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AN ANALYSIS OF ROTOR DOUBLE POINT GROUND FAULTSSHI SHIWEN aa Department of Electrical Engineering, Southeast University, Nanjing, ChinaPublished online: 07 May 2007.

To cite this article: SHI SHIWEN (1998) AN ANALYSIS OF ROTOR DOUBLE POINT GROUND FAULTS, Electric Machines & PowerSystems, 26:2, 141-154, DOI: 10.1080/07313569808955813

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AN ANALYSIS OF ROTOR DOUBLE POINT GROUND FAULTS

SHISHIWEN

DepanfnentofElectncalEngineenngSoutheast UniversityNanjing, China

ABSTRACT

This paper analyses the second harmonic component which appears in stator emf

during rotor double point ground faults or interturn faults. The second harmon­

ic component can be used to realize rotor double point ground protection for

large generators. Starting from basic physical concepts, the paper discusses the

rotor winding mmf and gap flux density distribution during normal and fault

conditions as well as corresponding stator harmonic emf. A method of "calcula­

tive flux density curve" is proposed, so that analysis process is simplified and

more convenient.

1. INTRODUCTION

With the increase of capacity of synchronous generators more perfect protective

relaying of rotor against double point ground faults or interturn faults is re­

quired. A novel type of rotor protection based on response to second harmonic

component of stator voltage is proposed. This article, starting from basic physi­

cal concepts, analyses the distribution of rotor winding mmf and flux density at

normal case, the distortion of gap flux during rotor double point ground or in­

terturn faults and the corresponding second harmonic component or stator in­

duced emf. A method of /I calculative flux density curve" is proposed so that the

analysis is simplified and more convenient.

2. ROTOR FLUX DENSITY DISTRIBUTION

On salient pole synchronous generator distribution curves of magnetic flux densi­

ty and that of mmf can be regarded to be similar to each other. Mmf distribu­

tion curve of a pair of poles is a trapezoid, it is the summation of all slot

windings' mmf distribution curves, which are rectangulars, Flux density

(rnmf) distribution curve of a pole for two-pole machine is shown in Fig. 1. It is

not symmetrical. Resultant distribution curve of flux density of two poles is 8

trapezoid 8S shown in Fig. 2.

Request reprints from Shi Shiwen. Manuscript received in final form September 5, 1996.

Electric Machines and Power Systems, 26:141-154, 1998Copyright e 1998 Taylor & Francis0731-356X/98 $12.00 + .00

141

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142 S. SHIWEN

Then we analyse the geometrical relations of flux density distribution curve of

one pole. According to the principle of equality of flux - in and flux-out, areas

of positive and negative parts are equal, and from Fig. 1 we get

1 1 I-mZ-B' (m+m+Za)It=Z-B"[Z+(-Z--a) XZ]1t

that is:

d' 9' d 9 d l

I I0000005

rjo.JOOooo

I I

(A) I I. I I

I :ml "I· f-m'l I

I -z-, II. I I

I• I I

I , I I I

I I I

I I

I II

(8)

9/-

'... 7T -1------'J

II

I

(D)

(e)

d'FIG. 1

ZB' (m+a)=B"(3-Za-m)

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ROTOR DOUBLE POINT GROUND FAULTS

again from Fig. 1, it is obvious that

B' B"-=

143

(4)

arc I-m-2--a

Solving (1) and (2), we get

a=1.(m'-4m+3)8

From Fig. 2 we have B=B'+B". Resolving the resultant flux density curve byFourier series method[l] we get amplitude of fundamental component as B1

B 4B. Q

1 = {3rc sm"

WhereI-m

{3=--rc2

9 d el'II

IIIII

," ~8' 1~'-----LII

FIG. 2

Hence amplitude of fundamental comonent of flux density curve of one pole is

1 B 2B. Q ( )2" 1 = {3rc sm" 6

Substituting B=B' +B" in (2), we get

B'=~B (7)I-m

Substituting (4) in (7) we get

B'-~~B-1-m4sin{3 1

Substituting (5) in above equation, we getarc'

B'=-4'QB1 (8)smp

Substituting B= B' + B" in (7), we get

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144 S. SHIWEN

B"=(l-~)B!-m (9)

3. CALCULATIVE MAGNETIVE FLUX DENSITY CURVE

Flux density distribution of one coil is shown in Fig. 3 (B), in which positiveflux density is b' and negative, btl. If there are W turns of winding on a pole,by superposing we get B'=Wb' (see Fig. I). Let b'+b"=b, we get B=Wb.Shifting the flux density distribution curve upward by a displacement b", we geta distribution curve as shown in Fig. 3 (C) which is defined as "calculative fluxdensity curve". It is valid for calculating second and other even harmonics ofmagnetic flux.Adding the" calculative flux density curves" of W turn coils of one pole altogeth­er we get a distribution curve which is identical with the resultant rotor flux dis­tribution under this pole at normal case.

