Chapter 0005

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Transducers - I

The primary objective of process control is to control the physical parameters such as

temperature, pressure, flow rate, force, level etc. The system used to maintain these

parameters constant, close to some desired specific value is called process control system.

These parameters may change because of internal and external disturbances hence a

constant corrective action is required to keep these parameters constant or within the

specified range.

The Fig. 8.1 shows the general arrangement of a process loop. It consists of four

elements,

1. Process 2. Measurement 3. Controller 4. Control element.

Parameter

P

Measurementof parameter

For the proper feedback, it is necessary to measure the value of the actual parameter P.

Most of the controllers are electronic in nature and hence require electrical input. Hence

feedback signal required is in electrical form in most of the practical process loops. But

actual parameter is temperature, pressure, level etc. Hence a device is required in the

feedback path which will not only measure the output parameter but will produce

proportional analog signal in the electric form. Many times the device is required to

measure the physical parameter and produce the proportional signal which is also

nonelectric such as pneumatic pressure. So in broad sense a transducer converts one form

of energy to another form. But the electrical transducer produces an electrical signal

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proportional to the nonelectrical quantity to be measureQ. But as we are interested in the

electrical instrumentation, a transducer can be defined as,

A device which converts a physical quantity into the proportional electrical signal is

called a transducer.

The electrical signal produced may be a voltage, current or frequency. A transduceruses many effects to produce such conversion. The process of transforming signal f rom one

f orm to other is called transduction. A transducer is also called pick up.

Actually, electrical transducer consists of two parts which are very closely related to

each other. These two parts are sensing or detecting element and transduction element.

The sensing or detecting element is commonly known as sensor.

Definition states that sensor is a device that produces a measurable response to a

change in a physical condition.

The transduction element transforms the output of the sensor to an electrical output,

as shown in the Fig. 8.2.

Non-elect

quantity

ricalSensing

SensorTransduction

Elec

element response element sig

trical

nal

The common range of an electrical signal used to represent analog signal in the

industrial environment is 0 to 5 V or 4 to 20 mA. In industrial applications, nowadays, 4

to 20 mA range is most commonly used to represent analog signal. A current of 4 m A

represents a zero output and current of 20 mA represents a full scale value i.e. 5 V in caseof voltage representation. The zero current condition represents open circuit in the signal

transmission line. Hence the standard range is offset from zero.

Many a times, the transducer is a part of a circuit and works with other elements of

that circuit to produce the required output. Such a circuit is called signal conditioning

circuit.

A transducer is a device that receives energy from one system and transmits it to

another in diff erent form. Basically there are two types of transducers; namely electrical

and mechanical. The mechanical transducers are those primary sensing elements that

respond to changes in the physical condition of a system and gives output in diff erent

f orm. For example, when a bimetallic strip is subjected to a temperature change then the

output is the mechanical displacement of the strip. The mechanical transducers are

distinguished from the electrical transducers on the basis of the output signal generated.

The mechanical transducers generate output signal which is mechanical by nature. The

electrical transducers respond to non-electrical quantities but generate output signal which

is electrical by nature. It is practically always possible to use either mechanical or electrical

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iii) Resistance of corrosion,

iv) Accessibility of the transducer for later repairs.

4. Measuring System Compatibility : The transducer selected and the electrical

system used for measurement should be compatible. The output impedance of the

transducer and the impedance imposed by the measuring system must be such

that one does not adversely affect the other.

5. Cost and Availability: General factors involved in selection are cost, availability,

basic simplicity, reliability, and low maintenance.

While selecting transducers of comparatively equal merits f or a given application, the

one that is most simple in operation and contains minimum number of moving parts

would usually be selected.

Transducers are selected which do not require excessive repair or continuous

calibration checking.

The selection of a transducer for a given application is normally a compromise

beh'een a number of factors discussed above.

In electrical circuits, there are combinations of three passive elements : resistor,

inductor and capacitor. These three passive elements are described with the help of the

primary parameters such as resistance, self or mutual inductance and capacitance

respectively. Any change in these parameters can be observed only if they are externally

powered. We have studied that the passive transducers do not generate any electrical

signal by themselves and they require some external power to generate an electrical signal.

The transducers based on variation of parameters such as resistance, self or mutualinductance capacitance, due to an external power are known as passive transducers. Hence

resistive transducer, inductive transducer and capacitive transducer are the basic passive

transducers.

