Fundamentals of
Measurement Technology
(9)
Prof,Wang Boxiong
Inductive transducers are based on the voltage
output of an inductor (or coil) whose inductance
changes in response to changes in the measurand.
A classification of inductive transducers,
1,Variable self-inductance
a,Single coil (simple variable permeance)
b,Two coil (or single coil with center tap) connected for
inductance ratio
2,Variable mutual inductance
a,Simple two coil
b,Three coil (using series opposition)
3,Variable reluctance
a,Moving iron
b,Moving coil
c,Moving magnet
4.6 Inductive transducers
1,Simple permeance-varying
4.6.1 Self-inductance arrangements
A sim p le p e r me a n c e - v a r y in g tr a n sd u c e r is c o m p o se d o f a n i r o n
c o r e,a c o il a n d a n a r ma t u r e,A n a ir g a p? is a r r a n g e d b e tw e e n the
c o r e a n d the a r m a tur e,Whe n a c u r r e n t i f low s thr o u g h the c o il,a
ma g n e tic f l u x
m
is g e n e r a te d w ithin it,w h o s e mag n itud e is
p r o p o r tion a l to th e c u r r e n t
i
,
LiW
m
（ 4.22 ）
w h e r e
W
= n u mb e r f o tu r n s
L
= s e lf in d u c ta n c e (
H
)
4.6.1 Self-inductance arrangements
Fig,4.15 Principle of permeance-varying transducer
1－ coil
2－ core
3－ armature
A lso a c c o r d ing to O h m’ s la w o f m a g n e tic c ir c u it
m
m
R
iW
（ 4.23 ）
w h e r e
Wi
= m a g n e tic m o tive f o r c e ( A )
m
R = r e luc ta n c e (
1?
H )
Sub stitu ting E q,( 4,2 3 ) in to E q,( 4,2 2 ) y ie lds
m
R
W
L
2
（ 4.24 ）
4.6.1 Self-inductance arrangements
N e g le c ti n g the iro n loss in the ma g ne ti c loop a nd a ssuming the a ir g a p?
is sma ll,the n the tota l r e luc ta nc e of the ma g ne ti c c irc uit,
00
2
AA
l
R
m
（ 4.25 ）
w he r e
l
= le ng th of t he ir on c irc uit (
m
)
= pe r me a bili ty of the ir on c or e (
mH /
)
A
= c r oss - se c ti ona l a r e a of the iro n,)
2
mbaA （
= le ng th of the a ir g a ps (
m
)
0
= pe r me a bilt iy of f r e e spa c e ( va c uum)
)/(104
7
0
mH
0
A = c r oss - se c ti ona l a r e a of the a ir g a p (
2
m
)
4.6.1 Self-inductance arrangements
Since th e f ir st te rm at the ri gh t - h and side of Eq,(4.25),th e
re luc tance o f the ir on,is m uch s m a ll e r than the second ter m,the
re luc tance o f the air g a p,then the to tal reluctan ce
m
R,w it h th e
f ir st t er m neglec ted,i s approxim ate ly
00
2
A
R
m
（ 4.26 ）
4.6.1 Self-inductance arrangements
S u b stitu ting E q,(4,2 6 ) into E q,(4,2 4 ) giv es
2
00
2
2?
AW
L? （ 4,2 7 ）
T h e se lf - ind u ctan ce is p rop o rtion al to th e cro ss - sectio n al area o f
the air g ap,
0
A,an d is in v ersely p rop o rtion al to th e len g th o f th e
air ga p,
,
4.6.1 Self-inductance arrangements
Th e se nsiti vity of the t ra nsducer
2
00
2
2?
AW
d
dL
S （ 4.28 ）
Sensiti vity
S
is i nversel y proporti onal to
2
,As? is not a
constant,non li near it y w il l occ ur,This kind of tr ans ducers operat e
of ten over a r ange of sm all change in t he ai r ga p,
4.6.1 Self-inductance arrangements
From E q,( 4,28 ) w e ob tian
)21(
2)(22
0
2
0
00
2
2
0
00
2
2
00
2
AWAWAW
S
If there is a very sm all a ir gap,that i s,
0
,the sen s itiv ity
S
can b e f urther app roxim ated as
2
0
00
2
2?
AW
S? （ 4.2 9 ）
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
T h u s th e se n sitiv ity,S,is n o w a co n atan t,an d th e in p u t
an d th e o u tpu t h av e an ap p rox im ately line ar relation sh ip,
In p ractice,w e o f ten selec t 1.0/,T h is k ind o f
trans d u cer is s u itab le to th e m easu rem en t o f sm all
d isp lacem en t,u su ally w ithin
mmto 1001.0
,
Ind u ctan ce ca n also b e ch an g ed b y ch an g ing
0
A an d
W
,
Fig,4.16 Permeance-varying inductive transducers
Varying-permeable-area-type Solenoid-type
Linear variable differential transformer(LVDT)
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.17 Self-inductance transducers for various applications
2,Eddy-current transducers
When a metal conductor is placed in a changing
magnetic field or is moving in a magnetic field,
an electrical current will be induced within the
conductor,This current is self-closed within the
metal conductor,and is therefore referred to as
electrical eddy-current or,simply,eddy-current.
