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The process of self excitation ininduction generatorsJ.M. Elder, B.E., J.T. Boys, M.E., Ph .D., and Prof. J.L. Woodward, M .A. Sc , B.E., Fellow IPEN Z

Indexing terms: Synchronous generators, Asynchronous generators, Induction motorsAbstract: The phenomenon of self excitation in induction machines is examined and a physical interpretationof how self excitation occurs is presented. The important parameters are shown to be the shaft speed, theresidual magnetism, the reduced permeability at low magnetisation, and the size of the capacitor connectedto the machine. The results are important in applications where induction machines are used as stand-alonegenerators for a power system, as the reliability of self excitation in such cases must be very high.

List of principal symbolsa = factor determining variation in amplitude withtime of asynchronous currents, 1/sC = terminal capacitance, FIm = magnetising curren t, Air,is = instantaneous rotor, stator current, AIr, Is = rotor, stator complex current, Ak = a constant proportional to the strength of theremanent magnetism in the rotorLm = magnetising inductance, HLr,Ls = roto r, stator leakage inductance, Hm = generalised angular velocityP = rotor electrical speed, rad/s, purr, rs rotor , s ta tor resis tance , "2Vo c = open-c i r cu i t vo l t age p roduced a t r a t ed f r equency ,Vco = asynchronous electrical stator frequency, rad/sco0 = rated frequency, rad/scor = synchronous electrical frequency proportional torotor speed, rad/s

= circuit reactances, Q,

1 IntroductionThe phenomenon of self excitation in induction machinesis nowadays well known. In most practical circumstances,however, such self excitation is undesirable as it can causesevere overvoltages [ 1 , 2 ] thereb y stressing the insulation ofthe machine, or it can cause torque and machine speedfluctuations [3 , 4] which detract from the performance ofthe machine and may cause significant overheating.With increased use of power-factor correction equipment,methods for determining the precise operating limits, beforeself exci tation occu rs, have been developed and refined [36]In these situations, self excitation occurs in an already excitedmachine operating initially under nominally correct cond itions.On the other hand, the conditions under which self excitationcan occur in an unexcited machine, spun up to speed fromrest with only the residual magnetism of the rotor to induceany initial currents, are not well understood but are ofincreasing importance with today's trends in power-generatingequipment.De Mello [7] discusses the concept of induction generatorsas the principal generation source in a power system and listsa number of advantages. The machines are of intrinsicallysimpler constru ction, they are more robu st, and offer potentialPaper 2292 B (Pi), first received 7th April and in revised form 7thOctober 1982The authors are at the Department of Electrical Engineering, Uni-versity of Auckland, Auckland, New Zealand

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improvements in power-system dynamic performance. Inadd ition , the fault current level drops to zero as the ex itationcollapses with the terminal voltage.Casel and Knitterscheid t [8] discuss the use of inductiongenerators as high reliability generators for critical areas suchas fire-fighting equipm ent.In many of these applications, an induction generator mustoperate in a stand-alone application, and consequently it isimperative that self excitation both occurs and can bemaintained. This paper discusses the mechanism by whichself excitation occurs and discusses how the terminal voltagecan, under favourable conditions, start from the relativelyinsignificant voltage casued by remanent magnetism in therotor and build up to the full rated voltage of the machine.The physics of the process is developed by considering themachine to behave at first as a synchronous machine with aweak permanent-magnet rotor and then as an asynchronousmachine as the terminal voltage rises to some useful value.For the theoretical development, the two possible statesof the machine are represented by the simple equivalentcircuits shown in Fig. 1. The machine starts as a permanent-magnet alternator (Fig. la) and switches to an asynchronousmachine (Fig. Ib), the transition being dependent on themachine parameters, the rotor speed and the size of theterminal capacitors. The circuit parameters are all assumedto be constant except for the magnetising inductance whichvaries in the way shown in Fig. 2 (this curve was obtainedexperimentally as outlined in Appendix 7, on the machinedescribed in Section 3). The drop in inductance at highmagnetising currents is well known and determines the finalsteady-state voltage. The importance of the low inductanceat low currents is less well recognised, but it is this feature

Fig. 1 Equivalent circuitsa Synchronous mode of self-excited induction generatorb Asynchronous mode0143-7038/83/020103 + 06 $01.50/0 103

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that largely determines the initial self-excitation charac-teristics of the machine.This paper presents a theoretical analysis of the way inwhich self excitation occurs as a change in the state of themachine. The theory is complemented with several experi-mental results showing possible failure modes for suchexcitation. Finally, a start-up sequence under which selfexcitation is most likely to occur is presented and evaluated.

