximum falls in the pole-axis and the quadra-ture-axis respectively. In the following pages we shall have to deal chiefly with the pole-axis component.

For the pole windings or field coils which are assumed to be uniformly distributed in t h e interpolar space (see fig. 4a), the combination with a fictitious winding system lying in the plane of the inner face of the pole shoes and uniformly distributed between these shoes, produces the pole core leakage flux. The fictitious pair of windings shown in fig. 4b generates the leakage flux between the pole shoes.

a logical system, so that there is no arbitrary element of significance. Actually, the slot leakage of the stator winding and the pole and pole shoe leakages of the rotor winding are found from the evident displacements of the windings in question towards the air gap. Tooth-tip m.m.f. and space-harmonic m.m.fs. should be eliminated from the armature m.m.f. in order to obtain a field rotating in synchronism with the rotor winding.

-, I I S » '""MUMSMMimmSJillSi""" " " • • — ' I

Fig. 4. Fictitious winding systems for differentiating:

a. pole core leakage field; b. pole shoe leakage field; c. pole face leakage field; d. Sinusoidally distributed armature pole-axis winding system and

However, the following considerations are bound to be more or less arbitrary. For practical reasons, we prefer to transfer the rotor winding from the irregular pole surface to the smooth arma-ture surface, as shown in fig. 4c. The thus differentiated pole face leakage flux takes mainly a tangential path. This component wiU only have some significance when the air gap increases towards the pole tips; in this case, however, its flux path closes chiefly through the rotor and may be considered as an extension of the pole shoe leakage. In fig. 4d the pole winding thus obtained is shown again, together with the sinusoidally distributed pole-axis com-ponent of the armature winding.

We may assume the calculation of the leakage field components as differentiated above to be generally known, so for this we do not intend to provide any formulae. We are now going to consider the two windings shown in fig. 4d.

2.3 The Simplified Synchronous Machine

2.31 For the determination and differentiation of the magnetic fluxes generated by the winding systems of fig. 4d, we may use graphical methods as developed by LEHMANN and others or we may utilize analytical methods. In this paper we shall favour the analytical method because it can provide us with general formulae; however, in order to evaluate these formulae we shall also make use of the result of graphical methods.

The disadvantage of the analytical method is that an exact calculation of the magnetic fluxes in the salient-pole type synchro-nous machine is practically impossible, even when iron saturation is neglected. Though field theory and Maxwell's equations provide a formal solution to the problem, it appears in practice that only in very simple cases — much simpler than the one under con-sideration — formulae for evaluating the principal quantities can be developed from this formal solution ^^) ^^) ^^).

In order that we may yet apply the analytical method without abandoning exactness and obtain in a simple manner intelligible results (in which the departure from the orthodox methods of cal-culation will be evident), we shall make use of a simplified syn-chronous machine. I t then remains to be demonstrated that the results thus obtained are valuable for the calculations in the actual salient-pole t j ^ e synchronous machine.

2.32 Our simplified synchronous machine shall have an air gap of constant width. In this air gap there will be no magnetic per- I

meability in tangential direction, so that the magnetic lines of force are straight lines at right angles to the stator and rotor surfaces. The winding systems of fig. 4d, i.e. the sinusoidally distributed stator winding and the semi-uniformly distributed rotor winding, are placed along the smooth armature periphery. Though the pole winding is assumed to cover the space between poles as exists in the actual synchronous machine, in the simplified alternator the pole shoe width is assumed to be equal to the full pole pitch and the distance between two adjacent pole shoes is assumed to be negligible as far as its influence on the air gap flux is concerned. Since there is no permeability in tangential direction, the radial distribution of stator and rotor windings is irrelevant and we may assume the winding systems to be distributed along the stator surface or along the rotor surface or to be uniformly distributed along the air gap.

The air gap is filled with a medium whose permeability varies locally, but only in tangential direction, whereas it is constant in radial direction. The effect of this variable permeability will be accounted for by introducing an effective air.gap 6 which is a func-tion of the electrical angle 6. We assume that ó (0) is symmetrical with respect to the pole centre and is the same under all poles. It may be noted that for the simplified synchronous machine, the calculation of the normal component of the air gap induction from the product of air gap m.m.f. and air gap permeance is per-fectly correct. The results obtained with the simplified machine will be of practical value for the actual synchronous machine with salient poles, if the proper permeance functions are chosen (if necessary, different ones for the stator and the rotor windings).