4.6.1 Self-inductance arrangements
D u e to th e f lux b y th e ed d y - cu rr en t,t h e eq u iva len t resis tan c e o f
the co il w il l ch an g e,w h o se m ag n itud e i s dep en d e n t n o t o n ly on
the g ap? b u t also o n th e res ist ivity o f the m et al co n d u c tor,
,it s p e rm e ab il ity,?,an d the f requ en cy o f the ex ci tin g
cu rren t,
,Ch an g ing an y o n e o f the se p aram eter s y ield s the
ch an g e in co il ’ s eq u iv alen t res ist a n ce,an d thi s can b e u s ed
f o r dif f eren t ap p licatio n s
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.18 Principle of eddy-
current transducer
Fig,4.19 Equivalent circuit of the eddy-current
transducer and the measured object
4.6.1 Self-inductance arrangements
T h e m etal co n d u cto r i s repres en te d b y a sh ort - c ir cu it co i l
co u p led lin e arly w ith th e trans d u c er co il,A m u tua l ind u ct an ce
M is d ef ine d as the d eg re e o f co u p ling b etw een the elem en ts,
an d M d ecrea ses w h en t h e g ap
in creas es,1
R
an d 1
L
are the resi s tan ce an d in d u ctan ce o f the co i l,an d 2
R
an d 2
L
are th e resis tan ce an d in d u ctan ce o f th e co n d u cto r resp ectiv ely,
By K ir chhof f ’ s la w of cur re nt and voltage,w e have
0
.
22
.
22
.
1
..
2
.
11
.
11
ILjIRIMj
EIMjILjIR
（ 4.30 ）
Solving the above group of equati ons yields
2
2
2
2
2
22
12
2
2
2
2
22
1
.
.
1
)()(
L
LR
M
LjR
LR
M
R
E
I
（ 4.32 ）
2
2
22
2
.
12
.
12
2
22
.
1
.
2
LR
IMRjILM
LjR
IM
jI
（ 4.33 ）
4.6.1 Self-inductance arrangements
Further w e ca n obtain the equivale nt r esis tance of the c oil,Z,after
being af f ect ed by m eta l c ond uctor in the f ollowing re lat ion,
2
2
22
2
22
212
2
22
2
22
21
LR
M
LLj
LR
M
RRZ
（ 4.34 ）
Th e equival ent i ndu cta nce of the c oil i s then
2
2
22
2
22
21
LR
M
LLL
（ 4.35 ）
4.6.1 Self-inductance arrangements
The second term of Eq,(4.35) is dependent of the
eddy-current effect,The eddy-current generates a
magnetic field,which opposes the original magnetic
field and thus reduces the inductance of the coil.
4.6.1 Self-inductance arrangements
1L i s r e l a t e d w i t h t h e s t a t i c m a g n e t i c e f f e c t,
Eddy-current transducers are divided into two types,
High-frequency reflective ones,usually used for
measurement of displacements and vibrations,etc.
Low-frequency transmissive ones,mostly used to measure
thickness of materials.
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.20 Low-frequency transmissive eddy-current transducer
Fig,4.21 Voltage-dividing AM circuit for eddy-current vibrometer
4.6.1 Self-inductance arrangements
Th e r esonant frequency
CL
f
'
2
1
（ 4.36 ）
W hen the re sonant f r equency f r eac hes the oscilla ti ng
f re quency o f the oscill a tor,the out put
e
has it s m axi m u m
value,
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.24 Applications of eddy-current transducers
4.6.1 Self-inductance arrangements
1—coil
2—moving
shorted-circuit ring
Fig,4.25 Eddy-current transducer arrangements for the measurement of
(a) displacement and (b) angle
Eddy-current transducers can also be used to measure the
temperatures of magnetic materials or mediums (liquid or
gas),The principle for the measurement is based on the
relationship in which resistivity of conductor varies with
temperature.
4.6.1 Self-inductance arrangements
)](1[
0101
tt （ 4,37 ）
w h ere
1
—— resistiv ity o f th e co n d u cto r at th e tem p eratu re
1
t
0
—— resistiv ity if th e co n d u cto r at th e tem p eratu re
0
t
—— resi stan ce - tem p eratu re co ef f icien t o f th e co n d u cto r
4.6.1 Self-inductance arrangements
Fig,4.26 Configuration of eddy-current thermometer
4.6.1 Self-inductance arrangements
Fig,4.27 Temperature-frequency characteristics of magnetic and nonmagnetic
materials
4.6.1 Self-inductance arrangements
Fig,4.28 Eddy-current temperature transducer
1－ compensating coil
2－ coil frame
3－ measuring coil
4－ dielectric thermal isolated washer
5－ temperature sensing element
Fig,4.29 Mutual induction
Mutual-inductance transducers,also known as differential
transducers,utilize the mutual induction as the operating
principle.
dt
di
Me
1
12
（ 4,38 ）
w h ere M =th e p rop o r tion al co ef f icie n t,al so k n o w n as the
m u tua l ind u ctan ce (
H
),w h ich is a m easu re o f the co u p l i n g
b etw een the co ils 1
W
an d 2
W
,w h o se m ag n itud e d e p e n d s o n
the re lat ive p o sitio n s o f the tw o co i l s an d the p e rm eab ility o f the
su rrou n d ing m ed ium,
4.6.2 Mutual-inductance transducers
This type of transducer is essentially a transformer,
whose primary coil is fed with a stable alternating
exciting current,and an output voltage is then
induced in its secondary coil,When the measured
parameter causes a change in the mutual
inductance M,so the output voltage changes too,
Since the secondary stage uses two coils
connected in a differential way,this transducer is
also called the differential transformer transducer,
The most frequently used transducers are the
solenoidal differential transformers.
4.6.2 Mutual-inductance transducers
4.6.2 Mutual-inductance transducers
Fig,4.30 Differential transformer
4.6.2 Mutual-inductance transducers
Fig,4.31 Rotational differential transformer
Fig,4.35 illustrates a phase-sensitive detection
circuitry used in differential transformers for the
measurement of small displacements.
