Chatper 4 Analog Sensors and Transducers

Sensor – A device that measures or detects a real-world condition, such

as motion, heat or light and convert the condition into an

analog or digital representation

– Thermocouples, photodiodes, …

– All of the electrical energy at the output is derived from

physical input (self-generating, passive)

Displacement VoltageResistance

Electrical excitation

Transducer – A device for converting energy from one form to another

for the purpose of measurement of a physical quantity

Displacement

Acceleration

Force

Pressure

Temperature

Flow

Voltage

Current

Frequency

– Potentiometers, resistance strain gauges, semiconductor Temp

Resistance

Capacitance

Inductance

– Has a physical input, an electrical output, and an electrical

excitation modulated by the physical input (active)

1

Motion transducers are often used to measure the following physical

variables:

• Displacement (including position, distance, proximity, size or gage)

• Velocity (rate of change of displacement)

•Acceleration (rate of change of velocity, such as vibration test)

• Jerk (rate of change of acceleration, earth-quake induced structural • Jerk (rate of change of acceleration, earth-quake induced structural

damage detect, high-speed elevators, wind hazard) 5000M/S3

2

Resistive Displacement Measurement -- Potentiometer

• a displacement transducer of both rectilinear and rotatory type

(carbon, platinum, or conductive plastic)

(mounted at a fixed position)

(attached to a movable object

Device

terminals

1

2

3 (attached to a movable object

for displacement measurement x )

3

• works as a voltage divider circuit giving

= ref

lox

= kx

vv

• device output at terminal 2

• reference voltage connected to terminals 1

and 3

lx

3

Performance considerations on the use of potentiometers

• use a regulated reference voltage source to maintain a stable ov

• choose a device with a reasonably high resistance to reduce

thermal effect

• linearity of device can be affected by the load impedance (of the

measuring circuit)

Rc

a

Apply Kirchoff’s current law at node

xl

a

RL

Displacement x / l

4

Neglect loading effect

Rotary Potentiometer

Wlpar

Shalt

Sl'lart seal and bearlng1

TABLE 2. 1 Comparisons of Typical Potentiometer Elements

Potentiometer Resistances Resistance element uvnilable tolerance Linearity

Wire wound 50 ohms to 5% to 1% 1% to 0.1% 250K ohms

Cermet 250 ohms to &% to 1% o. 5% to 0.15% 1 megohm

Conductive 250 ohms to 10% to 2% 0. 5% to 0. 05% plnstic 250K ohms

Hybrid (plastic 500 ohms to 10\ to 5% 0.5% to 0.1\ plus wire) 30K ohms

Resistive Element

Supply

vref

Resolution

o. 25 to o. o:l% (best at higher resist-ances)

Essentially infinite

Essentially infinite

Essentialty infinite

Temperaturo Rotational coefficient life

±20 ppm/°C 2,000 . 000 revolutions

±100 ppm/°C 10,000,000 revolutions

±.300 ppm°C 25,000,000 revolutions

±70 to :!:150 10,000,000 ppm/°C revolutions

5

Motor Closed Loop Control Using a Potentiometer

Advantages Disadvantages

Contactless Potentiometer:

Megatron MP1618 - analog, MAB25A – 10/12 bit resolution

Advantages Disadvantages

-- inexpensive

-- easy to use

-- no amplification required

other than buffering

-- mechanical & electrical loading

-- wear

-- electrical noise

-- self heating

-- Stable Vref required

-- Limited to low frequency

applications

HTM File

6

Closed Loop DC Motor Control Using a Potentiometer

(for Lab #1)

+

-

Differential

Amplifier

Error

amplifier

G = 20

Power

amplifier

G = 1

DC

motor

Rotary

potentiometer

+ 5V

- 5V

Position

command

input

- 5V

7

Displacement Measurement -- Variable-Inductance Transducers

• Based on the principles of electromagnetic induction

• Three broad types:

1. Mutual-induction transducer ( transformer )

2. Self-induction transducers ( inductor )

3. Permanent-magnet transducers for velocity sensing ( motor ) 3. Permanent-magnet transducers for velocity sensing ( motor )

8

• Non contact linear displacement sensor

1. Linear Voltage Differential Transformer (LVDT)

refV

• Consists of a primary coil driven by an

ac carrier signal (several to tens of KHz)

• Secondary coils are wired out of phase wrt

oV +-

• Carrier signal is inductively coupled to 2

symmetrical secondary coils using a

ferromagnetic core Core

displacement• Secondary coils are wired out of phase wrt

each other

HTM File

• Coil on left gets induced a higher voltage level

than that on the right when core is displaced

to left from null position, and vice versa

• Magnetic core is attached to movable object

displacement

X

• Both secondary coils get induced the same

level of voltage when the core is centered

between them ( X = 0, null position )

9

Synchronous demodulation output

]

• . , .......

Primary excitation Secondary 1 Secondary 2

Secondary 1 • Secondary 2

•

Primary excitation Secondary 1 Secondary 2

Secondary 1 • Secondary 2

10

l

11

12

DC output proportional to displacement (cannot resolve direction)

+ DC output proportional to displacement and direction+ DC output proportional to displacement and direction

Phase compensation due to

non ideal characteristics of device

13

• Linear position sensor with a high level of linearity ( typically < 0.5%

over full stroke)

• Typically stainless steel in construction and environmentally sealed

• Can be designed with a long stroke or can be sensitive to very small

position changes ( + 0.5mm to + 550mm)

• Resolution only limited by conditioning circuitry

• No sliding contacts to corrode or wear; low output impedance (~100 )Ω

• Used in harsh industrial, aerospace or military environments where

reliability and performance are crucial

• Typically stainless steel in construction and environmentally sealed

submersible sealing available

• DC- LVDT -- signal conditioning circuitry

built into the device package

(reduced operating temperature range)

HTM File

• As a rule, the excitation frequency should be chosen to be at (in radian)

Maximum speed of operation

Stroke of LVDT10 X

14

Rotary Voltage Differential Transformer

• For measuring rotational movement

• Similar to LVDT in operational

concept except cylindrical core is

replaced by a rotating cam attached to

shaft

• Differential output voltage proportional

to shaft rotation angle

HTM File

• Linearity limited to + 40o

-- severe limitation!

15

• Based on mutual induction principle

Resolver

• Can detect object rotation up to 360o

• Consists of 2 sets of orthogonal pickup

coils affixed to the stator

ref av = v sin ωt

• The coil wounded on the rotor is to

be energized by an ac signal

; o1 ref o2 ref

sinv = av cos θ v = av θ

• Signals picked up by the two stator

coils are dependent on the displacement

angle of the rotor:

16

;o1 a

v = av cos θ sin ωt o2 a

v = av sin θ sin ωt

refv

θWant to derive a single output signal relating to !