We shall further assume t h a t the rotor diameter is large in relation to the pole pitch, so t h a t the effect of the curvature of stator and rotor peripheries on the air gap field may be neglected. Furthermore we shall make our calculations per 1 cm length of the machine, neglecting the effect of the coil-end connections. 2.4 Differentiation of Armature Pole-axis Field and Rotor Field.

The Electrical Analogue

2.41 To get an insight into the properties of a fairly compHcated magnetic circuit, it is recommendable to make use of the electrical analogue of this circuit because:

(a) on account of its more frequent use, the electric circuit is much more familiar to us than the magnetic circuit;

(b) the electric circuit allows more accurate calculations, due to the virtually perfect behaviour of materials used, as good conductors or good insulators. On the other hand, in almost any magnetic circuit there are leakage fluxes which cause a complicated flux distribution.

In order to make the proposed differentiation of armature pole-axis field and pole field quite clear and to arrive at the necessary formulae, we will use as an analogue the electrical equivalent of the magnetic circuit in our simplified alternator. This electrical analogue is shown in fig. 5. We are now going to explain this dia-gram and derive from it some properties and formulae, which we shall afterwards "translate" into the magnetic circuit of the alternator.

2.42 The electrical analogue consists of n e.m.fs. of different but constant values, E^, E^. . . Ej . . . E„ (generated, for instance, by galvanic elements), which together with their series-connected resistances R^, R^. • . Rj • • • Rn (conductances G^, Gg. . . G,-. . . G„) are parallel-connected between two busbars of zero resistance. These busbars are, moreover, interconnected by an external resistance R (conductance G).

I TJinjii 1

1 . 7R

' arUTJTj 1

Fig. 5. Electrical analogue.

For this electric circuit we can write the following equations: (a) First of all we assume R= co,G = 0. Supposing the unknown voltage between busbars to be E„, then the currents in the n parallel branches are:

I, = [E,~-E„)G,

/ , = ( £ , - £ „ ) G , (2.1)

The summation should be extended from ƒ = 1 to ƒ = n. Since 21^ = 1 = 0 (2.2) it follows _E{E,G,)

^'--EGT- ^-^

(b) Next, we assume R = 1/G to be finite. Supposing the busbar voltage to be E, then

A = ( £ , - £ ) Gi

I2. = (-^2 — E) Gg

(2.4) / , = (£, - E) G,

i„={E„-E)G„

I ^ EIj = E(Efi,) — EEG,

Now

/ = £G from which follows

P ^(^A)

(2.4a) (2.5)

^2.6^ G + EG, ' ' This formula can also be derived directly from (2.3) by adding one more parallel connected branch and then reducing its e.m.f. to zero. Substituting (2.6) in (2.5) yields:

/z

(2.7)

(2.8)

'"G + EG,^'^''''

With (2.3) this results in:

I =

From (2.4a) follows also:

^ = EG, GEG, G + EG, " EG, " • I EG, (2.9)

The equations (2.3), (2.8) and (2.9) can be interpreted as follows: Every e.m.f. E, can be differentiated according to:

All e.m.fs. E„ act in parallel upon the outer circuit (R). The residual e.m.fs. e, generate equalizing currents via the busbars, which do not pass through the external resistance R.

Accordingly, every current /,• should be divided up as follows:

I, = {AI), + i, (2.11)

where {AI), contributes to the external current and i, is the equaliz-ing current which is independent of R. To demonstrate this independence, we derive from (2.9):

I={E,-E)EG, (2.12)

from which follows:

{AI),= {E„^E)G,. (2.13)

The second component:

i, = I, - (AI), = {E, - E) G, - (£„ - E) G, = ( £ , - E„) G, (2.14)

is independent of R, as stated before.