4.6.2 Mutual-inductance transducers
Fig,4.35 Phase-sensitive detection circuitry for small
displacment measurement
The st r oke o f nor m al commer ci al transl a ti ona l L VD T’ s var ies f r o m
about m?100? to a bout cm25?,while li n ea r it y is usua ll y of the
or der of
250,?
per ce n t,Sensit ivity depen ds on supply and st r oke,
and ca n v ar y f r om
cmV,10
to
mmV?40
,Resol uti on is
infini tesim al and am pl ific ation of the output voltage al lows
det ection of m oti ons down to about
m,?10
,Rota r y L VD T’ s have
about
1?
per c ent li n ea r it y f or tr ave l o f
40
,with se n si ti v it y of
the or der of
r e ed e gmV10
,
4.6.2 Mutual-inductance transducers
4.6.2 Mutual-inductance transducers
Fig,4.36 LVDT force transducer
Ferromagnetic materials,especially Nickel-iron
alloys,change their permeability in the direction,
in which a tensile or compressive force is applied
on them,This ―magnetic-elastic effect‖ has been
already used to force measurements,
The pickup is composed of a ferromagnetic
pressure body (force-receiving element) 1 and a
coil 2,The ferromagnetic pressure body is usually
a nickel-iron alloy,which deforms under the
action of an external force,Thus the permeability
increases,while for a compressive stress,the
permeability decreases.
4.6.4 Magnetic-elastic force pickups
F r o m E q,( 4,2 5 ),th e r e luc ta n c e c h a n g e s to
A
l
R
m
,s o
l
AW
L
2
（ 4.47 ）
Whe n the w ind ing
W
,the c r o ss - s e c tion a l a r e a
A
a n d the
p e r me a b le le n g th
l
a ll r e ma in u n c h a n g e d,the n the se lf - ind u c ta n c e
l
is a u n iqu e f u n c tion o f the p e r me a b ilit y,a n d the c h a n g e o f
l
c a n b e u se d a s the m e a su r e o f s tr e ss,
4.6.4 Magnetic-elastic force pickups
4.6.4 Magnetic-elastic force pickups
Fig,4.37 Magnetoelastic force pickup
(a) change in permeability of a Ni-Fe alloy in accordance with
the normal stress
(b) cross-sectional configuration
1—Ni-Fe alloy pressure body
2—coil
3—magnetic field lines
Capacitive transducer employs a capacitor as the sensing
element to convert the change in a measurand into the
change in capacitance.
4.7 Capacitive transducers
F
A
C
0
( 4,4 8 )
w h e r e
A p la te a r e a (
2
m ),
0
d ie le c tr ic c o n sta n t in v a c u u m,mF
12
0
1085.8
,
d ie le c tr ic c o n sta n t f o r the me d ium b e tw e e n t h e p la te s,
a n d
1
f o r a ir,
d ista n c e b e tw e e n th e p la te s (
m
),
4.7 Capacitive transducers
Fig,4.41 Parallel-plate capacitor
A
C
0
1
(4.49 )
E xp and ing E q,(4,49 ) in T aylor ’ s series yields
2
0
2
0
0
1
1
1
A
A
A
C
(4.50 )
4.7.1 Distance-varying
4.7.1 Distance-varying
Fig,4.42 Operating principle for a distance-varying-type
capacitive pickup
4.7.1 Distance-varying
Th e re lation ship betwee n the capac itance C and the distance?
is a non li n ear one (Fig,4.42 (b)),W hen is sm all,an
approxim ate re lation shi p can be o btai ned betw een
CC?
an d
,W hen
1.0
,the li ne ar d eviation is 10%,W hen
01.0
,the deviat ion w il l r educe t o 1%,
N eglec t the ter m s of h igher than s econd orde r to sim plify Eq,
(4,50 ) i n the f ollowing f orm,
2
0
1
A
CCC ( 4,51)
T he sen sitivity of the capacitive p ickup f rom E q,(4.5 1),
2
0
AC
s
(4.52 )
D if f erential arr ang em ent
2
0
21
2 A
CCC (4.53 )
T hu s the p ickup ’ s sens itivity
2
0
2
AC
s
(4.54 )
4.7.1 Distance-varying
4.7.1 Distance-varying
Fig,4.43 Differential capacitive pickup
Distance-changing-type capacitive transducers are
used in measurement of displacement and all other
quantities which can be converted to displacements.
Major disadvantages are their nonlinearity and the
higher inner impedance.
The stray capacitance of the transducer also affects the
measurement accuracy
4.7.1 Distance-varying
Its n o n l in e a r ity i s a b o u t 1 % ~ 3 % o f f u ll s c a le,th e
f re q u e n c y r a n g e o f m e a s u re m e n t is Hz
5
10~0,
4.7.2 Area-varying
Fig,4.44 Area-varying-type capacitive pickups
4.7.2 Area-varying
T h e ch an g e in p late area is
xbA (4.5 5 )
So the ch an g e o f cap acitan ce w ill be
x
b
C
0
(4.5 6 )
Its sen sib ility
b
x
C
s
0
(4.5 7 )
4.7.2 Area-varying
A n g u lar arrang em en t
2
2
r
A
(4.5 8 )
w h ere
cen ter an g le co v ered b y th e co m m o n area,
r
radiu s o f the h alf circu lar pla te,
W hen the r e is a cha ng e in angle,,the cha ng e in
ca pac it anc e will be
2
2
21
r
C ( 4.59)
S ens it ivi ty
2
2
21
rC
s?
