unit iii force, magnetic & heading sensors · strain gauge strain is the change in dimensions...

UNIT III FORCE, MAGNETIC & HEADING SENSORS

SENSORS AND TRANSDUCERS UNIT III

Strain Gauge

Strain is the change in dimensions of a solid body when subjected to a force or stress.

Stress is the force applied to a material per unit area.

So when stress is applied to a solid body, it gets strained leading to changes in

dimensions. Hence stress is the cause and strain is the effect.

Principle of Strain Gauge:

Whenever a metal conductor is stretched or compressed, its resistance changes since both its

length and diameter change. There is also a change in resistivity of the conductor when

strained and this property is called piezoresistive effect. Hence the resistance strain gauges

are also called as piezoresistive gauges.

Theory of strain gauge / Derivation of Gauge Factor for Strain Gauge:

Consider a strain gauge made of a circular wire as shown in fig.6(a).

Fig.6. Circular wire (a) Before application of tensile force (b) After application of tensile

force

The dimensions of the wire before being strained are:

L length of the wire

A area of cross section of wire

D diameter of the wire

ρ resistivity of the wire

R resistance of the wire

Hence, resistance of unstrained gauge is:

𝑅 =𝜌𝐿

𝐴

When a tensile force or stress is applied to the wire, the length of the wire increases and the

diameter decreases as shown in fig.6(b).

Let,

ΔL be the change in length of the strained wire

ΔA be the change in area of cross section of strained wire

ΔD be the change in diameter of the strained wire

ΔR be the change in resistance of the strained wire

To determine the dependency of change in resistance (ΔR) on the applied stress (s), we

differentiate the expression of R w.r.to. s.

∴𝑑𝑅

𝑑𝑠=

𝑑

𝑑𝑠(

𝜌𝐿

𝐴)

(a)

F F

Tensile Force

L

D

ΔL

𝐷 − 𝛥𝐷 F F

L

(b)



=𝜌

𝐴

𝜕𝐿

𝜕𝑠−

𝜌𝐿

𝐴2

𝜕𝐴

𝜕𝑠+

𝐿

𝐴

𝜕𝜌

𝜕𝑠

Dividing the above equation on both sides by 𝑅 =𝜌𝐿

𝐴, we get,

1

𝑅

𝑑𝑅

𝑑𝑠=

1

𝐿

𝜕𝐿

𝜕𝑠−

1

𝐴

𝜕𝐴

𝜕𝑠+

1

𝜌

𝜕𝜌

𝜕𝑠 (1)

From the above equation, it is clear that, the per unit change in resistance (ΔR/R) is dependent

on the per unit change in length (ΔL/L), per unit change in area (ΔA/A) and per unit change in

resistivity (Δρ/ρ).

The area of circular wire is given by:

𝐴 = 𝜋𝑟2 = 𝜋 (𝐷

2)

2

=𝜋𝐷2

4

∴𝜕𝐴

𝜕𝑠=

𝜋

4∗ 2𝐷 ∗

𝜕𝐷

𝜕𝑠

=𝜋

2𝐷

𝜕𝐷

𝜕𝑠

⇒1

𝐴

𝜕𝐴

𝜕𝑠=

4

𝜋𝐷2∗

𝜋

2𝐷

𝜕𝐷

𝜕𝑠

∴1

𝐴

𝜕𝐴

𝜕𝑠=

2

𝐷

𝜕𝐷

𝜕𝑠 (2)

The Poisson’s ratio is given by:

𝜈 =𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛

𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛= −

𝜕𝐷 𝐷⁄

𝜕𝐿 𝐿⁄

∴𝜕𝐷

𝐷= −𝜈

𝜕𝐿

𝐿 (3)

Substituting (3) in (2), we get,

1

𝐴

𝜕𝐴

𝜕𝑠=

2

𝐷

𝜕𝐷

𝜕𝑠= −

2𝜈

𝐿

𝜕𝐿

𝜕𝑠 (4)

Substituting (4) in (1), we get,

1

𝑅

𝑑𝑅

𝑑𝑠=

1

𝐿

𝜕𝐿

𝜕𝑠−

1

𝐴

𝜕𝐴

𝜕𝑠+

1

𝜌

𝜕𝜌

𝜕𝑠

=1

𝐿

𝜕𝐿

𝜕𝑠+

2𝜈

𝐿

𝜕𝐿

𝜕𝑠+

1

𝜌

𝜕𝜌

𝜕𝑠

For small variations in parameters, the above equation can be written as:

Δ𝑅

R=

Δ𝐿

L+ 2𝜈

Δ𝐿

L+

Δ𝜌

ρ

Gauge Factor of Strain Gauge:

The gauge factor is defined as the ratio of per unit change in resistance to the per unit

change in length.