In general, the resistance of a metal conductor is given by,

pL

A

Resistivity of conductor (D m)

Length of conductor (m)

Area of cross-section of conductor (m 2)

The electrical resistive transducers are designed on the basis of the methods of

"arintioll o f anyone of the qnantities in above equation; such as change in length, change

in iueil of cross-section and change in resistivity.

where p --

L

A

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The sensing element which is resistive in nature, may be in different forms depending

u p o n the mechanical arrangement. The change in pressure can be sensed by Llsing

~nsitive resistive elements. The resistance pressure transducers may use Bellow,

Diaphragm or Bourdon tube.

Resistance

lead

Resistance

leads.

Resistancecontact.

Fig. 8.17 Resistance pressure transducer

8 .9 Resistance Position Transducer

In many industrial measurements and control applications, it is necessary to sense

position of the ob ject or the distance that object travels. For such applications, simple

resi~tanceposition transducer is very useful.

It works on the principle that resistance of the sensing element changes due to the

wiations in physical quantity being measured.

A simple resistance position transducer is as shown in the Fig. 8.18 (a).

Shaft ~: : : = = = = = = Wiper- R 1

W

V in+

R2

W Vout

B

(b) Equivalent circuit

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The transducer consists a sliding contact or wiper. A resistive element is mounted with

the sliding contact which is linked with the ob ject whose position is to be monitored.

Depending upon the position of the object, the resistance between slider and the one end

of resistive element varies. The equivalent circuit is as shown in the Fig. 8.18 (b). T h e

output voltage Vout depends on the position of the wiper. Thus depending upon positionof the wiper, the output voltage is given by,

Thus Y o u I is proportional to R2 i.e. wiper position. The output voltage is measured

using voltmeter which is calibrated in centimeters and allows direct readout of the object

position.

)1). Example 8.1 : A r esistance posit ion transducer uses a sha f t with a stro ke of 50 CI11.T h e

tot al resistance of t he po t ent iomet er is 5ill. Calculat e out put voltage when w iper is 1 0 e m

f rol11 e xtreme end if a p plie d v oltage is 5 V.

Solution: The equivalent circuit is as shown in the Fig. 8.19.

R2

= 10 cm x 5 K = 1 kn50cm

VOUl

~

The strain gauge is a passive resistive transducer which is based on the principle of

conversion of mechanical displacement in to the resistance change.

A knowledge of strength of the material is essential in the design and construction of

machines and structures. The strength of the material is normally characterized in terms of

stress, which is defined as the force experienced per unit area, and is expressed in pressure

units. Stress as such cannot be directly measured. It is normally deduced f rom the changes

in mechanical dimensions and the applied load. The mechanical def ormation is measured

with strain-gauge elements. The strain is defined as the change, ( t d ), in length, (I), per unit

length and is expressed as t : . ; { in microstrains.

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Breaking

stress

It is observed that the curve is linear as long as the stress is k ept below the elasticmits.Strain measurements are usually carried out on the free surface of a body. Normally

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The most common materials used for wire strain gauges are constantan alloys

containing 45% Nickel and 55% Copper, as they exhibit high specific resistance, constant

gauge factor over a wide strain range, and good stability over a reasonably large

temperature range (from O°C to 300°C). For dynamic strain measurements, Nichrome

alloys, containing 80% Nickel and 20% Chromium are used. They can be compensated for

temperature with platinum.

Bonding cements are adhesives used to fix the strain gauge onto the test specimen.

This cement serves the important function of transmitting the strain from the specimen to

the gauge-sensing element. Improper bonding of the gauge can cause many errors.

Basically, the cement can be classified under two categories, viz, solvent-setting cement

and chemically-reacting cement. Duco cement is an example of solvent-setting cements

which is cured by solvent evaporation. Epoxies and phenolic bakelite cement are

chemically-reacting cements which are cured by polymerization. Acrylic cements are

contact cements that get cured almost instantaneously.

The proper functioning of a strain gauge is wholly dependent on the quality of

bonding which holds the gauge to the surface of the structure undergoing the test.

8.10.2 Derivation of Gauge Factor

The gauge factor is defined as the unit change in resistance per unit change in length.

It is denoted as K or S. It is also called sensitivity of the strain gauge.

S = ~RjR

~l/l

Length of the gauge wire in unstressed condition

~l = Change in length in stlessed condition.