ciII

CJ'II

d'II

9 d'

I 1_-(j)-- -0-

" W'i,,:lS ~IN slotsI If

'=:::====~;::;b'=I ~==:::::::J! ih'I II II____----'D~b _

(B)

(C)

(A)

FIG. 3

cJ'I

II

FIG. 4

Generally there are several coils in a rotor slot. The analysis method is same as

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ROTOR DOUBLE POINT GROUND FAULTS 145

(0)

above by regarding them as a concentrated winding and changing W to ~,

where N is the nember of slots of one pole. Flux density distribution and its cal­

culative curve of a winding in a pair of slots are shown in Fig. 4. Let its positive

and negative height be b', and b", respectively, thus the height of calculative

flux density curve is b,= b', + bIt,. Below we shall prove that the height of calcu­

lative flux density curve of every pair of slot winding are all equal.Let coil width be ylt and permeability of iron core path is regarded to be infinityp.,=OO, therefore slot winding mmf drops in two air gaps at ylt and (2-y)lt

parts. Owing to neglecting saturation, flux density is proportional to mmf andboth have the similar distribution curve. According to principle of equality offlux-in and flux-out of rotor, we have

yb'.= (2-y)b".

hence

b', 2-yb",=-y-

and mmf's also have the same relation, that is

F' 2-yF"=-y-

Because F'+F"=F, we get

mmf for air gap along width y equals F' =F 2~y

mmf for air gap along width 2-y equals F"=F ~

b' _IloF,_IloF 2-y,- S - S 2

b",= ~F"=~F ~

b.= b'.+b".= ~F

hence b, is independent of y.

4. FLUX ANALYSIS OF SLOT WINDING DURING TWO POINT GROUND

OR INTERTURN FAULTS

First, short circuit of one slot winding of a salient two pole-machine is consid­

ered. By principle of superposition the resultant flux density curve equals normal

flux density curve superposed by the reversed flux density curve of this slot

winding. Obviously the second harmonic component is produced by latter part.

For analysing second harmonic component we can use • calculative flux density

curve". Fig. 5 shows the construction of rotor slots and teeth.where m-width of large teeth in fraction value of a pole pitch.

N-number of pairs of slot on one pair of poles.

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146

FIG. 5

S. SHIWEN

(2)

(See (5»

n-slot number, enumerated from axis qq' .We regard that widths of slot and small tooth are equal. There are (2N-1) slotwidths in the area 1-m, Suppose that n - th slot winding is short circuited,theny 1 1-m2=2- 2N-1 (2n-I)

y=1_ 20 - m) (2n-I) OI)2n-1

Gap flux density curve is shown in Fig. 6. In Fig. 6. the calculative flux densitycurve of short circuited slot winding is also shown. Its height b.:

B 2b'=N/2=N B

which is superposed on the normal flux density curve. By using Fourier series

analysis to calculative flux density curve, the second harmonic component is ob­tained. the amplitude value of which is2B 1.!if' -;rsmyn

and second harmonic emf E, will be induced in the stator winding2B 1 .

E K -N-smyn, ., -O..:---,n.;,;-__E, =K" B,

where E1--emf of fundamental frequency at normal caseK" ,K'l--winding factor of second harmonic and fundamental compo­nents.

By using Eq. (4). we get2B 1 .--smyn

E, K, N nE, = K" -c-4-=B,....-. -(:l­

-SlOt-'

~'"1-m

where ~=-2-n

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ROTOR DOUBLE POINT GROUND FAULT5 147

The superposed flux cure also offers fundamental component, which is2B 2 . y1rN-;;sm "2For 50 MW typical turbogenerator , N=14, m=O. 3, K'I=O. 88, K,,=O. 485.

1-m~=-2-1r=0.351r

henceB = 4Bsin~= 4BsinO. 351r

, ~lt O. 351t'thus

21.E, K" N-;;smY1rE, =K" 1.03

n=l, y=O. 95, s : 180'=171', and i:=o. 38%

For different values of n , calculated results are shown on the table 1.

TABLE 1

n y y • 180' i: X 1OO( % )

1 0.95 1710 0.382 0.845 152' 1. 143 0.74 133' 1. 784 0.636 114.5' 2.25 0.532 96' 2.46 0.43 77. 5' 2.377 O. 325 58. 7' 2.07

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148 S. SHIWEN

When a pole winding is short circuited entirely. adding the data on the 4th col­

umn of the table we get ~: = 12.34%.