O.Ar

0.2

1.0 2.0Fig. 2 Variation in magnetising inductance with magnetising current

2 Theoretical considerationsFor the tw o simple equivalent circu its shown in Fig. 1, weconsider the relationships between the terminal voltage,current and rotor speed.2.1 Synchronous modeThe equivalent circuit for the synchronous mode, Figure la ,has the form of a resonant circuit in which the forcingfunction is the remanent magnetism in the rotor, inducingsynchronous currents in the stator.The amount of remanent magnetism present in the rotoris measured in terms of the open-circuit (capacitors discon-nected) voltage at rated frequency. To determine the responseto the forcing function, it is incorporated into the equivalentcircuit as a voltage source, the magnitude and frequency ofwhich are dependent on the r otor speed, (as shown in Fig. 3).This model of machine behaviour may be compared withthe equivalent con stant excitation machine discussed by Mainer[9] in his theoretical treatment of permanent-magnet alter-nators. However, factors which rule out a direct applicationof Mainer's methods in this case include the ill-defineddistribution of remanent flux in the rotor, the relativelyhigh values of recoil permeability which apply to the softiron, and departure from linear operation. The practical

j r s i n u >r t

Fig. 3 Synchronous-mode equivalent circuit incorporating remanentmagnetism as frequency-dependent voltage source

permanent-magnet alternator has marked saliency and mustbe analysed by the 2-axis method. Round-rotor analysis isused here, because although different levels of saturationalong, and perpendicular to, the axis of remanent magnetismdo create magnetic saliency , the effect is only significantat rotor speeds well below that which results in a synchronoustransition.Possible analysis of the machine as a hysteresis generatorhas also been considered. However, such treatments of thehysteresis machine as those by T eare [10] and Copelandand Slemon [11] have been concerned principally withmotor operation, and have considered the use of hard magneticmaterials with hysteresis loops of regular shape incorporatedin rotors of simple geometry. That the peak airgap flux densityis not explicitly known in the case of the self-excited generatorimposes a further major difficulty.From Fig. 3, the magnetising curre nt Im can be determined:

where co r is the synchronous electrical frequency proportionalto the rotor speed, k = VOC/CJ0 (a constant proportional tothe remanent magnetic flux density in the rotor) and Voc isthe open-circuit voltage produced at rated system frequencyco 0.

p,puFig. 4 Possible operating points of induction generator in synch-ronous mode for three levels or remanent m agnetismIn this circuit, the normal response of a resonant circuitto a forcing function is modified by a nonlinear magnetisinginductance and the variation in magnitude of the forcingfunction with rotor speed. Solving eq n. 1, using the valuesfor the magnetising inductance and current from Fig. 2 andthe machine parameters from Appendix 7, a set of possiblesynchronous operating points can be obtaind. Plotting

these points produces curves of the form shown in Fig. 4,for values of remanent magnetism varying form an open-circuit voltage of 0.5 V to 0.1 V at 50 Hz. The curves risegradually until increasing current causes a rapid rise inmagnetising inductance (see Fig. 2), and an associated suddendecrease in synchronous resonant frequency which causesthe curves to bend back upon themselves. If the machinespeed is increased montonically from zero, the response willfollow a curve of the form shown until the knee is reached(point X for instance), at which point, increasing rotor speedmust cause the operating point to jump past synchronousresonance (to point Y). It must be noted that these curves,along with those for the asynchronous mode, can be shiftedwith respect to machine speed by simply varying the amountof capacitance connected to the machine. The range 1.0 to1.4 per unit was convenient for the equipment used to drivethe induction machine. This speed range is simply reducedby increasing the capacitance.10 4 IEEPROC , Vol. 130, Pt. B, No. 2, MARCH 1983

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2.2 The asynchronous modeUsing the circuit of Fig. \b , and following the method ofWagner [3], the current supplied by the machine to the capa-citor (the stator current), is is given b yis = Re(Isemt) (2 )

where Re means the real part, / s is the com plex stator currentand m = u+jojSimilarly, for the rotor circuit, the rotor current ir isi r = Re(Ire{m~JP)t) (3 )

where Ir is the complex rotor current and p is the rotorelectrical speed.The use of Kirchhoffs loop equations with standardphasor notation yieldsls(rs + m(L s + Lm) + 1/mC) + IrmLm = 0 (4)

and-jp)Lm = 0 (5)r(r r + (m -jp) (Lr + Lm)) +

Solving eqns . 4 and 5 for Ir gives[{m 2 C(L S + Lm) + mCrs + l}{-rr-(m-jp)(Lr + Lm)}m (m-jp)CLl]I r = 0 (6 )