Derivation: Consider that the resistance wire is under tensile stress and it is deformed

by ~I as shown in the Fig. 8.14.

= Length of the wire in m

A = Cross-section of the wire in m2

When uniform stress (J is applied to th.is wire along the length, the resistance R

changes to R + ~R because of change in length and cross-sectional area.

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I I

: d :~I I

J - l - J

rR changes

1to

R+LiR

F

Fig. 8.21 Deformed resistance wire

f ' . . 1

a = Stress = -I

f ' . . I I I =Per unit change in length

f ' ..A / A =Per unit change in area

f ' ..p/p =Per unit change in resistivity

(specific resistance)

Now R =p i

A

d( ~ IdR A ) P 0 1 pi oA I op

.. - --=-----+--da da A o a A20 a A o a

Note thato~( ~ )

1 oA=---A20 a

Multiply both sides by i,1 dR p 01 1 pi o A I op

--------+--Rd a RA oa R A 2 o a RA ca

Using R ~ on right hand side,

1 dR 1 al 1 oA 1 (lp=--- _ ._+--

Rda l oa A O G P o a

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dl dA a p---+-

I A p

/ ' ) . 1 / ').A / '). p---+-

I A p

Key Point : ThliS fo r finit e str ess, total change in resistance is dl le t o fractional cha nge in

len gth , area 17nd resi stivity.

For a circular wire, A = ~d24

a A~(2d) ad=

as 4 . as

1 a A~~(2d)ad=

A as A 4 as

1 a A 1 (2d) ad=A as (d2) a s

Cancelling as,a A

~ad i.e. / ').A 2 / ').d

... ( 2)----

A d A d

Now the Poisson/s ratio 1 1 for the wire is defined as the ratio of strain In lateral

direction to strain in the axial direction.

/ ').d/d1 1 = - / ' ) . I j I = Poisson' s ratio

/ ' ) . d d = -11 ( ~/ )

Using (2) and (3) in (1)/

t.R .t .l_ nd + t.p = t .1_2[ _ 1 1 t .l]+ t.p

Rid p I I P

t.R / ' ) . 1 t.pR = [[1 +21 l]+p-

Neglecting piezoelectric effect, / ') .p can be neglected.p

t . R j R

Gauge factor = -- /- = 1 + 2 1 1 / ' ) .1 I

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S = G = 1 + 2 f . l + t:.p/pt : . l / 1

where last term is due to piezoelectric effect.

))). Example 8.2 : A res istance strain gau ge with a ga ug e f ac to r o f 4 is cemented to a steel

member which is subjec t ed t o a s tr ai n o f 1x 10-6. If t he or iginal gauge resist a nce i s 15 0 n,calculat e the change in r es istance .

Solution : The gauge factor is given by,

S = t :.RjR

t : . l /l

t : . l

SRi

8.10.3 Types of Strain Gauges

Depending upon the principle of operation and their constructional features, strain

gauges are classified as mechanical, optical, or electrical. Of these, the electrical strain

gauges are most commonly used.

1. Mechanical Gauges : In these gauges, the change in length, t :.l , is magnified

mechanically using levers or gears. These gauges are comparatively larger in size,

and as such can be used in applications where sufficient area is available on the

specimen for fixing the gauge. These gauges are employed for static strain

measurements only.

2. Optical Gauges: These gauges are similar to mechanical strain gauges except that

the magnification is achieved with multiple reflectors using mirrors or prisms. In

one type a plain mirror is rigidly fixed to a movable knife-edge. When stress is

applied, the mirror rotates through an angle, and the reflected light beam from the

mirror subtends an angle twice that of the incident light. The measurement

accuracy is high and independent of temperature variations.

3. Electrical Strain Gauges : The electrical strain gauges measure the changes that

occur in resistance, capacitance, or inductance due to the strain transf erred fromthe specimen to the basic gauge element. The most commonly used strain gauge is

the bonded resistance type of strain gauge. The other two, viz., capacitance and

inductance type are used only in special types of applications.

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8.10.4 Basic Forms of Resistance Wire Strain Gauges

The resistance wire strian gauges of metallic type are available in two basic forms;

b o ,r / ded and l I n b on d ed t ype Ti r e b a n ded metallic strain gauges are f urtner cIassified as f iat

grid, helical grid and thin foil type strain gauges.

