Lehigh UniversityLehigh Preserve

Theses and Dissertations

1970

Asynchronous operation of round rotorsynchronous machinesWilliam F. HechtLehigh University

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Recommended CitationHecht, William F., "Asynchronous operation of round rotor synchronous machines" (1970). Theses and Dissertations. 3848.https://preserve.lehigh.edu/etd/3848

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.ASYNCHRONOUS .. OPERATION OF . ,. ' ~ ..., '

ROUNl) -ROTOR SYNCHRONOUS l~lACHirlES

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Willi-amF. Hecht .- -- -_•

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A Thesis ).

Presented to the Graduate Committ·ee .. -: . ,· ·- .

of Lehigh University • 1n Candidacy of: for the Degree

Master. of Science • .

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·E.1ectrical Engineering

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'

Title Page \ • •

... • • • • • • • • • • • - - . • • • • • • C'-

Certificate.· of Approval • • • • • • • • • • • • • •

AcknovJledgement-s • • • • • • • • • • • • • • • • • • • Table ,of- -Conte11ts · -, ,,,, - ~ ; .. '-1 j • · et

L -· • • • • • • • • • • • • • • •

I

. . . . . . . . . . . . . . . . '.. . I

.- --~- -

-ABSTRACT- -~ . ' • • • • • • • . . . . . . . . . ". . Chapte~

1 J

I. INTR0DUC'I.'_!·O'J1· •• •• • • • • • • • • • • • • • • • . \ -

II. EDDY CURRENT ·EFFECTS • ~ •• • • • • • • • • •

A. Introduction • • . ' • • • • • • • • • ' . . B. ·Electric an-d Fields / ~. Magnetic • tl1e Iron. .,. ' in • • • e • • • • • • • •

c. Eddy Ctlrrent Power Loss-. • • •• • • 0 n. Rotor Eddy Currents. • • • • • • • •

,1

III. EFFECTIVE ROTOR IIv.tPEDANCE REFERRED TO THE STATOR i • • • • • • • • I • • • • • • • .r

.. A. Introduction • • • • • • • • • • • •

I

·-·····----··- .. · ... ---- .-.---- ·-·._ ... ·--.,.·-------···--. ----·-· -·-'······---·--·'·-~----··-·--·--·11. ............ Stator ........ 11:ffi1£ ..... Due ______ t_o ____ I_2-----· _. __ ..• • ..• --· •··----•---·----~-----•

. ·-~. , ...... , '1

.· c. -Cal.cula .. tion· of R2 1 · and x2 • ••••• 'C" r'

·rv. THE EQUIVALEl'J'I' CI·RCOIT • • • • • ·- • • • _ .....

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--

Classical Equivalent di·rquit of an Inducti6n Machine ••••••••• • • • •

Assumed -Semi-infi.ni te ,· c·onducting FerrofI1a·gnetic Solid_ • • • ._ • • • • • • • •

' ' Idealized Rectangular B-H Curve • •

Distribution of Magnetic Intensity

\ • •

B ~nd Position of Separating Surface ••

• • •

• •

Distribution of Magnetizing Force

Distribution of Ei-+ec-tric: ·Field.

• • • • • • ~

• • • • • • -

E and z:; as Functions of -~t".:e· ,.- ••• •• • • • •• • f ' . --- I Eddy Current Paths • Rotor ·Sur.f:a-ce in • • • •

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Position of Separating Surface and d , ·Distribution of B • l;lach:ine Rotor. in • • • •

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l'1rnf Due to a Si·ngle Coii of Pitch p • • • •

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11 M.mf Corresponding to·the Assumed Applied Magnetic Field. • • • • • • • • • • 28·

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ABSTRACT ·I

............ ·····-•·--- _, ...... -··--·-·-· ·- .. - --- ....... - ·-- ........ -·--- -·· --------- ··----·--···-----·-··-- ---··---·---·····--'" -·· ,- .... ···---·-·--·-· --- ··----··· ---·-·-.. -.. ----------·-----·-·--·----------- ·--- ------·--·- ·-·-·--··------------ ___ , ___ ,-·.-~- __ ._: ______ , .. ·.·-----···--·--·-,0 - -·

After a ,gene3?ating. unit l.oses .. s_ynchro11ism, . -~t is

usua+ly disconnected from the system •. However, tests have· ...

shown .that if an unsynchronized round rotor machine· is1

left cpnnected to the system, its speed will increase to

some st'eady s~ate asynchronous value at ·which the ele~trical f..

p_ower output of the machine ~lus windag.e and friction losses)

is equ·al to the turbine mechanical _power input. The machine •e • ·~

~'-

out put is due to induced eddy'purrents tn the rotor surf~ce

causing the unit to act as an indtiction gene~ator • .J ...

')

A methbd is· presented for predicting the as:.ynchronous

operatiiig char.acteristics of a, round rotor·. synchronous ma-

chine whose field circuit is open~ The method is tb .adapt ~ .

"the.classical equivalent circuit of an iJduction machine to

represent a_s·ynchronous operation of a synch~onous machine.

The. stator resistance, stator leaka~e ~eactance,.

magnetizing reactanc·e and conductance. representing s.tat.o; '

•, .

'

iron losses ~rt~ generally known· for a large synchronous ma-.

chine. The equiva-lent-.. -rotor-·impedance- referred· to··· the

·f: . - . . . . - _____ ............ .... . - - _,. ________ _

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· .. -- - ·- ---- -- ·-·-- -. _,._~ -~-·-- ---'-------·------·-··- .

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. ...

and ,an a.i~r gap power trans~er of

·i.

, . . / ../

;,. -~-- - ... · .. .,.: - ... , '. .. 2

.,

• I

8 Ho N. - -- 3,r Y·O f

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The mmf produced by that component of armat.ure current. .• J

ptoducing flux linkages with the rotor.eddy currents-is \.

calculated as

·: .....

mmf -- f2 q Ii N Kp Ka p 1T

..

..

The m.rnf corresponding to the ass urned· applied magnetic field

H0 is calcu~ated, and used with the above equations to cal-•

culate the rotor resistance r~ferred to the stator ~Y~P/I 2 • . "t.,,,,._.,

The resultant'value • 1s:

! - r-:

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, . . 64 L q (N K0 Kd) 2 . . . .

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. ~

The. analysis of rotor eddy currents also results in,

thus completely deterrni.n,ing· the equivaJ.e:n:t, ·, .,. ;..,

f :\ . ·J

circuit •.

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.... .

.

INTRODUCTION [14,. 27, llJ ·

. { The accep"t~cl_ p;~~~iv~e ___ !A _P9W~~- __ $ystem. \operation in . ---.. ~---·--. ,-- ----------- . --·--·-.....:...-·--··-----------------·----·-··----·----- . -----------------·-·

. . "'

most Western countries _is that, in the event of a generating r

.~ unit J.osing synchronism due to. loss of exci·tation or due to . ., .

' . a system disturbance,. the machine is to be disconnected \ from the system. Operating practj.ces ·i·n the Soviet UniQ~i ·-- --- ·-··, - ;· "'" ... ··"'··- ......... .

. 1 -however, peimi t asynchronous ~ation of round rotor ma- , 6 - chin.es for up to thirty minutes after loss of .ex.citation.

'-. •

a If excitation can be restored during ehis perio.d, resynchro-nization is · attempted by the method of self-~ynchtoni.:z'at16n. -·------·-----~------- ··

Power engineers in the Soviet Union believe this practice significantly improves povrer system reliability by reducing

tr' ,'}

• ,• -• ., .• •· ·••· •• •••• • •·r

the forced· outage rate of generation. . Operating experience in the Soviet Union 126] and numerous tests there and in Great Britain [13, 14, 16, 17,. 18, 23, 28] have shown that •

------------------------------- -~--------

a significant real power outpu_t (about o.s to· 0.8 per unit)· .

can be obtained at slips of only about ·1% or.2%. T·his out-:-~ -, .