( 4.60)
This se ns itivity is a ls o a cons tant,which shows that the
input - out put r el at ion is a l ine ar one,
4.7.2 Area-varying
Distance-varying capacitive transducers can be
also used to measure pressures,accelerations,etc.
4.7.2 Area-varying
Fig,4.45 Capacitive pressure sensors,using
(a) a diaphragm (evacuate and barometric pressure),and (b) and (c),
capsules which expand with pressure
Advantages of capacitive pressure sensors,
+ High sensitivity
+ Fast response
+ Good resistance to adverse atmospheres
+ No self-heating
+ Wide operating range
Disadvantages,
– Nonlinear response
– Measurement errors due to stray parallel capacitances
– The circuit sophistication required
4.7.2 Area-varying
4.7.2 Area-varying
Fig,4.46 Capacitive transducers with movable middle electrode
(a) movable-plate capacitor (b) movable-cylinder capacitor (c)
differential capacitor in cylindrical structure (d) Capacitive level
measurement of a conductive liquid using isolated electrode
1 — electrode
2 — isolation
3 — liquid
2
2
1
1
021
1111
rr
aa
ACCC
(4.6 4 )
So
2
2
1
1
0
rr
aa
A
C
(4.6 5 )
W h ere
A
the p late area o f the trans d u cer,
4.7.3 Medium-varying
4.7.3 Medium-varying
Fig,4.47 Medium-varying-type capacitive transducers with (a) one plate
covered with a dielectric medium,and (b) movable dielectric medium
A ssum e tha t the m ed ium 1 i s th e a ir,tha t is,1?
r
,the n E q,(4.6 5 )
ch an g es to
2
2
10
0
2
2
1
0
rr
a
aa
A
a
a
A
C
(4.6 6 )
It i s fo u n d f rom E q,(4,66 ) tha t the to tal c ap ac itan ce
C
d ep en d s on
the d iele ct ric co n s tan t 2r
an d the m ed ium thic k n ess 2
a
,W h e n
eith e r o f the tw o p aram eters is k n o w n,the o the r o n e c an b e
d eterm ine d b y E q,(4,6 6 ),
4.7.3 Medium-varying
The method is often used to measure thickness of different
materials,such as paper,plastic film,synthetic fiber,etc.
lll
a
b
a
lb
a
llb
CCC
rr
rr
101
0
00
0
020
0
0010
21
(4.67 )
L etting the m edium 1 b e the air,and the capacitance o f a capacitor w ith
the air m edium be
0
C,then
0
000
0
a
lb
C
,
l
l
l
l
l
ll
C
CC
C
C
r
r
0
2
0
2
0
0
0
0
0
1
1
(4.68 )
Medium can be also changed by use of the configuration of
Fig,4.47 (b),
4.7.3 Medium-varying
This principle is often used in measurement of
level of nonconductive liquid or bulk goods.
4.7.3 Medium-varying
Fig,4.48 Capacitance transducer for measurement of
nonconductive liquid or bulk goods
Since the output of transducer and the
change in capacitance are very small,
suitable follow-up amplification is
necessary to convert them into suitable
voltages,currents or frequencies.
4.7.3 Medium-varying
1,Op-amp circuit
4.7.3 Medium-varying
x
io
C
C
ee
0
(4.69)
For a di stance - va rying c apacitive pi ckup,subst ituting Eq,( 4,48 )
into Eq,(4,69 ) yi elds
A
C
ee
io
0
0
(4.70)
w here?
i
e input voltage,
o
e o utput voltage,
0
C f ixed ca pacit ance
x
C equivale nt ca pacit ance of the pickup,
The output voltage is proportional to the
distance,
4.7.3 Medium-varying
Fig,4.50 Op-amp circuit
2,Bridge circuit
4.7.3 Medium-varying
Fig,4.51 Capacitance measuring bridge in terms of
Wien’s principle
1
1
1
1
1
3
2
2
2
2
11
R
Cj
R
Cj
R
R
Cj
R
Cj
R
(4.71 )
or
242141132132
CRRRjRRCRRRjRR (4.72 )
4.7.3 Medium-varying
For th e real p art is
1
3
4
2
R
R
R
R? (4.7 3 )
F o r the im ag ina ry p art,th ere is
1
3
4
2
C
R
R
C?
(4.7 4 )
W h en th e cap acitiv e p ick u p is ad jus ted,th en the b ridg e
g ive s o u tpu t,
4.7.3 Medium-varying
3,Frequency modulation circuit
As the capacitance is changed by the input,the
frequency of the oscillator changes accordingly,
which is then converted into voltage output after
being amplitude-limited,frequency-detected and
amplified.
4.7.3 Medium-varying
A capaci tive picku p f or m s part of the reson ant circ u it of
an FM - osc illator,T he resonan t f requen cy of the
FM - oscillator
LC
f
2
1
(4.77 )
w here?L ind uctance o f the os cilla tor cir cuit,
4.7.3 Medium-varying
Fig,4.53 Frequency-modulation circuit
The stray capacitance is caused by the
potential difference between the connecting
wires of the two electrodes of the capacitor,
To remove stray capacitance,
1,The pre-amp stage of the follow-up circuit
must be placed very close to the
capacitive pickup to reduce the influence
due to the changes in wire length and
position
2,Employ the so-called equal-potential-
transmission (or driving cable) technique
4.7.3 Medium-varying
4.7.3 Medium-varying
Fig,4.54 Operating principle of powered cable
Measurement Technology
(9)
Prof,Wang Boxiong
Inductive transducers are based on the voltage
output of an inductor (or coil) whose inductance
changes in response to changes in the measurand.