Assuming/,. =0, and simplifying eqn . 6 in terms of m, yieldsm3C[(Lr + Lm)(Ls+Lm)-L?n]+ m2C[(Ls+Lm)r r

m [Cr8rr + (L r + L m) -jpC(Lr + Lm)r s]{r r-jp(Lr + L m)} = 0 (7)

At this point, the equation may be normalised by convertingto per-unit values. Then, if p is expressed as a fraction ofcoo, we have from eqn. 7:-Xl] + ( m / c o 0 ) 2

[(X s + Xm)rr + (X r + Xm)r s -jp{(X r+ Xm)(X s + X m) -Xl} + (m/coo) [r srr + Xc(Xr + Xm)- jp{Xr + Xm)r8] + X c [r r -jp(Xr + X m)} = 0

(8 )The roots of this cubic equation define the conditions underwhich asynchronous machine operation is possible. Solving

Q02

Lm=

=0.24

2.0

-0.02

Fig. 5 Plot of a against machine speed for two values ofmagnetisnginductance, Lm = 0.4 Hand 0.24H

for m = a + jto always produces two roots that have anegative and are of no further interest here. However, thethird root has an a that can be negative, positive or zero,and it is this root which determines whether the rotorcurrent, the stator current and voltage, decrease, increase,or are sustained. Fig. 5 shows a plot of a against machinespeed for two values of Lm. A resonant speed occurs forwhich a = 0. For asynchronous currents to grow, and there-fore for self excitation to occur, the resonant speed mustbe exceeded. The speed at which resonance occurs can bereduced by increasing the terminal capacitance.1.5

1.0

05

CC=-O.15 cr=o.o cr=o.2

1.0P,pu 2.0Fig. 6 Contours of a repre senting exponential growth and decayof asynchronous currents in the induction machine having parametersgiven in Appendix 7

From the values of magnetising inductance shown inFig. 2, conto urs of mach ine current and speed for differentvalues of oc can be obtained, as shown in Fig. 6. The steady-state operation of the asynchronous machine must be at apoint of resonance; i.e. a = 0. As can be seen, at a particularspeed there are two operating points which satisfy thiscriterion, A and B. Point A represents the point at which themachine is fully excited. Point B, however, represents anunstable condition, in that any change in speed will cause themachine to drop out of resonance, an increase in speed tendingto cause self excitation with resonance being regained at pointA, whereas a decrease will cause the machine to drop back tothe synchronous mode. It is point B which is of interest, andFig. 7 shows the curves expanded about this area.

a=0005cr=0.0025 / or=0.0075150

,100

50

=0.01

cf=-O.OO!

1.0 1.2 1.4P.PUFig. 7 Contours of a for area of machine operation critical toinitiation of self excitation. Positive a correspon ding to growth ofasynchronous currents leading to excitaton

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2.3 Interaction between modesThe interaction between the synchronous and asynchronousmodes can be seen by combining the contour a = 0, forasynchronous operation, with the curves for synchronousresponse as shown in Fig. 4. For a given value of remanentmagnetism, a machine started from rest will have a capacitorcurrent that follows the appropriate synchronous curve (A)up to a knee point (X). If the speed is further increased,the cu rrent m ust jum p discontinuou sly t o point Y, which iswithin the area where asyn chronous operation is possible.Under these conditions, an asynchronous component willgrow rapidly and the terminal voltage will increase to theappropriate value. The fixed rotor poles due to remanenceare destroyed and the synchronous component of currentdecays to zero. If the synchronous operating point is closeto point X, only a small disturbance is required to initiateself excitation.

Fig. 8 Vector diagramsa Corresponding to point X in Fig. 4 for circuit shown in Fig. 3b Corresponding to point Y

The effect of this transition on the synchronous machinecan be seen from the vector diagrams in Fig. 8 (for circuit,see Fig. 3). Below resonance, the load angle 6 is such thatThe MMF Fx produced by the current flowing, reinforcesthe remanent MMF F2 to produce the resultant MMF Fo.While above resonance, the current acts to reduce theremanent magnetism. The change in phase of the currentbetween the two operating points is of the order of 150degrees. For th e resonant-frequ ency line to be crossed andself excitation to occur, the synchronous machine mustsustain this large change in phase effectively a large load-angle swing.3 Experim ental observationsUnder most operating conditions, the remanent magnetismwill persist for a sufficient time to guarantee self excitationwhen the change in state occurs the rate of change of rotormagnetism will be limited by the highly conductive rotor.If, however, the transition is attempted at very slow speeds,it is considered likely that problems may occur.To investigate the consequences of slow load-angle swings,a number of experiments were performed. These experimentsinvolved accelerating the induction machine from rest atdifferent ramp rates. Fig. 9 shows typica l system responsesfor a 3-phase, 4-pole, 2.25 kW induction machine for rotor-speed ramp rates ranging from 1.0 rad/s2 down to 0.12 rad/s2 .The responses are plotted as machine current againstrotor speed with the contour a = 0 shown (see Appendix 7for comment on the position of this contour). Curve a, at thehighest ramp rate, shows a typical response where the initialsynch ronou s build up leads to a jum p through synchronousresonance causing the m achine to self excite.106

