Zero-sequence impedance in single-phase induction motor
by H C Chopra
A THESIS Submitted to the Graduate Committee in partial fulfillment of the requirements for the
degree of Master of Science in Electrical Engineering
Montana State University
© Copyright by H C Chopra (1948)
Abstract:
The purpose of this project was to bring to light more facts regarding the zero-sequence impedance in
single-phase induction motors.
It had been observed in the laboratory that while a three- phase induction motor (wye-connected and
neutral available) is running at certain speed, if we change the connections such that power is applied
between one line terminal and the neutral as shown in Fig. 1, the motor continues to run at one-third the
previous speed. This is the customary method of measuring the zero-sequence quantities and forms the
basis of this project.
In the following pages it has been shown that the zero- sequence impedance definitely exists in the
stator circuit, as well as in the rotor circuit of a single phase induction motor.
« The value of the. winding pitch is found to affect to certain extent the actual value of the
zero-sequence impedance. Z2R 0-8BQBBNCE IMPEDABGB IB
SlMGLE-PHASE IBDUCTIOB'MOTOR
by
8 * CHOPRA
A TBE8I8 Submitted te the Graduate Committee
In
partial fulfillaeht of the requirements■
" for the degree of
' .
'
'
Easter of S e i e n d e .i n rElee.trl^al.;Engineering
- . Montana State'College
I n Charge of M h J o r W p f k
a i r m a n / G n ^ n a t^Corniaittee
iontana
y
3
7
cJ f
P
i
f
2
TABLE OF CONTENTS
Page
Abstract
3
Nomenclature
4
History and Introduction
5
Definition of the Zero-Sequence
Impedance and the related theory
5
Description of the Equipment ana the
Test Procedure
7
Assumptions
12
Mathematical Equations
15
Apparatus Specifications and the Test
Data
18
Connections
28
Calculations
43
Summary and Conclusion
47
Reference
48
K!
%
x
x
V' X I,'
■ ■>.
S=
3
ABSTBAG2
The purpose o f ■this project was to bring to light'more
facts regarding the zero-sequence impedance .in single-phase in­
duction motors »
Xt -had been observed In the laboratory that while a threephase induction motor (wye-connected and neutral available) is
running at certain speed. z if we change the connections such
that power is applied between one line' terminal and the. neutral
a s shown in Flg» I s the motor continues to run at one-third the
previous speed,.
This is the customary method of measuring the
sere-Seqmeiice quantities and forms the basis of this project.®
In the following; pages it has been shown that the: hero-'
sequence impedance definitely euis.ts in the stator circuit5 as
w e l l as in the rotor circuit of a single.phase induction motor,
The value, of the. winding pitch is found to affect to certain
extent the actual value of the zero-sequence impedance^
4
ROMBNCLATmE
Meanlne-_______ . . / Superscript
■
Positire-sequeuee
- Superscript
Regatire-sequince
■ o Superscript
Zero-sequence
1 Subscript
Stator circuit
2 Subscript.
Botor circuit
e Subscript, or without any
subscript
Stator and rotor circuit
combined
an? bn? en. Subscript
GorrespeBding terminal to
neutral •
S Subscript
InduptlTC
.
'
.lmpedanee measured by single
■ phase- supply
3-0%
:
Impedance measured by three
pha.se supply ■
Operator for rotation of
vector by 120°
S (in equations)
Slip
S and M
South and Sorth pole res­
pectively
(in. Winding Diagram)
tj_5 tg? tgg X in .Winding Diagram)
Starting point of the three
phases respectively
n
Weutral Feint
F
(in Winding, Diagram)
.
The factor a s .defined o n .
Page 24
.
Z m o ^ E Q i m N O E IBPBBAEOE IB
EISIORY A m
lNm<a)OCl%OB
_
.
The general concept of -the gero«seqnenc.e ImperJance of a
single-phase induction motor% these days^ is that Ittis present
only- In the stator.circuit and none I n the rotor oireuit,. 'B h t ,
the previous, record does-, not shew m u c h work having been done
to give'a sufficiently positive, proof to this belief and.--&oreover certain observations In the lahcratory work did not agree
with this conception*.
It was w i t h this aim of establishing.'a
..satisfactory interpretation of sero-sequence Impedance pertain-=
log to single-phase induction motors* that the wor k of this
project has been attempted*
Ero m the experimental data and
the discussion thera-upoB, It w i l l be seen that th e %ere»sequenee
impedance does not consist of that i n the stator circuit only
'
hut Is also present in the rotor circuit o f the single-phase
induction motor*.
OEQDSNOE
self-e
It seems desirable to- explain the term yEERO-a to make the paper more explanatory in it-
.
op zEEBsi z m o ^ ^ m ^ c E 1 # # ^ #
A m 3 #
The term *^Zero-8equencey .owes its origin to the application
Of Symmetrical-Components theory to the solution of unbalanced polyphase circuits or balanced circuits with unbalanced terminal
conditions* in Electrical Engineering*
Without going into much
detail* it would suffice to mention a theorm of the ^Symmetrical
Components1’ l*e*
11Apy system of u n b a > a m e d 5 three phase currents, or voft^
. a g e s «, whose vector sum Is not zero may be represented by. two
symmetrical component systems .and one single phase system which.,
.have been given the- specific names as .belows
1#
Po sitive-sequence component*
2*
Begative-sequence component*
3*
Zero^seqtaenee component*
A system of three symmetrical vectors I s .one in whi c h the three
vectors are equal In magnitude and are displaced from each other
b y equal angles*0
.
