A PROTECTION BOHEME FOR ALTERNATOR FIELDS by James …

A PROTECTION BOHEME FOR ALTERNATOR FIELDS

by

James Torey Lancaster

Thosia submitted to the Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy for the degree of

MASTER OF SOIENOE

in

Electrical Enginoering

May, 1965

Blacksburg, Virginia

TABLE OF CONTENTS

CHAPTER

I.

II.

III.

IV.

v. VI.

VII.

VIII.

INTRODUCTION • • • • • •

THE REVIEW OF LITERATURE

THE INVESTIGATION ••••

• • • • • • • • • • • • • •

• • • • • • • • • • • • • •

• • • • • • • • • • • • • •

Object of the Investigation.

Method of Procedure •••••

Discussion of the circuit.

• • • • • • • • • • •

• • • • • • • • • • • •

. . . . • • • • • • •

Discussion of statistical analysis of data . . . Results. • • • • • • . . . . . . . . . . . . . . .

DISCUSSION OF RESULTS. • • • • • • • • • • • • • • •

CONCLUSIONS. • • • • • • • • • • • • • • • • • • • •

SUMMARY ••• • • • • • • • • • • • • • • • • • • • •

ACKNOWLEDGE:viENTS • • • • • • • • • • • • • • • • • •

BIBLIOGRAPHY • • • • • • • • • • • • • • • • • • ·• •

VITA • . • • • • • • • • • • • • • • • • • • • • • • •

APPENDIX • • • • • • • • • • • • • • • • • • • • • •

PAGE

5 6

7

7

7

7

1:;

24

28

29

:;o :;1 :;2 :;4

:;5

LIST OF FIGURES

FIGURE

l. SC I-ID;I!JATIO DIAGRAM OF RELAY CIRCUIT • • • • • . . . . . . . 2. BLOCK DIAGRAlt. OF TIMING TEST CIRCUIT ••• . . . . . . . .

PAGE

8

9

:,. SCHEMATIC DIAGRAM OF TRIGGER CIRCUIT •

4. TYPICAL DISPLAY OF OPERATING TIME •••

• • • • • • •••• 11

. . . . . • • • • • 12

5. REGRESSION LINE FOR TEN PER-OE.l~T UNBALANCED CONDITION. • • 18

6. REGRESSION LINE FOR FI FTEEl'i PER-CENT UNBALANCED CONDITION. 19

7. REGRESSION LI1"E FOR T\'lENTY PER-CENT UNBALANCED CONDITION • 20

8. PREDICTION LINES FOR TEN PER-CENT UNBALANCED CONDITION • • 25 9. PREDICTION LINES FOR FIFTEEN PER-CENT UNBALANCED CONDITION · 26

10. PREDICTION LINES FOR TWENTY ?ER-CENT UNBALANCED CONDITION. 27

LIST OF TABLES

TABLE PAGE

1. Observed Times of Operation For Ten Per-Cent Unbalanced Condition In Milliseconds. • • • • • • • • • • • • • • • 14

2.

:,.

4.

5.

Observed Times of Operation For Fifteen Per-Cent Unbalanced Condition In Milliseconds. • • • • • • • • • •

Observed Times of Operation For Twenty Per-Cent Unbalanced Condition In Milliseconds . • • • • • • • • • • • • • • •

Data For Calculation of Regression Line For Ten Per-Cent Unbalanced Condition. . • • • • • • • • • . • • . • • • . Data For Calculation of Regression Line For Fifteen Per-Cent Unbalanced Condition .• • • • • • • • • • • • • . • •

6. Data For Calculation of Regression Line For Twenty Per-

15

16

21

22

Cent Unbalanced Condition. • • • • • • • • • • • • • • • 2;

5

I INTRODUCTION

An unbalanced condition on a three-phaae power system can_·

generate a voltage ncroaa tne field of an alternator which 11 pro-

portional to the degree of unbalance and of twice the system fre-

quency. Thia voltage is due to negative-phaae-aequence currenta

in the stator winding of the machine during the unbalanced condition.