A. Bonded Resistance Wire Strain Gauge :

In bonded resistance wire strain

gauge resistive element is cemented

to the base which may be a thin

sheet of paper, a thin sheet of

bakelite or a sheet of teflon. The

resistive element may be in the

form of wire, foil or film of the

material. The most common types

of bonded strain gauges are as

shown in the Fig. 8.22.

In metallic bonded strain gauge

a fine wire element about 25 ~m or

less in diameter is looped back and

f orth on a base (carrier) or mounting plate. The base is cemented to the member subjected

to stress. The grid of fine wire is cemented on a carrier which may be a thinsheet of paper,

bakelite, or teflon.

A tensile str ess tends to elongate the wire and thereby increase its length and decreases

its cross-sectional area. The combined effect is an increase in resistance.

In this gauge, the strain is detected using a metal foil. The metals and alloys used f or

the f oil and wire are nichrome, constantan, isoelastic (Ni + Cr + Mo), nickel, and platinum.

On account of their larger surface area, foil gauges have a much greater dissipation

capacity. Therefore they can be used at a higher operating temperature range. The

characteristics of f oil type strain gauges and wire type strain gauges are sim.ilar, includingalmost the same gauge f actor.

The advantage of f oil type strain

gauge is that they Cflnbe fabricated on

a large scale, and in any shape. The f oil

can be etched on the base.

The Fig. 8.23 shows the foil type

strain gauge.

The etched foil strain gauges can be

made thinner than comparable wire

units. Also they are mOTe f lexible.

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Because of these properties, the etched foil can be mounted in remote and restricted

placesand especially on curved surfaces.

The resistance film formed is typically 0.2 mm thick . The resistance value of

commercial foil gauges is between 50 and 1000Q.

B. Unbonded Resistance Wire Strain Gauge

In general, the basic usage of the unbonded strain gauge is as displacement transducer.

It can be constructed in variety of configuration. The unbonded strain gauge consists of a

stationary frame with an armature supported at the centre of the frame as shown in the

Fig. 8.24.

Movingarmature

Supporting

pillars

Movement of

armature

Stationary

frame

Resistancewire

Fig. 8.24 Unbonded strain gauge

The strain gauge is constructed such that an armature can move only in one direction.

Its travel in that direction is limited by four filaments of strain sensitive wires. These wires

are wound on the rigid insulators. These insulators are mounted on the frame and on the

armature.

On the application of the external force, the armature moves in the indicated direction.

The length of elements A and D increases, while the length of elements Band C decreases.

Then proportional to the change in length, the resistance of the four strain elements

change. To detect this change in resistance, we can use Wheatstone bridge as shown in theFig. 8.25.

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Resistance temperature detector is a primary electrical transducer which is used to

measure the change in the temperature. It is commonly known as resistance thermometer.

The resistance thermometers are based on the principle that the resistance of the conductor

changes when ~he temperature changes. Basically the resistance thermometer determines

the' change in the electrical resistance of the conductor sub jected to the temperature

changes.

The temperature sensing element used in this thermometer should exhibit a relatively

large change in resistance f or a given change in temperature. Also the sensing element

should not undergo permanent change with use or age. Another desirable characteristic for

the sensing element is the linear change in resistance with change in temperature. When

the sensing element is smaller in size, less heat is required to raise its temperature. This is

suitable for me~surement of rapid variations in temperature. Platinum, nickel, and copper

are the metals most commonly used to measure temperature. The relationship between

temperature and resistance of conductor is given by equation:

I Rt = Rref [ 1 + a L\t ]

Rt : Resistance of the conductor at temperature t'C,

Rrl"f: Resistance of the conductor at the reference temperature, usually O'C,

a :Temperature coefficient of the resistance,

M : Diff erence between the temperature to be measured & reference temperature.

Almost all metallic conductors have a posit ive tem pera ture coef fi cien t so that their

resistance increases with an increase in temperature. A high value of a is desirable in a

temperature sensing element so that a substantial change in resistance occurs for a

relatively small change in temperature. This change in resistance [ L\ R] can be measured

with a Wheatstone bridge, the output of which can be directly calibrated to indicate the

temperature which caused the change is resistance.

Most of the metals show an increase in resistivity with temperature, which is first

linear and then increases in an accelerated f ashion. The metals that exhibit good sensitivity

and reproducibility for temperature measurement purposes are copper, nick el, and

platinum. Among these, copper has the highest temperature coeff icient with the most

linear dependence. However, copper is generally not used due to certain practical

problems. Because of its low resistivity, the size of the resistance element increases to get

reasonable sensitivity. In the range below 400 K, a gold silver alloy can be used which has

the same characteristicb as platinum.

