Fundamentals of
Measurement Technology
(10)
Prof,Wang Boxiong
A piezoelectric transducer is an active
transducer or self-generating transducer,It
utilizes the piezoelectric effect of some
materials,which generate electrical changes
on some of their surfaces when acted by
external forces,So piezoelectric transducers
are often used to measure pressure,stress,
and acceleration,etc.,and have wide
applications in engineering.
4.8 Piezoelectric transducers
Certain materials can generate an electrical charge
when subjected to mechanical strain or,
conversely,can change dimensions when
subjected to voltage,Pierre and Jacques Curie are
credited with its discovery in 1880.
The materials that exhibit a significant and useful
piezoelectric effect fall into three main groups:
natural (quartz,Rochelle salt) and synthetic
(lithium sulphate,ammonium dihydrogen
phosphate) crystals
polarized ferroelectric ceramics (barium titanate,
etc.)
certain polymer films.
4.8.1 Piezoelectric effect
W e use po lari z ation int ensity ve cto r to exp re ss the piezoe l ectr ic
ef f ect of a m ater ial,
zzyyxx
PPPP (4.78 )
W here
x
,y and z f or m a coo rdina te s ystem rela tin g to crys ta l
axes
4.8.1 Piezoelectric effect
Fig,4.55 Axis numbering system for piezoelectric constants
W ri ti ng the polar iza tion intensi ty P in the f orm of axial stre ss?
and shear str ess? gives
xyzxyzzzyyxxzz
xyzxyzzzyyxxyy
xyzxyzzzyyxxxx
ddddddP
ddddddP
ddddddP
363534333231
262524232221
161514131211
(4.79)
W here nm
d
,is th e piezoelec tr ic cons tant w it h it s sub scr ip t m as th e
axis di re cti on perpend icul a r to th e surface on w hich ele ctrical
char ges a re generated,a nd
n
as the a xis dir ection in w hich str ess is
applie d,
4.8.1 Piezoelectric effect
In Fig,4.55
subscript 1 corresponds to x-axis
subscript 2 to y-axis
subscript 3 to z-axis,
It is evident that the piezoelectric constants
are different when the directions of force
application and the generated deformations
are different,
4.8.1 Piezoelectric effect
Quartz crystal is a commonly used material,
A quartz crystal has a shape of hexahedral
structure with three mutually perpendicular
axes as its crystal axes
The longitudinal axis Z-Z is known as optical
axis
The axis X-X going through the edge of the
hexahedron and being perpendicular to the
optical axis is called electrical axis
The axis Y-Y perpendicular to both the axis X-
X and the axis Z-Z is called mechanical axis
4.8.1 Piezoelectric effect
The piezoelectric effect generated by the force
applied in the electrical axis X-X is called
longitudinal piezoelectric effect
The piezoelectric effect generated by the force
applied in the mechanical axis Y-Y is called
transverse piezoelectric effect
A force applied in the optical axis Z-Z doesn’t
generate any piezoelectric effect.
4.8.1 Piezoelectric effect
4.8.1 Piezoelectric effect
Fig,4.56 Quartz crystal
(a) quartz crystal shape (b) the coordinate system for crystal
axes (c) a cut of the crystal
There are three major piezoelectric effects,namely
the longitudinal,the transverse and the shear effect.
4.8.1 Piezoelectric effect
Fig,4.57 Acting directions of piezoelectric effects
T h e p o lar iz atio n inte n si ty
xx
P is p rop o rtion al to the str ess
xx
,
tha t is,
lb
F
ddP
x
xxxx 1111
(4.8 2 )
W h ere?
x
F p ressu re ap p lied in the d irection o f X - X ax is,
1
d
p iezo e lect r ic co n st an t,
112
1
103.2
CNd f o r
q u artz cry stal,
l
len g th o f the cu t,
b
w idth o f th e cu t,
4.8.1 Piezoelectric effect
T h e p o la riz atio n int en s ity
xx
P is a lso eq u al to th e el ect ric al
ch ar g e d en sity p rod u ced o n cu t su rf ace,
lb
q
P
xx
xx
(4.8 3 )
W h ere?
xx
q th e ch ar g es p rod u ced o n the su rf ace
pe rpen d icu lar to the X - X ax is,
F rom E q s,(4,82 ) an d (4,8 3 ) w e g et
xxx
Fdq
11
(4.8 4 )
4.8.1 Piezoelectric effect
E q,(4,84 ) show s that the prod uced char ge
xx
q,w hen a qu artz
crystal cut is app lied w ith a pressu re in the X - X direction,is
prop ort ion al to the app lied f orce
x
F,bu t is indep end ent of
cut ’ s geo m etri cal size,Co ntrary to this,w hen the f orce
x
F is
app lied in the Y - Y direction
4.8.1 Piezoelectric effect
Th e produce d char ge stil l appears on the surface per pendi cular to the X - X
axis,with t he m agnitude
y
x
y
y
x
y
xy
F
l
l
dF
bl
bl
dq
1212
(4.85)
W here
12
d
piezoel ect ri c consta nt fo r t he str ess a pplied i n the Y - Y dir ect ion
xy
ll,
length and the w idth of the quart z cut,re specti vely
A ccor ding to the sym m etr ica l pr operty of the quart z cr ystal axes,ther e i s
1112
dd
Th us Eq,( 4,85 ) c hanges to
y
x
y
xy
F
l
l
dq
11
(4.86)
4.8.1 Piezoelectric effect
The produced charge of a quartz crystal cut,when
it is applied with a compressive force in the
mechanical axis Y-Y,is proportional to the
geometrical size of the cut,the polarity of the
charge is then opposite (negative sign in Eq,(4.86))
to the polarity of the charge produced when it is
applied with a compressive force in the electrical
axis X-X.
4.8.1 Piezoelectric effect
When a piezoelectric cut is applied simultaneously
with many forces,a complex stress field will be
generated inside it,Thus both the longitudinal and
transverse effects may occur.
4.8.1 Piezoelectric effect
L D FQ? (4.87)
W here Q,D,and F ar e m atr ice s,and L is a
colum n vector,wh ose m agnitudes depend on bo th the
dif f er ent forces a nd the cut s ize,
A quartz crystal is a kind of crystallite of SiO2,As
is shown in fig,4.58,each crystalline unit contains
three Si atoms and six O2 atoms,and the O2 atoms
lie in pairs against each other,Each Si atom has
four unit positive charges,while each O2 atom has
two unit negative charges,In the crystalline unit,
the Si and O2 atoms form a hexahedral structure
and they reach a state of charge equilibrium,
4.8.1 Piezoelectric effect
4.8.1 Piezoelectric effect
Fig,4.58 The piezoelectric effects in crystalline quartz,(a) Longitudinal
effect and (b) transverse effect,each showing the charge displacement Q
caused by crystal deformation resulting from force F,Also shown are
typical electrical connections and sketches of typical transducers.