Ihe zero-seqnetice component consists of three equal vectors
i*e»* of the same'magnitude, a n d in the same direction*
•
-
Also 11'
-
•
■
.
■has been shown by EeSsers6.Wagpef and: Bvans in their book
I1S Z m m i R i a A L o m p o m a M l S 0 that I n symmetrical systems the
different sequences do not react upon -each other i*,e*.5 positive^
sequence, currents produce only positive sequence ‘voltages..^
. .
Pegative-Sequenee currents produce only negative' sequence volt'a g e s ;and zero-sequence currents produce only zero-sequence volt­
ages*
Sow when voltage of a given sequence is applied, to a piece
of apparatus* a very definite current of the same, sequence flows
and the apparatus is said to -have a definite amount of Impedance
to the particular sequence*
Ihus the impedance to zero— sequence
currents is called the zero-sequence impedance*
In a similar
manner t h e .positive-sequence .impedance and the ne gative-sequence .
impedance are' defined*
7
- Xt had .been observed I n the laboratory Shat while a .threephase induct ion motor, star-connected w i t h its neutral available „
9
is r u n n i n g i f the connections are changed, such that the power
is applied between on© line terminal and Bentral9 as in Fig? I 9 '
the motor continues to r m 9 but,, at one-third the previous speed*
then the Question, arose what did make the motor keep rimming?
Fr o m "where did it get its torque to continue to run if there are
n o 2ero-sequence currents in the rotor circuiti
A s Fig* I -ShGWS9 the three phases, of f he stator windings
are now in parallel and a single phase supply has been put across
its two e n d s 9 one' being the neutral a nd the other being one of
the" terminals,,.
The three.'phases-'are' 120 degrees apart in space
but under these circumstances the same voltage vector i s .across
each of those and as such only the zero-sequence currents are'
in the circuit*
As the motor keeps running*, it shows t h a t :there
is current flowing in the rotor circuit*
Still sometimes it is
argued on the basis, of Fig. I and F i g , la that as t h e three;m.m.fs
are- 120 degrees apart and also equal in magnitude9 their sum is.
zero* which means there is nothing to. cause the currents to. flow|
and therefore zero-sequence impedance does not exist in the rotor
circuit..
All this led to the idea of making investigations as
to the actual presence of zero-sequence, impedance in the rotor
circuit or otherwise to find the cause of the motor having con­
tinued to r u n o
8
On the basis of these observations? many different types
of three-phase induction motors were taken to collect sufficient
data for the purpose of discussion=,
k single-phase winding is
considered to be a part of a h imaginary three-phase5 star-con­
nected winding under the conditions of Jhig=. 1»
Further it "is
known that power taken by the motor under "91Ioeked Botor Iesttf.=,. _•
is almost entirely due to copper losses, increasing as the- square
of the current,
Hie- iron losses are only- small evenb&t the maxi­
mum voltage employed In this particular test? because the iron
saturation takes place much below the normal working value of ,
the vdltase*
Thus the readings of voltage and, current- give the
egalValent total impedance of the motor, l»e», the
pedance o f the stator and the rotor*
im­
Because the current taken
b y the motor will (Jepnsnd chiefly on the resistances of the wind­
ings^ & slight lack of symmetry may produce a .considerable i m ­
balance,
be equal.
the currents taken b y the three phases may hot
It is therefore found desirably to employ am,ammeter
-and a watt meter in- each line.
,
fhe single-phase winding is being considered as an imsginary three-phase case, to comply wit h the theory of symmetrical
components and under these conditions I^ & Ief- o, whereas
wh e n we employ the blocked rotor test on three-phase connections,
these conditions are not exactly fulfilled, because I# and I<,
are not each equal to zero=*
Thne it looked logical to mea­
sure the impedance per phase of the motor b y applying single-
9
phase supply to one of :the phases only? ,which satisfies the -eonait I on . % vs. Ie. a. djk -it was thus observes that the- equivalent
Impedance per phase measured. I n 'these two-different'ways was
n o t 'necessarily always the s a m e lhese two values differed ■
to a lesser'or a greater degree? d e p e n d i n g ,upon the value' of ,.the ;
pitch?-.at full pitch the. two' values being lust the Sbmev
for:
the sake of comparison.? both the sets of readings are recorded
for all the motors*
As far a s ’,this project is concerned?, the
equivalent impedance Measured by the second method shall be
employed i n the calculations^
It may?, however? be noted that
the equivalent impedance obtained, by the. three-phase blocked?rotor test? itself is. equal to the positive sequence impedance
and. the negative sequence Impedanes.respectively under the -stand­
still conditions? Ipev9 when S.