A voltage induced in this manner may reach a magnitude of such order

aa to exceed the insulation level of the field winding and cause

tho inaulation to break down thua damaging.the machine.

Exiating protective lilYstema which arc deaigned to protect

an alternator against auch a condition arc often too alow in their

operation to be of much value. Speed ia essential in auch an

operation, .both for the protection of the machine and for the stability

of the entire power aystem.

6

II THE REVIEW OF LITERATURE

While the literature pertaining to the application, construc-

tion, and operation of relays is quite extensive, the literature

pertaining to this particular aspect of protective relaying is

practically non-existent.

Several sources listed methods of determining negative-

sequence impedance, and Henderson(;) outlines a negative-phase-

sequence scheme of protection for an alternator. Several other

sources listed protection against loss of generator field, but this

is only remotely related to this discussion.

Mason (7) mentions protection against rotor overheating

because of unbalanced stator currents, but makes no mention of

protection against voltages generated in the field due to an un-

balance. This also applies to Westinghouse (2) (8), Kimbark (6), and Skrotzki (lo).

7

III THE INVESTIGATION

Object of~ Investigc.tion

This study was set up with the following objectives in minds

(1) To design and construct a very fast, dependable, solid state

relay capable of protecting a synchronous machine from the voltage

induced in the field winding due to negative-phase-sequence currents

caused by unbalanced conditions; (2) To find a method of detecting

such unbalanced conditions as swiftly as possible; and(;) To test

this protective scheme in the laboratory.

Method of Procedure

Discussion of the circuit. Several devices were designed,

constructed, and tested before one was found which seemed to give

consistently good results. The circuit of the device which was

used ia shown in figure 1. The test circuit that was used to obtain

data is shown in block-diagram form in figure 2.

Since a synchro generator is very similar to a synchronous

machine, it was decided that such a device could be used as a de-

tector. The synchro generator was operated as a synchronous motor

and the relay was connected to the field by means of a current

transformer. The current transformer was necessary because the

power sypply used to excite the field presented a very low impedance

to the 120 cycles per second frequency of the generated signal

voltage, which tended to short-circuit the signal voltage.

A trigger circuit-was designed and built to provide positive

AJI capacitor ratings are in microfarads.

INPUT

LEGEND: C1=5 C2.C3=100 C4.C5=3 C6.C 7=0.001 C8=1

+22v

R1,R 2.R7 = 10 k R3,R10,R12 = 3,3 k R4= 6.8k R5= 22k R6= 5 k

R9:330 0=1N673 R13=1k R8= 1 k pot R11= 2 k pot

Figure 1

SCHEMATIC DIAGRAM OF RELAY CIRCUIT

O>

70 v 3(1) N:. Source FIELD DC Source

FAULT

TRIGGER ELAY LOAD OSCILLOSCOPE

\()

Figure 2

BLOCK DIAGRAM OF TIMING TEST CIRCUIT

10

triggering of the oscilloscope sweep. The oscilloscope was trigger-

ed by the unbalance, and the vertical input to the oscilloscope was

the voltage across the relay load. Thus the difference in time be-

tween the appearance of the unbalance and the relay pick-up was

displayed on the oscilloscope. A diagram of the trigger circuit

is shown in figure;. A picture of a typical display is shown in

figure 4. A Tektronix type 564 Storage Oscilloocope was used.

Existing and readily available equipment and supplies were

used in this study. It should be pointed out that the synchro-

generator used as a detector was not designed for this type of ser-

vice, and that much more consiatent results could have been obtained

along with faster operating times if a machine had been available

whicµ was more auited to the taak. Also, such a machine could have

resulted in the elimination of the amplification stages of the relay

circuit. Both the relay and trigger circuits had to be designed

according to the following criteriaa

l. The circuit must reapond to a fault which was either

positive or negative going at the instant of starting.

2. The circuit must be extremely fast in its operation,

preferably on the order of micro-seconds.

;. The circuit must be extremely stable throughout its

operating temperature range.