To measure the generated electrical charges on
two faces of the piezoelectric crystal,electrodes
must be made on these two faces,A metal film,
usually silver or gold one,is evaporized on the
crystal surface to form an electrode
4.8.2 Operating principle of piezoelectric transducer and
associated circuitry
A p iezo electric tran sd u cer can b e co n sid ered as a ch ar g e g en erato r
o r as a cap acito r,w h o se a m o u n t o f cap acitan ce is,
A
c
0
W h ere
d ielectric co n stan t o f p iezo electric m aterial,
5.4
f o r
q u artz,
0
d ielectric co n stan t f o r v acu u m,an d
112
0
1085.8
mF?
,
d istan ce b etw een tw o electro d es,(
m
),
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.60 Piezoelectric cut with electrodes and its equivalent circuit
(a) piezoelectric cut with metal films as electrodes (b) connected in
parallel (c) connected in series (d) the equivalent circuit
The measured force is proportional to the
generated amount of electrical charges
A piezoelectric transducer can be considered as a
charge source
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
a
a
C
q
e? ( 4,8 8 )
A piezoelectric transducer can be also
considered as a voltage source.
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.61 Equivalent circuits of a piezoelectric transducer
(a) as a charge source (b) as a voltage source
If a p iezoelectr ic tr ansd ucer is con nected w ith a m easuring
circ uit,the inf luences of the cable capacitance
c
C,the inp ut
im ped ance of the circ uit,
i
R,the inp ut capacitance
i
C,and
the leakag e resista nce o f the transdu cer,
a
R,sho uld be taken
into co nsid erat ion,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.62 Actual equivalent circuits of a piezoelectric transducer
(a) as a charge source (b) as a voltage source
Two kinds of pre-amplifier are employed,
The voltage amplifier with resistance feed-back
The charge amplifier with capacitance feed-
back
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.63 Equivalent circuits for a piezoelectric transducer
connected with a voltage amplifier
A voltage amplifier
where
q= amount of charge produced by the piezoelectric
transducer,
C= total capacitance of the equivalent circuit,
C=Ca+Cc+Ci,where Ci is the input capacitance of
the amplifier,Cc is the equivalent capacitance of the
piezoelectric transducer,and Ca is the stray
capacitance due to the connecting wires.
e= the voltage established across the capacitor,
i= the leaked current.
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
id tecq (4.89)
A n d Rie?,w h e r e R is the e q u iva le n t r e sista n c e f o r the
inp u t imp e d a n c e o f th e a mp lif ie r,
i
R,a n d the le a k e d r e sista n c e
o f the tr a n sd u c e r,
a
R,tha t is,
ai
RRR //?,
I f tFF
00
s i n,
tLqtL D FL D Fq
0000
s i ns i n ( 4,9 1 )
W h e r e
the c ir c u la r f r e q u e n c y o f the f o r c e,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
L e tt ing 1?L to simpli f y the a na ly sis,w e ha ve
tqq
00
s i n ( 4.9 2)
the n
tqi d tC R i
00
s i n ( 4.9 3)
or
tqi
dt
di
CR
000
c o s ( 4.9 4)
Th e ste a dy - sta te solution f or E q,( 4.9 4) is
)s i n (
)(1
0
2
0
00
t
CR
q
i
RC
ar c t g
0
1
( 4.9 5)
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
T h e v o ltag e acro ss the cap acito r
)s i n (
)
1
(1
1
0
2
0
0
t
RC
c
q
Rie (4.9 6 )
If the am p lif ier is a lin ear o n e,th en its o u tpu t
)s i n (
)
1
(1
1
0
2
0
0
0
t
RC
c
q
ke (4.9 7 )
W h ere
k
is the am p lif ier ga in,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
T h e lo w - f requ en cy resp o n se o f th e p iezo electric tran sd u cer
is d ep en d en t o n the circu it ’ s tim e co n stan t,RC,d eterm ine d
by the trans d u cer,the co n n ectin g w ires,an d the lo ad,A n d in
p erf o rm ing a d y n am ic m easu rem en t,to estab lish a ce rta in
o u tpu t v o ltag e an d to en su re an accu rate m easu rem en t,the
m easu ring circu it of th e p iezo electric tran sd u cer m u st h av e a
h igh in p u t im p ed an ce,a n d a ce rtain a m o u n t o f cap acitan ce,
i
C,m u st b e co n n ected in p arallel to the in p u t term in als to
inc rease the tim e co n stan t
RC
,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Solution,shorter wire connections and the driving
cable technique
E q,(4,97 ) tells us that the ou tpu t vo ltage
0
e is closely
rela ted to the capacitance C w hen a vo ltage am plif ier i s
em plo yed,T he w ho le m easuring sys tem is very sen sitive to
the capacitance
c
C,V ery lon g co nn ecting w ires and their
po sition v aria tions w ill ca use v aria tion in transdu cer ’ s
ou tpu t,and th is af f ects the sens itivity,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
A charge amplifier is a high-gain operational
amplifier with a capacitance feed-back
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
fyiifyiicai
CeeCeCeeCCCeq )()()( (4.9 8 )
W h ere
i
e inp u t v o ltag e,
y
e o u tpu t v o ltag e,
f
C f eed - b ack cap acitan ce,
A s
iy
Kee,w h ere K is the o p en - loo p g ain o f the ch ar g e
am p lif ier,th en
ff
y
KCCC
Kq
e
)(
(4.9 9 )
W h en
K
is l ar g e en o u g h,the n
ff
CCKC,an d E q,
(4.9 9 ) is sim p lif ied as
f
y
C
q
e
(4.1 0 0 )
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
The output voltage of a charge amplifier is directly
proportional to the generated charge amount,and is
independent of the stray capacitance caused by wire
connections.
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.64 Equivalent circuit for a piezoelectric transducer
connected with a charge amplifier
Applications,the measurement of force
(torque),pressure,vibration (acceleration),
etc
According to their applications,they are
mainly divided into two types,
– Piezoelectric accelerometers
– Piezoelectric force transducers
4.8.3 Applications
1,Piezoelectric accelerometer
Piezoelectric accelerometers are in wide use for
shock and vibration measurement,In general,
they do not give an output for constant
acceleration because of the basic characteristics
of piezoelectric motion transducers,But they do
have large output-voltage signals,small size,and
can have very high natural frequencies.
Intentional damping is provided,with material
hysteresis being the only source of energy loss,
results in a very low (about 0.01) damping ratio,
but this is acceptable because of the very high
natural frequency.
4.8.3 Applications
A seismic- (absolute)
displacement pickup are
used almost exclusively
for measurement of
vibratory displacement in
those (many) cases where
a fixed reference for
relative-displacement
measurement is not
available.