Multiply each quadrature signal by to get

1

X

X

o1v

o2v

ref av = v sin ωt

m1v

m2v

LPF

LPF

f1v

f2v

1-tan θ

V

( )2 2 211 cos2 ,

2a a-

m1 o1 refv = v v = av θ sin ωt = av θ ωt cos cos

( )2 2 211 cos2 ,

2a a-

m2 o2 refv = v v = av θ sin ωt = av θ ωt sin sin

10ω

ω θ

21,

2 af1v = av θ cos

21

2 af2v = av θ sin

Choose a carrier frequency to be 10 times the rate of change of

and low pass filter the above modulated signals with a cutoff frequency

of , we get

Thus 1- f2

f1

vθ = tan

v

17

Using the same resolver, we can alternatively drive the two sets of

stator coil with a quadrature sine wave (as a function of time) :

rv

The phase shift of wrt equals to the angular position of

the rotor itself. 1

vr

vθ

18

Primary advantages of the resolver:

• Fine resolution and high accuracy

• Low output impedance (high signal level)

• Small size (e.g., 10 mm in diameter)

• Direct availability of the sine and cosine functions of the measured

anglesangles

Limitations:

• Nonlinear output signals

• Bandwidth limited by supply frequency

• Slip-rings and brushes would be needed if complete and multiple

rotations have to be measured

19

Synchro Transfomer

• Consists of 2 identical rotor-stator pairs

(Tranmsitter and Receiver)

• Each stator consists of 3 coils separated

by 120o in space

• Transmitter rotor driven by either

50/60 Hz for industrial or 400Hz for military or aeronautical use

• Each coil in the transmitter picks up a signal whose amplitude dependent

• The 3 signals are used to drive their 3 corresponding stator coils in the

receiver

• Each coil in the transmitter picks up a signal whose amplitude dependent

on the angular position of the rotor t

θ

• If the rotor at the receiver assumes an angular position of , then

its output becomes ( ) - t ro ref

cosv = av θ θr

θ

= a if - = 0 o

o ref t rq q v v

• At the receiver end, control electronics can be applied to rotate the

rotor to the desired angle wrt to e.g. tθ = 0 if - = 90 o

o t rq q v

20

• Flexible, eliminates mechanical linkage between control unit and target

• Controlling unit be a long distance from the target unit

• Provides reliable, continuous and accurate control over the target unit

• Small in size and uses very little electrical power for its operation

• Used in position control for satellite dish

and military

gun order signal gun mount

The application of a synchro

control transformer to a gun

mount power drive

gun order signal gun mount

• Disadvantages ?

21

• Two secondary coils wired in

series and in phase (additive,

2 end limbs of E core)

• Magnetic flux developed in E core

dependent on distance (magnetic

reluctance) of separation

Mutual Inductance Proximity Sensor

• Carrier signal connected to primary

winding (middle limb of E core)

reluctance) of separation

X of the ferromagnetic target object

• non-contact, negligible mechanical

loading

• Change of magnetic flux causes changes

in output

• is a nonlinear function beyond a

short displacement X, say 5 mm

ov

ov

• Measure transverse displacement only

22

Detecting Broken Drill Bit

Detecting Set Screws on Hub for Speed or Direction Control

Detecting Presence of Can and lid

Detecting Full Open or Closed Valve Position

Detecting Broken Drill Bit on Milling Machine

23

• Inductance can be measured with

an AC bridge circuit

• Can also be connected to a tuned

circuit and check for resonance

2. Self- Inductance Transducer

• Change of displacement of object

alters the reluctance and therefore

inductance of the circuit~

AC

Supply

vrefInductance

Measuring

Circuit

Ferromagnetic circuit and check for resonance

c g o gc o

= = +

+

r o c o c r o c

N NL

ll l

A A Amm m mm

  Â

2 2

2++

• Assumption: cross section area of core, air gap, and object = cA

both core and object have same relative permeability rm

x(Measurand)

Target Object

24

Eddy Current Transducers

A circular flow of electronics (eddy current) will

be induced within the metal plate when it

• moves across a constant magnetic field

• is stationary and exposed to a time vary

magnetic field

Strength of eddy current increases with strength of magnetic field and

frequency of the magnetic flux

Metallic object presents as a load to the magnetic field sourceMetallic object presents as a load to the magnetic field source

The metallic object in the proximity will be

induced an eddy current, thereby loading

down the magnetic field from the coil and

changes its inductance

In an eddy current transducer probe, a coil is

is driven by a RF oscillator ( 1 to 100MHz) to

generate a radiating magnetic field

25

• Non contact

• Non ferrous

(Measurand)

x

The inductance of the active coil is compared to a second reference coil

(also located in the probe) with the use of a bridge circuit

Output

vo

Coaxial

Cable

Impedance Bridge

Demodulator

Low-Pass Filter

Radio Freq.

Converter

(Oscillator)

Calibrating

Unit

Target

Object

Conducting

Surface

Active

Coil

RF Signal

(100 MHz)

Compensating

Coil

• Non ferrous

• Output not linear with

displacement

• Sensitivity increases

with resistivity of object

• Calibration unit

generally required (linearized output

possible)

(Oscillator)

20 V DC

Supply• Requires a bridge and

demodulation circuit

RF

Generator

Bridge Output

(to Demodulator)

R1

L + ∆L

L

C

~

C

R1R2

R2

Compensating

Coil

Active

Coil

HTM File

26

3. Permanent-Magnet Transducers for Velocity Measurement

• For measurement of both rectilinear and angular velocities

• Can be either a DC or AC type

HTM FileHTM File

• Application of Faraday’s law for linear

velocity measurement

• DC output linearly proportional to velocity

• Normally comes in a stainless package

for shielding from stray magnetic fields

• An alternate implementation:

use a stationary magnet to generate

a constant magnetic field and

attach the coil to the moving object

• Sensitivity ~ 0.5V / inch per second

27

• DC tachometer – for angular

speed measurement

• Coil mounted on rotor which is

connected to object through the

rotating shaft

• Permanent magnet mounted inside

housing to generate a fixed magnetic

field

vo

Commutator Speed

Rotating

Coil

Permanent

Magnet

SN

• Coil voltage made available to outside

world with use of a brush and commutator

• Ripple voltage appears at the output

due to brush / commutator arrangement

• Sensitivity 5 ~ 10 V / 1000 RPM

• For a constant magnetic field of flux density

B, the output voltage of the tachometer is

Rotating

Coil

2r

h

28

AC Induction Tachometer

• 1 primary and 1 secondary stator

winding orthogonal to each other

• Shorted rotor coil coupled to rotary object

• AC carrier drives primary winding ref

=

sinωt a

V

V

• With a stationary rotor, = o

V 0

• With rotor spinning at , flux distribution is changed ωc

• No brush/commutator is needed

• Sensitivity up to 10V / 1000RPM

• Typical carrier frequency

60 and 400Hz > 10

29

Variable-Capacitance Displacement Transducers

The capacitance made up of 2 parallel plates is

kAC =

xx

AA – overlapping areax – distance of separationk – dielectric constant (permittivity) ε ε εr o=

voCapacitance

Bridge

vo

CapacitanceBridge

DCOutput

vo

CapacitanceBridge

kA KC = =

x x

MovingPlate(e.g.,

Diaphragm) Positionx

Fixed Plate

kwh + ε w l - hC =

xo ( )

k

Fixed Plate

LiquidLevelh

l

Rotating Plate

A

Fixed PlateRotation

θ

30

• Vref can be up to 25KHz forhigh-bandwidth measurement

ref refv Cv = x

Ko -

• An inverting amplifier can be used to linearize the output voltage for a capacitive transverse displacement sensor

