Transcript 2 Modeling of DC Machines

Modeling of DC Machines
By
Dr. Ungku Anisa Ungku Amirulddin
Department of Electrical Power Engineering
College of Engineering
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Outline
 Introduction
 Theory of Operation
 Field Excitation
 Separately Excited DC Motor
 State-Space Modeling
 Block Diagrams and Transfer Functions
 Measurement of Motor Constants
 References
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Introduction
 DC 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
 if produces f
 Armature winding –on rotor
 ia produces a
 f and a mutually
perpendicular
 maximum torque
 Rotor rotates clockwise
 For unidirectional torque and
rotation
 ia must be same polarity under
each field pole
 achieved using commutators
and 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 if (f ) and ia (T)
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DC Machine – Field Excitation
 Shunt Excited
 Field winding parallel to
armature winding
 Variable-voltage operation
complex

Coupling of f (if ) and T (ia)
production
 T vs  characteristic almost
constant

AR = armature reaction
(as T , ia , armature flux
weakens main flux  f , )
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DC Machine – Field Excitation
 Series Excited
 Field winding in series with
armature winding
 Variable-voltage operation
complex

Coupling of f (if ) and T (ia)
production
 T ia 2 since if = ia
 High starting torque
 No load operation must be
avoided (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

Long-shunt
connection
Short-shunt
connection
shunt and series field
opposes each other.
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DC Machine – Field Excitation
 Permanent Magnet
 Field provided by magnets
 Less heat

No field winding resistive
losses
 Compact
 Armature similar to
separately excited
machine
 Disadvantages:


Can’t increase flux
Risk of demagnetisation
due to armature reaction
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Separately Excited DC Machine
Ra
ia
+
vt
_
Armature
circuit
va  Raia  La
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Lf
La
Rf
+
if
+
ea
Field
circuit
vf
_
dia
 ea
dt
_
v f  Rf i f  Lf
Te  Kia  Kbia
Electromagnetic torque
ea  K  Kb
Armature back e.m.f.
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dif
dt
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Separately Excited DC Motor
 Motor is connected to a
load.
 Therefore,
d
Te  J
 B  TL
dt
where
TL= load torque
J = load inertia (kg/m2)
B = viscous friction
coefficient (Nm/rad/s)
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DC Machine - State-Space
Modeling
 DC motor dynamic equations:
dia
(1)
va  Raia  La
 ea
dt
d
(3)
Te  J
 B  TL
dt
 Therefore,
Dr. Ungku Anisa, July 2008
ea  K  Kb (2)
Te  Kia  Kbia
dia
R
K
1
  a ia  va  b 
dt
La
La
La
(5)
d K b
B
1

ia    TL
dt
J
J
J
(6)
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(4)
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DC Machine - State-Space
Modeling
 From (5) and (6), the dynamic equations in state-space
form:
 Ra
 sia   L
a
 s   
   Kb
J


 1
i
La   a    La
   
 B    0
J 

Kb
0  v 
 a
T 

1
 L

J
(7)
where s = differential operator with respect to time
 This can be written compactly as:
  AX  BU
X
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(8)
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DC Machine - State-Space
Modeling
 Comparing (7) and (8):
X  ia
U  va
 T - - - - - state variable vector
TL  - - - - - input vector
 Ra
 La
A
 Kb
J

1
B   La
 0

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T

La 

B 
J 

Kb
0 

1 
J
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DC Machine - State-Space
Modeling
 The roots of the system are the eigenvalues of matrix A
 Ra
 La
A
 Kb
J


La 

B 
J 

Kb
1  Ra B  1
1 , 2      
2  La J  2
2
 Ra B Kb 2 
 Ra B 

    4



 La J 
 JLa JLa 
(9)
 1 and 2 always have negative real part, i.e. motor is
stable on open-loop operation.
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DC Machine – Block Diagrams
and Transfer Functions
 Taking Laplace transform of (1) and (3) and neglecting initial
conditions:
V s   K b ωs 
I a s   a
Ra  s La
ωs  
(10)
K b I a s   TL s 
B  s J 
(11)
 These relationships can be represented in the following block
diagram
Va(s)
TL(s)
+
-
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Ra sLa 
Ia(s)
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Te(s) Kb +
Kb
1
B  s J 
(s)
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DC Machine – Block Diagrams
and Transfer Functions
 From the block diagram, the following transfer functions can be derived:
Kb
ωs 
G ωVa s  
 2
Va s  s JLa   sBLa  JRa   BRa  Kb 2


(12)
 Ra  s La 
ωs 
G ωTL s  
 2
TL s  s JLa   sBLa  JRa   BRa  Kb 2


(13)
 Since the motor is a linear system, the speed response due to simultaneous
Va input and TL disturbance is:
ωs  GωVa sVa s  GωTL sTL s
(14)
 The Laplace inverse of (14) gives the speed time response (t).
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DC Machine – Measurement of
Motor Constants
 To analyse DC motors we need values for Ra, La and Kb
 Armature Resistance Ra
 DC voltage applied at armature terminals such that rated ia
flows
Vdc  Vbrush  Vcontact resistance
Ra 
ia, rated
 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 Rf of field circuit
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DC Machine – Measurement of
Motor Constants
 Armature Inductance La
 Apply low AC voltage through
variac at armature terminals
 Measure ia
 Motor must be at standstill
(i.e.  = 0 and e = 0)
La 
 Va

 I
 a
2

  Ra 2


2f
(variac)
 f = supply frequency in Hz
 Ra = ac armature resistance
 Similar procedure can be
applied to find Lf of field circuit
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DC Machine – Measurement of
Motor Constants
 EMF Constant Kb = K
 Rated field voltage applied
and kept constant
 Shaft rotated by another dc
motor up to rated speed
 Voltmeter connected to
armature terminals  gives
value of Ea
 Get values of ea at different
speeds
 Plot Ea vs. 
 Slope of curve = Kb
 Units of Kb = [V/rads-1]
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Ea (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.
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