Variable Capacitance Transducers
• The Capacitance of a two plate capacitor is given by
A – Overlapping Area
x – Gap width
k – Dielectric constant
kA
C
x
k   0 r
0 
Permitivity of vacuum
r 
Relative permitivity
• A change in any one of these parameters may be used for sensing
• Examples - Transverse displacement, rotation, and fluid level
• A capacitance bridge can be used to measure the change in the capacitance
• Other methods include measuring a change in charge
• Charge – charge amplifier
• Voltage – high impedance device in parallel
• Current – low impedance device in series
• Or inductance capacitance oscillator circuit
Capacitive Rotation Sensor
• One plate rotates and the other is stationary
• Common area is proportional to the angle
C  K
• The relationship is linear and K is the sensor constant
• Sensitivity is
C
S
K

Capacitance
Bridge
DC Output
vo
Fixed
Plate
Rotation
A
θ
Rotating
Plate
Capacitive Displacement Sensor
• One plate is attached to the moving object and the other is kept stationary
• Capacitance is
and sensitivity is
K
C
x
S
C
K
 2
x
x
• This relationship is nonlinear but can be linearized by using an op amp circuit
vo 
vref Cref
K
x
C = K/x
Capacitance
Bridge
vo
Cref
A
Moving Plate
(e.g., Diaphragm)
Position
x
Fixed
Plate
Supply
Voltage
vref
+
−
−
+
+
Op amp
Output
vo
−
Displacement Measurement by changing
Dielectric
• Displacement can be measured by attaching the moving object to a solid
dielectric element placed in between the plates
• Liquid level as shown below can be measured as the dielectric medium
between the plates changes with the liquid level
Capacitance
Bridge
Fixed
Plate
Level
h
k
Liquid
vo
Displacement Measurement
i
R
+
Supply
Voltage
vref
vref  vo
d
i  Cvo  
dt
R
+
Output
vo
−
−
K
C
x
vo
1

vref 1  RKj x 
From phase
From magnitude
x
RK
1 M 1
2
x  RK tan 
Capacitive Angular Velocity Sensor
i
+
Supply
Voltage
vref
Current
Sensor
+
C 
−
−
i


d
dC
Cvref  vref
dt
dt
d
i

dt
Kv ref
Capacitive Sensor Applications
• Mechanical loading effects are negligible
• Variations in dielectric properties due to humidity, temperature,
pressure, and impurities can cause errors
• Capacitance bridge can compensate for these effects
• Sensitivity – 1pF per mm
Capacitance Bridge Circuit
Compensator
Z1
AC
Excitation v
ref
Sensor
Z2
Bridge
Output
vo
v 
+
Z3
Z4
Bridge Completion
vref  v
Z1
vo 

vo  v
0
Z2
vref  v
Z3
0v

0
Z4
 Z4 / Z3  Z2 / Z1  v
1  Z4 / Z3
For a balanced circuit
Z2 Z4

Z1 Z 3
ref
Bridge output due to sensor change
vo  
vref
Z1 1  Z 4 / Z3 
Z
Piezoelectric Sensors
• Substances such as BaTiO3 (barium titanate),SiO2 (quartz in crystalline), and
lead zirconate titanate can generate an electric charge when subjected to stress
(strain)
• Applications include
• Pressure and strain measuring devices
• Touch screens
• Accelerometers
• Torque/Force sensors
• Piezoelectric materials deform when a voltage is applied. Applications include
• Piezoelectric valves
• Microactuators and MEMS
Charge
Source
q
Equivalent
Capacitance
C
Z
1
jC
• Output impedance of a piezoelectric sensor is very high
• It varies with the frequency ~MΩ at 100Hz
Sensitivity
• Charge sensitivity
q
Sq 
F
For a surface area A (pressure applied – stress)
1 q
Sq 
A p
• Voltage sensitivity – change in voltage due to unit increment in pressure per
unit thickness (d is the thickness)
Sv 
1 v
d p
q  Cv
S q  kSv
• k is the dielectric constant of the crystal capacitor
Piezoelectric Material Sensitivities
Material
Charge Sensitivity
Sq (pC/N)
Voltage Sensitivity
Sv (mV.m/N)
Lead Zirconate Titanate (PZT)
Barium Titanate
Quartz
Rochelle Salt
110
140
2.5
275
10
6
50
90
Piezoelectric Accelerometer
Spring
Direction of
Sensitivity
(Input)
Inertia Mass
Output
vo
Piezoelectric
Element
Electrodes
• Inertia force caused by the acceleration produces a voltage
• Light weight, high frequency response (1MHz)
• High output impedance – small voltages ~1mV
• High spring stiffness – natural frequency or resonant frequency is high
(20kHz)
• Useful frequency range – 5kHz
Accelerometer
Signal (dB)
Resonance
Useful Range
1
5,000 20,000
Frequency (Hz)
Frequency response curve of a piezoelectric accelerometer
• Typical accelerometer sensitivities – 10 pC/g (pico Coulomb per gravity) or 5mV/g
• Sensitivity depends on the piezoelectric properties and the way the inertia force is
applied
• Large mass would result in a large force and a large output signal but
• Load the measurand
• Lower the resonant frequency
Charge Amplifier
Rf
Cf
A
−
vo/K
q
C
Piezoelectric
Sensor
+
K
Output
vo
Cc
Cable
+
Charge
Amplifier
−
• Impedance matching
• Reduce speed of charge leakage
qC
vo
v
 Cc o  C f
K
K
vo  vo  vo K

v

0
 o K 
Rf


Rf C f
dvo
dq
 vo   R f
dt
dt
Rf s
vo  s

q s 
Rf C f s  1


R f j
vo  j

q j
R f C f j  1

c  Rf C f