Outputvo

SupplyVoltage

vref

+

−

Cref

C = K/x

+

−A

+

−

From magnitude: From phase:

ov - vd CV

i = dt R

ref o( )=

A simpler circuit to calculate x (slow changing)

31

• Sensor capacitance may change due to a change of it operating environment, such as humidity, temperature, pressure ..

22

ZjωC

= 1

11

ZjωC

= 1

ref av = v sinωt

Capacitance Bridge Compensation Circuit

−−−−

+

AC Excitation

CompensatorZ1

SensorZ2

Z3Z4

Bridge Outputv

ref o

1 2

v v v v + = ,

Z Z

- -0

ref

3 4

v v v + = ,

Z Z

- 0 -0

4 3 2 1o ref

4 3

Z Z Z Zv = v

Z Z

1 +

)( -

• when bridge is balance ov = 0 4 2

3 1

Z Z =

Z Z

o4 3

ref

1

Zv = Z Z

v

Z + d d

( )-

1• If then = + ZZ Z d2 2

Bridge Completion

32

Differential (Push-Pull) Capacitance Displacement Sensor

• A more convenient way of measuring displacement using the bridge

• when displacement = 0 = Z Z4 3

• If center plated is moved to the right by , then

1 x δx 1 x δx- +

xd

• Consists of 2 fixed plates and 1 movableplate centered in between

3 4

1 x δx 1 x δx ,

jωC jωK jωC jωKZ Z3 4

- += = = =

• We can connect and to the bridge circuit with as compensating elements

Z Z3 4 = Z Z1 2

4 3 2 1 4 3o ref o ref

4 3 4 3

Z Z Z Z Z Z

v = v v = v Z Z Z Z

1

1 + 1 + Þ

) )( - ( -

o ref o ref4 3

4 3

- x

+ xv = v v = v

Z Z

Z Zdé ù

ê ú Þê úê úë û

33

Capacitive Sensor Applications

• Probes come in different sizes and shapes fordifferent applications

• Measurement of thickness of an automobile brake pad

• Liquid nitrogen level measurement

• Measurement of chest wall movement

• Sensitivity in the order of 1pF /mm

34

A Crystal structure of a material is a unique arrangement of its molecules in a special way

Cubic crystal structure of the PZTlead zirconate-titanate molecule

Above the Curie temperature, the positively charged Ti/Zr ion shifts lead zirconate-titanate molecule

below the Curie temperaturepositively charged Ti/Zr ion shifts from its central location

Groups of molecular dipoles alignwithin small areas, or domains, to form large dipole moments after sintering

Domain dipoles align in the directionin line with the external electric fieldat elevated temperatures

35

• When a mechanical stress is applied (when squeezed or stretched), to a piezoelectric substance, it produces an electric charge

• A mechanical deformation (shrinking or expansion)

Piezoelectric characteristics exhibited by the treated PZT:

• Piezoelectric sensors make use of these effects: pressure andstrain measurement, touch screens sensing, accelerometers, ..

• Actuator applications : piezoelectric valves, micro linear motor

• A mechanical deformation (shrinking or expansion) will result when an electric field is applied

36

• High output impedance – several M at around 100Hz and decreaseswith frequency

W

= jωC

Z1

Piezoelectric Sensor Model

q∂Charge sensitivity --F

qSq ∂

∂=

With a fixed area A -- SA

q

pq = 1 ∂∂

Voltage sensitivity – change in voltage due to a unit increment in pressure (or stress) per unit thickness of crystal

Sd

v

pv = 1 ∂∂ vq kSS =

37

Piezoelectric Material Sensitivities

Material Charge Sensitivity Sq (pC/N)

Voltage SensitivitySv (mV.m/N)

Lead Zirconate Titanate (PZT)Barium Titanate

QuartzRochelle Salt

1101402.5275

1065090

38

Piezoelectric Accelerometers

• For measurement of acceleration,shock or vibration

• A mass when subject to an externalforce will experience an accelerationf = Ma a = f/M

• Many different accelerometer types Piezoelectric

• Sensing element is a piezoelectricSpring

• A spring is used to keep the sensingelement under compression at all times

• A metal enclosure provides protection to sensor

• When subject to a vibration, the charge developed q qq S F S Ma= =

Inertia Mass

• Sensing element is a piezoelectriccrystal bonded to a mass

Outputvo

Spring

PiezoelectricElement

Electrodes

Direction ofSensitivity(Input)

39

• Typical accelerometer sensitivities 10pC/g 5mV/g

• Lower frequency limited by signal conditional system, mountingmethod, charge leakage, SNR, etc

• Very high resonant frequencydue to high stiffness of spring andlow mass

Acc

eler

omet

er

Sig

nal (

dB)

Frequency (Hz)

5,000 20,000

Useful Range

1

Resonance

• Air bag deployment

• Predictive machinery maintenance

• Computer hard drive protection

Some applications

Frequency (Hz)

40

A charge amplified is used to :

• Convert the charge generated by the accelerometer to an voltage

• Eliminate load effect of the sensor by converting its high outputimpedance to a low output impedance for subsequent signal processing

Cf

−A

RfAssume open loop gain of opamp = K

0o o o o oc f o

f

v v v v v Kq C C C v

K K K R

+ + + + + + =

& & && &

Outputvo

−−−−vo/K+

−

+

−Charge

Amplifier

CcC

PiezoelectricSensor

Cable

q

K

f

For K in the order of 105 and larger

R Cdv

dtv R

dq

dtf fo

o f+ = −

( )( ) [ ]

v s

q s

R s

R C so f

f f

= −+1

( )( ) [ ]

v j

q j

R j

R C jo f

f f

ωω

ωω

= −+1

41

( )f

f f

R jωv jω =

jω R C jωo

q

( )-

( ) + 1• First order system with a time constant

c f fτ R C=

• A piezoelectric sensor cannot be used for measuring a constant (DC)signals

• Output = 0 when = 0 ω

• Output approaches to at very high frequencies fC1-

42

- Lε =

'L LL L

d=

Stress – the force acting on a unit area in a solid

s = FA

The electrical resistance of a conductor l

resistivitylengthcross section

ρ

lA

Strain Gages

Strain -- amount of deformation per unitlength of an object when a load isapplied.

