Transcript Prezentacja programu PowerPoint

4. Microsystems in measurements of
mechanical quantities- displacement,
velocity and acceleration
Mechanical quantities important in measurements with sensors:
x(t), (t),  = x/x , - position (linear, angular), displacement, elongation (strain)
v = dx/dt,  = d/dt, a = dv/dt – velocity, acceleration (linear, angular)
F = ma,  = dF/dA, M = F d - force, pressure, torque
dm/dt, dV/dt - mass flow, volume flow
For quantities varying in time we measure:
t
1
A   A(t )dt
t0
average values
t
rms values
2
rms
A
1
2 dtperiodic motion t = T
  A(t )for
t0
The unknown parameters can be determined from the basic relationships
between quantities, e.g. from the knowledge of acceleration one obtains
succesively
v( t )   a ( t )dt  C1
x ( t )   v ( t )dt  C2
Bearing in mind determination of integration constants, it is necessary to do
additional measurements.
Recent MST technologies allow to fabricate low cost but precise acceleration
sensors.
Accelerometers, regardless of the conversion technique, require the
existence of a seismic mass, which displacement with respect to the housing
is rgistered. Taking into account the conversion technique of a displacement,
one deals with different kinds of accelerometers: piezoelectric, piezoresistive,
capacitive, thermal.
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Mechanical model of vibration sensor
An equation of motion of mass m with respect to the reference frame with
coordinate y, under the influence of spring force – ky, damping force – b dy/dt
and inertial force – md2x/dt2 can be written as:
d 2 y b dy k
d2x

 y 2
2
dt
m dt m
dt
1. m – large
b – small
k – small
2.
Sensor case moves relative to
the Earth along a coordinate x
m – small
b – large
k – small
3. m – small
b – small
k – large
Adjusting the oscillator
constants one can neglect
selected terms in the equation,
thus obtaining different sensors
y  x
position sensor
b
dx
y
m
dt
velocity sensor
k
d2x
y 2
m
dt
accelerometer
In reality the sensor mass should be small enough to avoid influence on the investigated
object. In this case one can build a sensitive accelerometer and other3vibration
parameters can be obtained by integration of acceleration.
Piezoelectric accelerometer
Piezoelectric plates are sandwiched between the casing and the seismic mass, which
exerts on them a force proportional to acceleration.
1. – seismic mass
2. – piezoelectric plates (with magnification)
3. – tension control
4. – FET preamplifier
5. – cable attachment
In MEMS technology a silicon cantilever
with deposited piezoelectric film, e.g.
BaTiO3 is used.
Experimental setup with piezoelectric accelerometer used for
investigation of vibration parameters.
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Piezoelectric effect
y axis
(mechanical)
Fy
b
Fx
a
x axis
(electrical)
Section of the quartz crystal in a plane
perpendicular to c axis (z-axis).
There exist 3 mechanical axes
(perpendicular to crystal planes)
and 3 electrical axes
(drawn through edges).
A plate cut from the crystal is also shown.
Applying a force to the plate along the electrical axis one generates the
charge on the surfaces, to which a stress is applied (longitudinal piezoelectric
effect).
Acting with a force along mechanical axis y we induce a charge on the
surfaces as before (transversal effect).
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Piezoelectric effect, cont.
Quartz crystal structure
(the first atomic layer is shown,
in the second layer there are 3 O2- atoms,
the third layer is identical as the first one,
aso.)

x1
x1



Longitudinal effect
Transversal effect
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Piezoelectric effect, cont.
Application of stress σ generates a charge with density q
dq
d
 kp
dt
dt
kp –piezoelectric module
(for quartz 2.2 ·10-12 C/N, for ferroelectrics ca.100 times higher)
For the longitudinal effect (force Fx) one obtains for surface Ax a charge
Q’ = Axq’ = Axkpσ = kpFx
- independent of Ax
For transversal effect (force Fy) one gets
Q’’ = -kpFy b/a
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Piezoelectric effect, cont.
Generated charge on capacitance C gives the voltage
U = Q/C = Q/(Ck + Cm) = kpFx/C, Ck, Cm – cap. of crystal and cable, resp.
For n parallel connected plates one gets
U = nQ/(nCk + Cm)
This gives piezoelectric sensitivity
Sp = dU/dFx = nkp/(nCk + Cm)
Sensor discharge time constant is then equal:
τ = (Ck + Cm)/(Gk + Gm)
G – conductance
This time constant limits fmin.
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Capacitive sensor
l
Capacitance of a flat-plate capacitor
C = ε0εr A/ l
A - plate area, l – distance between plates
C
C
Sensitivity
dC/dl = - ε0εrA/ l2 , changes with l
dC/C = - dl/l , high relative sensitivity
for small l (nonlinearity)
0,1
l
l
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Differential capacitor
ΔC = C2 – C1 = ε0εr A·2Δl/(l2 – Δl2)
for Δl << l
ΔC = ε0εrA·2Δl/l2, hence
ΔC/C = 2Δl/l
C1
l
l
l
C2
C
Therefore one obtains increased
sensitivity and linearity.
-1
-0,6
0,6
1
l
l
Differential technique decreases
the temperature error and the
influence of ε drift.
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Capacitive
displacement sensor
Cylindrical capacitive sensor with
movable dielectric shaft in ratiometric
configuration:
W – common electrode
S – fixed electrode
R – variable electrode
The displacement causes moving of the shaft and is calculated from the ratio
of capacitances
CWR/CWS
In practice the S electrode needed carefull screening to avoid the inflence of
air humidity variations on CWS capacitance
(change in configuration of the electric field).
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Angular position sensor

In the simplest case a rotating capacitor
can be used
C
C  C0   C
C0

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12
Angular position sensor, cont.
Practical realisation of a differential
rotating capacitor
(Zi-Tech Instruments Corp.)
Stators 1 and 2 form with a rotor separate capacitances.
The difference of those capacitances varies linearly with movement of the rotor.
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Capacitive accelerometer
From discussion of the mechanical model of vibration sensor it follows that
for the system with high spring konstant k we can cosider the deflection,
i.e. also the change in capacitance, as proportional to the acceleration.
In this case one obtains a capacitive accelerometer.
A capacitive accelerometer with a
differential capacitor fabricated in silicon
bulk micromachining technology.
The movable mass is sandwiched between upper and base fixed electrodes
forming two variable capacitances.
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Examples of MEMS accelerometers: Analog Devices ADXL250 (on the left)
and Motorola dual-structure microsystem before encapsulation (on the right)
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Tilt (inclination) sensor based on
capacitive accelerometer
Construction of integrated Analog Devices
accelerometer:
(a) scheme of interdigitated differential capacitor,
(b) upper view of the sensing structure.
The central moving belt forms with static belts
the interdigitated structure (46 capacitors) and
deflects from the central position by inertial
forces.
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Inclination sensor based on
capacitive accelerometer, cont.
Determination of the tilt angle θ
from measurements of the
gravitational acceleration
Tilt angle determination for
single axis and dual axis
sensors (g = 1)
single axis sensor
dual axis sensor
For single axis sensor the
sensitivity decreases with θ
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