2.43 We shall now translate the above relations established for the electric circuit of fig. 5 into terms relating to the magnetic circuit of our simplified alternator. A portion of the magnetic circuit covering half a pole-pitch is shown in fig. 6. The correspondence between the electric circuit of fig. 5 and the magnetic circuit of fig. 6 can be seen as follows:

1 4 1

i / '

1

Fig. 6. Magnetic circuit of salient-pole type synchronous machine.

The parallel connected branches of fig. 5 correspond to the infinite number of infinitely small flux tubes perpendicular to the armature and rotor periphery shown in fig. 6. The width of any

R

of these tubes will be designated hy — dO (R = air gap radius, p =

P

number of pole pairs), the effective air gap by (5 (6); the permeance of a tube corresponding to the value G, in the electrical analogue

Rdd

is then: . The e.m.fs. correspond to the sinusoidal m.m.f. of the armature pole-axis field and the trapezoidal m.m.f. of the pole winding.

The external circuit is formed by the pole core and the yokes. These also need only be considered as far as they cover half a pole-pitch. As far as the yokes are concerned, it follows from the symmetry that a magnetic short-circuit between stator and rotor iron can be established in the imaginary plane between two poles, without disturbing the flux distribution. We shall assume t h a t the magnetic reluctance of the stator iron is negligible, consequently, t h a t only the rotor is saturated, and t h a t there is no leakage flux between poles. The effect of a possible saturation of the stator iron is dis-cussed in par. 2.91 sub (c).

The busbars with zero resistance are represented by the armature surface and the pole-shoe surface. If there are assumed to be slots in the stator, then the "armature busbar" is formed by the armature yoke.

We shall now proceed to translate the electrical formulae relating to' fig. 5 into the magnetic equivalent of fig. 6. Integration should take the place of summation, because the magnetic equivalent comprises an infinite number of branches of infinitely small per-meance in parallel arrangement.

The m.m.f. M{6) varying with Ö along the armature periphery may be divided up into an amount M which is independent of 0, and a residual m.m.f. m{0):

M{0) = M + m{6). (2.15)

All m.m.fs. M act in parallel on the external magnetic circuit which is closed through pole core and yokes. The residual m.m.fs.

m{d) produce equalizing fluxes which close through the armature

and pole-shoe surfaces, outside the yokes. The magnitude of M is found from the formula (2.3) for £„ in the electrical analogue by substituting:

M{d) for E,

^ ^ for G,

p m

so that: M = r RdB C

J ^^^^PW) J ""^'^

(Rdjd) r

J pöid) J

de

de

pd{e)

The residual m.m.f. is found from:

w(Ö) = M ( e ) — M . (2.17)

Hence, the m.m.f. of the armature pole-axis field and that of the rotor field may be divided up each into two components:

(a) a m.m.f. of constant value throughout the pole-pitch generating a flux which closes through the poles and yokes;

(b) a residual m.m.f. which generates an equahzing flux through the pole-shoe and armature surfaces.

The flux producing effect of the component sub (a) is influenced by the saturation of the iron. We shall therefore call this component the saturation component. The flux producing effect of the com-ponent sub (b) is almost exclusively dependent on the permeance of the air paths so that these fluxes are proportional to the current. We shall call this component the linear component.

2.5 A Test Model

2.51 The differentiation into a saturation component and a linear component described in par. 2.4 can be illustrated and

r ~ " ~ ~ »—• A . i

*N

1

tested by means of a simple test model. This consists of a magnetic circuit as shown in fig. 7a. The central leg has an air gap of a length

d, in which a coil I of M uniformly distributed turns is fitted, see

fig. 7b. This coil energizes the magnetic circuit whose legs A and C constitute parallel paths. The lower part of leg B is smaller in section and it is mainly in this part the magnetic saturation is found.

We shall compare this model with the magnetic circuit of a salient-pole type synchronous machine. The lower half of the central leg corresponds to the pole iron *). The energizing winding of the model corresponds to a fictitious winding in the machine uniformly distributed over the armature periphery and rotating along with the poles in such a way that the centre line of the winding always coincides with that of the poles. The legs A and C have no equi-valents in the synchronous machine, but they would be present if magnetic short-circuits were established between stator and rotor iron in the imaginary planes between two poles. This might be done without influencing the magnetic fluxes, because in the synchronous machine there is no magnetic potential difference in these spots between stator and rotor yokes.