4.8.3 Applications
Fig,4.66 Seismic-displacement transducer
(a) translatory (b) rotary
)()(
000
xxmxmxCkx
im
(4.1 0 3 )
W h ere
m
x an d
i
x are th e a b so lute d isp lacem en ts a n d w e h av e
ch o sen o u r ref eren ce f o r
0
x is z er o w h en the g ravi ty f o rc e
(w eig h t o f M ) is a cti n g alo n g the
x
ax is stat ica lly,
M an ipu lation g ive s
i
xmkxxCxm
000
(4.1 0 4 )
4.8.3 Applications
F u rtherm o re
inn
xxxx
0
2
0
2 (4.1 0 5 )
W h ere
m
k
n
,
km
C
2
,
4.8.3 Applications
4.8.3 Applications
T w o k ind s o f seis m ic p ick u p s,the seis m ic - (ab so lute )
d isp lacem en t p ick u p,w h ich receiv es the m easu red v ibratio n
d isp lacem en t
i
x an d v elo city
i
x? an d o u tpu ts the relative
v ibratio n d isp lacem en t
0
x an d v elo city
0
x?,an d th e
seis m ic - (ab so lute ) acceleratio n p ick u p,w h ich receiv es the
m easu red v ibratio n acce leration
i
x an d o u tpu ts the relative
d isp lacem en t
0
x,
4.8.3 Applications
S u b stitu ting tXx
ii
c os? into E q,(4,1 0 5 ) yield s
)(
)(
1/2)/(
/)(
)(
)(
2
22
jX
jX
jj
j
jX
jX
i
o
nn
n
i
o
(4.1 0 6 )
T o k eep size (and th ereb y lo ad ing o n the m easu red sy stem ) to
a m inim u m,so f t sp ring s are p ref erred to lar g e m ass es,
Inten tion al d am p ing in the
0,6 to 0,7 is o f ten em p loy ed
to m inim ize reso n an t resp o n se to slo w trans ien ts,
4.8.3 Applications
Fig,4.67 Seismic-displacement-pickup frequency response
(a) Amplitude-frequency response (b) Phase-ferquency response
If the inp u t o f the ar ran g em en t in F ig,4,6 6 i s n o w the acc e lera tion
i
x,th en trans f o rm ing E q,(4,1 0 6 ) giv es
1/2/)()(
)(
)()(
)(
222
nn
n
i
o
i
o
jj
K
jX
jX
jXj
jX
(4.1 0 7 )
W h ere
2
1
n
n
K
,
4.8.3 Applications
4.8.3 Applications
Fig,4.68 Characteristic curves of seismic-acceleration pickup
(a) Amplitude-frequency response (b) Phase-frequency response
So an accelerometer has the characteristics of zero-
frequency-response,If the mechanical-electrical
conversion part of the accelerometer and the
measuring circuit also have zero-frequency-response,
the whole system will have the zero-frequency-
response too and can be thus used to measure
vibrations with very low frequencies and movements
of constant accelerations.
4.8.3 Applications
T he op era t ing f req ue nc y ran ge of a piez oe le ctr ic
ac ce lero m eter is w ithin a f lat in terv al o f 1~0?
n
In thi s f lat inte rva l,t he vib ra tio n disp la ce m en t
0
x i s
pro po rt ion al to the m e asu red a cc e lera tion
x
,a nd th e
am plitud e is u nity w he n 0?
n
,
The actual transfer characteristics of a piezoelectric
accelerometer will be a combination of Eqs,(4.102) and
(4.107) because of the utilization of a charge amplifier for
the measurement:
4.8.3 Applications
]1/2)/)[(1(
)()/(
)(
)(
2
nn
n
i
o
jjj
jKK
jx
jE
(4.1 0 8 )
w h ere?K the sen sitiv ity o f the cir cu it ry,
fq
CKK /?,an d
q
K elastic s tif f n ess co ef f icien t,
Due to the influence of the successive measuring circuit,it
doesn’t have a zero-frequency response and cannot be
used to measure static displacement.
4.8.3 Applications
T h e a c t u a l f r e q u e n c y r e s p o n s e,w h o s e l o w - f r e q u e n c y
r e s p o n s e i s p r a c t i c a l l y d e t e r m i n e d b y )1/( jj,
4.8.3 Applications
Fig,4.69 Piezoelectric accelerometer frequency response
The design details of piezoelectric accelerometers
can be varied to emphasize selected features of
performance desired for particular applications.
4.8.3 Applications
Fig,4.70 Piezoelectric accelerometer designs.
(a) Peripheral-mounted compression design,
(b) Center-mounted compression design.
(c) Inverted center-mounted compression design.
(d) Shear design.
4.8.3 Applications
Figure 4.73(a) shows construction details of two
piezoelectric load cells,The smaller one is
permanently preloaded to measure in the range
1,000-N tension to 5,000-N compression without
relaxing the built-in compressive preload,The
larger unit comes with external preloading nuts
which may be adjusted to cover the range 4,000-N
tension to 16,000-N compression.
4.8.2 Piezoelectric force transducers
Fig,4.73 Piezoelectric load cells and dynamic error analysis (a) Sectional
drawing of the force transducer,T—top; P—piezoelectric discs; GP—
guide pin; S—preloading screw; N—preloading nut; B—base.
4.8.2 Piezoelectric force transducers
4.8.2 Piezoelectric force transducers
Figur e 4.73 ( b ) shows su ch a tr ans du ce r be ing us ed to m ea sur e
the f or ce applie d by a v ibr a ti on sh a ker to som e st r uc ture b ei ng
vibr a ti on - te s te d (
1m
K and
2m
K r epr esent the stif f n es s of
mounting sc r ews),I f the im peda nce of the s truct ur e ( including
2m
K ) is ca lled
s
Z,the n s inc e
sss
vfZ?,the f or c e ac tually
appl ie d to the struc tur e is
ss
vZ,The f or ce
m
F m e as ur ed,
however,is tha t in
t
K,which is p r opor ti on al to the r elat ive
dis placem en t of
1t
M and
2t
M si nce this is the defle ction of
the pi ez oel ec tr ic e le m ent,
Under dynam ic condit ions
m
F is not necessaril y equal to
s
F,and a dy nam ic analys is as sho wn below is useful in
developi ng crit eri a f or accurat e m easurem ent,
stssm
DvMZvF
2
( 4,1 1 0 )
and
1
2
DZ
DM
D
F
F
s
t
s
m
(4.1 1 1)
where
dt
d
D?
4.8.2 Piezoelectric force transducers
If 0
2
t
M,the re w o u ld b e n o erro r,thu s tr an sd u ce r s
w ith s m all
2t
M are clearly p ref erab le,
F o r a ― sp ring - like ‖ stru ctu re, DKDZ
ss
,an d
22
1
s
t
s
m
K
M
i
F
F
(4.1 1 2 )
4.8.2 Piezoelectric force transducers
As is stated before,the piezoelectric effect is reversible in
that the piezoelectric plate will change dimension,when
subjected to voltage,This inverse piezoelectric effect can be
used in constructing piezoelectric actuators,
– High-frequency vibrating tables
– Ultrasonic generators
– Microphones
– HF-switches
Precision micro-displacement devices,piezoelectric
bimorphs are used in,
video tape head positioners
dot matrix printer heads
relays
piezoelectric fans and many other applications.