1»
'
:Ihe resistance of the stator windings can be -measured conveniently by. passing small? direct^current through them and then
determining the current and voltage»
for this set of observations
a commercial, potentiometer was used to make the matter easy*
-
' 1
It
1
is necessary to find the resistance of the stator circuit to find
separate stator impedmice and the rotor impedance from the equiva lent impedance of t h e circuit*.
W h e n the M o c k e d Botor Test* is performed on th e 3ame
motor- with the connections -as shown..jn .Fig* I? and the current
and. voltage readings are recorded;.?."then: the equivalent zerosequence impedance of the motor c a n be calculated*
Ihus the .
.
3.0
-nLocked Botor Testn was performed under two different conditions
o n different motors, (three-phase % star* connected and with the
neutral ,arallahle) :. .
-
The mathematical treatment which Is ..'recorded on pages 15 to 17
also goes to prove the existence of zero^sequence: impedance §
both in the stator and the rotor circuit of a ;sliigle-»pha'se in­
duction Tnotgr5 w h e n considered on.the basis of an imaginary
three phase= '-It .was-first intended to make calculations for
the rotor reactance from the design, data of different motors
and compare w i t h :the experimental data, but the design data ,
Conld not b e procured from the manufacturers, quickly enoughc
■ For every motor'which .had beam picked up for collecting
experimental data in connection w t h ,this project,, a set of wind*-/
ing diagrams has been prepared* ■- Qne diagram shows the flow of
insta'nteneous current in the stator windings, whe n the connections
are as- shown in Big* I, i 6.e,».5 single*phase connections and the
other diagram shows the flow of current when the Connections'
are normal three-phase star - connections»
These diagrams very
clearly show how the change in number of. poles takes place, i n '
case of every motor 9. when three-phase star-connections are •
changed to those of' .-Figs- I*
.'
It ma y be .-noted that the value of
pitch marked on both the diagrams for the same, motor, i s ;thevalue of pitch as in the ease of three-phase winding*
fp
"
tp, tg, t3 in the diagrams represent the starting points
of the three phases and n represents the neutral point«
The
,
A s c r i p t i o n of the connections in these winding diagrams is
given on page ^8 9 fox ready reference to trace and check the
direction of current in the different phases of the windings*
The dash <3> stands for the particular coil' and { $ } represents
-the end connection between two different eol3s
the windings.
in one phase of
Another point is worth mentioning here* regarding
these winding diagrams* I =e»,9 in single-phase connections 9 the
amount of current in all the coil arms is the same at all vhe
instants, with the result that where in a particular slot tha
upper and the lower coil arms carry current in opposite directions* they cancel each other’s effect completely*
fhis is true
at any Instant* but in the case of three-phase connections*
.
the picture is a little bit different* because the current In
the different phases very differently from instant to instants
However* it should be noticed that w h e n the situation is con­
sidered over a certain length of time* the average •net effect
of the current flowing in opposite directions in upper and lower
coil arms in certain slots* is zero* in tnis ease also*
Io ascertain the presence of current in the rotor circuit 9
under the conditions of ?lg. I, w i t h the rotor blocked, the
problem was attacked from a different angle 9 f ° f t
was
planned to measure current and voltage across one of the rotor
bars.
It came to be noticed that there w a s g u i t e an appreciable
amount of current* but the voltage was only a few millivolts. fhe current wave shape showed up very well on the cathode ray. ■;.
12
oscilloscope screen^ but a little ',further working proved ttial
there was not power enough i n the. rotor bar circuitf. as to g i v e .
" a 1r a w "shape of sufficient amplitude ? o n the G 0 E« oscillograph o''
I n short this part of the -circuit needed power s as well a s volt­
age amplification and-in the absence' of some handy amplifie#^ •
under the circumstances3 it was decided to take, pictures of
the wave shape on the oscilloscope screen*
.
Io place the timing
wave shape and the one o f rotor bar circuit, together side by side $
an electronic switch w a s used#
■■
A s would be noticed from photographs Io». 2 and Be-* 4 S the
60 cycles timing wave shape is slightly distorted*.
■ '
This was
Observed'to be caused by the electronic switch^ but the., rotor ,
. b a r wave shape- was however not effected in any w a y by it,
lhese
pictures clearly show that the wave shape of the rotor bar Current
under both the conditions, Ioeti9 three-phase and the simple phase
is definitely of the fundamental, frequency and that n o third .or
fifth harmonics are involved#
This Is thus in support of .our- .
considering the' vectors of the fundamental frequency*
.. A SSUMPTION
..
The assumption that the equivalent reactance o f 'the motor
consists that of ■stator and the rotor, as equal part., is fairly
correct for a n average Slse motor=.
The Westinghouse SlecttlC’
Company has some.different values, for the ratio of the statori
reactance ah', to- tiie rotor reactance.
These, as noted below,, are.
taken.from their b o o k ? Factory Testing of- ,Sleetric .Apparatus=#-;
.
wT&e value of %2.
