Analysis of helical armature windingswith particular reference to

superconducting a.c. generatorsA.F. Anderson, B.Sc, Ph.D., C.Eng., M.I.E.E., J.R. Bumby, B.Sc, Ph.D., and B.I, Hassall, B.Sc.

Indexing terms: Generators, Superconductivity

Abstract: An unusual form of polyphase armature winding is described in which conductors lie on a helixof constant radius as they pass from one end of the machine to the other. There are no end windings. Ananalysis of the winding is performed which allows analytical expressions for the flux densities and machinereactances to be found. Measured and calculated values of synchronous reactance agree to within 2%. Someof the more unusual properties of helical windings are discussed.

List of symbols

H

Hz

H6

In(nkr)

I'n{nkr)Kn(nkr)

K'n{nkr) =

magnetic field vector, A/mradial component of magnetic field, A/maxial component of magnetic field, A/mtangential component of magnetic field, A/mpeak phase current, Amodified Bessel function of first kind, ordern, argument nkr(dldiX!n(nkr))modified Bessel function of second kind,order n, argument nkr

LM

-™01 > -^02

-4 0 5 , ^ 06

ft

abf"•wn

£ken

= tangential component of linear current den-sity, A/m

= axial component of linear current density,A/m

= self-inductance, H= mutual inductance, H= number of turns in series per phase = Z/2a= number of armature conductors per phase

^AQ-J^A^, \ = Constants in general solution of?2 > Dn, Fn, j Laplace's equation>M, v )= magnetic flux linkage= magnetic scalar potential= Kn{nkrt)Un(nkri)

number of parallel paths per phasenumber of phasebands per phasesystem frequency (50 Hz)nth harmonic winding factor = kbnksn

nth harmonic breadth factor«th harmonic skew factor«th harmonic radial-flux-density environ-mental screen factor«th- harmonic tangential-flux-density environ-

Paper 675C, first received 9th October 1979 and in revised form25th February 1980Dr. Anderson is with the Electromagnetics Group in the ElectricalResearch Department of C.A. Parsons Ltd., Newcastle, England;Dr. Bumby is with the Department of Engineering Science, Uni-versity of Durham, Science Laboratories, South Road, DurhamDH1 3LE, England; and Mr. Hassell is with the University of Leeds,England

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

kImnqr

rf

6

Po

Mo

Suffixesgnphrzd

nth harmonic axial-flux-density environmentalscreen factorTT/2/

armature winding half-length, mnumber of phasesspace harmonic ordertime harmonic orderradius of magnetic field point, mradiation screen radius, mouter radius of outer rotor, mradius of superconducting field winding, mwinding radius, menvironmental screen inner radius, m(27r/m)(i- l ) , /= 1,2,---msynchronous reactance, £1, p.u.transient reactance, ft, p.u.subtransient reactance, ft, p.u.subtransient reactance associated with radi-ation screen, ft, p.u.phaseband displacement, radconstant = [Kn(nkri) — yn(nkri)In(nkri)]circumferential progression, radresistivity, ftmphasespread, radpermeability of free space, H/mangular frequency 2nf, rad/s

phasegroupspace harmonic orderphaseradial componentaxial componenttangential component

1 Introduction

Since 1940 the unit size of two-pole 3000rev/min turbo-generators has increased from 60 to 660 MW. At the sametime, the specific power output has increased from 0-3 kW/kg to 2kW/kg;2 in order words, there has been almost asevenfold improvement in the power/weight ratio. This hasbeen made possible by the introduction of hydrogencooling and, more recently, water cooling, of the statorconductors, which has enabled more effective use to bemade of the iron and copper within the machine.

Within the framework of the maximum magnetic and

129

0143-7046/80/03129 +16 $01-50/0

electric loadings used today, it appears that there may bedifficulties in increasing the size of conventional generatorsbeyond 2000 MW; difficulties which arise because of twolimits. These are: first, the limit on maximum indivisibleweight which is set by power station cranage and bridgeand ship loadings; secondly, the limit on the length betweenrotor bearings, which is set by the critical speeds.2 Onepossible way of overcoming weight limitations would beby subdividing the inner stator, which is usually the heaviestitem, and building it on site from smaller factory-builtassemblies.3 Other ways of overcoming, for instance, theproblem of rotor length, are by developing slotlesswindings4'5 or alternatively, superconducting machines.6"9

In both cases, higher specific power outputs becomepossible, albeit at the cost of introducing new technology.

One advantage of either slotless or superconductinggenerators should be increased efficiency. One of the mainreasons why the efficiency of the superconducting generatorcan be increased8 is because of the elimination of the rotorexcitation loss, which is typically 6 MW for a conventional1300 MW generator.

The work of the present authors has been confined tothe superconducting generator; the essential componentsof which are shown in Fig. 1. A superconducting rotorwinding at 5 K, supported in slots in a stainless-steel rotorforging, is surrounded by an ambient temperature rotorscreen to protect the superconductor from transient andnegative sequence armature fluxes. The ambient temperaturestator consists of two parts: the first is a self-supportingstructure in which the water-cooled copper armature isembedded; the second is an iron core, or environmentalscreen, which screens the environment from magneticfields and increases the radial flux density and conse-quently the induced armature e.m.f. per metre length.This paper is concerned with the properties and design ofthe novel helical stator winding in particular.

Helical windings are not new and found limited favourfor dynamos in Germany in the late 1880s12>13 and havesince been revived for small d.c. machines of the ironlesstype, where they offer the advantage of low inertia andsmooth torque at low speeds. They have also been used forlinear motorsls and transverse flux tubular motors.16 Theuse in large a.c. machines was first proposed by Ross,Anderson and Macnab10'11 who. suggested it as a suitablearmature winding for superconducting a.c. generators.More recently, a 4MVA helical armature winding has beenused by Watanabe et al17 as the armature winding for usein either a slotless or a superconducting generator, andConley et a/.18 have used a helical winding in an experi-mental 10MVA superconducting a.c. generator at MIT.

-laminated iron environmental screenarmature support structure

70K /

inner-rotortorque tube

\

drive end L

i[ Ii

/JH

\> ,

/

/

=r^ / ' ' ' '

/' ' ' ; ^T

I

water- cooled^armature

winding

\z& k4•^—Unon-drive\ end\ outer rotor.

N.325Kinner rotor,4K

superconducting field winding

Fig. 1 Basic components of a.c. superconducting generator

130

The helical winding has no distinct division into activeand end regions as has the conventional winding, becauseeach conductor follows a helical path of constant pitchfrom end to end of the machine, see Section 2. In theconventional winding on the other hand, each conductorfollows an axial path in the active region and only has acircumferential component to its path in the end regions.Thus the helical winding is characterised by a circumfer-ential component of current density, which only decreasesgradually with increasing axial distance from the ends;instead of being confined to the end winding region as inthe conventional winding. This suggests that there might beaxial flux problems which would not occur with a moreconventional winding. However, work by Tavner et al.19

has shown that, contrary to what might be expected, theend-region axial fluxes in the iron core produced by ahelical air-gap winding are less than those produced by acorresponding conventional diamond winding. Such obser-vations are also supported by the work of Watanabe et al11

The synchronous and subtransient reactances are funda-mental to generator design and performance and, in orderto design a helical winding, these reactance expressionsmust be developed in the simplest possible form. In thepresent paper the magnetic field distributions within thegenerator which are produced by a helical winding areobtained in terms of Bessel functions and reactanceexpressions are then derived, which account for all thearmature space harmonics. In this way the lowest harmonicsum that gives satisfactory reactances can be used. Analternative to the above method is to use a transmission-line analogue.20 However, although this method yieldsmagnetic field values without the introduction of Besselfunctions, the solution is numerical. This is a disadvantageof the method, since analytic inductance expressionssimplify iterative generator design techniques.8 Morerecently, Alwash28 has developed an analytic 3-dimensionalmethod which is expressed in Fourier-Bessel expansions.The solution requires extensive computation for the bound-ary matchings, although it is capable of achieving highaccuracy.

