Transduction Based on Changes in the
Energy Stored in an Electrical Field
Lecture 3
Department of Mechanical Engineering
Variable Area parallel Capacitor
◆
◆
Besides the gap, the other variable which can be
used for transducers are area and permittivity
Recall
εA ε oε r A
C=
=
d
d
If the plates are maintained at a fixed
separation, and one plate moves in a
direction normal to the field.
C=
ε wy
w is the width of the plates and y is the
length along which they overlap
d
The energy stored by the capacitor
Q2
Q 2d
W =
=
2C
2ε wy
Clearly the energy varies inversely with the region of overlap
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Variable Area parallel Capacitor
◆
Force and voltage related to overlap y
dW
Q 2d
=−
F=
dy
2ε wy 2
V =
dW
Qd
=
dQ
ε wy
Note that y is varying with time. These two equation are nonlinear
There are a number of ways we might choose to use the variable
effective area capacitor in transduction:
• Relating y and V
• Relating F and V
• Relating F and Q
In each of these cases, a choice must be made to work with the
nonlinear relations or to produce a nominally linear transduction
relation for restricted ranges
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Variable Area parallel Capacitor
◆
Voltage related to overlap y
Consider V =
Qd
ε wy
Assume that the lateral motion of the moving plate can be written as
y = yo + ~
y
Then we have V0 + V~ = d
εwyo
Assume that
~
Q Qo
~