. put is t~e· result of induced eddy currents in the-roto~

' . '

··surface which · cause the machine. to act a_s ___ an~.-~induct.ion-- ---~···,------..-- - •.II", ,•.'.·.·,,o~,,: """'",.,-,,,, ,.•.·, ,_.,,,,., _ _. , •. ,,,,,~ ,,,.,·.•,, .,,;.·,

~ ... "'i-h".\' '1'-"•"''""·11,"..,_., 1~ ....... ,.,.._ """""l.u....,.uu.,=-..,..,w,1"~~,!l'•·i..\-W.)',,"l\.',_....,.,.\(,",111,,.,,H.i.'>.~""""'"'~•·~w"'"'•V·"~·,-,,,·~,,,"

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----.----·------------~~

•

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• . ill

-------'·· ···-- --- -- ---·---~--- ~-'·-"--- -----------a-pp a rent.···· The-- principal risks ---invol ved ___ in asynch_ronous .,.

ope.ration are:

A. excessive rotor l1eating due. to induced------- ---- ------ -- - ------·------------------ ---- ------ -- ---------- --...r---- ---

.-------·---· '"' --

·,,

eddy currents •. I

B •. -excessive armature heating caused by high - .'.i' .

armature ,currents. due to operation at a

very lo~JI, power .factor. (Under asynchronous. )- .

condition~, the machine must draw all its· ' magnetizing current from. the sys tern·.)

C~• abnormally lov1 terminal vol tqge causing

reduced output of station au~iliaries, .,.

.~gain due to the machine drawing its mag-

t . . l t f h ne 1z1ng curren- -rom t e,sy~tem. '

D. in.sufficient pov1er :output at -lo\1v slip i

to produce the necessary braking torque

to prevent ovar-s-peed-c -trippins of the-___ ·_;,___. _____ - ·· · ·. --- ... · --· · ·----un1t ~ • -~· ·-

This thes·is prese.nts a method· for predicting the, . .

4

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·-~ ~·· - -· .

.. , , · asynchronous operating-~ characteristics of ·a round_ rotor syn-•.• • •• ·•. • ·, . .1/f> - - ------:- _..,._ -~

chronous ·machine whose field circuit_ is ___ o_pen-•. -~~hJ~_~ ____ will_be ___________ ~--~~- ~-: ______ : , ------~ ' . ----------------~-·:-----.- -- . ' .

. done by determining -values for the el·emehts of tl1e classical ~-

- . . eglli \Ta!_e~i:- ci!'f\.l~;',of · an induction maChine, shown !::1 __ :.~~1:"_':-.-!.!.-----· . : -. :; . - · . '. . . . .. . . ' -~·"-"='"""=n

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6

' ED-DY·---etJRREN'.r·-EF·F~CTS c-12-, ZO 1 24, 25, 10] - :, .. ···-. - - ------- ---------...,. __ . ---- --- -------- - ---------~ II. , . .,

..,.. . . __ A_._ _ ____ In.:tr.o.duc.ti.on--- . -----~-. -· _____ ;_ _________________ . -------------~----------.. ----- ---- ----- ---~--=-----

,, ' . When iroh--is subjected_ to an alt~rnating ~nunf, · -en~rgy is lost in the iron due to eddy curre~ts and hyster-

esis. ..The difficulty in accu~ately calculating these losses

is due to the extremely nonlineaJ:".relation.between· Band H.

An approximate solution, valid for, small values of H, may ..

be obtained by assuming the magn~tic permeabili~y to be .,.

constant. Hysteresis.losses, which may predominate at (

' . ' -these low ~alues of H, must be estimated and added to the ' . . .

. ---- ··----~·---- ---· -···----' -~·1-·-------- --. . .

- ·------------ --· --------- -- ~ ~--· .. ----. -- - ~ . ·-· .. - .... - ... ··-. . ,... ... . ~ .

calculated eddy current losses~ For large values of H

(those well into the ·saturation region) eddy. curre1ht losses predominate ~nd hysteresis losses become less-significant.·

/ - I

This is particularly true of the rotor losses i-n a synchro-'/

. '

nous machine operating asynchronously, since the rotor ~iron

.·surface is saturated· and the· frequency of ·applied mmf ·(which

is equal to the slip frequency) is low· • . ) . '

.,.

•. .,

Electric ~nd Magnetie ~ields in th~ Irbn .·-,. • . . . · - ------~----T-o---ea-lculate fields ~within the iron, consider a_. I

' ' semi-infinite, -conducti~g, f·erromagnetic- .solid ,extending in· t - '

--~--:· .. -=--== =-~=-·=:en':C±Z~~aI="i@{;:tTO-n··w1:fli·~·:a-=·prane-.. ·faee ··rn·=ilie·x=··-~-·Y·"piane · •... ·· ..... ·-ts~~-- --~-~=-=-==--=-.---~ Figure_ 2. )· · Since the rotor iron can be expected Oto operate

~·. . .•.

•.

--·,------·. ---- ---·------- ---- --- _.,._ --- .,___ --- - ~- - -- - -- '--·- ·- ;. ' ' ~( ,,-; .

I

_,_.· __ ._._~_· ...... _ .. _

' - J

- - -- ··-- -· -- - ·-·. ,., ... _! -·-·---

i , ... {'i

~- '· ...

..

... •

I-. ·, '

.•

ii,"•

,·

- --- - --------- - . ..-

we 11 into the saturation region~ the idealiz~d ~ctangµlar ...

B-H curve of Figure 3 will be as_sumed.

. . - -··· . .. . - ·---·· .. -·-·· ---··· ___ .. _ ---- ·---- -•-------- ---------·-

..•... - ·- ··- ····«.·---·----

7

-- ·- - . ----~---, .., .

-----~ At ________ z____;;_ __ o_, ____ le_t ____ a ______ ma:g_ne_ti c fie -ld .. b·.e~~app~l i e·!L~11h i ch._Jl __ a __ s __ - ___ _

only a y-99n,.pon.~n.t ___ eq_ual-~.to: __ .,. ~~ .. ~ . ' .... -.---~--·---~~- ,.~------· , ... -·· -- -·--··· '""" - ' ~ - - - ·------~--~------ ----;-------· --• •• _..,,-~ • ...,_-~ti°".·'• - -- • • - ----~-----·,.·----- --· ~--. •I=----- ' -- ... -·-·--

·,,

..

'• :, . ~

·,.

(1) ·:t·'.'".'.·

~. '

-· ' .·,·. I --1

The corresponding electric field will be J1orma.1 to the . mag-

'\ netic field and in the,x-direction •. MaxweJlis equatioris

then ·reduce to: '

' ( 2'} · . .:·- az = ye: ·wh·e.:re,.: · ·y --~ GOnducti vi ty - -

. ( 3) ---- = - ----az at

Since, in this case, with a rectang~la~ saturation curve .

ab/at may be infinite, define: ~

-00

~ = b dz

. Z

. (·S) ah I - - ye - ' az

'

'-.,.., . \ .

- , ,,. --- ~. ·,· . : ..

. :r._· ..

i . ' '.JI'"

. . ,,r,'.';l':'F,,9:'-.J:'(:J:;t,,;:-';;"'t ,,;,_'>'j~:;,";m,,, , -·• ''' S• · 'd·> ·' ,_.,,_., .. ,., '"w~,,~.,,-,,, '""""~"'"'"~~-"·"•••••'-'"'" ""' ·. ,;,. '"'.'~·'"·•-·•·-»~, ·-'"' ''"·""'"·,,.-,,,_.,,,.,,,.,,.;-.• -,,, .. w., ~• ,·• •··"''-'••,·.,-., , ,, •• c••·J·c.,_,,,,",·-·, .'-.oe:,· , •. ,, .e< c.,· ·,,,._. .,,. ,._,._, .,_., , ,, ... , .• ,, .• ·•

... i'

'/,

---------· --··--- -------··-···

..