Crack growth measurementon foundation

The electrical resistance of a conductor lR ρ

A= cross section

areaA

When a conductor is subject to stain, the deformation will change itselectrical properties stain gage

Direction of

Sensitivity

FoilGrid

BackingFilm

Solder Tabs(For Leads)

43

• For linear deformations, the strain applied

s S s s

R L = S ε = S

R Ld d gage factor or sensitivity

RA

= ρl

• Change in resistance in material = change in resistivity + change in shape

( )dR

R

d d A

A= +

ρρ

l

l

s S s s = S ε = SR L

gage factor or sensitivity

• 2 ~ 6 for most metallic and 40 ~ 200 for semiconductor strain gagess S

44

• Diverse measurement applications, can be used for measuring variablessuch as displacement, acceleration, pressure, temperature, liquid level, etc

• Can be used with a bimetal strip fortemperature measurement

Direction ofSensitivity

(Acceleration)

• Strain-gage accelerometer – inertia force on mass is measured with a strain gage at end of cantileverelement.

Outputvo

Strain Gage

Housing

SeismicMass

m

Strain Member

Cantilever

temperature measurement

Different foil-type gages

Base MountingThreads

45

1 ref 3 refo

1 2 3 4

1 4 2 3

R v R vv =

R + R R + R

R R R R = v

- ( ) ( )

( - ) R R

• Any one or more resistor may be astrain gage

• A Wheatstone bridge is often used tomeasure the change in resistance ofthe strain gage

1 4 2 3ref

1 2 3 4

R R R R = v

R + R R + R

( - ) ( )( )

R R =

R RÞ 1 3

2 4

= 0 when the bridge is balanced

• Current to load is 0 for all RL

o

ref

R Rv =

v R R

dd

dÞ non linear!

(4 + 2 )

o

ref

v R =

v R

d d4

• For small , R Rd

• Case 1: 2 3 4 1 R = R = R = R R = R + Rdand

o ref

R + R R Rv = v

R + R + R R + R

dd

d

é ùê úë û

2( ) -

( )( )

(

46

o

ref

v R =

v 2R

d dÞ linear!

• Case 2:

3 4 1

2

R = R = R R = R + R,

R = R R

d

d

and

-

o ref

R + R R R - R Rv = v

R + R + R - R R + R

d dd

d d

é ùê úë û( ) - ( )

( )( )

(

ref

o

ref

v R =

v R

d dÞ linear!

• Case 3: 3 1

4 2

R = R - R R = R + R,

R = R + R R = R R

d d

d d

and

-

47

ref

δv R = = = R + R,

v Ro d

d1 4 when R R

For maximum bridge sensitivity:

• The general case:

refv R

= = R - R

d2 3and R R

k = = bridge output in the general case

1, 2, 4bridge output when only 1 strain gage is active

Bridge constant : o

ref

v Rk =

v R

d d4

o

ref

v R = k

v R

d d4

For the bridge with a single active element

48

t t a

t aa

v = = T L

T L

d dε- -ε

Poisson’s ratio (0 ~ 0.5 for most materials)

aL'

• When a piece of material is stretchedin one direction, it tends to get thinnerin the other two directions.

Poisson’s Ratio (Example 4.12)

tT

aL

T 't

Let’s say we want to measure the axial strainapplied to a square rod:

• 1 opposite pair of strain gages mounted axially

1 4δR = δR = δR

• a second pair mounted in the transverse direction

2 3δR = δR = -vδR

o

ref

v R = v

v R

d d4

2(1 + )

k v = 2(1 + )

applied to a square rod:

49

o

ref

δv = Cε

v

Calibration constants

C calibration constant of the strain gage bridge

s

δR = S ε

RSs strain gage sensitivity

ov R = k

v R

d d4

k bridge constant refv R

4

k C = sSÞ

4

ob

δvS =

δRBridge sensitivity

o

ref s

δv δR 1 = ε =

v C R S1

o

ref s

δv ε

v k SÞ

4 1=

50

• Much higher sensitivity thanmetallic foil type ( 40 – 200)

• High resistivity (several K ) lower power consumption +

lower heat generation

W

Semiconductor Strain Gages

• Sensing element made of piezoresistive material such as germanium orsilicon doped with impurity (boron)

Single Crystalof

Semiconductor ConductorRibbons

• Low hysteresis they return morereadily to their original shapes.

lower heat generation

• Brittle and difficult to mount on curved surfaces

• Higher temperature coefficient

• Maximum strain less than 0.001m/m

• More expensive than metal foil type

Gold Leads Phenolic GlassBacking Plate

51

• Nonlinear strain-resistance relation – can be approximated by

δR = S ε + S ε

R2

1 2

P-type SC gages N-type SC gages

52

• Assume a circular shaft torsionmember

rε = T

2GJ ε principal strain at 45o to shaft axis

Strain-Gage Torque Sensors

• Strain gages can be used to measurethe torque developed on a torsion member in a rotary drive system

2GJ

ε2GJ T =

r

o

s ref

δv8GJ T =

kS r v

r shaft radius

T torque transmitted through member

G shear modulus of shaft material= shear stress/shear strain =

x h

F A

d

J polar moment of cross section area of torsion member used to predict resistance to torsion

rp=4

2

53

Strain-gage configuration for a circular shaft torque sensor

Axial loads and bending loads are compensated in all cases

BridgeConstant K 2 2 4

54

Deflection Torque Sensors

• Torque can be determined by measuring the twist angle on the torsion member

GJT θ

L=

• Use a synchro-transformer andcouple the 2 rotors to the torsionmember

• Phase shift between the 2 pulse trains represents the twist angle

• Both magnitude and direction oftorque can be determined

• Use two variable inductance probesto measure the positions of the teethin the 2 ferromagnetic wheels

A direct-deflection torque sensor

55

• Signal coupling changes with positions of the 2 slits controlled bydeveloped torque

• Shaft made of non-magnetic material with a 3-section ferromagnetictubing attached

• Widths of gap openings depend on magnitude and direction of torque

• Reluctance changes with gap width and therefore signal coupling

Variable-reluctance Torque Sensor

• Operations similar to that of a LVDT

56

Strain-gauge Force Sensors (Load Cells)

• Convert the force acting on them into electric signals

• Contains a number of strain gages bonded ontoa mechanical structure (typically stainless steel)for complete connection to a Wheatstone bridge