Then we analyse short circuit conditions of winding of a pair slot of a salient 4­pole machine. Normal flux density curve and superposed Wcalculative flux densi­

ty curve" of slot winding are shown in Fig. 7(B). in which abscissa is in electricdegrees X.

Superposed reversed flux density curve is not sinusoidal. but it is periodical. the

period of which is 41t in electric degrees. For convenience of Fourier series analy­

sis, the abscissa is changed to mechanical degrees Xm • and the period of super­

posed curve is changed to 21t. As shown in Fig. 8. coil width of faulty winding

is Y21t

in mechanical degrees. The meaning of y is same as in 2-pole machine.

Fourier series analysis is discussed as below.Period of fundamental flux density curve is 21t in mechanical angle degrees. ob­

viously this flux will induce a 25Hz emf in stator winding.In China. fundamental frequency is 50Hz.

Period of 4 - th harmonic flux density curve is ; in mechanical angle degrees

(it corresponds to 1t in electric degrees). obviously this flux will induce a 100Hzemf in stator winding.By Fourier series analysis C1 ] we get

Amplitude of fundamental flux density= b, ; sin Y41t

= ~, ; sin Y41t

A I, d f h fl d . b 2 1. 2B 2 1.mp itu eo 4-t ux enslty= 'x3smy1t= /if ' x . ,smy1t

hence2B 2, ylt-. -sIn-

E,!, K,,!, N It 4~= K" B,

where EI/,-25Hz emfK,,!,-winding factor of 25Hz emf2B 2.- • -smylt

E, K" Nlt 4-=E, K" B,

The above equations show that E, is half of that for 2 - pole machine. that is

right in physical sense.

5. FLUX ANALYSIS DURING A POLE WINDING SHORT CIRCUITED

First. consider a salient 2-pole machine. When the winding on a pole is shortcircuited entirely. gap flux density distribution is equivalent to that normal fluxdensity curve is superposed by a reversed flux density curve of the faulty pole

winding. The latter produces second harmonic flux. Gap flux density distribu-

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ROTOR DOUBLE POINT GROUND FAULTS 149

d d cd d d9 ~ q <j

(A) J I I I I-.21T -l! -TT -

1T1 0 TT,i 11' is ~X

2 2.

ci

(C)

cf__

FIG. 7

d'

d d cI d d-, 'i 9 I bs q q, I ,; I I

-1T 77 "TTo I 0TT --x....2. I yrr I T

2-

FIG: 8

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150 S. SHIWEN

(14)

(6)

05)

tion curve is shown in Fig. 9. Superposed flux density curve in Fig. 9 is plottedby its calculative flux density curve. It has been pointed out in § 3 the calcula­tive flux density curve of a pole winding is identical with the resultant rotor flux

distribution under this pole at normal case (here in Fig. 9, it is reversed in direc­

tion. )

Resolving the calculative flux density curve into Fourier series

bCx)=80+ B',sinx+B',sin2x+ .. ·+B".cosx+B",cos2x+"·

There are no sine terms because of symmetry to ordinate (d -axis), that is B'.

=B',="'=O

2 I'B", =- b(x)cos2xdxIt •

It... . --x=~ IT Bcos2xdx+~ IT B 2 cos2xdx

It • It T ~O-m)2

B 1o ) 0 -l-cosmn)It -m It

(refer to appendix 1)

therefore1

E K ( )-, 0 -i-cosrnn )

, "l-m It

E, =K,. 1. 03

put m=O. 3

O_~)lt,O+cosmlt)= O. ~lt'[l+cos(O. 3X180')]=0. 229

hence ~:= 00.4:: .°i.2; ; =0.1225=12.25%

it is identical with the result of previous paragraph. which verifies that the

method used is correct.Next for salient 4 pole machine, analysis method is same as for 2 - pole ma­chine. Fig. 10 shows flux density curves of individual pole windings and resul­

tant curve.Positive height B' and negative height B" of flux density curve of a pole winding

can be obtained as follows1 1 1-m2B' (m-l-m-i-Zc) =2B-[3+3+2(-2- -a)]

I. e.2B' (m+a)=B"(7-m-2a)

From geometrical relation of Fig. 10. we haveB' B"a-=l-m

---a2 .

Solving OS) and (6) simultaneously we get B' and B".It is obvious from Fig. 10 that B= B' + B" where B is height of resultant flux

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ROTOR DOUBLE POINT GROUND FAULTS 151

g'

I

II J

d' d

III

I

I

I

I~L--~f---_l__+;;__=-~_;__-....>...::::_-rrl X

d

jf-------;;;;;;;;;;;;;;;;;;;j;;;;;;;;;;;;---+--------~iiiiiiiiOiiiiim'lIIII

FIG. 9

density curve (trapezoid). In Fig. 10(F), reversed calculative flux density curve

of a pole winding is shown by dotted lines, which is superposed to the normal

resultant flux density curve and produces 2-nd harmonic component. Abscissais in electric degrees.