4. The operating temperature range must be broad.

5. The circuit must be rugged and reliable.

11

A-C INPUT

___ ...:;~----~---e--o+30VDC

R1

LEGEND: C=1 microfarad R1=1 k pot R2=1 k

C

R3= 130 R4= 47

D= 1N673 UT= 2N489A

Figure 3

UT

SCHEMATIC DIAGRAM OF TRIGGER CIRCUIT

PULSE OUTPUT

0

12

-.... _,_ -_ ... _,_

-_,_ _,_

----...

I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I I I I I I

I I I I I I I I I I I ' I I I I I I I I I I I I I I I I I I ------

-

----... _,_ ... _,_

5 10 15 20 25 TIME IN MILLISECONDS

Copy of oscilloscope trace of voltage ncroas relay load. Sweep= 0.5 milliseconds/division. 100 volts across load. 15% unbalanced condition.

Figure 4

TYPICAL DISPLAY OF OPERATING TIME

The circuits shown accomplish these conditions, being temperature

compensated over a range of -4o0 o to 100°0 (9). An additional test was performed to determine whether the

relay was too zensitive. This test consisted of operating a syn-

chronous motor connected to a D. 0. generator which served 1u a load.

The relay was placed in parallel with the motor. Load was suddenly

removed or added, and the operation of the relay was observed.

Discussion_£! statistical analysis_£! data. It was decided

to prepare a atatistical study of the data obtained from the pro-

cedure given above, in order to predict the maximum operating times

of the device. These data are given in tables 1, 2, and, for 10%,

15%, and 20% unbalance, respectively._ Attempta were made to obtain

data for 5% and 25% unbalances, but difficulties were .encountered.

The 5% unbalance resulted in erratic behavior due to the very poor

wave form present in the laboratory three-phase supply. This wave

form also created some inconsistencies in the data used, notably

in the 10% unbalance. The 15% and 20% unbalances were large enough

to compensate £or the poor wave form. The 25% unbalance resulted

in the machine losing synchronism. It also resulted in very fast

operating times, on the order of one millisecond or less for the

majority of operation&. It may safely be said that a 25% unbalance

will result in operating times that are fast enough to eliminate

all relay time for practical purposes of system stability.

It was found that the data used fitted a linear regression

14

Table l

Observed Times of Operation For Ten Per-Cent Unbalanced Condition

In Milliseconds

0.15 ;.oo 0.95 0.05 0.05 0.05 1.10 1.15 1.40 1.35 0.05 0.05 0.05 4.20 0.05 0.05 0.05 1.30 1.70 0.05 0.10 2.65 0.05 1.20 1.20 0.05 2.10 2.90 1.20 0.05 0.05 1.20 1.15 0.20 0.05 1.45 0.05 0.05 5.10 4.30 0.05 0.05 0.75 5.30 0.15 0.05 1.45 0.05 4.10 0.10 0.05 1.10 0.05 1.60 4.25 0.05 2.90 1.05 0.90 0.95 0.05 0.05 0.05 0.05 1.00 0.10 0.20 0.10 1.25 1.35 3.30 0.05 0.10 0.05 0.10 0.05 0.10 0.10 1.15 1.15 3.15 3~45 3.80 1.25 0.10 0.05 0.15 2.40 1.25 3.05 0.15 1.00 1.90 0.05 0.90 0.05 0.05 0.10 1.60 0.55 0.15 0.05 0.20 0.10 0.,:05 0.10 1.00 4.60 ;.10 0.05 o.ao 0.05 0.05 1.;o :t.15 0.05 0.10 2.;o 0.95 0.05 0.05 0.20 0.05 0.05