4.8.2 Piezoelectric force transducers
The electromagnetic transducer is a device
which converts the measurand into an
induced electromotive force (e.m.f.),It is
also called the electromagnetic-induction or
electromotive force transducer,
4.9 Electromagnetic transducers
Faraday’s law of induction
the moving-magnet variable-reluctance tachometer
the d.c,tachogenerator and moving-coil velocity
pickup
the magnetoresistive pickup.
4.9 Electromagnetic transducers
dt
d
We
(4.1 13)
w here e is th e generate d voltage,W is the num b e r
of coil turns and
dt
d?
the ti m e ra te of c hange of f lux
li nkages w it h the coi l,
Fig,4.75 Electromagnetic transducers
(a) translational velocity pickup
(b) angular velocity pickup
4.9.1 Moving-coil and moving-magnet pickups
s i ns i n
y
W B l vyW B le (4.1 1 4 )
w h ere
B f lux d en sity ( T ),
l
len g th o f co il (
m
),
W
n u m b er o f tu rns w h ich cu t the f ield,
y
v rela tive v e loc i ty o f co il an d m ag n et (y d ir ectio n )
(
sm
)
an g le f o rm ed b y the co il - m o v ing d irec tion an d f ield
d irection,
4.9.1 Moving-coil and moving-magnet pickups
Since B and l are constant,the output voltage follows
the input velocity linearly and reverses polarity when
the velocity changes sign,Such pickups are widely
used for the measurement of vibratory velocities.
4.9.1 Moving-coil and moving-magnet pickups
For 90?,tha t is,the c oil - m oving dir ec tion is
per pendi cul ar t o t he f ie ld,t her e is
y
W B l ve? ( 4.1 15)
The transducer shown
in Fig,4.75(c) uses a
permanent-magnet
core moving inside a
form wound with two
coils connected as
shown.
4.9.1 Moving-coil and moving-magnet pickups
Fig,4.75(c) Velocity pickup
Fig,4.75 (b) shows the structure of an angular-
velocity pickup,in which the coil rotates in and cuts
the magnetic field,thus generating the induced e.m.f.
4.9.1 Moving-coil and moving-magnet pickups
k W B Ae? (4.1 1 6 )
w h ere
an g u lar f requ en cy o f co il,
A cros s - sectio n al area o f co il (
2
m ),
k
co n stan t related to th e stru ctu re,
1?k
,
W h en
W
,
B
an d
A
are f ixe d,th e o u tpu t v o ltag e
e
is u n iqu ely p rop o rtion al to th e an g u lar ve loc ity o f th e
co il relative to the f ield,
Fig,4.76 Equivalent circuit of the moving-coil electromagnetic transducer
4.9.1 Moving-coil and moving-magnet pickups
o
Z is t he im pedance of the coil,
L
R is t he load r esis tance
(i ncluding the i npu t r esis tance of the a m plifier ),
c
C is the
str ay capac it ance of the wire
mR
c
03.0,mpFC
c
70?,an d LjrZ
o
,r is ab o u t
k2~300,L is in the o rder o f m H s an d can b e n eg le cte d,S o
the o u tpu t v o ltag e
oC
L
o
L
ZCj
R
Z
ee
1
1
(4.1 1 7 )
If th e w ire is n o t lon g,th en
c
C can b e n eg lected,A n d if w e let
cL
ZR,th en E q,(4,1 1 7 ) can b e sim p lif ied to
ee
L
,
4.9.1 Moving-coil and moving-magnet pickups
Electromagnetic transducers can be divided into
two types,
absolute velocity transducers
relative velocity transducers.
4.9.1 Moving-coil and moving-magnet pickups
Fig,4.77 Absolute vibration transducer
1-diaphragm
2-permanent magnet
3-damping cup
4-frame
5-center shaft
6-housing
7-coil
Analyzing the characteristic curve in terms of
the criterions for accurate measurement,we can
get the following conclusions:
1) To realize an accurate measurement,the
amplitude response must be a constant.
4.9.1 Moving-coil and moving-magnet pickups
w h en 1?
n
,th e am p litud e resp o n se c u rve
ten d s to u n ity w ith the in crease in
n
A s the am p litud e resp o n se s f o r dif f eren t d am p ing
ratios ap p roac h a co n stan t (stead y - state) v alu e in
d if f eren t rates,a n d th e m ax im u m rate o ccu rs at
707.0,u su ally a d am p ing ratio 7.0~6.0
is ad o p ted
Theoretically,the upper limit of the working
frequency range of the transducer is infinite,
But in reality,since the transducer will be
affected by the factors such as mounting
stiffness and local resonances of its
components,so the working frequency range
is finite.
4.9.1 Moving-coil and moving-magnet pickups
In gener al,if 707.0,and if a
m easur em ent e rr or %5? is r equire d,then
the frequency ra nge f or m easur em ent is
about
n
7.1?,
2) Although damping can improve the flatness of
the part of the response curve near resonant
frequency,it also increases the phase-shift.
4.9.1 Moving-coil and moving-magnet pickups
w h en
n
,if th e f requ en cy co m p o n en ts o f
inp u t an d o u tpu t h av e a p h ase - sh if t o f ab o u t
1 8 0,th at is,the trans d u cer acts n o w as an
inv erter,the m easu rem en t resu lt can b e still
co n sid ered as an accu rate o n e,
It is k n o w n f rom th e p h ase resp o n se p lot th at
sh o u ld b e se lected as
n
8~7? to o b tain
an ap p rox im ate p h ase inv ersio n ch aracteristic,
3)
4.9.1 Moving-coil and moving-magnet pickups
T h e n atu ral f requ en cy
n
o f v elo city
trans d u cer is a n o the r im p o rtant p aram eter,It
d eterm ine s th e lo w er lim it of th e f requ en cy
rang e m easu red,T o ex p an d the trans d u cer ’ s
w o rkin g f requ en cy ran g e,
n
sh o u ld b e
d esig n ed as low as p o ssib le,T h e lo w er lim it of
the w o rkin g f requ en cy ran g e f o r m o st
co m m o n ly - u sed v elo city trans d u cers is u su ally
w ithin 1 0 ~ 7 5 H z,
With the coil and magnet fixed,when the
measured object (usually of permeable material) is
in motion to influence the reluctance of the
magnetic circuit,induced voltage will be
generated because of the change in magnetic field
(Fig,4.79),Transducers based on this principle are
called magneto-resistive transducers
4.9.2 Magneto-resistive transducers
Fig,4.79 Principle of magneto-resistive transducer and its applications in
(a) measurement of frequency number (b) measurement of rotating
speed (c) measurement of eccentric amount (d) measurement of
vibration
,4.122 Example of a moiré structure
4.9.2 Magneto-resistive transducers
Measurement Technology
(10)
Prof,Wang Boxiong
A piezoelectric transducer is an active
transducer or self-generating transducer,It
utilizes the piezoelectric effect of some
materials,which generate electrical changes
on some of their surfaces when acted by
external forces,So piezoelectric transducers
are often used to measure pressure,stress,
and acceleration,etc.,and have wide
applications in engineering.