T em perat ur e se nso r s made out of nick el and nick el alloy wires can be used with good

reliability in the temperature range of 100 to 450 K. It is less expensive than platinum and

has a somewhat higher tel.1perature coeff icient that increases with temperature. It is less

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stable. Nickel wire element is less expensive and easier for manufacturing than platinum

wire elements. They are often used in low range industrial applications.

Platinum wire is used for most laboratory work and for industrial measurements of

hig~ accuracy. The platinum resistance element is the best choice for many applications,

because of its inherent accuracy and reproducibility. Since the resistance to temperature

I'ariation is extremely slf lble and reproducible, the sensor is recommended as an

international standard for temperature measurement. The platinum element can be used

formeasuring temperatures upto 1000 K

Metals commonly used for resistance thermometers are listed in Table 8.3 along with

their salient properties.

Metal Resistance temperature Temperature range °C Melting point °C

co-efficient-t °C

Platinum 0.39 - 260 t a 110 1773

Capper 0.39 o ta 180 1083

Nickel 0.62 - 220 ta 300 1435

Tungsten 0.45 - 200 ta 1000 3370

The wire resistance thermometer usually consists of a coil wound on a mica or ceramic

former, as shown in the Fig. 8.28. The coil is wound in b ifilar form so as to make it

noninductive. Such coils are available in different sizes and with diff erent resistance values

ranging from 10 ohms to 25,000 ohms.

To avoid corrosion of resistive element, usually elements are enclosed in a protective

tube of pyrex glass, porcelain, quartz or nickel, depending on the range of temperature

and the nature of the fluid whose temperature is to be measured. The tube is evacuated

and sealed or f illed with air o r any other inert gas and kept around atmospheric pressure

or in some cases at a higher pressure. \

Mica strip

/'

\ . ;r" Platinum wire Mica strip

/.~g ?~~

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8.11.2 Advantages of RTD

1. The measurement is accurate.

2. Indicators, recorders can be directly operated.

3. The temperature sensor can be easily installed and replaced.

4. Measurement of differential temperature is possible.

5. Resistance thermometers can work over a wide range of temperature from -200 O(

to + 650 0c .

6. They are suitable for remote indication.

7. They are smaller in size .

8. They have stability over long periods of time.

8.11.3 Disadvantages of RTD

1. A bridge circuit with external power source is necessary for their operation.

2. They are comparatively costly.

3. There is a possibility of self heating.

4. Large bubsize as compared to thermocouple.

Basically thermistor is a contraction of a word 'thermal resistors', The resistors

depending on temperature are thermal resistors. Thus resistance thermometers are also

thermistors having positive -temperature coefficients. But generally the resistors having

negative temperature coef f icients (NTC) are called thermistors. The resistance of a

thermistor decreases as temperature increases. The NTC of thermistors can be as large as

f ew percent per degree celcius change in temperature. Thus the thermistors are very

sensitive and can detect very small changes in temperature too.

Thermistors are composed of a sintered mixture of metallic oxides, such as manganese,

nickel, cobalt, copper, iron, and uranium. Their resistances at ambient temperature may

range from 100 n to 100 ill. Thermistors are available in a wide variety of shapes and

sizes as shown in the Fig. 8.29.

Smallest in size are the beads with a diameter of 0.15 mm to 1.25 mm. Beads may be

sealed in the tips of solid glass rods to form probes. Disks and washers are made by

pressing thermistor materia~ under high pressure into Hat cylindrical shapes. Washers can

be placed in series or in parallel to increase power dissipation rating.

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Leads ,-Glass coated

:1 (:) bead

Lea~

(e) Washer type

Fig. 8.29 Different forms of construction of thermistors

Thermistors are well suited for precision temperature measurement, temperature

control, and temperature compensation, because of their. ver y large change in resistance

with temperature. They are widely used for measurements in the temperature range

-1000 C to +2000 C. The measurement of the change in resistance with temperature is

carried out with a Wheatstone bridge.

8.12.2 Resistance Temperature Characteristics

The m<lthematical relationship according to which the resistance of thermistor behaves

as temperature is given by,

Rn =RTl =RT2 =

0

RT2

e[~( ~1-;2)]

Resistance at T1 0K

Resistance at T2 0K

Constant depending on thermistor material ranging

between 3500 to 4500 0K.