𝐺𝑎𝑢𝑔𝑒 𝐹𝑎𝑐𝑡𝑜𝑟, 𝐺𝑓 =Δ𝑅 R⁄

Δ𝐿 L⁄

Consider the equation,

Δ𝑅

R=

Δ𝐿

L+ 2𝜈

Δ𝐿

L+

Δ𝜌

ρ



Dividing the termΔ𝐿

L on both sides, we get,

Δ𝑅 R⁄

Δ𝐿 L⁄= 1 + 2𝜈 +

Δ𝜌 ρ⁄

Δ𝐿 L⁄

∴ 𝐺𝑓 = 1 + 2𝜈 +Δ𝜌 ρ⁄

Δ𝐿 L⁄

If the change in resistance due to change of resistivity is neglected, then,

𝐺𝑓 = 1 + 2𝜈

The gauge factor for various materials is listed in Table.1.

Material Gauge Factor

Nickel

Manganin

Nichrome

Constantan

Soft Iron

Platinum

Carbon

Doped crystals

– 12.1

0.47

2

2.1

4.2

4.8

20

100 – 5000

Table.1. Gauge Factor of various materials

Types of Strain Gauges:

The major types of strain gauges are:

(a) Unbonded metal wire strain gauge

(b) Bonded metal wire strain gauge

(c) Bonded metal foil strain gauge

(d) Vacuum deposited thin metal film strain gauge

(e) Sputter deposited thin metal film strain gauge

(f) Bonded semiconductor strain gauge and

(g) Diffused metal strain gauge

Unbonded metal wire Strain Gauge:

An Unbonded metal wire strain gauge is shown in fig.7.



Fig.7. Unbonded metal wire strain gauge

This gauge consists of a wire stretched between two points in an insulating medium

such as air.

The wires may be made of copper nickel, chrome nickel or nickel iron alloys.

The wires are about 0.003 mm in diameter and have a gauge factor of 2 to 4 and can

sustain a force of 2mN. The length of wire is 25mm or lesser.

The force to be measured is applied to the diaphragm, which in turn conveys it to the

spring element via a force rod.

When the spring element is subjected to the force to be measured, two strain gauge

windings experience compression whereas the other two experience elongation. As a

result, resistance of two wires increases whereas that of the other two decreases.

These four wires are made part of the Wheatstone bridge which gets unbalanced on

application of the force to be measured.

R4

R3

R2

R1

ei

e0

Mounting

Rings

Spring Element

Strain Gauge

Windings

Force Rod

Diaphragm



The electrical resistance of each arm is 120Ω to 1000Ω, the input voltage to the

bridge is 5 to 10 V, and the full scale output of the bridge is typically about 20mV to

50mV.

Bonded metal wire Strain Gauge:

Bonded wire strain gauges are shown in fig.8.

Fig.8. Bonded wire strain gauges

A resistance wire strain gauge consists of a grid of fine resistance wire of about 0.025

mm in diameter or less.

The grid is cemented to carrier (base) which may be a thin sheet of Bakelite or a sheet

of Teflon.

The wire is covered on top with a thin sheet of material so as to prevent it from any

mechanical damage.

The spreading of wire permits a uniform distribution of stress over the grid.

The carrier is bonded with an adhesive material to the specimen under study, which

permits a good transfer of strain from carrier to grid of wires.

The wires cannot buckle as they are embedded in a matrix of cement and hence

faithfully follow both the tensile and compressive strains of the specimen.

The nominal values of resistance for these gauges range from 40Ω to 2000 Ω, but 120

Ω, 350Ω and 1000 Ω are common values.

For excellent and reproducible results, the resistance wire strain gauges should have

the following characteristics:

They should have a high value of gauge factor. If gauge factor is high, the

change in resistance corresponding to the input strain will be high. Hence

sensitivity would be more.

The resistance of the strain gauge should be as high as possible.

The strain gauges should have a low temperature coefficient, to ensure that

errors due to temperature variations are minimal.

