Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 1 of 43
PIEZOELECTRIC TRANSDUCERSPIEZOELECTRIC TRANSDUCERSPIEZOELECTRIC TRANSDUCERSPIEZOELECTRIC TRANSDUCERS
DIRECT PIEZOELECTRIC EFFECT:
An electric polarization is produced by mechanical
strain in crystals belonging to certain classes, the
polarization being proportional to the strain and
changing sign with the strain. As a result of this
polarization, electric charges appear at the surfaces of
the crystal.
Charge q that develops, can be determined from the
output voltage, since,
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 2 of 43
q = C E0
where C is the capacitance of the sample of
piezoelectric material.
Materials such as quartz, Rochelle salt, tourmaline,
lithium sulphate (LS), ammonium dihydrogen
phosphate are inherently piezoelectric.
There are other materials (ferroelectric ceramics)
e.g. barium titanate, which can be made to have
piezoelectric properties by artificial polarization.
Polling:
Strong electric field is applied to the material,
while it is heated to a temperature above Curie
point (125°°°°C for barium titanate).
Then it is slowly cooled up to room temperature,
with the field still applied.
When the electric field is removed from the
cooled material, there is a remnant polarization
and the material exhibits piezoelectric
properties.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 3 of 43
Such materials are known as polarized
piezoelectric materials. Other examples are lead
zirconate and lead metaniobate.
A piezoelectric crystal has two sets of constants:
(a) The charge sensitivity or piezoelectric constant ‘d’
defined as the charge generated per unit force
applied.
(b) The voltage sensitivity ‘g’ defined as the electric
field produced per unit stress.
Both the ‘g’and ‘d’ constants depend on the direction
of application of force, and also on the direction of
measurement.
force applied in x and measurement in y direction.
force applied in z and measurement in z direction
xy
xy
zz
zz
d
g
d
g
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 4 of 43
Similarly the crystal has constants (dxx ,gxx ) , (dzy ,gzy )
etc.
Let us consider a rectangular slab of piezoelectric
material subjected to a compressive force f. the
thickness is h and ∆h is the deformation. The
measurement is carried out in the direction of
compression. Let A be the surface area on which the
force acts, and ε be the absolute permittivity of the
piezoelectric material.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 5 of 43
According to definitions, the constants in the direction
of compression are,
0
0
0
0 0
C/N
/
zz
zz
CEqd d
f f
EE A
Vmhg gf fh
A
E EA qC
f h fg
f
d
N
ε
ε ε ε ε
= = =
= = =
∴ = × =
→
=
→
=
Typical g values are 312 10 Vm/N
−× for barium titanate
and 350 10 Vm/N
−× for quartz.
The permittivity of quartz is about 114.06 10 F/m
−× and
that for barium titanate is 111250 10 F/m
−× .
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 6 of 43
Then for quartz the d value corresponding to the g
value given above, is
3 34.06 10 50 10 / 2.03 pC/Nd g C Nε − −= = × × × =
Similarly the d value for barium titanate is 150 pC/N.
Sometimes it is necessary to express the output charge
or voltage in terms of the deformation (rather than
force or stress) of the crystal, since it is really the
deformation that causes the charge generation. To do
this, the modulus of elasticity (ΕΕΕΕ) of the piezoelectric
material must be known.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 7 of 43
0
0
0
0q
h
Voltage sensitivity with respect to deformation
= h
Charge sensitivity w.r.t. deformation is,
K
fA
hh
EEhg
fA
K
Eg
CEqCg CK
h h
Ε =∆
= =Ε ∆
∴ =
= Ε∆
= = = Ε =∆ ∆
EQUIVALENT CIRCUIT:EQUIVALENT CIRCUIT:EQUIVALENT CIRCUIT:EQUIVALENT CIRCUIT:
A piezoelectric transducer can be represented by the
following equivalent circuit.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 8 of 43
C = Capacitance of the sensor 10 pF to 1000 pF.∼
R = Leakage resistance of the sensor 1110 .Ω∼
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 9 of 43
0
=angular frequen
Z =output impedan
cy of temporal variation
of deformat
ce of sensor
1 =
1
ion.
1R
where,
R
j CRj C ω
ω
ω=
++
0E =Open circuit output voltage of PZT.
When ω=0, i.e. for static measurement, 11
0Z 10 .R= Ω∼
For the transducer alone, due to a static
deformation xi , E0 leaks off slowly through the
leakage resistance. However the decay will be very
slow since R is very large.