I 1.05

15

Table 2

Observed Times of Operation For Fifteen Per-Cent Unbalanced Condition

In· Mi 11 is econds

4.oo 0.01 ,.15 ;.45 ,.so 0.25 0.15 0.65 0.75 0.85 2.,5 1.15 0.01 1.95 0.85 2.70 1.15 0.95 ;.60 0.95 1.35 0.01 1.75 0.80 0.01 0.65 2.65 0.70 2.00 o.80 0.05 1.15 1.50 0.85 ,.80 1.,5 1.20 3.60 3.55 3.75 2.45 0.10 0.75 0.75 1.55 0.75 1.75 2.25 0.75 0.80 1.50 0.15 0.75 3.30 1.50 1.05 0.01 1.05 1.95 4.10 0.95 1.05 1.15 2.65 3.75 0.95 0.01 2.65 0.95 0.75 4.10 2.15 1.05 0.01 0.15 1.00 0.05 ,.80 4.95 4.75 0.85 0.05 0.01 2.05 3.15 0.10 0.05 5.15 0.75 0.90 0.15 0.10 o.80 0.95 2.70 1.00 0.25 3.60 0.70 0.85 0.95 0.70 2.50 0.85 2.00 2.25 0.01 3.53 1.30 1.20 o.80 0.75 5.25 ,.40 3.90 4.10 0.65 1.90 1.45 1.15 0.01 0.70 0.15 4.35 1.70 1.60 0.95 2.35 2.50 3.80 0.95 2.00 4.80 ,.25 1.10 1.10 ;.25 1.20 1.55 ;.20 2.00 1.95 o.80 0.80 0.95 0.75 1.00 1.25 3.65 ;.15

16

Table;

Observed Times of Operation For Twenty Per-Cent Unbalanced Condition

In Milliseconds

1.;o 1.25 0.05 0.10 o.40 0.15 1.15 1.00 0.65 0.60 1.05 0.05 1.00 0.75 0.70 0.75 1.10 5.00 0.05 1.;o 1.75 0.10 0.75 1.45 ;.55 0.15 ;.55 1.80 1.25 1.15 0.60 1.60 4.50 0.20 2.45 1.;o 0.25 0.10 1.65 0.50 2.65 0.65 0.10 0.05 o.;5 0.95 4~40 0.85 ;.05 0.50 0.75 0.10 1.60 ;.65 2.90 2.40 ;.60 ;.15 0.05 4.45 2.90 ;.55 1.00 2.40 0.05 0.05 0.80 0.85 2.20 0.20 1.10 0.05 ;.20 ;.20 1.20 0.10 0.15 5.,;o 0.85 0.05 ;.40 1.;o o.;5 0.05 2.60 1.50 2.50 0.05 o.;o o.;o 0.80 4.80 2.40 1.85 0.15 0.10 0.05 1.;o 1.05 0.05 1.05 1.10 0.10 0.75 0.05 2.;o 1.15 0.15 ;.10 0.85 1.15 2.80 o.;5 :;.;o 0.20 0.20 0.55 0.85 1.;5 0.10 0.85 0.05 0.60 2.80 0.10

17

in each case, according to the theory presented by Spiegel (11).

The plots of these regression lines are shown in figures 5, 6, and 7. These regression lines were obtained from data given in tables 4, 5, and 6 in the following manner:

X=g N

- IY Y=N

where N = number of operations observed.

According to Hoel (4) a regression line is given by:

Y'-Y= bCx-x> , where b=I:(xi-x) Yj

I(x. -x) 2 I

The chi-square test was used t9 determine the goodness of

fit for each of tho three unbalanced conditions used •. According to

Spiegel (11): 2 k (o- - e-) 2

X=r J 1 j=1 ej

The f'it was determined to be good by comparison Qf x 2 f'or the do.ta.

with X 2 as given in tables for six degrees of' freedom.

The standard error of' estimate, Sx~ , and the coefficient of

correlation, r, were determined f'or each case by use of' the equation

from Spiegel (11):

where tho negative sign indicates negative linear correlation.

100

80 Cf) z 0 I-

60 w Q._ 0 LL 040 n:: w d) 2 :) z 20

18

"'-.

' ......

" " "'-~I" r--....

1 2 3 4 5 TIME 1N MILLISECONDS

Figure 5 REGRESSION LINE FOR TEN PER-CENT UNBALANCED CONDITION

50

40 Cf) z 0 I-

30 w 0.. 0 LL 0 20 0:: w co 2 :) 2 10

"

19

"' ' "' '"" '""-

I~ 1 2 3 4 5

TIME IN MILLISECONDS

Figure 6 REGRESSION LINE FOR FIFTEEN PER-CENT UNBALANCED CONDITION

100

80

(/) z 060 I-<( 0:: w 0... 040 LL 0 0:: w ~20 :::, z

I~

20

I~ I"' ~. "'. II,.