The strain gauge should be free from hysteresis problem.

The variation in resistance should be linear function of strain to ensure

constancy of calibration.

(a)Linear Strain

Gauge

Carrier

(Base)

Wired

Grid

Termina

ls

(b)

Rosette

Carrier

(Base)

Wired

Grid Termina

ls

(c) Torque

Gauge

(d) Helical

Gauge



Their frequency response should be good, as they are frequently employed for

dynamic measurements. The linearity should be maintained over the entire

frequency range of operation.

Some materials used for construction of resistance wire strain gauges are listed in

Table.2.

Material Composition Gauge

Factor

Resistivity

(Ωm)

Resistance

temperature

coefficient

(/°C)

Upper

temperature

(°C)

Nichrome

Ni – 80%

Cr – 20% 2.5 100*10–8 0.1*10–3 1200

Constantan

Ni – 45%

Cr – 50% 2.1

48*10–8 ±0.02*10–3 400

Isoelastic

Ni – 36%

Cr – 8%

Mo – 0.5%

etc.

3.6 105*10–8

0.175*10–3 1200

Nickel – –12 6.5*10–8

6.8*10–3 –

Platinum – 4.8 10*10–8

4*10–3 –

Table.2. Materials for strain Gauges

The base or carrier materials used to support the wires are selected based on the

operating temperature. For room temperature applications, Impregnated paper is used.

Epoxy is used for a temperature range of –200°C to 150°C. Bakelite cellulose or fibre

glass materials for a range up to 200°C in case of continuous operation and up to

300°C in case of limited operation.

Adhesives act as bonding material between the surface to be strained and the plastic

backing material on which the gauge is mounted. Some common adhesive materials

are Ethylcellulose cement, nitrocellulose cement and epoxy cement. These adhesives

can be used up to a temperature range of 175°C. The main features expected out of

the adhesive material are that it should be quick drying type and it should be

insensitive to moisture to ensure good adherence.

The leads used should be of materials which have low and stable resistivity and also a

low resistance temperature coefficient.

Bonded metal foil Strain Gauge:

Bonded metal foil gauges are just an extension of the bonded metal wire gauges.

Foil type gauge use the same material as that of the wire gauges.

The foil type gauges have a much greater heat dissipation capacity as compared with

wire wound strain gauges because of their increased surface area for the same volume.

Hence they can be used for still higher temperature range applications. The bonding is

also improved because of larger surface area.



Some common metal foil strain gauges are shown in fig.9.

Fig.9. Metal Foil Strain Gauges

Foil type gauges are mounted on a flexible insulating carrier film about 0.025mm

thick made of polymide, glass phenolic etc.

Typical values of the foil type gauge’s resistance are120 Ω, 350Ω and 1000 Ω, with

the allowable gauge current of 5 to 40 mA. Gauge factors range between 2 and 4.

The manufacturing process of foil type gauges provides convenient soldering tabs,

which are integrated with the sensing grid.

Evaporation deposited (Vacuum deposited) thin metal film Strain Gauge:

The evaporation process forms all the strain gauge elements directly on the strain

surface; they are not separately attached as in the case of bonded strain gauges.

In the evaporation process, the diaphragm is placed in a vacuum chamber with some

Insulating material. Heat is applied until the insulating material vaporizes and then

condenses, forming a thin dielectric filmonthe diaphragm. Suitably shaped templates

are placed over the diaphragm, and the evaporation and condensation processes are

repeated with the metallicgauge material, forming the desired strain gauge pattern on

top of the insulating substrate.

Resistance and gauge factors of film gauges are identical to those of foil gauges.

Since no organic cementing materials are used, thin film gauges exhibit better time

and temperature stability.

Sputter deposited thin metal film Strain Gauge:

Thesputtering process forms all the strain gauge elements directly on the strain

surface; they are not separately attached as in the case of bonded strain gauges.

In the sputtering process, a thin dielectric layer is deposited in vacuum over the entire

diaphragm surface. The detailed mechanism of deposition is, however,

entirelydifferent from the evaporation method.The complete layer of metallic gauge is

sputtered on the top of the dielectricmaterial withoutusing any substrate. The

diaphragms are now removed from the vacuum chamber, and microimaging

techniques using photo masking materials are used to form the gauge pattern. The

diaphragms are then returnedto the vacuum chamber. Sputter etching techniques are

used to remove all unmasked metal layer, leaving behind the desired gauge pattern.