\.

... , ..

\ -

/"t,,

-1

. - _·'. ':.;,.._ ,~-o. .•... ,)._.-

H=H .0 • sin

: . I

,.__,, __ _

X

wt

··-.. ·-···-··---------··--··- -·- ----- ··-- ---·-- ---~--···11-- - --------- ...... -.!. •..• _; __ _

• ••

---"---------------.

z

Figure 2

Assumed Semi~infinite, Conductin9 •

b 1

f

~ . _J t .u.

• B ' -0 :.'·

- _+.

Figure 3

Idealized Rectangular B-H Curve '

I

'I

•

J

8 J .. '

-.. .,

,,'

H-, • •

Solid .. •'

Ile C

[ C

.. ...

( h

- '" ··----·.----i • . I .

.~ .

; . ..

i

I; i aqi

e --(6') at

.. ,

. -.. ,. --: .. ..

. - ' . '

- . . . . - ---'----')

c\, .. - ..... , .. ~.

.,

l t : r l

I ' f

I

t

t . ' . i ' t

. - _. -.. . . ---.- ---------- .-~. ------ ------------·--"7.,;.._ ·------_____;~-~~-----The approa

-· --- .. ····-··-- ·-·-- '~- - -- ... ~--~--~--a---------·--- - ---·----- ·------·---·------- --•

-~- ..... ~ ... ~ ..... ,_ ... -. -·· -·-------- - -~----·-· .. ., .. ---------·------·---·--·-,· I..,

. (. .. ,. I

. '• ________ ---~-~--- -·- --··-·-·-· . ··-- - ---·. -' .

10

... ~ ' surface. ·At any given time, let ·the ,

~

separating ~surface be at z = z;. (See , .... I ·,

Figure 4 .,) ---------------·--:--'--··-·----------------. -· ~-·:---- ... :. - . ---- ---·-·· - ---·· ----·· - --- -- '41:'• _______.:...------- .---·~·-· ----·-·----~------··-·--·-- . : . . . ..- .---:-:..,.--. . .

I

•

.~;.1 .,

:· ..

-_..,_,.

\

linearly (in s~ace) to zero ~t.z = t.

(See. Figure 5.) .~

4. e is equal to·E to-.the left o·f

z = z; (o< z< ~) and e is zero to the right

of z = z; (z>z;) •. · (See Figure 6.)

.... · .. --·· . .:: .... • ... _ ..

The proposed solution -meets' the boundary condi)b±·,o·ns :· "' ·.,. t - ... ,· ..

\; ?'··

~

B B • H'. - sig - 0 and .H H • wt - sin - • 0 . .,

- It remains only to check" it a·ga_inst equations ·cs) ~and_ :[6): :•

From equation (5): ..

I

(7) II - - y~: -' r;

.. ,.: ··'

,r .•• .

From the assume·d .so_lution: . . ·..; ., "'It • ..

- • ·-~,~. .-.- -.. - ...• - .•. -.,..---.~~.- ... ·-.-->" .•.. --~ '.\.,_ • -• ·.'., • '• ,;,n ,,,-. .- . • p •• -=;,~-=---,~-· • --- ~ -,., - " -- ~ - - • - - - - =7-- ·c - - , ,

-

(8) a~ az;. ..

(o

("·'

,·r

..

..

...

}:

, ..

-·-·-·--·-·---·-·- . --··-· ·-·-.-·--··· -- . . --~:~ - .......

•

. c

, ---~-. --. ---·-····--"--. ' . . ' . . - ---,

b,y

-B 0

J

)

J

.. ·

\.

\ ,

• f

,. r-,, .•

Separating ···Surface

I

.._ --....

V

'-

j}

z=T;

dt . - -- dt

'

z=o

Figure 4

Distribution of Band Position of

h,y

H

Magnetic Intensity Separati~g Surf~de

I I

z=z; z=-o ,

Figure 5

Distribution

z

e

E.

,i ,.

r'1agneti zing

I

Force

)

\

z

/

.. z

11

',

/

•. - .. . . ·... .•• -··

'i i i f ! ---·-t·

- ' -----.------------------

·,r

..

--·--------···-----------------·-------- -----·------'----'---~----\1-ilS!---~ -----

----~--·-- •• ··--·-·--··-·---·· - .•.• 0, ..... , .• - .... ----~- ··--··---- - ••.•• ,----~---·-···-·------- ·- ---·=·-· - ----··········-·"··-·

.... _, _____ ....... . .

Figure 6

Distribution of ••• -i~

J·

z=z;

E , ·ec···t-·· • -.1. .. · _ .··r:1.c

I

z=o

]?i_e.ld

.. - . .,....... - -- ·-·--·--·----······------- --·---· ·-----.-·-----·-· .•.. -1-.

z

--~--,---. ---~-'"·-------' "'

_j

( 10) '· e

' ..

. - r .. ! --;.j

--

I..~---

~-

E

.. ~--- .... ' "·-;- ,. . ,· ·.· -,. -· ·.,, · ...... , ... ".-.--.- _·,,,.,,;._--., .. ;,-_ ... ·( ..... ,: __ ~-.-":".";'• .. ·.', .. ~~--·-----/- ,.- -. '·~.'-'_:·,:· _.

1, ..

I ..

' 1-2

.... -

(o

i-a----.... -.. _-___ -____ -_._• ____ --._• ____ • ___ • ___ .•. __ .•_._._• ___ •. _____ •_._•,. __ • ___ • __ ._-.., •.•._-.._,· __ • __ ..... __ •._: ..... ----.--. ___ .... ___ ... _· .-------·-··_-, -.. -___ .... -_-. ·--·---·--.. -__ -.. _ .. __ ....... -. -__ ....... _ ............... :· ........ -.. -._:: .. ...... __ .......... _-._ ... ·_: ..... ,_ .... _._ ... _ ... ____ .... __ --__ -.. _ --_,_---.-_ ~~~~~~i~~~I?, · -"

/'

I •

.~.:

·.t:.

. ... t.. - . _ ...

- I -

,,I ,:

(16) t = 1/2 2H0

wyB0

. L

i

. .,.. 13-.1.·,:

_,· :.~

' wt . . "

. . . sin 2

. . - . . . - . ' . '.. . . ... ., . --- - -·------ -~--~--------------·· ·----···. ··---------- . ·-----·~·-----··-·----------···· -·

.;

-\ .

I ...

-..

--- .· .. . : . _- . . . ' .

.,

Since the maximum depth of penetration ( t=o) . . 1S ...,

attained at th~ e~ of.each half~period:

(17)

:n .

.. . To find E(t), rewrite equation (7) as:

:ii#'

\ ~ • - I

H l Ho • wt -... sin (18) E - - - -- -

y z; y 0 • wt sin ' 2

. ' ····- . ,... :. ---·· __________ ,, __ --· ·- -- - ·····---- ·----· -- ·---- ···-------· -·· ---- - . ---·-··· ......

.. n

[

-

2110 --wt· - ...... E - COS-_ 2 yz;

.\ •.• ,, ..

(.o

•• !_ ' • ~ . - .

. . . ,4 ' ·-

T O •• . - ---- -·~--- -- ---- ., .. ______ --·· ----·-· ---- - - - --·-·-------------· -·- -,·- ---- ------ - •',· --~-- --------- ---------· ·--~ ·-------------- .. ---~----------·------ -----·-.or.',--.------------·--------• ' - ,, ' .

