Sugata Munshi Department of Electrical Engineering Jadavpur University Page 1 of 43 PIEZOELECTRIC TRANSDUCERS PIEZOELECTRIC TRANSDUCERS PIEZOELECTRIC TRANSDUCERS PIEZOELECTRIC 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,
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
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
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
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
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....