A classification of inductive transducers,
1,Variable self-inductance
a,Single coil (simple variable permeance)
b,Two coil (or single coil with center tap) connected for
inductance ratio
2,Variable mutual inductance
a,Simple two coil
b,Three coil (using series opposition)
3,Variable reluctance
a,Moving iron
b,Moving coil
c,Moving magnet
4.6 Inductive transducers
1,Simple permeance-varying
4.6.1 Self-inductance arrangements
A sim p le p e r me a n c e - v a r y in g tr a n sd u c e r is c o m p o se d o f a n i r o n
c o r e,a c o il a n d a n a r ma t u r e,A n a ir g a p? is a r r a n g e d b e tw e e n the
c o r e a n d the a r m a tur e,Whe n a c u r r e n t i f low s thr o u g h the c o il,a
ma g n e tic f l u x
m
is g e n e r a te d w ithin it,w h o s e mag n itud e is
p r o p o r tion a l to th e c u r r e n t
i
,
LiW
m
（ 4.22 ）
w h e r e
W
= n u mb e r f o tu r n s
L
= s e lf in d u c ta n c e (
H
)
4.6.1 Self-inductance arrangements
Fig,4.15 Principle of permeance-varying transducer
1－ coil
2－ core
3－ armature
A lso a c c o r d ing to O h m’ s la w o f m a g n e tic c ir c u it
m
m
R
iW
（ 4.23 ）
w h e r e
Wi
= m a g n e tic m o tive f o r c e ( A )
m
R = r e luc ta n c e (
1?
H )
Sub stitu ting E q,( 4,2 3 ) in to E q,( 4,2 2 ) y ie lds
m
R
W
L
2
（ 4.24 ）
4.6.1 Self-inductance arrangements
N e g le c ti n g the iro n loss in the ma g ne ti c loop a nd a ssuming the a ir g a p?
is sma ll,the n the tota l r e luc ta nc e of the ma g ne ti c c irc uit,
00
2
AA
l
R
m
（ 4.25 ）
w he r e
l
= le ng th of t he ir on c irc uit (
m
)
= pe r me a bili ty of the ir on c or e (
mH /
)
A
= c r oss - se c ti ona l a r e a of the iro n,)
2
mbaA （
= le ng th of the a ir g a ps (
m
)
0
= pe r me a bilt iy of f r e e spa c e ( va c uum)
)/(104
7
0
mH
0
A = c r oss - se c ti ona l a r e a of the a ir g a p (
2
m
)
4.6.1 Self-inductance arrangements
Since th e f ir st te rm at the ri gh t - h and side of Eq,(4.25),th e
re luc tance o f the ir on,is m uch s m a ll e r than the second ter m,the
re luc tance o f the air g a p,then the to tal reluctan ce
m
R,w it h th e
f ir st t er m neglec ted,i s approxim ate ly
00
2
A
R
m
（ 4.26 ）
4.6.1 Self-inductance arrangements
S u b stitu ting E q,(4,2 6 ) into E q,(4,2 4 ) giv es
2
00
2
2?
AW
L? （ 4,2 7 ）
T h e se lf - ind u ctan ce is p rop o rtion al to th e cro ss - sectio n al area o f
the air g ap,
0
A,an d is in v ersely p rop o rtion al to th e len g th o f th e
air ga p,
,
4.6.1 Self-inductance arrangements
Th e se nsiti vity of the t ra nsducer
2
00
2
2?
AW
d
dL
S （ 4.28 ）
Sensiti vity
S
is i nversel y proporti onal to
2
,As? is not a
constant,non li near it y w il l occ ur,This kind of tr ans ducers operat e
of ten over a r ange of sm all change in t he ai r ga p,
4.6.1 Self-inductance arrangements
From E q,( 4,28 ) w e ob tian
)21(
2)(22
0
2
0
00
2
2
0
00
2
2
00
2
AWAWAW
S
If there is a very sm all a ir gap,that i s,
0
,the sen s itiv ity
S
can b e f urther app roxim ated as
2
0
00
2
2?
AW
S? （ 4.2 9 ）
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
T h u s th e se n sitiv ity,S,is n o w a co n atan t,an d th e in p u t
an d th e o u tpu t h av e an ap p rox im ately line ar relation sh ip,
In p ractice,w e o f ten selec t 1.0/,T h is k ind o f
trans d u cer is s u itab le to th e m easu rem en t o f sm all
d isp lacem en t,u su ally w ithin
mmto 1001.0
,
Ind u ctan ce ca n also b e ch an g ed b y ch an g ing
0
A an d
W
,
Fig,4.16 Permeance-varying inductive transducers
Varying-permeable-area-type Solenoid-type
Linear variable differential transformer(LVDT)
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.17 Self-inductance transducers for various applications
2,Eddy-current transducers
When a metal conductor is placed in a changing
magnetic field or is moving in a magnetic field,
an electrical current will be induced within the
conductor,This current is self-closed within the
metal conductor,and is therefore referred to as
electrical eddy-current or,simply,eddy-current.