;
'i
^nd •Zg depend upon the class of the
.
motor.
Glass. A « Motors are normal torque 5. norma I" st a.rting-- cur :eent s '
sauirrel-cage motors and are frequently called general purpose
motors
Glass -By- Motors are normal, torque 9 low-starting-current ?
;.squirrel-cage motors^
Glass CmjMotors are high starting torque 9 ■low-startlng-curren t g
■ squirrel-cage, motors*,’.
motors are high-slip squirrel-cage motors
Two and Three Fhase Motors
Oiass
GlassClass
Class
Sound'
A
B
G - D
' .
Rotbr- '-"
%2_
8
^
S:l£
wwaiii-rmn
This assumption-of allocating the reactance in ,equal part,
to -.both the stator and the rotor can further be extended to
apply to the. component parts (positive sequence^ the negative
sequence and the zero-sequence ) equally well as to the equivalent
impedance o’ For. this purpose,," the actual measured value of the
zero-sequence impedance .will "he employed. ■
The winding diagrams of. the different motors, under the two different -conditions that are being considered in this proIect, show that .with the single phase connections in some eases,
the Effective, a mpere’turns which produce the flux, are consider-
14
ably reduce# when t&e meter is throws over to the single^phase
'cGnBeetions^' ■Frdm- this it can be reasoned that the reactance
P B T t .of Sero-Seqnence impedance is affected t o - a ■certain extent^
©h this basis certain factor isferlsed which takes care of the
.change Sn-. the effective ampere turns and the change in the
number of poles with the change of the connections to single
phase.
This helps os to find the aero-sequence impedance straight
off from the positive sequence impedance*
These values are com­
pared with the ones obtained b y other methods*
Shls factor is found purely by physical .reasonings, based
o n the study of the winding diagrams of the particular motors
on which experimental data has been collected*.
This factor whe n
attached t o the reactance part of the positive sequence Impedance
component* gives the .reactance part of the aero-sequence Impe^
dance of the motor ? as shown below*
The- value of the factor
depends mostly on the value of the pitch*
Ihus
F .a ( S ^ ^ ^ S c a ^ e r e ^ ^ ^ S p e r ^ ^ S H l o T ^ p o l e s J r n l - p h a s e
The factor is observed to apply in cert a i n eases fairly welly
a n d the results obtained In this w a y are compared wi t h the ones
obtained by other methods.
15
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cpF,CIPICAY11 KS
I.
YBST DATA
Induction Kotor
Ihree phase> star Connected
Volts 220.
Cycles 60
Amps 2.7
No-Load Speed 36OO
H . F . 3/4
Cell Pitch . 1/2 Pole Pitch
No. of Slots 24
(Stator)
LINL \OLTS
Table III
Table II
Table I
Z./'vevcfTT
LINE. VOLTS
Zj = R"+ J
-------------- ,
I
H
CO
ZO
2,<5 35 H-J 34.05
35
21.25
+J
19.35 + J /6 92
41
26. 3 H- J 2 7.73
45
22 9
H J<2 6.^5
40
13.26 + J <5.36
IOO
26. /5 4 J 26 3
5o
21.9
+ J 25.7
10
21.57 +J /4-955
/40
26 /9 H-J 25-S
60
2 3 .6 5 +J 2 4 9
go
Zl- 6 3 + J /4-6/
2g./ H- J 26.05
10
22.75
+JZ4-5
90
20. 8 2 + J 14-67
30. 3-h J 25.2
30
2 Z ,9
-hJ Z4 -Z,
IOO
20.43 -hj /j.9&
90
23.25
Hj 23.5
ZO
/3.05 4 J 18-15
30
•
I 160.
ZOO
2 7.6
:
L
___ I
T
19
I. b.
The following set of observations was taken on the above
motor, but with the rotor removed and the stator circuit hav­
ing been connected as In Fig. I.
Table IV
LINE VOLTS
Z s„=
IO
/ 4- 66 -I-J /0.38
15
/4-44
Zo
/4 64 + J / 3. 95
30
14-73 + J lo.y
4o
/4 g3
// 6 8
/0.76
I. c •
A cathode ray oscilloscope was connected across one of
*
the rotor bars and the wave shape was observed, with two dif­
ferent connections, i.e., first, with connections as shown In
Fig. I, and second, with the normal three-phase connections.
The photographs of, the different wave-shapes that were observed
are attached herewith on page
20
PHOTOUhAPh MO. I
ROTOh BAR CURRENT WAVE-ShAPB
Connections as in Fig. I.
21
PHOTOGBA P H NO. 2
COMPARISON OF WAVE-SHAPE PHOTOGRAPH
NO. I W I T H 60 CYCLES WAVE-SHAPE
(1)
Top W a v e Shape
60 cycles timing wave
(2)
Lower W a v e Shape
Rotor-bar current w ith single
phase connections
i
22
ROTOB B AB CUBBENT WA V E- SH AP E
Connections Three Phase
23
COMPARISON OF WAVE-SHAPE PHOTOGRAPH
NO. 3 W I T H 60 CYCLES WAVE-SHAPE
(1)
Top.Wave Shape
Rotor bar current connections
(2)
Lower F a v e Shape
60 cycle timing wave
2.