'' . - f ··-- -·

.. ! ..

·- ,;f' --- ..

-·-· -~-- - --- . , ...

• ;: "1, 14

I ...

.J

· in each successive half-peri·od. _ Equation (19) also shows

that E is not a f.unction of z for o\.ll':

E, r;

,

--\

l .....

'E ~ \

E

\ \

1T

and

/

I,

I

/ IE

/"'

I I

I

1

Figure 7

......

21f

as .Functions-- of

\

'l

I t

• I r' f

,,, .15 I ••

-~----····-···

z; "'

I

\ \E

\

' ' 31T wt

wt

''"·"'''· ,,, ~.-, ,,,, w< •:.v,,y _,._.,.,, ,~r .-,, ,._,., •'~'-"' ~ ., ~-•"'"'''' .• ,,..,, :•,•, .. ;, ,,.., ,0. ,,.• .. · ,, ''''" .,, • .,,-, .•• , .. ;;, ", ... , • ._, .. ,, ,,. .. v.- ,,.,.., ,.,, ,, ,,.._, ._, •. ..., ·~·· ,., ,, ., , .~,- •. ~ •• ,,~~-, ,,, ,,,., ,·:e .·.< ,1, •. • ,·._,.,,,._,_, • .,, ., ,J,.,', •. ·"'·"-'.·•···''·' ,:,;,, •. ,.,~,« .. •.·,,, ,:,.r.,,,.•.u ,., •fr•'

---·------£::,

.,..

;. : ''l•

·•·

·\:.. I

.}

. .. --···-------· _. -----·-·-'···-·-··---- : ...... ·.' ·--- - ·- ·. •- ,.-.,--.. -·····-·-···-·- ...... ,. .. - .. -,..-·,,--.· ·- ' ·-·-----·.-··-··-·'-·--•··-···

-'

·'\.·_ .. · .. 1

.;

..

Calculating the fundainental of E:

(24)

/

(25}

_(26)

{27) ·:

• ..

(2 8)

.....

-----· -ext--·· ... • ---------·- -

/

-· . ·--·· ~-

E = 2 l 1

(An cos nwt + B sin nwt) n /

. "'\ .

I A -- -n

T

·...:

1 B . ,n --- -·

,T

A. -- --"'- 1-----

)

Ho Al --

yo1r

---···-·· r O i.'

Bl ::: 2H0

yo '

·T

E (t)

0

T E (t)

0

,,..

1 2

2TT

2 -- + 2 3

1 2

_,,-

cos nwt dt

sin ·nwt dt

1T

0

1T

0

cos

cos

(wt)

2-

(wt)

2

"

..

--~ - '

""'·

cos ·rwt)· d{wt)

..

sin (wt) d (wt)

....

.,

16

,, ·- ·-- -~· :_~ :

;/,.

.!,'

/

..

- -···...---· - ' - -..;. .. :

~ . . . I 'I ·"' •·"'"-1:·

- t

•:·. ·.-····~---

I : I :

. ~ : '

·~

·'

'

.. ----·-------~---- ---~· ·--··· . -·--·-··_._ ___________ _

'

'

:( ,·

,\

.i

_ .......

(29) Bl - Ho ·2 + - -·yon 3 -- -- --- ~-- - -----

2

--- ---- ---- ---- , _____ _;:_·._· -·· .·_. _·_:c .. ·---'--.. -·--·~---~~-.--.. '· · ..... .

\. /'

· ...

(30) . E = -- (cos~ wt + 2 sin wt) /,-

8i-I0 .. ~ (31) - (1 2j) E - -- 3yoir

' ~

.......

l Calculating the real part r of the complex surface impedance

(32) n.·

(

''Q,,

hence:

-n • •

... :' ...

H --

·'-

H sin 0

\·

wt

,,,

j.

-~-·-

·.-~

-- ~ --~~-- - ,......... ___ -·--

.. --·------- ··-···•-·-.. ··---·------·----- . ···- ~----- -· -------------. ·--- ---------- -________ :__ --·-·····-·- - ------- ;~,

" -·E 16 ' (34.) - (1 • 1/2) \ n - - + J - -l.· - 3nyo (,.--. H

i.

;i. ·-- ••• , .. --··-·- _..,,... ... ,.,.., .... -.~-- ~---~-------~---" • _ ... ________ _

17

•";;•

•

• "··

.,

r :{

' I

~--

-· ~,--,.,.,...,.--, . -_..,.·._·~> .·.·--~.·.:·-· .. ,.· .·.:.- .. < .• ------~-... ·----·· -··· ·- ,· '".·,;-, ••

•

,. .

0

• • ,1.'· ' -

16 1· 16 r - ·- - - -- - -, ·(35)

31T yo ' 3'JT )

-·~-

-: :i

,~ .- .

wB0 1/2

2yif0

; 18

.,

-· -·-----·-.. ·----·- . . .. .. . . .... - . . -- . -·-- . ·- .. --.. ~·-·"' ........

,~ :.

•.

(36)

. ,.

becomes:

1 .

(37)

( 38) -

( 39)

•:,-.

'?·

. 8 1 8 ~wB0 112· X - - ·' . 1:1. - - y5 -31T 31T 2ylI0

r ,,,.

~

·The eddy current loss per unit surface area then

. ' •

N

• j .

1 - r H 2 ·2. 0

•' - -·· ---·- ... ·- -- .

•

1 16 t~· -- -2 3,r

8 H 2 N· = - o

.:,,·

·,-.

--: -:-· . ' - . -

-

wB0 1/2

H -2: 0 .f .. 2yH0

•: ...

'

---~· _ · ______ __J_lf_ ---¥-0 -_··--. _, - ;• _ ___,,____ -~-~~~~==-====-"=........-=·===

.... J ,_

,The eddy current an.alysis 9iven above is for a

'(

\'

l

surface subjected to. a mag.netic field varying sinusoidally

with time. In th~ case·bt .a machine·operating asynchronously, •

..,..

.,

I

I! . -·- i ii

I

..

-- ------- --- ·-· - . ---~- -- . . \

.. ,/

• . ......... , ..... ' . ., • •• C

·/-· ·-,· •''":" /

~:

~-· 19

sr.

f· the rotor surface·is·subjected to a field rotating-at ... -., " ..

. slip frequency and sinuso1dally distributed ·in space~

Thus, any point on the rotor·surface sees a· magnetic·

' l--· ,·,."

---·-----~~--

--- -· -------~~-·-' .· ·- ·-

~'

.,

. ,·

a,a·

·- ------· --'--.--~----,,-----,...----,_ -~-.· . . .... -. -field varying sinusoidally in tim·e •

.

'Figure 8 shows qualitati-vely the. eddy current paths •...

in the .surface of a rotor at one point/in time. These eddy

currents p~oduce a field rotating at·slip speed relative to ,·

the rotor and at . sy~chronous · spe~p. ~elati ve to the stator.· • Figure 9 sho,.,s, for.· a two-pole mach.ine, :the position of the

separating surfa6e and the distribution of the magnetic

intensity Bin a machirie rotor.

) /

.... . h ··: ' .. -···---,--.. ····- :·~ . -"----.--- ', .... - ... ·-------- ..... ·. -,-·-·····-· " __ ,, ___ ·.---, ... - ·.·•·· . ·--"( -,,,.,

C'·

-~.: .;,:

:, •.I -

-... . ~,, .. 1. . .

. ,

.·1;· .

r

~-· ...