Fig. 8a, M.m.f. distribution of distributed coil; 86. m.m.f. of saturation component; 8c. m.m.f. of linear component.

The m.m.f. of the winding in our model has the triangle distri-bution shown in fig. 8a and for an alternating current with an

*) Again we assume that with this type of synchronous machine the saturation is chiefly concentrated in the pole cores.

r.m.s. value of I amperes, its maximum value, which occurs in the centre, is: _

M „ „ , = 0,4 TT w / V 2 . (2.18) Although a divided winding producing a sine-shaped m.m.f.

would show a closer resemblance to the synchronous machine, we prefer our winding with triangular m.m.f., not only because it offers the advantage of an easier construction, but also because it is better suited for the test we propose to make, on account of its greater amount of leakage.

According to the method dealt with above, the m.m.f. is differ-entiated into two components. The air gap being constant, the first component corresponds to the average value, see fig. 8b:

M,.,, = 0,2 TT M / V i (2.19) The residual m.m.f. has a distribution according to fig. 8c. The first

component generates a flux which can close through the parallel paths formed by legs A and C and which is affected by the saturation in the part of leg B with reduced section. The residual m.m.f., on the other hand, produces a flux which closes through paths running close to the air gap surface and which is therefore unaffected by the saturation in leg B.

For unsaturated iron, the flux hnkages due to the first component, per centimetre length in the direction perpendicular to the plane of the drawing, are, see fig. 7a:

w =

ƒ.,„.».2j(.^)(.»f).i'V-^. ,2.0,

The flux linkages per cm due to the residual m.m.f., independent of the saturation of the iron, are:

0 id

(2.21) The total flux linkages with unsaturated iron are:

M fid

xu _ xff \ xu _ 1 ""'» in 09\

• ^iot — ^ave 'T ^res — 5 T • y^.^^j

which may also be determined directly from fig. 8a.

For our demonstration model, the uniformly distributed winding offers the advantage over the sine-shaped winding of having a

comparatively greater leakage flux. The residual flux which is independent of saturation induces a voltage equal to 1/4 of t h a t induced by the total flux in unsaturated condition. Saturation of the iron increases this ratio.

2.52 In fig. 7a is drawn another winding II, whose turns lie outside, though as close as possible to, the air gap, and are con-centrated into a compact coil. Supposing this winding has | « turns, it will excite the same flux in the iron as winding I if the current has the same value I. The concentrated winding, however, does not produce a leakage flux except for the very slight flux which closes around the winding outside the iron.

We shall now consider the characteristic lines of both windings which indicate the relation between the voltages E induced by the air gap flux and the current I in the windings; we shall call these the saturation characteristics. They may be determined b y measuring the terminal voltage as a function of the current and subtracting from the measured terminal voltage the ohmic voltage drop and the small leakage reactance drop, due to the leakage fluxes outside the air gap.

Hence, the two saturation characteristics must show the following relation: with equal currents the flux in the iron and, consequently, the degree of saturation, will be the same in both cases. The voltages induced in both windings by this flux are also the same, because for the distributed winding the effectiveness is reduced by a factor \ and therefore the effective number of windings is \n. However, the voltage in winding I is increased by the flux compo-nent due to the residual m.m.f., which flux is independent of the saturation of the iron.

Consequently, the saturation characteristic of winding I may be derived from t h a t of winding II by adding for any value of the current I an amount \ IXjj to the voltage E^j induced in winding II, Xjj being the reactance of winding II in unsaturated condition, and determinable from the linear part of the characteristic.