2 Development of an helical armature winding

As generator sizes increase, the length between bearingsbecomes a limitation, see Section 1. Increased output inlarger machines can only be obtained by increasing theoutput per unit length by an amount which keeps thelength between bearings down to an acceptable level. Inthe case of the conventional machine, electric and magneticloadings can only be increased marginally and thereforethe only remaining variable is the effective stator-windingdiameter. Stator-winding diameter is however closely tiedto the rotor diameter because of the need to closely couplethe two windings and so give an acceptable rotor m.m.f.and stator magnetic loading. Thus since the rotor diameteris limited by rotational stresses, so too indirectly is thestator diameter.

On the other hand such restrictions no longer apply tothe superconducting machine, and a considerable decouplingbetween rotor and stator windings becomes possible with-out incurring the same penalties that would arise in theconventional machine. This is because the superconductingwinding can operate at current densities at least ten timesgreater than is current turbogenerator practice and this, inturn, permits magnetic loadings at least as high as thoseobtained in the conventional machine, i.e. 0 9 T , but at a

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

substantially increased stator diameter; even though therotor diameter is still limited by rotational stresses to avalue not far different from current practice. Hence thestator diameter can be increased from 1 -6 m in the conven-tional machine to 2 0m in the superconducting machine,with a consequent increase in the possible output per metre.Furthermore, in the conventional turbogenerator theavailable winding space is limited by the need to supportthe windings in slots in the iron core, whereas in the caseof the airgap-wound superconducting machine the pro-portion of the circumference available for the windingincreases, as does the possible winding depth; this allowsthe electric loading to increase from typically 200kA/mto 300kA/m in a 1300MW design. The increased armature-winding diameter and electric loading combine in the caseof the superconducing generator to produce a short, large-diameter armature winding with a power output of300MW/m instead of the lOOMW/m typical for theconventional turbogenerator.

A secondary result of the absence of clearly definedclosed magnetic circuits to constrain the flux is that thedistinction between active and end regions becomes blurred.

Fig. 2 Evolution of helical winding from diamond lap winding

a Diamond lap winding, long machineb Diamond winding, short machine with greater specific machineoutputc Diamond winding, with straight portion removed to give semi-helical windingd Full helical winding

Because of the way in which the active and end regionsmerge, it becomes apparent at a fairly early stage in thedesign that the conventional winding might not necessarilybe the best and that alternatives should be examined. Ifa helical winding is used, then it can be uniformly sup-ported from end to end; a feature which is particularlyimportant now that the armature conductors are no longershielded from the main flux by the stator teeth and musttherefore bear the full machine forces.

To understand how a helical winding works, considerthe winding length of a diamond winding to be progressivelyreduced, as in Fig. 2, by eliminating the so-called centralactive region and allowing the two diamond-end windingsto coalesce and form a helical winding. One phase of atwo-pole helical winding is shown in Fig. 3, where eachphase consists of two parallel-connected helical loops,each of which is termed a phase group. Each phase groupconsists of one right-hand (inner) helical phaseband con-nected in series with a returning left-hand (outer) helicalphaseband; a right hand phaseband being defined as thatwhich traces a right hand screw.

Because the development of large generators in the UKis generally concerned with two-pole 3000rev/min2

machines, only two-pole helical windings are consideredin the present paper.

3 Method of analysis

3.1 Notation adopted in the description of a helicalwinding

Each phaseband is described by three parameters:(a) + 1 or — 1 depending on whether a right-hand (r.h.)

or left-hand (l.h.) helixed phaseband is considered(b) the circumferential angular position a at which the

centre of the phaseband crosses z = 0, Fig. 3(c) the radius of the phaseband r{.

e.g. Kz(rh oc, + 1) refers to the axial component of linearcurrent density for the /th right-hand phaseband centredat angular position a and radius rt.

The angular position of the two phasebands in a phase-group are related as

<*/(-1) = (1)

while the angular position of a phasegroup is defined as theangular position of its right-hand phaseband,-

ocg = <*(+ 1) (2)

and the angular position of a phase as

phasegrocentre Oc

lefthandphaseband

righthandphaseband

Fig. 3 One phase of helical winding

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980 131

or

(3)

because the complete phase is symmetrical.

3.2 Representation of a helical winding

Each phaseband of the helical armature winding is repre-sented by a linear current density sheet of negligible radialthickness, with axial and tangential linear current-densitycomponents as described in Section 11.1. In order to yielda closed analytical solution to Laplace's equation, aperiodicvariation is introduced in the axial direction in a similarmanner to that suggested by Hammond,21 by assuming thehelical winding to be one of an infinite number of helicalwindings placed end to end. Such a winding is termed an'infinite periodic helical winding'.

3.3 Representation of the environmental screen

A superconducting a.c. generator requires an environmentalscreen which is normally constructed from laminatedmagnetic iron, in order to screen the environment from themagnetic fields produced inside the generator. However,conducting environmental screens have been suggested,6

particularly for airborne applications,22 and this analysiscontains solutions for both types of screen. The generaleffect of the iron environmental screen is, as expected,to enhance the main magnetic field, whereas the con-ducting screen has a demagnetising effect.

The environmental screen is assumed to be infinitelylong in the axial direction. If it is constructed from lami-nated magnetic iron it is assumed to have infinite per-meability and zero conductivity, i.e. there are no eddycurrents. If on the other hand it is constructed from non-magnetic conducting material, it is assumed to have arelative permeability of unity and infinite conductivity. Ineither case the effect of the screen is included in themagnetic-field analysis by using an image winding with theappropriate current polarity placed outside the actualhelical winding. In the case of the iron environmentalscreen, the image winding assists the radial field and there-fore is of the same polarity as the armature winding. Inthe case of a conducting screen, the image winding mustoppose the radial field and therefore is of opposite polarityto the armature winding.

In practice, the iron environmental screen tends to beshorter than the helical armature winding, as indicated inFig. 1, and it is likely too that a conducting end screenwould be used. Consequently the assumption of an infinitelylong environmental screen, whilst necessary to enable aready solution to the problem, gives a high value of reac-tance and is mainly responsible for the difference betweencalculated and measured reactance values. However, in themodel results of Section 6, the difference in the synchron-ous reactance is less than 2%, indicating how little the endscontribute to the reactance.

3.4 Mathematical solution method

The infinite periodic helical winding is analysed in cylin-drical polar co-ordinates with the origin of the co-ordinatesystem taken at the axial centre of the winding such thatthe actual winding extends from z = + / to z = —I andconstitutes one half-period of the 'infinite periodic helicalwinding'. The magnetic field due to an individual phase-band at the general circumferential position a,- is obtained

from a solution of Laplace's equation, expressed in terms ofmagnetic scalar potential, by the method of separation ofvariables. Magnetic scalar potential terms produced by theimage winding which represents the environmental screenare combined with those from the helical armature windingand magnetic-field distributions are then obtained in themanner described in Section 11.2.