When an external voltage measuring device of
relatively low internal resistance is connected across
the sensor for measuring 0E , the charge q leaks off
rapidly, preventing static measurement.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 10 of 43
m
m 0 0
00
If Z input impedance of voltmeter,
1V
1
m
m
m
ZE E
ZZ ZZ
=
= =+ +
For mV to be close to 0E , we should have Zm >>Z0
which may be difficult to achieve. The situation is
particularly complicated for static measurement, since
then ω=0 and Z0 →∞ .
The output impedance of piezoelectric sensors ranges
from infinity (ideally) for static applications of force,
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 11 of 43
to about 10KΩ for very high frequency applications (~
100 KHz).
To overcome the problems discussed, the device for
measuring E0 should be preceded by a unity gain
buffer amplifier which offers a very high input
impedance.
BUFFER AMPLIFIER CIRCUIT FOR BUFFER AMPLIFIER CIRCUIT FOR BUFFER AMPLIFIER CIRCUIT FOR BUFFER AMPLIFIER CIRCUIT FOR
PIEZOELECTRIC TRANSDUCERS:PIEZOELECTRIC TRANSDUCERS:PIEZOELECTRIC TRANSDUCERS:PIEZOELECTRIC TRANSDUCERS:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 12 of 43
Special-purpose operational amplifiers known as
electrometer op-amps, are used as buffers. The
electrometer op-amps have extremely high input
impedance. Typical example is AD515 having an input
impedance 1015
Ω || 0.8 pF, manufactured by Analog
Devices.
CIRCUIT ANALYSIS:
ae
a
e C a C
RRR
R R
C C C C C C
=+
= + + +
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 13 of 43
0 0
(1)
(2)
q i
iq
C R
iq e
e
q K x
dxi K
dt
i i i
dx dv vK C
dt dt R
=
=
= +
∴ = +
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 14 of 43
00
0
0
Taking Laplace transform of both sides,
( )s ( ) ( )
,
(1 ) ( ) s ( )
System transfer function is
s( )( ) (3)
( ) (1 ) 1
where,
= Time constant of enti
q i e
e
e e q e i
q e
i e e
V sK X s sC V s
R
or
sC R V s K R X s
K RV s K sG s
X s sC R s
τ
τ
τ
= +
+ =
∴
′= = =
+ +
q
e
re circuit.
KK = Voltage sensitivity of entire circuit
C
w.r.t. deformation.
′ =
The frequency response function is:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 15 of 43
1
2 2
2 2
1
( ) tan1 21
( )1
( ) tan2
KK jG j
j
KG j
Arg G j
ω τωτ πω ωτ
ωτ ω τ
ω τω
ω τ
πω ωτ
−
−
′′ = = ∠ −
+ +
′=
+
= −
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 16 of 43
0 0
2 2
1
force is considered as input, the system function is,
V ( ) V ( ) ( )H(j )= (4)
( ) ( ) ( )
H(j )1
H(j )= t
(
an
) 5
)
2
( )(
i
i
i
q
e
X j h h
If
j j X j
F j X j F j
d
C
Ar
g d
F j A A C C K
g
g
ω ω ω
ω ε
ωω ω ω
ω τ
ωω τ
ε ε
ω
ω ωτ
ε
π −
= = × = =
= ×
−
Ε Ε Ε
+
=
=
Ε
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 17 of 43
For static deformation, SS gain=0 not suited
for static measurement.
For LF sinusoids low gain & considerable
phase shift between xi and v0.
Suitable for HF measurements.
=3, G(j ) 0.95For Kωτ ω ′≈ . Thus for ωτ > 3, i.e. ω >
3/τ, G(j )ω lies within 95% of K’. So ω = 3/τ sets
the lower frequency limit of transducer.
Circuit is not suited for slowly varying
deformations but works well when deformation
changes rapidly.
Quasi-Static Measurement:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 18 of 43
0
i
0
( )( )
( ) 1
x ( ) ( )
,
( )
( ) ( ) ( )11
i
m
mi
m mi
V s K sG s
X s s
Let t X u t
Then
XX s
s
X K X KV s G s X s
ss
τ
τ
τ
ττ
′= =
+
=
=
′ ′∴ = = =
+ +
0
inverse LT of both sides,
v ( ) ( ) ( )t tm q
m
e
Taking
X Kt X K e u t e u t
Cτ τ− −
′= =
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 19 of 43
If τ = Re Ce is large, decay is slow eanables
quasi-static measurement.
τ can be increased by increasing Ce by
connecting an external capacitor across sensor.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 20 of 43
Voltage sensitivity is sacrificed since q
e
KK
C′ =
.
This can be tolerated since K ′ is usually large.