• Temperature compensation over a certain range

• Available in different mechanical styles for• Available in different mechanical styles fordifferent (typically harsh) force measurementapplications

HTM File

57

Tactile Sensing (4.10)

• For force distribution measurement over an area

• In robotic grasping, object has to be held without slipping nor falling

• For object identification, including shape, location, orientation andsurface properties

• Closely spaced force sensor array can be used

Typical industrial tactile sensing requirements

• Spatial resolution of about 2 mm • Spatial resolution of about 2 mm

• Force resolution of about 2 gm

• Force capacity of about 1 kg

• Response time of 5 ms or less

• Low hysteresis (low energy dissipation)

• Durability under harsh working conditions

• Robustness and insensitivity to change in environmental conditions

• Capability to detect and even predict slip

58

Principle of optical proximity sensor

Optical Tactile Sensors

• Distance of separation x can be determined from optical receiver output

• Flexible tactile element consists ofa reflecting surface sandwiched

• Optical fibers uniformly and rigidlymounted normal to inner layer padfor light projection

• Light projected onto reflectingsurface through beam splitter andoptical fiber

• An image camera picks up the reflected light beam for tactile force distribution analysis

a reflecting surface sandwiched between a touch pad and a trans-parent rubber layer

59

• Movable pin moves down and blocks thelight path when tactile force is applied

An Optical Tactile Sensor with Localized Light Source and Photo-sensors

• Light source / receiver pair and movablepins distributed on tactile sensing pad

60

• Concept to demonstrate how to determine the magnitude and locationof a point force applied to a sensing surface

P R + R + R + R 1 2 3 4= Force equilibrium in the z axis:

Px = R + R x = R + RPa

a a Þ2 3 2 3( )Y-axis moment balance:

( )Py = R + R y = R + RPa

a a Þ3 4 3 4X-axis moment balance:

• Use an array of semiconductor strain gages mounted under the touch pad on a rigid base

Piezoresistive Tactile Sensors

P3 4 3 4

61

• The other part of the beam travels an extra distance of 2x caused by reflection of target object

Laser Interferometer

• For accurate displacement measurement down to sub-micron level

• Part of the laser beam is reflected back from front face of beam splitter A

• Phase different between these 2 components x = π

λf

22

HTM File

62

Laser Doppler Interferometer

• For accurate measurement of speed of a moving object

• A light source with frequency f (say, 5 x 1014 Hz), when reflected by a moving target will experience a shift of

f

2f = f + f∆

• For v = 10 m/s , f »∆ 32MHz

• can be determined by observingthe fringe pattern due to light waveinterference

f∆

2f = f + f∆

f v = kvf

c

2∆ =

interference

v = v + v = a sin f t + sin f t

= a sin f + f t cos f - f t

p p

p p1 2 1 2

2 1 2 1

( 2 2 )

2 ( ) ( )

Þ2

∆fbeat frequency =

ππ

ππ

Let the 2 waves

1 1v = asin f tp2

2 2v = asin f tp2

π

π

63

• Sound waves echoed by an object

converted back to an electric signal

Ultra Sound Sensors

• Use a piezoelectric disk to transmit

sound waves when pulsed with an

electric signal burst

• Distant between sensor and target

is ctx =

is x =

2

t - time of flight of sound pulse

from generator to receiver

c - speed of sound in media

64

• Blind zone caused by sound waves

transmission in progress

• Come in different

packaging styles

• Most applications use ultrasound

sensors in transceiver mode

65

• Fish finding

• Collision avoidance

• Seabed mapping

66

Pressure Sensors

p - p = ρgh

Manometer Counterbalance piston

p F

=

Bourdon tube

refp - p = ρgh p

F

A=

DiaphragmBellow Helical tube

67

Flow Sensors

Orifice flow meter Pilot tubeAngular-momentum flow meter

Coriolis velocity meter Rotameter

68

Temperature Sensors

Thermocouple

• Electron configuration due to heat transfer produces

voltage-Seebeck effect

• Two metals –Fe and Constantan, Cu and Constantan,

Chrome and Alumel

• Sensitivity 10mV/oC

• Fast measurement (e.g., 1ms)

•measures from -250oC to 300oC•measures from -250 C to 300 C

•Metal element in a ceramic tube –resistance changes with the temperature

•Metals used –Platinum, Nickel, Cu

• useful temperature rage from -200oC to 800oC

Resistance Temperature Detector (RTD)

•Unlike an RTD, thermistor is made of semiconductor materials

• Has a negative change in resistance with temperature

• Has better accuracy, more robust, faster response, and higher sensitivity than

than RTD, but far more nonlinear

Thermistor

69

Rating Parameters of Sensors and Transducers

Transducer Measurand Measurand Output Typical Acc.uracy Sensitivity Frequency Impedance Resolution

Max Min

Potentiometer Displacement 10 Hz/ DC Low 0.1 nun 0.1% 200 mV/nun

LVDT Displacement 2,500 Hz/ DC Moderate 0.001 nun or 0.1% 50 mV/uun less

Resolver Angular 500 Hz/ DC Low 2 min. 0.2% 10 mV/deg displacement (limite.d by

excitation freq.)

Tachometer Velocity 500 Hz/ DC Moderate 0.2 nmlfs 0.5% 5 mV/nmJ!s (50Q) 75 mV/rad/s

Eddy ctm ent Displacement 100 kHz/ DC Moderate O.OO!nun 0.5% 5 V/nun

proximity sensor 0.05% full sc.ale

Piezoelectric Acceleration (and 25kHz/ 1Hz High l nun/s2 0.1% 0.5 mV/m/s2

accelerometer velocity, e.tc.)

Semiconductor Strain 1kHz/ DC 200Q l -lO)t€ 0.1% I V/€ max

strain gage (displacement, (limited (l)t€=10-6 2000)t€ acceleration, etc.) by fatigue) unity strain)

Loadcell Force 500 Hz/ DC Moderate 0.0 1 N 0.05% l mV/N (10 - 1000 N)

Laser Displacement/ 1kHz/ DC I OOQ 1.0 pm 0.5% I V/nun Shape

Optical encoder Motion 100 kHz/ DC 500Q 10 bit ±Yl bit 104 Pulses/rev.

70

Chapter 5 Digital Transducers

• A measuring device that produces an output in a digital format, or in

form of a pulse train, or a frequency, that is proportional to the physical

input

• For an analog transducer, both the sensor stage and the transducer

stage are analog in nature

• For a digital transducer, the sensor stage is typically analog. The

transducer stage generates the discrete output signal

Key Advantages of Digital Transducers

• Digital signals are less susceptible to noise or other operation parameters,

such as temperature, supply voltage, aging, etc higher reliability!

• Output signals can be processed with the use of high speed computers or

digital electronics more flexibility compare to analog processing!

