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CHAPTER IIFU NDA MEN TA L PRIN CIPLES

1. T h e E lectromagnetic Ma ch in e. At voltages that can be developedand used by normal means, the electrostatic forces are very weak.On the other hand, even with comparatively small currents a con-siderable mechanical force can be produced by electromagneticmeans; in consequence the machines in common use as generatorsand motors are electromagnetic in type. Only in very high-voltagegenerators and transformers have the electrostatic field and itseffects to be considered as regards their secondary influences on theoperating characteristics, apart, of course, from the question ofinsulation.2. Induction and Interaction . There are two related principlesforming the foundations upon which are based all electromagneticmachines concerned in the conversion of electrical energy to or frommechanical energy. These are (a) the law of induction and (b) thelaw of interaction.(I) LAW OF INDUCTION The essentials for the production of an

electromotive force are electric and magnetic circuits, mutuallyinterlinked. The summation of the products of webers of magneticinduction with complete turns of the circuit is termed the totalflux-linkage: if it is made to change, an e.m.f. is induced in theelectric circuit. This e.m.f. persists only while the change is takingplace, and has a magnitude proportional to the rate of the changewith time. The instantaneous e.m.f. is

e =- (dN/dt) voltswith the linkage N in weber-turns. The e.m.f. has a directionsuch as to oppose the change. Thus if the electric circuit wereclosed on itself, and the number of line-linkages formed by it andsome externally-produced magnetic field were reduced, then thee.m.f. induced would produce a current in the closed circuit, generat-ing a self-magnetic field superimposed upon the external field andtending to make up the deficiency.For engineering purposes the induction law is generally used inthe simplified form-

e =- To(dfJ>/dt) volts . . (I)Here To is the number of turns in the electric circuit, all of whichare linked completely with all the webers of induction of a givenflux. For this purpose a gross flux may be resolved into (i) a mutualor working component, and (ii) a leakage component.The electromagnetic method of producing an e.m.f. in a circuit

6

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FUNDAMENTAL PRINOIPLES 7(in order that the e.m.f, shall produce a current and thus enableelectrical energy to be delivered) is therefore to provide a magneticfield linked with an electric circuit, and to change the number ofline-linkages N=T/P. Considering for simplicity that the electriccircuit comprises a coil of Tc turns, then the change of line-linkages.may be accomplished in a variety of ways-. (i) Supposing the flux constant in value, the coil may movethrough the flux (relative motion of flux and coil) ;(ii) Supposing the coil stationary with reference to the flux, theflux may vary in magnitude (flux pulsation) ;(ill) Both changes may occur together; i.e. the coil may move

through a varying flux.In case (i) above the flux-cutting rule can be applied. The e.m.f.in a single conductor of length l m. can be calculated from the

Field due to current iI 1 \ \ ' \ \ \ \ 1I I \ , \ \ \ \ I II ror7'e \ \ ')\ \ \ 1 II flfJ~/JJIIII1III1I1I1

1 f a t + f II f f + t(a) (lI) Resulfanf ofComponent fj'eMs. fields in (aJ.FIG. 3. ELEOTROMAGNETIC INTERACTION

(C) Erred of current'reversal

rate at which it cuts across a magnetic field of density B webers perm.2 when moving at um. per sec. at right-angles to the direction ofthe flux- .e=Blu volts (2)This is referred to as the e.m.J. of rotation. It is always associatedwith the conversion of energy between the mechanical and electricalforms. The e.m.f. in a coil in case (ii) is found directly from eq. (1)as the e.m.f. of pulsation or transformation. No motion is involvedand there is no energy conversion. For case (ill) both e.m.f.'s areproduced: this case is treated generally in Chapter XXVI.(2) LAw OF INTERACTION.When a conductor of length l m.,

carrying a current iamperes, lies in and perpendicular to the direc-tion of a magnetic field of density B webers per m.", a mechanicalforce is developed on it of magnitude /

f =Bli newtons . (3)in a direction perpendicular to both current and field. In thediagram (a) of Fig. 3, B represents the density of an originalmagnetic field. The introduction of a conductor carrying a current

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8 DESIGN OF ALTERNATING CURRENT MACHINESbrings at the same time a new magnetic field. The original, field _andthe conductor field combine to form a resultant field (Fig. 3 (b)).The field density actually existing round the conductor is not B.It is greater than B on one side and less on the other. c The distortionis an essential feature in the production of mechanical force.GENERATORSANDMOTORS.In a generator, an e.m.f. is producedby the movement of a coil in a magnetic field. The current producedby the e.m.f. interacts with the field to produce a mechanical forceopposing the movement, and against which the essential movementhas to be maintained. The electrical power ei is produced thereforefrom the mechanical power supplied.In a motor, we may suppose a conductor or coil to lie in a magneticfield. If current is supplied to the coil. a mechanical force is mani-fested and due to this force the coil will move. Immediately thatrelative movement takes place between coil and field, however,an e.m.f. is induced, in opposition to the current. To maintain thecurrent and the associated motor action, it is therefore necessaryto apply to the coil, from an external source, a voltage sufficient toovercome the induced e.m.f. Thus the motor requires electricalpower to produce a corresponding amount of mechanical power. . .The directions of flux, current and movement in generator andmotor action are given in Fig. 4. The coil is free to move about the