4.6.1 Self-inductance arrangements
D u e to th e f lux b y th e ed d y - cu rr en t,t h e eq u iva len t resis tan c e o f
the co il w il l ch an g e,w h o se m ag n itud e i s dep en d e n t n o t o n ly on
the g ap? b u t also o n th e res ist ivity o f the m et al co n d u c tor,
,it s p e rm e ab il ity,?,an d the f requ en cy o f the ex ci tin g
cu rren t,
,Ch an g ing an y o n e o f the se p aram eter s y ield s the
ch an g e in co il ’ s eq u iv alen t res ist a n ce,an d thi s can b e u s ed
f o r dif f eren t ap p licatio n s
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.18 Principle of eddy-
current transducer
Fig,4.19 Equivalent circuit of the eddy-current
transducer and the measured object
4.6.1 Self-inductance arrangements
T h e m etal co n d u cto r i s repres en te d b y a sh ort - c ir cu it co i l
co u p led lin e arly w ith th e trans d u c er co il,A m u tua l ind u ct an ce
M is d ef ine d as the d eg re e o f co u p ling b etw een the elem en ts,
an d M d ecrea ses w h en t h e g ap
in creas es,1
R
an d 1
L
are the resi s tan ce an d in d u ctan ce o f the co i l,an d 2
R
an d 2
L
are th e resis tan ce an d in d u ctan ce o f th e co n d u cto r resp ectiv ely,
By K ir chhof f ’ s la w of cur re nt and voltage,w e have
0
.
22
.
22
.
1
..
2
.
11
.
11
ILjIRIMj
EIMjILjIR
（ 4.30 ）
Solving the above group of equati ons yields
2
2
2
2
2
22
12
2
2
2
2
22
1
.
.
1
)()(
L
LR
M
LjR
LR
M
R
E
I
（ 4.32 ）
2
2
22
2
.
12
.
12
2
22
.
1
.
2
LR
IMRjILM
LjR
IM
jI
（ 4.33 ）
4.6.1 Self-inductance arrangements
Further w e ca n obtain the equivale nt r esis tance of the c oil,Z,after
being af f ect ed by m eta l c ond uctor in the f ollowing re lat ion,
2
2
22
2
22
212
2
22
2
22
21
LR
M
LLj
LR
M
RRZ
（ 4.34 ）
Th e equival ent i ndu cta nce of the c oil i s then
2
2
22
2
22
21
LR
M
LLL
（ 4.35 ）
4.6.1 Self-inductance arrangements
The second term of Eq,(4.35) is dependent of the
eddy-current effect,The eddy-current generates a
magnetic field,which opposes the original magnetic
field and thus reduces the inductance of the coil.
4.6.1 Self-inductance arrangements
1L i s r e l a t e d w i t h t h e s t a t i c m a g n e t i c e f f e c t,
Eddy-current transducers are divided into two types,
High-frequency reflective ones,usually used for
measurement of displacements and vibrations,etc.
Low-frequency transmissive ones,mostly used to measure
thickness of materials.
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.20 Low-frequency transmissive eddy-current transducer
Fig,4.21 Voltage-dividing AM circuit for eddy-current vibrometer
4.6.1 Self-inductance arrangements
Th e r esonant frequency
CL
f
'
2
1
（ 4.36 ）
W hen the re sonant f r equency f r eac hes the oscilla ti ng
f re quency o f the oscill a tor,the out put
e
has it s m axi m u m
value,
4.6.1 Self-inductance arrangements
4.6.1 Self-inductance arrangements
Fig,4.24 Applications of eddy-current transducers
4.6.1 Self-inductance arrangements
1—coil
2—moving
shorted-circuit ring
Fig,4.25 Eddy-current transducer arrangements for the measurement of
(a) displacement and (b) angle
Eddy-current transducers can also be used to measure the
temperatures of magnetic materials or mediums (liquid or
gas),The principle for the measurement is based on the
relationship in which resistivity of conductor varies with
temperature.
4.6.1 Self-inductance arrangements
)](1[
0101
tt （ 4,37 ）
w h ere
1
—— resistiv ity o f th e co n d u cto r at th e tem p eratu re
1
t
0
—— resistiv ity if th e co n d u cto r at th e tem p eratu re
0
t
—— resi stan ce - tem p eratu re co ef f icien t o f th e co n d u cto r
4.6.1 Self-inductance arrangements
Fig,4.26 Configuration of eddy-current thermometer
4.6.1 Self-inductance arrangements
Fig,4.27 Temperature-frequency characteristics of magnetic and nonmagnetic
materials
4.6.1 Self-inductance arrangements
Fig,4.28 Eddy-current temperature transducer
1－ compensating coil
2－ coil frame
3－ measuring coil
4－ dielectric thermal isolated washer
5－ temperature sensing element
Fig,4.29 Mutual induction
Mutual-inductance transducers,also known as differential
transducers,utilize the mutual induction as the operating
principle.
dt
di
Me
1
12
（ 4,38 ）
w h ere M =th e p rop o r tion al co ef f icie n t,al so k n o w n as the
m u tua l ind u ctan ce (
H
),w h ich is a m easu re o f the co u p l i n g
b etw een the co ils 1
W
an d 2
W
,w h o se m ag n itud e d e p e n d s o n
the re lat ive p o sitio n s o f the tw o co i l s an d the p e rm eab ility o f the
su rrou n d ing m ed ium,
4.6.2 Mutual-inductance transducers
This type of transducer is essentially a transformer,
whose primary coil is fed with a stable alternating
exciting current,and an output voltage is then
induced in its secondary coil,When the measured
parameter causes a change in the mutual
inductance M,so the output voltage changes too,
Since the secondary stage uses two coils
connected in a differential way,this transducer is
also called the differential transformer transducer,
The most frequently used transducers are the
solenoidal differential transformers.
