Download Electromechanical systems

Lecture 9:
1. Finish purely electrical systems
• Modeling in the Laplace domain
• Loading of cascaded elements
2. Modeling electromechanical systems
• Introduction
• Sensors and the measurement system
• Actuators (continue with DC motors next
class)
ME 431, Lecture 9
Modeling Electromechanical Systems
1
Modeling in the Laplace Domain
E ( s)
complex impedance  Z ( s) 
I ( s)
• Complex impedance is treated like
resistance
• Ohm’s law
• Equivalent components in series and parallel
ME 431, Lecture 9
• Idea: model circuits directly in terms of
Laplace transformed equations
2
Modeling in the Laplace Domain
• Resistors:
• Capacitors:
E ( s) 1
1
1 I ( s)
 Z (s) 

L [e   i dt ]  E ( s) 
I ( s) Cs
C
C s
• Inductors:
E ( s)
di
 sL
L [e  L ]  E ( s )  LsI ( s )  Z ( s) 
I (s)
dt
ME 431, Lecture 9
E ( s)
R
L [e  iR ]  E ( s )  I ( s ) R  Z ( s) 
I ( s)
3
Example
• Find the transfer function Eo(s)/Ei(s)
Example (con’t)
• Note: this approach is only valid if initial conditions are zero
Loading of Cascaded Elements
ME 431, Lecture 9
• Consider the two loops separately
6
Loading of Cascaded Elements
• Consider the two loops separately
• Note that
G1(s)
Ec(s)
G2(s)
Eo(s)
Ec Eo Eo
G1G2 

Ei Ec Ei
ME 431, Lecture 9
Ei(s)
• The second loop loads the first loop
• Can add an isolating amplifier to decouple
• This is addressed by some simulation
software
7
Electromechanical Systems
• Most control systems (including automobiles)
include electrical and mechanical components
• Need components that convert between the
domains
Example:
mechanical domain
electrical domain
angle
Transducer
R+
E
-
Controller
voltage
Actuator
voltage
voltage
Sensor
U
torque
Y
Plant
speed
Electromechanical Systems
• Piezoelectric materials produce charge when deformed
(ex: accelerometer, microphone, load cell, etc.)
• Electrical properties of many materials change with
temperature, deformation, etc.
(ex: thermistor, pressure transducer, strain gage)
• Motion of a conductor through a magnetic field can
induce an emf (ex: LVDT, resolver)
• Special materials produce charge in response to light
(digital camera, optical encoder)
ME 431, Lecture 9
• Sensors/Transducers:
often convert mechanical quantities into electrical
ones
9
The Measuring System
measurand
Stage 1
Sensor
transduced
signal
Stage 2
Signal
Conditioning
filter
amplify
integrate
differentiate
DAC
ADC
analogous
driving signal
Stage 3
Readout/
Computation
ME 431, Lecture 9
• Measurement process generally involves
multiple elements that each may have
dynamics that need to be modeled
gauge
LED display
LCD display
speaker
computer
10
The Measuring System
Example
Stage 2
Stage 1
accel
signal
w/o noise
voltage
sensor
filter
Stage 3
analogous
to velocity
integrator
increased
voltage
amplifier
ADC
digital
Computer
Numerical Integration
• Error accumulates if
there is a bias in
the measurement
ME 431, Lecture 9
• Simple approach
calculates the area
of a series of
rectangles
12
Numerical Differentiation
• Large error when
noisy measurements
are differentiated
ME 431, Lecture 9
• Simple approach
calculates the slope
between two points
13
Analog to Digital Conversion
• Can introduce time
lag and
quantization error
ME 431, Lecture 9
• An analog signal is
sampled at discrete
intervals of time
and is held
460
455
450
445
14
440
1.47
1.48
1.49
1.5
1.51
1.52
Electromagnetic Induction
• Converts electricity into force/torque
• Examples: solenoids, speakers, electric
motors
• Lorentz’s law: A current carrying conductor in
a magnetic field generates a force
ME 431, Lecture 9
• Like many sensors, some actuators employ
electromagnetic induction
15
• Current through coil
generates a magnetic
field (Ampere’s law)
• Magnetic field imparts a
force on the iron core
V
• Spring return
• Examples include valves
and switches, like a car
starter
ME 431, Lecture 9
Solenoid Actuator
16
• Current moving through a magnetic field
induces a force F  i  B
ME 431, Lecture 9
DC Motor
• Parts:
• Stator: stationary part (includes the magnet)
• Rotor: rotating part (wire coil)
• Commutator: half rings that connect to the
current source through brushes
17
• Multiple coils and
multiple magnetic
pairs ensure current
carrying wire near
magnet for higher
proportion of time
• Armature wrapped
around iron core so
that magnetic field
doesn’t have to cross
a large air gap
ME 431, Lecture 9
DC Motor
18
DC Motor
1. Armature control: change torque by
changing current in the armature (rotor)
2. Field control: change torque by changing the
strength of the magnetic field (by changing
current through an electromagnet)
ME 431, Lecture 9
Two approaches to DC motor control ( F  i  B )
19
• Armature-controlled DC motor model
• Model resistance and inductance of the
coil as lumped parameters
• Same with mechanical inertia and friction
Fixed
field
Ra
La
+
ea +
_
Armature
circuit
ia
ME 431, Lecture 9
DC Motor
T
eb
θ
20
J
_
Rotor
b