Lect 8,Ritik

8/14/2019 Lect 8,Ritik

1/23

Lecture 8 - EE743Lecture 8 - EE743

Direct Current (DC)

Machines - Part II

Professor: Ali KeyhaniProfessor: Ali Keyhani


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2

s Shunt-connected DC Machine

DC Machines


3/23

3

DC Machines

s The dynamic equations (assuming rfext=0) are:

fafra

alaaaaa

f

fffff

iLedt

diLireV

dt

diLirV

=

++=+=

Where Lff = field self-inductance

Lla= armature leakage inductance

Laf = mutual inductance between the field

and rotating armature coils

ea = induced voltage in the armature coils

(also called countercounter orback emfback emf)


4/23

4

DC Machines

)(

)1(

:

)1(

:

rpmLe

aaaaa

a

aa

a

fafraa

laaaaa

ffff

f

ff

f

f

fffff

JBTT

eirV

r

L

iLe

dt

diLireV

windingarmaturetheFor

irVdt

dp

r

L

dt

diLirV

windingfieldtheFor

+=

++=

=

=++=

+=

==

+=


5/23

5

DC Machines - Shunt DC MachineShunt DC Machine

s Time-domain block diagram

The machine equations are solved for:

11

1

GVp

ri f

f

f

f =+

= ( ) 2

1

1

GiLVp

ri fafra

a

af =+=

( )

faafe

Le

m

r

iiLT

GTTBpJ

=

=+= 31


6/23

6

s Time domain block diagram


G1

G3

G2

Laf X

X

r

ia

if

if

ia+

-

+

-

Va

Vf

Vf


7/23

7

s State-space equations


Let

raf

t iiX ,,= +=t

rrr dtt

0

)0()(

Re-writing the dynamic equations,

;

Laf

af

rmr

a

aa

rf

aa

af

a

aa

aa

f

ff

f

ff

ff

TJ

iiJ

L

J

B

dt

d

VL

iLLi

Lr

dtdi

VL

iL

r

dt

di

1

1

1

+=

++=

+=


8/23

8

DC Machines - Permanent MagnetPermanent Magnet

s The field flux in thePermanent MagnetPermanent Magnet

machines is produced by a permanent

magnet located on the stator.

s Therefore,

s

Lsfif is a constant determined by thestrength of the magnet, the reluctance of

the iron, and the number of turns of the

armature winding.

Constant== vfaf KiL


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9


s Dynamic equations of a Permanent Magnet

Machine


10/23

10


s Dynamic equations,

dt

dP

r

LeiPrV

a

aaa

aaaaa

==++=

,

)1(

rvaave

rmLe

KeiKT

JPBTT

==

+=

,

)(


11/23

11


s Time domain block diagram

The equations are solved by,

( ) 11

1

1

1G

p

r

pre

i

a

a

aaa

a =+

=+

=

2

1G

BJPTT mLe

r =+

=


12/23


13/23


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14


In a matrix form,

+

=

+=

L

aaa

r

a

mv

aa

v

aa

a

r

a

T

V

J

Li

J

B

J

K

L

K

L

ri

P

BUAXX

1

0

01


15/23

15


s Transfer Function,

s Let

, 21L

r

a

r

TG

rG

==

1

1

2211

1221

1211

Jb

Lb

J

Ba

J

Ka

L

Ka

L

ra

aa

mv

aa

v

aa

a

==

==

==


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16


s The, we will have

Re-arranging the equation,

Lrar

araa

TbaiaP

VbaiaPi

222221

111211

++=

++=

( )

( ) Lra

ara

TbaPia

VbaiaP

222221

111211

=+

=


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17


s In a matrix representation,

BAX

TbVbi

aPaaaP

L

a

r

a

=

=

)()(

22

11

2221

1211


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18


s Solving for ia

( )

))((

)(

))((

)(

21122211

2211

21122211

2222

1211

aaaPaP

aPVb

i

aaaPaP

aPTb

aVb

i

a

a

L

a

a

=

=


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19


s Let m be,

s The equation is then reduced to,

2v

am

K

Jr=

111

)(11

2

+

+

++

+

+

=

J

BP

J

BP

T

K

V

J

BPr

im

ma

m

a

Lma

v

am

a

a

a


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20


shown thatbecanit

,letand,b,...,aforngSubstituti

))((

)(

))((

)(

2m1111

12212211

21112211

12212211

2221

1111

v

a

aLr

L

a

r

K

JV

aaaPaP

aVbTbaP

aaaPaP

Tba

VbaP

=

+

=

=


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21


( )

0Vwith

Condition)Load-No(0Twith

111

111

a2

L1

ma

2

a

2

a

ma

2

==

==

+

+

++

+

=

L

n

a

n

mm

La

v

r

TG

VG

J

BPJ

BP

TPJ

VK

n


22/23

22


s The characteristic equation (orforce-freeforce-free

equationequation) of the system is as shown below,

1

factordamping

sytemtheofnoscillatio

frequencynaturalundamped

factordecayinglexponentia

02

2

21

22

=

==

=

=

=++

nn

n

n

n

,bb

PP


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23


s If < 1 , the roots are a conjugate complexpair, and the natural response consists of an

exponentially decaying sinusoids.

s If > 1, the roots are real and the naturalresponse consists of two exponential terms

with negative real exponents.

Lect 8,Ritik

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