4.8 Piezoelectric transducers
Certain materials can generate an electrical charge
when subjected to mechanical strain or,
conversely,can change dimensions when
subjected to voltage,Pierre and Jacques Curie are
credited with its discovery in 1880.
The materials that exhibit a significant and useful
piezoelectric effect fall into three main groups:
natural (quartz,Rochelle salt) and synthetic
(lithium sulphate,ammonium dihydrogen
phosphate) crystals
polarized ferroelectric ceramics (barium titanate,
etc.)
certain polymer films.
4.8.1 Piezoelectric effect
W e use po lari z ation int ensity ve cto r to exp re ss the piezoe l ectr ic
ef f ect of a m ater ial,
zzyyxx
PPPP (4.78 )
W here
x
,y and z f or m a coo rdina te s ystem rela tin g to crys ta l
axes
4.8.1 Piezoelectric effect
Fig,4.55 Axis numbering system for piezoelectric constants
W ri ti ng the polar iza tion intensi ty P in the f orm of axial stre ss?
and shear str ess? gives
xyzxyzzzyyxxzz
xyzxyzzzyyxxyy
xyzxyzzzyyxxxx
ddddddP
ddddddP
ddddddP
363534333231
262524232221
161514131211
(4.79)
W here nm
d
,is th e piezoelec tr ic cons tant w it h it s sub scr ip t m as th e
axis di re cti on perpend icul a r to th e surface on w hich ele ctrical
char ges a re generated,a nd
n
as the a xis dir ection in w hich str ess is
applie d,
4.8.1 Piezoelectric effect
In Fig,4.55
subscript 1 corresponds to x-axis
subscript 2 to y-axis
subscript 3 to z-axis,
It is evident that the piezoelectric constants
are different when the directions of force
application and the generated deformations
are different,
4.8.1 Piezoelectric effect
Quartz crystal is a commonly used material,
A quartz crystal has a shape of hexahedral
structure with three mutually perpendicular
axes as its crystal axes
The longitudinal axis Z-Z is known as optical
axis
The axis X-X going through the edge of the
hexahedron and being perpendicular to the
optical axis is called electrical axis
The axis Y-Y perpendicular to both the axis X-
X and the axis Z-Z is called mechanical axis
4.8.1 Piezoelectric effect
The piezoelectric effect generated by the force
applied in the electrical axis X-X is called
longitudinal piezoelectric effect
The piezoelectric effect generated by the force
applied in the mechanical axis Y-Y is called
transverse piezoelectric effect
A force applied in the optical axis Z-Z doesn’t
generate any piezoelectric effect.
4.8.1 Piezoelectric effect
4.8.1 Piezoelectric effect
Fig,4.56 Quartz crystal
(a) quartz crystal shape (b) the coordinate system for crystal
axes (c) a cut of the crystal
There are three major piezoelectric effects,namely
the longitudinal,the transverse and the shear effect.
4.8.1 Piezoelectric effect
Fig,4.57 Acting directions of piezoelectric effects
T h e p o lar iz atio n inte n si ty
xx
P is p rop o rtion al to the str ess
xx
,
tha t is,
lb
F
ddP
x
xxxx 1111
(4.8 2 )
W h ere?
x
F p ressu re ap p lied in the d irection o f X - X ax is,
1
d
p iezo e lect r ic co n st an t,
112
1
103.2
CNd f o r
q u artz cry stal,
l
len g th o f the cu t,
b
w idth o f th e cu t,
4.8.1 Piezoelectric effect
T h e p o la riz atio n int en s ity
xx
P is a lso eq u al to th e el ect ric al
ch ar g e d en sity p rod u ced o n cu t su rf ace,
lb
q
P
xx
xx
(4.8 3 )
W h ere?
xx
q th e ch ar g es p rod u ced o n the su rf ace
pe rpen d icu lar to the X - X ax is,
F rom E q s,(4,82 ) an d (4,8 3 ) w e g et
xxx
Fdq
11
(4.8 4 )
4.8.1 Piezoelectric effect
E q,(4,84 ) show s that the prod uced char ge
xx
q,w hen a qu artz
crystal cut is app lied w ith a pressu re in the X - X direction,is
prop ort ion al to the app lied f orce
x
F,bu t is indep end ent of
cut ’ s geo m etri cal size,Co ntrary to this,w hen the f orce
x
F is
app lied in the Y - Y direction
4.8.1 Piezoelectric effect
Th e produce d char ge stil l appears on the surface per pendi cular to the X - X
axis,with t he m agnitude
y
x
y
y
x
y
xy
F
l
l
dF
bl
bl
dq
1212
(4.85)
W here
12
d
piezoel ect ri c consta nt fo r t he str ess a pplied i n the Y - Y dir ect ion
xy
ll,
length and the w idth of the quart z cut,re specti vely
A ccor ding to the sym m etr ica l pr operty of the quart z cr ystal axes,ther e i s
1112
dd
Th us Eq,( 4,85 ) c hanges to
y
x
y
xy
F
l
l
dq
11
(4.86)
4.8.1 Piezoelectric effect
The produced charge of a quartz crystal cut,when
it is applied with a compressive force in the
mechanical axis Y-Y,is proportional to the
geometrical size of the cut,the polarity of the
charge is then opposite (negative sign in Eq,(4.86))
to the polarity of the charge produced when it is
applied with a compressive force in the electrical
axis X-X.
4.8.1 Piezoelectric effect
When a piezoelectric cut is applied simultaneously
with many forces,a complex stress field will be
generated inside it,Thus both the longitudinal and
transverse effects may occur.
4.8.1 Piezoelectric effect
L D FQ? (4.87)
W here Q,D,and F ar e m atr ice s,and L is a
colum n vector,wh ose m agnitudes depend on bo th the
dif f er ent forces a nd the cut s ize,
A quartz crystal is a kind of crystallite of SiO2,As
is shown in fig,4.58,each crystalline unit contains
three Si atoms and six O2 atoms,and the O2 atoms
lie in pairs against each other,Each Si atom has
four unit positive charges,while each O2 atom has
two unit negative charges,In the crystalline unit,
the Si and O2 atoms form a hexahedral structure
and they reach a state of charge equilibrium,
4.8.1 Piezoelectric effect
4.8.1 Piezoelectric effect
Fig,4.58 The piezoelectric effects in crystalline quartz,(a) Longitudinal
effect and (b) transverse effect,each showing the charge displacement Q
caused by crystal deformation resulting from force F,Also shown are
typical electrical connections and sketches of typical transducers.