For convenience of harmonic analysis we change the abscissa into mechanical de­grees xm. Calculative flux density curve is plotted alone in Fig. 11.

Amplitude of fundamental flux density curve=1. f" b(xm) cosxsdx;7r •

2 fT 2 fT B 4xm= - Bcosxgdx.,+- -1--0- - )cosxmdxm1C 0 1t' T -m 1C

2 B 4 mn 1=----(cos ----) (7)7rl-m7r 4.f2

it induces 25Hz emf in stator winding (see Appendix 2).

Amplitude of 4th harmonic flux density curve = 1. f" b(xm>Cos4xmdxm7r •

2 fT 2 fT B 1- 4xm=- cos4xmdxm+- -1--(--->Cos4xmdxm1t' 0 1t' T -m xB 1=-, • -1-0+cosm7r) (8)

27r -mit is half of that in (4), that is right in physical sense.

SUMMARY

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152 S. SHIWEN

x-

d

II

/Ita'~I

I

II

I___I

\\E/~

III

9

d

I

d

d

IIII

I

IIIIl------.J"r--=------,l~rI_*__*:____;:;;_r__j_~-27T1

II

1<1

(E)

(Bl

( D)

ee)

FIG. 10

-TT

11-;\--,----r--1c-/-!~ : \ i

O!:l!l!l TI/.2.-'%! ~. 4 hTT -- Xm

FIG. 11

1. During the rotor double point ground or interturn faults on a generator sym­metry of flux density curve is distorted and second harmonic component appears

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ROTOR DOUBLE POINT GROUND FAULTS 153

in it. On this condition 2-nd harmonic emf is induced in stator winding. The 2

- nd harmonic comonent of stator voltage is used to realize rotor double pointground or interturn fault protecton of large generators[J··J.2. On salient 2- pole machine when one pole winding is short circuited entirely.2-nd harmonic stator emf may be up to nearly 13 percent of fundamental emf

at normal case. It depends on the width of large tooth of rotor. If only one coilin a pair of slots is short circuited. it may be lower than one percent.3. On salient 4 - pole machine when a coil of a pair of slots or a pole is short cir­

cuited. 2 - nd harmonic stator emf is less than that in 2- pole machine by halfvalue.4. By using method of N calculative flux density curve", analysis of rotor diuble

point ground or interturn faults is simplified and more convenient.

APPENDIX 1

Derivation of Eq. (14)

B"I=1. f. b(x)cos2xdxJ1: •

In interval 0- ~J1:. bCx)=B (See Fig. 9)

In interval ~J1:_ ; • b varies with x I from Fig. 9 we have

b B

that is

hence

J1:"2-x

b(x)=B -J1:-=---"2 0 - m )

J1:-u' .. --x

B"I =1. IT Bcos2xdx+1. r B -2=-----cos2xdxJ1: • J1: T ~O-m)

221 m• 2 1 1 • 2 (-2) 1=n' "2B[sin2x]7+nB-l_-m' "2[sin2x]¥+nB O-m)J1:' 4"[cos2x

+2xsin2x]tI

= ~ • _l-O+cosmJ1:)"It I-m

APPENDIX 2

Derivation of Eq. (7)

Amplitude of fundamental flux density = 1. f. b(xm)cosxmdxmJ1: •

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154

mlf0-+7'mlf If--+-.4 4 •

b

b(xm)=B (see Fig. 11)

b varies with Xm I from Fig. 11. we have

BIf-O-m)4

S. SHIWEN

that is

b(xm)=-lB 0- 4xm)-m If

Hence amplitude of fundamental flux density = 1. IT Bccsxsdx..+1. IT -1B1C 0 1C T -m

0- 4:.. >Cosx..dx,

2 Bs' mlf+ 2 B (si If ,mlf) 2 B 4 [ +. J;'=- m - --- sm --SIO - ----- cosx X SIOX ••If 4 n:l-m 4 4 n:l-mlf ...... T

2 B 4 mlf 1=----(cos ----)lfl-mn: 4.,f2

REFERENCES

[IJ Netushil A. V.• Straxov S. V.• 1955. Principles of Electrical Engineer­

ing Vol. 11. Russian. Mossow.

[2J Shi Shiwen , Qiao Huanru and Zhou Lihua , 1980. Rotor Protection Re­

sponding to 2 - nd harmonic component of stator voltage - Selection of

1979 conference on relays and protection systems. Chinese Institute ofElectric Engineering. Bejing,

[3J Shi Shiwen , 1987. Protective Relaying for Large Generator-Transformer

Units, Hydroelectric Press. Bejing.

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