"'· "'~ "' "' 1 2 3 4


Figure 7 REGRESSION LINE FOR TWENTY PER-CENT UNBALANCED CONDITION

5

21

Table 4

Data For Calculation of Regression Line For

Ten Per-Cent Unbalanced Condition

Time (X) Operations (Y)

X-X (X - X)Y (X - X)2

0.80 70 -2.40 .. .:.168.00 5.76 1.60 ;; -1.60 - 52.80 2.56 2.40 5 -0.80 - 4.oo o.64 ;.20 7 o.oo o.oo o.oo 4.oo ; o.80 2.40 o.64 4.80 5 1.60 8.00 2.56 5.60 2 2.40 4.80 5.76

22.40 125 o.oo -209.60 17.92

22

Table 5


Fifteen Per-Cent Unbalanced Condition

Time Operations x-x (X - X)Y (X - X)2 (X) (Y) o.ao 48 -2.40 -115.20 5.76 1.60 .',9-5 -1.60 - 72.00 2.56 2.40 17 -0.80 - 1;.60 o.64 ;.20 12 o.oo o.oo o.oo 4.00 19 o.80 15.20 o.64 4.80 6 1.60 9.60 2.56 5.60 ; 2.40 7.20 5.76

22.40 150 o.oo -168.80 17.92

Table 6


Twenty Per-Cent Unbalanced Condition

Time Operations X-X (X - X)Y (X - x,2 (X) (Y)

0.80 57 -2.40 -1;6.80 5.76 1.60 ;; -1.60 -217.80 2.56 2.40 9 -0.80 - 7.20 o.64 ;.20 1; o.oo o.oo o.oo 4.oo 7 0.80 5.60 o.64 4.80 4 1.60 6.40 2.56 5.60 2 2.40 4.80 5.76

22.40 125 o.oo -;45.00 17.92

24

Figures 8, 9, and 10 were then constructed to form time pre-

dictions on the basis of the observed data and the statement:

••• if we construct lines parallel to the regression line of Yon X at respective vertical distances a x' 2s , and 3Gy x from it, we should find, if N is largeyenough;•trui.t there would be included between these lines about 68%, 95%, and 99-7% of the sample points. (11).

The same reasoning holds true for horizontal distances plotted for

The actual calculations involved in this study will be found

in the appendix.

Results

The actual results are given in graphical form in figures 8,

9, and 10.

The load-changing test on the synchronous motor resulted in

a slight decrease in the sensitivity control, but the relay was still

sensitive to unbalance.

100

~80 0

t5 60 Q_

0 LL 040 0:: w aJ 220 :::> z

,~ "i

25

G' ~~--

'¾ I

2 4 . 6 8 10 TIME l N MILLISECONDS

No Predictions Possible.

Figure 8

PREDICTION LINES FOR TEN PER-CENT UNBALANCED CONDITION

12

100

(/)

Z 80 0 I-<(

f5 60 0... 0 LL 0 40 cc w (l)

2 20 :) z

26

', ' ' ' ' '~

" ' " "'' ' '" ~, '" '" ' ' " > ...

... 0

... ,~ ' ' ' ' ' ' ' ' 2 4 .6 8 10 12


Figure 9

PREDICTION LINES FOR FIFI'EEN ?ER-CENT UNBALANCED CONDITION

100

(./') z 80 0 I-<(

5 60 0... 0 LL 0 40 er: w m

20 :::::> z

\ \ \

I--

27

\ \ \ \ ' \ \ \ \

\ \ \ \\ \ \ , t,

1,, -;; 0

'f \ \ \ \ \ \ \ \

\ \ \ \ ' \ ' \

2 4 · 6 8 10 12 TIME IN MILLISECONDS

Figure 10

PREDICTION LINES FOR TWENTY PER-CENT UNBALANCED CONDITION

28

IV DISCUSSION OF RESULTS

It was shown that a more pronounced unbalance resulted in

shorter operating times, and tho.ta definite negative correlation

exists between time and number of operations. Statistical predic-

tions were made, which indicated that 99.7% of all operations would

occur below 10.94 milliseconds for 15% unbalance and below 8.51

milliseconds for 20% unbalance, which is a distinct improvement over

existing relays.