To measure the generated electrical charges on
two faces of the piezoelectric crystal,electrodes
must be made on these two faces,A metal film,
usually silver or gold one,is evaporized on the
crystal surface to form an electrode
4.8.2 Operating principle of piezoelectric transducer and
associated circuitry
A p iezo electric tran sd u cer can b e co n sid ered as a ch ar g e g en erato r
o r as a cap acito r,w h o se a m o u n t o f cap acitan ce is,
A
c
0
W h ere
d ielectric co n stan t o f p iezo electric m aterial,
5.4
f o r
q u artz,
0
d ielectric co n stan t f o r v acu u m,an d
112
0
1085.8
mF?
,
d istan ce b etw een tw o electro d es,(
m
),
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.60 Piezoelectric cut with electrodes and its equivalent circuit
(a) piezoelectric cut with metal films as electrodes (b) connected in
parallel (c) connected in series (d) the equivalent circuit
The measured force is proportional to the
generated amount of electrical charges
A piezoelectric transducer can be considered as a
charge source
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
a
a
C
q
e? ( 4,8 8 )
A piezoelectric transducer can be also
considered as a voltage source.
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.61 Equivalent circuits of a piezoelectric transducer
(a) as a charge source (b) as a voltage source
If a p iezoelectr ic tr ansd ucer is con nected w ith a m easuring
circ uit,the inf luences of the cable capacitance
c
C,the inp ut
im ped ance of the circ uit,
i
R,the inp ut capacitance
i
C,and
the leakag e resista nce o f the transdu cer,
a
R,sho uld be taken
into co nsid erat ion,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.62 Actual equivalent circuits of a piezoelectric transducer
(a) as a charge source (b) as a voltage source
Two kinds of pre-amplifier are employed,
The voltage amplifier with resistance feed-back
The charge amplifier with capacitance feed-
back
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.63 Equivalent circuits for a piezoelectric transducer
connected with a voltage amplifier
A voltage amplifier
where
q= amount of charge produced by the piezoelectric
transducer,
C= total capacitance of the equivalent circuit,
C=Ca+Cc+Ci,where Ci is the input capacitance of
the amplifier,Cc is the equivalent capacitance of the
piezoelectric transducer,and Ca is the stray
capacitance due to the connecting wires.
e= the voltage established across the capacitor,
i= the leaked current.
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
id tecq (4.89)
A n d Rie?,w h e r e R is the e q u iva le n t r e sista n c e f o r the
inp u t imp e d a n c e o f th e a mp lif ie r,
i
R,a n d the le a k e d r e sista n c e
o f the tr a n sd u c e r,
a
R,tha t is,
ai
RRR //?,
I f tFF
00
s i n,
tLqtL D FL D Fq
0000
s i ns i n ( 4,9 1 )
W h e r e
the c ir c u la r f r e q u e n c y o f the f o r c e,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
L e tt ing 1?L to simpli f y the a na ly sis,w e ha ve
tqq
00
s i n ( 4.9 2)
the n
tqi d tC R i
00
s i n ( 4.9 3)
or
tqi
dt
di
CR
000
c o s ( 4.9 4)
Th e ste a dy - sta te solution f or E q,( 4.9 4) is
)s i n (
)(1
0
2
0
00
t
CR
q
i
RC
ar c t g
0
1
( 4.9 5)
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
T h e v o ltag e acro ss the cap acito r
)s i n (
)
1
(1
1
0
2
0
0
t
RC
c
q
Rie (4.9 6 )
If the am p lif ier is a lin ear o n e,th en its o u tpu t
)s i n (
)
1
(1
1
0
2
0
0
0
t
RC
c
q
ke (4.9 7 )
W h ere
k
is the am p lif ier ga in,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
T h e lo w - f requ en cy resp o n se o f th e p iezo electric tran sd u cer
is d ep en d en t o n the circu it ’ s tim e co n stan t,RC,d eterm ine d
by the trans d u cer,the co n n ectin g w ires,an d the lo ad,A n d in
p erf o rm ing a d y n am ic m easu rem en t,to estab lish a ce rta in
o u tpu t v o ltag e an d to en su re an accu rate m easu rem en t,the
m easu ring circu it of th e p iezo electric tran sd u cer m u st h av e a
h igh in p u t im p ed an ce,a n d a ce rtain a m o u n t o f cap acitan ce,
i
C,m u st b e co n n ected in p arallel to the in p u t term in als to
inc rease the tim e co n stan t
RC
,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Solution,shorter wire connections and the driving
cable technique
E q,(4,97 ) tells us that the ou tpu t vo ltage
0
e is closely
rela ted to the capacitance C w hen a vo ltage am plif ier i s
em plo yed,T he w ho le m easuring sys tem is very sen sitive to
the capacitance
c
C,V ery lon g co nn ecting w ires and their
po sition v aria tions w ill ca use v aria tion in transdu cer ’ s
ou tpu t,and th is af f ects the sens itivity,
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
A charge amplifier is a high-gain operational
amplifier with a capacitance feed-back
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
fyiifyiicai
CeeCeCeeCCCeq )()()( (4.9 8 )
W h ere
i
e inp u t v o ltag e,
y
e o u tpu t v o ltag e,
f
C f eed - b ack cap acitan ce,
A s
iy
Kee,w h ere K is the o p en - loo p g ain o f the ch ar g e
am p lif ier,th en
ff
y
KCCC
Kq
e
)(
(4.9 9 )
W h en
K
is l ar g e en o u g h,the n
ff
CCKC,an d E q,
(4.9 9 ) is sim p lif ied as
f
y
C
q
e
(4.1 0 0 )
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
The output voltage of a charge amplifier is directly
proportional to the generated charge amount,and is
independent of the stray capacitance caused by wire
connections.
4.8.2 Operating principle of piezoelectric transducer and associated
circuitry
Fig,4.64 Equivalent circuit for a piezoelectric transducer
connected with a charge amplifier
Applications,the measurement of force
(torque),pressure,vibration (acceleration),
etc
According to their applications,they are
mainly divided into two types,
– Piezoelectric accelerometers
– Piezoelectric force transducers
4.8.3 Applications
1,Piezoelectric accelerometer
Piezoelectric accelerometers are in wide use for
shock and vibration measurement,In general,
they do not give an output for constant
acceleration because of the basic characteristics
of piezoelectric motion transducers,But they do
have large output-voltage signals,small size,and
can have very high natural frequencies.
Intentional damping is provided,with material
hysteresis being the only source of energy loss,
results in a very low (about 0.01) damping ratio,
but this is acceptable because of the very high
natural frequency.
4.8.3 Applications
A seismic- (absolute)
displacement pickup are
used almost exclusively
for measurement of
vibratory displacement in
those (many) cases where
a fixed reference for
relative-displacement
measurement is not
available.