-100 100 200 300

Fig. 8.30 Resistance temperature characteristics of a

thermistor

Temperature

°C

The equation consists of an

exponential term and shows

that it is highly nonlinear in

nature. It has high negative

temperature coefficient

characteristics. The resistance

temperature characteristics of a

thermistor is shown in the

Fig. 8.30.

Resistance

in n

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Though the characteristics is highly nonlinear, for small range of temperatures, canb t :

assumed to be linear.

Negative

resistance

region

The voltage current

characteristics of thermistor are

shown in the Fig. 8.31.

It can be seen that as current

increases, the voltage across

thermistor increases, attains a peak

value and then decreases, when i t

decreases, the negative resistance

region starts.

Voltage

in VD O C

(

When small voltage is applied tothermistor, small current flows. This

does not produce heat so as to

change resistance of thermistor.

Under this condition, it follows Ohm's law and V and I are proportional. But large

currents produce large heat. This increases temperature to such a value where resistance of

thermistor decreases and draws more current. The current continues to increase till heat

dissipation of thermistor equals the power supplied to it. This is called self heat

characteristics of thermistor. This makes it suitable to measure flow, pressure, liquid level

etc. If rate of heat removal is fixed then thermistor is sensitive to power input and used

for voltage or power level control.

Current

in A

8.12.4 Current Time Characteristics

At low voltages, the thermistor takes long time to reach peak current. As voltage level

increases, the time to reach peak current decreases. These characteristics are called current

time characteristics and are shown in the Fig. 8.32.

Current

in mA

12345678

Fig. 8.32 Current time characteristics

Time

in sec

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Hence the transducers are made with two outputs. Out of these two outputs, one

corresponds to the inductance (either self or mutual inductance), while other corresponds

to the chnnge in inductance (either l:.L or Mv1). Then the subsequent stages measure the

diff erence between two outputs which is commonly known as differential output. The

tr'1llsducers based on this principle are called differential output transducers.

A typical differential output transducer consists a coil which is divided into two parts.

Thl' output terminals "re taken out from each part of the coil. There are two basic types 01

thl' dif f erentIal output transducer namely, linear differential output transducer and

angular differential output transducer as shown in the Fig. 8.38 (a) and (b) respectively.

Fig. 8.38 Differential output transducer

In case of a diff erential output transducer based on the variation in self inductance,

only one coil is used as shown in the Fig. 8.38 (a) and (b). While f or a diff erential output

transducer based on the variation in mutual inductance, multiple coils are pref erred.

When the displacement is applied to the transducer, the wiper moves and the

inductance in one part becomes (L + l:.L) while in other part becomes (L - l:.L). The change

is nw"sured by tak ing the dif f erence between the two. So the diff erential output measure

corresponds to 2 l:.L change in inductance. As compared with inductive transducer based on

chnnge in inductance, the output corresponds to l:.L change ill inductance. Thus accuracy

and sensitivity increases while errors are eliminated in case of the differential output

transd LiceI'.

There is another type of dif f erential

output inductive transducer based on

variable reluctance as shown in the

Fig. 8.39.

In such transducers, the dif f erential

output corresponds to change ill the

self inductance due to change in the

reluctance. In transducer, all the cores

are iron cores.

Fig. 8.39 Differential output inductivetransducer based on variable reluctance

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reduce eddy current losses. The displacement to be measured is applied to an arm

attached to the core. With the core in the center, or ref erence, position, the induced emf s in

the secondaries are equal, and since they oppose each other, the output voltage will be

zero volt.

\ .t\

"P E

"

j

" i<--

" -

"'\

-f--T-.

Corew

A.C. input

~

Core Arm ~ _

Displacement .•...• ~ CORE I

~FFig. 8.40 Linear variable differential transformer

When an externally applied f orce moves the core to the lef t-hand position, more

magnetic flux links the left-hand coil than the right-hand coil. The emf induced in the

left-hand coil, ES], is theref ore larger than the induced emf of the right-hand [oil, Es2' The

magnitude of the output voltage is then equal to the diff erence between the two secondary

voltages and it is in phase with the voltage of the left-hand coil.

Outputvoltage

= 4 @ JCoreatA

I~I @ ¥ FCoreat B

Core ato Null position

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