Induction Kotor
Three Phase, Star Connected
Cycles 60
Volts 220.
Amps. 3.2
No-Load Speed 900
H.P. 1/2
Coil Pitch s I
No. of Slots 48
(Stator)
■
LI NE VOLTS
z° =
Table VII
Table VI
Table V
un£
R*+ J
vMrs
34>Z€ "
15
4.63 + J 7-55
7 .4 4
ZO
4.76 + J 7.59
25
4.G 2. +J 7. 36
5 . 0 g + J 7 . 21
50
4-4 + J 7 01
GO
5 - 2 ' + J 7.o7
35
4 . 3 6 + J 6.97
70
5 J S + J
40
4 41 +J A 9/
3 5 7 + J 5 .5 9 2
20
5 .IZ
G
3-3 6 + J 5 . 3 o 2
3c
5.15 + J
IO
3.3 9 9 + J 5 - 6 5 5
4-0
5 . 0 6 + J 7 .2 5
15
3 .3 8 7+ J 5 - 6 7
5o
ZO
3 .5 4 9 + J
5 .5 6 2
23
3 .5 Z 5 + J
5 .4 3
•
H-Ze -
8 .0 4
4
I
LlHE VOLTS
Re + J * c
+ J
7 . 1Z
-!
■
j
4
3I
Induction Motor
Three Phase, Star Connected
Cycles 6C
Volts 220.
H.P. 2
No-Load Speed (Variable)
No. of Slots 36
(Stator)
Coil Pitch •
Acps 2.8
(Variable)
Coil Pitch - I
Table X
Table IX
Table VIII
—
.
UNF V O t r s
zf =
Ll NF
VOLTS
10
9 - 9 2 5 + J /5. 4 6
Lo
15
8- 6 (
4 0
20
'
25
.
/ Xd
+ J 14.13
3
,
Re + J * e
Ll NF VOLTS
9. I Z
+ J L 7.1
20
7-35 + J 1 6 -3 8
J 1641
30
7 .8
J 15-56
40
7. 6 1 + J
50
7.4 Z S + J >3.57
CO
7. 6 -h J ' 3 . 5
8 .5
+
8-4 + J
II. 8 8
CO
'•»8,. 4 1 +
€ .9 6 + J
(2.33
80
8-28
-+J
8 .2 5 + J
'4 99
35
6 . 8 7 +Jtl-IS
I OO
4 0
7 .0 3 5 + J '0.59
n o
8. 5 4
6 - 3 9
/20
8 -8 2 -5 + J '3.31
56
_______ _______
+ J 10' C Z
,-^ z e =
=
+ J
/ 4 . 1G
■
13. 8
—
+ J 16.05
'4 .57
f
ro
vi
#
3.
(Continued) Coil Pitch « 2/3
Table XI
Line
volts
Ze0=
Table XII
UNE V O L T S
JX°
3
Table XIII
Ze = Rg +J
LINE
VOLTS
i f Zf =
\
+ J x e
Z
5 '7 6 + J 0
30
8.OZ-hJ /8-74
/5
7.4
4
5. 5$ + JO
4o
S- 6Z + J 15.9
ZO
7-G J i-J
/3./3
6
5 . 5 5 + JO
50
S - 6 7 + J 14.5
30
7 '4 8 + J
10.95
6
5 * 4 3 + JC
70
9. Z + J 12. 8
40
7. 0 3 + J 3 . S 5
SC
$.9
4 5
6.16
no
9.7 > J W 92
3 85 +Jf/.S
.
150
_____I
+ J IZ .5 Z
5 0
-h J 1 4 . 9 Z
+ J S
6 . 3 5 + J
5 5
£-43
Gz
S 4 4
+ J S- 3 5
-
In this motor the ends of all the coils have been brought out and are arranged on
a circular board to facilitate the study of induction motor winding in the laboratory.
**
'
This proved very handy for the purpose of this particular project, in obtaining dif­
ferent coil pitch values on the same machine.
On this machine the toil pitch values
tried, are I and 2/3#
2.
4.
Induction Motor
Three Phase
Cycles 60
Volts 220— 440.
H.P. 10
No-Load Speed 1800
No. of Slots 48
(Stator)
Coil Pitch « 3/4
Table XIV
UNE VOLTS
Amps 26/13
Z° - R0+ J %°
Table XV
LJA/£1^0^75
3_^Ze=
Table XVI
R^+JX-e
U/V£ VOZ.73
Zc-
Io
ZJI +
J'.758
IO
'.Z5 4 / '.57
O. G9 + J 0 - 5 6 S
ZO
1.7 3 5 + J Z Z9
ZO
A/3 f //'57
S'
0 63
3o
/ 4 97+ JZ 5
30
/.09/4
II.Z
0. 631 + J O . 6 8 7
50
1.413 t J
40
/.03
'3.9
0 . 6 39 +J 0.6&I
Go
/.4 2.3+ J 4.4/
50
1.0 78 -I- J
/£.45
O. 6 3 7+J 0.693
So
/.4/5 4 J a.46.