4 Magnetic field of a helical winding

The magnetic field distributions for an individual phase-band of an infinite periodic helical winding with an environ-mental screen are shown in Table 1. The environmentalscreen modifies the magnetic field distributions of theunscreened winding by the geometric factors defined inTable 2.

The magnetic field distributions described in Table 1include terms that are independent of axial or tangentialposition. The effect of these zero-order terms is reflectedin the magnetic field distributions as a solenoidal fieldwhich is produced by the zero-order component in thetangential linear current density A"0(rf, a,-, ±1). Becausethe winding is part of an infinite helix, a magnetic fieldcomponent is produced only inside the winding, just asthough it were an infinite solenoid. The zero-order com-ponent of axial linear current density, Kz(rt, a{, ± 1),produces a tangential magnetic field outside the windingthat varies, in accordance with Ampere's law, inverselywith radius.

Using superposition to combine a left-hand and a right-hand phaseband to give a phasegroup, as in Table 3, elim-inates the zero-order tangential magnetic field. This isbecause the zero-order component of axial current densityin the right-hand phaseband, Kz(rh a,-, +1 ) , opposes theequivalent component in the left-hand phasebandKz(rit at, — 1), as shown in Fig. 4. However, because thecorresponding zero-order components of tangential linearcurrent density, K9(rh oct, + 1) and Kgj/i, a,-, — 1) are bothin the same direction their effect is additive and the sole-noidal magnetic field is increased. This is also shownschematically in Fig. 4. When two phasegroups are com-bined to produce a phase, the relative phaseshift of 180°which exists between the two phasegroups ensures that thissolenoidal magnetic field is eliminated. The influence ofthis effect on measured and computed inductances isdescribed in Section 6.

5 Reactance calculations

5.1 Synchronous reactance

Reactance expressions are obtained by assuming that thearmature winding is concentrated at the geometric meanwinding radius rs, and by only taking account of thefundamental time harmonic.

Consider the two helical loops shown in Fig. 5; onecentred at 0 = 0 and the other at 0 = a, and assume thereto be a phasegroup of Z/b turns centred about the secondloop. The flux from this phasegroup which links the firsthelical turn is, by Gauss's theorem, equal to the flux thatcrosses normally any surface bounded by the first, like thatshown in Fig. 5. Evaluating the resulting surface integrals asdescribed in Section 11.4 allows the inductance betweenphasegroups to be expressed as in eqn. 51. If the phase-group inductance values are combined appropriately, phaseand generator inductances are obtained, see eqns. 53 and

132 IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

Table 1: Magnetic field distributions produced by a helical phaseband

Linear current-density distributions

±Y2-nabri nabrt £?t

inside the winding r < r,-

+Z!EhB y k-Mkenyninkri)ln(nkr)s.n +

4a*>/ nab ^ n bn(nkrt) l

HrUi.oii.t 1] =

He[ri, a,-. ± 1 ] =

Z,\'phq

7; °°c'phq V1

6n(/7Ar/-f)

n - * /„- ,ln(nkr)

8n{nkrt)sin n(0 — a,- + Arz)

cos n(d — a,- + Arz)

_

Outside the winding r,- < r < r^

Sl[ri,ai.f\], , Z'phqe Zlp

2abn nab ^ n 8„(/?*:/-,)s\n n{d-a{ +kz)

kKn.^nkr) +

sin n (6 — a: + kz)

nab

Hglrt.at.tU =Z' phg2lab 6n{nkri)

cos /? (0 — a,- + kz)

Environmental screen factors krn, kQn, kzn defined in Table 26n(nkr{) = [Kn(nkrt) - )

55. The synchronous reactance of a helical winding, givenin eqn. 55, can be expressed in the more usual form as

A J Mm

nn

n = 2rm ± 1, r any integer

with the winding factor

k2 = k7

(4)

(5)

kjn is the skew factor defined by eqn. 48 and krn the wthharmonic radial flux environmental screen enhancementfactor defined in the first column of Table 2.

5.2 Subtransient and transient reactance

Under fault conditions the armature flux is initially deflectedaround the outside of the outer rotor, as if this outer rotorwere a perfectly conducting screen. This effect is includedin the analysis by representing the outer rotor screen by animage winding at the surface of the outer rotor, which isequal in magnitude, but opposite in phase, to the armature

current. This current introduces an additional magneticscalar potential term into the magnetic field solution asdescribed in Section 11.4.4, such that the resulting inte-gration over the Gaussian surface of Fig. 5 gives for thesubtransient reactance:

xdn —m•nn

1 -

1 -

ohms

(6)

where the factors yn(nkr) and ($n(nkr) are defined in eqns.43/and 44b, respectively. This expression is similar to thatfor the synchronous reactance except for the geometricmodifying factors, both of which depend, through Besselfunctions, on the armature length. The two modifyingfactors relate to the different generator components as

1 -

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980 133

a geometric factor relating armature and outer rotor

1 -yn(nkrD)\

a geometric factor relating the iron environmental screenand the outer rotor.

(Note: For a conducting environmental screen yn(nkrx)replaces pn(nkrx). Table 4 details all the permutations).

As the fault period progresses flux will penetrate theouter rotor and two further reactance expressions requiredefinition; the first being associated with the radiationscreen, when flux is deflected round the outside of thisscreen, and the second, the transient reactance, when flux isdeflected round the outside of the rotor field winding.These can be obtained from eqn. 6 by substituting theouter rotor radius rD, by the radiation screen radius rc, andthe superconducting field winding radius rf, respectively.The transient reactance is unusual in the superconductinggenerator because it is associated with a transient open-circuit time constant greater than 100 s compared with oneof less than 10 s in the conventional turbogenerator.

5.3 Effect of harmonics on reactance calculation

The reactance between phases and the synchronous reac-tance are computed for the 1300 MW design of Table 5.This generator has a six-phase armature winding withopposite phasegroups connected in parallel. The resultsof the reactance calculations are shown in Table 6 first forthe fundamental harmonic only and secondly for theharmonic sum of n — 1 to 50.

The Table shows that when considering a phase or

phasegroup, the space harmonics form an important con-

Table 2: Expressions for nth space harmonic environmental screenfactor

Inside statorr < rB

Outside stator

Ironenvironmental Arm

screen

k6n

fc Conductinga> environmental kr

E screen

ken

1 - 1 * r

1 - 1 *

Ironenvironmental kr

screen1 —

1 pn(nkrx)}

yn{nkr) I

1 -yn(nkrs)

1 -

Jc Conducting.y environmental kT

o screen1 -

yn{nkrs)1 —

yn{nkr)

1 —yn{nkr8)

1 —Pn{nkr)

If no environmental screen krn, kQn, kzn = 1yn{nkr) always negativeQn (nkr) always positive

134

stituent of the reactance and consequently must be takeninto account. However, once all three phases are combinedto give the three-phase synchronous magnetising reactance,eqn. 4, many of the space harmonic terms cancel out andonly those terms with n = Inn ± 1 are left to contributeto the reactance. The effect of individual harmonics isreflected into any reactance expression through both theskew and environmental screen factors but, more domi-nantly, by the factor kbn, kbn being the breadth factor.

