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Modeling of DC MachinesBy
Dr. Ungku Anisa Ungku AmirulddinDepartment of Electrical Power Engineering
College of Engineering
Dr. Ungku Anisa, July 2008 1EEEB443 - Control & Drives
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Outline Introduction
Theory of Operation
Field Excitation Separately Excited DC Motor
State-Space Modeling
Block Diagrams and Transfer FunctionsMeasurement of Motor Constants
References
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IntroductionDC motor in service for more than a century
Dominated variable speed applications before
Power Electronics were introduced
Advantage:
Precise torque and speed control without
sophisticated electronics
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Introduction Some limitations:
High maintenance (commutators & brushes)
Expensive Speed limitations
Sparking
Commonly used DC motors Separately excited
Series (mostly for traction applications)
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DC Machine – Theory of Operation Field winding - on stator pole
i f produces f Armature winding –on rotor
i a produces a f and a mutually
perpendicular maximum torque
Rotor rotates clockwise
For unidirectional torque and
rotation i a must be same polarity under
each field pole
achieved using commutatorsand brushes
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DC Machine – Field Excitation Depends on connections of field winding relative to
armature winding
Types of DC machines: Separately Excited
Shunt Excited
Series Excited
Compounded Permanent Magnet
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DC Machine – Field Excitation Separately Excited
Field winding separated from armature winding
Independent control of i f ( f ) and i a (T )
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DC Machine – Field Excitation Shunt Excited
Field winding parallel toarmature winding
Variable-voltage operationcomplex
Coupling of f (i f ) and T (i a)production
T vs characteristic almostconstant
AR = armature reaction
(as T , i a , armature fluxweakens main flux f , )
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DC Machine – Field Excitation Series Excited
Field winding in series witharmature winding
Variable-voltage operationcomplex
Coupling of f (i f ) and T (i a)production
T i a
2 since i f = i
a
High starting torque
No load operation must beavoided (T = 0, )
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DC Machine – Field Excitation Compounded
Combines best feature of
series and shunt
Series – high starting torque
Shunt – no load operation
Cumulative compounding
shunt and series field
strengthens each other.
Differential compounding
shunt and series field
opposes each other.
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Long-shunt
connection
Short-shuntconnection
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DC Machine – Field Excitation Permanent Magnet
Field provided by magnets
Less heat
No field winding resistivelosses
Compact
Armature similar toseparately excited
machine Disadvantages:
Can’t increase flux
Risk of demagnetisationdue to armature reaction
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Lf Rf
if
aaaaaa edt
di Li Rv
+
ea
_
La Ra
ia +
vt
_
+
vf
_
Separately Excited DC Machine
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 12
dt di Li Rv f
f f f f
abae i K i K T Electromagnetic torque
ba K K e Armature back e.m.f.
Armaturecircuit
Fieldcircuit
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Separately Excited DC Motor Motor is connected to a
load.
Therefore,
where
T L= load torque
J = load inertia (kg/m2)
B = viscous friction
coefficient (Nm/rad/s)
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Le T Bdt
d J T
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DC Machine - State-Space
Modeling DC motor dynamic equations:
Therefore,
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 14
aa
aaaa edt
di Li Rv
Le T Bdt
d J T
a
ba
a
a
a
aa
L K v
Li
L R
dt di 1
Lab T
J J
Bi
J
K
dt
d 1
abae i K i K T
ba K K e (1) (2)
(3) (4)
(5)
(6)
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DC Machine - State-Space
Modeling From (5) and (6), the dynamic equations in state-space
form:
where s = differential operator with respect to time
This can be written compactly as:
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 15
L
aaa
b
a
b
a
a
a
T v
J
Li
J B
J K
L
K
L
R
s si
10
01
BUAXX
(7)
(8)
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DC Machine - State-Space
Modeling Comparing (7) and (8):
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J B J
K
L
K
L
R
b
a
b
a
a
A
vector variablestate-----T
ai X
vector input-----T
La T vU
J
La
10
01
B
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DC Machine - State-Space
Modeling The roots of the system are the eigenvalues of matrix A
1 and 2 always have negative real part, i.e. motor isstable on open-loop operation.
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J
B J
K L
K
L
R
b
a
b
a
a
A
a
b
a
a
a
a
a
a
JL
K
JL
B R
J
B
L
R
J
B
L
R 22
21 4
2
1
2
1, (9)
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DC Machine – Block Diagrams
and Transfer Functions Taking Laplace transform of (1) and (3) and neglecting initial
conditions:
These relationships can be represented in the following block
diagram
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 18
aa
b
L R
K
s
sωsVsI aa
J B
K b
s
sTsIsω
La
(10) (11)
aa L R s
1
J B s
1
K b
TL(s)
Te(s)Ia(s)
Va(s)
K b
(s)+
+
-
-
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DC Machine – Block Diagrams
and Transfer Functions From the block diagram, the following transfer functions can be derived:
Since the motor is a linear system, the speed response due to simultaneous
V a input and T L disturbance is:
The Laplace inverse of (14) gives the speed time response (t).
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 19
22a
ωV
sssV
sωsG
a
baaaa
b
K BR JR BL JL
K
(12)
(13)
22L
ωT
ss
s
sT
sωsG
L
baaaa
aa
K BR JR BL JL
L R
sTsGsVsGsω LωTaωV La (14)
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DC Machine – Measurement of
Motor Constants To analyse DC motors we need values for Ra, La and K b
Armature Resistance Ra
DC voltage applied at armature terminals such that rated i aflows
This gives the dc value for Ra
Need to also correct for temperature at which motor is
expected to operate at steady state
Similar procedure can be applied to find R f of field circuit
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 20
rated,
resistancecontact
a
brushdca
i
V V V R
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DC Machine – Measurement of
Motor Constants Armature Inductance La
Apply low AC voltage throughvariac at armature terminals
Measure i a
Motor must be at standstill(i.e. = 0 and e = 0)
f = supply frequency in Hz
Ra = ac armature resistance
Similar procedure can beapplied to find L f of field circuit
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 21
f
R I
V
L
a
a
a
a 2
2
2
(variac)
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DC Machine – Measurement of
Motor Constants EMF Constant K b = K
Rated field voltage appliedand kept constant
Shaft rotated by another dcmotor up to rated speed
Voltmeter connected toarmature terminals givesvalue of E a
Get values of ea at differentspeeds
Plot E a vs.
Slope of curve = K b
Units of K b = [V/rads-1]
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E a (V)
(rad/s)
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References Krishnan, R., Electric Motor Drives: Modeling, Analysis and
Control , Prentice-Hall, New Jersey, 2001.
Chapman, S. J., Electric Machinery Fundamentals, McGraw Hill,
New York, 2005.
Nik Idris, N. R., Short Course Notes on Electrical Drives,
UNITEN/UTM, 2008.
Ahmad Azli, N., Short Course Notes on Electrical Drives,
UNITEN/UTM, 2008.
Dr Ungku Anisa July 2008 23EEEB443 - Control & Drives