Go
1 . 0 % 7 - h J 1.94
18.5
0.645 4 J 0 6&7
30
/.4' 4/2.475
lo
/-063
3-Z
0-699+
4-3
J
O.S'ZS
t J 0.66 3
Z.42.
J /.33
+ J 1.948
A 93
4j/. 92
V
28
C oa /,v / c r/o/v s__
D/ 1SH
—
’
RE P f i E S E f v r s
COIL
REPRESENTS
END
COMMA
FI Q
3.U
F/6 J.&
PHASE
PHASE
I.
t (— —
2.
t2
C O N N E C TfO/\
"
3 - 3 , 4 - V , 16- 16, 1 5 - 1 5 , 1 4 - 1 4 , 1 3- 1 3— n
3-*>,I O - I o ' , n - n ytZ-l2,?A-Z'\, 2.1-Z?),ZZ-Il,
,Zl-Zl----
PHfISE
FIQ
3.
t 3 - ^ l 7 - i 7 /1 » 8 - / 8 , » 3 - l 9 ^ 2 0 - 2 0 /, 5 ? - 6 , 7 - 7 , 5 - 5 ----- »->l
eI-.cl /I D 4. b
i,
PHHSE
t(— *“ l - l ' , Z
- 2 , 3 - 3 , I Z - I Z r I I - U , I O- I O , 1 9 -1 9 ', Z O - P o ' j Z I - Z I ,
, 3 0 - 3 0 , 2 9 - 2 9 , 2 8 - 2 8 ------- ► n
PHASE
2.
n - ^ 4-4,5 -5', 6-€, 15'-15,14-14,13- i3rzz-2Z'Z3-z^
, Z 4 - z 4 y
PHASE
3.
t 3— *- 7 - 7 , ' 8 ,
Zi-Zl',
33 —3 3 , 3 2
3 2 , 3 » - 3» — —
S - a ' , ' g ' /S , / 7
/ 7 , / 6 - / 6 , 2 5 -2 5,26-26^
3 6 - 3 6 , 3 5 - 3 5 , 3 4 - 3 4 -----
F / G . 5. a A/vz) / ^ /G 5- 6
PPASE
I.
t,—
*-l-l',Z-2' 8- S,l'-1,13-13',14-14,Z0-Z0,19' 13,25-25'
yZG-Ze' 32- 3Z, 31' 31-------
29
FIG S.^ ,4/VO F/C, 5. &
Ph a s e
z
.
(Co n t d )
t z— ^ 3 3 - 3 %
, 1 6 - 1 6,
PH/)SE
3,
34-3V, 28-^6 ,zv-z?,zi-zi'za-za^ 15-15,§-S,tO-lo',4r'-4,3-3 ------- -
n
t3 *- 5- 5^ G-G,tl-lZ,n~n,l7-nf
,l&-l&/,Z^/-Z^,Z3,-Z3f
, 2 9 - 2 3':, 3 0- 3 o ' 3 6 - 3 6 , 3 5 - 3 5 --------- n
FIG 6. Q-,,AHD FIG> 6 . &
PHASE
I.
t,-- ^ H f z,2 - 2 Z, 8-8,7- 7, / 3 - / 3 ' , IA-MjZO-IOfI^-IS,
, 2 5 - 2 5 Z 2 6 —2 6 , 3 2 - 3 2 , 3 1- 3 1, 3 7 - 3 7 ' 3 8 - 3 8 '
, 4 4 - 4 4 , 4 3 - 4 3 ------- - n
PHASE
2.
fc2
- 4 5 - 4 5 ' 4 6 - 4 6 ' , 4 0^-40, 3 9 - 3 9 , 33-33' 3 4 -3 4 '
, 2 8 - 2 8 , 2 7 - 2 7 , 2 / - 2 l ' 2 2 - 2 2 ' /6- / 6,
15-15, 9 - 3 ,
, 10-10 , 4 — 4 , 3 —3 ------ *-71
PHASE
3.
t,
* - 5 - 5 , 6 - 6 Z / 2-/2
, 2 3 - 2 3 , 2 3 - 2 9' 3 o - 3 o '
/S-/<S, 2 4 - 2 4 ,
3 6 - 36, 35-35, 4 /-4 ^
, 4 2-42 , 4 8 - 4 8 ,4 7 - 4 7 —
FIG
7. a A/vj? F / G
PHASE
I
7. 6
t f— *- M , 2 - 2 , 3 - 3 , 4 - 4 , / 6 - / 6 , / 5 - / 5 , / 4 - / 4 , / 3- / 3,
, 25-25Z
, 2 6 - 2 6 ' 2 7*-2 7 ^ 2 8 - 2 8 ' , 4 0 - 4 0 , 3 3 - 3 9 ,
) 3 8 - 3 8 , 3 7 - 3 7 ' ------- «►n
30
F /0,___7. A / I r / D f / 6 y. A
P"A?F
Z
(c'orvr o )
t , ----- ~
9-3, !O-ta', H-n', iz-iz', Z4~
Z^, ZZ-ZZ7
, Z / - <?/, 3 j - 33 , 3 4 - 3 4 , 35-35' 36-36,4 8 - 4 6,
i 4 7— 47, 4 6 — 4 6 1 46 — 4 7 ---
/
PH4 5 f
3
/
/
,
I
, PHf SE
HLt-sfL
t t ,C 1 /IHD 1 3
L .~
,
/
t'J-lj', 'Z-'8-1 !S-'S’Z-O Zo\
, 30 - 3 0 . Z 9 - 2 9 ,
A
,
* 4 / - 4 f , 4 Z - 4 2 . 4 3 - 4 3 , 4 4 - 4 4, 3 Z - 3 2 i 3 / - 3%
Z, ANj) PHASE
Po I n r .