For a six-phase winding, with 30° phase spread, har-monics introduced are the 11th, 13th, 23rd, 25th etc., forwhich the variation of k\n/n is shown in Table 7. Thefirst harmonic is dominant such that the synchronousreactance calculated with n = 1 shows negligible error whencompared with the value calculated using the harmonicsum.

The variation of kbn/n for the three-phase winding witha phase spread of 60° is also shown in Table 7, the har-monics now being 5th, 7th, 1 lth, 13th etc.. Again, the firstharmonic is dominant, indicating that in calculatingsynchronous, sub transient and transient reactance, onlyfirst-harmonic terms need be considered.

5.4 Comparison of reactance of a helical winding withthose of a straight winding

By assuming the armature winding to be infinitely long andeach phase to be represented by a current sheet of the form

2TrhkwniphsinndA/mirr8

the synchronous reactance is

_ mX*n = ~im , 2/ ohms

where kwn, the winding factor is

(7)

(8)

(9)

Fig. 4 Zero-order components of phaseband and phasegrouplinear current-density distributions

a Zero-order components of righj-hand phasebandb Zero-order components of left-hand phasebandc Zero-order components of phasegroup

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

Table 3: Magnetic field distributions produced by a helical phasegroup

Linear current density distributions

KZg[ri, <*,-] = — ^ ^ Y kbn cos ft(0 —a,) cos ftArz + ^ £ kbn S"n "& ~a^ s i n nkz

•nabri n_( nabrj n = 2nodd neven

sin «(*-«,•) sin bn cos ft(0 -a , ) cos ft*z

n odd

Inside the winding r < /-,-

n,[^.«i] = -Ziphq'2/ab

r- *-. t« -* —COS rtArZ Sin n(0 —a,0

r | ] —cos nkz sin ft(0 —a,)

nab £, 6n{nkri)

2ZiphQ f kbnkdnyn(nkri) ,n(nkr)\ «» " ^ cos »«» ~a.»

f

6n(ft/r/-,) r j s i n nkz s i n o ( 0 —otj),

sin ft/rz sin ft(0 —a,)

cos ft/rz cos ft(0 — a,-)

odd

n odd

n even

AJ odd

n even

ft odd

ft even

Outside the winding r,- < r

— cos nkz sin ft(0 —a.)

KAnkr)

7ra/j

V ^bn Kdn„=, ft 8n(ft/rr,)

£

n s l 6n(n/rr,) r j s i n ft/tz sin ft(0-a,-)

5n(ftAr/-,)

?n Kn\nkr)

r \\ —cos ftArz cos ft(0—a,)

sin nkz sin ft(0 — a,-)

cos ft*z cos n[0 -a,-)

ft odd

ft even

ft odd

ft even

ft odd

ft even

ft odd

ft even

Environmental screen factors Arrn, Ar^n, Ar2n defined in Table 25()

Table 4: Expressions for the ftth space harmonic combined environmental and rotor screen factors

Inside stator rD < r < rs Outside stator rs<r <r~

k' 1 +yn{nkr8)

1 +yn (nkr)

^-a-b

-1

1 -ln(nkr8)

ab

1 —yn[nkrs)

1 -Pjnkr)

ab

1 -

1 -yn(nkr8)\

a = (in{nkrx) for iron environmental screena = yn(nkrx) for conducting environmental screena = 0 for no environmental screenb = Pn(nkrD) for iron rotor screenb = yn(nkr£f) for conducting rotor screenb = 0 for no rotor screen

IEEPROC, Vol. 127, Pt. C, No. 3, MA Y1980 135

kpn being the nth. harmonic pitch factor. This equation is kjn, of the helical winding does not tend to 1 as the arma-of the same form as that for a helical winding, eqn. 4,except for the winding factor, kwn, and environmentalscreen factor krn. The environmental screen factors obtainedby the two dimensional analysis outlined above, see Ref-erences 23, 24 and 6, are shown in Table 2 along with thecorresponding values for the helical winding, the funda-mental difference between the environmental screenfactors being the dependence on winding length introducedby the helical winding. However, finite-length effectsbecome less prominent as the length/diameter ratio of thehelical winding increases. Hence, as this ratio tends toinfinity (i.e. as the winding length tends to infinity) theenvironmental screen factors for the helical winding tend tobecome the same as those for a long straight winding. Inthe same way, the modifying factors associated with thesubtransient reactance of the helical winding, defined ineqn. 6, tend to

[••ferii-r2n

(+ I iron environmental screen_ | conducting environmental screen

as the winding length tends to infinity. This is identicalwith the expression obtained for the long straight windingwith a conducting environmental screen by Appleton andAnderson6 and an iron environmental screen by Miller andHughes.25

Unlike the environmental screen factors, which tend tothe same value as winding length increases, the skew factor,

ture length tends to infinity, but to 0-5. This arises because,whatever the length of the generator, a proportion of theflux must be lost due to skewing of the conductors through180° over the total generator length. The variation of skewfactor kln

w i t n length/diameter ratio is shown in Fig. 6.One effect of skewing the armature conductors is to

reduce the amount of rotor flux linking the armature wind-ing, and consequently the induced e.m.f.. However, as therotor flux density at the armature winding falls off towardsthe ends of the machine, the majority of the generated e.m.f.is towards the centre of the machine. Hence in practicaldesigns this loss of flux is found to be less than 10%.

This loss of flux in the helical winding is compensatedfor by the superior winding support and by the reductionof synchronous reactance (and in consequence the reducedarmature reaction). This drop in synchronous reactancearises because the skew factor for a helical winding is lessthan unity and because the environmental screen factorkrn is also less than it would be for a straight winding.krn has a value of 1-54 for a straight winding and 1-38for the helical winding design of Table 5.

6 Comparison of inductance calculations with modelresults

To assess the accuracy of the inductance expressions of

Table 5: 1300MW a.c. generator design parameters

Parameter Value

MVA rating 1530MVAPower factor 0-85 laggingTerminal voltage (r.m.s./phase) 16 kVArmature current (r.m.s./phase) 15-94 kANo. phases 6No. parallel paths/phase 2No. phasebands/phasegroup 2No. phasegroups/phase 2No. conductors/phase 48Field-winding radius 0-423 mRadiation-screen radius 0-5 mOuter-rotor inner radius 0-515 mOuter-rotor outer radius 0-65 mArmature mean winding radius 1 -03 mInner radius of iron environmental screen 1 -4 mArmature half-length 2-9 m

Fig. 5 Surface of integration for calculation of inductance

136

Table 6: Calculated reactance values for helical winding with ironenvironmental screen, showing effect of space harmonics

Reactancep.u.

