1@2modeling asaof an electric machine

8/17/2019 1@2modeling asaof an Electric Machine

1/24

Modeling and Operation ofan Electric Machine

Dr S.P. Singh

Professor

Department of Electrical Engineering

IIT Roorkke


2/24


3/24


4/24


5/24

TORQUE AND VOLTAGE IN NONSALIENT – POLE MACHINES

• The machine is regarded as a circuit elementwhose inductances depend on the angular positionof the rotor.

• – lu! linkages "λ # and magnetic $eld energ% are

e!pressed in terms of currents and inductances.

–

– Tor&ue can 'e found from the partial deri(ati(e of

magnetic $eld energ% or co)energ%. –

• The result is a set of non)linear di*erentiale&uations


6/24

VIEW POINTS :

• Parallel de(elopment of following two (iewpoints help in o'taining an appro!imatee!pression for inductance parameters. The% are+

• To&ue is e!pressed e!plicitl% as the tendenc%for two magnetic $elds to line up in the samewa% as permanent magnets tend to alignthemsel(es.

• ,enerated (oltage is e!pressed as the result ofthe relati(e motion 'etween a $eld and awinding


7/24

Assumptions

• Stator and rotor are concentric c%linders and the slot openings are neglected.

• Therefore- the stator and rotor self +inductances ss and rr respecting areconsidered to 'e constant.

• ut

•

• Stator)rotor mutual inductance depends on the angle /θ0 'etween themagnetic a!es of stator and rotor winding.

• • Mutu!" In#u$t!n$%

•

• The ma!imum positi(e (alue when θ12 or 3π

• is 4ero when θ 1 ± π53.

• The ma!imum negati(e (alue when θ1±π

• • 6ence 7

•

• sr "θ# 1 sr.8osθ "i#

•


8/24

Mathematical Model of the Machine

• lu! linkages of rotor and the stator in terms of

inductances- are 7• λs 1 ss.is 9 sr"θ#.ir 1 ss.is 9 sr.8osθ. i r

• λr 1 sr"θ#.is9 rr.Ir 1 rr 8osθ .is9 rr.ir

• Terminal (oltages :s and :r are +

• (s 1 rs .is9pλs

• (r 1 rr .ir9pλr

• p 1 d5dt 1 time)deri(ati(e operator


9/24

T%&min!" 'o"t!(% E)u!tions *o&Dou+", E-$it%# m!$.in%

• The d%namics and statics of Electromechanicalenerg% con(ersion- tor&ue and (oltagee&uations in non salient pole machine

forrotor pSini L piCos L pi Lir v forstator pSini L piCos L pi Lir v

sr sr s sr r rr r r r

r sr r sr s ss s s s

−−−−−−++=

−−−−−++=

)()()()(θ θ θ

θ θ θ

pθ - instantaneous speed in electrical radians per second.

In a P-pole machineθ = P/2 θ

m

• The d%namics and statics of Electromechanicalenerg% con(ersion- tor&ue and (oltagee&uations in non salient pole machine

The a'o(e e&uations are the fundamental e&uations of themachine and (alid for dou'l% e!cited ac as well as dcmachines. ;ith a little modi$cations in the a'o(e e&uationsare (alid for di*erent t%pes of ac and dc machines


10/24

θ Cosii Li Li LW r s sr r rr s ss fld ...21.

21 22

++=

m

r sr sm

m

fld

ed

d ii

d

dWfLd ii

W T

θ

θ θ

θ θ

θ )..,(),,(

+=

∂

∂+=

Therefore Te for a P-pole machine is :-

mr s sr r s sr e

P Sinii L

P Sinii L

P T θ θ

2..

2...

2−=−=


11/24

(s 1 rs< is 9ss."p is#9sr 8osθ "p ir # + sr ir Sinθ "pθ# ====">#

(r 1 rr< ir 9rr."p ir#9sr 8osθ "p is# + sr is Sinθ "pθ# ===="3#

Basic Equations

mr s sr r s sr e P Sinii L P Sinii L P T θ θ 2

..2

..2

−=−=

)(1

Lr me

r T BT J dt

d −−= ω

ω

And usin torsional – mechanical equation:

Electric tor&ue is 9 (e for motor and + (e in case of generator.

or a an% dc machine ?rd term of the (oltage e&uations is alwa%sa'sent as $eld and armature mmf in case of dc machine are at&uadrature.