/
Z - ?,
7 - 7, 6 ~ C
/V/-
r«f
,
5 - 5 --
s r^/ir/N G
RES Pf C T/Vf L Y, /I,VS
Zl
Po/A/r o f
HE PKL Sf/v P5
PHfiC L
THt
31
]
a
Ofl
= M.M.F
QG =M M-F
OC
DUETO
DUETO
= M-M-Zr DUE
TO
C U R R E N T IN OCl F I G . I
C U R R E N T I N OC- FIGr.I
C UR.R B N T
FIG La.
I N OC FIG.I
i
32
FIGJ
'i r
r M Z W ^ V V n n ^ A y v W h
13
© ®
*3
PITC H
I ® ® 7
CONNECTIONS
I-+
>
F I G d J r
I
SC
®
©
3?
®
®
®
®
S
27
,
50
, 29
1
2
®
®
3''
32'
3
®
,5
33'
n
4
®
®
34'
28
®
5
6
®
30®®
.
t,
/m ®
2'® ® 8
30 ® ® 24'
s '® ® 3
2 9 ® 0 23'
28 ® ® 22Z
POLES
NO
4'® ® IO
FORMED
t
5 ® ® H
27 ® ® 2i
26 @ ®
®
18
9
®® ^
2)
®
“
PITCH
l6 '
'
15
21
®
14
I
0
20
13
®
®
f9
Z
/
f2
®
®
12
I'
®
®
17
io'
®
®
8%
®
14
15
'c
CONNECTIONS / - ?
= f
FfG 4 a
i.
3
© ®
4 ® ® »o
^ I 28 ® ® 2Z
/ IO
PITCH
CONNECTIONS
3*?
PITCH
= / (m
l
CONNECTIONS 3-f
)
FfG
50
. .
(sftV
S
s
N
PiTCH
CONNECTIONS
= I (full)
FfG 5.&
/-/
I
PITCH= I(full)
CONNECTIONS Z ~ f
FIG
6.a
40
.® ©
Q ©
33
^'®^3or
© © /4
0
©
©
©
25
CONNECTIONS i-4>
P I T C H = I (“ 'LL)
R G G.&
I__ I
/6
47 4 8 ,
z
3
@
0
® ,© ©
® © ©
V
□
../'",I"
" " " A ' ' ,
6 0©
C O N N E C T IO N S
PITCH = f
FIG 7a
l-j>
42
38 ® ®
4 ® ® '3
r\ I 37 ® ® z8
36 © ® 27
PITCH
C O N N E C T IO N S
=
FlGTfe
2>-j>
43
CALCULATIONS
MOTOR
NO.
I
=
PITCH
z _
Z
cJ
T/lh/fva
THE
JHE
STfiTCn
CO/VI P O N E N T S
AND
SE
3 (i- <^>
^ (j
T=
14-65
/1 ^
3 ( z i . 9 t J Z 5 - 7 ) - Z ( 2 6 - n + J 2'S' 8)
=
/33?.+;/^.
/ *
,
+ J *
+
* F )
H 60 3 T J H,t> g
-
Z6-13 T J Z5 ■S
TO
SE
> f
THE
Z L fiO -S tR V E N C E
4- J.X(
° =
KfiSfS
, Wf
//V fP E D A N CE
_
i, 4
»4 -3 -5 +J 7.
/?ZVD
g 7 ^ -hj 7 4 8
I
value
\
-
V
MezisuRf1D
5 7 XJ /4-955
VEfLOE
Re T OH
I
=
=
SuHED
PARATE L
F =
zz HI
Z
M E /!
X-
Hf i
V£
44
MOTOk
HO.
2
P'TCW = I
Z
Z
=
3
3 ( 4 - 4 + J 7-01} -
=
3.06
—
v\
Ahd
Motor
ho
. 3
Zj
(
3 -^
z
(5 .