How to Increase RHow to Increase RHow to Increase RHow to Increase Reeee ????
Effect of placing an external resistance REffect of placing an external resistance REffect of placing an external resistance REffect of placing an external resistance RSSSS
in series with amplifier input lead:in series with amplifier input lead:in series with amplifier input lead:in series with amplifier input lead:----
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 21 of 43
( )
1 2
00 0
0q 0 0
0
,
1
L.T. and arranging,
( )K ( ) ( ) ( )
,
( )
( )1
C
i S a S aq e
a a a
S a S ai e
a a a
q a
S aie S a
i i i i
or
dx R R v R RdK C v v
dt dt R R R R
Taking
R R V s R RsX s C sV s V s
R R RR
Finally
K R sV s
R RX ssC R R
R
= + +
+ += + +
+ += + +
∴
=+
+ + +
Case-I:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 22 of 43
( )[ ]
( )[ ]
( )
S a a
S a S a0
S a
a
If R R ; i.e. if R is large.
R R R R( ) =
( ) 1 1 1
,
&
R R
R is large w.r.t. R no significant increase in
is achieved.
qa e aq
e
i e e
q
e
e ae
R
KRR s C RR sK
CV s K s
X s sC R sC R s
Kwhere K
C
C RRC R
τ
τ
τ
τ
+
×+ + ′
= =+ + +
′ =
=+
∵
Case-II:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 23 of 43
( )
( )
( )
S
S a
S a0
S a
S a
S a
If R is small compared to R i.e. if R R ,
R RR R( )
( ) 11 R R
,
; = R RR R
a a
q ae
e
i e
q a ae e
e a
R
K RC s
CV s K s
X s ssC
where
K R RRK C C
C R R
τ
τ
τ
+
× +
+ ′ = =+ + +
′ = × + >
+ +
Charge Amplifier Circuit:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 24 of 43
b is at ground potential, & a is virtual ground.
Hence eab (t)≈0 no currents flow through
C,R,CC , Ra , Ca .
0
0
0
,
( )
( )
f ai
f
iq f
q
i f
i i i
or
i i
dx dvK C
dt dt
KV s
X s C
+ = ≈
= −
= −
∴ = −
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 25 of 43
Static system v0 is instantly &linearly re;lated
to xi .
Problem---- input bias current
0
0
this voltage drivesamplifier into saturation
0
Integrating and rearranging,
1v ( ) ( )
ai
f ai
iq f ai
q
i ai
f f
i
i i i
dx dvK C i
dt dt
Kt x t i dt
C C
≠
∴ = − +
= − +
= − + ∫
iai charges Cf steadily , until amplifier is driven to
saturation.
Remedy---- A resistance Rf is connected in parallel
with Cf .
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 26 of 43
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 27 of 43
1 2
0 0
0 0
00
1 1( )
ai
iq f ai
f
iq f ai
f
q
ai
f f f f
i i i i
dx v dvK C i
dt R dt
dx dv vK C i
dt dt R
K vv t i dt dt
C C C R
+ + =
+ + =
∴ = − + −
= − + −∫ ∫
Cf gets a discharging path through Rf .
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 28 of 43
1 2
0 0
0
2 2
f
0
0
,
1( ) ( )
( )( )
( ) 1 1
,
response
G(j )=1
G
; =C
(j )1
G(j )= tan2
q
f
ai
iq f
f
f q i
f
q f
i f f
f
i i i i
dx v dvK C
dt R dt
or
sC V s sK X sR
sK RV s K sG s
X s sC R s
where
Frequency
K j
KK
Ar
RC
j
K
g
τ
τ
ωτω
ωτ
ωτω
ω τ
π
τ
ω −
+ + =
+ + =
+ = −
− ′−∴ = = =
+ +
′−
+
′=
+
−
′ =
− 1 ωτ
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 29 of 43
Magnitude response is identical to that of a PZT-
buffer amplifier combine, and exhibits the same loss
of static and low-frequency response.
Advantages---
1. K’ and τ are independent of sensor, cable &op-
amp parameters.
2. Long cables can be used without affecting K’.
3. τ can be made large with large Rf, improving LF
response.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 30 of 43
Disadvantage—Poor signal-to-noise ratio since
high value of Rf is used.
Example:Example:Example:Example: With quartz PZTWith quartz PZTWith quartz PZTWith quartz PZT, C, C, C, Cffff ~10 pF to 10 pF to 10 pF to 10 pF to
101010105555 pF and RpF and RpF and RpF and Rffff ~1010101010101010 ohms to 10ohms to 10ohms to 10ohms to 1014141414 ohms.ohms.ohms.ohms.