• Fast data transmission and high data volume storage possible

• More compact and lower cost in general

• Resolution inferior compared to its analog counterpart

71

Shaft Encoders

• Digital transducers for measuring angular displacement or velocities.

• Can be either a standalone unit or integrated in a motor assembly

• Output is a pulse signal and can be either the absolute encoder type

(digital word output) or incremental type (pulse train output)

• Different implementation techniques: -- Optical

-- Sliding contact

-- Magnetic saturation-- Magnetic saturation

-- Proximity sensing

$27 - $67 each

Anaheim Automation

Optical encoder

72

Optical shaft encoder

• A code disk with a fixed number of

transparent windows is fixed on the

motor shaft

• Optical path between LED and light

light senor is chopped to produce a pulse

train as the disk rotates

• Angular displacement can be determined by counting the number

of pulsesof pulses

• Resolution dependent on number of transparent windows: 40, 200,

400, …. 12000

• Angular velocity can be determined by counting the number of pulses

within a fixed timing window

• A transparent disk with opaque spots are common for high resolution

encoders

73

• All conducting segments are

connected by wires to the DC

supply source via the shaft

Sliding Contact Type

• Transducer disk made of

insulation material with

coated with metal conducting

segments evenly spaced to

make up the encoder track

• A sliding contact makes a connection each metal segment as the disk

rotates

supply source via the shaft

and brush commutator

• Advantages are high sensitivity and low cost

• Disadvantages are friction, wear, contact bounce and contact oxidation

74

• High strength magnetic strips

are imprinted around the

disk to form the encoder track

• A micro-transformer is used as

the pick-off element whose

primary coil is driven by an

Magnetic Encoders

• Encoder disk made of non-

ferromagnetic material

primary coil is driven by an

AC carrier (100KHz typical)

• The presence of a magnetic strip will saturate the core of the

transformer and reduce its reluctance causing its output voltage to drop

• Non-contact but more expensive than other types due to pick-off element

and demodulation circuitry

75

Proximity Sensor Encoders

• Use a magnetic induction probe

or an eddy current probe as the

signal pick-off element

• Encoder disk has imprinted raised

spots to form the encoder track

• For magnetic induction probe,

disk and raised spots are made of

ferromagnetic material

• For eddy current probe, raised spots are plated with conducting materials

ferromagnetic material

• Raised spots decrease probe reluctance with an increase in output voltage

76

• Reference pulse for finding the“home” position of the encoder

• 2 photodiode sensors (pick-off 1 and 2)are spaced half the window length(quarter pitch) apart

• CW rotation if V 1 lags V2 by 90o

Incremental Optical Encoders

• 2 possible configurations :1) Offset sensor 2) Offset track configuration

• CW rotation if V 1 lags V2 by 90(V2 is high when V1 goes high )

• CCW rotation if V 1 leads V2 by 90o

(V2 is low when V2 goes high)

77

• We can also use a high speed counterto determine the direction of rotation

• Counter starts to count from zero whenV2 goes high and stops when V1 goes high (count value = n2)

• Counter starts to count when V1 high until V 2 goes high (count value = n1) (n1)

(n2)high (count value = n2)

• CW rotation if n1 > n2

(n2)

(n1)• CCW rotation if n2 > n1

78

Internal hardware of a high resolution incremental encoder

• To avoid errors (reading light as no light)due to instabilities and changes in supply voltage another photosensor is placed halfpitch away

• Masking disk enhancessignal reception strengthand discriminates noise

79

Displacement Measurement

If M is the number of pulses that represent the maximum angulardisplacement of ( +180o or 360o typical), then for a pulse count n, the displacement is

max θ±

maxn

θ = θM

Suppose the pulse count is stored in a register with r bits with 1 bit usedas the sign bit,

-1r M = 2

For max θ = ± o± 180 d r r

θ = = 180 360

2 2

o o

-1∆

Minimum count minθ Þ all zeros; Maximum count maxθ Þ all ones

max min dθ = θ M θ + ( - 1)∆ rmax min d θ = θ θ Þ -1+ (2 - 1)∆

Displacement resolution Digital resolution

Digital resolution

80

Physical Resolution

• For a single track encoder disk with N windows, the angular separation between 2 windows is o360 N

• Can be detected by counting 2 consecutive rising edges of the encoder output pulse

• By counting both the rising and falling edges of two encoder output

• By counting both the rising and falling edges of the encoder outputpulse, the angular resolution is o360 2N

• By counting both the rising and falling edges of two encoder outputsignals that are offset by half a window, the physical resolution

pθ N

o360∆ =

4

• If the moving object whose displacement to be monitored is geared up toto the encoder by a ratio p,

pθ pN

o360∆ =

4

• Output resolution of an encoder can further be increased by usinginterpolation technique (about 20 times increase in resolution)

81

Velocity Measurement and Resolution

2 Methods in measuring angular velocity using an incremental encoder:

1) Count the number of encoder pulses elapsed over a fixed period of time

n pulses

Fixed period = TIf there are N windows on the encodertrack and n pulses have been detectedover a period T, then the angular speedis

= =

π n πnNω T NT

22= = ω

T NT

= =

ππfNω

m f Nm

22

The velocity resolution in this counting mode corresponds to a change of n from 0 to 1

= cπ

ω NT2

∆

Velocity resolution

= = = tNωπf

πf πf πf πfω

Nm NmN m + 1 Nm m + 1 2

2-

22 2 2 2

∆( ) ( )

;

2) Measure the time duration of a encoder pulseusing a fixed, high frequency clock (of f Hz )

m clock periods

82

Absolute Optical Encoders

• Generates a coded digital word to represent each discrete angularposition of its code disk

Example of a binary absolute encoder with4 encoder tracks representing 16 sectors

• Use a separate reflective type of LED/photodiode pair for reading the code oneach track

83

• Gray code -- a binary numbering scheme in which two successivevalues differ by only 1 bit

• Avoids faulty interpretation of encoder disk position due to different switching speeds of optical sensing devices

• Encoder resolution is nθ = o360

∆2

• Angular velocity can be determined by measuring the time intervalbetween 2 successive code values

• A reading indicates absolute angular position -- error in missing pulse not accumulative (like the incremental encoder)

• Position reading still maintained after power failure

• Code matrix on disk more complex and therefore lower resolution than the incremental encoder

• More expensive than incremental encoder due to the need of morelight sensors

84

5. Ambient effects (vibration, temperature, light noise, smoke, etc)

Encoder Error

1. Quantization error (due to digital word size limitations)

2. Manufacturing tolerances ( inexact positioning of the pick-off sensor,inaccurately imprinted code patterns, irregularities in signal generation