4.6.2 Mutual-inductance transducers
4.6.2 Mutual-inductance transducers
Fig,4.30 Differential transformer
4.6.2 Mutual-inductance transducers
Fig,4.31 Rotational differential transformer
Fig,4.35 illustrates a phase-sensitive detection
circuitry used in differential transformers for the
measurement of small displacements.
4.6.2 Mutual-inductance transducers
Fig,4.35 Phase-sensitive detection circuitry for small
displacment measurement
The st r oke o f nor m al commer ci al transl a ti ona l L VD T’ s var ies f r o m
about m?100? to a bout cm25?,while li n ea r it y is usua ll y of the
or der of
250,?
per ce n t,Sensit ivity depen ds on supply and st r oke,
and ca n v ar y f r om
cmV,10
to
mmV?40
,Resol uti on is
infini tesim al and am pl ific ation of the output voltage al lows
det ection of m oti ons down to about
m,?10
,Rota r y L VD T’ s have
about
1?
per c ent li n ea r it y f or tr ave l o f
40
,with se n si ti v it y of
the or der of
r e ed e gmV10
,
4.6.2 Mutual-inductance transducers
4.6.2 Mutual-inductance transducers
Fig,4.36 LVDT force transducer
Ferromagnetic materials,especially Nickel-iron
alloys,change their permeability in the direction,
in which a tensile or compressive force is applied
on them,This ―magnetic-elastic effect‖ has been
already used to force measurements,
The pickup is composed of a ferromagnetic
pressure body (force-receiving element) 1 and a
coil 2,The ferromagnetic pressure body is usually
a nickel-iron alloy,which deforms under the
action of an external force,Thus the permeability
increases,while for a compressive stress,the
permeability decreases.
4.6.4 Magnetic-elastic force pickups
F r o m E q,( 4,2 5 ),th e r e luc ta n c e c h a n g e s to
A
l
R
m
,s o
l
AW
L
2
（ 4.47 ）
Whe n the w ind ing
W
,the c r o ss - s e c tion a l a r e a
A
a n d the
p e r me a b le le n g th
l
a ll r e ma in u n c h a n g e d,the n the se lf - ind u c ta n c e
l
is a u n iqu e f u n c tion o f the p e r me a b ilit y,a n d the c h a n g e o f
l
c a n b e u se d a s the m e a su r e o f s tr e ss,
4.6.4 Magnetic-elastic force pickups
4.6.4 Magnetic-elastic force pickups
Fig,4.37 Magnetoelastic force pickup
(a) change in permeability of a Ni-Fe alloy in accordance with
the normal stress
(b) cross-sectional configuration
1—Ni-Fe alloy pressure body
2—coil
3—magnetic field lines
Capacitive transducer employs a capacitor as the sensing
element to convert the change in a measurand into the
change in capacitance.
4.7 Capacitive transducers
F
A
C
0
( 4,4 8 )
w h e r e
A p la te a r e a (
2
m ),
0
d ie le c tr ic c o n sta n t in v a c u u m,mF
12
0
1085.8
,
d ie le c tr ic c o n sta n t f o r the me d ium b e tw e e n t h e p la te s,
a n d
1
f o r a ir,
d ista n c e b e tw e e n th e p la te s (
m
),
4.7 Capacitive transducers
Fig,4.41 Parallel-plate capacitor
A
C
0
1
(4.49 )
E xp and ing E q,(4,49 ) in T aylor ’ s series yields
2
0
2
0
0
1
1
1
A
A
A
C
(4.50 )
4.7.1 Distance-varying
4.7.1 Distance-varying
Fig,4.42 Operating principle for a distance-varying-type
capacitive pickup
4.7.1 Distance-varying
Th e re lation ship betwee n the capac itance C and the distance?
is a non li n ear one (Fig,4.42 (b)),W hen is sm all,an
approxim ate re lation shi p can be o btai ned betw een
CC?
an d
,W hen
1.0
,the li ne ar d eviation is 10%,W hen
01.0
,the deviat ion w il l r educe t o 1%,
N eglec t the ter m s of h igher than s econd orde r to sim plify Eq,
(4,50 ) i n the f ollowing f orm,
2
0
1
A
CCC ( 4,51)
T he sen sitivity of the capacitive p ickup f rom E q,(4.5 1),
2
0
AC
s
(4.52 )
D if f erential arr ang em ent
2
0
21
2 A
CCC (4.53 )
T hu s the p ickup ’ s sens itivity
2
0
2
AC
s
(4.54 )
4.7.1 Distance-varying
4.7.1 Distance-varying
Fig,4.43 Differential capacitive pickup
Distance-changing-type capacitive transducers are
used in measurement of displacement and all other
quantities which can be converted to displacements.
Major disadvantages are their nonlinearity and the
higher inner impedance.
The stray capacitance of the transducer also affects the
measurement accuracy
4.7.1 Distance-varying
Its n o n l in e a r ity i s a b o u t 1 % ~ 3 % o f f u ll s c a le,th e
f re q u e n c y r a n g e o f m e a s u re m e n t is Hz
5
10~0,
4.7.2 Area-varying
Fig,4.44 Area-varying-type capacitive pickups
4.7.2 Area-varying
T h e ch an g e in p late area is
xbA (4.5 5 )
So the ch an g e o f cap acitan ce w ill be
x
b
C
0
(4.5 6 )
Its sen sib ility
b
x
C
s
0
(4.5 7 )
4.7.2 Area-varying
A n g u lar arrang em en t
2
2
r
A
(4.5 8 )
w h ere
cen ter an g le co v ered b y th e co m m o n area,
r
radiu s o f the h alf circu lar pla te,
W hen the r e is a cha ng e in angle,,the cha ng e in
ca pac it anc e will be
2
2
21
r
C ( 4.59)
S ens it ivi ty
2
2
21
rC
s?