τ~ 101010107777 secondssecondssecondsseconds,,,, enabling practically dc enabling practically dc enabling practically dc enabling practically dc
response.response.response.response.
Piezopiles
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 31 of 43
To increase the sensitivity, more than one
piezoelectric elements can be sandwiched between to
constitute a transducer system referred to as
bimorphs or multimorphs or piezopile.
Even if the elements are mechanically in series, they
can be electrically in series or parallel.
The series electrical connection increases the voltage
sensitivity but decreases the transducer capacitance.
Parallel connection increases both charge sensitivity
and capacitance.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 32 of 43
PIEZOELECTRIC ACCELEROMETERPIEZOELECTRIC ACCELEROMETERPIEZOELECTRIC ACCELEROMETERPIEZOELECTRIC ACCELEROMETER
PZT sandwiched between seismic mass and PZT sandwiched between seismic mass and PZT sandwiched between seismic mass and PZT sandwiched between seismic mass and
base of casing.base of casing.base of casing.base of casing.
CasingCasingCasingCasing→ rigidly fastened to workpiece rigidly fastened to workpiece rigidly fastened to workpiece rigidly fastened to workpiece
in motion.in motion.in motion.in motion.
Proof mass Proof mass Proof mass Proof mass →free to vibrate ( 1 degree of free to vibrate ( 1 degree of free to vibrate ( 1 degree of free to vibrate ( 1 degree of
freedom.freedom.freedom.freedom.
No intentional dampingNo intentional dampingNo intentional dampingNo intentional damping→low damping ratio.low damping ratio.low damping ratio.low damping ratio.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 33 of 43
Rarely, casing filled with silicone oil for Rarely, casing filled with silicone oil for Rarely, casing filled with silicone oil for Rarely, casing filled with silicone oil for
dampingdampingdampingdamping→viscosity depends strongly on viscosity depends strongly on viscosity depends strongly on viscosity depends strongly on
temptemptemptemp→heater installed in fluheater installed in fluheater installed in fluheater installed in fluid to haveid to haveid to haveid to have const. const. const. const.
temp.temp.temp.temp.
Hemispherical spring kept under tension by Hemispherical spring kept under tension by Hemispherical spring kept under tension by Hemispherical spring kept under tension by
screwing cap.screwing cap.screwing cap.screwing cap.
Spring preloaded by screwing down cap to Spring preloaded by screwing down cap to Spring preloaded by screwing down cap to Spring preloaded by screwing down cap to
prestress the PZT.prestress the PZT.prestress the PZT.prestress the PZT.
Why ?Why ?Why ?Why ?
Ans:Ans:Ans:Ans:
To work the piezomaterial in the linear portion of To work the piezomaterial in the linear portion of To work the piezomaterial in the linear portion of To work the piezomaterial in the linear portion of
chargechargechargecharge----strain characteristic.strain characteristic.strain characteristic.strain characteristic.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 34 of 43
Allow measurAllow measurAllow measurAllow measurement of both +ve & ement of both +ve & ement of both +ve & ement of both +ve & ----ve acceleration ve acceleration ve acceleration ve acceleration
without putting PZT in tension, without putting PZT in tension, without putting PZT in tension, without putting PZT in tension, since it is very since it is very since it is very since it is very
difficult to have proper adhesion mechanism to difficult to have proper adhesion mechanism to difficult to have proper adhesion mechanism to difficult to have proper adhesion mechanism to
put PZT in tension as mass moves up.put PZT in tension as mass moves up.put PZT in tension as mass moves up.put PZT in tension as mass moves up.
• Preloading results in output voltage Preloading results in output voltage Preloading results in output voltage Preloading results in output voltage →allowed allowed allowed allowed
to leak off. to leak off. to leak off. to leak off.
Subsequent aSubsequent aSubsequent aSubsequent accln.ccln.ccln.ccln. resultsresultsresultsresults in electric charge in electric charge in electric charge in electric charge
whose sign depends on sign of accln.whose sign depends on sign of accln.whose sign depends on sign of accln.whose sign depends on sign of accln.
THEORY:
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 35 of 43
• Zi = Displ. of workpiece ( & hence
accelerometer) w.r.t inertial frame of
reference.
• Zm = Displ. of mass w.r.t inertial frame
of reference.
• Z0 = Zi Zm = Displ. of mass w.r.t
casing.
• xi = deformation of sensor= Z0
Equation of motion is:
onseismic
2
002
2 2
0 002 2
mass
0
,
Input
m
Netforce forc
i
e
d Z dZm b CZ
dt dt
or
d Z dZ d Zm b CZ m
dt dt dt
+ + =
− − =
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 36 of 43
2
2
2
0 002
where,
b=Damping Constant.