HarmonicSum

n = 1 to

n = 1

Table 7:

6-phase

n

1

11

13

23

50

wLph wMph(n/6) wMph[n/3) xs

0-224 0-128 0056 0-5018

0-167 0-145 0084 0-5013

Effect of harmonics on the breadth factor of 6-phase and3-phase windings

kln/n

0-97736

000073

000044

000008

3-phase

n kln/n

1 0-91289

5 000730

7 000266

11 0-00069

IEEPROC, Vol. 127, Pt. C, No. 3, MA Y1980

Section 5.1, extensive measurements have been made usinga 1:9 scale model of a 1300MW turbogenerator26'28 atLeeds University. The helical winding model has a meanradius of 105 mm and length of 406 mm and is shown inFig. 7. To simplify construction, it employs comparativelyfew conductors, 24 per layer, which are disposed in threephases with opposite phasegroups connected in series tominimise errors due to circulating currents. The inductancebetween individual turns can be measured. The iron environ-mental screen model, shown in Fig. 8, is manufactured fromlaminated iron and has an inner radius of 134-5 mm andlength of 356 mm. The ends of the environmental screenare manufactured from aluminium with an inner radius of150 mm and extended 88 mm to the aluminium end platesof the model. The relevant dimensions of the model aresummarised in Table 8.

Measured and computed values of interturn mutualinductance, both for the helical winding in air and insidethe iron environmental screen, are shown as a function ofcoil separation in Figs. 9 and 10, respectively. Computedinductance values are obtained by assuming a phasespreadof 15° (the total spread of each conductor) and summingto a harmonic order of 81; this being the maximum orderaccomodated by the computation.

M

0 9

0-7

0 50 5 10 15

nrsS=TT

2 0

Fig. 6 Graph of skew factor fcjn against kr8 for helical winding

Fig. 7 Model helical winding

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

Fig. 8 Iron environmental screen

08f

solenoidal inductance

,180°

-0-2 L

Fig. 9 Interturn mutual inductance as function of separation formodel winding in air

1 Or

computed (n=1 to 81)

computed (fundamental componentonly)

solenoid inductance

- 0 2measured

Fig. 10 Interturn mutual inductance as a function of separationfor model helical winding inside iron environmental screen

137

Table 8: Model helical winding dimensions

Helical winding:I.h.-layer radiusr.h.-layer radiuswinding lengthconductor diameter

mm100110406

7

Iron Core:inner radiuslength

134-5356

The presence of the 'solenoidal' inductance term in thephasegroup inductances, and interturn inductances, makescomparison of measured and computed phasegroup, orinterturn, inductance values difficult because the solenoidalinductance value computed by the 'infinite periodic helicalwinding' is that of an infinite solenoid, whereas thatmeasured is of a solenoid of finite length. The latter can besubstantially different from the former. The interturnmutual inductances measured in air (Fig. 9), however,show very close agreement with the calculated values, beingon average higher by 10% of the solenoidal inductanceterm. These results suggest that the inevitable overestimationof solenoidal inductance has been countered by main fluxlinkage around the winding ends. When measured in thelaminated iron core (Fig. 10) a more pronounced dis-crepancy, on average 50% of the solenoidal inductance, isshown. This arises because the model core is shorter thanthe winding and the boundary extended to the machineend plates by aluminium rings of inner radius 150 mm.Consequently, any measured results may be expected to beless than those predicted by the infinite periodic modelwith the infinitely long environmental screen assumption.Also shown in Fig. 10 is an interturn inductance com-putation using the fundamental harmonic component only.This clearly demonstrates the high space harmonic contentof interturn, or equally phasegroup, mutual inductances.

A comparison of self-inductances is shown in Table 9and gives two measured and two calculated values ofinductance for the winding in air and inside the laminatediron core. The first column gives directly measured values,whilst the second gives values derived from measurementson the previous subgroup. For example, the phase self-inductance was derived from the measurements on 60°spread phasegroups, showing a discrepancy of 2-5% withthe directly measured value. This discrepancy, absent frommutual values, is attributable to the conductor necessaryto form the series connection spanning 180°. The promi-nence of this effect is because of the air-cored nature of the

geometry, even in the iron core, and exaggerated in themodel by the small number of conductors, 16, that form aphase. The computed values of self-inductance depend onconductor size. The first column in Table 9 assumes a fullspread for the current sheet of 15° per turn, whereas in thesecond column a spread of 7-5° has been used. The trueangle subtended by the conductor is approximately 3°, butif this spread is used with infinitesimal depth then excess-ively high inductance values result. In other words if acontinuous current sheet is used to represent discreteconductors, a fair model results.

The error between computed and measured values inTable 9 is in all cases less than 10% and the synchronousreactance values agree to within 2%. The final row of Table9 uses only the fundamental component in eqn. 4 to com-pute synchronous reactance and gives results to within2%; indicating that only the fundamental component needbe considered when computing synchronous and sub-transient reactances.

7 Design of a helical armature winding for use in asuperconducting a.c. generator

For the generator design of Table 5, the radial flux densityproduced by the superconducting field winding at thearmature in the presence of an iron environmental screenwas computed by GFUN27 with account being taken ofenvironmental screen length, and the length, depth andslotting arrangement of the superconducting field winding.The computed radial flux density variations are shown as afunction of angular position and axial position in Figs. 11and 12, respectively.

Analysis of the variations in radial flux density withtangential position at the armature radius indicates amaximum harmonic content at the axial centre of themachine of under 1%. Hence the e.m.f. induced in a phasecan be assumed to be due to the fundamental harmoniconly and is given by

scjkbZ2 Cl *. HZBr{z) cos — dz volts (10)Jo 21

where the variation of radial flux density with axial positionBr(z) is shown in Fig. 12.

Computation of the e.m.f. requires numerical integrationof the axial variation of radial flux density produced by thefield winding, Br(z), while the length and diameter of thehelical winding determines the machine inductances

Table 9: Self-inductance values, measured and computed, for model helical winding

Helical turn30° phasegroup60° phasegroup60° phase La

phase mutual Ma

Synchronous inductance L8

Synchronous inductance(fundamental approx., 60°phasespread)

Self-Inductance, juH

ex-core

measured

direct

0-82-366-68

15-14 08

19-1

derived

2-346-65

14-74 08

19-1

computed

a = 15°

0-712-266-72

14-64-44

1 9 0

18-8*

a = 7-5°

0-8552-517-17

15-54-38

19-8

18-9"

in core

measured

direct

0-8922-677-64

20 06-12

26-1

derived

2-647-55

19-66-12

25-7

computed

o= 15°

0-8272-718-24

19-66-60

26-2

25-9*

a =7-5°

0-9742-968-70

20-56-53

27-1

26-1 * •

Notes: * This is equivalent to a = 60°, identical to vectorially adding 4e.m.f.s with a= 15°, separated by 15°** Vectorial addition of 4 e.m.f .s with a = 7-5°, separated by 15°

138 IEEPROC, Vol. 127, Pt. C, No. 3, MA Y1980

through eqns. 4 and 6. For a correct design, at a preselectedterminal voltage and current, the two quantities of inducedvoltage and synchronous reactance will satisfy the generalphasor diagram shown in Fig. 13.