At a slightly lower ramp rate (curve b) , self excitation isalmost achieved but fails as the synchronous machine losessome remanent magnetism. Self excitation then occursat a higher speed. At a still lower ramp rate (curve c) thereis considerable interference between synchronous andasynchronous components, resulting in substantial currentfluctuations until self excitation occurs. At extremely slowrates (curve d) , the remanent magnetism is steadily reduceduntil finally self excitation occurs. This demagnetisationwas verified by halting and reversing the ramp in machinespeed. The machine's response did not retrace curve d bu tdropped onto a curve corresponsing to a lower remanentmagnetism.

200 200

\

700

c 1.0i

\ \\Vp1>\

P.PU

's

1.0

200

1.0 1.0

20 0

1 0 PFig. 9 System responsesCurves a to d show response to reducing ramp rates in machine speed.Curve e is the response to a very low level of remanent magnetismRamp rates, rad/ s2:(a) 1.0, (6 )0. 82 , (c) 0.44, (d) 0.1 2

There appear to be two mechanisms by which the remanentmagnetism of the machine can be altered. At very slow ramprates, the demagnetising armature reaction caused by the loadangle is sufficient to keep the machine below t he area ofpossible asynchronous operation up to a speed of approxi-mately 1.24 pu in this case. At higher ramp rates, asynchronouscurrents do appear but may 'beat' or interfere with thesynchronous currents. Such interference may cause theoperating point to vacillate in and out of the area of possibleasynchronous operation (curve c) . Yet again, the asynchronouscomponent may degauss (either partially or completely)the rotor of the machine as it drives it through many cyclesyet fails to self excite if the collapse of the synchronouscomponent is faster than the growth rate of the asynchronousone (curve b) .

In practice, of course, all these effects are present at thesame time and it is difficult to distinguish betwee n them ina real case. At high ramp rates, no interference effect isobserved, as the rate of growth of the asynchronou s compo nentis too high.The machine response for an almost completely de-magnetised ro tor is shown in Fig. 9 curve e, with self excitation

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occurring at a speed of approximately 1.44 pu. Even with acompletely demagnetised rotor, self excitation will occurif the speed exceeds that resonant speed determined bythe minimum value of magnetising inductance and the valueof terminal capacitance. Demagnetisation of the rotor wasachieved by using a critical ramp rate which produced themaximum reduction in remanent magnetism after the machinefailed to excite. The load angle effect alone cannot producethis level of demagnetisation as, at extremely low ramp rates,self excitation occurs at speeds below 1.24 pu.4 Practical implica tionsA major problem in starting stand-alone induction generatorsis that of guaranteeing self excitation when a machine havingan unknown amount of remanent magnetism in the rotoris accelerated from rest. An understanding of the mechanismof self excitation, as presented in this paper, permits recom-mendations to be made for improving the reliability of start-up.

Q3

1 0 . 2

0.1

100 200Im,m AFig. 10 Plot of magnetising inductance showing error bars

30 0

A reduction in the remanent magnetism can occur atreduced acceleration rates, as previously shown, if the terminalcapacitor remains connected while the machine is run up.Similarly, although it is beyond the scope of this paper toexamine this subject in greater detail, any load connected tothe machine demonstrably reduces the ability of the machineto excite. Thus, to improve the reliability of start-up, itis recommended that, in all situations, the capacitor remaindisconnected until the final machine speed is attained, andfurthermore that any major load remain disconnected untilthe m achine is fully excited (th ere will usually be some loadimposed by the control devices connected to the generatoroutput). In addition, the reliability of starting can be madevery high by one of the four following methods:(a ) by passing a DC current through the machine beforeit is run up to speed, sufficient remanent magnetism m aybe guaranteed(b)by switching in charged terminal capacitors. If thecapacitors are charged to a high voltage, say rated machinevoltage, the discharge current is normally sufficient to causeself excitation even with a degaussed rotor(c ) by increasing the machine speed above the rated value,causing the resonant speed at low magnetisation to beexceeded, and thereby initiating self excitation (note that