It was further determined that the relay could be desensitized

slightly, and that it would not be sensitive to sudden load changes

within the·normal range of loads.for the machine, but that it re-

mained sensitive to unbalanced conditions at its terminals. The

design is such that it will allow the operating voltage to be

changed, and its sensitivity to be controlled.

29

V CONCLUSIONS

The detection circuit used in this study could be vastly im-

proved by the design of a machine specifically for this purpose.

It is the autr~r•s belief that this method of detection and relay

circuit show promise of both increased speed and sensitivity over

existing types. Also, the relay circuit is quite rugged and reliable,

with no moving parts. The detection devico has the disadvantage

of being a rotating machine, but there is no reason why a machine

designed for this purpose could not be made highly reliable.

It is also the author's belief that consistent results could

have been·obtained for all conditions of unbalance from very small

up to the point where loss of synchronism occurred if the laboratory

wave form had been pure sine waves of the type normally encountered.

It has been shown by Thompson (12) that wave form distortion also

affects the operation of conventional relays.

;o

VI SUMMARY

Voltages appearing across the field of an alternator due to

negative-phase-sequence currents in the stator may be of such magni-

tude as to cause insulation failure in the field. A fast method of

detecting and relaying such a condition was devised.

Readings were taken of operating times for various conditions

of unbalance. Statistical time predictions were made for various

percentages of total operations.

It was concluded that the scheme offers the possibility of

faster operating times than existing methods, with no loss of de-

pendability. Also, the relay portion of the study offers increased

simplicity and ruggedness over existing types.

VII ACKNOWLEDGEMENTS

The author wishes to express his appreciation to Professor

George R. Powley for his guidance and encourQgement; to Professor

George C. Barnes, Jr. and Dr. Mansell H. Hopkins for their interest

and suggestions; to Professor Ralph R. Wright, Head of the Electrical

Engineering Department, for making available facilities in the

Electrical Engineering Laboratories; and to Dr. Albert L. Duke for

the loan of special equipment.

The author also wishes to express his appreciation to Mr.

Rudolph P. Hensley, M.r. Fred Bower, and Mr. Marvin Surface for their

assistance·.

Finally, the author wishes to express his gratitude to his

wife, Patricia P. Lancaster, for her inspiration, cooperation,

patience, help, and typing.

1.

2.

4.

VIII BIBLIOGRAPHY

Clarke, Edith. New York:

Circuit Analysis :lf_!:::::2._ Power Systems, Volume II. John \·iiley and Son:s, 19.50. Pp. 328-362.

Electric Transmission and Distribution Reference Book. Fourth Edition. East Pittsburgh, Pennsylvania: Westinghouse Electric Corporation, 1950. Pp. 349-352.

Henderson, John. Automatic Protective Gear for!:£ Supply S¾stems. London: Sir Isaac Pitman and Sons, 1934. Pp. l 1-143.

Hoel, Paul G. Edition. 128.

Introduction to Mathematical Statistics. New York: JolmWiley and Sons, 1954. Pp.

Second 127-

5. Hopkins, Mansell H., Jr. 11A Method of ~1easuring Negative-Phase-Sequence Currents in a. Three-Phase System." Unpublished Master's Thesis, Virginia Polytechnic Institute, Blacksburg, 1958. Pp. 26-28.

6. Kimbark, Edward Wilson. Poi1er Svstem Stability, Volume III. New York: John Wiley and Sons, 1956. Pp. 220-226.

7. Mason, O. Russell. The Art and Science of Protective Relaying. New York: John \•/iley and Sons, 195b. Pp. 221-225.

8. Silent Sentinels. Newark, New Jersey: Westinghouse Electric Corporation, 1949. Pp. 42-43.

Silicon Controlled Rectifier Manual. Third Edition. Auburn, New York: 'General Electric Company, 1964. Pp. 55-60.