. -· .. ; ./_..,.. ~:

·.: ':'\""

·"'··-·····':

'. .... --:-.---~-... -··· ~~~

' · . . ,l"' ----:-·"-· . ·_... "---'-· .. ...:..:...c----;--~--;--:-:-~--'--"--'':-:----~----,., . :\. -----~-----~------,_ ·---~···--·

'·· ar..,~«'W"--'••·~=·~,,t.uv~,,~,u,-,•.,.,,.,.,.,..~-.,.-,;11.110\•-~•--W"-""'1.-.,..U-:.....M'!IO>'

' .. • • !

,' I ,,

.f f;

"l f I I ! j

I ,

i j

I i r I ;

\

t: .> i

·- ······-··-

i ...

.... -... j,

I

---~···,.,,__ __ ._._._. -·~--·----·. --------·--·----- - -- -----··- ---------------------- ---------------------··---~--~.· .. ·-~==-- - .. - -- . •- . . ----. - -- . . . --- . -- ·. ----.- ____________ J II

I

,· •

~.J

1,11

L i

l

Figure 8

·' \ E.¢Idy Current Paths • in otor Surface 1: ·,

./

·,

... ··•.

~-:

( •.I;

0 . -----·--·------ --- -'. ------';--· . r---~---, . . -. ··-·-·.' .. --~.--- ·- _,,_ - . ·.,,. ,_+_..,_.cq: _________ . .. ..

,_,

.... , .. ,,,, •. ~.,-.,.,,.,. ... ····· ., .. _,. -.-.. ·-.' ·---·--- .. -:------··· .:··- ... ··. ---· ---~- ·-·--- -- .·

"

Figure 9

Posi~ion of Separatin~ Surface and Distribution t B • in Rotor :,.: of Machine

'if.

"

I . ., ...

. \ . , .. \ ' .ff'"' .

'"\

I ' '"' . 21.

' ,. ,:i·

·_.·,t '

. ' '

III. EFFECTIVE ROTOR IMPEDAJ.'lCE REFE.RRE_D. TO. THE STATOR

[ 1 , 2 .. , 1 7 , 5 , 6 ] ,.

...... -·-· ----··-··- ---,-~-,. , __ ---~--:-,-- .--,~.--... --~----·~··-~··,---.·--- .... ----·- ... ,,, __ _ ••' • • •- •••·--· ,. •-- •••--•••-•••• .. ---·--·- •• - -- • ----••·----::--·••-"".:"'"""'"":-:-::~•--..,;-•-"•••~•' -:•.-,'••-:~,•-. - ___ : ........... ...:.__.'".,--...;-=-..c-,.......---'--,\';c...~·-·.•,-.-~~-;_..o'---.,••• ....... __ -"--:---.."'- '---•S-- .,.-. .!._._ •• ..'.'.:..!, __ ·•r-;::"--.,•;•.-=-·--• • •·•----•••••••-

-. -.~

•

•, .,;~ ;

"

A. Introduction ' ' '

To. find the ef.fective ·rotor resistance and reactance

referr~d to ~he stator (:R.2 ' and X2' in Fi,gure 1), the ap-

proach will be as follows: J

l. ~-..

1. _By conventional induqtion madhine

theory, the mmf produced: ·b:Y the . load

/ current in the sta·tor {_I 2: :in Figure ~)·-.

must be equal and opp-osite to the mmf

prod~ced by eddy cur;ents in the rotor.

In other \*lords, I 2 in th.e equi va1ent

circuit is that comoonent of stator .. . . ',fl.,.

current producing flux linkages with . ,

the rotor.

2 •. Determine the nunf in terms of r2 and

the stator winding configuration •

3.- Dete.rrnine the mmf correspondi-ng _ to

equation (1) ,· the assumed applied -

maanetic field '.

-____ )

- J"' - ;- .· .• . ' . . . : . . ·--.. ---· - ···-~~.-:.:_· .:___-.:..~.

4. Eliminate the mmf between the equa-

tf

for r2 • Knowing r2 and N (the power

' . . . \'

: i

-"'"''-·-· ••• ~ ... p

·---· .--·---·--··---- __ ,____ . ,, ___ "'

·. l

·'i

'

;.

, I

---. .... .,,(. __ - - -

: ____ _j I

' - _22 i

\•

• transfer per unit area in equation ..

. . - .- .. . -·· . . -··· .. ··~·. ••: - •• ·: -'··· < .. , ••••••• ··-- • - •• -· • - -· - --

' ,-... (39)) we-can solve for ~2 •. From

J equations ( ~?.) _____ §11}9 __ (__J§) , we _know _______ -------------- ----.---~------C-= ___ .... _ -~--- · ·-----.. -----~------ · ___ ---'-·

I •

----·-·····- . - -- .......... ------. ----- . ___ ...,__________ . :·-.'· __ ...... --···· ---- - - ~ - .

. •• :. • • ..!,. ... -- _:... - . -· ,

.. - -B. Stator mrnf Due to I 2

Figure 10 shows,··, in a developed view, the-· ntrnf. o·f .

a single coil of pitch p carfying DC. For this case, the

mmf is:

(40) mmf = -pi-I

\

.. -.-·-.

(41) I nun£ - ( 2~p) ~1 -/":

. - . ·- .,. :· /~i:.:f' . ,. -· - --·

(42) mmf = -pM ;.

.. r

1:

\.

_f

. --- ---- ..

1,. ·. ·' ... . ..

' ; ',~'" .

r· ..

..

. .

. (1r

...... ---~-------·- . . ... ~ _...::..__ ______ ' . -.. .:;·_ . ... - - .. -~ . . . --

• Air • gap ~

j " A.

' -4 ~ . St ator

• 1 ron ·1

~ -\

. .,

·I

··-' ' --------- -----------------------· ~----·

'

·- .---------·---------

..

,.

-,

..

-

f'Jinf

.·.-..

'/ / ,,

... ·-

·,:

2,r

(2-p) 1T

Due.to

~-,. -,

•

........

"

"" ""' _'1 ,,

..._ r- ~

Figure

a Single

-·-·----·---· - .. /. ··-··---·-··-·--·-·--------·--·-·-·-·---·--·--~·--·------·-···-----

.. · , ... ·-·· ·--- ... -... -·---, ·- ---····-----"···-· .... -·~--- . -·--. --·.

,.

.A· :t_

'-"'·

\.

~

p,r

10

Coil

23

·,_:.

- - "

. ·'j . . ., '· ~-: .

--. -;-. - • ~-·-- ---~ -~--.-_·-·.- --.--""="":"· .. -- .• --.-- •, ·-··.·--.------,_·--------·-·-~ :--·-- -----··:-·-------------··-··-

\

·-! . •:

>

1

I \.

' 'ti

.. r-

r/ ,

;;.,,,,

j "

(2-p)M

' . ~ ,

I I, ~

pr-1

-~ ,,

of Pitch p

. .

-

-·,

.. .,

-. -

r.

•

"'

i

...

. . . i

l· -24

'-! ..

. .. !

' . I ".t .. ////_ . ~-. No-Ee that the terill sin kp1r iS the pi teh factor for. the__~-:~-~--:_~--~~--~--=-------· -- ---- - ·- ---2--- . -- ---- = . . .' -·-----·-----: ~---·~------~-------- ··--------- --------

- . kth harmonic, K k • P ~ . i _j • --·--·------·--···--·---·- - -----_ -------- --------------·---··----------------------·-··------) _________ , ___ --------·-·--· ---------·----- ·----- -- -· - - - - ------------- - - --- -·---· ·------··----- ------- --- - - -- --- -------- ------- ···---·----

- - ------------ ··--····--· -~-------: ··-- . ·- . .--·-. -

.,

. (

... - .. _·, --

..

¢

_Applying a sinusoidal current in place of DC to

the single -coil ··of pitch p:

k=oo (44) nunf = l

k=l

4M. . nk sin

fl

. fQ

f

1J

. .