 ~ Qo ~
y   Qo ~
y 2 Q~
y




Q
+
Q
−
higher
order
terms
+
−
+
 o 

  y2

y
y
o 
o 

 o


~
~
d
Q
dQ
~
~
oy
and y yo are small, V =
−
2
εwyo
εwyo
If we resort to electrical isolation of ac system portion from the dc bias
dQo ~
y
~
V =−
εwyo2
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Variable Area parallel Capacitor
◆
A rotational position sensor exploiting
the variable area transduction
mechanism by GMC Instruments Inc.
– This sensor uses a pair of parallel
plate capacitors in an electrically
differential arrangement
– As the wedge-shaped rotor moves,
the effective capacitor area in one set
of plates increases while that for the
other decreases
– The spiral shape of the electrodes is
chosen because it produces a linearly
increasing or decreasing electrode
effective area/rotational position
relationship
Position
Voltage
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Variable Area parallel Capacitor
◆
Angle Displacement Sensors
Capacitance is proportional
the angle of displacement
Am= the area corresponding to α=0
Function C = f(α ) is linear here
1. Moving plate
2. Fixed plate
3. Congruous area
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Variable Area parallel Capacitor
◆
Voltage related to Force
dW
Q 2d
=−
F=
dy
2ε wy 2
dW
Qd
Using V =
=
dQ
ε wy
Since
Q = CV =
εwyV
QV
F =−
2y
F, Q, V, and y are all time varying parameters
We have
d
F =−
εV 2 w
2d
A quadratic relation between force and voltage
Again we could linearize this relationship by assuming the force to consist of a large
dc and a small ac component, while the voltage similarly consists of a large dc and a
small ac component. The resulting zeroth and first-order relations would be
Fo = −
εVo 2 w
2d
~
εVoV w
~
F =−
d
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Variable Area parallel Capacitor
Electrostatic motor in which torque
results from variable area capacitors
By Jafimenko.
Example of a silicon electrostatic micro-motor
By Brysek et al.
The voltage applied is at a frequency that matches the rotational speed
Department of Mechanical Engineering
Variable Area parallel Capacitor
A comb-drive electrostatic micro-actuator
◆
Force related to Charge
dW
Q 2d
=−
F=
dy
2ε wy 2
This mechanism is not straightforward because the
lateral motion influences the force and charge relation
This mechanism is typically not exploited in
electromechanical sensors and actuators
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Parallel capacitor with
Varying permittivity
For each segment of the capacitor
Qi = CiV =
ε i AiV
d
Total charge Q = (ε1 A1 + ε 2 A2 )V
d
Comparing with Q=εAV/d, the effective
permittivity
ε eff =
(ε1 A1 + ε 2 A2 ) = ε1 y + ε 2 ( y − 1)
A
Capacitive transducer in which the
permittivity varies spatially
l
The voltage constitutive equation is
Qd
V =
[ε1 y + ε 2 ( y − 1)]w
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Varying permittivity
◆
The thickness of a dielectric film can be measured by
pulling it between two plates of the capacitor.
If εf>>εo,
1. Film
2,3: plates
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Varying Permittivity
◆
Large Displacements and Level
Sensors
Sensitivity
1. Dielectric element
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Varying Permittivity
◆
Liquid level sensor
The elementary capacitor C’
with two circular imaginary
electrodes at radius r,
separation dr from each other,
and length L
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Varying Permittivity
Therefore the capacitance between
the electrodes with radii R1 and R2:
(A)
With liquid level h, the
overall transfer function for
the level sensor will be:
Equation (A)
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Varying Permittivity
Thus for a small gap:
Sensitivity of C to a change of h will be:
◆
We can modify the above results for measuring large
displacement of a moving rotor element
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Capacitor’s Equivalent Circuit
◆
An ideal dielectric has a zero conductivity. This means that
the electrical circuit of the element with this dielectric is
pure capacitance, which does not dissipate any real power.
The reactive power of capacitor is
Qr = ωCV 2
◆
◆
And its impedance is
ω: angular frequency, rad/s
Qr: reactive power
1
jω C
A real dielectric between the capacitor plates contributes
to losses. The power dissipated in the capacitor:
Zc =
P = VI cos θ
I: current through capacitor
V: voltage applied to the capacitor
cosθ: power factor
θ: power factor angle
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Capacitor’s Equivalent Circuit
◆
The losses are also evaluated in terms of phase angle δ
cos θ = sin δ ≈ tan δ ≈ δ
◆
Usually δ is very mall
The actual element carrying the real dielectric can be regarded with a
perfect capacitor Cs or Cp with the series or parallel resistances
1
cos θ
For “pure” capacitor:
R
=
Rs =
p
ωC p cos θ
ωC s
θ=90o
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Capacitor’s Equivalent Circuit
◆
Relationships for the two equivalent circuits of real capacitors:
(
C s = C p 1+ tan δ
2
)
Rp
Rs =
1
1+
tan 2 δ
For Real capacitor, its impedance
Z = R − jX
◆
Introducing the capacitive reactance X for parallel capacitor
1
X=
ωC
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Differential Displacement Sensors
◆
The output signal of a sensor element
can be difference between two
capacitances or their reactances.
If the middle plate is displaced to the right by ∆δ
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Differential Displacement Sensors
◆
To calculate the difference for the singlecapacitor element, we assign
X 2 − X1 = −
1
∆δ
ωε A
Obviously, the output signal from the element is doubled for
the differential type and its transfer function C2-C1=f(∆δ) is
more linear
Department of Mechanical Engineering
Example: Capacitive Flow Microsensors
◆
A silicon micro-flow sensor, shown in the figure, using
parallel plate transduction mechanism was fabricated by bulk
silicon micro-machining, and boron etch-stop
◆
Gas flows into the inlet at pressure P1, passes through a
silicon flow channel and leaves with outlet at pressure P2
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Example: Capacitive Flow Microsensors
◆
The gas flow-rate Qm generates the pressure differential (P1P2) across the fluidic conductance Gf of the channels,
The differential pressure is measured by the parallel plate
capacitive sensor
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Example: Capacitive Flow Microsensors
◆
◆
The boron-doped membrane forms the deflection plate
The capacitance is measured using an integrated CMOS switchedcapacitance circuit which simply integrate the charge difference
on the sensing capacitance Cs, and a reference capacitance Cref,
Where Cf is the feedback capacitance in an
op-amp circuit and Vref is the voltage
reference pulse height
A resolution of 1 fF was reported
which corresponding to a change in
pressure of 0.13 Pa
Cho and Wise, 1993
Department of Mechanical Engineering