3.3 93 + J 5 - 6 5 5
G+J7-Z5J
measured
value
5 . O G T *T 1 . 2 . 5
=
0 9 3 3 + J 2.-6 3
= I
t , = 4-57
F = I
=
3 ( i-<f> Z c ) -
Z (
-
3 ( 7 S f T j 1 4 - 5 7 ) - 2 -(6 -4 8 + J '4-69)
=
6 - 3 7 + J 13.33
Ze ^
\
0
y
o
7 .4 5 x I
2 •4 6 6 T J 4 - S 3
Z
^ e)
6 ’ 53
—
pitch
a;
+ J
5 - oG > /
Z° =
c)
^
=
=
I, = 2 .4 6 6
F = I
€ . 9 G -HJ
12-33
=
2 - Z 8 + J
f4-83 x 1
=
% .Z 8-tJ 14-69
Z
=
Z°
measured
value
M O TO R
(Co
NO,
n t q
)
Z, = 4-57 + J &'IG
NOW
F on
4/vj
Z
PITCH=
4d
= Z.3S+J 6,16
*L, - 4 5"7
A -O
o
Z =
3 (^i3 (7 0 3-^/9.^5)
(9 6 7 ^ Jf
3.75 + J c «55
c)
N U J fiS U R a o
S-S 8 + J p
0
Z -
U f iL U C
8-674 Ji4-f7*o
3.674J0
0
Z:,1 — -4-57 + J o
d
An O
Mo t o r
n o
, -4
/ +0! F J 0
P itch =
i
Z
=
0(
Z e)
— 2 (3—
7, - 0. 4 9
Zt )
-
3 (i o o i + J i - 5 3 ) - Z ( F 4 i 3 + J Z-4Z)
_
0 - 4 4 7 -F J 0 - ^ 5
46
M o T O fi
no. 4
(co/vrs)
IJ
Z 0=
C)
Z 0 -
A 4/3
-
1-413
Z,0=
and
x
° l=
O '631 + J O ' £ 6 7
+ J
Z . 4 2 » - L
+
O-
J
&07
0.49+JC.345
o - H / - f J o. 3 4 3
m e
*
so*
£
d
v /u - u a
47
AmmABY m D oQNcmsrm
it s h o n M be stated here that .the -field a f M Y esti^atioh
o f this project is- limited to the -standstill eond it ions c Gnly~»:
The careful examination <f the calculations..shows that
in some eases# the. directly.-measured, value of zefolsequehce?
compares very, closely to the one. obtained-by ;the method ex­
plained under the, heading "Mathematical Equations" and one thing
however is absolutely -clear from ou r data and the calculations
thereupon# that the value of winding -pitch ,definitely comes
into the picture as far- as the actual amount of zero-sequence v
Impedance of the motor concerned*. -. Some of the self-evident
points in-' this connection are noted belowy .
1» ' Eero^sequenee impedance is present in'the rotors of.
-
most single-phase-induction motors, ■ '
The Winding pitch affects, the amount of zero-sequence
impedance.*
3*
Eor 2/3 pitch there is no zero-sequ'ence reactance’in
either-' stator or rotor*
The zero-sequence'impedance for this
condition consists.of stator resistance- ,only*
4v ■ A W y e -connected three-phase motor having 2/3 pitch W i t h
'
- q
-its three terminals connected to form -one line -connecfibn and. its
neutral t h e d t h e r ^ A l l not ti .
torque*
-
o n single-phase*. It develops no
-'
To - Zero-sequence impedance- at -standstill for full pitch
motor is# according to. test resu l t s , essentially equal to the-
48
standstill positive-»«*and standstill negative— sequence impedance*
•A study.of current distribution charts would indicate that.this
is to be expected* .
6p
•
I t appears that positive-* negative+-* and zero-sequence
impedance for use i n the equations developed in this thesis refers
■ to value measured 'with three-phase current*
7-*
In contrast, with the statement above the positive-*
and negative—sequence values of impedance to be used in the
conventional two-phase-symmetrical component theory refer to
values obtained by actual single-phase measurement*
8,
' lhe zero-sequence current-in the rotor at standstill
is of fundamental frequency*
The flux distribution in space
around the periphery of the rotor is a third harmonic of the
fundamental*
'
.I
49
BEFEEBMQB.
I cl
Sjnmnetrieal Components (a Pook) ) C= F, W ag n e r and Eo .P 0
Bvansl9 IIcSrawrEill -Book Company*' Ine01* Bew I o r k s
1933» ■
' .■■■■'. '
/2=. . .Induction- Motors .on Unbalanced Boltages5 S 0 P» See d and. ;--E 0 J >0 W 0 Koopman0.
Transactions s Volume 55s Bovember_s
I9369 pages .1206.^13!O ...
3»
Split-phase Starting o f .Three-Phase Motorss G. F. Tracyj
and W*. 1». Wysss AlEB Transactionss Vol '54s October, 1935?
-v , . p a g w ,1069^72^ . "
.
'
5V
FacteiyTesting of .Electric Apparatus (a handbook) 5 Westlnghouse Bleetrie and Bannfaetnring Company? East Pittsbnrggh9/
MONTANA STATF
iiiutvcbcttv,
-__
J i/b 2 10013327
N378
__
045z
uop. it.
Chopra, H 4 C .
single-phase induction motor
N37&
045%
Cop. 2
34370