Resolving this pulsating field into forward and backward

rotating fields, equation (44) can be rewritten:

(45)

\.

k=co rnmf = }:

k=l

/

. 2r~ kpn iTK sin 2 [cos(kx-wt) + cos(kx+wt)]

_ . .,,

. ~·In a ~ractical po~yphase'machine, 1 let:

J . '

······----- .. ·-·--·------·----- ·,;- _·, .. . -. .- .··---.- -•• - _ . g = n~er of .Eh~_§e· .pelt~ per pole. . . \ .

______ >---- __ P_ ----~-- number __ a:f __ p_o.le .... ~p-ai rs • -----· -~---:-------~--------------------- / I

n = number of s·lots per phase belt per . ;. - -·---- -- - -.- -- ---·-·· . -~····-·--,-------·-·:··-····

'·

,,

..

- ·--· - " . . .

;

'I I

~---···--~-_:.-··---·~--~,·~·--··-···£2.:!.~ .......

'-· ~--···,~' .,;.,_;.-,,-· .· ~· .... : ..... ,. ~ -.. ,-,=--···--"'·--,.,, ... ., -~ ............ ·~'-· ··- _,..,, .... ,.~-------.-~------- .~,-- -- - -- -----·- - --- - - - ---- ----- - --- ------- ----- -- ~-·---·- . --- .------. :_ ' -------=--------·~---~--··---· - --- -----· ------------~--~--. '.v --

- - - --- - - - ' - - - --.-:-:I' . !

. ·--,

·;,· --

'!"'- - .•

I

.J• ~ ..

·rn a q-phase win·ding·, the phase belts are 2,r/q I electrical ~ --i

l

----------·---·---·· -- .- ---· ·ra·a-ians~···apa:rt:; ·· aria ·the time ··phase

. . . - - -- -- -· ----- ------'-------- ------------------·-- ---·""·------ ----· -~-

)

). ·-~·-

"'\ \ '"' '·

26

and ,.t

--- -·,----. -~--- - - ~----. >.. . ... -· - -~- . :- .. ·-· ...

' - - . - --L

.. --· ... -.~ , .. . ---- ,· . . . . ....... ,, ····· . - - -· -- ...... -·· •... - - - "'·-"··· - - - --- ····-- ..... _.,_._ .. ,. -- ----• ........... - ....... -· - __ ,,_

(49) N = 2PnT

-----····-·-- ........... ···-- -~-----·---- --· ---·--·--·--------·-- -··-··-··----·------~----···-···-·- --~--~~--------------------' . - -~-----···--- -- ·-·····----~-- --· ,____________ - ;-

there.fore \

(50) M = -./2 Pn

)

~· ,; .

-Substituting equation (50) in equation (46):

-1···

. ·. ~ ' . -~ .,. ·r· /" .

~ .

J

(51) mmf - (fundc'..mental) = cos (x-wt) ~

Pn , .• , .... _,. .• .,._ ."-'', ."."' , •,• • ,.,,.,,, ••>,~••

---- - ---

' -

- ~ ·· •..

....... (_.,

,. ~ .

:.,

f

- .t. .

(52) H = ~o _$j_n wt - --- -.1.,. : . .:· ~-- • : - ---~- ··- ,.:.. • ' .. ,. --·. ··~ . ,·· ... --- ·-· .. - . . . ',·· - .... ;- :·· . --;-, .... ...

---- ---- ·'''"'"• . ' This relation holds- for any given_ .. ·point on the rotor

surface. For the.entire rotor circumference at a given ' -,

p~int in time, the magnetizing force is

(53)

•

H = H sin (Pe) 0

·,.

,

where Pis the number of pole pairs and e is the position· - .

on the rotor circumference· in radians. The peak mmf at the

... . ,. . ---- ----'j; -

,t

·. ·~ ..... -·- -~ -· " - - . -., ..

- .

.,._

.....

•

'i

II

ii

I" II

:I '111

~ :1

I ~ II ' :: I

'~

I~

f '~

ID I

,D

I u

-rotor surface is therefore the integral of Hdx over one-h~lf

pole pitch.· (See Figure 11.) ' Ill I

I

I - • ··········-··- :·-··.--.-- ..... ';i'-. ?--~- _., . ............ .. -.. -""""""'" ..... _ .. -. _ ... , .. -·. ·- ..... _ ....... _. __ .. ,_ .. .,., ...... ., ...... ___ .. ___ .... ,, .... --- ---· .. -,~'""""· .. _,__' •«·-·· ....... - ·- --- ------ .... _ ... " _._ .. _.~ .. .:· -~. _., -. ~-~.---_ .... _ - . - . --- .. -- -- ---- '),

··'1

(54)

.·:.-

----- - .• .... ·;,..;1

.. :t'·

... : ~ . ·i

' --------·

(55)

' •. \,.

I '

peak. nunf =

7f'D X=4p

110 sin_(:PS)dx

'! .. .,,.:,.,v:_.,,.- ,•, -•-~-•'"-~-------·------L ___ ..

'• . .. ~ . ...

x:::.o

.... ·- -- -- .. - -~ - ---·--·-···-· - .

peak mmf = H • 0

·sin

x=o

• I. t,

'

-·· "if -, ' '!• ,';J

·-~:--_ --~~J--·-'··--·-·. -· ~-··-·-· ...::. __ ... -·-=-------·-:··--·-·--_---·

. '

...

....

·:..r .,

':L

r/ .,~

.. · ...

,

.".•

.v.

'4

'·

I.

i I 'I : i : i . \ i ii

H

; I ! I I i ] . I

. I .. ,.!

:(J

\

. l

'~

I'\ ~ t

I I

j•_[ 'I

l·.1 i• "li ,;J. i • ; I

11· I I i

..

,•'t:

.·l

•

. r

I I 1·

r: I;

I I

I

Mm£

r r

-~ !

·:'-:

ff::;:::Ho • sin P8

• '",

;,,

,-. ~·· ii:

'

X = 7TD 4p,

e = 7T 2P

" i . • ,

./

Figure 11

Corresponding· to the Assumed Applied

/

.... r, \. . :,· ..

-.. :} ::,

/l

·-1

"-

~ ..• ,~ - i

i I

\, X= trD

p

e 2,r --"'

p

'

•,

.:

·• ..

J;..,:

'f

.,,.

. ,

~ -~ l'l it;; rr 1,-';

lt k1 ;~·'c. :~!\ ~ i

!,l{i"'·;

&;re

i'~\ ts;\\

~~--·· ,tr i'c''.t (-·,.',",

\j; .

~ . ( 56 )· r

, ...

peak mrn_f

-.

. .

I ; ~

_ H0 D ..

·,$_

-

' . . . ~ 1

- -- -----;-. -_ ·~ ,

·, 29

~ ; ; ·-· - :

I

. i

• • - ••- -•.•·•- ., '••,-•·•: ·-,' ~.,,:,.:· • • : • ._ • "C' .• : .••. :,:.:, '"• ,:.,- • ,,-. • •·•• • - -_ 2P

' -~-

' ---· --···-··-- - - - -·- - -- -- - - - ------- - 1·

... ---------· -

.. ';_,.,, .. ,

.,

"'" '}'·

(

f

---- -----~·-------'----~-~-.-.;-----·-·-_ _,....::.:.... _. __ ._._. __ . __ .. _____ ( 6-1) X · , ·-··---·-----------··- --------- ---------··-- .. ·2--

.-:. ~- ... . ·------·-· _, _____ ._. --·~---

0,

'

·:~

- -- ------; ·--·----·-··-:

·'

,.

'"

..

.}:;.

·-~

. . ~~-·--:- -·-----.--··---.......

where 0

··~·

~-:;;.

Ho

'/

··- .. ··. ----,-: .

::-""'" ___ ,

;;,-.-:;. _.,. ..

