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Unformatted text preview: ME375 Handouts ElectroElectroMechanical Systems
• DC Motors
– Principles of Operation
– Modeling (Derivation of Governing Equations (EOM)) • Block Diagram Representations
– Using Block Diagrams to Represent Equations in s  Domain
– Block Diagram Representation of DC Motors • Example School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  1 DC Motors Motors are actuation devices (actuators)
that generate torque as actuation.
torque • Terminology
– Rotor : the rotating part of the motor.
– Stator : the stationary part of the
the stationary part of the
motor.
motor.
– Field System : the part of the motor
that provides the magnetic flux.
– Armature : the part of the motor
which carries current that interacts
with the magnetic flux to produce
torque.
– Brushes : the part of the electrical
th
th
circuit
circuit through which the current is
supplied to the armature.
– Commutator : the part of the rotor
that is in contact with the brushes. School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  2 1 ME375 Handouts DC Motors  Principles of Operation
• Torque Generation v
B
v
df B v
dL i Force will act on a conductor in a
magnetic field with current flowing
through the conductor.
v
v
v
d f = ia ⋅ d L × B
Integrate over the entire length: i f= Total torque generated:
B τ Coil = School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  3 DC Motors  Principles of Operation
Let N be the number of coils in the motor. The total torque
generated from the N coils is: τ m = N ⋅ (2 ⋅ ia ⋅ B ⋅ L ⋅ R)
=
For a given motor, (N, B, L, R) are fixed. We can define
KT = 2 ⋅ N ⋅ B ⋅ L ⋅ R
Nm / A
as the Torque Constant of the motor.
Torque
The torque generated by DC motor is proportional to the
The torque generated by a DC motor is proportional to the
armature
armature current ia : τ m = KT ⋅ ia Large KT :
– Large (N, L, R).
(N, L, R) is limited by the size
and weight of the motor. For a DC motor, it is desirable to have a large KT . However, size
and other physical limitations often limits the achievable KT .
School of Mechanical Engineering
Purdue University – Large B:
Need to understand the
methods of generating flux ...
ME375 ElectroMechanical  4 2 ME375 Handouts DC Motors  Principles of Operation
• BackEMF Generation
BackElectromotive force (EMF) is generated in a conductor
v
moving in a magnetic field:
vvB
v
vv
v ×B
de = ( v × B ) ⋅ dL
emf v
v Integrate over the entire length L: e em f = Since the N armature coils are in series, the total EMF is:
E em f = 2 N ( R ω ) B L =
13
2
v Kb = 2 ⋅ N ⋅ R ⋅ B ⋅ L Define the BackEMF Constant Kb :
Back V / (rad / sec) The BackEMF generated due to the rotation of the motor armature is opposing the applied
Backvoltage and is proportional to the angular speed ω of the motor: E em f = K b ⋅ ω
Note: KT = Kb is true only if consistent SI units are used !
School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  5 DC Motors  Modeling
Schematic
+ eRa −
+
ei(t)
_ RA + eLa −
iA LA +
Eemf
_ Element Laws:
Electrical Subsystem θ, ω
τm
JA τL B
Mechanical Subsystem FBD: Interconnection Laws: School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  6 3 ME375 Handouts DC Motors  Modeling
Derive I/O Model: I/O Model from ei(t) and τL to angular speed ω :
and FG
H
FG
H IJ
K
IJ
K FG
H
FG
H IJ
K
IJ
K b &
LA τ L + RA τ L
LA J A
LA B
RJ
RA B
&&
&
ω+
+ A A ω+
+ K b ω = ei ( t ) −
KT
KT
KT
KT
KT
I/O Model from ei(t) and τL to angular position θ :
and
&
LA τ L + RA τ L
L A J A &&&
LA B
RJ
RA B
&&
&
θ+
+ A A θ+
+ K b θ = ei ( t ) −
KT
KT
KT
KT
KT
School of Mechanical Engineering
Purdue University b g g ME375 ElectroMechanical  7 DC Motors  Modeling
Transfer Function: Ω (s) = ⋅ Ei ( s ) − ⋅Τ L ( s ) θ (s) = ⋅ Ei ( s ) − ⋅Τ L ( s ) Q:
Q: Let the load torque be zero (No Load), what is the steady state speed (NoLoad Speed) of the
(Nomotor for a constant input voltage V ? Q: Let the load torque τL = T, what is the steady state speed of the motor for a constant input
voltage V ? School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  8 4 ME375 Handouts Block Diagram Representation
• Differential Equation → Transfer
Transfer
Function (System & Signals)
Y (s) = G (s) ⋅ U (s) G ( s) Input
Signal Y (s) = U1(s) ± U 2 (s)
U1(s) Y(s) U(s) • Signal Addition/Subtraction Output
Signal Y(s) +
±
U2(s) Ex: Draw the block diagram for the following
DE: Ex: Draw the block diagram for the following
DE: &
Jω + Bω = τ &
Jω = τ School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  9 Block Diagram Representation
• Transfer Function in Series • Multiple Inputs Y ( s ) = G 2 ( s ) ⋅ Y1 ( s ) , Y1 ( s ) = G1 ( s ) ⋅ U ( s ) b g U(s) G1 (s) Y1(s) G2 (s) Input
Signal = G1 ( s ) ⋅ U 1 ( s ) ± G 2 ( s ) ⋅ U 2 ( s ) Y (s )
Output
Signal • Transfer Function in Parallel