Resistance and gauge factors of film gauges are identical to those of foil gauges.

Since no organic cementing materials are used, thin film gauges exhibit better time

and temperature stability.



Semiconductor Strain Gauge:

It is always desirable to have high sensitivity of strain gauges, which can be achieved

easily if the gauge factor is high.

Semiconductor strain gauges have a high value of gauge factor about 130.

Semiconductor strain gauges depend on their operation completely on the

piezoresistive effect, which is the change in resistivity of the semiconductors when

strained.

Metals depend on change in length and area with strain whereas the semiconductors

depend on change in resistivity with strain.

A typical semiconductor strain gauge consists of silicon or germanium as the resistive

material.

A strain sensitive crystal material with leads sandwiched in a protective matrix is

more common and is shown in fig.10.

Fig.10. Semiconductor Strain Gauge

Semiconductor wafers or filaments of thickness 0.05mm are bonded on to suitable

insulating substrates such as Teflon.

Gold leads are generally employed for making the contacts.

Some advantages of semiconductor strain gauges are:

They have high gauge factor of about ±130. Hence small strains of order of

0.01microstrain can be easily measured. (improvement in sensitivity)

Hysteresis characteristics are excellent.

Fatigue life is excellent of the order of 10*106 operations and the frequency

response is up to 1012Hz.

They are very compact, and can be made in sizes of 0.7 to 7 mm, hence are

very useful in measuring local strains.

Some drawbacks of semiconductor strain gauges are:

o They are very sensitive to temperature variations.

o Linearity is very poor. The equation of the fractional change in resistance is

given by∆𝑅 𝑅⁄ = 𝐴𝑠 +

𝐵𝜀−4, 𝑤ℎ𝑒𝑟𝑒 𝐴 & 𝐵 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝑎𝑛𝑑 𝜀 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑟𝑎𝑖𝑛. The gauge factor

varies with strain. The gauge can be made linear by proper doping.

o They are more expensive and difficult to attach to the object under study.

Diffused Strain Gauges:

Terminals

Base

Semiconductor



These gauges use the diffusion process used in IC manufacture for constructing the

strain gauge.

A silicon diaphragm would be used and impurities would be deposited on the silicon

diaphragm to realize it as a strain gauge.

The diffused strain gauge result in low manufacturing costs as large number of

diaphragms can be made on a single silicon wafer.

Rosettes:

Rosettes are a combination of strain gauges, designed to overcome the difficulty of

measuring the strain resulting from the problem of locating the direction of principle

stress.

In practical situations, an element may be subjected to stress in any direction and it is

not possible to locate the direction of principle stress. Hence Rosettes were designed

which could measure the value of principle stress without actually knowing their

directions.

Some common rosettes are shown in fig.11.

Fig.11. Some forms of rosettes

Applications of Strain Gauge:

Some applications of strain gauge are:

Measurement of strain

Measurement of stress

Used as secondary transducer with load cell, torque meters, diaphragm type pressure

gauges, temperature sensors, accelerometers and flow meters.

Hall Effect Transducer

Principle of operation:

Hall Effect transducers are based on Hall Effect. The Hall Effect is stated as “When a

magnetic field is applied to a current carrying conductor at right angles to the direction of

current, a transverse electric potential gradient is developed in the conductor.”

Working:

Consider the Hall Effect element shown in fig.7.

3 – Element Rosette

45° stacked (wire type)


90° stacked (foil type)


60° planar (foil type)



Fig.7. Hall Effect Element

The working of Hall Effect transducer can be explained as follows:

Consider that current is passed through leads 1 and 2 of the strip.

The output leads connected to edges 3 and 4 are at the same potential when no

transverse magnetic field passes through the strip.

When a transverse magnetic field passes through the strip, an output voltage appears

across the output leads.

The magnitude of this voltage depends upon the current, the strength of magnetic

field, and the property of the conductor known as Hall Effect.

The output voltage is given by:

𝐸𝐻 =𝐾𝐻𝐼𝐵

𝑡

Where KH is the Hall Effect coefficient in V – m /A – Wb m–2

t is the thickness of the strip

I is the current in Amperes and

B is the flux density in Wb/m2

The Hall Effect emf (EH) is very small in conductors and is generally difficult to

measure.

In semiconductors, the Hall Effect emf is quite large and can be measured easily by

PMMC meter.