C=Stiffness of Spring.
Acceleration of workpieceid Za
dt
d Z dZma m b CZ
dt dt
= =
∴ = + +
STATIC MEASUREMENT:
For a constant acceleration input ‘a’ ,
under SS condition,
0
2
0 0
2
0
Constant.
0 ; =0
i.e.
Z
dZ d Z
dt dt
ma CZ
=
∴ =
=
0 i
mZ x a
C∴ = =
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 37 of 43
DYNAMIC MEASUREMENT:
2
0 002
2
0 0 0
0
2
2 22
Transfer function of accelerometer p
Taking L.T.,
( ) ( ) ( ) ( )
( ) ( )( )
( ) ( )
1 1
r
oper is
2
,
i
n n
d Z dZma m b CZ
dt dt
mA s ms Z s bsZ s CZ s
Z s X s mH s
A s A s ms bs C
b C s ss s
m m
where
ξω ω
ω
= + +
= + +
∴
= = =+ +
= =+ ++ +
Undamped natural frequency.
Damping Ratio.
n
C
m
ξ
= =
=
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 38 of 43
( )
0
22
2
2
0
2
2
2
0
2
2
210
2 222 2 2
2
0
( ) 1( )
( ) 21
( ) 1
( ) 21
,
( ) 1
( )2 1
r= ,
( ) 1 1 2tan
( ) 1 2 11 4
( ) 1
( )1
n
n n
n
n n
n
n n
n
n
n
Z sH s
A s s s
Z s
A s s s
or
Z j
A jj
Plugging
Z j r
A j r j r rr r
Z j
A jr
ξω
ω ω
ω
ξ
ω ω
ω ω
ω ω ωξ
ω ω
ω
ω
ω ω ξ
ω ξ ξ
ω ω
ω
−
∴ = =
+ +
=
+ +
= =
− + +
= = ∠ −− + −
− +
∴ =
−( )2
2 2 24 rξ+
For an accelerometer with no intentional damping, For an accelerometer with no intentional damping, For an accelerometer with no intentional damping, For an accelerometer with no intentional damping,
ξ≈0.01 in a good instrument0.01 in a good instrument0.01 in a good instrument0.01 in a good instrument....
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 39 of 43
2
0n
Then,
( )1.05 at r=0.2, i.e. at =0.2
( )
n Z j
A j
ω ωω ω
ω
Over ω=0 to 0.2ωn ,
Mag resp is constant (deviation≤ 5%).
0Z a∝ for r ≤0.2(approx), i.e. ωn ≥ 5ω.
For hf applications ωn should be large.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 40 of 43
DYNAMICS OF COMPLETE ACCELEROMETER
SYSTEM:
TF of PZT-cable-buffer combination is
0 0
0
0
22
2
2
0
( ) ( )( )
1 ( ) ( )
TF of complete accelerometer system is,
( ) 1( ) ( ) ( )
( ) 1 21
1( )
( ) ( ) ( )( ) 1 1
i
n
n n
n
V s V sK sG s
s X s Z s
V s K sT s G s H s
A s s s s
V j K jT j G j H j
A j j
τ
τ
τ
τ ξω
ω ω
ω ωωτω ω ω
ω ωτ
′= = =
+
′ = = = + + +
′ = = =
+ ( )
( )
2
2
22 2 22
2 2
2
1
( ) ( ) ( )1 1 4
n
n
r j r
K rT j G j H j
r rr
ξ
ωω ω ω
ξω τ
− +
′∴ = = ×
− ++
At low frequencies:
Mag of 2nd
order response ≈ constant.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 41 of 43
Mag response of total system dominated by 1st
order response.
At hi frequencies:
Mag of 1st order response ≈ constant.
Overall mag response governed by 2nd
order
response.
1st order response saturates to 5% of K' at
ω=3/τ.
For ξ ≈ 0.01, 2nd
order response starts
deviating from const. value by more than 5%,
from ω=0.2ωn .
Usable linear range: 3
0.2 nω ωτ
≤ ≤
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 42 of 43
For 3/ 0.2n
τ ω ω≤ ≤ the phase shift between V0
and A is very small.
Sugata Munshi
Department of Electrical Engineering
Jadavpur University
Page 43 of 43
Salient points:
Typical shock accelerometer →0.004 pC/g→fn
=250KHz.
Accelerometer for low-g
measurement→1000pC/g→ 7 KHz.
Size can be as low as 7 mm3.