10. Skrotzki, Bernhardt G. A. Electric System Operation. J',~cGrs. w-Hi 11, 1954. ?p. 64-65.

New York:

ll. Spiegel, Murray R. Theory and Problems of Statistics. New York: Schaum, 1961. Pp. 202~42-244.

12. Thompson, Frederick W. 11The Effect of Voltage Wave Form on the Operation of Two 1",rpes of Current Overload Relays. 11 Unpub-lished Master's Thesis, Virginia Polytechnic Institute, Blacksburg, 1951. P. 17.

13. Transistor Manual. Seventh Edition. Syracuse, New York: General Electric Company, 1964. Pp. 300-528.

14. Wagner, C. F. and R. o. Evans. York: McGraw-Hill, 19:;;.

Symmetrical Components. New Pp. :;0-99, 289-292.

Wright, Ralph R. and H. Richard Skutt. Electronics Circuits and Devices. New York: Ronald Press, 1965. Pp. 140-145, ;9:;-;95.

The vita has been removed from the scanned document

APPENDIX

CALCULATIONS

Regression line: y 1 - y = b(x - i) where b = I (X - X)Y

I (X - x) 2

Chi-square test for goodness of :f'i.t:

,,.2 K (o· - e->2 A =L I I

j=1 e j

where the number of degrees of freedom vis given

by v : k - l

For ten per-cent unbalanced condition:

Time Observed Estimated

0.80 70 46 1.60 ;; ;6 2.40 5 27 ;.20 7 18 4.oo :; 9 4.80 5 - l 5.60 2 -10

125 125

For fifteen per-cent unbalanced condition:


o.80 48 41 1.60 45 .?.? 2.40 17 25 :;.20 12 17.5 4.oo 19 10 4.80 6 2 5.60 -2 - 6

150 122.5

For twenty per-cent unbalanced condition:


0.80 57 64.5 1.60 33 49 2.40 9 33.5 3.20 15 17.5 4.oo 7 2 4.80 4 -15 5.60 2 -28

·· 125 125.5


b = -11. 7, from datu in table 4.

Regres3ion line is y 1 = -11. 7x + 55.44 -whel"e y = 18.

The statistic X 2 = 9.0

2 From tables, X =18.5, .995

Since O .676 < 9 .O < 18.5, degrees of fl"eedom.

and 2 X .0 5 = 0.676.

the fit is good for six


b = -9.4, from data in table 5. Regres3ion line is y 1 = -9.4s + 48.1

where y = 18. 2 The statistic X = 12.0

Since 0.676 < 12.0 < 18.5, the fit is good for six degrees of freedom.


b = -19.3, from data in table 6.

Regression line is y 1 = -19.;x + 79.76

where y = 18.

The statistic X 2 = 17 .05

Since 0.676 < 17 .05 < 18.5, the fit is good for six degrees of freedom.

Standard error of estimate and coefficient of correlation:


sx.Y = o.o r = -1.0

Therefore hypothesis is statistically significant.


sx.Y = 1.98

r = -0.146



SX.Y = 1.4;7

r = -0.50


.A. PROTECTION SCHEME FOR ALTERNATOR FIELDS

James Terry Lancaster

Virginia Polytecbnic Institute

Department of Electrical Engineering

MASTER OF SCIENCE THESIS

Abstract, 1965

A synchro-generator and a unijunction transistor oscillator

in conjunction with a silicon-controlled rectifier form a detection

and relay circuit for voltages appearing across the field of a three-

phase alternator generated by negative-phase-sequence currents due

to unbalanced conditions.

Readings were taken of the times required for operation of

the relay for various conditions of unbalance. Statistical studies

were made to enable time predictions to be offered.

It was concluded that the scheme offers the possibility of

faster operction, and hence better protection and system stability,

than is possible with existing types. In addition, the scheme will

result in no loss of dependability over existing types, and the relay

portion offers outstanding ruggedness and sensitivity.

A PROTECTION BOHEME FOR ALTERNATOR FIELDS by James …

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