4.8.3 Applications
Fig,4.66 Seismic-displacement transducer
(a) translatory (b) rotary
)()(
000
xxmxmxCkx
im
(4.1 0 3 )
W h ere
m
x an d
i
x are th e a b so lute d isp lacem en ts a n d w e h av e
ch o sen o u r ref eren ce f o r
0
x is z er o w h en the g ravi ty f o rc e
(w eig h t o f M ) is a cti n g alo n g the
x
ax is stat ica lly,
M an ipu lation g ive s
i
xmkxxCxm
000
(4.1 0 4 )
4.8.3 Applications
F u rtherm o re
inn
xxxx
0
2
0
2 (4.1 0 5 )
W h ere
m
k
n
,
km
C
2
,
4.8.3 Applications
4.8.3 Applications
T w o k ind s o f seis m ic p ick u p s,the seis m ic - (ab so lute )
d isp lacem en t p ick u p,w h ich receiv es the m easu red v ibratio n
d isp lacem en t
i
x an d v elo city
i
x? an d o u tpu ts the relative
v ibratio n d isp lacem en t
0
x an d v elo city
0
x?,an d th e
seis m ic - (ab so lute ) acceleratio n p ick u p,w h ich receiv es the
m easu red v ibratio n acce leration
i
x an d o u tpu ts the relative
d isp lacem en t
0
x,
4.8.3 Applications
S u b stitu ting tXx
ii
c os? into E q,(4,1 0 5 ) yield s
)(
)(
1/2)/(
/)(
)(
)(
2
22
jX
jX
jj
j
jX
jX
i
o
nn
n
i
o
(4.1 0 6 )
T o k eep size (and th ereb y lo ad ing o n the m easu red sy stem ) to
a m inim u m,so f t sp ring s are p ref erred to lar g e m ass es,
Inten tion al d am p ing in the
0,6 to 0,7 is o f ten em p loy ed
to m inim ize reso n an t resp o n se to slo w trans ien ts,
4.8.3 Applications
Fig,4.67 Seismic-displacement-pickup frequency response
(a) Amplitude-frequency response (b) Phase-ferquency response
If the inp u t o f the ar ran g em en t in F ig,4,6 6 i s n o w the acc e lera tion
i
x,th en trans f o rm ing E q,(4,1 0 6 ) giv es
1/2/)()(
)(
)()(
)(
222
nn
n
i
o
i
o
jj
K
jX
jX
jXj
jX
(4.1 0 7 )
W h ere
2
1
n
n
K
,
4.8.3 Applications
4.8.3 Applications
Fig,4.68 Characteristic curves of seismic-acceleration pickup
(a) Amplitude-frequency response (b) Phase-frequency response
So an accelerometer has the characteristics of zero-
frequency-response,If the mechanical-electrical
conversion part of the accelerometer and the
measuring circuit also have zero-frequency-response,
the whole system will have the zero-frequency-
response too and can be thus used to measure
vibrations with very low frequencies and movements
of constant accelerations.
4.8.3 Applications
T he op era t ing f req ue nc y ran ge of a piez oe le ctr ic
ac ce lero m eter is w ithin a f lat in terv al o f 1~0?
n
In thi s f lat inte rva l,t he vib ra tio n disp la ce m en t
0
x i s
pro po rt ion al to the m e asu red a cc e lera tion
x
,a nd th e
am plitud e is u nity w he n 0?
n
,
The actual transfer characteristics of a piezoelectric
accelerometer will be a combination of Eqs,(4.102) and
(4.107) because of the utilization of a charge amplifier for
the measurement:
4.8.3 Applications
]1/2)/)[(1(
)()/(
)(
)(
2
nn
n
i
o
jjj
jKK
jx
jE
(4.1 0 8 )
w h ere?K the sen sitiv ity o f the cir cu it ry,
fq
CKK /?,an d
q
K elastic s tif f n ess co ef f icien t,
Due to the influence of the successive measuring circuit,it
doesn’t have a zero-frequency response and cannot be
used to measure static displacement.
4.8.3 Applications
T h e a c t u a l f r e q u e n c y r e s p o n s e,w h o s e l o w - f r e q u e n c y
r e s p o n s e i s p r a c t i c a l l y d e t e r m i n e d b y )1/( jj,
4.8.3 Applications
Fig,4.69 Piezoelectric accelerometer frequency response
The design details of piezoelectric accelerometers
can be varied to emphasize selected features of
performance desired for particular applications.
4.8.3 Applications
Fig,4.70 Piezoelectric accelerometer designs.
(a) Peripheral-mounted compression design,
(b) Center-mounted compression design.
(c) Inverted center-mounted compression design.
(d) Shear design.
4.8.3 Applications
Figure 4.73(a) shows construction details of two
piezoelectric load cells,The smaller one is
permanently preloaded to measure in the range
1,000-N tension to 5,000-N compression without
relaxing the built-in compressive preload,The
larger unit comes with external preloading nuts
which may be adjusted to cover the range 4,000-N
tension to 16,000-N compression.
4.8.2 Piezoelectric force transducers
Fig,4.73 Piezoelectric load cells and dynamic error analysis (a) Sectional
drawing of the force transducer,T—top; P—piezoelectric discs; GP—
guide pin; S—preloading screw; N—preloading nut; B—base.
4.8.2 Piezoelectric force transducers
4.8.2 Piezoelectric force transducers
Figur e 4.73 ( b ) shows su ch a tr ans du ce r be ing us ed to m ea sur e
the f or ce applie d by a v ibr a ti on sh a ker to som e st r uc ture b ei ng
vibr a ti on - te s te d (
1m
K and
2m
K r epr esent the stif f n es s of
mounting sc r ews),I f the im peda nce of the s truct ur e ( including
2m
K ) is ca lled
s
Z,the n s inc e
sss
vfZ?,the f or c e ac tually
appl ie d to the struc tur e is
ss
vZ,The f or ce
m
F m e as ur ed,
however,is tha t in
t
K,which is p r opor ti on al to the r elat ive
dis placem en t of
1t
M and
2t
M si nce this is the defle ction of
the pi ez oel ec tr ic e le m ent,
Under dynam ic condit ions
m
F is not necessaril y equal to
s
F,and a dy nam ic analys is as sho wn below is useful in
developi ng crit eri a f or accurat e m easurem ent,
stssm
DvMZvF
2
( 4,1 1 0 )
and
1
2
DZ
DM
D
F
F
s
t
s
m
(4.1 1 1)
where
dt
d
D?
4.8.2 Piezoelectric force transducers
If 0
2
t
M,the re w o u ld b e n o erro r,thu s tr an sd u ce r s
w ith s m all
2t
M are clearly p ref erab le,
F o r a ― sp ring - like ‖ stru ctu re, DKDZ
ss
,an d
22
1
s
t
s
m
K
M
i
F
F
(4.1 1 2 )
4.8.2 Piezoelectric force transducers
As is stated before,the piezoelectric effect is reversible in
that the piezoelectric plate will change dimension,when
subjected to voltage,This inverse piezoelectric effect can be
used in constructing piezoelectric actuators,
– High-frequency vibrating tables
– Ultrasonic generators
– Microphones
– HF-switches
Precision micro-displacement devices,piezoelectric
bimorphs are used in,
video tape head positioners
dot matrix printer heads
relays
piezoelectric fans and many other applications.