( 4.60)
This se ns itivity is a ls o a cons tant,which shows that the
input - out put r el at ion is a l ine ar one,
4.7.2 Area-varying
Distance-varying capacitive transducers can be
also used to measure pressures,accelerations,etc.
4.7.2 Area-varying
Fig,4.45 Capacitive pressure sensors,using
(a) a diaphragm (evacuate and barometric pressure),and (b) and (c),
capsules which expand with pressure
Advantages of capacitive pressure sensors,
+ High sensitivity
+ Fast response
+ Good resistance to adverse atmospheres
+ No self-heating
+ Wide operating range
Disadvantages,
– Nonlinear response
– Measurement errors due to stray parallel capacitances
– The circuit sophistication required
4.7.2 Area-varying
4.7.2 Area-varying
Fig,4.46 Capacitive transducers with movable middle electrode
(a) movable-plate capacitor (b) movable-cylinder capacitor (c)
differential capacitor in cylindrical structure (d) Capacitive level
measurement of a conductive liquid using isolated electrode
1 — electrode
2 — isolation
3 — liquid
2
2
1
1
021
1111
rr
aa
ACCC
(4.6 4 )
So
2
2
1
1
0
rr
aa
A
C
(4.6 5 )
W h ere
A
the p late area o f the trans d u cer,
4.7.3 Medium-varying
4.7.3 Medium-varying
Fig,4.47 Medium-varying-type capacitive transducers with (a) one plate
covered with a dielectric medium,and (b) movable dielectric medium
A ssum e tha t the m ed ium 1 i s th e a ir,tha t is,1?
r
,the n E q,(4.6 5 )
ch an g es to
2
2
10
0
2
2
1
0
rr
a
aa
A
a
a
A
C
(4.6 6 )
It i s fo u n d f rom E q,(4,66 ) tha t the to tal c ap ac itan ce
C
d ep en d s on
the d iele ct ric co n s tan t 2r
an d the m ed ium thic k n ess 2
a
,W h e n
eith e r o f the tw o p aram eters is k n o w n,the o the r o n e c an b e
d eterm ine d b y E q,(4,6 6 ),
4.7.3 Medium-varying
The method is often used to measure thickness of different
materials,such as paper,plastic film,synthetic fiber,etc.
lll
a
b
a
lb
a
llb
CCC
rr
rr
101
0
00
0
020
0
0010
21
(4.67 )
L etting the m edium 1 b e the air,and the capacitance o f a capacitor w ith
the air m edium be
0
C,then
0
000
0
a
lb
C
,
l
l
l
l
l
ll
C
CC
C
C
r
r
0
2
0
2
0
0
0
0
0
1
1
(4.68 )
Medium can be also changed by use of the configuration of
Fig,4.47 (b),
4.7.3 Medium-varying
This principle is often used in measurement of
level of nonconductive liquid or bulk goods.
4.7.3 Medium-varying
Fig,4.48 Capacitance transducer for measurement of
nonconductive liquid or bulk goods
Since the output of transducer and the
change in capacitance are very small,
suitable follow-up amplification is
necessary to convert them into suitable
voltages,currents or frequencies.
4.7.3 Medium-varying
1,Op-amp circuit
4.7.3 Medium-varying
x
io
C
C
ee
0
(4.69)
For a di stance - va rying c apacitive pi ckup,subst ituting Eq,( 4,48 )
into Eq,(4,69 ) yi elds
A
C
ee
io
0
0
(4.70)
w here?
i
e input voltage,
o
e o utput voltage,
0
C f ixed ca pacit ance
x
C equivale nt ca pacit ance of the pickup,
The output voltage is proportional to the
distance,
4.7.3 Medium-varying
Fig,4.50 Op-amp circuit
2,Bridge circuit
4.7.3 Medium-varying
Fig,4.51 Capacitance measuring bridge in terms of
Wien’s principle
1
1
1
1
1
3
2
2
2
2
11
R
Cj
R
Cj
R
R
Cj
R
Cj
R
(4.71 )
or
242141132132
CRRRjRRCRRRjRR (4.72 )
4.7.3 Medium-varying
For th e real p art is
1
3
4
2
R
R
R
R? (4.7 3 )
F o r the im ag ina ry p art,th ere is
1
3
4
2
C
R
R
C?
(4.7 4 )
W h en th e cap acitiv e p ick u p is ad jus ted,th en the b ridg e
g ive s o u tpu t,
4.7.3 Medium-varying
3,Frequency modulation circuit
As the capacitance is changed by the input,the
frequency of the oscillator changes accordingly,
which is then converted into voltage output after
being amplitude-limited,frequency-detected and
amplified.
4.7.3 Medium-varying
A capaci tive picku p f or m s part of the reson ant circ u it of
an FM - osc illator,T he resonan t f requen cy of the
FM - oscillator
LC
f
2
1
(4.77 )
w here?L ind uctance o f the os cilla tor cir cuit,
4.7.3 Medium-varying
Fig,4.53 Frequency-modulation circuit
The stray capacitance is caused by the
potential difference between the connecting
wires of the two electrodes of the capacitor,
To remove stray capacitance,
1,The pre-amp stage of the follow-up circuit
must be placed very close to the
capacitive pickup to reduce the influence
due to the changes in wire length and
position
2,Employ the so-called equal-potential-
transmission (or driving cable) technique
4.7.3 Medium-varying
4.7.3 Medium-varying
Fig,4.54 Operating principle of powered cable