To design an armature winding, the generator dimen-sions, reactances and electric and magnetic loadings areobtained using an iterative design technique8 whichassumes the generator windings to be infinitely long and tobe represented by fundamental harmonic current sheetsat their respective geometric mean winding radii; thisreduces the magnetic field problem to two dimensions. Afield-winding length is then selected and a flux plot such asthat shown in Fig. 12 obtained for the three-dimensionalgenerator configuration. An iterative computer programthen computes the induced e.m.f. using numerical inte-

=00m

30° 60° 90°angular position from polar axis, degrees

Fig. 11 Circumferential variation of radial component of fluxdensity produced by superconducting field winding at rs = 1-Omascomputed by GFUN at 100% excitation

•o ?o c

a*

n 1 4

• 1 2

iron environmental^ ,\screenx s X. N

two-dimensional

r | 1 °

!l°-8h;:r3-m-2 §

i ! °2,

approximation

rotor endwindings

14

12

10

0-8

0-6

0 2

10 15 20 2 5 3 0 3 5axial position, m

Fig. 12 Graph showing axial variation of radial component offlux density produced by superconducting field winding (as com-puted by GFUN at 100% excitation) with relative position of rotor-end windings and environmental screen shown to scale

IxQ

Fig. 13 Phasor diagram for superconducting a. c. generator

IEEPROC, Vol. 127, Pt. C, No. 3, MA Y1980

gration of eqn. 10 and an estimate of the induced e.m.f.E* is obtained from the phasor diagram as

E* = + 2x8 sin 0 + JC| p.u. (11)

For a correct design the e.m.f. E and its estimate E* mustbe equal. If E and E* are not comparable, then armaturelength and/or the number of armature conductors arevaried until a solution is obtained and the subtransientreactance obtained by eqn. 6 is equal to, or greater than,the value computed in the two-dimensional design. Thisensures that outer rotor stresses during a short circuit areacceptable. If varying both armature length and the numberof conductors will not yield a satisfactory solution, thenthe length of the rotor field winding must be changed andthe iteration repeated.

For an armature of given length the helical winding hasbeen shown to have a lower synchronous reactance (highershort-circuit ratio) than that predicted by two-dimensionalanalysis for a section of an infinite straight winding of thesame length. Hence armature reaction effects are reducedwith an helical winding.

8 Conclusion

A closed analytical solution has been presented for themagnetic field distribution and machine reactances of asuperconducting a.c. machine with an helical armaturewinding. There is a significant harmonic content in thephasegroup and the phase inductances, but when all threephases are combined to give the synchronous, subtransientand transient inductances many of the harmonics cancel,except those of order n = Irm ± 1, where m is the numberof phases and r any positive integer. Hence the importantsynchronous, subtransient and transient inductances arelargely dominated by the fundamental and therefore can bereadily calculated. The reactances can be expressed in asimilar form to those for a section of an infinite straightwinding, but are modified by two factors, namely a skewfactor k\n

a nd an environmental screen factor krni both ofwhich contain Bessel functions that depend on the ratioof length to diameter (kgn = 1 for a straight winding).Laboratory measurements on a model helical windingindicate that the synchronous reactance expressions areaccurate to within 2%.

9 Acknowledgments

The authors would like to thank IRD Co. Ltd., and NEIParsons Ltd., for permission to publish this paper. Theythank their former colleagues in the Electrical EngineeringDepartment of IRD Co. Ltd. for their help and advice andin particular J.S.H. Ross and M. Reay.

The authors would also like to thank Professor P.J.Lawrenson of the University of Leeds for making availablethe facilities for the model winding measurements andDr. T.J.E. Miller, formerly of the University of Leeds, whoinitially designed the model helical winding.

The assistance of C.W. Trowbridge and J. Simpkin of theRutherford Laboratory in producing Figs. 11 and 12 isgratefully acknowledged.

B.I. Hassall is much indebted to the Science ResearchCouncil for financial support.

139

10 References

1 VICKERS, V.J.: 'Recent trends in turbogenerators', Proc. IEE,1974, 121, (11R), pp. 1273-1306

2 STEEL, J.G.: 'New designs of large generators', Engineering,1978, 218, pp. 43-45

3 HORSLEY, W.D.: The high speed generator - Eighty yearsof progress'. 29th Parsons Memorial Lecture, 1964

4 DAVIES, E.J.: 'Airgap windings for large turbogenerator', Proc.IEE, 1971, 118, (3/4), pp. 529-535

5 SPOONER, E.: 'Fully slotless turbogenerators', ibid., 1973,120(12), pp. 1507-1520

6 APPLETON, A.D., and ANDERSON, A.F.: 'A review of thecritical aspects of superconducting generators'. Presented at theApplied Superconductivity Conference Annapolis, 1972,Paper M2

7 ROSS, J.S.H., ANDERSON, A.F., and APPLETON, A.D.: 'Adiscussion on large superconducting a.c. generators'. Presentedat the International Conference on Electrical Machines, London,1974, Paper A3

8 ROSS, J.S.H.: The engineering design of large superconductinggenerators', ibid., Vienna, 1976

9 APPLETON, A.D., ROSS, J.S.H., MITCHAM, A.J., and BUMBY,J.R.: 'Superconducting a.c. generators: Progress on the design ofa 1300 MW, 3000r/min generator'. Presented at the AppliedSuperconductivity Conference, San Francisco, 1976

10 ROSS, J.S.H., ANDERSON, A.F., and MACNAB, R.B.: 'Alter-nating Current dynamo-electric machine winding'. BritishPatent 1395152, 1975

11 ANDERSON, A.F.: Discussion on'Fully slotless turbogenerators',Proc. IEE, 1975, 122, (1), pp. 75-79

12 FRITSCHE: British Patent 13080,188713 The Electrician, April 12, 1889, pp. 655-65714 SWAINTON, R.: Des. Eng, Feb. 1972, pp. 40-4215 EASTHAM, J.F., and LAITHWAITE, E.R.: 'Linear motor

topology', Proc. IEE, 1973,120, (3), pp. 337-34316 EASTHAM, J.F., and ALWASH, J.H.: Transverse-flux tubular

motors', ibid., 1972,119, (12), pp. 1709-171817 WATANABE, M., TAKAHASHI, M., TAKAHASHI, N., and

TSUKUI, T.: 'Experimental study of a practical airgap windingstator arrangement for large turbine generators'. IEEE WinterMeeting, New York, 1979, Paper F 79 190-0

18 CONLEY, P.L., KIRTLEY, J.L., HAGMAN, W.H., and ULA,A.H.M.S.: 'Demonstration of a helical armature for a super-conducting generator'. IEEE PES Summer Meeting, Vancouver,1979, Paper F79 716-2

19 TAVNER, P.J., PENMAN, J., STOLL, R.L., and LORCH, H.O.:'Influence of winding design on the axial flux in laminated-stator cores', Proc. IEE, 1978,125, (10), pp. 948-956

20 EL-MARKABI, M.H.S., and FREEMAN, E.M.: The electro-magnetic field and shielding of a three phase helical windingplaced coaxially inside a multi region cylindrical conductingstructure'. Presented at the International Conference on Elec-trical Machines, Brussels, 1978, Paper Gl

21 HAMMOND, P.: The calculation of the magnetic field ofrotating machines. Part 1 - The field of a tubular current',Proc. IEE, 1959,106, Part C, pp. 158-164

22 SOUTHALL, H.L., and OBERLY, C.E.: 'System considerationsfor airborne, high power superconducting generators', IEEETrans., 1979, M-15, pp. 711-714

23 HUGHES, A., and MILLER, T.J.E.: 'Analysis of fields andinductances in air-cored and iron-cored synchronous machines',Proc. IEE, 1977, 124, (2), pp. 121-126

24 BRATOLJIC, T., FURSICH, J., and LORENZEN, H.W.:Transient and small perturbation behaviour of superconductingturbo-generators', IEEE Trans., 1977, PAS-96, pp. 1418-1429

