lect 8,ritik
TRANSCRIPT
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Lecture 8 - EE743Lecture 8 - EE743
Direct Current (DC)
Machines - Part II
Professor: Ali KeyhaniProfessor: Ali Keyhani
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s Shunt-connected DC Machine
DC Machines
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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)
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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
+=
++=
=
=++=
+=
==
+=
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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
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s Time domain block diagram
DC Machines - Shunt DC MachineShunt DC Machine
G1
G3
G2
Laf X
X
r
ia
if
if
ia+
-
+
-
Va
Vf
Vf
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s State-space equations
DC Machines - Shunt DC MachineShunt DC Machine
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
+=
++=
+=
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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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DC Machines - Permanent MagnetPermanent Magnet
s Dynamic equations of a Permanent Magnet
Machine
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DC Machines - Permanent MagnetPermanent Magnet
s Dynamic equations,
dt
dP
r
LeiPrV
a
aaa
aaaaa
==++=
,
)1(
rvaave
rmLe
KeiKT
JPBTT
==
+=
,
)(
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DC Machines - Permanent MagnetPermanent Magnet
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 =+
=
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DC Machines - Permanent MagnetPermanent Magnet
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
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DC Machines - Permanent MagnetPermanent Magnet
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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DC Machines - Permanent MagnetPermanent Magnet
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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DC Machines - Permanent MagnetPermanent Magnet
s In a matrix representation,
BAX
TbVbi
aPaaaP
L
a
r
a
=
=
)()(
22
11
2221
1211
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DC Machines - Permanent MagnetPermanent Magnet
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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DC Machines - Permanent MagnetPermanent Magnet
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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DC Machines - Permanent MagnetPermanent Magnet
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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DC Machines - Permanent MagnetPermanent Magnet
( )
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
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DC Machines - Permanent MagnetPermanent Magnet
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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DC Machines - Permanent MagnetPermanent Magnet
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.