Curves I and I I in fig. 9 give the saturation characteristics derived from measurements made on a model according to fig. 7a. These curves refer to a distributed and a concentrated winding, respectively, whose numbers of turns are in a ratio of 2 to 1. Curve

I ' has been derived from curves I and II in the following manner: The tangents to the linear unsaturated parts of curves I and I I were drawn and the reactances derived from the slopes of these tangents were found to be 9.1 and 6.2 ohms respectively. The

33 v / / ^

A

Fig. 9. Measured (I and II) and calculated (I') saturation characteristics of the test model.

difference of 2.9 ohms between these two values was assumed to be the reactance due to the air gap leakage flux of the divided winding. Increasing the ordinates of curve I I by an amount equal to 2.9 times the corresponding amperage, yielded curve I ' which, as can be seen, follows the measured curve I fairly well. The differential reactance of 2.9 ohms does not equal the third part of 6.2 ohms, the unsaturated reactance of curve I I . This may be explained b y the presence of butt joints in the yokes A and C of the model, which joints are indicated in fig. 7a by dotted lines. These joints reduce the main flux but not the equalizing flux of the distributed winding. The ampere-turns required for the iron circuit, which on account of the relatively great length are also effective in the unsaturated condition, cause a similar effect.

The results obtained in the measurements of the model confirm the soundness of the principles on which our discussion of the salient-pole type synchronous machine will be based, namely, that of differentiating the air gap flux into a portion influenced by the saturation of the iron and a portion which is independent of this saturation.

2.6 Formulae for the Differentiation of Armature Pole-axis Field

and Rotor Field. — Armature Reaction and Armature Leakage Reactance.

2.61 For the m.m.f. of the armature pole-axis field we write:

M(Ö) = M„ cos e (2.23) where Ö represents the electrical angle, 0 = 0 being assumed to

be the pole centre, while for the m.m.f.-amplitude of the armature pole-axis field M^ may be written:

M„ = 2,4 V2 nj^ sin W *) (2.24)

n = number of effective turns per armature phase and per pole I^ = r.m.s. value of the armature phase current

!F = internal phase angle between pole e.m.f. and armature current.

When the assumed symmetry of the pole is taken into account, the saturation component of the pole-axis m.m.f. according to formula (2.16) is: 7r/2

r

de

c

de

J

d{d)

reciprocal value of the relative air gap: (5(o)

Mr '

*) This formula may be derived, for instance, from a relation given by RICHTER '*) for the "Feld-Erregerkurve" (field excitation curve) from which the m.m.f. follows by multiplication by a factor 0.4 -n.

35

We shall usually refer to the y-function as the air gap permeance function.

The m.m.f. of the linear component of the armature pole-axis field is:

w,(e) = M , ( c o s Ö - ^ J (2.29) the corresponding induction is:

M

M ö ) = ^ ( c o s 0 - ^ J (2.30) and the amplitude of the space-fundamental induction:

w/2

^-ai = - ƒ ^ (cos e - k„) COS e de. (2.3i)

o

If the effective air gap had a constant width throughout the pole pitch, equal to its value in the pole centre ó(ö) = d{o), then the space-fundamental induction due to the armature pole-axis m.m.f. would have an amplitude:

• M

Bai = — - • - (2.32)

"^ d{o) ^ '

We assume

Ki = ^a B,, (2.33)

the coefficient a^ also being equal to the ratio linear component reactance fictitious main reactance

where the linear component reactance is the armature reactance corresponding to m^{d), and the fictitious main reactance corres-ponds to the total armature pole-axis field in the unsaturated machine with a constant effective air gap of ö{o).

For the coefficient a^ we may then write: 7r/2

. a„ = - I (cos e — k^) cos e y{d) de (2.34a)

0

w/2 ff/2

2.62 The saturation-component formulae of the field coü, corresponding to (2.23), (2.24), (2.25), (2.27) and (2.28) are:

0 ^ e ^ i 0„ M {e) = M, (2.35a) i e , ^ 0 ^ ^ M(0) = M , ^ ^ ^ (2.35b) j> Mf = 0,4 :/T Hflf (2.36) Jflp 7r/2 7T — 2 9 öie 0(0) ' J - - ^ - 0 ^ ^ w/2 M = ? ^ : (2.37)

r de

J 0(0)

Jr(e)

r20 — 0

Jy(ö)

where n^ = number of turns per pole

IJ = excitation current

0 = pole width, expressed as an electrical angle. For the linear component, the space-fundamental induction has the amplitude:

iflp ir/2

^n = -] U T ^ ( l - ^ , ) c o s 0 i 0 + - ^ ^, cos0rf0 .

-i^m i V ( ^ ) ^ " - ^ ^ ^ (2.40)'