-·,:

;- '

--~.-----. .....,., __ ..,.,.._,,_, __ _

'

32 L q --(N K K ·) 2 p d

t'.

y D ·, ··;;.;·------- ----------.----

2H0 1/2

-··- ~yB0 ·, . '

•

I,

2 12 q I2 -N KP K d,, -

D 1f t

~- 1.,

r

.. . --· ·-· ·····i - ........ ,... .. , .....

,., c.·

, ...... \;,"

. ·. ·,;;

"·

~-~

t.:

··,:.:,•!l;;'

"

- ••• _ _,;,,,...!a;...-

--~----

-- -~- ~-· ---

30

... ,

;,

_________ . ___ · __ _

.,

~

.~

. ...... ·· . .

. . . .

. . . .-· . . ..

,,, .. ,,,,•,.;·-J·'·V

________ ..,_____ ________ - ---- -----

"

...

' . . ... .._ .....

• b .. . --

''

31 r'

. . I

----~- --·~--~~•--- r-- ·--- .. - .. ~-- ---- ___ .__. __ ---·-·-- ~··--- - ---- ."_,I .

-- - --- - ···---

\

.....

IV. THE EQUIVALENT CIRCUIT [ 1, 2, 17, ~·,. 6] ,. ,,

._;. ·---·-· ,".: ... ·- ..... - - ···- ..

,i'f~ . g

In the equivalent circu~t of F~gtire.1: ---- ~ - -- ---------- ---- ------ ---- -----------------------~--::. - -- ·- ··-- --·---- ---------- .•. ..... -------· .•.... :: ------ ______ -, - ___ .;;=:;--::::--_ - ~ -··--- •--·· - --'---:-- =~~=-----,----...--,. '-----,.··---: • •·-·=-=·- ._, ._ "'----.---+--------~---,C:..----=C ...... c;:-_.

,.

,. . ' . ·- ...... ,1:~-, - ' . ' ,·,·· ........ -

A. R1 - stator resistance

B. x1 ·-, stator leak~ge reactance r

c. Xi1 - magnetizing reactance of stator .

' -D. . 9h - conductance representing losses

""

in stator due to ·eddy currents I l

ancl hvsteresis -.,, '

,E,. R2 ' - equivalent rotor resist·ance,

referred to the stator, given

by equation (60)

F. · X2' - equivalent rotor leakag~ reactance,

referred to the stator, given by

equation (61) ~. ~- --- ---. ____ ... __ ,_ ---- ·-----------------·---------~~--~-- y ,,,J

The equivalent circuit (and hence operating char~c-

teristics) for asynchronous operation of a round rqtor

~i-nchronous machine has ·been comp~.~tely. determined,. since . .

A· through D are .available for a give;n machin.e from design·

.·,:

·.~.

....

" and test data, and expres.si_ons .have been .. derived for E and .. f. · In a machine operating above sync,h.ronous speed, the.

I '.)

,·' ,;

-lu ,;I II

I

'1

J .. ---i . i

.. ·-- -- ,,

~···-~-=~==~~=~=-:. ___ -~=~=~~~~=·-~"··-········-~: .. ·.· .. . ~sli:p-~-~-.i·~·-···-:neg.a.tiue-.,~,·--;-S~i\~c,e_. _____ i_t __ . __ is ....... ~de_fine d_.~ __ hy~:--~---·-·-·--··:-·----·-·--·---··--~~---·--------~·------· ··-------------~--,-c-a----~-.--~=-~---. --·-- .~-.:-~.~• i • • I

. ~ ' ~ . I . . ' '

(&2) s =

'

f - '

(l) -0

·w 0

- r·--

·-'-~~-·-·_....,._ , . ·,·, ·.·, .. ;, ".,_. ·---. .. : .. ·_:._, ; ... ·

. -------~---·. __ : __ - .-.,·_ ··------- -- .· .

,t.•

----. (' .·· ' '

. :• :--::...._ ,. ... .. .... ' - " i ... l • 'I

.~."" ' ":i, ••

...... ,;,. j

\' ·'•. ·, ------····---------·-··-----·-·- --·--·- __ . .,_c ---~- -----~-

'. where s = s_lip · .;. [,_ ..... :·

-.. = synchronous speed -- rotor speed .

. ----------------- - - ------~----~-·----~-------- -----------------·-·- - -·- --~ ---.~ . ------- ---·"' ··--~------- .. -·-------~--··---·-----··-- ·- -- --

:or .

·I

·.~-·-.

Therefo.re, R2.' (l-ss) · t. • B · t d d · d ct· is nega 1ve. y s an ar in u ion

machine theory:

.. . •• .".#

. '

~·

(64)

j

where PT - turbine power -mach.ine shaft,

and friction .• .. ,, .. • ,,.-. : .• fl"

PRL - I2 2 R2 ' -

wher~ P = rotor loss RL

..

input

less

:,·~

I.,

;~

,.l

on\ t·he

\iindage

-.......

• < ./

•

n_

Eq~ation (64), is of primary importance·, since its ' -· . ,t: '

,I

32· ,

-·- '"1·

- ---- .--~· - ~·-·-~ - .

.. ,,:

..

..

- .' ·. - ..... - ...... ~~

• ,:

-. . ... results ·for different operating conditions can be compared

•

,•

........ --f

- .

with the rotor circuit .power loss under convent~~-~~l _()P.~F.-~-: ........... --------- ---·----

-- ',,----- -, --- •. -·· ----- --__ - -- - .. ··--=-· .....

tion. •

,/.,

'·..t.·~-

.. J . '

ti

I.. '

..

-J •

:c

.. · .. _, ·' ,•: ', ~ ,• :,:· ,;_ ' ,. ' .... '-_-....._

" ...

-- ~ .. - . :.f

.• . J.:, .. ...

... - -~ ._, .. )'. . -----· ... -·-···-··---'·- ··- ·----·-··-··---------------~- ··- - --- -- - -- -- . -··-·- -- ---·- -- . -- -~--- -------- .. .. ,

-.:

., ·,

-Similarly: '{ :"-·

...

) i' . '

.• ~ '.,', ~ ~~--- · ----~~-------------(_6 5_). · - - - ---~-~ --- --- .- . .-.- :· -,, ·-t·· '· ....

/i'

/

..

. \

•.

where PsL = stator winding, power loss. ;t -

' 1

The results of equation (65) can be c·ompared · directly with

the stator winding loss for normal synChronous · opera\ion.

_Hence, the proposed method: ~

A. permits calculation of tl1e asyn·chronous

operating characteristics of a round -rotor synchronous machine with the field

B.

circuit open._

permits prediction of the thermal risks

to the machine involved~in asyn9hronous

operation. ., _q:._

.,.. :~.· , .. , -r- .. . -.. ~· '_,,"

•'. .. ~~ - _._: __ -.. -.-. ·"c-.

'

,,.

...

33

. :. :')_

'-:-=-

" .... - .•,: .: ... - ·~ ·-. . •- ...... ~,...,..,, ... , . "'"":: o,,a•o••..:.,; •••" •••;4->-•''"""f·•,··-••>'-"l-~,.,f,.~-!,,.,_ .... ', '"" ,-,-••• ~~ ·'•" ,.~.-,,.'

... --- .. ·--- -·----- -·· ··-··-----·--·~----·-·--·-·~.:..--~-----... ~--:-·...;, ................. ~-·:-,·· .... _ .. ·-· ;;_.,: ·----~---· ....... -- •, ---- .:;.._ ......... - .. , . .-.. ·' .... . ~--'"---~-: ·~- _.,.,..~ .. _.-- ····-'\:·:.,:::::-,:"=::··?~-7''.:··- ~--- ... :-··-:::--·.-:_ ......... ___ ., ... -•. --"---_:·· .... : ___ -::___,__.:..,:, ... _,.-___ ,_--_~---:-.. ··-~· ... --·-···-~··-·-·- '1.: •·,· .. ~~·'".