3. Coupling error (gear backlash, belt slippage, loose fit, etc)

4. Structure limitations (disk deformation and shaft deformation dueto loading)

a) Shaft eccentricity – rotating shaft does not line up with its geometric axis (es)

b) Assembly eccentricity -- center of code disk does not fall on the shaft axix (ea)

c) Track eccentricity -- center of track circle does not coincide with geometric center of the disk (et)

d) Radial play -- looseness in the assembly of the disk and the shaftin the radial direction (ep)

6. Assembly error (eccentricity of rotation, etc.)

85

sm

am

tm

Let = mean value of shaft eccentricity error es

= mean value of assembly eccentricity error ea

= mean value of track eccentricity error et

= mean value of radial play error eppm

The root mean square for these 4 sources of error is

s a t p = + + + m m m m m2 2 2 2

The probability that the eccentricityerror of an encoder that falls within

is 95.5% µ σ µ σ - 2 and + 2

The standard deviation of the overall eccentricity is

s a t p = + + + s s s s s2 2 2 2

86

Miscellaneous Digital Transducers

Digital Resolver -- operations like that of the analog resolver usingthe principle of mutual induction

• Fine electric circuit patternsare etched in copper bondedto the surface of the disks

• Rotor disk circuit is driven

• Consists of a rotor disk and a stator disk

• Rotor disk circuit is drivenby an AC carrier throughslip-ring and brush

• Stator disk has 2 separate printed patterns that are offset by a quarter pitch

• AC carrier is inductively coupledto the secondary foils to produce quadrature signals when the rotordisk rotates

V1

V2

87

Demodulation necessary to convertoutputs to digital signals

• Resolution up to 0.0005o

• Slip ring and brush needed to connectcarrier to primary foil

• More expensive that optical encoders

• Well suited to severe industrial• Well suited to severe industrialenvironments

88

• Ferromagnetic teeth wheel when used magnetic induction probe

Digital Tachometers

• Use a tooth wheel instead of an encoder disk

• Magnetic induction probes mountedradially facing the teeth and offset by half a tooth position

• Teeth lowers reluctance of magnetic probe and increase its inductance

• Can use eddy current probes instead of magnetic induction probesteeth surface will have to be coated with conducting material

• Non contact, simple, immunity to environmental effect (except magnetic effect)• Poor resolution, mechanical errors due to loading, hysteresis, and irregularities• Requires signal conditional and demodulation circuitry

89

• Hall voltage n

IB =

ρ qtHV

Hall Effect Sensors

• An output voltage (Hall voltage) will begenerated across a current-carry piece of semiconductor material when it is placedin a magnetic field perpendicular to the current flow in the material

nρ qt

I -- currentB – magnetic

field densityq -- charge of an

electront -- thickness of material

nρ -- number of charge carriersper unit volume

A Hall-effect shaft encoder or digital tachometer

90

• Available in form of an IC package for generating either a linear or digital output

A linear Hall effect sensor IC

• Non contact, rugged, relatively insensitive to change of common environmental factors except for stray magnetic field, high reliability

• Output not linearly related to the displacement of a magnetic object

• Applications: vehicle speed sensing, ABS brake systems

91

-- mask is in form of a narrow stationary strip mounted alongthe direction of movement ofobject

Rectilinear Optical Encoder

• Operations similar to the opticalshaft encoder except:

-- a linear encoder track (codeplate) plus 2 LED / photodetector pairs are mounted on the moving object as one assembly

• The 2 photodetectors are offset by a quarter pitch window to produce the quadrature signals as the object moves

• Requires protection from dusty environment !

92

Cable Extension Sensors

• For measuring rectilinear motionespecially with large excursions (up to 50 meters)

• Uses a spool and cable coupled to an angular motion sensor with the end ofcable attached to the moving object

• When the object moves away from the sensor, the cable extends, causing the spool to rotate causing the spool to rotate

• Sensor can be a multi-turn potentiometer or an optical encoder

• Cable is retracted without slack by the auto-tensioner when the objectmoves towards the sensor

• Disadvantages include mechanical loading, time delay in measurements, cable slack, andtensile deformation

• Applications: lift control, automobile crash testing, manufacturing automation, underwater depth measurement?

93

Binary Transducers

Electromechanical limit switch Ultrasound(5 to 1000cm)

Photoelectricdevices

Capacitive deviceEddy currentInductive device

94

1. Sensing range 2. Response time3. Sensitivity4. Linearity5. Size and shape of the object6. Material of the object7. Orientation and alignment8. Ambient conditions

Digital Transducer Selection Criteria

8. Ambient conditions9. Signal conditioning considerations10.Reliability, robustness, and design life

95

Chapter 1 (Sen) Magnetic Circuits

When a conductor carries current, a magnetic field of intensity H is produced by it. (Right hand rule). H

Ampere’s circuit law: the line integral of the magneticfield intensity H around (tangential to) a closed pathis equal to the total current linked by the contour.

Or

i H =

2πrA / m

Or

where is the angle between H and dlθθθθ

For a single conductor carry a current i, the magnetic field intensity generated by it at adistance r is

96

When the closed path is threaded by the current Ntimes,

Ni is called the magnetomotive force (mmf)And its unit is Ampere-turn (At)

As the magnetic field intensity H exits, it produces a magnetic fluxdensity B, so that

B – H Relationship

density B, so that

B = µH 2weber / ( tesla)m-- permeability of material

4π10-7 Henry/meter

»r oµ µ B = H 2 weber /m -- relative permeability of

material 2000 -- 6000 for ferromagnetic materials

-- permeability of free space

97

H l Ni mmf= = F F F F

N H = i /

lAt m

Magnetic Equivalent Circuit

Applying Ampere’s law to the circuit:

Ni B =

lT

m

µNi Ni

A l l µA

= =

l

Assuming zero leakage flux, the flux cutting the cross section area A

B dA BA ò = = WbΦ

= Ni =

ÂÂFFFF

98

There is an equivalence between an electric circuit and a magnetic circuit:

Electric Circuit Magnetic CircuitMmf (F)Emf (E)

i = E / R

R l As =

Mmf (F)

We therefore tend to model a magnetic circuit with its correspondingelectric counterpart when performing magnetic circuit analysis

-- conductivity of resistiveelement

G = 1 / R -- conductance

s -- permeability of magnetic element

P = 1 / -- permeanceÂ

99

Magnetization Curve

When the current driving the toroid is increased, the resulting flux density becomes saturated atan high H Reluctance depends on flux density!an high H Reluctance depends on flux density!

Shape of curve dependent on properties ofmedium.

Note: to develop a certain level of flux densityB*, you need a different amount of current for adifferent material.