X 1 ( s ) = G1 ( s ) ⋅ U ( s ) , X 2 ( s ) = G 2 ( s ) ⋅ U ( s )
Y ( s ) = X1 ( s ) ± X 2 ( s ) b Y1 ( s ) = G 1 ( s ) ⋅ U 1 ( s ) , Y2 ( s ) = G 2 ( s ) ⋅ U 2 ( s )
Y ( s ) = Y1 ( s ) ± Y2 ( s ) Y ( s ) = G 2 ( s ) ⋅ G1 ( s ) ⋅ U ( s ) U1(s)
U2(s)
Input
Signals G1 (s)
G2 (s) Y1(s) + Y (s )
± Output
Y2(s) Signal g = G1 ( s ) ± G 2 ( s ) ⋅ U ( s ) U(s)
Input
Signal G1 (s)
G2 (s) X1(s) + Y (s )
± Output
X2(s) Signal School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  10 5 ME375 Handouts Block Diagram Representation
• Feedback Loop
U(s) + X (s) Y (s )
G (s ) Input − Output
Signal Signal
H(s) School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  11 Block Diagram Representation of DC Motors
Schematic
+ eRa −
+
ei(t)
_ RA ElectroElectroMechanical Coupling: + eLa −
iA LA +
Eemf
_ θ, ω Take Laplace Transform of the Eqs. τm τL JA B Governing Equations:
d
LA iA + RA iA + Eemf = ei (t ) (
dt
&
J A ω + Bω = τ m − τ L τ m = KT ⋅ i A ⎫
⎬
Eemf = Kb ⋅ ω ⎭ (
⎛
⎜
⎝ ) )
⎞
⎟
⎠
School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  12 6 ME375 Handouts Block Diagram Representation of DC Motors
Electrical Subsystem: Mechanical Subsystem: Take Laplace Transform of the Eqs. Take Laplace Transform of the Eqs. Q: Now that we generated a block diagram of a voltage driven DC Motor, can we derive the transfer
function of this motor from its block diagram ? ( This is the same as asking you to reduce the multiblock diagram to a simpler form just relating inputs ei(t) and τL to the output, either ω or θ .)
School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  13 Block Diagram Reduction
From Block Diagram to Transfer Function
• Label each signal and block +
− − 1
LA s + RA KT + 1
JA s + B Kb
• Write down the relationships between signals School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  14 7 ME375 Handouts Block Diagram Reduction
• Solve for the output signal in terms of the input signals • Substitute the transfer functions’ label with the actual formula and simplify
Ω (s) =
θ (s) = KT
LA s + RA
⋅Τ L ( s )
⋅ Ei ( s) −
LA J A s 2 + ( BLA + RA J A )s + ( RA B + Kb KT )
LA J A s 2 + ( BLA + RA J A )s + ( RA B + Kb KT ) KT
LA s + RA
⋅ Ei ( s ) −
⋅Τ L ( s )
s( LA J A s 2 + ( BLA + RA J A )s + ( RA B + K b K T ))
s( LA J A s 2 + ( BLA + RA J A )s + ( RA B + K b K T ))
School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  15 Model Reduction
Q: Physically, what do we mean by neglecting armature inductance ?
By neglecting the armature inductance, we are assuming that it takes no time for
the current to reach its steady state value when there is a step change in input
voltage, i.e., a sudden change in input voltage will result in a sudden change in
the armature current, which in turn will result in a sudden change in the motor
torque output. This is equivalent to having direct control over the motor current.
output.
current.
From Block Diagram: Mathematically:
d
L A iA + RA iA + Eemf = ei ( t )
dt
&
J A ω + Bω = τ m − τ L τ m = K T ⋅ iA
Eemf = K b ⋅ ω +
− 1
LA s + RA KT U
V
W School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  16 8 ME375 Handouts Example
Media Advance System in InkJet Printers
The figure on the right shows the media
advance system of a typical inkjet printer.
dv
sys
The objective of the system is to precisely
and quickly position the media such that ink
droplets can be precisely “dropped” on to the
media to form “nice looking” images. The
system is driven by a DC motor through two
sets of gear trains. You, as the “new kid” on
the development team, are given the task of
specifying a motor and designing the control
system that will achieve the desirable
performance. Some time your manager will
also walk by your desk and ask you if a
certain level of performance is achievable.
How would you start your first engineering
project?
School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  17 Example – DC motor with load
Schematic DC Motor + eRa −
+
ei(t)
_ RA + eLa −
iA LA +
Eemf
_ τm θ, ω θL, ωL Assumptions: JL N2 • Gears and shafts are
rigid and massless. JA
N1 BL B Block
Block diagram of the load inertia: Block diagram of the gear train: School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  18 9 ME375 Handouts Example – DC motor with load
Block diagram of the DC motor subsystem:
+
− 1
LA s + RA −
KT + 1
JA s + B Kb Reduce the mechanical portion of the block diagram: School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  19 Example – DC motor with load
Simplified block diagram:
+
− 1
LA s + RA KT Kb Transfer Function from input voltage Ei(s) to the angular velocity or position of the load: School of Mechanical Engineering
Purdue University ME375 ElectroMechanical  20 10 ...
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 Fall '10
 Meckle

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