4.8.2 Piezoelectric force transducers
The electromagnetic transducer is a device
which converts the measurand into an
induced electromotive force (e.m.f.),It is
also called the electromagnetic-induction or
electromotive force transducer,
4.9 Electromagnetic transducers
Faraday’s law of induction
the moving-magnet variable-reluctance tachometer
the d.c,tachogenerator and moving-coil velocity
pickup
the magnetoresistive pickup.
4.9 Electromagnetic transducers
dt
d
We
(4.1 13)
w here e is th e generate d voltage,W is the num b e r
of coil turns and
dt
d?
the ti m e ra te of c hange of f lux
li nkages w it h the coi l,
Fig,4.75 Electromagnetic transducers
(a) translational velocity pickup
(b) angular velocity pickup
4.9.1 Moving-coil and moving-magnet pickups
s i ns i n
y
W B l vyW B le (4.1 1 4 )
w h ere
B f lux d en sity ( T ),
l
len g th o f co il (
m
),
W
n u m b er o f tu rns w h ich cu t the f ield,
y
v rela tive v e loc i ty o f co il an d m ag n et (y d ir ectio n )
(
sm
)
an g le f o rm ed b y the co il - m o v ing d irec tion an d f ield
d irection,
4.9.1 Moving-coil and moving-magnet pickups
Since B and l are constant,the output voltage follows
the input velocity linearly and reverses polarity when
the velocity changes sign,Such pickups are widely
used for the measurement of vibratory velocities.
4.9.1 Moving-coil and moving-magnet pickups
For 90?,tha t is,the c oil - m oving dir ec tion is
per pendi cul ar t o t he f ie ld,t her e is
y
W B l ve? ( 4.1 15)
The transducer shown
in Fig,4.75(c) uses a
permanent-magnet
core moving inside a
form wound with two
coils connected as
shown.
4.9.1 Moving-coil and moving-magnet pickups
Fig,4.75(c) Velocity pickup
Fig,4.75 (b) shows the structure of an angular-
velocity pickup,in which the coil rotates in and cuts
the magnetic field,thus generating the induced e.m.f.
4.9.1 Moving-coil and moving-magnet pickups
k W B Ae? (4.1 1 6 )
w h ere
an g u lar f requ en cy o f co il,
A cros s - sectio n al area o f co il (
2
m ),
k
co n stan t related to th e stru ctu re,
1?k
,
W h en
W
,
B
an d
A
are f ixe d,th e o u tpu t v o ltag e
e
is u n iqu ely p rop o rtion al to th e an g u lar ve loc ity o f th e
co il relative to the f ield,
Fig,4.76 Equivalent circuit of the moving-coil electromagnetic transducer
4.9.1 Moving-coil and moving-magnet pickups
o
Z is t he im pedance of the coil,
L
R is t he load r esis tance
(i ncluding the i npu t r esis tance of the a m plifier ),
c
C is the
str ay capac it ance of the wire
mR
c
03.0,mpFC
c
70?,an d LjrZ
o
,r is ab o u t
k2~300,L is in the o rder o f m H s an d can b e n eg le cte d,S o
the o u tpu t v o ltag e
oC
L
o
L
ZCj
R
Z
ee
1
1
(4.1 1 7 )
If th e w ire is n o t lon g,th en
c
C can b e n eg lected,A n d if w e let
cL
ZR,th en E q,(4,1 1 7 ) can b e sim p lif ied to
ee
L
,
4.9.1 Moving-coil and moving-magnet pickups
Electromagnetic transducers can be divided into
two types,
absolute velocity transducers
relative velocity transducers.
4.9.1 Moving-coil and moving-magnet pickups
Fig,4.77 Absolute vibration transducer
1-diaphragm
2-permanent magnet
3-damping cup
4-frame
5-center shaft
6-housing
7-coil
Analyzing the characteristic curve in terms of
the criterions for accurate measurement,we can
get the following conclusions:
1) To realize an accurate measurement,the
amplitude response must be a constant.
4.9.1 Moving-coil and moving-magnet pickups
w h en 1?
n
,th e am p litud e resp o n se c u rve
ten d s to u n ity w ith the in crease in
n
A s the am p litud e resp o n se s f o r dif f eren t d am p ing
ratios ap p roac h a co n stan t (stead y - state) v alu e in
d if f eren t rates,a n d th e m ax im u m rate o ccu rs at
707.0,u su ally a d am p ing ratio 7.0~6.0
is ad o p ted
Theoretically,the upper limit of the working
frequency range of the transducer is infinite,
But in reality,since the transducer will be
affected by the factors such as mounting
stiffness and local resonances of its
components,so the working frequency range
is finite.
4.9.1 Moving-coil and moving-magnet pickups
In gener al,if 707.0,and if a
m easur em ent e rr or %5? is r equire d,then
the frequency ra nge f or m easur em ent is
about
n
7.1?,
2) Although damping can improve the flatness of
the part of the response curve near resonant
frequency,it also increases the phase-shift.
4.9.1 Moving-coil and moving-magnet pickups
w h en
n
,if th e f requ en cy co m p o n en ts o f
inp u t an d o u tpu t h av e a p h ase - sh if t o f ab o u t
1 8 0,th at is,the trans d u cer acts n o w as an
inv erter,the m easu rem en t resu lt can b e still
co n sid ered as an accu rate o n e,
It is k n o w n f rom th e p h ase resp o n se p lot th at
sh o u ld b e se lected as
n
8~7? to o b tain
an ap p rox im ate p h ase inv ersio n ch aracteristic,
3)
4.9.1 Moving-coil and moving-magnet pickups
T h e n atu ral f requ en cy
n
o f v elo city
trans d u cer is a n o the r im p o rtant p aram eter,It
d eterm ine s th e lo w er lim it of th e f requ en cy
rang e m easu red,T o ex p an d the trans d u cer ’ s
w o rkin g f requ en cy ran g e,
n
sh o u ld b e
d esig n ed as low as p o ssib le,T h e lo w er lim it of
the w o rkin g f requ en cy ran g e f o r m o st
co m m o n ly - u sed v elo city trans d u cers is u su ally
w ithin 1 0 ~ 7 5 H z,
With the coil and magnet fixed,when the
measured object (usually of permeable material) is
in motion to influence the reluctance of the
magnetic circuit,induced voltage will be
generated because of the change in magnetic field
(Fig,4.79),Transducers based on this principle are
called magneto-resistive transducers
4.9.2 Magneto-resistive transducers
Fig,4.79 Principle of magneto-resistive transducer and its applications in
(a) measurement of frequency number (b) measurement of rotating
speed (c) measurement of eccentric amount (d) measurement of
vibration
,4.122 Example of a moiré structure
4.9.2 Magneto-resistive transducers