25 MILLER, T.J.E., and HUGHES, A.: 'Comparative design andperformance analysis of air-cored and iron-cored synchronousmachines', Proc. IEE, 1977,124, (2), pp. 127-132

26 MILLER, T.J.E.: Transient magnetic fields in the super-conducting Alternator'. Ph.D. Thesis, University of Leeds, 1977

27 ARMSTRONG, A.G.A.M., COLLIE, C.J., DISERENS, N.J.,NEWMAN, M.J., SIMPKIN, J., and TROWBRIDGE, C.W.:'New developments in the magnet design computer programGFUN'. Presented at the 5th International Conference onMagnet Technology, Frascati, Rome, 1975

28 ALWASH, S.R.: Theoretical and experimental determinationof the electrical parameters of a superconducting a.c. generator'.Ph.D. Thesis, University of Leeds, 1980

11 Appendixes

11.1 Linear current density distribution

11.1.1 Notation: The total linear current-density distributionof a phaseband in terms of its axial and tangential linearcurrent density components is

K [/-,-, a,-, ± 1 ] = Ke [rh ah±l]ae+kz [rh ah ± 1 K(12)

while the linear current density distribution of a phase-group, with phasebands on different radii is

Kg[ru, at] = K[r,, ah + 1] + K[rh a, + TT, - 1] (13)

For a phasegroup centred at at + iT, the current flow isnormally reversed such that the linear current densitydistribution of a phase is

Kph[ru,«i] =Kt[rtJ,at] -K9[rUlcn + ii\ (14)

The phase current is given by

fphfl = Iphq sinq(wt—x)

x - — (i — 1), / = 1,. . . ,mm

and the total current per phaseband is

05)

b a(16)

A phaseband current distribution of magnitude K, spreado and centred at 0,- can be represented by a periodic pulsedsquare wave with Fourier series

/(«) = f°+ £ - s i n (52) cos „(» - A) (17)

Fig. 3 shows that a right-hand phaseband advances linearlywith 8 such that

<Pi = kz + a,-

Likewise for a left-hand phaseband

<t>i = —kz + a,-

(18)

(19)

Since the total machine length is 2/ over which a phasebandcircumscribes one pole pitch the helix periodicity is k = n/2l.

/ 7.7.2 Axial linear current-density distribution: If the phase-band subtends a phasespread o the circumferential distanceoccupied by the phaseband is ort and

1 z

-iphq

ort ab

(20)

Substituting in eqn. 17 and for/(0)

Ziphq= ±

luabfi

r(nab n =X kbn cos n(d — oci + kz)

(21)

The linear current density for the corresponding ith phase-group is, from eqn. 13, (with rt = rj)

140 IEE PROC, Vol. 127, Pt. C, No. 3, MAY 1980

ri> a;l = — ~ Z kbn cos «(0 — a,) cos nkz

n odd

ZZ,lnhn f̂,sin n(6 — a,) sin nkz

n = 2neven (22)

with the corresponding phase value

n = ln odd

c o s n{6 — a,) cos wfcz

(23)

/ / . 1.3 Tangential linear current density distribution: Tan-gential linear current density is again a pulse wave advancingin the z-direction for increasing 0, i.e. 0,- = ±kz + at, withwave period 41. The phasespread is Haiti implying

Z-n(24)

giving after substitution in eqn. 17 for the phaseband

llab n =Y.kbn cos nid-

(25)

and for a phasegroup

K r , _ ^p/»g

0 2/a6

ZiH—^-^ X kbn sin n (6 — oti) sin nkz

lab n = in odd

HZi

lab n = 2n even

cos « (0 — a,) cos nkz

(26)

and finally for a phase

^ [ a , / ; ] = — ^ L /rbnsin«(0 — a,-)sinnfczn odd (27)

/1.2 Solution of Laplace's equation

11.2.1 Winding in air: Laplace's equation in cylindricalco-ordinates is

i an i•I : 1—z

= 0 (28)

(29)

dr2 r dr r2 dd2 dz2

A solution of the formft = Fi{r)F2(6)F3(z)

can be obtained by the separation of variables such that

+ C2 cos (fid —vz + I / / 2 M)}

ju, v * 0 (30a)

(306)/i = ^ = 0

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

Introducing the boundary conditions of Section 11.3 gives(a) inside the winding

ln = AnIn(nkr)\ ]I;cos [n(

ln = BnKn(nkr)

(cos [n(6 — kz) + \p2n] rh . helix

i(d+kz)+\Jsln] l.h. helix

(31a)

n0 = A07z

(b) outside the winding

Icos [n(d —kz) + i//2n] r n -

cos [«(0 + kz) + i//ln] l.h. helix(32a)

ft0 = ^06^ (326)

before the introduction of the environmental screen.

11.2.2 Modification of the potential distribution by theenvironmental screen: The image winding produces apotential distribution

[cos [n(6 —kz) + \p2n] r-n- helixnn = DnIn(nkr)\

[cos [n(d +kz)+\pln] l.h. helix

(33)

which is incorporated in the potential distributions (eqns.31 and 32) as,

(a) inside the winding

(cos [n(d-kz) + \l/2n]

Icos [n(6 +kz) + \pln]

(346)

(6) outside the winding

(cos [n(0-«„ = [BnKn(nkr)+Dn Jn(nkr)]

[cos[n(d+kz)+ipln]

(35a)

£lo=Ao6d (356)

Dn is determined by the boundary condition (iv), Section11.3, and describes the type of environmental screen used.

11.2.3 Magnetic field distributions: The separate com-ponents are obtained from H= — \7fi to give:

(i) Radial fields

r < rs

(cos [n(6~kz)+ p2n]= -[An+Dn)l'n(nkr)\ (36)

Icos [n(d + k)+ l ]> r> rB

(d + kz)+

Hrn =

cos [n(d —kz)+ \J/2n]

cos [n(6 +kz)+ \pln]

(3D(ii) Tangential fields

r<r.

n

Hen = [An+Dn]In(nkr)-\ .

141

and for no environmental screen

n (sin [n(8 —kz)++ DnIn(nkr))-\

r

Heo — ~l 0 6

(39a)

(39b)

(iii) Axial fields

(sin [n(6 -kz)+ \J/2n]~[An+ Dn] In(nkr)nk\_ s i n [n

= ~Aon

r > rs

(-[DnIn(nkr)+BnKn(nkr)]nk\

where

* f

(40a)

(406)

sin [n(6-kz) +4/2n]

-sin [n(0+kz)+\pln]

(41)

dr

dr

= n*

(42a)

nkr I

(42ft)

Substitution of the boundary conditions of Section 11.3gives

(43a)

(43ft)

- Ziphq

2-nab

Mab

Bn[ri,och±l} =

(43c)

(43d)

nnab [Kn(nkri) - 7n(«fcri)/n

where

Kn(nkrx)

For an iron environmental screen

and

and for a conducting environmental screen

142

(43e)

(43/)

(44a)

(44ft)

(45a)

A, = 0 (45ft)

Substitution of these constants with the magnetic-fieldexpressions, eqns. 36 to 42, gives the magnetic field distri-butions for an individual phaseband listed in Table 1.Corresponding linear current-density distributions are alsoshown in this Table. Magnetic-field distributions for aphasegroup, phase and total machine phases can now beobtained by progressive use of superposition; the firststage, that of combining phasebands to form a phasegroupbeing shown in Table 3. Table 2 defines the environmentalscreen factors used in Tables 1 and 3.