axis O. The componentfields are shown, thedirection of the mech-anical force, and thedirections of rotationfor motor and genera-tor action. The direc-l' tion of the e.m.f. issuch as to maintain thecurrent in a generatorand to oppose it in amotor. The action isreversible: i.e. the same

arrangement may act either as generator or motor.TRANSFORMERS.Forces are developed in the transformer but arenot allowed to produce movement. Consequently there is no concernwith mechanical power, and only transformer e.m.f.'s are generated.3. Classification. The principles of 2 are applied as in Fig. 5.(a) Rotary Machines. Two magnetic elements Fig. 5 (a), onefixed (stator) and the other (rotor) capable of relative rotation, areseparated by a narrow annular gap. The stator is usually the outerelement for mechanical convenience. Each element carries one ormore windings, and a mutual magnetic flux crosses the gap to linkthem. Rotor rotation results in e.m.f. induction and in electro-mechanical power conversion through interaction torques.

ie Gent : ' - - - - -o to r eFIG. 4. ELEMENTARY ELECTROMAGNETIC

MACHINE

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FUNDAMENTAL PRINCIPLES 9(b) Transformers. The induction-regulator transformer, in whichthe relative position of the elements has to be adjustable, is identicalwith (a). For the normal static transformer the gap is not necessary,so the flux is established in a closed magnetic circuit, Fig. 5 (b ).In spite of constructional variety, the electrical differences

Stator

Rotor

D 1 1 ~,(a.) R.otatin,g machine (h) Static transformer, FIG. 5. ELEMENTS OF ELEOTROMAGNETIC MAOHINESbetween the several types of machine are only secondary, and resultfrom-

(i) the kind of power system, d.c. or a.c., on which the machineis to work; and(ii) the kind of connections made between the windings andthe power system: i.e. tapped phase windings or commutatorswith brushes.The following combinations are considered in detail in ChapterXXVI.

Flux

CLAsSIFICATION OF ELEOTROMAGNETIC MAOHINES

Type-nameoils I Connections IA Constant Moving Commutator D.C. machineB Constant Moving Tappings Synchronous machineC Pulsating Fixed Commutator TransverterD Pulsating Fixed Tappings Transformer, regulatorE Pulsating Moving Commutator A.C. commutator machineF Pulsating Moving Tappings Induction machine

Case I

This book is devoted to cases B, D and F: none has a commutator.Static transformers are treated first, then a general discussion ofrotating machines in Chapter VIII to precede a more detailed ex-amination of cases F and B.

4 . Three-p hase Comp lex or D iagram . The working flux < l> in an a.cmachine results from a current Irepresenting the combined m.m.f.'sof all windings linking the magnetic circuit. The e.m.f. E generated.in a winding in which < l> varies sinusoidally lags 9 0 on < l> and I.The complexor relationship is the same as that of I, < l> and Eo; inFig.2A.

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Ea

lO DESIGN OF ALTERNATING CURRENT MACHINESFor three-phase machines under steady, balanced conditions thecomplexor diagram is drawn with the voltage and current of phase A,but with the flux < l > m common to all phases. Consider phase A on noload when the axis of < l > m passes the phase-centre: this is theinstant of zero linkages and of maximum induced e.m.f. E a . It isreasonable to' draw < 1 > 0 1 1 along the pole-axis; and to preserve thetime-quadrature relation, E a must be drawn at right-angles (lagging)to < l > m ' as shown in Fig. 6. Thus for this instant E a lies in the

FIG. 6. CONVENTION FOR THREE-PHASE MACHINEm.m.f. axis of phase A. Peak current I' if in phase with peake.m.f. Ea, must then also be drawn in the axis of phase A. As shownby the dots and crosses, this will be the instantaneous m.m.f. axis of_the whole armature. The conductors of phase A produce interactiontorque with < l > m corresponding to the power EaIa.Thus the flux is coincident with the pole axis on no load, or with _the axis of resultant m.m.f., on load. E a lags 90 on < l > m ' and thecurrent Ia has the direction of the axis of phase A.SYMMETRICALCOMPONENTS.The e.m.f.'s generated by themachine in Fig. 6 have the positive phase-sequence ABC. Withunbalanced loading, the asymmetric voltages and currents may beresolved into symmetrical components with positive, negative andzero phase-sequence. The behaviour of machines under conditionsof unbalance can then be discussed in terms of these symmetricalcomponents.