The fourth term - speed (oltage which is due to relati(e motion incoils is there for stator (oltage e&uation 'ut a'sent in rotor(oltage e&uations. is

On the 'asis of a'o(e assumption the d%namic e&uations forseparatel% e!cited dc motor can deri(ed as follows


12/24

Model of D8 Shunt Motor

a

a

r f

a

af

a

aaa v L

i L

L

L

ir

dt

di.

1..

.+−

−= ω

f

f f

f f f v

L L

ir

dt

di.

1.+

−

=

Lr m

f a

af r T J J

Bii

J

L

dt

d .

1.. −−= ω

ω


13/24


14/24

Out put

0 0.2 0.4

-100

0

100

200

300

Time (second)

Armature currnt ia [A]

0 0.2 0.4

0

2

4

6

Time (second)

Field currnt if [A]

0 0.2 0.4

0

100

200

300

Time (second)

Angular velocity, wr rad/sec


15/24

SIMNI MODE


16/24

Separatel% E!cited dc Motor

a

a

r f

a

af

a

aaa v L

i L

L

L

ir

dt

di.

1..

.+−

−= ω

f

f f

f f f v

L L

ir

dt

di.

1.+

−

=

Lr m

f a

af r T

J J

Bii

J

L

dt

d .

1.. −−= ω

ω


17/24

Separatel% E!cited D8 ,enerator

• θ1K22

• T 1) T

• :a1 ):a

Separatel% E!cited D8 ,enerator

• θ1K2 2

• T 1) T

• :a1 ):a


18/24

Shunt connected dc motor

f a vv =

ff a f r vi /=

ar f af aa r wi Lvi /)( −=

)/(*)/1( 2 f aa f r af af e r ivr w L LT −=

θ=" 0


19/24

To&)u%/sp%%# )u!#&!nt o* op%&!tion

ω

T01

2 3

T 4'%ω 4'%Pm 4'%

T /'%ω 4'%Pm /'%

T /'%ω /'%

Pm 4'%

T 4'%ω /'%

Pm /'%


20/24

ITRODN8TIO TO EE8TRI8 DRI:ES ) MODNE >

3/)u!#&!nt op%&!tion

ωm

Te

Te

ωm

Teωm

Te

ωm

ω

T

• Direction of positi(e"forward# speed is ar'itrar%chosen

• Direction of positi(e tor&uewill produce positi(e"forward# speed

Quadrant 1

Forward motoringQuadrant 2

Forward braking

Quadrant 3

Reverse motoringQuadrant 4

Reverse braking


21/24

our Quadrant dc motorcharacteristic

5un$tion Qu!#&!nt

Sp%%# To&)u% Po6%&output

orwardMotoring"M#

I 9 9 9

orwardRegeneration"R#

I: 9 )

Re(erse

Motoring"RM#

III 9

Re(erseRegeneration "RR#

II 9


22/24

Grmature and (oltage and currentre&uirement of four &uadrant dc dri(e

5un$tion Sp%%# To&)u% 'o"t!(% $u&&%nt Po6%&output

orwardMotoring

"M#

9 9 9 9 9

orwardRegeneration"R#

9 ) 9 ) )

Re(erseMotoring"RM#

) 9

Re(erseRegeneration "RR#

9 9 )


23/24


R!tin(s o* $on'%&t%&s !n# moto&s

Torque

Speed

Power limit for

continuous torque

Continuous

torque limit

Maximum

speed limit

Power limit for

transient torque

Transient

torque limit


24/24


C"!ssi7$!tion o* IM #&i'%s (Buja, Kamierkowski, “Direct torque control of PWM inerter!fed "# motors ! asure$%, &''' Transactions on &ndustrial 'lectronics, ))*+

1@2modeling asaof an electric machine

Documents