11.3 Boundary conditions

(i) By comparison with the linear current densitydistributions of Section 11.1 for periodicity in 6 and z

H = n

the space harmonic number and

•nn» = n k = -

for a right-hand helixed winding

Ci = 0, C2 = 1

for a left-hand helixed winding

C, = 1, C2 = 0

(ii) As r -* 0, Kn(nkr) -*• °° while the magnetic fieldsare finite.

Bn = 0- for r < rt.

(iii) For a helical winding in air (no environmentalscreen) as r^-°°, In(nkr)-* °° while the magnetic fieldstend to zero. An = 0 for r > rt.

(iv)The boundary conditions at the environmentalscreen depend on the type of screen:

(a) No environmental screen. In this case no imagewinding is necessary and the previous boundary conditionsare sufficient.

(ft) Iron environmental screen. (jur = °°, p = °°). All theflux enters the iron core normally such that at r = rx,He=Hz=0.

(c) Conducting environmental screen (/ir = 1, p = 0).All the flux is excluded from the environmental screen suchthat atr = rx,Hr = 0 for all 6 and z.

(d) At the stator current sheet radial flux density iscontinuous and

^"outside ~ ^"inside = ^ z "

^"outside ~H™inside = ~K6n

11.4 Inductance calculations11.4.1 Phasegroup inductances: The total flux enteringthe Gaussian surface of Fig. 5 from a phasegroup of Z/ftturns centred at 6 = a is

+ 1 n-kz

(pgn(oc) = Uo j j -Hrgn[rs,oc]rsdddz-I kz

+ Ho J J H2gn[rs>cc]rd6dr (46)0 -n

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980

Evaluating this integral and replacing the first helical loopby a phasegroup of Z/b turns and breadth factor kbn

gives the mutual flux linkage between phasegroups as

H0Z2irr2 .

lab21 hh (47)

where k2n is defined as the skew factor, and

,2 _ 7n("fo"8)[frs/n-l-yn(nkr8)In(nkr8)]

(48)

fcrn is the radial flux environmental screen factor defined inTable 2, column 1. Mutual inductance is obtained from

giving2 2

2b2lH

(49)

(50)

2n0Z2k2

bnk2nkrn2l

= c o s

for all n

the total mutual inductance being the sum of the harmoniccomponents

Mg(ot) = Mg0(a) + (51)

/1.4.2 Phase inductance: Phasegroups may be interconnec-ted in a variety of ways to produce a phase, the most usualbeing:

(i) a phase and phasegroup are identical(ii) two phasegroups 180° phase displaced are connected

in parallel(iii) two phasegroups 180° phase displaced are connected

in series to give

r27,2Mphn{a) =

n odd

H0Z2k2

bnk2nkrn2l

2nna2 cos na H

(52)

(53)

11.4.4 Subtransient inductance: The image winding repre-senting the outer-rotor screen introduces an additionalmagnetic scalar potential distribution of the form

n = FnKn{nkr)cos [n(d — kz) + i//2n] r.h. helix

cos [n(d + kz) + \}sln] l.h. helix

(56)

which combined with eqns. 34 and 35 gives(a) inside the winding

{[An+Dn]In(nkr) + Fn

cos [n(d-kz) + \p2n]

cos [n(6 +kz) + \pln]

(57a)

(57b)

(b) outside the winding

{DnIn(nkr)+ [Bn+Fn]Kn(nkr)}\(cos [n(6 —kz) + i//2n]

fto = A,

[cos [n(d+kz) + \pln]

(58a)

(586)

Introduction of the additional boundary condition at theoutside radius of the outer rotor modifies the scalar mag-netic potential and field distributions in such a way that theenvironmental screen factors kdn, kzn and krn of Table 2are modified to those of Table 4. Consequently, with aniron environmental screen the radial field outside the statorwinding is given by

Hrsn =

K'n{nkr8)

miab [Kn (nkrs) - yn (nkrs)In (nkrs)}

lninkrs) \ [ yn(nkrD)\

1 -yn(nkrD)

sin n(6 —a,- * kz)

(59)

/1.4.3 Synchronous inductance: For an w-phase generatorwith no saliency the synchronous inductance is

L8n = ~Mphn(0)

n = 2rm ± 1

r any integer

giving2/,2m\XQZ2k\nk

28nkrn2l

Anna2 H

(54)

(55)

For nonzero n there is no net axial flux entering the Gaussiansurface of Fig. 5 and so the phase inductances are entirelydue to radial magnetic fields and the subtransient inductanceis

k2wnkrn2l

. 7n(nkr8)1

yn(nkrD)

1 -yn(nkrD)

(60)

IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980 143

A.F. Anderson was born in Edinburgh,Scotland, in July 1939. He receiveda first class honours degree in Elec-trical Engineering and the Ph.D. degreefrom the University of St. Andrews in1962 and 1966, respectively.

From 1965 to 1967 he worked asan application engineer on rolling-mill drives with the Plant ApplicationEngineering Department AEI Ltd.,Rugby. From 1967 to 1970 he was an

NRDC Post Doctoral Fellow at the University of Dundeeworking on axially laminated reluctance motors. In 1970 he-joined the Electrical Engineering Department of Inter-national Research and Development Co. Ltd., where heworked on superconducting a.c. generators and on timingmotors. In 1971 he was appointed Group Leader of thesuperconducting a.c. machines Group.

In 1974 he joined the Electromagnetics Group of theElectrical Research Department of C.A. Parsons Ltd.,Newcastle, and in 1976 he was appointed Group Leader,with responsibilities for the instrumentation of largemachines and experimental investigation into their electro-magnetic behaviour under service conditions.

Dr. Anderson is a Member of the Institution of ElectricalEngineers.

J.R. Bumby was born in Thirsk,England, in April 1949. He received afirst class honours degree in EngineeringScience and the Ph.D. degree in Elec-trical Engineering from the Universityof Durham, in 1970 and 1974, respec-tively.

He joined the Electrical EngineeringDepartment of International Research& Development Co. Ltd., Newcastle-upon-Tyne, in 1973 where he worked

on superconducting a.c. generators,- hybrid electric vehiclesand the generation of electrical energy from sea-waveenergy. In 1975 he was made responsible for the electricaldesign of superconducting a.c. generators and in 1977 hewas appointed Group Leader of the Special ElectricalProjects.

In 1978 he joined the Department of EngineeringScience, University of Durham as a lecturer in ElectricalEngineering. Dr. Bumby's current research interests aresuperconducting machines and power system stability andcontrol.

Dr. Bumby is an Associate Member of the Institutionof Electrical Engineers.

B.I. Hassall graduated in 1975 fromNottingham University with firstclass honours in Electrical and Elec-tronic Engineering. He was thenappointed Research Officer at Inter-national Research and DevelopmentCo. Ltd., Newcastle-upon-Tyne towork on superconducting a.c. gener-ators. In January 1978 he became aResearch Fellow at Leeds Universityunder part of the SRC's programme

investigating a.c. generators.Mr. Hassall is an Associate Member of the Institution

of Electrical Engineers.

144 IEEPROC, Vol. 127, Pt. C, No. 3, MAY 1980