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Last lecture
Today’s menu
Resistive sensing elements:
Displacement sensors (potentiometers).
Temperature sensors.
Strain gauges.
Capacitive sensing elements.
Inductive sensing elements.
Reactive Deflection bridges.
Electromagnetic sensing elements.
Deflection bridges.
Thermoelectric sensing elements.
Elastic sensing elements.
Piezoelectric sensing elements.
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Capacitive sensing elements
Capacitive sensing elements (cont’d...)
Examples
General principle
(A)
Consider two metal plates with areas A, separated by a distance d by
some dielectric medium:
(B)
x
d
x
x
E1
d
E2
l
A
E
(C)
(D)
(E)
d
The capacitance is then given by
C=
ε0 εA
.
d
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Capacitive sensing elements (cont’d...)
Capacitive sensing elements (cont’d...)
Examples (cont’d..)
Examples (cont’d..)
(A) Variable displacement sensor:
(D) Capacitive pressure sensor:
C=
ΔC
(1 − ν 2 )a2
P.
=
C
16Edt3
ε0 εA
.
d+x
(B) Variable area displacement sensor:
C=
(E) Differential capacitive displacement sensor:
C1 =
ε0 ε
(A − wx).
d
εε0 A
d+x ,
C2 =
εε0 A
d−x
(C) Variable area displacement sensor:
C=
ε0 w
[ε2 l − (ε2 − ε1 )x] .
d
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Capacitive sensing elements (cont’d...)
Inductive sensing elements
Capacitive sensing elements are incorporated in a.c. deflection
bridge circuits or oscillator circuits.
Variable inductance/reluctance sensors
First, some comments on magnetic circuits:
The sensor is not purely capacitive, but also has a resistance in
parallel to represent losses in the dielectric. The quality of the
dielectric is often expressed in terms of the loss tangent,
tan δ =
6
In an electrical circuit, an electromotive force (e.m.f.) drives the
current through the circuit
e.m.f.
1
ωCR
= current × resistance
In a magnetic circuit, the magnetomotive force (m.m.f.) which
drives a flux φ through a magnetic circuit is:
m.m.f.
= flux × reluctance = φ × .
The reluctance limits the flux through the circuit, just as resistance
limits current flow through an electric circuit.
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Inductive sensing elements (cont’d...)
Inductive sensing elements (cont’d...)
The reluctance of a magnetic circuit is given by:
The flux in one turn is given
by
ni
weber
φ=
=
i
The total flux is given by
where
l is the total length of the flux path,
n
turns
n2 i
N = nφ =
l
,
μμ0 A
μ is the relative permeability of the circuit material,
μ0 = 4π × 10−7 H/m is the permeability of free space,
The self-inductance is defined
as
n2
N
=
L=
i
A is the cross-sectional area of the flux path.
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Inductive sensing elements (cont’d...)
Inductive sensing elements (cont’d...)
The inductive displacement sensor
The inductive displacement sensor (cont’d...
L
Air
gap
core permeability
mc
radius
r
i
n
turns
R
air gap
d
armature permeability
mA
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Inductive sensing elements (cont’d...)
Reactive deflection bridges
The inductive displacement sensor (cont’d...
Typical capacitive bridge
The total reluctance is
R2
TOT = CORE + GAP + ARMATURE
R3
where
ETh
CORE
=
GAP
=
ARMATURE
=
R
μ0 μC r2
2d
μ0 πr2
R
μ0 μA rt
C0
Ch
VS
~
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Reactive deflection bridges
Electromagnetic sensing elements
Typical inductive bridge
R
These elements are used for measuring linear and angular velocity and
are based on Faraday’s law. This means that if a flux N linked to a
conductor is changing with time, then the back electromotive force
induced in the conductor is
dN
,
E=−
dt
R
ETh
L1
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i.e. proportional to the rate of change of the flux N .
L2
VS
~
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Thermoelectric sensing elements
Thermoelectric sensing elements (cont’d...)
Thermoelectric or thermocouple sensing elements are commonly used for
measuring temperature.
The reason for this is that, if two metals A and B are joined together, there
will be a difference in electrical potential across the junction. The potential
depends on the types of metal and the temperature.
ETAB = a1 T + a2 T 2 + a3 T 3 + . . .
For the temperature difference between two junctions, the potential
difference is then
ETAB
− ETAB
= a1 (T1 − T2 ) + a2 (T12 − T22 ) + a3 (T13 − T23 ) + . . .
1
2
A
T1
The junction potential can be described by a power series of the form
ETAB1
AB
ET2
T2
B
AB
AB
ET1 - ET2
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Elastic sensing elements
Piezoelectric sensing elements
The general principle of elastic sensing elements is to convert a force to
an output displacement, which is then described by a change in
impedance.
If a force is applied to a crystal, the atoms of the crystal are displaced
from their normal positions, as
x=
Elastic elements are often used to measure:
torque = force × distance
1
F,
k
where k is the stiffness of the crystal. The dynamics can be described by
a second-order system.
pressure = force / area
acceleration = force / mass.
In a piezoelectric crystal, the displacement results in a charge
The dynamic behavior of these can often be described by second-order
systems. See the text book for details.
q = Kx =
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K
F.
k
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Piezoelectric sensing elements (cont’d...)
Piezoelectric sensing elements (cont’d...)
The piezoelectric effect is reversible, which means that if we apply a
charge across a crystal, its dimensions will change accordingly.
In order to measure the charge q the faces of the crystals are coated with
metal electrodes, resulting in a capacitor, with capacitance
CN =
ε0 εA
,
t
The crystal can therefore be represented as a charge generator q in
parallel with a capacitance CN , or as an equivalent Norton circuit with a
current source iN in parallel with CN , where
iN =
dq
dx
=K .
dt
dt
In the Laplace domain this is (in transfer function form)
ΔīN
(s) = Ks.
Δx̄
where A is the area, t is the thickness ε the permittivity of the crystal, and
ε0 the permittivity of free space.
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Piezoelectric sensing elements (cont’d...)
Piezoelectric sensing elements (cont’d...)
If the crystal is connected to a resistive load using a capacitive cable, we
have the following system:
The transfer function of the system, including the piezoelectric crystal, a
capacitive cable, and a recorder is:
G (s) =
iN
CN
CC
RL
ΔV L
=
,
s (CN + Cc ) + 1
ΔiN (s)
and relating it to the input force we get
RL
ΔV L ΔiN Δx
ΔV L
(s) =
ΔiN Δx ΔF
ΔF
Piezoelectric
crystal
Capacitive Recorder
cable
See the text book for the details of this derivation.
The piezoelectric crystal generates a current, proportional to the velocity
of a force acting on its surface.
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Next lecture
Summary
Capacitive sensing elements.
The remainder of chapter 9. Have covered only deflection bridges so
far.
Ideal amplifiers are covered in other courses. Read up on this, and
we can focus on limitations and errors.
Inductive sensing elements.
Reactive Deflection bridges.
Electromagnetic sensing elements.
Thermoelectric sensing elements.
Elastic sensing elements.
Piezoelectric sensing elements.
There are more details and some other examples of sensing elements in
the book. Read this on your own. The elements presented at the lecture
are only examples. Make sure you get the big picture.
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Recommended exercises
Questions?
8.14 – 8.18
8.3, 8.4, 8.11.
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