the machine's rotor and bearings must be rated for the higherspeed)(d)by adding sufficient terminal capacitance to reducethe resonant speed to below the rated machine speed.In the last two methods, care must be taken to avoid largeovervoltages when the machine excites. In large machines,these two methods may not be practical owing to the largereduction in magnetising inductance at very low currents.Typically, the ratio between the inductance in the essentiallylinear operating range and the inductance for very smallcurrents is of the order of 5 :1 for the iron alone. The presenceof the airgap reduces the effective ratio. Thus a small machinemay have a ratio of 2 : 1 , whereas large machines will approach5 : 1 . With high ratios, very large capacitance or very highmachine speeds would be required to ensure self excitation byeither of the last two method s.Experimental work in this area is sometimes made ratherconfusing in our experience by initially unexpected behaviour.One example is that the small transient resulting from dis-connecting and reconnecting the capacitance to a nonexcitedmachine is often seen to be sufficient to start self excitation.Also small slip-ring induction machines may not excite owingto the discrete voltage drop at the brush contacts .5 ConclusionsThe feasibility of using an induction machine as a stand-alone generator is enhanced by the treatment of self excitationpresented in this paper. The actual method of starting mayvary from one installation to another. Using the methodsoutlined above, however, the reliability of starting can be veryhigh and advantage can be taken of the ruggedness, simplicityand low cost of the induction machine.6 References

1 WAGNER, C.F.: 'Self excitation of induction moto rs', AIEE Trans.,1939, 58, pp . 4 7 - 5 12 SMITH, I.R., and SRIHARAN, S.: T rans ients in induction ma-chines with terminal capacitors', Proc IEEE, 1968, 115, pp. 5195273 WAGNER, C .F.: 'Self e xcitation of inductio n m otors w ith seriescapacitors', AIEE Trans., 1941, 60, pp. 12 41-12474 LIMEBEER, D.J.N., and HA RLEY, R .G.: 'Subsy nchrono us res-onance of single-cage induction motors', IEEProc. B, Electr. PowerAppi, 1981, 128, (1) pp. 33 -4 25 BARKLE, J.E., and FERGUSON, R.W.: 'Indu ction generatortheory and application', AIEE Trans., 1954, 73, Pt III A, pp. 12-196 KITAEV, A.V., and ORLOV, I.N.: 'On the physical mechanism ofasynchronous self excitation and self oscillation', Electr. Technol.USSR, 1979, (2), pp. 19 -2 87 DE MELLO, F.P., and HANNETT, L.N.: 'Large scale inductiongenerators for power systems', IEEE Trans., 1981 PAS-100, pp.2610-26188 CASEL, J., and KNITTERSCHEIDT, H .: 'Indu ction generatorsfor generating and emergency power supplies of the future', Electro-tech. Z. ETZ, 1981, 102, pp. 139-1419 MAINER, O.E.: Theoretical treatment of permanent magnet alter-nators', Int. J. Electr. Educ, 1968, 6, pp. 41 7-42 510 TEARE, B.R.: T heory of hysteresis-motor torqu e', AIEE Trans.,1940, 59, pp. 907-91 211 COPELAND, M.A., and SLEMON, G.R.: 'An analysis of the hys-teresis motor YJEEE Trans., 1963, PAS-82, pp. 34-42

7 AppendixThe machine used in the experimental investigations wasa 3-ph ase, 4-pole, 2.25 kW induc tion mach ine. The machineparameters were rr = 2.1 S7, rs = 2.91 1, Lr = 13.5mH,Z,s =1 3 . 5 m H a n d C = 2 5 M F .These parameters were assumed constant. To measure themagnetising inductance Lm, the machine was driven atsynchronous speed and the applied terminal voltage varied.

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However, at very low machine currents, even with the useof precise electronic wattmeters, the measurement of mag-netising inductan ce was extrem ely difficult.To obtain more accurate values for Lm, the remanentmagnetism was utilised to provide the voltage required tomeasure Lm at very low machine currents. The machine wasrepresented by the circuit shown in Fig. 3 with the terminalcapacitor disconnected. From this circuit, the magnetisinginductance was calculated by measuring the open-circuit

voltage and the short-circuit current, each test being repeatedseveral times to ensure that a stable level of remanentmagnetism had been achieved. However, owing to the limitedmagnitude of the remanent magnetism, this method couldonly produce results for machine cu rrents below 30 mA.Fig. 10 is the plot of Lm against machine current, with theinclusion of error bars to give an indication of the uncertaintyin the values of Lm and therefore in the exact position of thea contou rs, at low machine currents.

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