The Hall Effect coefficient value for various materials is listed in Table.1.

Material Field Strength

(Wb/m2) Temperature (°C)

Hall Effect

Coefficient (KH)

As 0.4 to 0.8 20 4.52 ∗ 10−9

C 0.4 to 1.8 Room −11.73 ∗ 10−9

Bi 0.113 20 −1 ∗ 10−6

Cu 0.8 to 2.2 20 −52 ∗ 10−12

Fe 1.7 22 1.1 ∗ 10−9

Ge 0.001 to 0.8 25 −8 ∗ 10−3

Si 2 23 4.1 ∗ 10−6

Sn 0.4 Room −2 ∗ 10−12

Hall

Strip

E

H

4 3

1 t

Transver

se

magnetic

field

I A

2



Te 0.3 to 0.9 20 53 ∗ 10−6

Table.1. Hall Effect coefficient for various materials

Applications:

The applications of Hall Effect transducer are:

Magnetic to electric transducer

Measurement of displacement

Measurement of current

Measurement of power

Magnetic to electrical transducer:

A semiconductor plate is inserted into the magnetic field whose flux density is to be

measured. Depending on the value of flux density, the Hall Effect voltage KH is produced

since𝐸𝐻 =𝐾𝐻𝐼𝐵

𝑡. Hence the value of voltage produced is directly proportional to the value of

flux density. Hence this transducer can be used as magnetic to electric transducer.

Some advantages of this transducer are:

Requires very small space in the direction of magnetic field

A continuous electrical signal is obtained in response to the magnetic field strength

value.

Some drawbacks of this transducer are:

o Highly sensitive to temperature variations

o Since the hall coefficient value changes from plate to plate, recalibration is to be done.

Measurement of displacement:

The setup for measurement of displacement is shown in fig.8.

Fig.8. Setup for measuring displacement using Hall Effect transducer

The working of this transducer can be explained as follows:

A ferromagnetic structure having a permanent magnet is shown in fig.8.

The Hall Effect transducer is located in the air gap adjacent to the permanent magnet.

The field strength produced by the permanent magnet in the gap where the Hall Effect

element is placed varies with the position of the ferromagnetic plate.

E

H

Perma

nent

magne

t

Struct

ural

mem

ber

Hall

Effect

Eleme

nt

Ferromag

netic

Plate

Displace

ment



Hence with displacement, the field strength varies, which in turn causes a change in

the hall voltage (EH) produced.

This method can be used to measure displacement as small as 0.025 mm.

Measurement of current:

The circuit for measurement of current using Hall Effect transducer is shown in fig.9.

Fig.9. Measurement of current using Hall Effect transducer

The working of this transducer can be explained as follows:

The current to be measured is passed through the conductor.

This current sets up a magnetic field surrounding the conductor.

A Hall Effect element is placed in a slotted ferromagnetic tube that acts a magnetic

concentrator.

The voltage produced at the output terminals is proportional to the field strength,

which in turn is proportional to the current that passed through the conductor. Hence

the voltage produced is proportional to the current to be measured.

This technique can be used for measuring current less than one mA range to

thousands of amperes.

The advantages of this technique of current measurement are:

This is a contactless measurement technique, where the current is measured

without making any electrical connection between the conductor circuit and

the meter.

The circuit need not be interrupted for carrying out measurement.

Measurement of power:

The Hall Effect transducer used for measurement of power is called Hall Effect multiplier

and is shown in fig.10.

Fig.10. Hall Effect multiplier for measurement of power

The technique of measuring power using Hall Effect multiplier is described below:



The current is passed through the current coil which produces a magnetic field

proportional to the current ‘i’. This field is perpendicular to the Hall Effect element.

A current ‘ip’ proportional the voltage is passed through the Hall Effect element in a

direction perpendicular to the field. This current is limited by the multiplier resistance

Rs.

The output voltage of the hall effect multiplier is given by:

𝐸𝐻 =𝐾𝐻𝑖𝑝𝐵

𝑡

Here 𝐵 ∝ 𝑖 𝑎𝑛𝑑 𝑖𝑝 = (𝑣 𝑅𝑠⁄ ) ∝ 𝑣, hence𝐸𝐻 ∝ 𝑣𝑖 ∝ 𝑝

Hence the hall voltage is proportional to the instantaneous power.

unit iii force, magnetic & heading sensors · strain gauge strain is the change in dimensions...

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