'.

t

,.

:/_.· , ..

-"!-·.

·,•,.

. -. ' _.,_ .. ,__ . "\.. ,·.--: ....... . . { -··: --.'!

, ..

·o ": . .-:' ·: . '.· - . ,• - . . . - - ··_ ··: ' ... '•:-'•""

,c- f ·',"··'· • ~ • • I

! - - - '·--·· . ... • -·

\ .

•

_;:.

.• - "t

......... : ... __ ; .. ;, .... ~ ,, .. ,_ --- ... ·- .. --- .. -· ·- -

.,,.,.

.i i

, I

i

:I 1,1

~ '" if 'I ,II , j

f I I

\

'

f

. ..

. \ . . - ... ·~ - -· - -- - . .

\..

.....

· ..

BIBLIOGRAPiiY · .1.- -.-.:.::

......

·- ..,. Books

.,;.:._.

\

I - .

---• . -----·····-- --- --- -- -

[1] Alger~ P~ L. The Nature-of Polyphase Induction Machines. QNew York: John lililey & Sons, Inc., ·19 51 •.

(2) Bet"11ey, L. V. Alternating Current rvta,chinery. New - York: The Macmillan Company, 1949.

[3] Concordia·, Charles. Synchronous 1--1acl1ines--Theory and Performance. New York: John Wi~ey & Sons, Inco, l951.

[4]

[SJ

[6]

Crary, s. B. Power System Stability Vol. II. ~ York: John ~il~y, Sons, Inc., 1947.

New

Fitzgerald, A. E. and Kingsley, c. Electric i1achinery. _ New York: ~1cGrav,-Hill Book Co.·, Inc., 1961.

Kemp, Philipo Alternating Current Electrical Engineeringe 7th ed. London: Macmillan and Co., Ltd., 1949.

[7]· Kron,·G. E uivalent Circuits of Electric ·New Yor: Jon Wi ey & Sons, Inc.,

[8] L~nglois-Berthelot, R. Electro-Magnetic Machines. London: MacDonald & Co., Ltd., 1953.

(9) Langsdorf~ A. s. Theory of Alternatin¥-crirrent _. -. Machinerx. · New York: r-19Grav1-Hi l Book co·., : Inc·. , _ -19 5 5 • , . _ · . . _ ,

•

. (10] Ramo, s., Whinnery, J. R. and Van Duzer, T. Fields ~-----···------------------------------------·a -----------------...--------------~------.. -------------- -1 -- · ---------~:--~:~ - . _ an -1-iaves. 1.n Commun1ca~1on. E ·Jfct:·ron:rcs-;·--···---New

----- -·------ --- -York: John ~viley & Sons, Inc., 1965 •

,r

< \

f

/

. .-,

,.,

•

·.· ..

•

.;, " +------------~----1 ·:· ----- . --------·---

[ 11] Venikov, V. A. Transient Phenomena in Electrical ___ -~:--~~~--- --~--' -_ -~-~~=-~~----~--~~~~~-~~i~64: New. York: The Macmillan ·· · · · ····· ~- ,: _: .. :

·,·, ),I_

.,.

; b -·l

'· .;

. ..

. /

/

,.

--·--·---'--~-

:', -~ ;';i:, ·. :- ·.::_,,;.}·_. .. _:\:/_:_,_:,:: )):_':_'i\~ .:t.~.· .. -.. -~_-:}i ;_,}, ::\.;~ :;. .. · .. ··. i· :i'· ."'·.

·.::.:,-.>.::,·;-::i·"·,.·,.,.-. : -. ·- ~' . . ·. . ' . . "

'---- ---·- ------- - - --- ' . ---··-·--·---·-.--.,-..-,..._ ... ~~~~~-~~,.......,.=...._.._""~~=;1oa,;:.n,..~~s-::·s-~~~-~,,c.r.::f·;.:

,- ~---;---~-----·· -· . . . l:t,

/.

· (201

. _. ___ _J------....'.---i. '

, .

- --- -- .. -,-- - . ;· .... ......

--,,~ F

Jami~son, R. A., "Eddy-Current Effects in Solid,. Unslott~~ Iron Rotors," Proceedings of the Institution of Electrical Engineers, vol. 115, no. 6 {June, 1968}, pp~ 813-820. ·

36

"'

f .

---·-·-···-·····-·-·-·-·-------.-,----· ________ .[_2_l} _______ K_e_s_av armJrt.hy, · 1~_. c1.p.g _ :R9-j_ agq2_~_!c.111 , P. .. .~. , . "Edcly I -··-·-· -·-- --·--·······-·--·--·----- ... -. ----·---····----·-- --· ·11 . , ...

'J

-·-···--·--······ --··· .... ·-··--···-···· -·- -··~···--·---·-··-··~,-- -·- --

(22]

[23]

Currents in Solid Iron Due to Alternating Magnetic.Flux," Proceedings of the Institution of El~ctrical Epgineers, vol. lO~C (June~ 195q),

; pp •. 20 7-=213. ·

, "The Poly~hasi.induction Machine with ------Solid Iron Rotor," Transactions of the Ameri-can Institute of. -Electrical Engineers, vol. 78 part IIIB (December, 1959), _pp. 109~..:1098.

Kovacs, P. C. and Csaki, F. , "Asynchronous Running of Turbo.:.Alternators," International Confer-ence on Large High Tension Electric Systems, no. 115, (session 19_58J • . . -· _ ...... ___________ -------------·-----··

(24] MacLean, w., "Theory of Strong Ele_ctromagnetic ivaves in Massive Iron," Journal of Applied·Ph~sics, vol/ 25, no. 10 (October, l9S4), pp. 12 7~1270.

[25]

[26]

[27]

Malik,

/ J

o. P. and Cory, B. J., "Study of Asynchronous ~~ -Operation anq Resynchronization of Synchronous_ Machines by· Mathematical Models," Proceedings of-the Institution of Electrical Enq1neers,

. vol. f13, no. 12 (December, 1966), pp. 1977-1990.

M~~ikonyants, 0 L. G., Portnoi, M. G., and Khachaturov, A. A., ·"Generalization of.Experience of Slip Oper~tion in Power Systems," Elektrichestvo1 no. 6 (1965), pp. 1-5. __ \

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. , and Syromyatnikqv, .· I._ A. , ·~ Asynchronous · . -----......C-o-ndi tions of Operation of Synchronous t,,lachines, "· Elektricheski Stantsii (USSR), no. 7 (1960)i pp. 42-46, transiated ··in CIGRE, (session 1960),

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' . I ..

William F. Hecht is the son of r,1r. and Mrs.

William o. Hecht and was born on March 18, 1943, in New -

York_ City. After\graduation from Garden City_High School

in Jl.lne 1960_, he attended Lehig.h Univers-ity, receiving a

~Bachelor of Sc~enc~ degree in El~ctrical Engineering in

June, 1964.

· Since 1964, Mr. Hecht has be.~n employed by the

Pennsylvania· Power & Light Company in Allentown, PennsyJ.--

vania as an Engineer and, later, Project Engineer in the

System Planning and Lines and. Substations Departments.

His major responsibilities have been concerned with Dis-

tribution System Pl~nning~

The author is a member of the Institute of Elec-

· trical and Electronic Engineers. He has serve·d the Leh~gh

Valley S~tion of-that organization as Chairman of the \ .. . . Student Activities Com.rnittee, Assistant Treasurer, and .~

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Lehigh UniversityLehigh Preserve1970

Asynchronous operation of round rotor synchronous machinesWilliam F. HechtRecommended Citation

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