100

Magnetic Circuit with Air Gap

Composite structure – a magneticcircuit with more than 1 media

gcc g

c gc o

llA Am m

= =Â Â;

=

c c g gN i H l H lN ic g

= ++FÂ Â

;

gc FF

Motors and generators are designed with anairgap between the rotor and stator.

gcc g

c gB B

A A

FF= =; Fringing

Neglecting fringing effect,

g c g cc

A A B BAF= = =;

101

Example 1

A primitive relay, with a coil of 500 turns, a mean core path of 36 cm and a gap lengthof 1.5 mm. The core is cast steel. The relayactivates at a flux density in the core of .8 tesla.

36cm500 =

1.5mm

a. Calculatei.b. Compute of core.c. Find the coil current to maintain

a flux density of 0.8T when the air

rm mand

a flux density of 0.8T when the airgap is zero.

102

Example 2

Find .

36cm500 =

1 mmBg 4 A

103

Example 1.3

r m = 1200

Neglect magnetic leakage andfringing

All dimensions in cm

Square cross section for material

Find at the air gap.B H , , F

104

Inductance

An inductor can be formed by windinga coil with N turns on a magnetic core.

Flux linkage = Nl F

Inductancei

= L l

Therefore, = = = i i i

N HAN N B AL mF

Schematic diagram of a self-inductionproximity sensor

= = =Hl N

N HA N Nl A

mm Â

2 2

105

-- B-H curve starts from 0 to a and returns to c (where B = Br ) when i (H) 0.

-- Br – residual flux density

Hysteresis -- property of a magnetic material that does not return completely to its

original state after an applied force is removed.

Assume unmagnetized core att = 0 and apply i slowly to circuit

ab

0d e

a’

c

gf

-- Br – residual flux density

-- i (H) now reverse in direction to d whereB = 0 at H = -Hc (coercivity)

-- B-H curve forms an almost closed loop aftera few more cycles (hysteresis loop)

Magnetization curve – a plot of the tips of afamily of hysteresis loops from different valuesof H.

-- B-H curve forms an unclosed loop throughpath 0abcdefga’

106

Hysteresis Loss

Assume no resistance loss in the coil, the voltage e Across the coil is

= N dedtF

The energy transfer during the interval t1 to t2 is

= = t tt t

W p dt ei dtò ò2 2

1 1

i dt = = dN N i ddt

FF

F Fò ò 2

1

Since then and i = BA = HlN

F

Power loss due to hysteresis

f f

= = n

m ax

coreh hh B

P V WK

= BB

Hl AN

BW N dò2

1

lA B BcoreB B

VB BH d H dò ò2 2

1 1 ==

Energydensity

The energy transferred over 1 cycle is

= dB = cy cleW Hcore core hV V WòÑ X

107

Eddy Current Loss

A change of magnetic field density will induce a current ie (eddy current) to flow in it.

The eddy current loss in a magnetic core is

= fm axe e BP K 2 2

Heat generation as a result of loss i R2

To reduce eddy current loss:

1. Increase resistivity of core material.

2. Construct core with thin laminations with alayer of insulation between every two of them.Laminations should be orient in parallel to direction of flux.

108

If frequency of current change is low, losswould be mainly due to hysteresis.

Core Loss

Total core loss = hysteresis loss + eddy current loss

+ c eh = P P P

+ f = fnm ax m axeh B BK K 2 2

If frequency of current change is high, loss willbe due to both hysteresis and eddy current.

Usually, these 2 losses are lumped together when analyzing heat loss inmachine design.

109

Assume sinusoidal flux variation within the magnet core

Sinusoidal Excitation ( E -- )F

Assume sinusoidal flux variation within the magnet core

m ax( t ) s inwtF F=

Voltage induced in the coil m ax

m ax

de N N wtdt

E wt

wcos

cos

F F= =

=

The RMS value of the induced voltage is

4.44

Nf

m axrm s

m ax

m ax

wF

F

2 2= =

=

E NE

110

Assume -- R in coil = 0

-- magnetic core has a non linear B-H characteristics

-- no hysteresis loop

Sinusoidal Current Excitation ( i -- )

A sinusoidal voltage source e generating anexciting current will produce a sinusoidalflux in the core.

iF

F

iF

F

Let’s first rescale the original B-H curveto obtain the -i curve:

HlN

i =

How is related to e ?iF

111

1. Plot e and VS t. F

t

2. Plot VS t by extrapolatingfrom - i curve.F

i F

• can be represented by a pure inductor (no real power loss).

• is symmetrical with respect to e.i F

• the fundamental component lagse by 90o .i F 1

• is non sinusoidal, but is in phase with .Observations:

i F F

112

Same analysis as before except with hysteresis loop

• is non sinusoidal, nonsymmetrical wrte.

Observations: i F

• consists of 2 components: 1. accounts for core loss2. same as without hysteresis loop

i F

cimi

• circuit can be modeled with an ideal inductor in parallel with a resistor

F

Exciting current obtained from sinusoidal flux waveformand multivalued - i curve.

i F

113

Magnetization of Permanent Magnets

Permanent magnets are characterized by:

• Capable of maintaining a magnetic field without any excitation mmf

• Normally alloys of iron, nickel, and cobalt.

• Large B – H loop with a high retentivity ( Br ) and a high coercive force ( Hc)

• Mechanical hardness due to heat treatment hard iron

A

114

-- Assume hard iron initially unmagnetized

-- Gradually increase mmf from point 0 to A and remove source

-- Flux density will remain at point a with residual flux density Br

-- With mmf gradually reversed to –H1, flux density drops to point b

-- Minor loop operating path with mmf removed and reapplied.loop b-c (recoil line)

-- With mmf further reversed to –H2 and removed, flux density moves between point d and e

-- Hard iron completely demagnetized when H = - Hc

-- Slope of recoil line recoil permeability = 1.2 - 5 recm om

115

Want to determine size of permanent magnet to product across air gap!gB

= - = mm m g g g m

g

lH l H l H Hl

+ 0;

For continuity of flux, = B = m m g gA B AF

Volume of permanent magnet is

Since g o gB Hm ,= g g g m

m mm m g

=A Ao

B A A lB Hl

m=

m m mV A l=

Minimum volume of magnet required to produce Bg across the airgap (Vg)is when BmHm is a maximum. BmHm Energy product

=Bg g g g

m m

B A H lH

X

= g g

o m m

VBB Hm

2

where is the volume of the air gap g g gV A l=

116

Demagnetization Curves of Permanent Magnets

Alnico 5 Ferrite D (Ceramic) Rare earth

• High residual flux• Low coercivity

• Low residual flux• High coercivity

• Alloy of aluminum-nickel-cobalt

• Strontium or bariumferrite

• Excellent temperaturestability

• Mechanically brittle

• Low cost• Mechanically brittle

• Low cost

• Highest energy prodcut (BH) max

• Higher cost

• Sm-Co – higher T (300o) operation, more corrosion and oxidationtolerant than Nd-Fe-B

• Extremely brittle(Sm-Co)

117